NASA's Heliophysics Summer School - August 19, 2024 - Matthias Rempel

Hello, ah, wow, okay, great. So, good morning everyone. Hope you had a good weekend. And, uh, this morning, we're going to get started talking about, um, how magnetic fields and stars are generated and their importance. And we have Matias rmle. Uh, Matias is here at, uh, um, ukar. And, uh, he received his PhD in 2001 from, uh, The Kepler Institute for Solar Physics in Fredberg and the Max Planck Institute in Kenberg-Lindau with Ma Schler. He then came to the US with a postdoc first and then as a staff scientist and has been there, been here ever since. Apparently, you enjoy Boulder very much. Anyway, I'll leave it over to you, Maas. Thanks.

Okay, can you hear me okay? Okay, good. So, this morning, I would like to talk with you about the processes which lead to the creation and destruction of magnetic fields in the universe. And I have two lectures. First one is a little bit more equation heavy. That's the reason why it's a large generated PDF. So, I hope I don't overload you with that because, I mean, I have to get some fundamentals of dynamo theory through here. That's hard to do without equations. The second presentation is more about applications to the Sun, to other stars, to the geodynamo, and that one has very few equations. So, I hope, just bear with me through the first lecture here. Okay, so, so see, doesn't... Okay.

So, the scope of these lectures is really, um, introducing the general concepts of dynamo theory. So, I would like to give a brief introduction about magnetic fields in the universe. What do we know of magnetic fields at different scales? A very brief review of MHD and the induction equation. I, you had already a lecture about MHD, so this is just a few slides reviewing a few essentials we need for these, um, lectures. And then I will make some few general remarks and definitions about what we consider a dynamo in the astrophysical context. We talk about a few specific types of dynamos, small-scale, large-scale dynamos. Introduce mean field theory, which is a formalism how to essentially model the large-scale component of the fields by averaging over smaller-scale components. And then finish here with a few concluding remarks. And then, as I said, the second lecture will be much more about applications.

So, let's talk a little bit about magnetic fields in the universe. Of course, the one body we know the best is the Earth. It has about a field of 0.5 Gauss. And that it's around 0.5 Gauss is not a coincidence. Kai-Fridrich Gauss, he invented the unit Gauss to be kind of a meaningful measurement of the Earth's magnetic field. That's the reason why it's of order, of order one in that unit. We do know that the Earth had a magnetic field for a couple billion years. And the way we know this is essentially when you have, um, volcanic rock deposits, the magma is too hot, it cannot be magnetized because it's above the Curie temperature. And then when lava cools down, it kind of freezes in the magnetic orientation which was present at the time. Particularly if you look at locations like, I mean, the mid-ocean ridge in the center of the oceans where the plates kind of depart from each other, you kind of have a steady record of magma deposits which kind of give you a record of the history of Earth's magnetism. And through that, we know that the Earth had a field on, on the order of this magnitude for a pretty long time. And this is much longer than the ohmic decay time scale, which is more something like 10,000 years for the Earth. And it has also strong variability on shorter time scales, as short as 1,000 years, which kind of suggests that there must be some dynamic process maintaining this sphere. And also, many other planets and moons in our planetary system do have large-scale magnetic fields.

In the case of the Sun, we have magnetic fields from the smallest observable scales, and that is kind of something on the order of 100 kilometers and maybe a little bit less, to the full scale of the Sun. The large-scale part of the sphere has a fairly regular 20-year cycle. Now, the ohmic decay time scale, in the case of the Sun, if you really use the plasma resistivity, would be something like a billion years. So, you could kind of argue maybe there's some left over from the origin when the, the Sun formed about, um, five billion years ago. But of course, it's only such a long time scale in the absence of turbulence. And we touch on this later.

And then we have observations of other stars. And stars with outer convection zones, they have very similar magnetism to the Sun, different strengths. If you have faster rotating stars, they tend to have stronger fields. Other stars also have cycles. It depends a little bit on where the stars are on their rotational track what the cycles are. And I will come to this back in the, um, second lecture. There are also stars which have outer radiation zones. There, it's possible that since they don't have turbulence, that there may be some magnetic field left from the, um, formation process of these stars. And of course, the stars with radiation zones, they have a much shorter lifetime. So, it's completely possible that after a few million, hundred million years, there's still some, um, field left over from their formation process. But they do have likely active field generation in their convective core.

And then even if you go to the scale of galaxies, there is field on the order of a few microgauss. And the field structure seems to be coupled with the matter distribution in the galaxy.

So, talking a little bit more about specifics. If you look at the Sun, at the Earth, this is a map of the magnetic field of the Earth. And red and blue here indicates the different polarities. So, the dominant field is a dipole, but it has some distortion. So, it's actually a superposition of several multipoles, but dipole and quadrupole are the two dominant ones. And the magnetic field also has a shorter-term variation. Perhaps the most significant one around this time is that actually the North Pole is moving from northern Canada over towards the direction of Russia. And it's kind of at a rate of something like 40 kilometers per year. So, that gives you something like a millimeter per second. And that is believed to be roughly the flow speed of convective, um, flows in the outer liquid core, iron core of the, of the Earth.

And looking at this magnetic field longer term, actually the Sun, the, the Earth had some field variations more on a random scale. A given polarity is present or something on the order of a few hundred thousand years. Currently, the Sun, the, sorry, I always make this up. The Earth had its, um, field, the current polarity we have for about 800,000 years. But if you look in these, um, records, um, the, you have switches of the polarity on rather short time scales of about 1,000 years. And even if you are within a phase of a given polarity, there are strong variations of the dipole moment. There are even some indications where there were some attempted reversals which really didn't make a full reversal. So, the magnetic field is actually very dynamic on longer time scales. And if you go even longer in the past, there have been even phases where the Earth seemed to have kept the same polarity for much longer time scales. It's something like 100 million years. And that might have been related to some changes in the, um, core-mantle boundary condition due to, um, plate tectonics and, and convection in the mantle of the Earth.

When you look at the Sun, this is a magnetogram from SDO/HMI. You see everywhere these very small-scale fields. And we talk more about this later. This is really salt and pepper mixed down to the smaller scales we can observe. But if you look at the larger-scale fields, we do have the sunspots with up to 4 kilogauss field strengths. So, that's something like 8,000 times the strength of the Earth's magnetic field. And these strong field concentrations, they also have preferred polarity. So, if you, for example, look here, Northern Hemisphere, you have always a negative polarity here on the right side. If, if you go to the Southern Hemisphere, the other way around. And if you look 11 years later, then this switches. So, there are these large-scale polarity reversals which suggest it's a process which really fields the full scale of the system. And this magnetic field comes in cycles of about 11 years. If you look at the full cycle, of course, this polarity flip is 22 years. Most of these active regions show up in low latitudes early in the cycle, around 30 degrees. Later in the cycle, close to the equator, which leads to the this, um, butterfly diagram, which is just a plot of sunspot positions versus time. This is not a motion of an individual sunspot. It's just that sunspots, they typically live something on the order of a few weeks, some in extreme cases they can live more than a month. But for the most part, sunspots appear early in the cycle in high latitudes and later in the cycle close to the equator. The strength of the cycle varies with time by a moderate amount, something like a factor of two. But if you look much further back in time, there have been phases where the Sun had much lower activity levels, which are called Grand Minimum episodes, like shown here. The most famous one is the Maunder Minimum around 1650 to 1700, where sunspots were mostly absent, not completely, but there were very few observations of sunspots. So, the dynamo seems to have some switch between an active and a less active state. We know that the Sun had kind of this sort of activity pattern for something like the past 100,000 years. And that is mostly what we get from tree-ring data in ice core records of C14 and Beryllium 10, which is essentially caused in the Earth's atmosphere by the impact of cosmic radiation, which itself is moderated by the strength of the solar magnetic field. We do not have any longer records for the solar magnetic field. But of course, we have observations of other stars. And through that, we know that stars have magnetic fields on much longer time scales than this.

If you go even to larger scales, also galaxies have magnetic fields. And this is an example from M51, the two galaxies which interact with each other. If you have binoculars, you can see this galaxy with binoculars. It's near the Big Dipper. I can show you exactly where you can find it if you're interested. The magnetic field here is derived from polarization of radio emission. And people find typically microgauss field strengths. And the magnetic field clearly shows the structure of spiral arms, right? You kind of see here that the direction of magnetic field and the strength is kind of following the, um, the structure of the galaxy. What's kind of interesting about galaxies is it's the only dynamo really in the universe which is optically thin. In principle, you can see through the entire region. In the Sun, the biggest challenge with understanding the solar dynamo is that we can only see the outer boundary, the photosphere. We really barely have any information about magnetic fields inside the convection zone. I mean, I think these, um, short bars here, they kind of indicate the direction. So, yeah.

Okay, so to summarize, in the universe, we have objects from the size of a planet to galaxies or even larger scale galaxy clusters, which do have magnetic fields on the scale of the object itself. But the physical properties in these objects are vastly different, right? We're talking from scales of a thousand kilometers, as in the case of the Earth, to 100,000 light years. In Earth-like planets, we talk about liquid iron. But in other stars and galaxies, we are talking about partially ionized plasma. We have spherical objects, we have disk-shaped objects. All of these objects have rotation. And the, the influence of rotation varies vastly. For example, the, the Earth is very strongly rotationally constrained. Objects, the Sun is kind of so at the boundary where the, um, where transitions from weak to strong influence of rotation. But all of them are rotating. That turns out to be a very critical part, as I will show later. Another critical, um, dimensionless number here, the magnetic Reynolds number, changes vastly. It's something like on the order of a thousand, a little less than a thousand for the Earth. But if you go to galaxies and so, it can be 10 to the 18. The big question I really want to address here is a common origin to all these magnetic fields, and can we understand this on the basis of MHD? And you did have a lecture about the MHD equations, so I go only very quickly, um, through this, right? The MHD equation is basically a reality of continuum, um, approximation. If you have strong collisional coupling, which is not a requirement for MHD, but it makes life easy because you can just do single fluid approximations and have a scalar gas pressure. And you have non-relativistic motions, which is good enough for most of the objects we're dealing with. Of course, if you want to understand how magnetic fields are generated in accretion disks close to black holes, you have to go a little bit beyond this. And the MHD equation is a combination of fluid description, the Navier-Stokes equation, with the non-relativistic Maxwell's equations, as well as Ohm's law.

Now, in general, solving the full 3D MHD equations, while that is becoming more and more possible with more powerful computers, it's still very challenging. And I talk to you about this in the second lecture. So, just to understand on a very fundamental level, what ingredients do you need to make a dynamo? I will follow in this lecture, um, kind of a semi-analytical approach while just understanding how a velocity field leads to induction effects. And this is called the kinematic approach because you ignore the Navier-Stokes equation. You just assume you have a given velocity field and what properties does this velocity field have to have that we actually can have a dynamo operating? So, in some sense, the velocity field is assumed to be given as a background turbulence. The Lorentz force feedback is neglected. And so, in order to do this, what we have to look at is just the induction equation. So, a very brief reminder where this comes from. The one equation, this is based on this Ohm's law, which is following from the, um, equation of motion for electrons. Here are formulated in terms of the drift velocity between electrons and the ions in the plasma. So, you have here the, um, electron acceleration term. You have a collisional term, these collisions between electrons and ions. Then you have here the, um, electromagnetic force on the ions and you have the electron pressure. You can reformulate this equation in terms of the electric current, which gives you Ohm's law. And I just highlighted here two terms in red. This is the most simplistic form of Ohm's law where you just have the relation between the current and the electric field. And of course, the coefficient here is the, um, plasma conductivity. And for the most of this lecture, we only use an Ohm's law of this very simple form. But I also would like to highlight this additional term here, which depends on the electron pressure. I come back to this one a little bit later.

And then, so we have the simplified Ohm's law. And this, this is of course in the co-moving frame of the plasma. If you then have it in the, um, rest frame, laboratory rest frame, you have to add in here the additional V cross B field, which just comes from the Lorentz transformation of the electromagnetic field. And then you put this, um, expression for the electric field, right? I replace now the, um, the current here with curl V. You put this into the induction equation, dB/dt is minus curl E. And that way, you end up with the induction equation for the electric magnetic field. And the eta here, which is the inverse of the plasma conductivity, is the magnetic diffusivity. So, most of this lecture will just analyze this equation. You have to, for the most part, forget about the rest of the MHD equations. Typically, this equation can be made dimensionless. And then you end up with a parameter in front of the dissipative term here, which is the inverse of the magnetic Reynolds number. The Reynolds number essentially measures the amplitude of induction effects here from the V cross B term to diffusive effects coming from this curl term. So, it's a typical length scale times the typical flow velocity divided by the magnetic diffusivity.

If we have a very low magnetic Reynolds number, this is a diffusion-dominated regime. And the induction equation just gets reduced to a diffusion equation. And it has only decaying solutions with a diffusive time scale, which is something like the typical length scale of the magnetic field squared divided by the magnetic diffusivity. And I show here, but are typical time scales. So, for the Earth, the outer core, the diffusivity is something like 2 m^2/s. It is the diffusivity of liquid iron. Length scale is about 1,000 km. Velocity is about a millimeter per second. You end up with a magnetic Reynolds number of about 300. So, induction effects certainly dominate over diffusion, but not by a very large amount. Diffusive time scales about 10,000 years. For the Sun, if you do the same for just the plasma conductivity, kind of the Spitzer conductivity, you get in the absence of any turbulence, you end up with a substantially higher Reynolds number, somewhere here, here about 10 to the 10 or something like that, 10 to the 9. And the main reason these Reynolds numbers are so much larger for the Sun is because the length scale and the flow velocities are much larger compared to the Earth. The actually the value of the diffusivity is more or less the same.

And now, the Sun, at least in the convection zone, the outer one-third of the Sun, has strong turbulence. So, essentially, the effective diffusivity is much higher. So, then you actually end up again with a much more moderate Reynolds number. If you take this into consideration, you can also do liquid sodium experiments in the lab. And on my last, in my second lecture, very much to the end, I just give one example of those. And they have been used to actually do experimental tests of the dynamo theory. I present here in a lab. You typically have the biggest problem in the lab is that your length scale and flow velocity are very moderate. So, it's a big challenge to get substantial Reynolds numbers. And liquid sodium is a pretty nasty thing to deal with. So, you need a lot of liquid sodium and pump it with high speed through pipes or whatever containment you have. So, these are pretty tricky experiments. But people have done them to kind of verify the dynamo theory I will be presenting here.

Okay, yeah. That's, that's even more scary. Yeah, yeah. So, if you now consider the advection-dominated regime, you can expand these, um, curl V cross B term. And you get essentially three components here. You get an advection term. You get these field line stretching term, which is essentially the velocity shear in the direction of the magnetic field. And you have these, um, compression term. If you have converging flows, like, for example, a star is forming, right? You have a collapsing cloud. This term essentially would, in that case, um, tell you that the magnetic field gets amplified by that. There's, you can write these much more compactly in the form that the total, um, time derivative of the magnetic field, which is just a combination of these two terms here in the front, is just given by these, um, field line stretching term. If you actually have a compressible fluid, you can find a very similar equation. And in that case, the divergence of V, you just replace with the equation of continuity. And then you can get a very similar form by just looking at the quantity B overall. And for example, this equation very easily can tell you, for example, if you have vertical transport of magnetic field in a stratified medium, if you have no expansion in the direction of magnetic field, so the right-hand side is zero, then magnetic field strength scales proportional to rho. So, if you take something from the base of the convection zone, bring it up to the photosphere of the Sun, you have a density contrast of 10 to the six, you would weaken the field by that factor of 10 to the six. But if you do have expansion, at least partial expansion of these plasma along the magnetic field, which you always do, then the dependence on the density is less severe. And in the most extreme case, if all the magnetic, all the plasma is just expanding along the magnetic field, then the magnetic field strength actually would not be, um, changing at all.

One way to think about this induction equation, which kind of leads to a very intuitive understanding on how actually the flow fields induce with magnetic field, is Alfven's theorem. If you consider the flux, um, through a kind of a, through a surface which has the property that the boundary of the surface is moving with the plasma, then the flux through the surface is conserved. So, for example, if you have an example, you have kind of a confined magnetic flux bundle here, and you have a shear flow, then this central surface here is displaced more strongly by the shear flow than the other ones. So, the action of the shear flow will be just that it bends the field lines like this and stretches them out. That's a very intuitive way to understand how flows and, and fields interact. And people very often describe this as that flux is frozen into the fluid or field lines move with the plasma.

Okay, so this is just a little bit of background, um, about the induction equation and so on. So, let's move on to dynamos. For zero velocity, all magnetic field has to decay on the time scale of this diffusion time scale, which is typically rather short for most objects we're dealing with. But we do have evidence that many objects in the universe have magnetic fields sustained for much longer time scales. And most importantly, the magnetic fields seem to be very dynamic. They evolve on time scales which are very often even much shorter than the diffusion time scale. So, this, um, leads to the idea that there must be some active process happening maintaining this field.

So, this is a very abstract definition of what we're actually looking at. So, essentially, you want to look at a situation where you have a magnetic field which is maintained by currents which only flow within a very confined volume, which is here just called S, which could be, for example, as the star itself. Inside this volume, you want to look for a solution of the induction equation. And then outside the volume, you have a potential field. And you smoothly couple the field across the boundary to the potential field. The velocity field is zero outside. And inside this volume, you do have arbitrary fluid motions which do have a finite kinetic energy. And then you call it a dynamo if you essentially, if essentially this B=0 solution, which is always a valid solution for the induction equation, is unstable. Meaning, if you just put in a very weak magnetic field, you get an exponential growth of the magnetic energy inside this volume. This is very abstract. So, let's just think about it a little bit. What it means. You're very likely very aware with dynamos and power plants and so. So, what do you think is similar and different between this dynamo and the dynamo in power plants? I mean, normally when you do electrical engineering, you normally don't solve this equation at all, do you know what the reason for that is? I mean, what are the most important building blocks of dynamos in power plants which you don't have in stars? I mean, you do have motions in both. And in, of course, you don't have a liquid material in a power plant. But actually, you do have motions. And most importantly, right, the most important induction effect comes from shear flows where in the direction of the magnetic field, you have some variation of the motions. And all power plant dynamos at least have a rotating part and a stationary part. In that sense, you actually have these shearing motions. You do have differential rotation in some sense in there. But what is the other big difference? Certainly that is a very big difference. Yes, but how do you build a dynamo in a power plant? What is the most important building block? Electric wires, right? Stars don't have wires. But in some sense, that difference looks large, but it's not really. What it just means is, in, in a power plant, right, this eta here, which is for simplicity in stars just a constant, so currents can just flow anywhere you want, essentially, you have a short circuit all the time. In a power plant, this eta is highly inhomogeneous, right? You only allow the currents to flow in certain locations. That is the main difference. So, in some sense, the main difference really is this, that you're inhomogeneous versus homogeneous. Dynamo in a power plant, a short circuit is a major disaster. You absolutely don't want this. But in these astrophysical dynamos in stars, you have a short circuit all the time. The question is, can you even in that situation maintain still magnetic field? And of course, in power plants, since you control the currents, it's mathematically much more easy to formulate the mathematical description of these dynamos in form of current. That's the reason when you do electrical engineering, you will not solve the induction equation, you deal with the currents directly. But if you go to astrophysical dynamos where you have no control over the current, it's mathematically much more convenient to just deal with the magnetic fields and just have the current as a derived quantity. Once you have B, you know what the currents are by just computing the curl of B.

Okay, one other, um, aspect of these dynamos. You can decompose the magnetic field into small-scale and large-scale components and kind of split up the energy into the energy of the large-scale field and the small-scale field. And you remember, like the, the magnetogram of the Sun I showed you, they have some large-scale field components like these sunspots, and you have these very small-scale field components which look like a salt and pepper pattern everywhere. So, there is a distinction of dynamo based on where most of the magnetic energy is. If you have basically no large-scale field, no magnetic field on the scale of the system, you only have these small-scale fields, you call it a small-scale dynamo. But if you have at least a substantial amount of energy in the large-scale field, it typically does not become much larger than the small-scale field because it can only maintain these large-scale fields by also having small-scale fields. They come back to this later. Then you call it a large-scale dynamo. It turns out that almost all turbulent chaotic velocity fields have these small-scale dynamo property. But if you want to have a large-scale dynamo, you need additional parts, additional symmetries in the system.

So, what does large-scale versus small-scale mean? Large-scale is pretty much anything which is aware of the larger scale in the system. So, if you look, for example, in this magnetogram on the sunspots, right? They have these preferred symmetries. They certainly know on which hemisphere of the Sun they are. They are part of the large-scale field. But then if you look somewhere away from sunspots into these salt and pepper pattern, and this is a more detailed field of view from a HMI magnetogram, this is kind of a scale something like 100 times 200 megameters. So, it's kind of like a small box, something like this here. You kind of see these salt and pepper fields on very small scales. And if you just randomly choose a box like this in the Northern Hemisphere and the Southern Hemisphere, you couldn't really tear it apart. So, this is really, um, component of the field which doesn't really care about any large-scale properties of the system. That would be definitely the small-scale component. But there is a smooth transition between the two. You cannot really draw a very clear line what is large-scale, what is small-scale. Because, for example, if you're looking close to sunspots, some of these sunspot fields decay, and then you see more, for example, of the negative polarity near this spot. You see a little bit more of these, the positive polarity outside this spot. In that case, you have some mix here. You have some small-scale field, but you also have some of the larger-scale field mixing in this. It, so it's a smooth transition. You cannot just draw a very clear line between the two.

Let's first talk about the small-scale dynamo action. And I want to motivate it here by just getting a very rough idea what it is about. Let's assume you have a fluid velocity field B, and you just put in two Lagrangian particles which just follow the velocity field at a position X1 and X2. And then you consider the separation between these two particles. And you derive an equation by, you just take the difference between these two equations. And then, since this delta is assumed to be very small, you can do kind of a Taylor series expansion of the velocity field. Then you get this equation which tells you how the separation of the particles evolve in time. And essentially, when this equation has exponentially growing solutions, which is, this is essentially the chaotic flow has a positive Lyapunov exponent at least in one direction, that is essentially the definition of what one calls a chaotic flow. And turbulent flows have this property. Now, just by looking at the induction equation, you actually see that this form of the induction equation we derived earlier has exactly the same mathematical form. Which essentially means is, if you have a chaotic flow and you sprinkle in a weak magnetic field, that magnetic field has to grow exponentially as well, just because these two equations are mathematically the same. Now, of course, the induction equation also has some diffusive term here, which will prevent this exponential growth. So, typically, what happens is you only find this exponential growth if the magnetic Reynolds number is large enough, typically something on on the order of 100. And this is essentially the essence of a small-scale dynamo. Where any turbulent chaotic flow has this property at a high enough Reynolds number, you will see exponentially growing solutions. But life is very often a little bit more complicated. And over the last 10, 20 years, when people studied these dynamos, there was some concern regarding the role of the magnetic Prandtl number. The magnetic Prandtl number is the ratio of the magnetic Reynolds number over the fluid Reynolds number, or it's ratio of viscosity to magnetic diffusivity. And essentially, what it tells you is where does this small-scale dynamo actually operate relative to the turbulent velocity spectrum. So, the blue line here indicates the kinetic energy power spectrum. If you have a large magnetic Prandtl number, the small-scale dynamo, at least in the kinematic growth phase when the field is very weak, most of the energy, most of the induction really happens here in the dissipation range of the velocity field, where the velocity field is very smooth and it doesn't really have flow scales on scales smaller than the magnetic field. And this people studied these high magnetic Prandtl number dynamos first because it's most easy to do this in numerical simulations. But the problem is that stars, for example, the Sun has a magnetic Prandtl number of about 10 to the minus 5. So, you actually have to make this dynamo work inside the, um, inertial range. And what this means, you have certainly these large-scale shear flows which are very efficient in inducing the field. But you also have a lot of very small-scale turbulence which is much more efficient in adding a turbulent diffusivity, which again destroys the field. So, there was a concern that this could lead actually to a situation where the dynamo is much more difficult to excite. And when people studied the Prandtl number dependence early on in simulations, then they started coming from the high Prandtl number to the low Prandtl number regime. What they found is that actually the critical Reynolds number you need to exceed is increasing as you approach lower and lower Prandtl numbers. And for a long time, it wasn't clear if this just continues to go up or if it will turn over at some point. And only actually a year ago, people were finally able to do these, um, kinematic dynamos at a very high, at a very low magnetic Prandtl number, something here, a few times 10 to the minus 3. They actually found is that eventually this turns over again, and these dynamos become more easily excited. So, right now, it looks like this issue has been resolved that even for the low Prandtl numbers like 10 to the minus 5 in the solar convection zone, it's still possible to keep the small-scale dynamo operating. And I will talk more about small-scale dynamos in the second lecture.

Today, I want to just show here a very rough sketch how you actually make a dynamo work. I mean, in a very simplistic way, just think of a magnetic field line as a rubber band. How do you kind of stretch this and twist it up to actually have a cyclic process where you can induce more and more magnetic field? So, we start with one, um, magnetic field line here. We stretch it out. We twist it one time to a figure eight. And then we kind of fold this back on each other. And then we end up, you know, with these two magnetic loops. But not quite, because you have this crossover point here. If you have some reconnection here at this crossover point, in the end, you end up as two loops. So, you essentially, right, by stretching it, you double the field strength. And then you have, no, two loops. So, I also have twice magnetic flux. And then you can continue this process over and over again.

A different way to do this is, you can again stretch this out. Then you bring these, um, field lines together here in the center. You reconnect here. Then you end up as two loops. And then you just shift them on top of each other. So, you end up in the same process. The one important thing I want to point out here are two things actually. The one is, any of these two processes, you can only do in 3D. You cannot make this work in 2D. And that is a general property. I mean, pretty much all these dynamos, they require 3D. There might be some very exotic cases if you have some crazy anisotropic magnetic diffusivities where you can make something work under different conditions. But if you just have ordinary magnetic diffusivity, you need a 3D space to make this happen. The other thing is, you, you need, yeah, what do you mean in 3D? Yeah, I mean, in some, right, you have to kind of bring this, right, for instance, the twisting, you have to do, you need 3D. And then you have to kind of fold this up where you have to go to 3D. In this case, you kind of have to repack this by kind of shifting the two loops on top of each other. I mean, yeah, I mean, you can have this embedded in a 3D space. I mean, you can have much more complicated structures. But you can just have a magnetic flux tube kind of, right, the toroidal magnetic field loop closed into each other. This is just an illustration to just highlight some very basic properties. But the field in reality can be much more complicated than that.

One important thing I wanted to point out is you need somewhere in this process, either here or here, some magnetic reconnection to actually complete the process. But there is a very fundamental difference between these two ways to do it, right? If you take a rubber band and do this, and you do not reconnect the rubber band, you actually can continue this multiple times without ever having to reconnect the rubber band. It just, you induce some very small-scale twist on the rubber band which just builds up and makes it much more complicated. But you can continue this process. Where the second process here, you have to explicitly cut your rubber band and reconnect it. So, if you now consider the situation that you reduce the magnetic diffusivity set toward zero, this process here will become very slow and inefficient. Where this one, you can continue because these topological features, you have to reconnect away, they will kind of propagate to smaller and smaller scales. That leads to another distinction of dynamos. Essentially, a fast dynamo is where the growth rate is independent of the magnetic Reynolds number. There is a slow dynamo, the growth rate will be limited by resistivity because you have this explicit resistive, um, step in there, where you have resistivity on a large scale having to, um, reconnect the field. When we talk about astrophysical systems, we're always interested in fast dynamos because all astrophysical objects have these extremely large Reynolds numbers. Slow dynamo wouldn't work in the astrophysical context.

Okay, now look at a very basic example. Let's consider a star like the Sun. And let's just look at the axisymmetric situation. So, we have an axisymmetric magnetic field, which you can write in this way, that you have the toroidal field, where which is in the azimuthal direction, and the poloidal field, which you can write expressed through a scalar vector potential. And you can do the same for the velocity field. And the velocity field, we have a meridional flow, which is our theta component, and the differential rotation in the azimuthal direction. And we do know for the Sun, for example, what the differential rotation looks like. And the meridional flow, for the most part, what we know for sure is that it's poleward near the surface. There's some debate on what it looks like deeper in the convection zone. But if you now look at this most basic example and you put this into the induction equation, you get these two equations for the toroidal field B and for the vector potential of the poloidal field. And for the most part, the meridional flow component is just transporting these fields around within a meridional plane, within the r-theta plane, but it doesn't really do anything dynamo-wise and actually amplifying the field. So, we can for now ignore these terms. Then you have the red terms here, which are just the magnetic diffusivity terms, which lead to a decay of the magnetic field. And you have the differential rotation term here, which actually takes the poloidal field, which you get from the curl of the vector potential here, and this becomes a source term for the toroidal field. So, when you, for example, start with a very weak poloidal field, so only have an A in here, and you have a very strong differential rotation, you can have a situation where at least initially, you see a strong increase of magnetic energy because this term will build up a strong toroidal field. But this initial strong increase in energy does not mean you have a dynamo because eventually, since there is no term here which can maintain the magnetic field in this equation, the poloidal field will decay away. At that point, this term vanishes, and also the toroidal field will decay away at this point. So, that's one important thing is, when you want to evaluate if a certain flow field is a dynamo, you really have to run it for long enough. And long enough means multiple ohmic decay time scales before you can be sure if it's a dynamo or not. So, obviously, what is missing here is, we need an additional term which can regenerate the poloidal field from a toroidal field to actually make this a dynamo. But this is actually a little bit hard because the problem we are seeing here, um, can be actually proven much more formally. There are a lot of anti-dynamo theorems. And Kari's anti-dynamo theorem essentially says that if you have a stationary axisymmetric magnetic field with currents limited to a finite volume in space, you cannot maintain this field by having a velocity of finite amplitude. So, it's kind of a big no-go. The only way you can actually make this work, you need a much more complex field, you have to add non-axisymmetric magnetic field components.

And just a very brief history of how dynamo theory came along. In 1990s, there were the first suggestions that magnetic field, the solar magnetic field, may be maintained by motion of conducting fluid. But when people really got serious about studying this, the first things they found is essentially all these situations where it cannot work. And really, the idea which got this going was 1995 by Jean Parker. And he suggested to do a decomposition of the field into axisymmetric and non-axisymmetric parts. And then average over the induction effects from these non-axisymmetric fields to actually get additional induction effects which can maintain the axisymmetric field. And this was then later more formalized here by Braginsky, Steenbeck, and Krause in form of the mean field theory, which I would like to introduce, um, in the remaining slides in this talk. And only in the last two decades, we had essentially, um, 3D dynamo simulations. Yeah, yeah, okay, yeah.

So, in order to do this decomposition into mean field and fluctuating parts, you have to do some averaging of differential equations. And in order to actually do this, you want to only consider averages which, um, follow these set of rules, mostly because when you have derivatives, you want to be able to kind of exchange the averaging procedure, that the average of a derivative is the same as the derivative of the average quantity, and so on. That is just to be able to manipulate the, um, differential equations. So, for example, if you have the axisymmetric mean, like the longitudinal average, that fulfills these rules. Another average you could consider is an ensemble average. Right, you have a turbulent system, you average over a large amount of realizations of these turbulent systems, to define your mean. If you apply these averaging to your induction equation, you can derive an induction equation for the average field which looks pretty much the same, right, at least as far as these terms are considered. But you do get this term, which is the average of the fluctuating parts of the velocity field in the mean field. And this is a new term in the induction equation, and which is, um, called the turbulent electromotive force. And this is essentially the essence of mean field theory to kind of study under which conditions this additional term you get in here can actually enable a dynamo which maintains the large-scale magnetic field. So, in principle, how you go about this is, you also, when you subtract this equation from the induction equation, you can get derived an induction equation for the fluctuating parts. And if you have a mean field, you essentially have this term here, which is the generator of your fluctuating magnetic field. You essentially want to solve this equation for a given mean field to evaluate what the fluctuating field is. And once we have that, you can compute the electromotive force. And then you can evaluate whether this electromotive force actually is maintaining your mean field. The problem is, solving this equation is way more complicated than it might look like. But it's not an easy task to do.

So, in general, just to study kind of general properties, people do a strong simplification, essentially that they expand these, um, electromotive force in terms of components of the mean field and the first-order derivatives of the mean field. And the assumption which really goes in here is that you do have a strong scale separation, meaning that the scales of these fluctuating fields are much larger than the large scale which defines the mean field, and also the correlation time of the turbulence is much shorter than the large scale as the, um, evolution of the system on the large scales. So, the question is now, if you have an expansion of this form, what are the properties of these coefficients A and B? You need to be able to enable actually a large-scale dynamo. It makes sense to do a little bit of mathematical, um, nudging of these coefficients to write it in an equivalent form. Essentially, you separate these A coefficients into a symmetric and anti-symmetric part, which gives you these first two terms. And then you decompose this gradient tensor of magnetic field into the curl of B and some other terms, which I ignore here to keep things simple. And then you again do the same trick that you decompose this, um, tensor you have in front of the curl B into the symmetric part and the anti-symmetric part, which gives you these coefficients or tensors alpha, gamma, beta, delta. And this is, there's nothing new I put in here, it's just a rearrangement of terms. And if you put this now back into the induction equation, you see that some of these terms look similar to terms you already had. For example, the gamma term looks like a velocity, whereas the beta term looks like the magnetic diffusivity. So, if you had already trouble getting a dynamo to work with these gamma and with the velocity field and the eta term before, it likely won't help you here. Now, there's one exception, of course.

this better is now a tensor, and it can be very complicated. It can be anisotropic. It could be even not positive definite. So there are some very exotic situations where this one could um produce Dynamo action. But the most interesting term is actually this Alpha term, where you induce a magnetic field, or you have here a term which is just proportional to the toal field. And that gets back to what I showed when I did this decomposition into the axis and non-ais symmetric part: we needed this term to maintain the poloidal field for a given toal field. And as I will show later, this Alpha term will actually provide that. You can do some anal is of symmetry constraints, but for the sake of staying on time, I will skip over this, but just give you the summary of it. Depending on what sort of turbulance you have, if you just have isotropic turlings, for example, you only get this better term, which is something like an enhancement of diffusivity; that is called Turin diffusivity. If you then add ratification to the problem, you get have again this better term, but it becomes anisotropic with the direction of stratification. In that case, you can get this gamma term, which is a transport term which acts like a velocity. And if you have only rotation in the system but notification again, you can have these better term, but it now has some an isotropy with respect to rotation. It can get these additional Delta term, right? Which is this one here, which can be actually a very exotic lead to some very exotic Dynamo effects as well. But this Alpha term, which is actually the most interesting for Dynam action, only exists if you add stratification and rotation together. And you can actually derive what these Alpha, Gamma, and beta terms should be if you make some strongly simplifying assumptions, like the fluctuating field is very weak compared to the mean field, and you have quasi isotropic turbulence, but it is non miros symmetric, so it can have helical Flows In This turbulence. And if you do this, you actually can derive very simple terms for this expressions. And what you actually see is that this Alpha term in this particular case will be a scaler which is proportional to the kinetic helicity of the flow with to the negative kinetic helicity. The beta term, which I call here know the turbant diffusivity, is just given by the convective flow amplitude squared times the correlation times give us 1/3. This GMA term, which is a transport term like a velocity, sees actually the gradients of the turbulent intensity. And so for example, if this correlation times key is in variant of space, you can just write this as um the gradient of the turbulent diffusivity. And so let's first think about these first terms, the turbulent diffusivity. This one just is a term which enhances the dissipation of magnetic fields. You can write this expression essentially it's on the order of something like a length scale time times the turent arms velocity, because if you take the correlation time scale times the velocity, that roughly gives you something like the length scale of your turent edes. And this one you can now write something like the magnetic Rance number like the microscopic diffusivity. So this is the dramatic enhancement of the effective diffusivity of the plasma. The PA of the Sun at the grain can be something like 10 to the 10. So this is a very dramatic effect. And yeah, let's see it here. So this gamma term, it's some also called um turent diamagnetism, mostly because it has a transport velocity which sees a negative gradient of turent diffusivity. So essentially, when you have turent diffusivity with various in spaces, what this term is, it expels magnetic field from regions of strong turbulence. Another way to think about this is also turbulent pumping. In the convection zone of the sun, you have the fastest flow velocities near the Photosphere of the Sun. So these gradient right with a negative sign leads to a gamma term which is radially inward directed. That's the reason why people call this turent pumping. If you put a horizontal magnetic field somewhere close to the Photosphere, this effect describes a transport of magnetic field into the solar interior, downward transport. And now this Alpha term is not the most interesting one related to the kinetic helicity. So what this term is doing really is the following. In order to get this, you need both rotation and stratification at the same time. If you have stratification, what it means is that UPF flows have to expand and down flows they have to contract. At the same time, since you have rotation, you do have pois force acting on these flows. So in this case, the expanding up flows they will turn clockwise, whereas the Contracting down flows they turn counterclockwise. But that means in both cases it follows kind of the left hand rule. Here you have a negative helicity in there, which means in this case you would get a positive Alpha effect. What this Alpha effect is really doing, if you have no a horizontal magnetic field line threading through here, is that you first by the up flows and down flows you bend this magnetic field Fe line, but at the same time you twist it. And then when you twist these Loops, you bend up here. Essentially what you do is you add again a poloidal field to this toal field you started with. And that is the essence of the alha effect, and that is the reason um why it actually makes a Dynamo work. So you can look at this by first inspecting the induction equation for for the mean field. So you have in the you have this syc for the mean field, which is the omic disspation of the mean field, and then you have here a thing which is related to Lawrence Force work the mean field might do. But the other term here, when you know this new Electro mod Force comes in here, you have this J do e term, which is actually the induction by energy conversion by the alpha effect. And you see here that the alpha effect can actually pump energy in the mean field if you do have if this J dob isn't zero, which is also called the current helicity um of of the magnetic field. Now a different way to look at this is just the following. You put these Alpha B in your induction equation, and you get now an equation of the form that the induction effects are this way that you in induce a new mean field which is pro prop to the mean current. So for example, if you do have let's start, you have a poloidal magnetic field, the mean current you're having is actually a toal current. This equation now tells you if you have an alpha effect present that you actually induce a toroidal magnetic field B. The toroidal magnetic field B has no poloidal current, and so you again with this toal magnetic field you induce a magnetic field. So this Alpha effect kind of allows you to go constantly from a toal to a pidal field back and forth, and that way you can go through a Dynamo cycle and maintain the magnetic field. And at least from this very basic example, you can see that in general you would expect stationary Solutions; it can just maintain the magnetic field you have here in place if you add no differential rotation to it. Large SC she flow like we have it in the case of the solar convection Zone, and let's assume we do have a pidal magnetic field in here which points in the as musle direction of the Sun. The alpha effect creates the poloidal current Loop, but since you know have a Shear flow, you shear this Loop and you produce a new pidal flux system. But this loop on one side it adds to the magnetic field the total feet you had here on the left side, but on the opposite side it actually subtracts from it. So from this you can actually see if you add these Alpha effect together with a Shear, you actually can get propagating magnetic activity patterns, and they are called a Dynamo wave, even though it's formally if you look at the equation it's not really a wave equation but it's a traveling um activity pattern. And this activity pattern actually would propagate along the Contours of constant Ang velocity, and and this is essentially called the parka Yoshimura Ru based on Parker and Yoshi Yoshi mura who first um discovered this. So if you go back to these set of equations which described the um evolution of an exis mag magnetic field, now if you put in the alpha effect right, then we have now here a closed system. We have no a term which can regenerate the poal field from a toal field, and you can vice versa with the Omega effect maintain the mean magnetic field. And that way you can circumvent this problem we had earlier in maintaining um a large magnetic field which is exos symmetric. So the large SK magnetic field can be still exos symmetric, but we have no small SK magnetic field which is non exos symmetric; we did average over that is why this actually works. Um let skip over this, see how much what time. Yeah, so one interesting aspect of these dynamos is that they tend to produce helical magnetic fields. The magnetic helicity is defined as the product between the magnetic field and the vector potential, and the reason it's interesting quantity is that it follows the conservation law. Now if you integrate, I mean if you put in the induction equation um compute the time time derivative of the helicity, assume you have a bounded volume and you have no helicity fluxes over the system, what you find is that the time derivative of the elicity is given here by this dissipation term which has here the magnetic diffusivity and the current helicity. This is if you just take the standard induction equation to derive this. Now if you put in these if you do this now for the mean field induction equation, you get very similar Expressions, but you do get this additional term which depends on the meanfield electromotive force. So if you have a Damo operating and these the meanfield electromotive force has a component in the direction of the mean field, you essentially produce a helicity separation. You induce helical magnetic field of one sign on the large scales, and helical field of opposite sign on the small scales. So you need to get a separation of helicity. The total helicity if you add the large scale and small scale together, this term drops out, and you can use this in the ASM totic expression, for example, to estimate the helicity of the small scale field here which is Nega has the opposite sign of the alpha effect itself. And then you have also some additional term here which comes from the turent diffusivity on it. And I will come back to this expression just in a minute. But before that, I would quickly since we only talked about kinematic effects, talk about non-kinematic effects: how does a Dynamo like this actually saturate? Now the really the proper way to do this would be doing 3D simulation, so you have the Lawrence force feedback in there, and I will show some examples of this in the second lecture, but that's in general still very difficult. But so how could you get at least understanding how these dynamos saturate? So you can actually do meanfield models which you expand a little bit, where in addition to the meanfield induction equation you also solve some meanfield equation for the large scale flows like differential rotation and the M on the flow. And from the mean magnetic field you can compute the lawence force, and at least this green component here which is the large scale part of sorence force result from the mean feeds themselves, you can in principle plug this back, and through that have a feedback of the Lawrence force on the mean Flo. For example, you can have a situation where these right if you have an alpha omega Dynamo like in the sun, where you have a strong Shear from differential rotation, you would get the lawence force which actually tries to limit the shear of differential rotation and then reduce the amount of differenti rotation, and through that can actually saturate the dyam anal. And this I would consider here the microscopic feedback. But you can also have feedback on smaller scates, like you can change these turent induction effects like the ipha effect. And when people do meanfield modeling, very often they use an expression of this form where this kinetic Alpha effect is reduced once thetic field has a strength comparable to the equipartition feed strength. And equipartition feed strength is the strength at which the energy density of the magnetic field is um comparable to the kinetic energy density of the magnetic field. And this is an expression which is used quite often because it kind of does a job. But the problem with this this expression is if you actually do the mass, it turns out to be a little bit more complicated. And for that I have to come back one more time to helicity, and I just want to introduce this concept here through very easy symmetry argument. Um so I wrote down here very simplified induction equation. I only use this stretching term here. The momentum equation has a very Sim has a very similar term in here; this comes from the Lawrence force. And you kind of see that these equations have an interesting symmetry: if you formally replace the velocity with the magnetic field and vice versa, these equations are actually just stay the same; they just turn into each other. At the same time, we have the electromotive force which is V Prime cross B Prime. Obviously if U formly replace V with b and vice versa, this term has to actually change the sign. So far we only considered this contribution in the alpha effect, but this is obviously not consistent with the Symmetry in the equations. There's a term missing which depends on the magnetic field. And in order to make this actually consistent with the Symmetry, you need in the alpha effect an additional magnetic contribution which depends on these small scale parent helicity. And that now I come back to this term I earlier derived that you can estimate that an alpha effect introduces a small scale current helicity. And then you can add this term here into this Alpha effect term, which is the same as I show before; the alpha this is just the kinetic kinematic component coming from the velocity alone. If you do this now, you actually see that you expect to get an alpha quenching term of this kind. The only difference is that you have here the ratio of the Turin diffusivity to the magnetic diffusivity, which is a magnetic Rance number, and that is a big concern because that means you have something which people call catastrophic quenching of the alpha effect. They're already form magnetic fields much weaker than the acup partition value; this Alpha effect would disappear since the magnetic Rance number is very large. Now this only happens when you consider a situation where the large scale field is just a homogeneous B field. But if the large GE itself is maintained by a Dynamo, it's helica itself, and then you have some of these additional terms in here. If I go back right, you have this additional terms which do not necessarily vanish in this very catastrophic way. You also could have ficity loss. And one idea is how to actually prevent this cat catastrophic quenching is that the Dynamo has to constantly shed small scality through the boundaries of the system. And the Sun is obviously doing it; that is all the corona Mass ejections you see all the time happening. And if you estimate the helicity loss through all the mass ejections, at least it roughly adds up to a value where you could get a Dynamo um continue to work with the alpha effect. But I just wanted to highlight here that really all these dynamos saturate can be in detail a very complicated process, and this is still not a completely settled question; this comes up again and again. And what really makes it hard is you cannot simply do a 3D simulation and figure out how you get around this, because in 3D Sim simulations very often your magnetic rains numbers are not high enough that this really becomes a problem. So you we have a lot of 3D simulations which work fine, but that is maybe because they don't have these problem with catastrophic wrenching, but they would be if you could do a 3D simulation at the magnetic RS numbers of 10 to the 10, which we actually can't do. Um I will talk about 3D simulation in the second lecture, so let's just skip over this. One last thing I would like to um ask you the question: where actually did the first magnetic field come from? Because if you have an induction equation of this form, b equal Z is always the solution. So how do you actually get a magnetic field in the first place? And and when I introduced arms law earlier today, I had these addition term in there, the electron pressure gradient, if you remember this was highlighted in green. So if you keep this term actually in here in the equation for the relation between J and E, you actually get an induction equation of the form you have this additional term in here which is a source term for magnetic field which doesn't depend on magnetic field itself, and with that actually you can create magnetic field from scratch. And for example, some thoughts where the first magnetic fields came from is when you had kind of the early Universe, you had ionization fronts from kind of the first bright objects in the universe, either stars or even on scales of galaxies, which essentially were driving ionization fronts to an inhomogeneous medium, and that essentially led to a situation where this gradient of the electron density and electron pressure were not zero. In that case you would estimate it's a weak feed of 10 Theus 23 G. You can produce then you have collapse of this to form galaxies, which amplifies it a little bit. Then you can have a galactic Dynamo working on it. So in principle, over the 10 gig years or so, you could get to the 10 to Theus 6 go of el Galactic Dio have today. And then this magnetic field kind of collapses and that becomes the seed field for stars. But even if you forget about all of this and do a very extreme your duning experiment, just go to the Sun and set the magnetic field to zero, how long does it take for the sun to remagnetize? Now the interesting thing is that this term you can also compute this term in the Photosphere of the Sun, and you will see that this in particular strong at the edge of solar granules. And it was estimated by the paper by Alena Keno a couple years ago, you get actually a feet on the order of 10 to Theus 6 go. Then you have this very efficient small SC Dynamo that has a growth rate likely on the order of minutes. So if you set the magnetic feet to zero on the sun, you start with 10 to Theus 6 gos immediately. With this Dynamo growth rate of a few minutes, you just have to wait a few hours, and all these salt and pepper feet in the quiet sun is back to where it is today. Now to get the magnetic field organized on larger scates, you have to wait longer. If you want to get the field organized on a scale of super gulation, which has a overturning time scale of a day or two, you have to wait a day or two. If you want to get it organized on the scale of the convection Zone, you have to wait for overturning time scale of the convection Zone. But at least the small ski field will be back almost immediately. The large scale field likely you have to wait something equivalent to several solar cycle time scale, several hundred years for that to come back. But all these time skits they are extremely short to the life cycle of a star. So in some sense, it's completely unavoidable to have a magnetized star like the sun; even you set it to zero, it will reset itself very very quickly. Okay, that's pretty much it for this part of the lecture. So the next lecture I will talk about a lot about the solar Dynamo, and then make some comparison of solar Dynamo to dynamos in solar like stars, and also talk a little bit about the geodynamo, mostly by comparing what is actually similar and different. Okay, thank you very much.

Okay, yeah, there one I was wondering about: what I I think you were calling um diamagnetism, yeah, this uh I was wondering about the if the conclusion of that slide was that because the sun has a actually radial outward gradient in velocity, that means it's actually pushing Fields it generates inward. Yes, that is happening. Yes, that's called turbulent pumping. But that's only happening beneath the Photosphere of this. Actually, once you go above the peak of velocities in the granulation layer, pretty much at Optical depth Unity, this term actually changes the sign, and then it pushes field out. And some of that is actually seen in the sun, mostly in the quiet sun. People found that if you go above the Photosphere, you have a preference for horizontal fields, and that might be part due to that term. That horizontal Fields is a Photosphere actually pushed out, and the preferred location for that Fe to exist is in a velocity minimum which is about 400 km above the Photosphere. So yeah, there's kind of at the Photosphere you have a Divergent um Divergence in the transport by this term. And the fields that are getting pushed inward, do you think that that helps to sort of organize and grow um a large scale field? Yes, it's certainly the case. I mean, one problem is for at least some mean Dynamo models people have been um putting together for the sun is if you take the turbulent diffusivity for the magnetic field as you expected from mixing length Theory, that's actually a pretty large value, and that would actually very efficiently transport the magnetic field towards the boundaries of the convection Zone, which actually makes it sometimes very hard to make these dynamos work. So having strong turbulent pumping in the surface layers to keep the push in, push the magnetic feet in, can be very helpful there. Thank you.

Um, I don't know much about Dynamo; this could be a naive question. Um, but if you sort of need you know the the fluid to move around and create Shear flows to generate field, does that mean that plasma beta is a useful sort of parameter to characterize whether a Dynamo can exist or not? In general, yes, because you really want to do a lot of work on the fluid, and that mostly happens in situations where really right the magnetic field is kind of weak compared to the gas pressure, so that really you have a lot of kinetic energy around which can act on the magnetic field and Shear it up. When you go in a low better region, it's typically the opposite way that the magnetic field actually releases energy and does work on the fluid, like in the Solar Corona you would not necessarily expect to have a Dynamo operating there.

Uh, how much do we know about when a Dynamo action breaks down? Like what could break down a Dynamo? You mean break down in terms of saturating where it kind of kills of the kinematic Ross, or really in the sense that it completely kills the Dynamo? Kill killing of the well, I mean one big concern, that's the reason why I actually added it in this lecture, is this catastrophic Alpha quenching right which I had in here related to the magnetic helicity. You have a buildup of small scale helicity which through this additional term in the afha effect can actually completely offset the afha effect. And this has been a big concern because I mean the the magnetic raino number in the sun is something like 10 to the N9, 10^ the 10. So essentially it would mean that the alpha effect already should saturate when the feed of the Dynamo is something only 10 the minus 5 times the etition feed strength, which is extremely weak. So in that case, if if you're really having a situation where you cannot find any way to circumvent these catastrophic quenching of the Ia effect, at least a Dynamo using the Ia effect wouldn't work. Now there are other effects. I mean you remember maybe late early on I had this additional Delta effect which I didn't discuss much, but there are some other effects you can maintain magnetic feed which are non helical, which would not be subject to this catastrophic quenching. So there are other things. But in general, if you it looks like from everything we know, at least when you have a high magnetic rains number, it's close to impossible to avoid this Dynamo action, at least the small scale magnetic Dynamo. What we know right now is if the rux number is larger than a few hundred, and you cannot avoid it from happening. And people were early on concerned about the influence of the magnetic pront number on it, but at least the most recent research showed the magnetic prontal number if you go to low prontal numbers, you raise a little bit the critical Ros number, but not that much that you would avoid the Dynamo completely under the conditions you have in Stars. So in general, I would say the only situation where you can be sure that you don't have a Dynamo if your magnetic Rance number is to lower, and to lower means typically lower than a few hundred. Otherwise it's unavoidable to have this happening.

I have a relatively basic question: is the average 11-year periodicity of our solar cycle function of the rotation rate of the Sun, and do stars that rotate faster have shorter solar Cycles or Stellar um activity Cycles? Yes, in and I will talk a little bit about this in a second lecture. But in general, people see a trend that the cycle periods get shorter if you have a faster rotating stars. But as I will show in the second lecture, even for the sun, people don't necessarily 100% agree why we have a 11e cycle. So it's kind of a challenge, since we can't actually measure feet in the convection Z, we only see what boils up the Photosphere. Actually, even though we that close to the star, it's still a big mystery how it actually produces Las.

So you made it through most of the equations. I think this talk, this part of the talk, is mostly equation free; it's more figures and movies. So it's more about now actually applying this um to understand the sun, stars, and planets. So just to start with, talking a little bit about the solar magnetic field. I mean I already mentioned in the previous talk that the larka solar magnetic field has certain symmetries which are shown in this movie, which shows the evolution of the um Photosphere magnetic field over multiple Cycles. What you can see is that at any given time, the Sunspot in a given hemispheres have a preferred polarity, where always the leading and trailing spot have the same polarity, but then in the southern hemisphere it's the opposite. And every 11 years, um that polarity again changes. Then you kind of see is that from the Sunsport, you see some of these magnetic field actually being transported towards the PS, which then leads actually to the reversal of the Polar Fields. You look at the sun in white light, you do see the sunspots mostly As the dark features, since feet is strong enough that it can locally suppress the convective energy transport. If you look at these sunsports in great detail, you see these marvelous fine structure of sunsports. You have the dark umra, their convection is mostly absent, not completely but mostly, but then you have this regime of transition called the pin amra, and towards gation further out. So if you want to have a large scale dynamo theory, you want to explain how essentially at least the large scale patterns come about, not necessar all the detail of the Sunsport structure. But if you look at Large Scale flaws, you also do see some solar cycle dependence of large scale flaws. This first panel here shows something which is called the torsional oscillations, and this is essentially showing the changes in the differential rotation over the solar sign. And what you actually do see, you see these bands of faster rotating and slower rotating fluid which kind of follow the um butterfly diagram of the solar activity. You see the contour lines overlay here indicate the um butterfly diagram. Helos seismology can also look at these um Chang in different rotation throughout the solar convection Zone, and you see here in low latitudes kind of these um Ranch which kind of extends somewhat into the convection Zone. We don't know how deep it really goes because the sensitivity is very poor towards the lower parts of the convection Zone, and you have stronger changes in the difference rotation near the PS. And some of this is likely due to some feedback of the Dynamo generated field on the differential rotation. Since the differential rotation induces a lot of magnetic feed, you expect the laforce feedback actually to work against it. And if you do model which include the this Lawrence force feedback, you do get some of these variations. At least the low latitude branch is more um difficult to get. And thetion oscillations cannot just be seen by helos seismology; they also can be seen in surface Doppler measurements, which is actually done here. You see here this is again the zonal component, the different rotation modulation, but you also see this is a marional component where you do see flow was moving away from the equator towards the poort. There is also some imprint of the active region belt on the SLK flows. Yeah yeah yeah yeah. I think for the helos seismology, well these ones I thought isn't they normally do something like 100, 100 day averages or something like this at least to get these deeper reaching flows. And of course, if you do these surface measurements, I mean you have to you have to want to at least average over enough indivi individual sunspots, for example. Some of these flow patterns you see in the Mar in the flow, they are kind of maybe related to larger scale inflows into active regions. So you definitely have to average over multiple of those that you don't just see the individual flow into active regions, but really the impact on the mean flow pattern. But I thought the hos seismology is typically something like 108 days if I remember it correctly, but they typically do so.

If you want to construct the solar Dynamo model for the sun, what's really the goal is? I mean the goal is really the model should kind of explain the solar like activity pattern in terms of having the cyclic Behavior, the equator but propagation of activity, have surface flux Evolution consistent with observations, have the large scale flow variations with consist consistent with observations, and then if you want to go a little bit further, also show a solar like EMP ude variation from cycle to cycle, and maybe even go one step further and trying to actually predict the future activity. So but the honest answer is that most models already struggle a lot just with the part one. So in in this talk I will mostly um focus on part one, how we actually understand just the basic activity of the Sun. But nonetheless, trying to explore for example why different Cycles have different amplitude certainly provides you with some additional constraints on how these Dynamo models actually might operate.

So what are the basic ingredients we having? We have flar scale flows. Most obvious ones are the difference rotation, the medal flow which is kind of these large scale conveyor Bel like flow. You strictly only know that it's definitely forward in most of the near surface layers; where exactly it returns and if there's just one big structure in the convection zone or if there are multiple cells, that's still open to debate. And then in addition to having these mean flows, they also do have some cyclic variations. Then we have various turent um induction effects. We have transport effects like the turbulent diffusivity, advective transport is the turbulent pumping. Then we have um effects which enable the Dynamo, most prominent the alpha effect. But then if you look in more detail, I mean if you look in the Photosphere, you don't just see a large scale homogeneous mean field; you actually see that the field is actually made out of these individual field of sunspot. So there is some organization actually on smaller scales. So really what you also want to understand is how the Dynamo produces a large scale field, but how this actually then leads to flux emergence, that some flux loops move towards the Photosphere and form something like sunspots. And actually, as I will show, there are some theories which actually think that it's actually these process of having sunspots forming in the Solar Photosphere that that is actually a critical component to the Dynamo itself; it's not just the byproduct of the Dynamo. In some sense, these sunspots moving into the Photosphere the typically have a systematic tilt angle where the leading polarity is closer to the Equator than the trailing one, and this on its own leads to something like an alpha effect, since it requires some twisting motions, and that's called the de mechanism. And that alone can actually explain a lot of the activity we do observe.

The modeling approaches we have a meanfield models where we solve equations for the mean field, mean flows, and mean magnetic fields. They are much less expensive, but the problem is you need to have good models to actually um describe all these mean feied induction coefficients I just um talked about in the previous lecture. In some sense, these models are not typically from first principles because they have a lot of degrees of freedom; you need to have some underlying turbulence model to derive these meanfield coefficients. The other approach is 3D numerical simulation; they solve the full set of equations, but they're also not from first principles because they're very expensive. So you it's challenging to run them for long enough to understand something like a solar cycle right? The cycle is something like 11 years; you really want to run them over multiple Cycles, which is expensive. So you have to compromise on the resolution, or you do high resolution for short periods, but then you cannot capture a full cycle. They do give some understanding of Dynamo ingredients, but we not at the point is where you just do the most obvious thing: you just take a rotating convection Zone, you drive a solar Luminosity through it in terms of energy flux, you rotate at the solar rotation rate, and just something which looks like the sun comes out of it. That's typically not happening in these models. Now with advances in Computing, of course this balance is shifted more and more towards 3D simulations, but at this point we still needing both.

Let's go to meanfield models. So one thing you have to keep in mind is when you do mean feed models, you only consider average quantities. So in some sense, sunspots there a key feature of the solar cycle because it's the most obvious thing you see if you look at the Sun, but meanfield models just average over them. So you have to keep that in mind; you cannot really capture something on that scale. The other problem is really if you go back to these basic assumption meanfield theory that you can express this turent electromotive Force as some expansion of the mean field, even if you just go to these first order in the derivatives, formally you have here 36 mostly unknown functions of R and Theta in the convection zound. There's enormous amount of degrees of freedom here, and they're not trivial to compute from first principles. So typically mean feeld model they use strong simpli simplifications for this, like a scalar Alpha effect, just a scalar turbulent diffusivity, to explore kind of the type of possible solutions are there. But once you want to go to fine tuning, you're really getting lost in degrees of read them. You can compute these mean field coefficient from 3D simulations, and there have been a lot of work about this in recent times. And what they typically show is then you do this consistently, and what you essentially do is you run a 3D simulation, but instead of just solving one induction equation, you actually solve an additional number of induction equations with imposed mean field. And these induction equations they are just diagnostic equations to measure if you impose a third certain mean feed, what are the induction effects you getting. And by solving of these additional reduction equations, which is called the test feed method, that you can get in the end enough constraints to actually compute all these coefficient from your Dynamo simulation. And what they typically find is you pretty much need all of them. And if you do this, you can end up with a meanfield model which has solution very similar to what you had in a 3D simulation, but you really need all these coefficients, which makes it very impractical to really use this as an approach to model the solar Dynamo from first princi. But it can be also a very useful approach to actually understand what's going on in your simulation.

So when you look at the solar differential rotation pattern, it kind of became very obvious that there are some interesting regions on the sun, like you have these strong Shear layer at the base of the convection Zone, and you have this near surface sheer layer. So people very quickly Focus the research on kind of understanding it: what are the different contributions from these different Shear patterns? You can have dynamos which operate in the bulk of the convection Zone; they have mostly latitud Shear. You have D can have dynamos which operate at the base of the convection Z where they have radius Shear. Or you have can have near surface Shear layers; they again have rage here. So a lot of uncertainties are really where you have the location of the D DN, where does the alpha effect really work, and what is the um turent transport, and what is the role of the Marion flow right? This large conveyor build flow which kind of works together with the turbulent transport mechanisms to transport magnetic field in the convection zone.

So the first model I quickly want to talk about here are kind of these thin layer dynamos, like Dynamo which operates in the shear layer at the bottom of the convection Zone. And if you solve the Dynamo equations and put in a she like the solar like sheia, you end up with these sort of Dynamo Solutions where you see propagation in Latitude. But the problem is, since the radio is sheer and high latitudes and low latitudes is opposite sign, you actually get the propagation two bands of propagation: one propagates equator what, one propagates forward, which is not exactly what the sun is showing. Another problem you very often see with these thin layer dynamos is when you look here that the length of the Dynamo wave here is the wavelength is pretty short; it's roughly comparable to the depth of the thin layer. If you look at observation of the sun, you basically see these extended butterfly wings, and you typically see only one of them at any given time. But once you constrain a Dynamo to a thin layer, you can see a multitude of them, which is not exactly what the sun is doing. And this propagation is a big problem because depending on what you're doing, if you take the most basic kind of model with an alpha effect which is proportional to the coine of the latitude, you mostly get this pow polar branch which propagates equator wats, but you don't see much happening in low latitudes. You can if you constrain these Alpha effect a little bit closer to the Equator, you can get a low latitude Branch, but it propagates in the wrong direction. If you then switch the alpha effect around, you can in the end get something in low latitudes which look solar likee, but you still have the strong polar branch which goes the opposite way too. And so you kind of see here I mean this was kind of before there was any observation of differential rotation; people just made up a solar interior rotation pattern they thought is Meaningful, and everything looked fine. But when helos seismology really nailed down differ rotation, all these models got into a big trouble. Yes yes, typically this is in some dimensionless units here. So but I mean yeah this is kind of you see kind of this would be in the sun this would be 22 years for example from here to here. And so in some sense you can sort of make it work if you kind of squeeze the alha effect, restrict the alha effect to low latitudes, and you have to change the sign. Now the sign change is not necessarily unexpected. I mean the problem in the northern hemisphere, the kinetic helicity is negative, which means you have a positive I effect, but that is mostly in the bike of the convection soone. UPF flows expand, down flows contract. But if you get to the bottom of the convection Zone, because you have there a boundary condition, down flows eventually have to expand, and that changes actually the kinetic helicity in the convection zone. So if you do have a just operating at the base of the convection Zone here, it's not necessarily unexpected that maybe the alpha effect might be negative. But the problem is of course you have these high latitude branches, and very often you have a little bit too much overlap between neighboring Cycles, which isn't what the sun exactly is showing.

Now then there's a large amount of dynamos which are more distributed over most of the convection Zone, and the model which has really gotten a lot of attention over the last 10, 20 years is what people called the flux transport Dynamo. And here the idea is we have these M on the floor, this conveyor belt flow forward at the surface. So somewhere there has to be an equator flow. So people were um thinking how about you add a flow in there, and you can get Solutions like the one shown here where you have this forward flow its equator what at the bottom of the convection Zone; it's actually this advection effect which pushes the magnetic field um towards the equator. And this class of model has been actually quite successful in producing the solarik behavior. And most of the time they mostly have an alpha effect which is um constrained to the near surface layer to just iic what actually sunspots are doing. When sunspots are rising toward the surface, you have coris force acting on the rising um magnetic flux bundles, and what that typically means is you have these systematic tilt angle where the leading polarity is closer to the Equator than the trailing one. That is kind of indicated in this schematic. You do have differential rotation producing pidal field from zidal field. Then you have some of these flux rising to the surface, and you have the coris force tilting these spots. And then you have due to these tilt angle, the leading polarity has a higher probability to actually reconnect across the equator and partially cancel across the equator from the trailing polarity. So in the end, you have the M the flow preferentially transporting the magnetic field from the trading polarity to the boards, and that essentially regenerates then the polar field. But it has then after you go through a cycle it has the opposite sign, then you start the same process again with the opposite sign. And these entire class of Ms are called backc flux transport Dynamo models, and in general they do have a good agreement. But still a lot of models have kind of these activity Cycles which start in too high latitude; you still have to kind of come up with reasons why we only you see the lowermost tip of this um activity pattern. But as other mean feed models, you have a strong sensitivity to many ingredients. And really one concern in these models is you know what is really the structure of the medal flow in the Solar convection Zone? You know it's forward it's a surface, but the structure of the flows in the convection zone is still heavily debated. I mean typically meanfield models they assume these really nice flow sales like indicated here. If you do numerical simulation, this is actually one of the nicer looking ones where you have mostly a single flow, so but you already see there are some particularly near the base of the convection so here some patterns of counter cells, and also you have something here near the surface. But one big problem with these models is you have kind of in order to have the mar flow being able to transport the magnetic field, you do have you need the turbulent diffusivity which is low enough so that it's ation dominated. Turent diffusivity is something like a scale height in the convection zone times the turent RS velocity. So you typically these models have to assume a low turbulent diffusivity, which you can only get if the RS velocity in the convection zone is somehow lower than you expect it to be. But there's another problem: is in order to enable or maintain the merional flow velocity through some turent transport processes of Ang mum, you also get the M on a flow speed which depends on the RS velocity squared over the rotation speed. So if you bring this down, you also bring the M flow speed down. So kind of stuck in there; you cannot just lower the tur diffusivity without also getting a too low M on a low speed. Some models can partially compensate for this by having a strong r turent

Pumping which kind of keeps the magnetic field, um, kind of pushed towards the lower part of the convection zone, where the flow is assumed to be, um, equatorward. Now the problem is, if you use helioseismology to kind of infer what is actually the flow in the deeper convection zone, it's not that clear. I show here two helioseismic inversions. They all these surface layers have a prograde flow, that is the one thing they all agree on. But this inversion, for example, shows an equatorward flow here in the middle of the convection zone, again a prograde flow at the base of the convection zone there. This more recent inversion actually shows something which is more the single flow cell. But some of these models, for example, they also put in constraints like divergence r v equal zero. Some of them are just straight helioseismic measurement. This is an example of an inversion which actually put in this mass flux constraint. Then some of the inference in the deeper layer is not really a direct measurement; it's something which you get out of the, um, mass flux constraints. And even using different instruments, some of these flow structures are really even sensitive to details of the measurement. So this is still a big open question.

Now one other complication is, even if you wouldn't know exactly what the meridianal flow is, there is this additional turbulent transport, this gamma term I introduced in the last lecture. The dominant component of this term is the radial transport, but when you are in a rotating system, the gamma effect itself can also have latitudinal transport. So you also have these additional turbulent transport processes which can actually add to the transport of magnetic field. So there's a lot of uncertainty, a lot of assumptions you have to make to actually get these models to work.

So why not just doing 3D simulations, right? You just we know the equations, just don't make any assumption or shortcuts, you just solve the equations including differential rotation, meridianal flow, and the nonlinear effects are automatically included. But even that approach has some intrinsic limitations. You do have to deal with your boundary conditions in the radial direction. You have these bottom layer of the convection zone where you form the shear flow, you have the transition towards the radiative core, you go from a convective regime to a strongly subadiabatic regime where you suppress convection. That boundary layer is not easy to properly address. The top boundary, because these models they cannot typically deal with a density contrast of 10 to the 6 in the solar convection zone, they leave out the top layers. They're not only dealing with the photosphere. Big problem is you cannot capture the Reynolds numbers of the sun, like magnetic Reynolds numbers of 10 to the 9, fluid Reynolds numbers even 10 to the 12. So you have to do something to the small scales. You either use a direct numerical simulation where you actually use the correct expressions for the dissipation functions, but you have to use dramatically increased diffusivities to actually make this simulation work. Or you do some sort of implicit large eddy simulations where you mostly use numerical terms to stabilize your system, which means you introduce diffusivity just near the scale of the grid and keep the large scales mostly untouched as much as you can. But even with doing all that, they're very expensive, and they provided some understanding of ingredients of the dynamo, but no complete picture yet.

Now the first simulations in rotating bodies have been done 1981 by Gilman and Miller, and some of these really early simulations really at low resolution. I mean, at that time what you could do is something like 32 grid points cubed or maybe 64 on a good day. That something you can run could, in principle, run on your cell phone nowadays. And some of these early models actually looked pretty, pretty promising in the sense that they produced already large scale field and and periodic reversals, even though the propagation of field was not always, um, that, um, equatorward. There has been a little bit more success with these models, um, for the geodynamo, and I talk about this in the end of the lecture. During these 3D solar dynamos started kind of in 2004, and these early dynamos they mostly showed pretty much a big mess of field without too much solar-like, um, activity behavior. Instead of having nice cyclic behavior and a dipolar field on the large scales. But at later times, at least people found that by actually spinning up these simulations a little bit, by modeling a fast rotating star, they could get magnetic field a little bit more organized in the convection zone. Some of the models even included the tachocline, and they could find some more organized field at the base of the convection zone. This is an example where you have these large scale toroidal bands produced with opposite polarity in each hemisphere. But the interesting thing is they only found this activity pattern after speeding up the rotation in these simulations, not so much for solar rotation rate. And this is shown here, where studies were done changing the rotation in these simulations. It mostly found cyclic solutions for faster rotation, but there is kind of this area around the solar rotation rate where it was very hard to actually get a dynamo excited, which is a little bit odd. Why should it be particularly hard for something like a solar-like rotation rate?

This is another example of a simulation which did show a 33-year period field generation in the bulk of the convection zone. So most interestingly, it actually did show something which looks like a butterfly diagram. You had these equatorward propagation of activity belts. This, I mean, the symmetry is not exactly dipolar here, there's some phase shift, they have some superposition of a dipole and a quadrupole. But there was one interesting thing I would wanted to point out here: that when you do these 3D simulations, you have to be aware that sometimes these nonlinear feedbacks, once the Lorentz force gets strong, can have a dramatic impact. So this simulation was started from a weak seed field, and you see a dramatic difference in the behavior of the simulation in the early phases when the magnetic field was weak. You had these extremely short cycles, or not even cyclic behavior at all, and then as the field became strong, you suddenly transitioned to this regime where they had this nice equatorward propagation. So it's kind of a phase transition which was caused by feedback of the Lorentz force on the flows in the system, which is something which is very difficult to capture in any mean-field model. If the Sun is doing something like this, you really can only do this in a 3D simulation. Um, and more recently, there have been a couple simulations which were able to actually build up toroidal field in the convection zone of sufficient strength that they actually could see magnetic field rising in flux bundles towards the upper boundary of the simulation domain, which isn't the photosphere, it's typically at least 20 megameters down from the photosphere. But at least these dynamos they started to make some connection to what we actually observe on the Sun, that you have magnetic flux poking through the surface in form of sunspots. Now the kind of flux loops you get in these simulations, it should be a little bit too large for solar active regions, but at least the simulation started to move in the right direction.

So just to summarize a little bit: a lot of different groups have found in recent years cyclic dynamos with periods somewhere in the 10 to 60 year range. Some of them have equatorward propagation of activity, but there's no simple explanation necessarily for the cycle length and the magnetic patterns. And because the cycle length can be a nonlinear effect where really the Lorentz force feedback plays a critical role in establishing that. And even for those models which do have the equatorward propagation, looking in detail, you don't necessarily get this for the right reason. Essentially, analysis in the simulation, you had equatorward propagation had to do with the fact that the differential rotation in these simulations were not quite solar-like, and you wouldn't necessarily expect this for solar-like differential rotation. A big contrast to mean-field models in these 3D simulations is there's in general no single dominant Turing induction term. If you want to derive these mean-field coefficients from these simulations, you pretty much need all of them, and the nonlinear feedback is really critical to even get, sometimes, these cycles right. So in some sense, we have a situation where both mean-field models and 3D simulations have quite serious challenges in actually providing a consistent model for the Sun.

Well, there's another approach. We could just let's try to do the most minimalistic thing and just ask, actually, what do the observations tell us? The most minimalistic approach here. So the one thing we know that if you correlate the strength of the polar magnetic field to the strength of the next cycle, that has you find actually a pretty good correlation. So here the, all the, um, points on these, um, this plot are essentially the strengths of the polar field over all these different cycles, then related to the strength of the next cycle in sunspot number. The problem is we don't really have good measurements of the polar field except for the most recent cycle where you actually had could, um, measure magnetic fields directly. So what was done in this plot, they actually used, um, disturbances to the Earth's magnetic field due to the solar, um, wind, the AA index, which is related to the polar field of the sun, to actually infer what the polar field of previous cycles was. So in some sense, what this correlation shows is really that the, um, magnetic field at the polar caps, the poloidal field, you see that this is really the source for the toroidal field which is wound up by differential rotation to produce the next solar cycle. And then there is this nice work by Robert Cameron and Man Cha as they pointed out that actually you can, all you have to know is really what is the magnetic field, the radial magnetic field at the solar surface, and what is the near surface differential rotation, and you can exactly compute what is the net toroidal flux in the hemisphere of the Sun. And essentially, you take the induction equation: the dB/dt is a curl of E, you integrate this over the entire hemisphere, and that integral of the dB/dt gives you the time derivative of the poloidal flux in the entire hemisphere. And then you have essentially this integral of the electric field over the boundaries of the domain. Pretty much this polar part and the part at the base of the convection zone doesn't matter because you can move this far enough inward that there isn't much field. The part here at the equator needs just to some exchange between the hemispheres, which you can model easily. And then you just have the surface integral which you actually know exactly what it is, because that is pretty much given by observations. You just need the surface differential rotation, you need the radial field at the surface, and this decay term here that is just coming from the exchange across the hemispheres. And if you then just take the observed differential rotation, the observed radial field, you can use this equation to actually compute what is the expected toroidal field in the hemispheres of the Sun. And you compare it essentially to what you get from observations, which you can get from the magnetic field which actually, um, appears on the solar surface in form of sunspots. And what you see is that these two things actually look quite similar, which strongly suggests that really you can pretty much constrain the net toroidal field produced in the convection zone by just having knowledge of the surface term.

So the question is: is this all we need, or do you need some additional dynamo effects in the solar convection zone? Now the thing is, here this is the net toroidal magnetic flux. If you have any additional dynamo effect which is kind of buried somewhere in the convection zone without making any connection to the surface layers, that one will not give you a net toroidal flux because you get this net only by having this contribution from the surface term. It would essentially produce a mixed polarity for the flux in the convection zone. And there are some constraints against this because the Sun actually follows these polarity rules pretty well, like Hale's polarity rules. The polarity of sunspot groups in the hemisphere, you only see a very few exceptions from these rules, which strongly suggests that the toroidal field in the convection zone is dominated by one polarity. So you wouldn't expect too much contribution from something which is just buried in the convection zone. So essentially, you start with what the Sun tells us: which is the polar field is strongly correlated with the strength of the next cycle, so that must mean it must be connected to the poloidal field which is converted to the toroidal field via differential rotation. The Sun generally obeys here polarity rules with only few exceptions, which means the toroidal magnetic field in the convection zone is likely mostly unipolar. And the surface term, which is observational constraint, essentially this surface latitudinal source coming from the active region eruption pretty much is sufficient to produce a toroidal flux with a sufficient amplitude to explain the solar cycle we observing. Which strongly suggests maybe you don't need much additional alpha effects somewhere operating in the convection zone, and they anyway would only produce a mixed polarity field. It goes even further: there are some observed nonlinearities in the near surface layers, for example you have inflows into the active region belts. What these inflows do is they essentially keep the magnetic field constrained in the active region belts, which means the two polarities of an active region they can locally cancel each other more, which means there's less magnetic field left over to actually produce the new polar flux. So you even observe the nonlinearities. And the regularity of the cycle suggests that you have a very subcritical dynamo with just some noise, and the noise is mostly the active region evolution, the randomness in the way sunspots form. So if you actually, I don't show it here, but actually if you use kind of a low order dynamical system model, you pretty much can explain really the long-term variability at least in a statistical sense for the Sun. And this is all things the Sun tells us, you don't have to make any assumption. The only assumption you have to make is really that wherever this toroidal field is in the convection zone, you do have some transport process, may it be meridianal flow or some latitudinal turbulent pumping, which pushes the magnetic field towards the equator. That is the one assumption you have to do. So the question is: do we need anything else, or is the Sun just doing this? And well, if the answer is no, then you have to find reasons why all these other dynamos, which you very often can find in either doing mean-field studies or doing 3D simulations, they don't seem to operate on the Sun. So what is so special about the Sun?

And in recent years, people actually have found a few things actually called conundrums which actually puts the Sun in a somewhat special place. So one thing is differential rotation, right? You do these 3D simulations, you pump a solar luminosity through them and you spin up a solar rotation rate, you get turbulent angular momentum transport which creates some differential rotation in these 3D simulations. But the sign of the differential rotation you're getting actually depends on how strong the influence of rotation is. There's a dimensionless number which measures this, which is the Rossby number. And there seems to be in the strong rotation regime you do get that differential rotation which actually does have the sign as we have it on the Sun: you have a fast rotating equator, a slow rotating pole. But if you go to weak rotation, you get the swap to anti-solar differential rotation. And this happens roughly at a Rossby number of one. The problem is the Sun is pretty much right near this transition. So whether you get something solar-like or non-solar-like really depends on do you end up a little bit more here or do you end up a little bit more there. And there was another thing related to this: what people realized also in the last 10 years is that it looks like that the Sun actually has rather low convective flow amplitudes on the larger scales in the solar convection zone. So what I show on this plot is it's a velocity spectrum, which is you take the power spectrum of the velocity times the wavenumber square root of it, that gives you essentially velocity amplitude as function of scale. Here at a wavenumber of about 1,000, that is where granulation is here. You have a bump at wavenumber of roughly 100 or so, this is where supergranulation is. Then towards large scale, this velocity drops quite substantially: at L equal 10 you have roughly 10 m/s. Now I overplotted here a couple numerical simulations of convection. All these models, these are high resolution simulations of just photospheric granulation, they agree pretty well in general with these, um, observed flow amplitudes. But if you go to these larger scale models of the deeper convection zone, the concerning thing you find is that they typically have flow velocities on the order of 100 m/s, and about a factor of 10 more than the Sun seems to have. And that means is when you do these 3D simulations, that the flows are much less rotationally constrained than they should be. So in some sense you end up too much in this part of the regime, but you should be here. It's currently not fully understood why this is actually happening. There seems to be something very fundamental about highly stratified convection we do not get right. And also, if you just use mixing length theory, which typically stellar evolution codes use, mixing length theory is also somewhere up here. There are indications that for reasons we do not fully understand, convective flow amplitudes in the Sun on larger scales are at least an order of magnitude lower than they should be.

Okay, I leave it for now there. I come back to this. But now just I would like to talk a little bit about the quiet Sun magnetism. So really, if we go away from active regions and just try to understand really how does the magnetic field look like at the smaller scales and where does magnetic field come from, but also does these small magnetic field actually have any consequences for the convection, differential rotation, and even the large scale dynamo? And these simulations of quiet Sun magnetism they started about 25 years ago and have been improved over the years. And I just show here very quick timeline what has been done. And these are all 3D convection simulations. The early ones were just incompressible, um, but then starting here at least from 2007, these were more realistic simulations. We actually have radiation transport in the photosphere, we actually get granulation patterns which look pretty much like the granules on the Sun. They do have a realistic equation of state which accounts for partial ionization near photospheric levels. And these models have been compared really in detail to observations. And what you have to do is you have to do the full forward modeling: you start from the model atmosphere, you compute what you would see in spectral lines which are sensitive to magnetic field, either the Zeeman effect or the Hanle effect which is scattering polarization. And through this you can account for a lot of resolution dependent effects, like when you have the Zeeman effect you have partial cancellation of the polarization signal, you don't see all the field which is there. So through the forward modeling you have to account for this. So people kind of have converged that the strength of this quiet Sun field is something like 60 to 80 Gauss for the vertical field at optical depth unity, so kind of the deepest layer we can see on the Sun. So this seems to be actually very weak, at least compared to sunspots which have something like 3,000, 4,000 Gauss. But you have to keep in mind this is magnetic field which is present on the solar surface everywhere all the time. So if you just take these 60 to 80 Gauss and you integrate this over the entire surface of the Sun, so this hidden mixed polarity, this is no net flux, this is mixed polarity field, you just integrate the absolute value of the vertical field, you end up with something like 4 times 10 to the 24 Maxwell. And this is a very large number. A typical solar active region or a sunspot is something like 10 to the 22 Maxwell. So this is something like 400 sunspots. So this in a factor of a few, the mixed polarity field we see in the photosphere at any given time is something like you take the entire 11-year sunspot cycle, you take these sunspots, you shred them into tiny pieces, you mix both polarities together and sprinkle them over the solar surface. That is what the quiet Sun is doing all the time. And because this magnetic field exists actually on very small scales, it decays very quickly, so you have to do this really on time scales of minutes to hours. So in some sense it gives you an idea that actually what's going on in the quiet Sun energetically is much more dramatic than what's actually happening on large scales building the sunspots. And this tells you already right away that we really need an independent dynamo, like the small scale dynamo I introduced earlier this morning, to maintain this magnetic field. You cannot have this just as a leftover from the large scale dynamo.

So these are some examples of small scale dynamo simulations in a small box of convection. And this is a snapshot from these simulations when they're in their kinematic growth phase. Typically they initialize with some really weak field, 10 to the minus 3, 10 to the minus 6 Gauss, and you do get magnetic field on the smallest possible scales you can resolve. If you look at the power spectrum, it peaks somewhere towards the smaller scales. But as these magnetic fields saturate and become stronger, the magnetic field becomes organized on larger scales comparable to granulation downflow lanes. And you see this in the magnetic power spectrum kind of having their peak more near where also the kinetic energy spectrum peaks in granulation. These dynamos they grow very quickly in the kinematic growth phase, and the higher the resolution the faster they grow. This is a bunch of small scale dynamo simulations ranging from 32 to 4 km grid spacing. In this 4 km case, the growth rate is something on the order of minutes, so it's really fast. When we look at the Sun, we never see the kinematic growth phase because it's really fast that the dynamo grows through this. And once you reach a strength which is something like 10% of the average field strength of the quiet Sun, you already move out of this kinematic phase and go to a much slower saturation phase of the dynamo. And in the saturation phase of the dynamo, you also get a small but still significant fraction of magnetic field with a strength of about kilogauss. So this small scale field on average it's weak, but you can have some kilogauss field in there as well.

This is an example how some of these small scale dynamo simulations look in detail. This is a simulation which has 4 km grid spacing, which is way smaller than what we can currently observe, um, with DKIST, the solar telescope on Hawaii. You could theoretically observe down to scales of something 20, 40 kilometers if you really get diffraction limited, but very often you have atmospheric turbulence which actually brings the resolution down a little bit from that. So something on the scales like I show here has not yet observed on the Sun, but there is some hope that we're getting much closer with that in the near future.

Now these are all simulations which are just concerned with the top layers of the solar convection zone, which means they do have some bottom boundary here which is not the real boundary of the convection zone. And one big challenge with these simulations is you have to make some assumptions about what you do in inflow regions. Typically these simulations use open boundaries to mimic that actually the convection zone is 200 megameters deep. So you do have plasma crossing the domain. You know exactly what goes out, but you have to make assumptions about what's going in. This is what I show here: these are just two simulations where you made two different assumptions. The left side you assume the magnetic field strength in inflow regions is zero, and this is truly a dynamo which just operates in the uppermost few megameters of the Sun. Whereas in this case there was an assumption that the inflows are magnetized, and you expect this to happen because all these downflows which go out with strong field eventually they hit the base of the convection zone, they have to turn up, and they're still magnetized, and there's dynamo action even on deeper scales in this convection zone. And it turns out that it's actually critical: if you don't account for it, you saturate at a weaker field strength, but if you have these deeper recirculation component you end up with something which is comparable with observation. So what we essentially learned from these simulations is that the solar small-scale dynamo is really operating over a wide range of scales, not just the photosphere, really multiple scales throughout the convection zone. And what that means also is that the stratification leads to an organization of this magnetic field on larger scales that is actually observed in the quiet Sun. This is a sample of a HMI magnetogram from the last, um, solar cycle, that solar minimum around 2008. You kind of see these salt and pepper field on small scales, but then you also see these network field on scales of supergranulation. If you do a small scale dynamo simulation in a big box, right, this is now a 20 megameter deep box, here you drive convection structures which have supergranulation scales, you also in these small scale dynamos can get a mixed polarity network on larger scales. So there is certainly contribution from a small scale dynamo even on organization of magnetic field in the photosphere on the scale of supergranulation. And just to illustrate this one more time: if you make a simulation a small box, this is what you get. This is 6x6 megameter, you mostly see granulation with weak field and intergranular lanes with strong field. As you increase the box, you can get some larger scale voids of weaker field, you start producing some kilogauss strength network fields. And if you go even to 100 by 100 megameter domain here, you have some of these void-like structures even on scales of 10 megameters or more, and you have really these strong patches of kilogauss field. So even a small scale dynamo can produce this. And this actually matters. You can actually combine these sort of dynamo simulations with the simulation which goes all the way into the corona. If you do have these small scale dynamo with a deeper circulation which produces these, um, supergranulation network structure, you do get a quiet Sun corona on top of this. And here this shows synthetic AIA emission computed from a simulated corona. You can compute the total radiative energy loss you would get in the simulation, and the number is within a factor of two comparable to typical estimates of what the quiet Sun is actually doing on the Sun. If you don't have this deep recirculation, you constrain this dynamo only to the near surface layers, you still maintain a corona, but it's very wimpy. In this case the radiative energy loss dropped roughly by a factor of 60. So having these deep-seated small-scale dynamo on the Sun is sufficient to even maintain a quiet Sun corona.

Okay, so I mentioned before that really the quiet Sun seems to be very energetic. Kind of by just looking at this rough estimate of numbers, it looks like you do the 11-year solar cycle pretty much every hour in the quiet Sun, which raises the question: you know, how much energy do you actually need to do this? And you can use these 3D simulations to estimate this. So what I show here is for some of these near surface high resolution simulations they go just 1.5 megameters beneath the photosphere. I show here a couple of the terms which show up in the kinetic energy equation. So the primary driving term of convection is the pressure buoyancy driving, this is work done by pressure gradients and the buoyancy force, which is here the solid blue line. This peaks just a little bit beneath optical depth unity, which corresponds here to zero. Most of this energy goes into acceleration of downflows in the intergranular lanes, and that shows up as the negative divergence of the kinetic energy flux, which is the blue dotted line here. Here if you combine these two terms, what is left over is this solid black line. This is kind of the driving, the turbulent driving which is left to do anything else, like running a dynamo. The question is, how much of this energy is actually used up by the small scale dynamo? And what I show here is the negative Lorentz force work, this term here. And what you see here is it depends on the magnetic Prandtl number. If you have a high magnetic Prandtl number, the Lorentz force only takes this very small fraction of these total pressure buoyancy driving which doesn't go into acceleration of downflows. But if you lower the Prandtl number in this simulation, I could go to something like 0.1, it already gets pretty close. Now the Sun has a Prandtl number of 10 to the minus 5. You can likely safely assume that pretty much all of the available pressure buoyancy driving here is pretty much absorbed by the small scale dynamo and turned into magnetic energy. This Prandtl number dependence of this energy conversion rate has been found in more general magnetoconvection turbulent simulations including large and small scales. It seems to be pretty, pretty universal: if you have a high magnetic Prandtl number, most of the energy is dissipated through viscosity and very little through a dynamo and resistivity. But if you lower the Prandtl number, most of the energy is converted to small scale field through the Lorentz force and then dissipated through magnetic resistivity, very little left for viscosity to dissipate. Now we have these numbers here now which is something like 150 ergs per cubic cm per second, which is kind of being dumped in the dynamo. You can integrate this over the entire solar surface, and if you just do it for these uppermost 1.5 megameters, that's a third of the solar luminosity which goes into maintaining the small scale field on the Sun. This is a pretty large number, and this is just the upper 1.5 megameters. There's more happening in the rest of the convection zone. How much this adds depends in the end on what is the total pressure buoyancy driving of convection, which we don't exactly know. You could take a mixing length model and it would be something like three solar luminosities. But you might wonder, does it even makes sense that this number is actually larger than the solar luminosity? But keep in mind, this is not the sum of energy. This is energy which pressure buoyancy driving creates turbulence. The kinetic energy is converted to magnetic field through the dynamo process, then that energy is dissipated through resistivity, goes back to the internal energy, and then is again available through pressure buoyancy driving to drive convection again. That happens on every scale height in the Sun. You have stratification of 10 to the 6 in the solar convection zone, so these can add up to a large number. So energetically, it looks like the small scale dynamo is actually extremely powerful machine which pretty much takes all the available pressure buoyancy driving in the convection zone to turn into magnetic energy, which is then dissipated through resistivity.

This raises a question: if it's so energetic, does it actually have effect on even larger things in the convection zone? And this is now we really getting here to what is really very active research. So not all of this is completely settled. There are some simulations which show this, but not all of them do. But this is a very impressive simulation which has been done just a couple years ago by Hotta and Kanya Kusano. This is now a global simulation of a rotating shell, but it has high enough resolution that you have actually a very efficient small scale dynamo operating in there. In principle you could also have a large scale dynamo, but at least for the time the simulation was run, there was no indication that it actually produced a large scale field. And you kind of can see here already that it looks like you have a fast rotating equator and a slow rotating pole. I mean, this is in a frame which moves with the, um, mean rotation rate, so you see here the, I mean, the fast, the anti-um, the rotation the opposite direction because in these corotating frame. So one interesting thing which was found in this simulation that you only get these solar-like looking differential rotation when once you have high enough resolution to have a really efficient small scale dynamo in there. So they did the simulation at three different resolutions. And low resolution they pretty much found more a fast rotating pole, slow rotating equator. And only when they went to the high high resolution case it has a fast rotating equator and slow rotating pole. Interesting thing is if they stayed at the high resolution and the zeroed out magnetic field and did a purely hydrodynamic simulation, differential rotation swapped sign and you had again a slow rotating equator and a fast rotating pole. So there's some indication that all these small scale field produced by a small scale dynamo here is even essential for getting differential rotation right. And one other interesting thing the simulation showed is that it actually reduced the convective power on large scale. So here it shows solid lines are the kinetic energy spectra for these three different resolutions, the dotted lines are the magnetic energy spectra. There is a moderate increase of magnetic energy for the highest resolution case, but the most dramatic effect was that actually the kinetic energy on large scales dropped by about a factor of 10. And this is all from small scale field, and this is essentially what enables this dramatic difference between these two cases. There are two things happening: by having reduced energy on these large scale flows, they are more rotationally constrained. This leads to some deflection of the convective energy transport towards the pole, which leads to a pole which is about 5 to 10 Kelvin hotter than the equator. This temperature difference allows you kind of to avoid these contour lines of Omega being aligned with the rotation axis, and you get to something which is more looking like actually what the solar differential rotation shows. The other thing is that the small scale field produces Maxwell stresses which also transport angular momentum, and the transport of angular momentum by the small scale field has the opposite sign as the Reynolds stresses. So when you get in this regime of strong field, it's actually the Maxwell stresses which flip around the sign of the angular transport, and that way you get the solar-like differential rotation.

Now this drop of energy on the large scale is an interesting effect because I mentioned to you previously that we have these convective conundrum on the Sun, that it looks like that all simulations produce too much convective power on large scales compared to what we actually observe. And the interesting thing is, at least that these small dynamo simulations with a very strong small scale dynamo at least showed some drop of energy on large scales. You would have to go at least another order of magnitude to get consistent with the Sun, what the Sun is doing. But at least there's some indication that you actually may have to understand the small scale dynamo to actually get all the large scale behavior on the Sun correct. So really an example here how the smaller scales are really coupled to the larger scales in the system.

Okay, let's move a little bit from the Sun to stars. So you may wonder: if you don't even understand the solar dynamo, what's the point of going to other stars where we have even less detailed observations? But of course this is interesting because the Sun is only a single realization of a stellar dynamo in a certain corner of the parameter space, and it's worthwhile asking how typical it is actually. And we cannot easily extrapolate from the Sun to other stars because of all the uncertainty we have about how the solar dynamo works. But by looking at the stars, we see how these dynamos operate dependent on the stellar structure, like the convection zone depths, right? If you go to lower mass stars, the convection zone depths relative to the radius of the star increases in depth, and you get fully convective stars. And most importantly, stars have very different rotation rates, so some of them are much more strongly rotationally constrained than the Sun. And I showed you before one challenge is that the Sun seems to be very close to the Rossby number one transition where you even have the swap from solar-like to anti-solar differential rotation. So people have looked into this, and I just want to highlight a few critical results here and don't go into too much depth.

So one very solid relationship which has been established by looking at other stars is this activity-rotation relation. And what is shown on this plot, these are all observations of magnetic field on other solar-like stars. Bigger dots correspond to more massive stars, small dots to much more lightweight stars. And what people see is that when you are, and this is shown as function of Rossby number, the Sun is somewhere here, somewhere around here. As you go to faster rotating stars, these stars become more active. And the activity is here measured in terms of the mean field strength in the photosphere, which is derived from Zeeman broadening of spectral lines. So about the Sun become more active for about an order of magnitude in increased rotation rate, but then they seem to get into the saturation regime, because the activity does not depend that strongly on rotation rate anymore. At least this regime of getting more active stars as you increase the rotation rate, this is also seen in 3D simulations. Here they also studied Sun-like or solar-like stars with different masses, different rotation rates. They show generally the same trend that activity goes up something by an order of magnitude when you increase rotation roughly by an order of magnitude. That part is more or less consistent in general terms. Interesting thing is what happens as you go into the saturation regime, and I think this is not fully understood. This is really a saturation of the dynamo itself, that in some way Lorentz force feedback prevents building up of stronger field. It could be also simply an effect that the photosphere at this point here, once you hit this point here, right, this is an average field strength of 2,000, 3,000 Gauss, it's pretty much like having a sunspot umbra everywhere in the photosphere. Right at this point you produce so much large scale field that you pretty much filled up the entire photosphere with sunspots. It might be simply a saturation due to space constraints; you just cannot put more flux into the photosphere. That causes this, um, trend to kind of taper off here.

One other thing people also observed is, and that is kind of highlighted in this plot, is that as you go from slow rotation to fast rotators, that the magnetic field becomes actually more complicated. Kind of these big round symbols indicate more simple magnetic field configurations, whereas the star-like features here they indicate a more complex magnetic field topology. So two things happening here: you get more flux produced on the large scales as you go to fast rotation, but also field topology more complicated. The Sun has a property that during solar minimum the polar caps have pretty much more or less unipolar field. These fast rotating stars very often have a strong mixed polarity component even at their polar caps, which makes field much more complicated.

Okay, one other interesting part is the overall rotation evolution of stars throughout their lifetime. And this plot on the left here shows kind of the evolution, rotation evolution of stars kind of from the early star formation through their main sequence phase towards the end phase when they essentially turn into a giant and finally end their life. Early, young stars always start with very fast rotation, because when the gas cloud collapses you spin them up quite dramatically, having rotation rates of just a few days here initially. These stars keep their fast rotation in the first few hundred million years before they start slowing down. And that has to do with the fact that even though they have strong field, it looks like the field is very complicated. They have highly multipolar field which make very inefficient coupling of this dynamo field to the stellar wind, which makes the angular momentum loss less efficient. But after they get older, beyond a few hundred million years, they get into this very efficient spin down. And this follows some well established laws which were first discovered a little bit over 50 years ago by Andy Skumanich, actually here at HAO. But the interesting thing, and that is something which was very recently only discovered by actually looking at all the observations from Kepler, where they monitored the, um, properties of stars for a couple years. Very precise photometry to both study exoplanets but you also can study stellar oscillations. So through asteroseismology you know these stars pretty well, you can establish also their rotation pretty well. But people found is that at later times these deceleration actually becomes less prominent. They have some weakening braking phase. And this is actually shown here. This is kind of the standard model what you expect based on these Skumanich laws, but once you get for solar-like stars to something which is just 90% of the solar Rossby number, suddenly this braking becomes less efficient. So it looks like that the Sun just falls into this transitional regime. In that sense, it's not the most typical solar-like star. It's really a star which is at the end of this efficient braking phase, and the Sun already has an angular momentum loss which is an order of magnitude or so less efficient than it should be when it is already there. So based on this, you can already get an idea that maybe if you study the Sun, we see a star which is already in a kind of a transitional regime, and not the most typical phase. And this is also evident if you look, for example, the solar cycle period which is here in the context of all the other stellar cycle periods. Is this general trend that cycles become longer as the rotation period becomes longer? Some stars have multiple cycles. There is these two branches, but the Sun doesn't lie on any of them. It's kind of in between. And the current understanding is really what's happening as stars evolve, at some point they kind of move off this branch. Depends on the stellar type. If you have the more massive stars that turn off earlier, or the solar-like stars that turn off happens roughly here. And the Sun is roughly in this transition phase. But also it is found is when you look at the cycles of these stars, if you are this part of the domain, you do find more or less regular cycles on these stars, somewhat similar to the solar-like cycle. If you pick a star which is more close to this transition, we have now some observations which really show stars which transition from a cyclic behavior to something which looks like a grand minimum, like the Sun also had. And then if you go really to stars which are at the end of this transition, they show mostly flat activity. So again indication the Sun is in some intermittent phase here, this grand minima due to this transition in the behavior.

Okay, I have a few more minutes to quickly talk about something very different: the geodynamo. And mostly contrasted with all these stars, what do we know about the geodynamo? So let's first look at the structure of the Earth. We living out here on the Earth's crust, then you have the mantle which has a very low conductivity, then we have the iron core of the Earth, which consists of two parts: we have the outer iron core which is liquid, the inner core which is solid, and we have this transition phase between mantle and core which is corrugated somewhat, people find it might be with a thickness of 250 km. So compared to all these stellar dynamos which have magnetic Reynolds numbers of 10 to the 9, 10 to the 10, the magnetic Reynolds number for the geodynamo is actually pretty moderate, something like 300. Which is interesting because if you do the kind of 3D simulations we can afford these days, you can do a magnetic Reynolds number of 300. So when you do a geodynamo simulation, you don't have to worry about small magnetic scales you cannot resolve; you can resolve everything which is there. Now the other thing is that compared to the Sun, which is kind of at this transition between weak and strong influence of rotation, the geodynamo is completely in this regime of really strong rotational influence. So the Rossby number is 10 to the minus 6.

What you end up with in this regime is more or less convection rods, which aligned with the AIS of rotation, alternating spinning clockwise, counterclockwise. And you do have some flows which are aligned with these convection rods due to breakdown of this geostrophic flow balance between Coriolis force and, um, and pressure forces. They break down at these interface between the core and the, the mantle due to viscous stresses. It's the AMAL layer which is forming here, and it's people think that some of the longer term evolution we see with the geodynamo might be related to that. Actually, these boundaries evolving on time scales of the mantle convection.

The geody has very little differential rotation, so it's generally mostly an alpha-square dynamo resulting from these, um, rotating columns, this helicity. And another interesting thing is that the geody anal has a static field which is much stronger than the equipartition field strength. It's in a strong field regime, which can happen when you're in this fast rotating system because you get a balance between the Lorentz force and the Coriolis force, not any longer between Lorentz force and attraction forces.

So what sets up actually these flow patterns? And this is related to AMAL pumping. Like all of you know this experiment, you have a cup of tea with some tea leaves, and you spin it. The tea leaves end up kind of at the bottom in the center. The reason this happens is you do have the centrifugal force here, which is balanced by a pressure force. You have higher pressure outside, lower pressure inside. But these balance breaks down at the bottom where you have a viscous shear layer. The fluid is rotating at a slower speed in this viscous shear layer. We have the pressure driving the flow from the outside towards the inside, and that collects the tea leaves here. You have something similar in high and low pressure systems, the Earth's atmosphere, or these rotating columns in the outer core of the Earth. The only difference is here you have the Coriolis force. So clockwise and counterclockwise spinning, um, columns have opposite pressures. Here, the, um, clockwise spinning ones have high pressure, the counterclockwise have low pressure. So then you have this EGMA boundary layer, which is in this case located at both ends of these columns. And the coral boundary leads to a breakdown of this balance, and that essentially drives the secondary flow along these rotating columns, and that gives you a net helicity. And so that you have an alpha effect which keeps this dynamo operating.

And I would like to also make here a quick connection to liquid sodium experiments already mentioned earlier. There have been quite a few of them, but there have been this one experiment which was done in Karlsruhe in Germany. They actually made up this tank of these columns, this up and down flows which had helical motions. The way they did this, they had essentially these pipes going through here. They had an axial flow in the center of the pipe, and then they had these spiral patterns, metal patterns. So when they pumped the liquid through that, they get these spinning motions, and so that they get the helicity to kind of mimic a little bit what might be happening in the outer core of the Earth. What they found in this regime is, yeah, when they increase the flow speed of the liquid sodium above a critical threshold, they could see this increase of the magnetic field. In this experiment, the seed field is the magnetic field of the Earth, which is somewhere around a half G, but then these fast flowing liquid sodium led to a dynamo effect and increased actually the strength of the field, which kind of is a little bit similar to what actually happens in the geodynamo.

And this is an example of a geodynamo simulation, a 3D simulation. And unlike the sun, these simulations were already done here in '95. They produced magnetic fields which looked dipolar, similar to the geodynamo. Some of these simulations also show magnetic field reversals. Doing the reversals, the magnetic field doesn't disappear. It's more that the dipole disappears, and you have higher multipoles, and then you just build up a dipolar field of opposite direction. This is, of course, very interesting if you want to study, for example, what happens to the Earth to a geody reversal. Because I mean, the magnetosphere shields us from all the impact of the solar wind, so it's critical to know that the magnetic field doesn't disappear, but it gets much more complicated. So actually, I'm not fully aware of people actually studied how the magnetosphere would look like if you plug in something like this, but it's certainly something interesting to study.

Okay, that gets me towards the end of this talk. The first lecture, I gave you some fundamental ideas how dynamo processes, um, work in astrophysical bodies. And then I studied now about specific applications to sun and stars. And in these applications, we have to say there have been some limited success, right? The problem is, mean-field models can capture many aspects, but they require some tuning due to many degrees of freedom. 3D dynamo motion, they found many examples of dynamos, but not, not many of them really look solar-like. And one fundamental challenge is here, it's even a challenge to just get already differential rotation right on the sun, because that is such a critical flow for the large-scale dynamo, and has to do with the fact that the sun appears to be very close to two critical transitions which happen near Rossby number of one. The one is the transition from solar anti-solar differential rotation. And if you put the sun into the context of stellar dynamos, there seems to be related to this also transition where really the nature of the dyn changes from a rather strong dynamo which leads to very efficient loss of angular momentum to a less efficient dynamo, which is likely dominated more by all the small-scale field we observe on the sun, which has much less effective angular momentum loss. And the sun being close to this critical transition, of course, makes any simulation very critical, very dependent on details, which makes it very difficult to actually model the sun.

The case of a geod, more there has been more success, I think, with simulations. But I say this as a solar person who doesn't know all the dirty laundry in the geodynamic community. Maybe if you ask geodynamo people, they have different opinions about this. But at least you get some of the basic properties right. But there is some debate on whether really these models get it right for the right reason. Because even though the these simulations, they can get the magnetic Reynolds number more or less correct with the resolution we can afford, you cannot really simulate the low Prandtl number regime, and also the very low EGMA numbers, which are related to this critical boundary layer between the outer core and the mantle, which sets up these secondary flows along these rotating columns, which actually lead to the helical motions you need for dynamo action. And of course, another thing is for the geodynamo, we only know the lowest multipoles, something like L equal 1, 3, or so, are constrained because any higher multipoles, they cannot really observe because they're hidden by permanent magnetism in the crust of the Earth.

Okay, thank you very much. Okay, that's that's my, that's my. Yeah, yeah. Okay, there's so they're there. Did them as a kind of a toy. So they're there, but it's, uh, I'm, I'm deeply suspicious because it's, it's well into the the right. Whatever. Okay. Need some even lower, lower. A lot of those simulations kind of cool. There's all these like, you know, different. It's different multipolar things where you can imagine here's the, here's the, here's suddenly you see auroras in places we normally don't see them. Silly putty reconnection. No, no, I don't think I'll attach you a y. Yeah. And then you theop break the loop. Yeah. And then you got two loops. Yes. Yeah. Yeah. F E Le to. And I it for outreach purposes. And so the, and we there's a whole, um, Andreas Mun, uhar, y, um, is at Southwest Research. Does, uh, solar physics. Um, and he came up with this. And so take a, you take a, you take your Sol, you roll it out, you create a little. Right. So you can imagine magnetic field circulating around, right? You can stretch that loop all you want, the geometry doesn't change. But then if you pinch it in the middle, break it, you now got two loops, right? And they're two loops that are kind of attracted to each other because the orientation of magnetic field is they overlap. And so then they do the same thing, and then you break it again, and you break it again, and they break it again. I usually use that as a little outreach demo. Uh, right. So we'll go into our question period. Um, who's got questions from a test? I have a very naive question. Is there any fractal structure in the dynamo series, or just divide it into large scale and small scale? I mean, I mean, the mean field theory is pretty much large scale. More scale. You don't really have any plug-and-play structure in there. That's what I mentioned, right? If you look at the sun, really an integral part of solar activity is having sunspots. So mean-field theory doesn't only deal with sunspots but just averages over them and just tries to describe what they do on average. So if you want to get to get sunspots in there, what people have been doing in the past is kind of use mean-field models, for example, to get kind of the large-scale toroidal field, but then they put in explicit flux emergence mechanisms to actually have small flux tubes from the toroidal field forming sunspots. And then through the spatial latent alpha effect, you kind of rebuild the polar field, the poloidal field of the sun. So it's kind of an add-on to mean-field dynamo. The, but people have kind of created these 3D back-and-forth latent dynamos which kind of get, take some concepts of mean-field theory, but also put in the effect of individual sunspots as well. Um, I just wanted to ask about something you shared early in your presentation, which was that sort of the turbulent regions of the sun have tend to the eight times more conductivity than, I guess, the non-turbulent regions. Does that make a big difference in the overall magnetic structure of the sun, or is that, or is the time scale of diffusion still too long that it doesn't really matter? So actually, so the magnetic diffusivity is 10 to the eight times higher. So it's the opposite. Yeah. But I mean, it pretty much leads to very different regimes for the radiative interior of the sun and the convection zone of the sun, right? In the radiative interior, without any turbulence, you have essentially a diffusion time scale on the order of a billion years. So that is a thought that there is likely a remnant field left from when the sun actually formed, which was essentially whatever this molecular cloud had, which then eventually formed the sun, and the magnetic field it amplified from the original field of the galaxy. That something of that is still there, and that field could have survived this long. Whereas in the convection zone, since these time scales are down to time scales of years, whatever was there left from the beginning has been wiped out by turbulence since then. But it's actually a little bit more complicated than than that, because people also has have found even in the solar interior, there are some additional instabilities possible which might even enable dynamo action in these strongly subadiabatic regimes where you don't really expect convection, but you might have some turbulence on horizontal shells and so which could potentially enable some dynamo action as well. But that's the very active field of search there, not generally agreement really what's happening, but there's a possibility even have a dynamo operating there. Um, yeah. Okay. Um, do gas giants like Jupiter and like gas planets also have dynamos? And if so, how, like what's the difference between dynamos there, those that stars? Pretty much definitely. Jupiter and Saturn do have a strong field. And I think, I mean, there's not too much data for Uranus and Neptune because they were only two flybys, but they also seem to have a field, albeit more complicated. So the biggest difference is that the gas giants, they don't have a liquid iron core. So the region where they might have a dynamo is likely something like metallic hydrogen. And the exact equation of state and the properties of that isn't that well known. But it's generally believed they do have dynamos. The dynamo of Jupiter actually looks a little bit like the solar one in terms of the overall field properties. It's just scaled up, much more powerful. Saturn has this interesting property that it has an extremely exotic magnetic field, which is a little bit puzzling and it's not completely understood. But but people think is that the dynamo region of Saturn is surrounded by another region which has a high conductivity but doesn't have any active dynamo action. That that region might have a slightly different rotation than the core of Saturn where the magnetic field is produced. And then essentially, what this rotation of this outer layer with high conductivity does, it kind of kills all the higher multipoles and only the dipole survives. Um, you talked about that there were possible like different layers sort of in the sun's interior where the magnetic field could be generated. Did you talk a little bit more about how that would impact sort of the large and small scale structure of the field that's produced? Yeah, so I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have it only constrained to thin layers, you might get dynamo waves which have a relatively short wavelength, which is opposite to what we actually observe on the sun. So the observations show this butterfly diagram, I mean, it's pretty wide, right? So observations test, we don't have evidence for really short wavelengths. And so the one challenge with anything which is restricted to a thin layer is very often that you get these very short wavelengths. The sun doesn't seem to have them. You get several overlapping cycles at any given time, but the sun has very small cycle overlap. Nonetheless, I think having magneto-rotational instability in this shear layer, it's a very important, could be a very important ingredient to just maintain this shear in the first place. I mean, one general problem is typically you get the near-surface layers angular momentum transport which is inward directed, which in principle should set up a near-surface shear layer. But very often in simulations, it doesn't because what you also do, you drive a meridional flow which pretty much compensates these angular momentum transport and wipes out the near-surface shear layer. So you have to kind of slow down this meridional flow, and perhaps magneto-rotational instability could provide an additional source of enhanced viscosity which limits the flow speed of the meridional flow and through that enables actually the formation of a near-surface shear layer. So through that, I certainly think that might play an interesting role. Certainly should look more into this. But I always have the feeling having the dynamo completely restricted to this layer might not be what the sun is doing, which doesn't mean it doesn't play a role, but it's only part of the picture, that is my my impression.

So I mean, with these different layers, the main reason people looked at these different layers is mostly the profile of differential rotation, right? You have the latitudinal differential, mostly in the bulk of the convection zone. Then you have these shear layers at the top and at the bottom, which have more radial differential rotation. So the reason people put a lot of emphasis on these layers is because, for example, if you have a classic alpha-omega dynamo where you combine the alpha effect with shear, if you have radial shear, you get latitudinal propagation of activity out of this, which made these two regions very attractive. The problem is, what I showed you, that you might get these dynamo waves going into the wrong direction. The other problem is, if you have