Transcription
All right, so good afternoon. I think I introduced myself to, um, I think all of you. If I, I have not, um, uh, right. So I am an assistant professor at UT Austin. I moved there in 2019. Uh, and before that, in fact, I was working first as a postdoc at, at J JPF under Marco's supervision. Uh, and then, uh, UCLA. I love LA a lot, but then, you know, um, I got the job in Austin, um, and that's how I moved. Um, actually, I don't know what else I, I should say. Well, when I was your age, more or less, I was doing my PhD, um, in Europe. Uh, and I was actually working on magnetosphere. So I was working on simulations and observations of whistlers during reconnection in the magnetodisk. So I thought I was set on my, you know, magnetospheric path, but then, you know, you never know what can happen. I switched to study waves and turbulence in the solar wind, and since then, I think I more or less stuck with that.
Okay. Um, I guess the bottom line here is that, you know, there are so many, you know, opportunities and topics to be explored. Um, plasmas are plasmas. So, uh, I have here a few references for you, um, in case you want to know more. Um, these are pretty classic textbooks, and the last one is a review by Bruno and Carton about turbulence in the solar wind.
Okay. So, um, this is the outline of this afternoon's lecture. It's kind of dense. Uh, chances are I won't get to the end. So, um, since I probably won't get to the end anyway, please do interrupt me, uh, if you have any kind of question. I would rather, you know, slow down, uh, or spend more time to explain some concepts rather than, you know, uh, losing any of you halfway. Okay. So, um, I will start by giving you a general introduction of why we care about turbulence, although this morning we also had some compelling arguments. Um, and then I will, uh, you know, point to some, you know, specific reasons as to why we care about turbulence in the solar wind. Then I will introduce some, some basic concepts by using standard fluids as an example. In fact, the first, you know, um, studies of turbulence start with, you know, with fluids before moving to MHD turbulence. Okay. Um, and then discuss how, you know, how much of all of this work to explain observations. It's going to be maybe a lot of information. I guess the, you know, what I want you to learn really from, from, from, from this lecture is why turbulence is important, what are the main properties of turbulent flows, and be aware that, you know, turbulence in magnetized plasmas is different, has some fundamental differences with respect to turbulence in standard fluids. Okay. Ready? Let's go.
So, um, turbulence is a state of fluids. It's a phenomenon that we experience in our everyday life, and it affects, every time we set, uh, fluids into motion, unless they are very viscous, like honey or Nutella, as we were saying, um, the other day. Um, well, unless you have those very viscous fluids, um, turbulence sets in anytime you, you set, um, some motion. Okay. And turbulence, in fact, plays a crucial role in many aspects of our, of our life. I have here some, some examples. Okay. So turbulence is a major factor in the current aviation industry, for example. That's because when you have an object moving through air, um, you create a, a turbulent boundary around the object which increases drag. And so engineers study the best shapes, um, that allow the, the flow to streamline as smoothly as possible around the object to reduce this, this drag. Turbulence also, um, is associated with chaotic motions that enhance mixing. So that's why you, you know, you want to stir your coffee if you like it with milk, so that you, you know, you mix them, uh, uh, you, you can mix them well. Okay. And so for this reason, turbulence affects transport of mass, momentum, heat. Okay. So, for example, turbulence is important to track, you know, um, the, the, the spread of pollutants in the air. For similar reasons, turbulence also affects weather patterns, and for example, it can create, it can help to create conditions, uh, for the formation of hurricanes. Okay.
Now, in space, we care about turbulence because, uh, it can heat and accelerate the plasma, right? Um, it can scatter cosmic rays, for example. And, you know, besides our, our close, you know, space environment, in interstellar space, turbulence is a major factor, uh, for star formation. So I found this cool video on YouTube which shows, where's my? Okay. Which shows turbulence in the air. The way it is visualized is by shining a, a green laser light through a glass rod on smoky air. So the glass rod, um, deflects the laser light, and the smoke diffuses the light so that you can actually see. Okay. And believe it or not, there is not really a, um, on play there, not really a, a definition, uh, of turbulence. Um, but there are some canonical features, uh, that we can recognize in turbulent flows. They're generally unsteady, and they appear to be chaotic and intermittent, which means that you have, you know, some, you know, uh, uh, instances in which the fluid is very active, and some instances in which the fluid is more quiet. They are associated, like I said, with, with enhanced mixing, and importantly, they are multiscale. So you can see in this image that you have structures, vertices, on multiple scales at, at any given time. This movie here, yeah, which is developing slowly, um, is shows the development of turbulence in a magnetized plasma. Okay. So you can see that as time goes by, you are forming structures. So what you are seeing here in color code is the current density out of the plane, okay, in the XY plane. Okay. And you have a strong magnetic field coming out. Um, and so you can see that you, you are forming here, uh, structures that are qualitatively similar to what I showed you in the previous movie. So plasmas, especially collisionless plasmas, um, are not strictly speaking fluids. Okay. But they do display, um, similar properties. They display a collective behavior, which manifests itself, for example, um, in the form of propagation of waves, and turbulence is another example. So in this lecture, I'm going to use the fluid description for plasmas, and in particular, the magnetohydrodynamic description, that, as we discussed, works fantastically well.
So now let me tell you why we care about turbulence specifically in the solar wind. You have seen yesterday, um, the, you know, the, the Parker solar wind model. So this is just a quick, you know, like refresh. Um, so Parker put forward the first theory of a, of a solar wind, starting from the steady-state equations, um, of conservation of mass here, and the momentum equation is the second, uh, is the second equation in this slide. Rho is the mass density, U is the flow velocity, P is the pressure, C is not the speed of light here, it's the sound speed, G the gravitational attraction of the sun, R the radial distance from the sun. So you can solve these equations, okay, and he argued that, um, the, you know, the only possible physically acceptable solution, although I'm sure that you have been, you know, better than I do, okay, why, um, this, the, you know, the acceptable solution is the one corresponding to the transonic solution, so a wind that becomes supersonic.
So, um, this plot here shows the flow velocity as a function of radial distance, where R is normalized to a solar radius, and you have a family of curves, each corresponding to a different temperature imposed at the base of the corona. So in this problem, the, the corona is assumed to be hot, but it isn't explained, okay, why in the first place. And I call this laminar because this flow that's predicted by this theory is smooth, and the, you know, uh, the flow follows well-defined streamlines which are radial, as we saw yesterday. Um, and just one year after that, this, this paper came out, one year or so, okay, the solar wind was actually measured in situ. So although there was still, you know, like a, I would say, like a, um, a, um, I would say, fight, not really fight, but a, um, controversy on whether the solar wind existed or not, um, one year after it was, the solar wind was measured, okay, which, you know, confirmed the existence of the solar wind, like Parker said. So good.
Now, this theory is beautiful in its simplicity. However, it fails to explain a few important things. Okay. So let's take a look at how the solar wind looks like. Um, have you already seen this plot before? Yes, all of you in this school, probably. Um, okay. So I'm going to go quick. So this is a composite image, okay, and it's a polar plot, okay. It's a polar plot of the velocity measured, uh, as a function of, uh, of, um, latitude. This is many years of observations by the Ulysses mission, okay, which is so far the only one who really explored the polar, the polar polar regions. Okay. And this is superposed to a snapshot of, you know, a composition of white light images and ultraviolet images. Okay. This superposition is meant to guide the eye in finding correlations between wind structures and magnetic structures. So if we know, so well, we can see here that there is a relatively steady wind coming from the higher, highest latitudes, and then there is a highly variable, slower wind at the lowest latitudes, in correspondence of closed magnetic field lines. So this is, first of all, to point to the fact that the magnetic field structure plays an important role to determine what kind of wind you get. The second, uh, thing I want to point out is that the relatively steady-state wind predicted by Parker looks like the one coming from, uh, the highest latitudes. The color here, of this, the blue and the red represent the dominant polarity of the magnetic field. Okay. So what that means is that the wind here is coming out from a, say, open field region, open magnetic field regions. Now, monopoles do not exist. Okay. So these field lines will connect out in interplanetary space. Okay. So the problem is that this wind is much faster than what the theory predicts. Okay. It's about 800, up to 800 kilometers per second with the Parker wind, um, and reasonable temperatures at the coronal base, you can get beyond 400 or so kilometers per second. So there is an important mismatch between observations and predictions from the theory. The other problem is, uh, that if we adopt the standard laminar model of the wind, um, well, these kind of models underestimate the temperature of the solar wind that we measure in situ. So this is a plot of the temperature as a function of radial distance taken by the Helios mission. Okay. And the solid line, uh, represents the best fit to a power law, which is reported here for reference. Now, the specific number in this case is not particularly important. What's important is that if the wind was expanding adiabatically, and adiabatic expansion is represented in this equation here, well, then the temperature will fall off much faster than observed. So this, in short, uh, uh, provides evidence that there has to be ongoing heating in the solar wind as it expands in interplanetary space. So there are two problems. Okay. The solar wind is hot and faster than theory predicts, and one of the scenarios is that waves can actually contribute to both heating and acceleration.
Okay. So this class of models, in fact, takes fall under the category of wave-driven winds, and these equations here are the starting point to describe a, to describe this wave-driven winds. Okay. So this is, this is a, uh, upgraded version of the equations I showed you earlier. Okay. These are still steady-state, okay. Um, however, well, the, you know, the, the magnetic forces have been included, and, um, and the effect of fluctuations has been included as well. So these equations have been obtained by assuming that there is a mean, an average wind, okay, plus fluctuations that are represented with these delta quantities. Um, and these equations are obtained by taking the average. So this represents the average wind, okay. So if fluctuations have amplitude that are not negligible, okay, so if they are not small amplitude, the waves, and then they do contribute to the average wind and they can accelerate. Okay. And the last equation here is a simplified form of an equation for the temperature, okay, that combines the, the adiabatic cooling as before, and it includes a source here, a heating source that comes from, in this case, dissipation of waves. Okay. In fact, in this kind of model, this epsilon, which has the dimensions of energy per unit time per unit mass, um, well, it's like a fudge parameter where you kind of put everything, uh, that's not described by fluid models. Okay. Now, what lends support to these scenarios is that we do observe plenty of fluctuations in the solar wind. So by pure, I would say, causality, okay, I'm showing here a plot of typical fluctuations we observe in the solar wind. So there are three lines, uh, here, uh, that correspond to fluctuations in the three components of the magnetic and the velocity field. And, you know, for each component, there are two lines, one representing the magnetic field renormalized, and the other line for the velocity field. Now, take a look at this plot. What, how is there anything that rings a bell? Yes, who's speaking? Yes, and therefore is that, yeah, so it's the Alfvén waves. Now, we have talked a lot about Alfvén waves yesterday. Okay. So Alfvén waves are, um, uh, permeating the solar wind. Okay. Now, the fact that we observe fluctuations that are, you know, so what that they look like waves that look like Alfvén waves, so they are coherent in that sense, we shall see that poses, um, a challenge at a fundamental level to the concept of, of the turbulent cascade.
All right. So this is a recap. Okay. What is one of the big problems? Understanding what heats and accelerates the solar wind. Evidence suggests that fluctuations, slash turbulence, might play a role in accelerating and heating the solar wind. Okay. Well, then, if we have, you know, identified the possible solution, what is the problem? Well, the problem is that we have to understand how the energy stored in the fluctuating fields can be ultimately released to the plasma. Okay. To to answer this question, we need to go beyond the laminar description of flows and beyond the description of linear waves that do not interact among each other. Okay. So this brings us to the topic of turbulence. And like I said, um, you know, I'm going to start by introducing this topic, um, by, you know, by discussing turbulence in standard fluids. Questions so far? All good? You with me? Sharp? Awake? Yes? Good.
Okay. So early experiments on turbulence were performed by Reynolds, um, in 1883. Yeah, 1883. And Reynolds studied the behavior of, of, um, of flows in a pipe. Okay. And he studied this behavior by tracking in the evolution of an ink that would be injected into the flow itself. And Reynolds observed that the behavior of this flow would change dramatically, qualitatively, um, depending on a dimensionless number that later was called Reynolds number. So the Reynolds number is defined as the typical mean flow velocity U, okay, times the, uh, macroscopic scale of the system L, in this case, is the diameter of the pipe, divided by the kinematic viscosity. So what he found is that for low Reynolds numbers, lower than something like 2,000, uh, the flow was laminar. Okay. So what that means is that, you know, the ink that he was injecting would follow a well-defined straight streamline, and there would not be mixing. As the, you know, you crank up the, uh, the Reynolds number, I might have said Luni number earlier, no, okay, the Reynolds number goes up above this, uh, 4,000, and I don't think there is a really an explanation as to why there is this critical number of the Reynolds number around 4,000, then the, the, the ink would lose completely its coherence. It will start to form like eddies and spread across the cross-section of the pipe, uh, very rapidly, and that represents the turbulent regime. In between, uh, you have an intermediate regime with intermediate values of the Reynolds number, where the ink will start to form, form like wavy patterns. However, there still will not be mixing, really. And, you know, although in this wavy pattern, the ink would maintain its coherence. And what I found here is on YouTube, a, a replica, a reproduction of these experiments. So the top movie that I'm going to play corresponds to the laminar flow. Okay. So this ink remains relatively on a on a straight line. The reason why you see this, you know, kind of wavy things is because the ink is being, is not being injected uniformly, and in fact, doing this kind of experiment accurately is very challenging. All right. The movie below, instead, corresponds to the, to the turbulent case, high Reynolds number, which has been achieved by, you know, increasing the, the flow velocity. Okay.
Now, at this point, uh, you can stick a hot wire into your flow and measure the velocity. And what I'm showing here is two experiments. So the, the measured velocity as a function of time, uh, in two experiments performed under the same conditions. What you notice is that, well, if you compare point by point the signal, what do you, what do you find? And the experiment is the same, huh? It's no, yeah, that's right. What else? It's, it's different. It's point by point different, right? So, um, let me stop for a, for a, for a second here. So, um, this is in apparent contradiction with the fact that, um, the, this system is governed by deterministic, uh, equations. Okay. And so there was this kind of, you know, um, problem. So how do you explain that? Because for given initial, so this system is described by the Navier-Stokes equation. For given initial conditions, given boundary conditions, um, the system is deterministic. Okay. So the answer to this apparent paradox was given much later, in 1963, by Lorenz, okay, who demonstrated that there are some dynamical systems that are highly sensitive to those small perturbations. So if you have, you know, even tiny, tiny perturbations on the initial condition, you can get significantly different solutions. Okay. However, uh, even though these two signals are point by point different, um, they display the same statistical properties. Yes, the, what ending? Ah, so you're still understand, they're cheating here. It's possible. Well, I will have to re-source this, and I'm not sure that this is the continuation of the other one. What do the point? Okay. This, wherever you stick, okay. So if you have a given flow, okay, um, you start to analyze it, okay, and you stick your probe in different positions, okay, you will get statistically the same results. And you can do the same experiment, stick wherever your probe, and you will get the same statistical results. So it's not surprising that there are indeed some, some similarities. I will venture, I venture to say that they are exactly the same, spitting to, but, um, I might source this better. Okay. I guess check it, check it out. Um, what that, what that, so same statistical property, mean value, RMS, RMS, what else do you notice? Some, some already said, uh, what do you say, like very noisy? So there are multiple scales. So this is the multiscale nature I was saying before. Uh, by eye inspection, you can see that there are low-frequency fluctuations, okay, and superposed to it, there are, you know, uh, high-frequency fluctuations, short-scale fluctuations. Okay. So even if you started from a smooth, larger-scale flow, soon after, you end up having energy distributed over multiple scales. So the process of transferring energy from the largest scale to the small scales is also something that we, uh, that we experience in our everyday life. So let me go back to the example of the coffee with the milk. Um, you know that you have to generate motions at very small scales, microscopic scales, in order to, to mix, you know, your, your parcels of milk with, with parcels of coffee. And you start to stir with the spoon. Okay. However, the perturbation, uh, introduced by your spoon has a scale that's far from being microscopic, right? So there has to be some mechanism that kicks in and allows you to transfer this energy that you are injecting at the scale of the, of the spoon, okay, all the way down to microscopic scales. Okay. And at the basis of this phenomenon is the nonlinear, uh, transfer of energy, um, through the, um, formation, interaction, and destruction of vortices. Okay.
Now, um, there is no current framework that really, uh, captures, uh, this, how this process works. Actually, but the, the commonly accepted scenario is that there is an instability of these vortices, okay, okay, that breaks them, uh, scale by scale. So this slide, um, uh, represents this process pictorially. So you start with, you know, your large-scale, uh, eddies or vertices at scale capital L. They become unstable, they break apart, generating smaller-scale vertices of size L prime, a fraction alpha of the previous L. Okay. And then the same process takes place, so that the L prime vertices break apart, okay, giving rise to L prime prime vertices, a fraction alpha of L prime, okay, and so on and so forth. So this process, uh, proceeds self-similarly, um, until you generate small enough scales, so the dissipation becomes faster than this, um, instability process of the vertices, and that's where you end your cascade. So if you translate this from configuration space to Fourier space, this is what it looks like. So you have your energy injection, uh, scale here, and as you break apart your vertices and you're cascading the energy toward smaller and smaller scales, you're are transferring energy in Fourier space towards larger and larger wave numbers until you reach the dissipation scale, and that's where that's your source, okay, that's your sink, um, of energy. Oops.
Now, um, as we shall see, this is really described by Navier-Stokes. Okay. However, to understand how this nonlinearity leads to the formation of smaller scales, it is very useful to use a simplified model, okay, represented by the one-dimensional Burgers equation, which contains the main ingredients that we are interested in. Okay. We have a nonlinear term, the nonlinear advection here, U ddxU, and then there is dissipation on the other side. Okay. Now, let's assume that we have a periodic function for simplicity, and we can decompose it in in Fourier series. So if you take this decomposition and you plug it into equation one, you get equation two, which is an evolution equation for the Fourier mode, for the Fourier, for the amplitude of the Fourier mode K. Okay. The dot here represents the, the time derivative. Okay. Now, the, uh, the trick is that the nonlinear term, when you move to Fourier space, introduces a convolution. Okay. And so this term, as you can see, involves different wave numbers, whereas dissipation acts on, you know, a given wave number. So if we take a look a little bit more in depth to what dissipation does, so let's neglect the nonlinear term, okay, what we obtain is a simple evolution equation for these amplitudes that you can solve. Okay. And the solution is given by an exponentially decaying amplitude. Um, so again, and here it's clear that the dissipation acts on a single wave number. You start with a given wave number and you end with that same wave number. Okay. And the time over which it decays, okay, depends on viscosity, okay, and it depends on the scale, and it's faster the higher the wave number. It goes as K squared. So the smaller, um, the your, the smaller your scale, the faster it dissipates. Now, if we introduce nonlinearity, that changes completely the, the process. The nonlinear term acts as a source, if you like, for, um, for, well, okay, let me rephrase this. The nonlinearity couples different wave numbers. Okay. So if you, even if you started with a single wave number, let's say wave number one, one wavelength in your box, okay, this term here, it's at subsequent time steps, acts as a source for higher for for for exciting higher wave number modes. Okay. And so this term, that's how you start to to transfer energy across the scales. Okay. Now, does this continue at infinity? At infinity? So does this, does this nonlinear term awake all the modes accessible to the system? Well, the answer is already here, but what would you say? Yes or no? No spoiler. Why? Why I cannot hear? Well, yeah, I mean, dissipation becomes more and more important as K increases. Okay. And at some point, this will start to dissipate faster than, than, you know, uh, than the process, the, the nonlinear process of, of, of, um, transferring energy. So, um, we can visualize this process by plotting the solutions to Burgers equation. Okay. So I, what I show here, um, is the, um, the, okay, the solution to Burgers equation, okay, starting from the same initial condition represented by a, a sinusoidal function of one wave number into a box. Okay. That's the solid line. The top row corresponds to a, uh, higher viscosity, and the bottom, the bottom row responds to a, a lower viscosity. And for the Burgers equation, you can also, you know, you can write it in dimensionless form and define a Reynolds number, okay, that in this case is defined, um, through the length of the box. Okay. So low, low Reynolds number on the top, high Reynolds number on the bottom. So initial condition, solid line, and then the subsequent lines, the, dashed and the dotted line, are two, two subsequent times. Okay. And on the right column, I'm showing the corresponding Fourier modes, as a function, as a function of time, no, as a function of K, at different times. So initially, you only have one wave number, okay, in both cases, and then as time goes by, you see that you are forming a distribution, okay, you're waking up higher wave numbers. Okay. And if you compare the top with the bottom, what do you notice? Right. Okay. Which is the high Reynolds number case? Yeah. So in retrospective, if you think, if we think about the Reynolds experiment, what this is, you know, illustrating is that when the Reynolds number is low, then viscous forces dominate over inertial forces, and they, and so they inhibit, uh, this, this formation of smaller and smaller scales. On the contrary, the high Reynolds number, in the high Reynolds number case, um, the inertia term dominates, and it activates this, you know, this cascade, this cascade of energy. Questions? No. 2K? Yeah, no, this does not. Yep. Yeah, yeah. No, no, no, no. This is just for the purpose of, of, just representing. No, it doesn't have to. Yeah. So I mean, really the coupling of K, the coupling of scales really depends on the underlying system, okay, and is represented by this term. Okay. Now, one of the, yeah, I go back, I'll come back to this. Okay. So in general, so you have a nonlocal interaction, really, okay, of scales. One of the assumptions of turbulence is that really what's interacting is, um, is eddies that have nearly the same scale. Okay. That's kind of the underlying idea of the sub-phenomenology.
All right. All right. Now, um, like I said, the, the theory of turbulence was really developed starting from the incompressible Navier-Stokes equation, reported here, and even if, of course, they are more complex, um, you know, the, you know, the action of nonlinear inertial forces here and dissipation is really analogous. Okay. So you have the competition of two, of two effects. Okay. Dissipation on the one hand here, um, and the nonlinearities, which which tends to transfer energy across the scales. The important thing is that this energy transfer, in fact, happens conservatively. Okay. This can be, um, noticed or understood by looking at the energy conservation that you can get from the Navier-Stokes equation. So let us define a kinetic energy per unit mass, okay, which is the volume integral of one half U dot U. Okay. So in order to get a, a, a conservation, a law for this quantity, you do product with U your Navier-Stokes equation. Okay. Oops. Here. And so what you end up is that dK/dt has three contributions. Okay. On the right-hand side is the contribution from dissipation. Okay. And then we have contribution one and two. One comes from the pressure forces, and two is for the, the nonlinear advection term. And now you just, you know, from your toolbox of, you know, algebraic vector operator and theorems, okay, you can manipulate one and two, and you transform this in terms of volume integral of divergences. Okay. And then you assume periodicity, okay, so that there is really no net effect from the from the fluxes at the surface bound at the bounding surface. So contribution one and two integrate to zero. Okay. The third, the third term, which involves the viscous dissipation, is non-zero. Okay. Can be rewritten in terms of the volume integral, okay, of nu, kinematic viscosity, times the vorticity magnitude squared. So what this tells us, okay, um, that the, you know, this conservation law implies that the convective and the pressure terms do not alter the overall energy balance, but only affect its transfer across scales. And the only thing that affects energy balance is viscous dissipation. Now, you can have a forcing. You can force your system, just like when you're are steering your coffee, okay, you are injecting energy into the system, or in the case of the sun, the sun is continuously injecting waves within the solar wind. Okay. So if you add a forcing term here to the energy balance, this is what you get. So the wonderful thing here is that you can achieve a steady state where dissipation balances injection. So the implications of all of these are two. Number one, I already said that the nonlinearities are responsible for transferring energy across scales conservatively. And the other implication is that if a steady state is reached, what that means is that the dissipation, the mean dissipation rate per unit, per unit mass, does not depend on viscosity. And this is the fundamental property of turbulence. Okay.
Now, so far, okay, we have kind of presented a scenario, uh, with some quantitative elements, uh, of, you know, of what happens in, in, in a turbulent flows, which is quite faithful. Okay. However, without further hypothesis, it is not possible to, to derive, to, to develop a theory of turbulence from the Navier-Stokes equation. Okay. And there are many questions that remain unanswered. For example, what happens between the scales where energy is injected and the dissipation scales? And what, when we say that small scales need to be reached, okay, how small are these small scales? Okay. And so the answer to this questions were given by Kolmogorov, okay, in his theory of turbulence that we, uh, you know, I'm going to present here phenomenologically. Essentially, it is a phenomenological theory. And Kolmogorov developed this phenomenology of turbulence starting from three hypotheses that are, you know, motivated by empirical effects. Okay. That at very large, but finite Reynolds number, the turbulent flow is statistically homogeneous and isotropic. So wherever, and this, so this is also why two signals may appear actually, you know, the same. Um, so homogeneous and isotropic means there is no preferential direction, and homogeneous, it doesn't really matter where you are performing your analysis. Okay. Second hypothesis is that at very large, but finite Reynolds number, the turbulent flow is self-similar at scales smaller than the injection scales and larger than the dissipation scale. Ah, okay. So the self-similar thing that we saw with the hierarchy of eddies. The third hypothesis is that at very large, but finite Reynolds number, the turbulent flow has a finite, non-vanishing mean dissipation rate per unit mass. Okay. And so the main elements of this phenomenology are the following. Okay. We have Lambda, the scale under consideration. Um, we have U Lambda, okay, which is the typical value of the velocity associated to that scale, U Lambda. Okay. And T and L, the nonlinear time or eddy turnover time associated to scale Lambda. And the, you know, if you take a look at your Navier-Stokes equation, you can see that this T L is the, I'm going to refer to this also as the nonlinear advection time. Okay. So it's Lambda over U Lambda. L comes from, is, is U dot grad, essentially the inverse of that. Oh, okay. And so these are the three elements. Okay. And then the hypothesis that he makes is that eddies of nearly the same size interact through their gradients and distort and get, you know, and get destroyed. The, the intuition behind this this hypothesis is that if you have a gigantic eddy and and a smaller eddy, the gigantic one is going to advect the, the much smaller one, and the much smaller one won't really affect much the, the larger one. Okay. So the most efficient interactions are between nearly same-sized eddies. Okay. So under this hypothesis, okay, we can define the rate of energy transfer from scale Lambda to the smaller ones, which is well, the kinetic energy per, per unit mass that goes as U Lambda squared at that scale, okay, divided by the nonlinear time. So the time it takes for your advection scale Lambda to pull apart. Okay. And that's our, uh, rate, uh, of energy transfer. Now, because as we said in the inertial range, I don't think I defined inertial range. Do you want to know what that is? Yeah, we are okay. So we are looking at energy transfer rate. Okay. So we have U squared. Okay. That's proportional to energy per unit mass. There is a one half that's missing. But we have a, a nearly, so this wiggle, it's, with the wiggle, we indicate is nearly equal to, up to a factor of order one, okay, divided by a time. So we want a rate. No, it's an energy rate, an energy per unit time, right? So when I, so the, the inertial range terminology just, you know, made its way. So let me go back for a moment to clarify the terminology. So, uh, are there questions regarding this plot? All right. So the energy, sorry, the inertial range, okay, is the range of scales between the injection and the dissipation. So it's where you have this self-similar cascade. Okay. And so this is where, you know, the nonlinear essential inertial term determines the dynamics. Good. Half an hour. Plenty. Where were we? Okay. So, ah, so, so we are interested in what happens in this inertial range. Okay. And we wanted, and so, you know, if the state of turbulence is statistically steady, then the rate of energy transfer from one scale to the next must be scale-independent. So this Pi Lambda, the rate of energy transfer that we defined in the previous slide, and I'm just rewriting it here, must be constant, must be scale-independent, and we call this epsilon. Okay. Um, and now we can obtain how the amplitude of the velocity fluctuations at scale Lambda scales with this scale Lambda. Okay. So, um, this is really a result. This could not have been predicted without the hypothesis that we made. Okay. And what it tells us is that the smaller the scale, so the smaller your eddy, the smaller the associated velocity. Okay. Yes. Yeah. So, um, you remember this? Yes. So this is swirling structures, and, you know, these swirling structures are deformed, they have multiple sizes. So Lambda refers to one, a given, of this, a given length Lambda. Okay. Yes. Yes. I'm sorry, I didn't understand the question. Ah, yeah, yeah. Direct in this? Yes, correct. Yeah. Direct in this. So maybe I should rephrase this in the sense that the, the transit of energy is across the nearby scales, and there is no, so this scale here, per se, is not going to dissipate directly, but scales need to be smaller and smaller, smaller, until they can be dissipated. But I mean, yeah, perhaps it's not the best way to describe this. Direct, direct in the sense, local in, at that scale, local in Fourier space. Okay. So if you look at this process in Fourier space, energy injection is far away, and dissipation is far away. But it's indirect in the sense that energy is injected at a given scale, and then it cascades. Of course, there is the effect of injection and dissipation through the turbulent cascade. Um, it's semantic. Okay. Um, now, if we apply this concept of scaling invariance of this energy transfer rate at the top of our turbulent cascade, okay, what we get is an expression for epsilon in terms of the microscopic quantities. So epsilon is the energy injection rate, okay. And if we apply this at the bottom of our turbulent cascade, um, so we balance dissipation with, with epsilon, okay, then we find finally the scale eta. And I apologize terribly for the abuse of notation. Eta here is not the magnetic diffusivity. It's a scale that happens for historical reasons to be called eta. Um, and so when we balance dissipation with our injection rate, um, then, then we find the scale at which the, the, the viscosity becomes dominant over over the nonlinear term. Okay. This is this is called the Kolmogorov scale, and it is proportional to the macroscopic length L times the Reynolds number to the minus 3/4. So the higher your Reynolds number, the longer is going to be your, your inertial range, in such a way that you can achieve balance between injection and dissipation. Now, so, okay, maybe some of you have heard about the Kolmogorov turbulent spectrum. What's the scaling? What, who says 5/3? Yeah, guess who says 5/3? Well, guess it's here, okay, but you don't recognize it yet. So to make this result more, um, familiar, okay, we can derive the prediction from this theory of the energy spectrum. Okay. So we define an energy density in Fourier space, and you basically, you, uh, you associate Lambda to the inverse of the Fourier mode. Okay. So if you do this operation, then from the scaling of U, here you get an energy spectrum that scales as K to the minus 5/3. And this prediction of the energy spectrum is very well, you know, uh, it has been verified in plenty of cases in fluids. Okay. This is one of those, you know, one of the experiments that have confirmed the Kolmogorov spectrum. Different numbers like, yeah, this is incompressible isotropic turbulence. So it's very specific. It's a very specific system. Um, there is evidence of Kolmogorov-like, uh, turbulent cascade in space. Okay. So this is a typical example of, um, the magnetic field energy spectrum in the solar wind, one astronomical unit. You will notice that it's kind of much more complex with respect to the previous plot from standard fluids. We have what's defined here, the injection range. It has a power law to f to the minus one. So, okay, here it's frequency rather than space K. Okay. We use, you will see this perhaps later in the, in the, in the laboratory, how do you get go from F to K or not? Yes. Okay. Good. Good. 30 minutes. Plenty. Where were we? Okay. So, um, after f to the minus one, then here it is our inertial range. A lot of effort is going to determining the, the spectral slope of this. Okay. And then there is a breaking point here, and if it is followed from a sub-inertial range, this happens when you are hitting the intrinsic scales of plasmas, will be the Larmor ion Larmor radius or ion inertial length, whatever comes first. Um, so rather than a, a standard dissipation range, you have this sub-inertial range where you have, you know, other physics comes in, okay, kinetic effects. So what I'm going to focus on is in this inertial range, which is at scales larger than proton kinetic scales, and therefore it falls into the MHD regime. Before going to that, I'm going to show you a couple of other examples. How much time do I have? 30 minutes. Okay. Uh, okay. So, um, similar, uh, cascades are observed in the magnetosphere. This is the magnetotail, and also in interplanetary space, there is evidence of Kolmogorov cascade. So here are, these are old data. This, this, this spectrum refers to density fluctuations. So these have been measured indirectly through radio scintillation. And this on the right, in fact, they found the same result. These are direct measurements of density fluctuations in interstellar, I mean, from Voyagers, okay, when it supposedly in, in beyond our solar system. So all right. So we have seen that there is plenty of evidence, okay, of a turbulent cascade similar to what Kolmogorov has described for fluids in magnetized plasmas. Okay. However, there are fundamental differences. There are similarities, but also important differences. First of all, MHD contains two scales. Okay. There is the scale associated to wave propagation, specifically, we will see the Alfvén time. And the fluid time scale, like in, like in Navier-Stokes. Okay. Magnetized plasmas have also a magnetic field, so isotropy is broken. The presence of a mean magnetic field introduces a preferential direction. And because there is a magnetic field, now we have two fields, the velocity and the magnetic field, okay, that are coupled together. So these are, uh, the complexity. And just like for fluids, also in MHD, really what has been studied a lot is incompressible MHD, and the building blocks of incompressible MHD turbulence are Alfvén waves. Okay. And I'm going to skip through this because now you are experts, right? Yes. So here are for your reference, your incompressible MHD equations. Okay. Like Marco said yesterday, we can introduce instead of using U and B, we is convenient to use Z+ and Z- the Alfvénic fields. Okay. So you can combine the momentum equation with the induction equation. You can, uh, appropriately sum and subtract them, um, so that from two equations, one for U and one for B, you get other two equations, one for Z+ and the other for Z-, and so these equations are nice because we can kind of recover some symmetries of Navier-Stokes equations with with some differences. First difference is this term here. Okay. This is the propagation term. A linear propagation term. So if you forget about what's on the right-hand side, okay, you obtain a simple advection equation, uh, for Z+ and Z-. So now you can see that they really represent Alfvénic signals that propagate parallel or anti-parallel to the mean magnetic field. So VA is the Alfvén speed associated to the mean magnetic field. Okay. So this is the first, first, first difference. The second important difference is that what was the U, the nonlinear term, well, now you can see that it mixes the two fields, the Z+ and Z-. So what that means is that the nonlinearity in incompressible MHD is triggered by the interaction of counterpropagating Alfvénic modes. And so, um, so we have two time scales. So we have what I called earlier the nonlinear advection time, Z- dot grad, and then there is the propagation, linear propagation time. And so depending on what is shorter, okay, or what term here is bigger, well, then it determines a specific regime of turbulence. Okay. So turbulence is, um, said to be weak, okay, when the linear propagation term dominates over the nonlinear advection time. So what that means is that the Alfvén time associated, um, um, Lambda over VA is shorter, okay, than the non, than the nonlinear advection time. The opposite limit defines the strong turbulence regime. There are countless phenomenologies of MHD turbulence, which I'm not going to review, but I'm going to discuss two that serve, in fact, as the basis for many variations on the theme. So the first phenomenology, in fact, the first phenomenology of MHD turbulence was introduced by Iroshnikov and Sagdeev, and it falls into the category of isotropic weak turbulence. So they assume the hypothesis is that turbulence is driven by the interaction of Alfvénic eddies, okay, and the interactions are weak. Okay. So the shorter time scale is dictated by the Alfvén, the Alfvén propagation time, and during a given interaction, these eddies are not deformed much. Okay. So the, so the second hypothesis is isotropy. One second. So isotropy, which means that they do not distinguish between scales parallel and perpendicular with respect.
To the mean field, yes. I mean, okay. Well, that's not yes. Yeah, right. That's okay. No. Did so, okay. Forget about waves, like we're talking about like structures, okay? Which are repres- alanic. Well, we are talking about structures represented by fluctuations in velocity and magnetic field, okay? Which, if they were non-interacting nonlinear, so if the amplitude were very small, this, this would represent signals that propagate the Alfvén speed. Does it pass? Yes. Somebody? What? Yeah. No. So, okay. Ah, no. Yeah, no. The, when we are talking about like nonlinear situ- nonlinear systems, really there is a big discussion on whether we can use the even the name wave, uh, or not. Marco, you want to say something? It's a little, yeah. But right. So when we are talking about like turbulent, uh, settings, especially when we talk about strong turbulence, you really have to forget about the concept of a nice sinusoidal wave. It's large amplitude structures which are deformed and, you know, um, yeah.
All right. So, um, so the weakness of the hypothesis is here in this inequality, and the isotropy is in the second, uh, you know, uh, uh, is not an equality, okay? This they approximate lambda parallel of the order of lambda perpendicular. Okay? So, um, that makes the, the cascade isotropic. Now, because the, the dominant, the shortest time scale is the, uh, Alfvén time, okay? Um, this Alfvénic eddies, let me bear with me with this analogy just for the sake of communication. We interact during a time delta t, which is determined by the Alfvén time. That's the hypothesis, okay? But because, um, all right. So, so and so during this single interaction, during one of these interactions, there's going to be a, uh, a variation of the velocity, delta u, equal to delta t, u squared over lambda. This comes from the, the, this comes from the momentum equation, okay? You have delta u over delta t, okay? You equal of the order of u dot u, okay? That's, that's this term here. And then we plug in our estimated length of delta t. Okay? The, the weakness enters here in the fact that delta u over u is much smaller than one. And so to have a, a sufficient strong perturbation delta u, you need many of these single collisions. And it is assumed that the process, okay, what is it? Oops, I'm going back. All right. So you need n of this, n of these collisions of Alfvénic eddies, which are assumed to be random. Because they're random, then the, um, the standard deviation of a random process scales as the square root of n. This is again the central limit theorem of this morning. Um, all right. And so, well, how, how much is this n? Does this n need to be, okay? Where we impose the delta u over u would be one, okay? So that's big enough to ensure that the, the Alfvénic eddy has deformed not linearly. And so you impose that the ratio delta u over u is one, and that gives you a number of collisions that is approximately equal to v_A delta t over u lambda squared. Okay? So the nonlinear times therefore would correspond to the n, n of this collision times, the time of delta t of the single collision. Okay? So you end up having a much longer nonlinear time than Kolmogorov. Okay? And so this ends up affecting the turbulent cascade. So you have all of the elements, in fact, to derive this energy spectrum on your own once you have determined the nonlinear time. Okay? You apply the, the, you know, the constraint of, you know, scale-free energy transfer rate, and you get your spectrum. You can do that.
The major limitation of this model is that turbulence is assumed to be isotropic, which is not what we observe. Neither in observations nor in simulations, okay? If there is a mean magnetic field, it's easier to break filament across the magnetic field rather than along the magnetic field, okay? So the second phenomenology introduced, that's, that's one of the most famous, is by Goldreich and Sridhar, who, in fact, introduced anisotropy, okay? And their phenomenology is based on two assumptions: critical balance, okay? So the two time scales are of the same order of magnitude, and strong turbulence, so that the nonlinear time is postulated to be the analogous of the Kolmogorov turnover time, okay? With the exception that instead of lambda, you have lambda perpendicular. So again, you have at this point all of the ingredients to obtain the corresponding predicted energy spectra.
Now, how well does this phenomenology do to reproduce observations? Um, so this is another example of the, the magnetic energy spectrum at different radial distances from the Sun. So these are taken by Parker Solar Probe, okay? So what we find is that this spectral slope ranges from minus 3 to minus 5/3. So in between the Rachnikov and Lashkin weak turbulence and the Goldreich and Sridhar, that are strong, strong, strong turbulence, okay? And more or less, that, that's always been more or less the debate on whether it's minus 3 or 5/3. But it appears, in fact, that the turbulence is evolving with radial distance.
The other aspect that is, uh, found, in fact, is that anisotropy, I mean, that the turbulent cascade is, is anisotropic. And at, at least at one astronomical unit, it seems to tend toward a critically balanced state. In particular, this lower plot here shows the spectral index as a function of the angle between B and K, okay? So when you are nearly parallel, you have a minus 2, as predicted by the critical balance. And then when you are perpendicular, you are close to 5/3. There, you know, other scalings have also been reported, which points to the fact that there is not really a conclusive answer as to what theory, um, explains turbulence in the solar wind.
If I have a few minutes, I'm not sure, and I want to kill you also, but, um, there are many other issues that are not explained by this theory. The fact that velocity and magnetic field display different spectra, um, and how much time do I have? Okay, sure. Um, that's, that's, that's enough. So the other thing I wanted to point out, which is quite interesting, I want to close the circle at this point. There are two types of wind with two types of turbulence, okay? So this plot here, so this plot here shows the velocity, okay, of the solar wind. So you have fast wind and slow wind, okay? And above and below, there is the, the, the, the calculated spectrum of Z+ and Z-. So in the slow wind, what you find is that the Z+ and Z- are balanced. I mean, they have the same content of energy. These two lines are nearly superposed. Good. The fast wind, if we look at that, completely different. Yes, yes, yes. Because as we said, I say repetitively, repetitively, um, we have an Alfvénic wind. So we have a non-Alfvénic wind, which is this. This is standard. This, this, you know, reproduces the observations of this kind of turbulence. It is consistent with the theories, the phenomenologies, and it is consistent with the hypothesis underlying those phenomenologies. Alfvénic turbulence is highly imbalanced. So it shows properties that are consistent again with the phenomenologies, but it's not consistent with the hypothesis, okay? So that's the big, I guess, one of the big, uh, open conundrums, if you like, of Alfvénic turbulence. I think that's, that's the main bottom line. I have a few extra slides to reinforce this argument, okay? Um, so this is another way to look at the same, same thing. So this is the distribution of solar wind data points in the parameter space defined by cross-helicity and velocity of the wind. The cross-helicity is defined here as a scalar quantity. So we love scalar quantities, um, that measure the imbalance, okay, of Z+ and Z-. And so the slow wind is, so the, in fact, what's coming to light is that there are two types of slow wind. One that's the standard balance, zero cross-helicity, nearly zero. And then there is a slow wind that is also highly Alfvénic, so cross-helicity nearly one. And this is the standard Alfvénic wind, this blob here. Fast and highly Alfvénic. Sorry, what Sigma M? I don't know whether Sigma M is the residual, so the imbalance of kinetic and magnetic energy. Um, in principle, if you're very Alfvénic, you will not have, so Sigma M should be, um, zero, okay? Because magnetic and magnetic energy and kinetic energy are exactly balanced by definition, right? Um, and instead, it's magnetically dominated when you have U nearly zero cross-helicity. Is that what you're referring to? Maybe. Um, and so the devil is in the details, because when even when we are highly imbalanced, we observe an excess of magnetic energy, right? Which should not be there. And we're trying to understand what causes it. Is it, honestly, I don't know about the Sigma M. Magnetic is also not well-defined because it's gauge, it's also gauge dependent. What is gauge dependent? No, it's not that. Also part of the problem. Yeah, that's right. Yes. If you had multiple, yeah, yeah. So, right. On. Yes. I think so. I was just going to say, you know, reinforce the idea that there is turbulence, but yet coherence, which is kind of a very interesting aspect of this Alfvénic turbulence. I'll just conclude that, well, it's not surprising that the phenomenologists do not exactly work well because they assume homogeneity, for example. The solar wind is certainly not homogeneous because it's an expanding medium. And so these are really the starting point equations to study how these fluctuations evolve, um, in the solar wind. And reflection, which is this, sorry, the, the inhomogeneity of the flow, um, because it is expanding, it's important because it reflects waves, okay? And so one way, so we know now that we need counter-propagating waves to trigger the turbulent cascade, right? And so reflection can help, okay? Because it reflects the wave. And so there is also another model, which is the turbulent, sorry, reflection-driven turbulence, that's also an important paradigm to study, you know, to study turbulence in the solar wind. And I, I close, I close here. Thank you very much. Got questions? All right. Okay. So I understand there's different types of turbulence, and I imagine a significant reason is because of differences in key sort of scaling parameters of the plasma. Um, but are there sort of fingerprints of the mechanisms which cause the turbulence, say waves in the corona versus Kelvin-Helmholtz instabilities in Earth's magnetosheath? Would those lead to significant differences in the kinds of turbulence you observe? What's, thank you. What's the origin of turbulence? So there are many ways you can have perturbations. You can have injection, like in the case of the Sun. You can have an instability, like a Kelvin-Helmholtz, that would lead to, to, I'm sorry, the turbulence, uh, all right. Now, um, I don't think I have a good answer to your question. Um, so the question is, if the way turbulence is initiated leaves an imprint? Yes, okay. In the, in the spectrum? Wow. Um, well, possibly. In the case of the solar wind, for example, I mentioned the, um, the energy-containing range, okay? Well, that's where you, so the energy-containing range is where, you know, um, supposedly the source is, okay, of whatever triggers your turbulent cascade. So in the case of the solar wind, there is this f to the minus one, okay? So some people believe that that's an imprint of what's happening on the Sun, for example. It's a difficult question to answer, but that's an example where you can find. There are theories that seek imprints in the turbulent cascade. So that is possible. Yeah. Well, yeah, I guess what, what is the, um, no, the question is whether the different type of scale, turbulence that you mentioned has something to do with the scaling, actually, like, V. Kolmogorov scaling, which goes like minus 5/3, and others which go by minus 7/3, etcetera. So whether the different type of turbulence has to do with, is have some connection with the scaling, actually, the spectra? So different types of turbulence give to give rise to a different spectrum? Is that your question? Yeah. It's your question if we have a different type of turbulence, what kind of scale do we expect? Well, okay. Um, yeah, that's another good question. And it really depends on the system you're studying. So I, I don't have, on top of, on top of my head, um, numbers. I'm not a great, uh, uh, a great, uh, you say, I'm not passionate about the scaling laws per se. Um, but I showed you an example of turbulence in the magnetosheath, okay? And it, I, I chose it at, uh, that was showing a k to the minus 5/3. Okay? But in, if you analyze data in the magnetosheath, you can have, um, a spectrum, so signatures, let's say that a spectrum is really a, um, symptom that you have nonlinearities ongoing, okay? But if you have dominance of compressible modes, magnetosonic structures rather than incompressible fluctuations, the spectrum is, the spectral slope is different. I don't remember the number. Okay? Yeah. So, uh, I, I know you've done a little bit of work, um, on this. So I just wanted to ask you, since we have you in person, what do you think is the relationship between turbulence and switchbacks? Oh, oh, that was one of the, right, right. I didn't finish my talk. And my, my last slides were, um, challenges brought by the recent observations, uh, by Parker Probe. And so, as it ends up, this nicely both Alfvénic, coherent, and apparently turbulent state is permeated by switchbacks. So switchbacks are embedded in that turbulence. They're part of it. They're not, how do I say, they cannot pretend from the turbulence, they're in there. Um, what's their role? That's one of the questions that people are trying to figure out. Somebody claims that nonlinearities are enhanced during intervals where you have switchbacks. Others instead, uh, perform other types of analysis and say that no, there is nothing special happening in switchbacks. Um, the fact is, they're highly coherent. The nonlinear term, therefore, in principle, is quenched. Um, I don't have an answer. I mean, we're working on it, I guess it's an open field of research. Interesting. What happens? Um, it's possible that they, what I studied, for example, is that structures like, like those, become unstable. Inevitably, all of the Alfvénic structures, if you do simulations, um, or some, some, uh, stability analysis, you find that there's no way they're not supposed to be there. They're supposed to be unstable, and yet we find that they are nice, there, standing there, propagating. Um, and, um, so we are trying to put all of these pieces together. Yeah. Yeah. There's one question. Um, so for the Kolmogorov scaling, we had the assumption of incompressible and isotropic. Yes. So my question is, does it go both ways? So when we find the scale, Kolmogorov scaling in observations, can we directly say that it, the plasma behaves more, at least in a dominant way, as an incompressible fluid and isotropic? Um, I wouldn't do, I wouldn't do it. But you, what you can do is that, okay, you find this, let's say this symptom of, okay, it looks like a Kolmogorov nonlinear turbulent cascade. And then you, you have to go and take a look at how the signal looks like. As it turns out, in the solar wind, they are really, they're nearly incompressible, actually, right? So this, this part is consistent. However, there are other things that are, um, are, um, inconsistent, let's say, with the hypothesis, uh, that underlie Kolmogorov in the solar wind. At least the fact that it is not really isotropic, it's an inhomogeneous medium, okay? So, um, how does that come into play? Um, and I had, um, something else I wanted to say that now I slept through my mind. Right. And also the fact that we do observe that it is incompressible, so density fluctuations are very, very like 10%. Uh, the question is, how is it possible? Because that's not supposed to be the case. Uh, so one of the questions to understand also is how do you end up having this incompressible nonlinear state with incompressible waves, and yet this appears to be, um, you know, with ongoing nonlinearities? So that's, there are this kind of apparent contradictions that make, I think, the topic of Alfvénic turbulence in the solar wind quite fascinating. I don't know if I answered your question. So I mean, you do not assume always verify. Thank you.