And thank you for the chance to be here for the invitation and the opportunity to speak of the colloquium today.
The work I'm going to present is in some part very new, but I'm not going to present the latest,
most complicated and sophisticated thing that was built on the eight or ten years of graduate students and post-docs and scientists,
because it becomes rather impenetrable when you get too far down into the weeds.
I am going to try to tell you about a phenomenon in Plasmas, which is ubiquitous and known since the sixties,
but unappreciated in the field of turbulence until recently.
And. Alex has kind of led the way on this particular project in terms of identifying these echo phenomena that I'm going to talk about.
And then the people on the list, they include graduate students and other other colleagues from around the planet plasma physics community.
There are many other people who've worked on these projects and been very much involved, including Paul Deller, who's here.
But the way I decided which names to stick up or the slides I used is that if I actually were using something specifically from someone,
a slide they gave me, then I did that.
Otherwise this list would be very long because I actually John Maynard Keynes has a quote as saying he said something along the lines of.
There is no limit to the number of something like the number of mistakes a person makes when they work alone.
And I have enjoyed working very much with colleagues.
I'm a collaborative kind of scientist, so this list will be much longer.
Okay, so I'm going to presume that there are people in the room who already know this work, and I know who you are.
And then there are also people in the room who would like to know what is going on.
And so I'm going to try to present at least part of the time, material that should be accessible to everyone.
So it may seem slow, but I'm doing it on purpose. Let me set the stage with the General Physics Colloquium.
I'm going to work on the classical physics problem. There's no quantum mechanics, but it is a mini body problem.
So we're going to consider systems of charged particles, very large number ten to the 23, ten to the 25, those kinds of numbers.
And you know, since it's a non relativistic classical system,
the particles are newly created or destroyed and they just move around under the influence of electric and magnetic fields,
some of which are sometimes in problems applied from the outside world.
And sometimes or in all cases, there's also self-consistent,
electric and magnetic fields that arise from the charges and currents that arise in the plasma.
You calculate those electric and magnetic fields from Maxwell's equations, and then under the influence of these fields,
particles move around in classical trajectories with position and velocity changing in time, none disappear.
And so the basic equations to describe what's going on has to describe where all those particles are.
The probability of finding a particle x one, the one at the first particle at x and B the second particle x2 and B two and so forth.
And it's a very simple equation. It's just that the total probability of finding all these particles in the system doesn't change.
And starting from there, it's just a legal equation.
It's reversible, and one could imagine working directly with that.
But of course, in the capital it is a very large number.
So we want to we want to simplify things.
And the. There are two limits where people get very interested, the strongly coupled limit and the weekly couple of limits.
So I'm going to be showing you weekly, couple of limit and I'll tell you what that means.
Weekly couple plasmas are systems where in in a sphere called the device sphere,
which is lambda by radius within a sphere, the number of particles within the by sphere is extremely large.
And if you have that system, then you will have screening on this dividing scale and there will be no long range fields other than such as Arise.
And I'll show you in a moment.
So this number of particles in a in the by sphere is called the plasma parameter capital lambda and the collective effects
dominate the behaviour of your at large scales and then you have within the device sphere something else is going on.
I'm not interested in what happens inside the device sphere.
I'm interested in phenomena which are astronomically, literally large compared to these little device spheres.
So these collective effects are mediated by the electric and magnetic fields that arise from the charges and currents and going to
the central assumption that makes everything work or the central observation is that the frequency of two body collisions is much,
much lower than the plasma frequency, which is a typical frequency of oscillation.
If you just took a plasma and kicked it, there would be a normal mode.
It'll make a P, it's a very high frequency and the collisions happen much less frequently than that.
So there's is rapid plasma frequency vibration.
And so you take this little equation down to something that can be solved analytically or on a computer by using the PBGC y hierarchy.
I'm not going to go through that. It's something that many of you have done.
And it reduces the dimensionality of the problem radically to simply a Boltzmann equation where you now have
this this little F is the probability of finding a single particle at a position X one and b one at a time,
T rather than all six in plus one.
And this left hand side of this equation, which I'm not at this point, I'm just going to just work on the left hand side.
It's just the total derivative along the trajectories of the particles, if you look at that.
So that looks a lot like if that were the only thing, it would still be DFT t equals zero.
But on the right hand side you take into account to body interactions or collisions.
And so you go through this process and you you end up with this equation.
The velocities are just the velocities which arise from.
Pushing. You know, solving Newton's equations and the accelerations come from the.
E plus, b cross b force. So that's pretty much it.
Okay. If the right hand side were really large and dominate the equation,
then you could solve this equation order by order and say the collision operator is the dominant thing in the problem.
And so we look for solutions to zero order. The collision operator equals zero.
And when you do that, you find that the solution in the equation is the usual maximal Boltzmann kinetic distribution of particles.
So this distribution, instead of having n dimensions and velocity space, right three and dimensions V, Z, Y and z now only depends on the density,
the temperature and the mean flow, and then the functional form of the rest of it with the velocity here is comes from the collision operator.
So this is a fantastic simplification. Now we have a three dimensional system.
We have a few three dimensional circles, since we need equations for the density, the flow of the temperature in space as a function of time.
So when the collisions are large, you can go through that process.
You only have to keep track of those things. And when the collisions are large, the mean free path is short compared to whatever you're looking at.
In fact, I'm going to say that's what I mean about the collisions for large.
I'm going to look at systems where the particles are colliding on scales that are short compared to whatever it is.
I'm interested in the wavelengths of the turbulence or the size of the device and what whatnot.
So in that limit we get a fluid description.
In fact, if you carry this through for each species and attend together in the appropriate ways, this is exactly how you get magneto hydrodynamics,
which is an appropriate theory for describing large scale plasma motions in a when they're collisional.
So I did that slowly because now that's not the system I'm going to solve.
We don't do in our group. We're not looking at magneto hydrodynamics, but this is what you would do if you were.
Instead, what we do is the opposite limit.
We're interested. We still the divide length is tiny.
Let's let's say for some typical system, it's less than a millimetre.
And the system, the scales that we're interested in are our hundred metres kilometres, something like that.
So the Dubai sphere is this tiny thing reached in 100 metres size objects and the main free path is in between.
Or even longer. If you look if you calculate the mean free path for a typical magnetic confinement fusion experiment,
it can be hundreds of kilometres in a device that's three metres in radius.
So essentially the particles are not colliding. They're pursuing some other orbits mostly, and then slowly diffusing because of other things.
Okay, so this is the equation we're going to solve.
And in fact we're interested in the limit where the collision operator now is small and the the left hand side is dominant.
This is a mess. We're back to the six dimensional nonlinear problem.
We have light waves moving across the system, high frequency plasma oscillations.
This equation still includes things down towards the by scale.
So that's too much. We're not interested in all that. And there's another further single simplification that we want to take.
And what we do is we try to identify in the equation here another fast timescale that we can average over so that we can look at the slower,
bigger things. And that fast timescale is the is going to be the gyration of particles around the background magnetic field.
So instead of looking at non magnetised plasma, I'm going to look at Magnetised plasma,
which is just defined by the fact that the radius of gyration of the particles in the
plasma around the magnetic field lines is small compared to the to the plasma itself.
Okay. I may have dynamics smaller than the radius of gyration.
I'm not throwing away those scales. I'm just saying I'm going to make an assumption that the plasma is magnetised,
that there's enough magnetic field to give you a radio that are inside the device.
And as soon as you do that, there's an idiomatic invariant that you can take advantage of the magnetic moment
and you can really hone this equation down to something that can be solved.
So this radius of gyration is simply the, you know, all these things.
It's the V over the frequency of the cyclotron motion, which is QB over in C or or cube over M if you don't like my units.
And we're going to look at things which are slow compared to this gyration frequency and long mean free path.
And that is going to be the limit that we're interested in.
So a typical plasma looks like this. And this in this system, the particles are rapidly gyrating and kind of drifting.
They're almost staying exactly on the field lines. But if there's there are effects which can cause them to drift off of the field lines slowly.
But mostly they're going round and round and round.
And the theory, which was started in the late sixties and the formalism of this transformation I'm talking about,
was completed in the eighties, is replacing this helical motion with rings here.
These rings then move around and you get an integral differential nonlocal equation to describe the dynamics.
But that sounds awful, but the reduction of all the time and space skills is fabulous.
And so this is now attractive, tractable theory. Get some bullets here.
The plasma ends up being this kind of plasma.
It's highly anisotropic.
If you look if you had a magic microscope and you looked in the perturbations of the density or something like that along the magnetic field,
they would have very long correlation lengths, would all look like spaghetti.
And across the magnetic field, a very short correlation length and short kind of variations of things.
And this theory kind of naturally gives you the tools to work in that asymptotic limit.
Let's just keep going. So on this fundamentally, when you say that we're going to order out these rapid gyrations, we get more things from the theory.
You find that there are several asymptotic multi-scale time and space scales that drop out.
And so the first is that on large scales, very large scales compared to this radius of gyration.
Things are essentially static.
It's a minefield theory and the distribution function can be split into this very large scale part, plus perturbations that can be arbitrary.
They're going to vary and do things. It's you take this large scale part,
it has a Boltzmann factor you to the minus Q five over T you can absorb that into the you can
separate that out from the distribution function and h becomes a thing that we like to follow around.
As I said, that's just now we're going to add and replace these distribution functions with rings instead of particles.
I used to joke and say, this is ring theory, but I don't know, I did it.
Okay, so I still do it. You get an equation for H, which is going to be the equation that we try to solve.
H is a five dimensional equation. So H is a function of three position, three position coordinates, energy and magnetic moment of the particles.
And we've thrown away the information of where the particle is around its star.
Well, that. There are Poisson brackets.
Chi here is just the general. It's fine. A and all that kind of stuff.
It's the fields. So there's brackets of the fields with the distribution function.
There's this free streaming of the particles along the field. Line B denotes the direction of the magnetic field.
So this is particles moving along the field line and other kinds of turns, including the collision operator.
So you have to add to this the Maxwell equations.
I won't take you through that.
You solve that system of equations on a computer and you can go off and predict the behaviour of plasmas and a lot of interesting systems.
So the slow evolution is what I promised you thought.
You go through all the same topics and you see that the number of particles doesn't change.
So that's good. The temperature can change.
If there's something about this plasma, if I stir it, I can change the energy of the plasma and or if I cool it, I could.
And those things happen slowly. It turns out we can relate this heating in the third line there to the change in entropy of the
plasma on the microscale so that we can calculate the heating from from a proper positive,
definite quantity here in the in in the integral. So everything is nice and copasetic and you can go ahead and do do calculations.
This thing is conserved, but it's not relevant to this talk.
I just had it on this slide. Okay.
Now, look, the point of this talk is to encourage it is to show you that Landau damping can be overwhelmed by echoes.
But to make any sense out of that, I have to show you what Landau damping is.
Talk about it in case some of you don't know and then show you what an echo is.
So what is Landau damping?
Well, that's a problem that everyone in plasma, they're either the last year of undergraduate or the first or second year of graduate school.
It's it's kind of boring. So I'm going to do it a different way.
I'm just going to look at the linear dispersion relation. So if I went back to this equation here and threw away all the brackets, which is to say,
throw away this term, and then I'm actually going to throw away collisions.
Let's look at a forum only collision with plasma and just see what is the dispersion relation, what kind of waves live in that plasma.
So that means for each species we just have diversity free streaming and this this term, which is related to energy change.
I'm going to make even simpler. Just look at a hydrogen plot loops, hydrogen plasma.
The electrons have a tiny jar radius, the ions have a little bit bigger.
We'll just ignore the electron things and you get a dispersion relation which to most of you is still ugly and the red isn't showing up.
But that's not a fix that the yellow on the left.
Those are our fame waves. Most people have heard that there's alpha waves swimming around in plasmas.
The red are sound waves and slow magneto sonic waves, which are going to be the focus of the talk and the right is coupling.
The right isn't quite clear, but when you look at scales large compared to the ion gyration,
you can ignore the green and you just get alfvén waves and slow modes decoupled.
So you have compressible things and you have non congressional things. Alpha doesn't compress the plasma.
It's like if you had to feel lines, they move together like this in waves and the plasma isn't pushed between them.
The other mode of oscillation, the red ones, the planet,
the felines are doing this and you're squeezing and squeezing the plasma two modes of oscillation roughly.
Okay. So the red one, if you go if you look at those compression,
all things which have density fluctuations associated with them and you can centre a plasma where the magnetic field is in an appropriate sense weak.
And what you find is in the middle you get that the frequency compared to K parallel V thermal,
which is a typical wave number, frequency and wave number in the system is just that.
This is a pure imaginary object here.
So that these compress, if I, if I take a plasma and I look at scales large compared to the ions gyration and I stir up density perturbations,
they're supposed to be damped on the timescale of K parallel B thermal so far.
So now later I want to remember that fact. And when I go out and look at plasmas in the world, they should not have density fluctuations with short,
parallel, mean free path short wave numbers because those should be strongly damped.
Okay. So that's the thing to remember.
And now that's in the high beta limit and the low beta limit. There's another wave. It's also strongly damped and it's called the ion acoustic wave.
You can make it weakly damped by making the ions. You know, I wrote one thing.
You can make it weakly damped according to this formula. Let's look at that wave.
Okay, let's look at one of the waves that arises and see why it's strongly damped.
This is Landau damping we're talking about.
So we're going to do the collisional stamping of the ion acoustic wave with what's called a case van Kampen approach,
which is to say it's kind of a continuous eigen function thing.
So we have the original equation dft t. It was age before.
Sorry, it's f for a moment. DFT t v. Parallel feedback grid f and this is e parallel the.
Electric field, it arises because particles maybe get bunched up. So we can look at the long wavelength limit, just make things a little bit simple.
And what we need to solve is an integral equation.
And this is the solution. Jcvi has a principle part and the belt function part and these functions it arrives here
f of v capital F of V is just the Maxwell in and delta n is the density perturbation,
which is the integral of this velocity perturbation.
So it turns out you can, with these continuous eigen functions,
solve this problem and all you do is make the appropriate substitutions, match up the initial conditions and let it go.
And what you see is a really beautiful picture of how Landau damping works.
And so this picture that I'm going to show you is on the x axis, on the x axis, on the horizontal axis, I should say, is V parallel.
And on the vertical axis is the perturbation of the of the distribution function.
Okay. So this thing, what I want you to see here is it's highly oscillatory.
And the lesser importance is that the analytical solution is the blue curve and the dots are solutions from this code.
So we're capable of calculating highly oscillatory functions to find the density perturbation.
I integrate over velocity, and this is a lot like problems that you've seen in steepest descent and other kinds of things.
The function is getting highly oscillatory, so there's negative, positive, negative, positive, negative, positive,
lots of contributions to the in a role that almost cancel and overall you don't get much of a density perturbation from this still complicated.
So let me show you the movie. So this is the distribution function evolving in time.
And you see that it gets more and more oscillatory as time goes on.
In the beginning of this this before the movie starts.
This the integral of the is is the density and the integral is whatever is under that curve.
And so you have a finite density of perturbation. That density gives rise to an electric field.
So this thing is something you could measure in a plasma. So you start this wave in the plasma and you let it go,
and the distribution function gets more and more oscillatory so that the integral of it is going to zero very rapidly.
And that's what landau damping is. There's no irreversibility here.
You could turn time backwards and it would come back, but we don't know how to turn it on backwards.
So. Basically it just keeps getting more and more oscillatory.
Now, to get this solution, we turned off the collisions.
And if we stopped this. I guess we're not able to if we if you look at this thing, the collision, we just respond to sharp curvature in this function.
It's a low placing in velocity space. So collisions would tend to kill this thing, even if the frequency of collisions is very, very, very low.
The variation of F of V is getting very, very strong.
And so even a tiny amount of collisions will finally kill this thing off and give you the irreversibility.
So let's go through it one more time. Very slowly.
We take the equation, DFT plus V, DFT Z, and we put everything else on the right hand side and just call it a source and forget about it.
Okay. If I have a partial differential equation of that form and the source isn't problematic,
then the solutions are any function of z minus v t and that means,
you know, we've solved to solve this partial differential equation, we can put in a plane wave E to the AKC, a z and evolve on time.
And that actually courts horns are very closely to the picture I just showed you.
The heat of the AKC factors out and the e to the ak v t is just something getting more and more oscillatory.
Okay. If we took a source on the right hand side to be a kick a delta function in time,
and we gave it a maxwell in distribution and then we let it and one harmonic cosine.
KC In the z direction, you would in fact get this solution in the bottom right.
And this is where phase mixing as terminology comes from.
The phase is the of the i k v t the second term in this parentheses, and that phase is getting advancing in time.
It's a complex phase. So it gets the function gets more and more wiggly.
The mixing comes from the Maxwell part E to the minus B square.
And so the convolution of those two things gives rise to the overall damping.
Here's a picture of it. Now I'm looking at velocities in the vertical direction and X or Z is in the horizontal direction.
This is just a picture of a perturbation of the distribution function in phase space.
So in this equation, everything moves trivially. So in the upper equation,
everything below the dash line is going to move to the left because it has negative velocity
and everything above the dash line is going to move to the right has positive velocity.
So in the beginning, if I integrate, say it zero vertically, I get a density bump because everything is the same sign red.
Okay, so there's a density bump.
As time goes on, those contours turn over and I start to pick up a little bit of blue and later it tips over even further.
And it's red, blue, red, blue, red, blue. This is a different picture of the same thing I showed you in the movie.
Okay. And the density dies off very quickly as these as these things get twisted over.
That's phase mixing. That's essentially the process behind Landau damping.
And now if you go and solve that original equation in all its glory and try to find roots of the dispersion relation,
you know, you do you find on the left is the potential versus time from some way.
It's a solitary it's got some high frequency, some slow damping.
The ratio of the damping to the frequency is ten to the minus three.
And the codes happily find these things and that's landau damping.
And the right hand side is just you can drive the system and look at the resonance of the plasma.
So these are normal mode. Q Curves, right? So the narrow ones are weakly damped and the fat one is so strongly damped.
And then you can do that for zillions of parameters and understand how all the normal modes of the plasma evolve in the linear system.
But we're not in a linear system. We're in turbulence. So the amplitudes, I assume, to get all of that, that the nonlinear terms were irrelevant.
And we can only aim for the linear terms when we now include the fact that turbulence is going to mix things around in nonlinear fashion.
With order one fluctuations, we get a different picture.
And the typical story of turbulence is there's something happening at the large scales, sort of cartoonish at the top,
and those large scale eddies interact with one another,
break up into smaller scale and smaller scale and smaller scale until finally you get to tiny scales.
Now we're in space instead of velocity space. And this is a kind of a turbulent picture.
Here's how that would look in a in the literature.
If you look in the literature and you say, I'm interested in these collision loss plasmas,
the vertical axis, would we be on a log scale, the energy and the fluctuations?
So you would stir at some large scale and those eddies would break up and break up.
And after you wait and get into steady state, there's some scale where there's viscosity or something which damps out.
And in the middle you have in a the kamagra or spectrum of K to the minus five thirds.
So in a turbulent plasma where you ignore Landau damping, this is the kind of thing you're supposed to get.
You're supposed to get energy going like the flux.
You can predict the fluctuation amplitude of the turbulence where it's universal,
a way from the special way you're stirring it and the dissipation and you get to the minus five thirds.
But we just went through a bunch of work to say these slow modes, these compression perturbations,
any density fluctuations that we see should be damped by this phase mixing process.
And so, in fact, the blue curve, the slow modes or the compression of waves shouldn't be the same.
They shouldn't be minus five thirds.
We think in the presence of this landau damping, maybe they will be steeper, maybe it's non universal and not even a power law.
It's a complicated system. So, you know, if you look at the literature before Alex started this kind of.
Looking back through everything and straightening things out, what you would find is people thought that the slow modes would be completely absent.
And all we would do if we looked at one space or in an experiment, we would see our fan waves.
We would see no density. Perturbations, nothing. Compression all.
But when we look out in space or in experiments, there are tons of density perturbations over a large range of scales, particularly on the right.
This is data from the solar wind. This is normalised density fluctuations versus wave number.
And over two or three decades, there are larger, you know, K to the minus five, the density fluctuations.
So we have a serious kind of glitch. The literature literature says there should be no density perturbations.
Straightforward theory says there should be no no density perturbations.
But when you look out in space for a typical large plasma, there are enormous density fluctuations.
So what gives? What is the solution to this paradox?
That's the point of of this talk. Why why are there these fluctuations?
So we're going to redo that little calculation, including one more term, and that's some nonlinear, nonlinear, turbulent advection.
So we have the same thing DFT, free streaming V, parallel DFT Z, and now we're going to have some flows,
some turbulent flows, mixing things in the perpendicular direction.
This is advection by the e cross velocity of here in the no, but basically perpendicular flows added at unit amplitude.
So what do you get? Well, you don't have to do a lot of algebra to see something interesting if you put in plain waves.
You'll see that the nonlinear term, the grad f will couple these perturbations together and you'll get the wave numbers coupling such that,
you know, you can have a wave wave number P in a wave with the wave number.
Q They will come together to give you the one. K And this coupling is going to be the thing that produces echoes.
Echoes are going to be really interesting.
Okay, so all we're going to do is look at the importance of a quadratic nonlinear d in this land l damping problem.
Okay. Pretty straightforward. So what what's going to happen in the next few slides instead of these waves damping away, as I showed you,
that we can calculate analytically and numerically to to to nothingness,
the density perturbations are going to be continuously reconstituted by echoes from this wave coupling.
Okay, so how would that work? Well to skip ahead.
What's going to happen is we're going to create perturbations in this x v space which are tilted to the left.
Let's see, I should do it this way, which are tilted to the left.
So now we start with no density perturbation at some moment in time and free streaming reconstitute some density perturbation.
And later you get a density when the natural, you know, just free streaming of particles with their velocities.
Uh, leads to this kind of thing. So this is what's going to happen.
But the question is, why would you ever expect to get something coherent up there on the left that would unfazed makes to give you this echo.
Well, the answer to that is pretty straightforward. The skeleton of the idea goes like this.
We could solve the equation and we have to do this here.
And we would get that some plane wave with wave in a group we go to like E to the IP,
Z minus T, because remember the linear waves just have it's any function of Z minus v t.
Oops. Sorry. Oops. Wrong way.
So let's let that run for a time.
Capital T, we're just going to do a fake problem. So we imagine we have a perfect control of our plasma.
We put in a little perturbation. It starts to mix over right.
And it and we go until it has accumulated phase PBT.
Then we put another mode in each of the I choose the different wave number and we let that one go.
Now we know that we're going to get just broke the pointer.
We know that because of the coupling P plus Q has to be K.
And so this one is now going to evolve. Thanks. It's the instantaneously looks like this.
And now we let it run forward in time. And everywhere the Z appears, you get z minus v, t.
And so F of k now looks like. Even the ak z minus v, t minus I APB capital t.
So we have the opportunity to have something interesting happen when the phase is such that the amplitude of the phase is one.
And that happens when you've got a time T which is P over K times, the original time that we let things go.
If you do that, you get an echo. Well, the echo has to happen after the two perturbations, not before.
And that gives you a constraint. It says that P over K has to be bigger than one.
P has to be bigger than K. So certain kinds of wave numbers are going to interact to give you this wave number.
And in fact, when you're done, what you find out is the sign of K or P times.
The sign of Q has to be minus one.
So it's a certain kind of of one way of going this way and a way of going that way can interact to give you an echo.
Okay. So this is sort of abstract.
You can get an echo. The question will be, do we get echoes and are they numerous and interesting?
So again, this is the kind of perturbations you get. All we're doing is unwrapping phase space and then letting it go forward.
If we do that, we we're trying to find the fate of these elfin in slow modes.
So let me just remind you very quickly, the iPhone waves on the left, no density perturbations.
They're incompressible. All they are velocity and magnetic field fluctuations.
And they're described by a theory which you can get from what I've shown you.
The slow loads are compressive. They give density fluctuations. Also delta field strength, magnetic field strength fluctuations.
And they're basically the slow modes. Follow the alpha waves around.
They're slaved and evicted. So they are fame waves.
You know, you can work out you can work out the description of the waves.
It's these variables, Zeta plus and Zeta minus. I don't want to belabour the point.
Those are left going and right going alpha and waves in the box. So there's some going this way.
Some going that way. And they're independently conserved.
It's not too important right now. The slow modes are.
Let's see. I have to point with this. DVT.
It's the convective derivative with the velocity and the beat out grid is long.
The perturbed feline is what that means, but essentially developed as a non, as a linear and a nonlinear part.
And beta grad is a linear and a nonlinear part. And you get this equation down here at the bottom.
This is the key equation for the moment. It says that this is this is this is the slow motion part of all that very kinetics.
And we just have a it's it's a it's straightforward equation to solve.
And now it looks kind of ugly. I think the next slide, I show you the approach that we're going to do.
Well, yeah, that's to wake you up. So this equation describes the evolution of G, which is the perturbation of the distribution function.
So it's a function of velocity. Okay. And position.
The alpha waves here are described by the Zeta pluses and minuses.
So my grad student wrote a code he needed to solve for G and the Alpha Waves, and he called it Gandalf or Gandalf.
That's where he got the name. So he wrote this code. It runs on a GPU, so it runs at teraflop speeds.
It has lots of exciting numerical properties.
It's also all the equations I've shown you, and it does it for you in real space and hermie polynomials in velocity space.
And you know, we're going to investigate just because echoes could happen or can happen.
Do they happen and are they interesting to the evolution of the system?
And so before I do that, I want to show you the kind of what do we do?
Well, here's the equation without the nonlinear term.
And on the right hand side, this kind of t is the source.
So we're just going to take the linear equation instead of solving an initial value problem like I showed you before, where the wave decays away.
I'm going to do it a little differently. I'm just going to stir constantly and with a lines of an equation.
So you can think of it as as a driven problem instead of an initial value problem.
The properties of this source are just delta correlated white noise with some amplitude epsilon.
And when you have a problem like this is a different problem on the bottom.
When you have a problem where you have just a scalar with the launch of an equation, with some driving chi at amplitude alpha and some damping gamma,
then you can predict with the fluctuation dissipation theorem what the amplitude of five square will look like,
the energy in the in the electric field and it looks like this epsilon alpha squared over the damping.
Well, we're doing a different problem. This is for a scalar. We're doing a a continuous variable in velocity space.
And so the first thing to do is to see what that answer is.
And what you get is by square, the energy looks like some general function of alpha, the driving and instead of gamma,
it's absolute value k which is a lot like the damping gamma when you get down to it, but it's different.
And this function F of Alpha is a transcendental function that's a mass to calculate.
But the grad student did a good job. And so his code,
the analytical response is suppose this is the f of alpha function looks like this
as a function of alpha and his his Gandalf code is able to calculate that properly,
all those little red dots so he knows he can do a driven system and it's behaving right.
That's the linear equations. How did he do it?
He did it with hermy polynomials. And so you took that D that equation for G and expand instead of GSV.
Now we're going to have general hermit moments g sub them so that you get a hierarchy of o.d well one one
level simpler pdes and the bottom one is the is the case for all hermit mode numbers bigger than two.
It's kind of all the rest of the ends of the same after you get past the sort of junk at the beginning and the coupling of
the M plus one hermit moment to the and the M minus one to the end is the expression of phase mixing and landau damping.
So we got to watch those two guys carefully. And in fact, if we just go to the large now,
we're getting to a little more sophisticated view of the same thing I've been talking about for the last hour.
We do a transformation and we end up with these functions G plus tilde and G minus tilde, which I'm not going to justify.
It's appeared in Paul's papers and in Alex's papers. It's a good transformation.
And you look at a large m so we're looking at the vote.
We're looking at that distribution function when it's very wiggly. Right?
Very. The hermit mode number is large and we're seeing what happens.
It turns out you can write the equation in the form, in the box. And if you stop and look at that, it's it's fantastic.
The G plus solution,
you get a plus sign here and you have a flux in M in hermit mode number and some damping from the collisions on the right hand side.
So this plus solution manifestly represents a flux of energy from low hermit numbers to high hermit numbers.
And that's the picture I showed you. That's the distribution function getting starting off not wiggly and getting
more and more wiggly as the energy goes to higher and higher hermit numbers.
The minus solution is the opposite. The minus solution is energy coming from very wiggly modes.
The flux is negative to the low ends. That's the echoes.
So this formalism gives you the echoes right away.
And so you just form the appropriate look in the middle, you form the appropriate G plus squared as a function of hermit number and g minus square.
And this tells you the flux plus is the flux of energy too high numbers to be
Landauer damped away and the minus is the flux of energy to the low hermit numbers.
They have a bunch of constants and then this one is one over squared compared to this where S squared is just the hermit number.
These have the form of parasitic solutions that Paul showed me correspond to fermion doubling in quantum mechanical systems.
But those are fake and come from the democratisation scheme.
These are physical waves that are, you know, plus and minus and in hermit space.
So what do they look like? I just want to show you a picture.
If I take that linear system that I told you about in this space, the driven system, you get a spectrum.
Now, we the point of all those transformations is to separate these two things and give one of them no variation.
So the F plus solution is just constant as a function of square root of M, and the other solution goes like one over A squared.
So the top one is landau damping. And the bottom one is the echoes in the linear limit.
So if I stir at at low mode number, low put large scale perturbations in velocity space, then I'll fill up phase space.
It'll it'll fill up the black curve until I get to the collisions.
And in fact, if you let the collisions go to zero, you get an infinite amount of energy stuffed into the system.
It goes like one over new to the one third. The Echo Park.
You would have to store at large M and you would find energy going back.
And then the linear problem where we just put electric fields, you don't get perturbations that large.
M So we're not accustomed to seeing the echoes. The echoes have to come from something else.
So they're not electric field perturbations. And so if you wanted to see them, you could mark it up.
This is the same equation now it has the second term up here is the advection.
And if we so I put this slide in hesitantly,
but I wanted to show you remember we're still driving G with this chi function which is del function correlated in time coming from.
And then we're going to also stellar random perpendicular velocity across the field.
Okay. So we have some large scale electric field wiggles and we have some convection wiggles.
And this is the beautiful thing you go through and you do that transformation on the nonlinear term.
And what you find, what Alex found was the nonlinear term breaks into two parts.
G plus is coupled to g plus.
So that this this one represents.
Just advection. If Delta Plus if the sign of K and Q are the same, if they're opposite,
then you excite this term and you get G minus or looking at the other way for g minus a g plus x is a source.
So this means that when the wave numbers are opposite, just as I showed you in the toi problem,
the nonlinear t gives you a source and you will find that you're generating echoes.
So it's very natural. If you have wave numbers of all kinds, you're going to have some kind of echo.
The remaining problem is a numerical problem to say you get a lot of those echoes or do you get not much.
And so. My student worked out the slow modes and for the problem I just showed you, the blue is where there's.
I'll just tell you if you don't have to think about it. These are the different wave numbers.
The blue is where there's a lot of echo and the red is where there's a lot of landau damping or phase mixing.
And there's a line you can work out between them where and this all sort of works for the little fake problem that I just showed you,
where I'm fake, stirring everything. What we like to do is the real problem.
And. And we'd like to do this slow mode.
So now I'm almost done. Now we do the real problem.
We do stick the slow modes, the alpha waves. We put it all into the code.
We have a self-consistent motion of the field lines, self-consistent flows and everything is calculated.
Self-consistent.
The only thing we're doing is it's kind of weird as we're stirring the alpha waves instead of letting them arise from some natural thing.
So we stir the alpha waves. We've got some slow modes and we want know what happens.
And just so that you can see that we're not cheating,
this is the energy in the velocity perturbations and the perpendicular field perturbations that should be roughly K to the minus five thirds.
If our simulations are right, this is the alpha waves. This is an important point, but I'm not going to belabour it.
What we find is that the slow modes develop a parallel spectrum and it goes like K parallel to the minus to so the density.
So you see these modes have density fluctuations that have some k parallel spectrum.
What does that mean? Well, how how how do we have these? How do we have these modes when they're supposed to be damned?
Well, you know, we can look individually at the field compression and the density fluctuations.
At the end of the day, I can put those there for if you want to come back.
This is the key picture. Blue means that there were echoes dominating and any other colour is where it's got landau damping.
And so before we did these, before the theory, before the simulations, people assumed that this picture would be red.
And in fact, when you go through and you calculate, you find that there are as many echoes as there are forward.
Landauer damping kinds of processes going and and what does that mean?
That's it's it's quite surprising. It means you should get density fluctuations when you look out in space.
If I wanted to ask the question, is the echo happening?
Where things? Is it, you know, just because there's a lot of echo, maybe that little yellow bit Lando dampens down to everything away.
Okay, so this is the picture of F, the capital F that I showed you was flat when it's Landauer damping in the turbulent problem.
It's got a very definite slope. And that slope is, you know, we'll work.
Alex will do the theory and figure out if he can predict that slope.
I'm sure he will. Anyone else that wants to do it should that slope arises from all the self-consistent dynamics.
And remember, this is the linear result, the top CR.
If we had Landau damping dominating, which is what everybody said would happen, you'd have a flat curve in this picture.
And instead it's a steep slope of something like my minus five halves.
So that's really quite remarkable.
In fact, you can you can look and this is the curves of the phase mixing in the energy and modes that is beta phase mixing and unfazed,
mixing in the fact that they're almost on top of each other tells you that the echoes are balancing the rest of the problem.
So what does that mean? What it means is. These are all the different moments in the plasma and it gets very small.
The amplitude drops off quickly instead of being flat. It means this collision, this plasma is acting like a fluid.
The turbulence is somehow through this echo mechanism, causing the plasma act like a fluid instead of a collision the system.
So that's the end. These numerical simulations support what Alex has in a recent paper that several of us here are co-authors on.
The Echoes, in fact, do overwhelm the landau damping for the slo mo problem.
And you should expect to see density fluctuations in the interstellar medium and the solar, wind and things like that.
This low collision ality plasma in a turbulent state can be really well-described by only a few low hermit numbers of room moments,
which is to say it's behaving like a fluid, not like a a collision with complicated plasma.
Many of us over the years model this turbulent dissipation as a nonlinear cascade to give you the amplitudes and landau damping,
to give you the absorption and this is wrong.
There is no the plasma. It's not absorbing.
It's it's echoing. So this may changed the predictions of iron an energy on an electron heating in various plasmas.
And to really say what's going on, we have to better understand the physics.
Once you get down to the radii, which the theory, you know, that's the next step and we'll use the same machinery.
So that's my time. Thank you.