 Hello, my name is George, George Milosevic. I'm a graduate student at University of Texas at Austin. And my advisor is Philip J. Morrison. This work has been done in collaboration with him and his former student, Manas Vilingam, who is at Harvard University right now. It's called on Cascade Reversal and Extended Image D. So the audience will be briefly introduced to Extended Image D in this Hamiltonian topological properties. And then we'll start talking about the turbulence, flux, transfer rates, and direction of Cascades in 3D and 2D turbulence. So recently, attention has been focused towards the turbulence at small scales, at least in the context of astrophysical turbulence with the examples of Earth's magnetosphere and the solar wind. We are investigating the helicity Cascades. And it's known in image D has been known for a very long time that there is an inverse cascade of magnetic helicity, which is often linked with the dynamo mechanism. But the question that we are asking are what are the effects of the electron inertia that is oftentimes ignored when you do image D? And for this, we are using the model called Extended Image D, which is formally a single fluid model, but it's endowed with two fluid effects. And these are a whole effect in the electron inertia that I just mentioned. We're going to work in alphanic units normalized to a typical length scale, so everything will be dimensionless. And the important parameters are the skin depth that you can see here. So there's an ion and electron skin depth that are proportional to the corresponding masses. So if you ignore electron mass, you're ignoring the electron skin depth. And this is the Newton's law for the bulk velocity. And if you ignore the DE, so if you ignore the DE, then you basically recover the regular image D. Notice the star here is defined over here. It also depends on the DE. And of course, you need Faraday's law. And again, so if you ignore the DE and DI terms, you're recovering regular image D. But this system fully describes two fluid plasma. And the various models, the various Extended Image D approximations you get from here have been shown to be Hamiltonian by Abdelhamid Kawazura and Yoshida. And this was later confirmed by us. So the reason people care about Hamiltonian methods is because they allow you to show that if you have a reduced model in the nondisputative limit, it will conserve energy. And there are some models in the literature that do not. And what you work with is a Poisson bracket that gives you time evolution of general functionals. This is the Poisson bracket. It depends on variable psi that for Extended Image D will consist of the density velocity in the B star. The star is defined in the previous slide. And this is the Hamiltonian for Extended Image D, which consists of the kinetic part and the magnetic part and the thermal part. The nice thing is that this Poisson bracket allows you to find the Kazumian variants, which are basically like functionals that commute with any other functional. So if you commute it with Hamiltonian, that's the time derivative of that Kazumir that basically tells you the time derivative will be 0. Therefore it's integral of motion. So this is a way to find integrals of motion. There are some other perks of Hamiltonian methods that I'm not going to go into. And the Kazumian variants for Extended Image D turn out to be these, which look very similar to the A dot B in MHD. But they are more general because this is a definition of B plus minus, which has a B star here and the vorticity. And A is basically an uncurl of B. And the kappa that you see here, there are two solutions of this quadratic equation. So another property that the general two fluid model has is the lead ragging of the two generalized magnetic fields. And basically lead ragging is more for math people, for physics people. It's known as flux freezing. And I'm sure people are familiar with this. People in MHD are familiar with this. So I'm going to skip. Oh yeah, and by the way, if you're interested in topological properties of Extended Image D, this is our paper on this topic. OK. So now we are in position to start exploring turbulence. We're considering the homogeneous and compressible Extended Image D turbulence. And people in turbulence are familiar with the direct cascade where you have a driving range and a dissipative range. And there's an assumption that the same flux of energy or helicity flows through each wave number. This principle was incorporated in this other paper that was done by my collaborator on this work, where they found the Kolmogorov spectra for Extended Image D. Now, the first step that we are doing, the first question that we're asking, we want to find the dissipation rates of the helicities. And we're basically following in the steps that were outlined by Vanerjee and Gaultier, who did this calculation for Hall Image D. But we're doing the same calculation for Extended Image D. And you work with two-point correlation functions, where x prime is basically evaluated is displaced by an r from the x. And the brackets denote the ensemble average. And upon applying these lead-ragging equations, you can easily show that this is the evolution equation. So in a stationary regime, this should be 0. And therefore, the helicity flux transfer rate should equal the damping, the phenomenological damping term that we add, where delta here stands for the increment. And f prime is basically f evaluated at x prime. So it's important to point out that, while this expression looks very similar to the one obtained by Vanerjee and Gaultier, it has more physics in it. It has the electron inertia that wasn't present before. And of course, in the Hall Image D limit, when you set dE equal to 0, you recover their results. So what is different? Well, in Hall Image D, the Casimir invariants are these. So these are the helisties, the a dot b and this one. And in extended image d, it's more complicated. So this one looks essentially the same as the mHd magnetic helicity. And this is some sort of combination of cross-helicity and magnetic helicity. Here, everything is combined in extended image d. And these authors that I mentioned previously, they argue that in Hall Image D, if you just look at this expression, they are assuming that you will get an inverse cascade because in image d, you have an inverse cascade of magnetic helicity. While this one, they're arguing, will have both the inverse cascade if magnetic energy dominates and the direct cascade of kinetic energy dominates. But then we are led to a question, what happens if we work with this more general model, which seems to have magnetic and kinetic contributions that are more symmetric? So what are the effects of the electron inertia? And how is, because if you look at this expression, it's more symmetric. So you would expect both cascades. And so how is the direct cascade of magnetic helicity lost when you go to Hall Image D or end image d? OK, and so for this, we decided to basically calculate the absolute equilibrium states that would predict the direction of the cascades. This method has been used earlier. So basically, you can find this in this turbulence textbook on image d. Basically, what you do is you find the equilibrium state and that's given by this Gibbs distribution. And then you argue that these various spectral quantities would like to relax into that state. However, we're not letting them relax. We're constantly pumping the energy. So this is a partition function and the large multipliers in the integrals of motion, which are basically energy and the helicities. And this approach has been successfully invoked in hydrodynamics in image d by Kreikman and Frisch. So that's the reason we're using it. OK, so the thing is that we wish to compare results to those in the literature. This particular one was done in image d. And the problem is that the generalized helicities that I mentioned earlier, they both become magnetic helicity when you ignore the, yeah, I was on the previous slides, but they're very symmetric and they both become magnetic helicity if you ignore the electron skin depth and the ion skin depth. So what I did is I combined, I found a linear combination that's more suitable for my purposes because this one, if you ignore the electron skin depth then this becomes a dot b and this term disappears. And if you ignore the ion skin depth, then this term disappears and you get v dot b. So this one becomes like cross helicity and this one becomes magnetic helicity. So it's more consistent with the image d. So in whole image d, so yeah, the first thing that we do is we calculate the absolute equilibrium states for whole image d because the expressions that you get in general are very complicated. So these equilibrium states are found, by the way, in the Fourier space. So you have to plug this in and then find the expression. This is the spectral quantity that corresponds to the magnetic helicity. And we're, it's important to point out that we're interested in how much of it we have per wave number magnitude. So you have to multiply with this four pi k square in 3D and you get some complicated expression. And then we do some analytical queries that I'm not gonna go into. But eventually we analytically show that these are really representative plots for the spectral quantities of interest. So this is the energy, cross helicity and magnetic helicity and the idea is if a spectral quantity is peaked at high k, it means that it would like to flow, it would like to have a direct cascade whereas if it's peaked at the low k, it would like to have an inverse cascade. Remember that low k corresponds to large scales and the high k corresponds to the small scales. So this is basically consistent with MHD and this result has been obtained earlier actually by some survey view and some other authors. So this is basically known and consistent with numerical simulations. What is less known is the inertial image D limit. So inertial image D is a model that lacks the whole drift. However, it does have the electron inertia. So basically you drop the DI and it simplifies your calculations considerably. And in addition, we have to assume something else to get a nice result and we're assuming that we are in a very short scales. So much smaller than the electron skin depth. And in this case, this kind of expression is obtained from magnetic helicity and the corresponding expressions for the energy and cross-helicity. And the plots are here. So this is energy, cross-helicity, magnetic-helicity. So the colors, by the way, correspond to different values of total cross-helicity over the energy. And what you see is that there is a market difference in how magnetic-helicity behaves. So this is a direct cascade again because it's peaked at the high K. So it would like to flow into the smaller scales. And this is the main result of our paper in New Journal of Physics where we're claiming that we're predicting that there should be a cascade reversal when you cross this DE scale. Of course, it hasn't been confirmed yet numerically which we would like to do very much as soon as possible. Okay, so now before I go into the, yeah, let me go into the 2D results that we got. So these are not published yet. The thing about 2D, first of all, the numerical simulations here would be much easier. Also that in 2D, inertial image D is more justified because basically, if it's a pure 2D though, you can essentially drop the ion skin depth. It doesn't really, it doesn't, an equation to the motion don't depend on it. And what you do is you represent the fields in this clutch-like form. So this is reduced image D except this is reduced extend image D. You have flux functions and stream functions. And for simplicity, I'm assuming that Z direction is completely absent. So V and nu are zero so that we are in pure 2D. And recently there was a paper that also inspired me to do this 2D extend image D turbulence. There was a paper by Grasso who obtained a Hamiltonian formalism for reduced extend image D. And so the Casimir invariants were available to us. And so we have two continuous sets of Casimir invariants. In fact, they're a little bit more general but we are using quadratic ones because quadratic invariants will survive under truncation. And we do truncate our continuous case base, right? So that's why we would like to have quadratic invariants. And I need to define certain quantities here. So the psi is a flux function and the psi star is like the B star of extend image D. In fact, it very closely corresponds to it. Omega is the Laplacian of phi. So phi is a stream function. So this is like magnetic elasticity in some sense. And this is like a cross-elicity because it's a little bit like V dot B in this, yeah. It corresponds to it in some sense. And yeah. So next we do again the standard Fourier expansion and these are the spectral quantities for F and for the Hamiltonian. And again, I'm not listing cross-elicity because it would take too much space. And we seek the equilibrium states that are given by this Gibbs distribution and this is the equilibrium for F. It's a more simple expression than the one that you have in 3D case. So now you can consider to understand how it behaves when you change the K. It's convenient to again make some assumptions. First of them is pretty common in this kind of business is to assume that cross-elicity is zero. That leads to gamma equals zero. So this parameter is zero and then this part just disappears and consider the MHD limit. So the MHD limit has been handled by five and other authors a long time ago and then confirmed in numerical simulations. So basically that's if Kd is much less than one, then this quantity, okay, so you can actually do this in your head. If Kd is much less than one, then you can drop this term. So this denominator scales as one, and numerator goes as K square. This is zero, beta you can kind of ignore under, you also have to analyze. There are some normalizability conditions that restrain where these quantities can be. So roughly speaking, this term scales as K square and this is inverse, right? So it's one over K square, but remember you have to multiply by the volume of your space, which is two pi K in 2D and so this term scales as one over K and that's the typical sort of inverse relationship that suggests an inverse cascade for this quantity. And for the energy you have a linear relationship. So I'm not listing this expression but it's a similar sort of estimation and this has been known. So we are basically recovering their expression. However, this is a sort of more new result where we're looking at the small scale. So this is extend, image D limit. Scale is smaller than the electron skin depth. And this case, again, you can do it in your head. So Kd is very large, so one can be ignored. So K cancels out. So this is constant. This remember is zero because we're considering that the states are not, you know, like V is not parallel to B basically. So there's no cross-solicity. So this term is also not there and this is constant. And then you have to multiply the constant by K which means it goes linearly with K. So there's a different behavior here. The energy though behaves the same way. So it's a direct cascade, it's a direct cascade. This is an inverse cascade and this is a direct cascade. So there's a cascade reversal into D as well. Okay, so I think I'm doing well on time now. Okay, so to sum up, we have addressed recent interest in short scale astrophysical turbulence and mostly theoretically, we haven't really applied it anywhere. We're hoping the people will apply it somewhere. And this is a little table that kind of tells you how the different invariance behave as you go through the scales. So MHD governs the large scales, all MHD smaller and inertial MHD even smaller scales. And then there are new 2D results that also kind of behave in a similar fashion. So the energy is always a direct cascade, the cross-list is a direct cascade and in the inverse cascade becomes a direct cascade for the magnetic helicity. Okay, and so yeah. So also the dissipation rates were computed but they basically motivated us to look for the cascade reversal. And the lack of the inverse cascade at small scales may have impact in dynamo theory. We plan to test the direction of case case into D numerically and maybe in 3D. There are some new results by Servideo in 3D that I haven't really had time to look at. And yeah, so recently we have found a Hamiltonian description for relativistic extent MHD that hopefully will allow us to find the corresponding chasm here so that we can do the same spill for relativistic case. And yeah, so that we can calculate the turbulent relativistic cascades. So thank you for your attention, I'm done. Yes, sure. Which one, like this one is fine or the 3D one? The one before? Well, there is one here but it's essentially all the same for the 3D and this is a Gibbs distribution also. Yes. Yeah, yeah. Yeah, I suppose you're referring to your paper in 2002 with Mahajan? Yeah. Yeah, so I haven't really, I know about this objection but I haven't really found a solution to this. So really numerical simulations I think will show whether this will be equilibrium or not. Okay, so you can determine, there's a sort of formal way of determining them by taking integrals of these expressions, integrating them and then finding the total F and total G and total H and then you have to invert and after you invert you'll find the, but the problem is that the expressions are often, in fact, for this case will be transcendental and especially for the 3D case. So what I do instead is I can find them numerically so I cannot do this analytically and also in addition to these, you can use sort of normalizability conditions. So you have to make sure that this probability distribution function is, maybe that answers your question, I'm not sure. It needs to be positive definite. I mean like this quadratic form needs to be positive definite and that puts constraints on alpha, beta and gamma. If it's not positive definite, then you won't be, the probability distribution won't be integrable and square integrable. So we find those constraints and that reduces the possible. Otherwise if you look at previous equations, I'm not taking too much time. Yeah, so this is pretty horrible. However, it behaves mostly like this and the way we saw this is we basically constrained the possible values of alpha, beta and gamma by the normalizability conditions. I mean it's, it may be sensitive, but this typical behavior will not change. So it's gonna be inverse relationship here and it's gonna be direct relationship here. At least I haven't seen it change and according to, so analytically we considered sort of more two cases by being more precise to extreme cases and the plots basically match them. So I'm kind of confident that that's how things will go. Yeah, all right.