So when you hear standard sort of public talks about fusion, the picture that they show you at the beginning is this one.
So and they tell you either this is what we're going to build.
Right. So this is the machine. And you might add this in in at least in terms of sort of funding and and story, as it were, as it's unfolding,
this this image and this the thing behind this image kind of dominates the diffusion world.
Well, you might ask. The question, where's the theoretical physics here?
Right. So all this stuff, this is clearly going to be a great engineering feat of 21st century civilisation,
possibly, actually the way it's going 22nd century civilisation. But.
But. But. Well, despite appearances, this is.
No, not all quite so nice and control and well organised as you would think.
Somewhere deep inside that the you know, you might ask where is it that the that this sort of, you know, of wild, primal, basic things are.
And that, of course, is what theoretical physics does while primal things the basic instincts of nature, if you like.
Right. So this is in polite society. They call them fundamental laws of physics, universal physics, that sort of thing.
So where's all this? Well, here it's in pink. It's it's it's this.
It looks nice and and and thing, but it is the beast.
So, so, so, so, so I think I think the bottom line is that is that, you know,
theoretical physicists are only useful in the in this program if they can engage well, engage tame the beast.
So let's discuss the beast. Let's let's strip out all the engineering and let's realise some very, very basic thing about, about what's inside there.
Right. So you have a toroidal plasma. You have a plasma. And the basic feature of that plasma is that it's hot in the middle and cold at the edges,
and it has to be like that because it has to be hot in order to fuse.
And it has to be cool because, you know, it's it's if we're talking about it being at 100 million degrees in the middle,
it shouldn't be it certainly shouldn't be a hundred million degrees at the edges because we don't want it to melt the device.
Right. So so we have to separate themselves from the beast somehow.
And so we're maintaining this plasma in this state, which is if you're a theoretical physicist, you don't actually think in terms of pictures.
It will be lots of pictures in movies in my talk. But in fact.
Right, you have to realise that in, in, in the theoretical physics department everybody thinks in terms of plotting Y versus X fundamentally.
Right. So, so, so, so the really interesting picture here is not this one but that one, and that's temperature versus radius.
Right. So so radius goes from zero to out and the temperature goes from hot to cool.
And so there is a temperature gradient in the system.
The system is out of equilibrium. Remember your statistical physics.
If you still remember your undergraduate days, the system is out of equilibrium.
And so it's potentially a dangerous situation because because nature wants to meet, always wants to be in equilibrium.
So the question for a physicist is, well, how hot does it get?
Or more practically, how hot can we make it?
How much of a temperature contrast can you create? Well, let's.
And so this is where theoretical physics starts. And theoretical physics starts on the level of second year undergraduate physics,
which which I hope some of you have at least some sort of vague recollection of.
I assume this, but I'll remind you of all the all the relevant things.
So, so let's start with what we teach our students in the second year undergraduate physics course.
Right? We teach them the heat equation. Right? So so we tell them that, you know,
if you want to know what the profile of the temperature somewhere is and remember the reason it gets hot is because you heat it in some ways,
which I won't go into Steve might in in in his talk.
So so you you have to solve the heat equation, right? Well,
the heat equation is the temperature by the T equals two heat diffusivity times the plus in of T plus whatever sources and sinks of heat you have.
Right. So this is s stands for heating minus cooling.
Should there be cooling as well? Right. And you might say, all right.
Well, the problem is the problem is simple. We look at it in steady state.
So we eliminate the time derivative. We solve this in the Taurus. We know all our inputs, we know the boundary conditions.
So we get the temperature profile. We hand over the solution to the engineers, and then we move on to thinking of dark matter,
quantum entanglement, brief history of time, the meaning of life, and, and, and the rest of it,
which is what we're supposed to be thinking about as sort of, you know,
coastal in ivory tower theoreticians in the Rudolf or centre for threats, theoretical physics.
But before we do that and of course, once we have built the fusion machine and have the unlimited source of.
Energy. It's, you know, the whole world will be a centre for theoretical physics and it's, you know, everybody will be thinking of nothing of that.
But but that these all these mundane concerns would disappear.
But before that happens. Let's let's ask a quick still an undergraduate question.
And that question is, what's this? Right. Because clearly so if I knew how I heat the plasma and we all know what the blossom is.
But the interesting question here is this deep, right? Because depending on what that is it.
Tell me in the end. Right. What, what that those temperature gradients are.
Right. You see the D multiplies the gradients of T and so D effectively normalises links here.
So, so in the end. Right. You know, how big are, you know, if you tell me that, that, that the temperature contrast has to be 100 million degrees.
How, how big a device I have to build and so on will all sort of depend on what that is.
Well, how do we calculate this? Well, again, let me remind you how to calculate heat diffusivity.
It's the reason he diffuses. And that's what we tell undergraduates.
There is a it's true, but at the same time it's a lie. And then you'll still see in the moment.
Y Well, it's, it's, it's not a lie. It's not the whole truth.
So the not the whole truth is the it right? Are you have you have hot particles in the middle.
They random walk their way through the tokamak. They're, they're in thermal motion so they're moving randomly when they well each of them is in
the first instance moving the straight lines and they collide with another particle,
change their direction. And so it's like, you know, drunk in the famous story of a drunken man progressing from a pub out into the world at some rate.
Right. So so so the the particle random walks its way eventually out from the core to the to the edge and brings its energies with it.
And so and so there is a diffusion of energy now that is just the diffusion coefficient.
And the diffusion coefficient is simply the ratio of the mean square displacement
of the particle by the time over which that means square displacement happens.
In other words, you could write it as the mean square particle velocity times, the time between collisions and the when particles move.
Similarly, the velocity is basically the speed of sound in that in that medium.
And so this is like the speed of sound times, the mean free path of the particle.
This is the standard undergraduate estimate of the diffusion coefficient.
It's actually in the in applause in the magnetised plasma and talking about this is actually on true even on that level.
In fact, it turns out that because what Felix has explained to you, the particles actually gyrate around the magnetic field.
So they can't actually move very far across the magnetic field. So all of that is only really true along the magnetic fields.
So if you orient your magnetic fields in such a way that there is no magnetic field taking anybody out radially,
which is what you care about the radial transport, then actually your step size for the particle is the radius of the lawnmower generation.
This is that Felix talked about so in fact it's built to x is is the lawnmower radius and and this is still the time between collisions.
So so you get this is this is the basic estimate that you might get for for the plasma.
And you know since since since we're theoretical this is here we're certainly not going to substitute the numbers.
But I'll tell you that this turns out to be way, way, way, way, way, way, way too small to explain the transport that we see.
Okay. So in fact, when when fusion devices were first designed, Steve might say something about this in this talk by Lyman Spitzer, they, you know,
when they thought when they basically estimated things based on collisional transport, you know,
they came up with a device that would that would have been almost a tabletop arrangement.
So. Well, it's you know, the reason it's no longer a tabletop arrangement,
but that huge monster of engineering that you saw is is precisely because this does not work.
Now, why doesn't it work? So let's look closer.
So let's go back to this to this toroidal plasma, which is hot in the middle and cold on the outside.
This all looks nice and and and and smooth and and kind of pain, doesn't it?
Well, in fact, it's nothing of the kind. Let me show you a movie over.
This is admittedly a computer simulation of of of what happens to a plasma like this,
which was done for to model the operations of the 3D Tokamak in San Diego.
And this is it. I'm not showing it because that tokamak is is any way any better or worse than any other tokamak.
But they do have the absolute best movie making machine there in Southern California.
And and so their movies are unrivalled. And so here's the movie.
Let's let's see if this actually works. So so they start this simulation.
And this is this is actually this is actually density of the plasma in the Tokamak.
And you see that it all breaks up in this in this sort of seething, boiling, unholy, chaotic mess, which is called turbulence.
So so what this is meant to demonstrate is that when you when you put something out of equilibrium, what tends to happen is,
you know, nature dislikes temperature gradients and because they are they constitute a lack of equilibrium.
And so it can try to drive the system unstable and to to try to rectify this situation for itself.
I'll show you in great detail how it will in some detail, how it exactly does this.
The point that I want you to carry out of this picture for now is that the whole thing is moving rather violently and chaotically.
Right. So actually what we should do is not second year undergraduate physics,
but third year the third year undergraduate physics fails you what happens to temperature when it's not just being diffused by collision,
but when it's actually in a moving medium. Right.
And what happens is, is, on the face of it, a fairly fairly harmless what happens is that the the heat equation develops another term.
And that's the term. It's you don't guarantee what this describes to you is the is the velocity of the of the plasma motions,
not the velocity of the particles, but the mean velocity of the motions at any given point.
And what this term describes is the advection of the velocity by these motions.
Okay. These are chaotic motions. No, no.
Let's do the following thing. Right. We're going to see that temperature has a mean profile.
I mean, obviously, now that everything is is is fluctuating, the actual profile,
the temperature will be all that will be old jacket and have the little fluctuations and so on.
But on the average, we still want to have a situation when in the middle things are hot and on the outside the things are cool.
Things are cool, right? So, so there will be some mean profile of tension which, which I'll call, which I'll call T bar.
And so that and that will be the average of the temperature in the plasma over.
Let's see these fluctuations and these fluctuations will assume have are fast in time.
So what you saw there is that you have you have all these fluctuations things things are changing very,
very fast and will assume that they're changing faster than the mean quantities change in the class.
Okay. So, so if you like, I've just been working on constructing.
Those of you who have done fourth year undergraduates will remember what, what, what mean field theory is.
Right. So? So I've embarked on constructing some kind of an effective, meaningful theory of of of what's going on.
So I will think of my temperature as being composed of the mean profile and the
fluctuations and the fluctuations will be small compared to the mean temperature.
So so it doesn't it doesn't change very much, but they're fast and the mean profile is slow.
Now, that means that I can if I average that equation and also assume that the difference of velocity zero,
which I can do because these motions tend to be subsonic and therefore incompressible.
So. If I average that equation above, what I find is an equation for the mean profile.
And so this is just the average of that term.
This is the average of that term, the source, the heating of the plasma.
I assume that I control that and that and that that's all happening slowly.
And the only interesting new place here is this divergence of the average of the velocity times temperature.
Okay, so this thing here now to calculate this, we need to know about the fluctuations because if you think of T as being t border plus delta t,
then then the average of u t is the average of u times t bar plus the average of you delta t.
And if I assume for now that there is no mean motion of the plasma, Steve, in this talk,
we'll actually talk about situations when there is mean motion of plasma.
But for now, we can see that there's no mean motion of the plasma. And if there is no mean motion, this is zero.
And all of that average is the is the average between the velocity fluctuations and the temperature fluctuations.
Well, so let's calculate that average. How do we calculate this?
So I'm going to do a sort of a split of theoretical physics and then switch back to free quality.
So now how do I do this?
Well, here's a simple way of doing this. Not entirely rigorous, but I think this is good enough.
So so this is the it's the average of you times,
t and I'm going to take this T and get it out of this equation, which I will just integrate over time.
Okay. So there's the U.
And this integral is simply the integral of all those terms of time from zero to T and it's that's the reaction term, that's diffusion.
So that's sort of in now in under this integral everywhere, I will simply approximate temperature with its mean value.
Okay. I will also use the fact that it's changing very slowly.
So, so, so so what happens at previous times is not that different from what happens,
but at the current time you might even think of it as being constant if we ever manage to construct a fusion device that operates in steady state.
So if we do this, then all these terms become zero.
Because basically those are just in terms and it's the average of you and that's zero.
So the only known non-zero term here will be when we average these fluctuations that time T with the fluctuations at some previous time.
And so what I end up with is the following expression for you don't see this.
So, so what I find is that it's proportional to the gradient of the mean temperature times this time integral of the correlation function.
This is again this is I think fourth year undergraduate physics, the correlations or maybe even third.
So it's correlation function of the velocity attempted with the velocity time t prime this quantity if you
look at where it is in the equation is a heat flux and that that's what's called the turbulent heat flux.
Now I can stick this quantity into the air and get this equation.
And finally, I will remember that all my mean things only vary along the radial direction.
So it's all actually one dimensional. So all these divergences become derivative suspect radius.
And what I end up with basically is an additional bit in the diffusivity.
So I end up with an equation that looks just like the equation I had before,
which was the equation for, for, for the, for the diffusion of temperature.
But now, besides the collisional diffusion, I also have this thing called the turbulent diffusion in this D turbulent is equal
to this correlation function of the turbulent fluctuations integrate over time.
So this is what the effective mean field theory for our system looks like.
It's it's it's basically just the there are you know, I should say at this point,
there are all kinds of bells and whistles and so on, which I'm sweeping under an enormous rock.
So but fundamentally, that's the idea of what of what one does.
So so it looks like this. There is a there is an equation for the temperature profile.
You'll set this this to zero to get to get steady state.
And in order to solve this equation, you need to know what this is.
Now, this general idea, you know, we gave our morning of theoretical physics a rather grand title.
But that title has too important had two important components.
One was that plasma was an unlimited source of energy. The other was that it was a normal state of all matter in the universe.
Right.
So this actually, you know, because we're theoretical physicists, because we, you know, we want to have a kind of an ivory tower sight to our lives.
You know, you might say, well, is this is this just for the talk?
This is actually quite the universal idea to treat plasma systems, which is generally physical systems this way.
I mean, the world is full of systems where there's something happening in the mean.
And also and also there are some fluctuations on top in the evolution of the mean things
depends on how the only statistical properties of the fluctuations that they're embedded into.
For example, my favourite example, right, if you didn't want to be a fusion physicist.
But if you're instead you're a plasma astrophysicist or just an astrophysicist, even not a plasma physicist,
then you know that the largest plasma object you can find in the universe are called clusters of galaxies.
And clusters of galaxies, despite the name, contain mostly dark matter and hot diffuse plasma like galaxies.
Galaxies and clusters of galaxies are, you know, rather an afterthought.
So they look like this in optical, they look like that in X-ray.
They are megaparsec sized objects. But the important thing, though, is not that they're so grand and big and so on.
This is actually most of the visible. I mean, apart from the dark matter, this is most of the matter in the universe.
And this is actually just a blob of plasma. Right?
So this is the normal form of matter. It's rather exclusive that we are here, not plasma.
So and it's also rather presumptuous of us to to then feel we can claim it, but we are presumptive question so we're going to do it.
So anyway. Well, the again for a theoretical this this is a nice pretty picture, right?
So astronomers like that kind of thing.
But but actually for a theoretical physicist, the interesting fact about this is that this this is roughly radially symmetric.
And this has a temperature profile. This is what its temperature profile looks like.
It's hot in the middle. It's actually not as hot as a tokamak. So, you know,
but with only just three tens of millions of degrees still fully ionise and it's colder in the middle and colder on the outside and hot in there.
And why that is, is also a mystery. And the way you solve that mystery is basically by doing that kind of theory.
So and of course, you know, you think it's laminar. No, it's not.
Remember, it's all very turbulent. If you look at the temperature fluctuations in that in that environment, they look like this.
It's again, in what we missed. So so it's you know, these are rather universal methods.
Let's go back to, uh, to, to focus on the future a bit more.
And well, first of all, the turbulent transport that I've derived through the turbulent diffusion coefficient turns out to be much,
much larger than the collisional diffusion coefficient, and it's much faster.
And, you know, fundamentally on a sort of a wishy washy level, it's faster.
I mean, of course collisions will relax you to equilibrium. You know how people teach you in statistical physics.
People say, oh, you know, if things are out of equilibrium, things will eventually go into equilibrium.
And, you know, the universe will eventually go into a boring, homogeneous state and etc., etc.
Right. The reason this is not as depressing as it sounds is because, you know, you can still hope that it might happen slowly.
Right.
So and and collisional relaxation does happen quite slowly, but nature actually tends to be rather impatient with with slow equilibrium and processes.
And so you won't typically wait for slow collisions to relax the system.
And so and so the transport becomes, becomes fast.
It's the same kind of situation as, you know, if I switch on the radiator in the back of this room.
Right, and try to and try to ask how fast the equilibrium of temperature will happen between the hot radiator and the I don't know, cold outside.
Right. The answer to this is not based on Collisional transport, because there will be convection, there will be motions.
Right. So, you know, like, again, nature will be impatient and and, you know,
because I'm standing here waving my hands, I'm generating turbulence, which will help to transport as well.
So so what we need to know, so this is the primary thing we need to know, right?
This, this quantity. And what we basically wanted to be able to do is to predict what it is as a function of everything.
Right? Everything being local equilibrium quantities, for example, the gradient of temperature.
You know, this might depend on how much temperature gradient there is.
In fact, it will depend on how much temperature gradient there is because if there is no temperature gradient,
things will not be turbulent and single things will just be boring and stay put and so on.
It is the temperature gradient that makes flows more unstable.
I won't describe exactly how that happens, but that that turbulence happens because that the energies,
the free energies injected into plasma via this temperature green.
So what tends to happen experimentally. And on some level, theoretically and also in numerical simulations,
is that this token diffusivity is a function of the temperature gradient basically behaves like this, it starts off well.
Sorry, this I should have said to bring Diffusivity plus collisional is of the total transport.
So it starts off as collisional transport which is which is very tame and slow and harmless and well understood and will give us a tabletop device.
And then there is some critical temperature gradient. And beyond that critical point, it just shoots up like crazy.
This is cool. This is turbulent transport. It's fast through in this poorly understood.
Not understood. I'm sorry. It's so and it's this is this is a property that that in the language of the game is called
stiff transport in the sense that the moment you go beyond a certain critical gradient,
it just shoots up like crazy. You can't really go very far beyond that.
So the name of the game practically is to move that critical gradient around anyway.
So Timmins is the enemy. In order to kill your enemy, you have to understand it first.
And we also have to understand it because you have to keep your self-respect as a Homo sapiens.
So anyway, so since we have to understand it, let's, let's take a look at it again.
Right? So what I showed you before was, was a movie of simulated turbulence, right?
And there is always there is always a degree of how should I put it, wariness that you should have.
When you look at simulations of anything, who knows what the computer is showing you?
I mean, you know, but in the end, physics is an experimental site.
So here's what turbulence looks like experimentally.
This is a measurement of turbulence using something called the Blue Emission Spectroscopy System on the Moscow Air column.
And this is done by Anthony Field, who is one of my collaborators of column.
And the movie actually was done by my student, Yong Kim, who was in Oxford until recently,
and he's now a professor in South Korea who are also serious about turbulence and fusion.
In fact, they're very serious. So. All right. So this is so this is what experimental evidence looks like.
This is density condensed. This is like a little window inside. You can't do the whole tokamak, right?
So this is a tiny sliver inside the Tokamak and well, let's let's see what they look like.
Well, these basically look like blobs that move around right randomly.
So. Well, what does this tell us about this turbulent diffusion coefficient, the fact that they look like blobs that move around?
Well, first of all, let's let's look at this expression and let's try estimate on a very hand-waving level what this is actually equal to.
Well, you know, it has two powers of velocity and it has a time here.
Right. And you see, this is a correlation function of resources.
So so if you if you basically take if I if I take some point here and if I take the velocity at this point,
the time T and multiplied by the velocity at this point that the of time multiply one by another.
If the difference between the times is long enough, that would be zero, right. Because they'll be uncorrelated from each other.
Right. So, so this function peters out to zero over some timescale which we will call the correlation time of the turbines.
And so the two event diffusion coefficient is the roughly speaking, the size, the square of the velocity times, its correlation time.
Well, what is its correlation time when again, if you looked at this movie,
you might have convinced yourself that, you know, you have these blobs, or we call them ages sometimes.
And the time that each of these blue globes moves is sort of comparable to its size, divided by the velocity at which it moves around.
So in the way it breaks up, you know, it's, you know, you didn't see any coherent things.
It just went like that, right? You know, what you saw is right.
Something like that. Right. So so it's a basically the correlation time is something like the turnover time of the globe.
So if the correlation time is that so l will be a disease agent.
I call them edges and globes and whirls and twirls and swirls and and so these are all the same things.
So so this is the research. This is this is the velocity. If I plug that in, I can come up with a couple more expressions for this for this to happen.
Transport, for example, that can be the square of the size of the 80 divided by the correlation time.
And it is actually evocative of what I explain to you about the Collisional transport, right?
Remember how the collision transport, the particle carrying energy goes a distance, collides with another particle,
gets deflected, goes another distance in the random direction and so on, and so diffuses well.
What happens here? This is again, the random walk argument, but now a particle carrying energy is carried by this motion until that motion
breaks up and then it's picked up by another motion that's random and and moves.
Moves off. And then it's being. Another motion and so on.
And so the step so is L and the correlation and, and the time it takes is the correlation part.
So again, it's, it's a, you know, I could have actually admitted that derivation just showed you that derivation to show you.
I'm a theoretical physicist, but fundamentally, I put it just written this and said, well, it's make sense, doesn't it?
So in other words, you could write that as velocity times this random walk.
And so and so again is there is a diffusive process here.
Well so I have was sort of surreptitiously surreptitiously started introducing all sorts of notions.
To do with the nature of turbulence. I talked about Globes and AIDS and all that.
So what I'm going to do in the rest of this lecture is I will attempt a very basic and non-readers introduction to what turbulence is.
Right, because because even if you weren't interested in fusion or indeed even plasma physics, turbulence by itself is quite a fascinating phenomenon.
It's a routinely it's almost become banal to refer to it like this.
But it's true that it's the last great unsolved problem of classical physics.
I think it's was it Heisenberg who reportedly told that, you know, answer the question as to what he will ask God when he meets him.
If you can only ask one question, and I think the question was about the structure of turbulence and was more precise in that he
wanted to know the precise scaling of and what the correlation functions of turbulence with,
etc., etc. But you know, I don't know if God would have time for that, but but there are insights to be gained from God and nature,
or rather nature pending our meeting with God, but about, about the structure afterwards.
So let me let me let me talk to you about turbulence generally.
Not even necessarily implausible. I'll come back to plasma later. How did it all start?
Well, turbulence, like everything else. It was first discovered by Leonardo da Vinci.
Right. You know, if you've heard enough of these of these talks, you know that that everything goes back to.
Then after that, every every engineering thing goes to Leonardo da Vinci.
Everything goes. In fact, Leonardo da Vinci invented the Tokamak, right?
I mean, you can fly, you know, if you leave through his notes, you will see a picture of a tourist with with, with.
In fact, yes, he did so and he tried to build it.
But the Duke of Milan instead wanted an enormous equestrian statue of himself.
And so and so they couldn't afford to take him up. And so there we are right now.
You know, governments at all times fundamentally want the same thing.
And that's an enormous question, equestrian statue of themselves.
So our goal is to persuade them that the Tokamak would be an enormous equivalent statue of themselves.
So anyways, this is this is from the notes of Leonardo da Vinci.
Right. So he he's been Leonardo was very good at observing reality.
And one of the things he noticed this, I suspect, is a is a picture of a sewage pipe discharging its contents into into,
I don't know, River Island or something like that. So so but what he noticed was the following.
He said, observe the motion of the surface of the water, which resembles that of here.
It does resemble that of here. Remember, he had a lot of hair.
It's it's it's it has two motions of which one is caught is caused by the weight of the hair.
The other by the direction of the curls or eddies.
Thus the water has eddy motions, one part of which is due to the principal current.
The other to the random and reverse motion. So he noticed for the first time that there was this separation of scales between there was
a mean feel theory and fluctuations in both in the in the in the in the motion of water.
He had met several of these sketches. There are all these he was pretty good at this sort of thing.
So so anyway, he noticed all these all these eddies.
And this, of course, is. So the basic idea that he came up with was that there is a mean laminar flow, which then breaks up into disordered motions.
Somehow it doesn't stay stable. Right. So this actually is a universal phenomenon, these these ageing motions being created in in various fluids.
Right. Here's a spectacular picture of an eddy. This is turbines in the wake of a of an aeroplane descending to Heathrow.
So that's what they look like. These turtles. This is this is an air.
This is this is the same thing.
This is a close up of the of the great red spot of Jupiter, observed by by not by the other man, but gathered in this spacecraft.
Galileo could not see in near-infrared. So and you see, there are these there are these overall currents.
And then there are all these there are all these eddy motions on top of them.
So this is very close by. These are called the giant radio loaves of flux, a million light years across like that.
Right. So so very little example, a very large piece of turbines.
Here's another spectacular astro picture. This is this is a so in the middle is something like a supernova,
not quite the supernova that something like a supernova went off and there was a dust cloud around.
And and there's a shockwave that hit this dust cloud. And they thought it all went spectacularly chaotic.
And so there you can see the there is an overall emotion there, also all these emotions.
So this is the universal. It's been seen in millions of places.
This is a familiar. A picture? Probably. Right. So. So. Cosmic turbulence.
Van Gogh could see it with a naked eye. Most of us need, need, radio, X-ray, infrared, that kind of thing.
But anyway, this is how it happens in the Tokamak.
Great. So this is a movie which was made by Edmund Haycock,
who is who did his Ph.D. here in Oxford and still here as a as a junior research fellow at the modern college.
So this is this is how turbulence happens in a in a plasma.
This is a simulation of a small piece of plasma. You see, it's first.
First, there are these irregular flows that are created and they all break up into into a mess.
So, Miss, let's talk about this mess and let's let's let's try to understand this almost sounds like a strange thing to say, the structure of chaos.
Right? But there is structure to chaos.
So let's talk about the structure of these fluctuations.
So this is a very famous picture. This is the biggest numerical simulation of terms ever made.
A while ago already it's it was done in Japan on a machine called Earth Simulator,
which at the time was the biggest machine in the world by Yukio Haneda.
This is just fluid turbulence being simulated. So and this is sort of a field of turbulence.
This is this is know fluid being turbulent in a box. And so you see all these messy fluctuations.
And one of the interesting things about these fluctuations that you notice is that if you take a piece of that box and zero in on that piece,
what will happen? I'll show you a succession of frames as you zero in further and further and further.
Right. So so if you just look at this box, a box, this sub box looks like this, and then we'll take another box inside it and it'll look like that,
and then we'll take another so box inside it and it'll take and it'll look like this.
And what you basically see in all of that, then it then runs out of resolution at that point.
But what you see in this succession of pictures is that it's not just a mess with a correlation length, it's a mess on multiple scales.
Right. So what happens is, isn't just chaos. Chaos is actually a relatively tame and easy thing to do.
Not to have is because turbulence is disorder on multiple scales.
Right? That's that if you like. If you, you know, if you want the definition of two, that's it.
Disorder of multiple scales. So, so, so what you see is this multi scale disorder and now we have two.
Oh, well, since I'm overdosing on movies these days, here's, here's, here's what it looks like in plasma.
So this is a very spectacular this is also a movie from Southern California.
So this is a this is a picture of a sort of a simulation of plasma turbulence on multiple scales.
Look look at the it's quite interesting to stare at look at the variety of structures that create you see you see, you have you have large structures.
Then you have small structures and top of them. Then you have small structures on top of small structures and so on.
So it is a, it is, it is a truly multi scale beast.
So how do you characterise such a thing.
I mean if you just think, think of it for a moment, not as a theoretician but even as an experimentalist, right?
So imagine you're handed a system like that. And you have to do something about it.
You have to you know, you know, before you start understanding something, you have to decide what is it that you're going to understand about it.
You have to measure something about it, right? Because in the end, what are we doing?
We're predicting some kind of Y versus X dependence of something, right?
So what should we measure? Well, one of the standard things about turbulence is,
is that if you measure the energy of these motions and if you then look at this energy scale by scale, what you tend to find is a power low.
Right? So there is a so it's power laws galore and in intervals.
This is this is this is the energy spectrum in the in an air jet.
So this is this is disturbance in air has this power low spectrum.
This is low look. So that's what power laws look like. Straight lines, the slope is minus four thirds.
And then this is density.
This is called this picture is famous. It's called the Great Power Low in the Sky.
This is a density fluctuations in the interstellar medium on the range of scales that goes that goes.
This is about ten orders of magnitude of scale separation.
And over that scale, scale. So there's an overall parallel.
There are many more in the solar wind spacecraft measure a turbine fluctuations find a point actually tends to be the same power load that you find.
You always find this 5/3. This is a measurement from a galaxy cluster, right?
So so this is just outside the Earth's orbit and this is this is a megaparsec away and the in the megaparsec across as well.
So and here's here's here's a picture from a tokamak.
So this is a simulation by Michael Michael Barnes and and Felix Parra, which was done here in Oxford in 2011.
So this is this is turbulence in the plasma. It's slightly different quantities plotted.
That's point seven, four and 5/3. And this is actually a measurement of turbulence.
These these blue lines, the red line, the simulation, the the blue lines is a measurement on the French document go towards the graph.
So again, power loss. So so what let's let's let's try to make sense of that.
Why is it that turbulence is multi scale and why does it have these power laws?
So fundamentally, this has to do with the way in which a nonlinear system will process energy that's injected into it.
Right. So that's a that's a sort of a key executive summary, if you like, of what I'm about to say.
So let me let me give you a simple example of how that works. Let's start again with third year undergraduate physics, right?
So if we're just talking about turbulence in water or gas, it satisfies something called the Nova Equation,
which is which is actually just a momentum equation for moving fluid. Right.
This is velocity, the rate of change of velocity acceleration.
This term says that the velocity of the moving fluid elements that direct each other, then move each other around, that's a pressure gradient.
There is viscosity, which is dissipation of energy. And I will put in some unspecified F, which is which is going to stand for injection of energy.
You know, I put a spoon in it. Let's start doing something like this.
That's f so if I do this well, what's the energy of this system?
The energy of the system is just the kinetic energy density times the times the square of velocity integrated over space.
Right? So that's, so that's the energy of the system. And I can derive from that equation, I can derive an evolution equation for this energy.
So I can, I can ask what is devoid of that.
It turns out that if you tried to. So you do that by multiplying that equation by those two.
And it so it turns out that if you do that,
all the contributions from the nonlinear terms and from the pressure disappear and what you find is a simple balance.
What you will find is that the change, the rate of change of energy is equal to the power that that you inject,
which is just you don't have time to integrate that over the volume minus the power you dissipate,
which is viscosity times integral over the square of the gradient of the velocity.
So that that's this term multiplied by velocity. Okay.
So, so that's that's that's the and it's unsurprising, isn't it?
I mean, you know, you put it in, then it dissipates, right?
That's that's the balance of energy in steady state.
These things have to have the balance right. Whatever whatever you put in has to be dissipated.
All right. Well, let's estimate these terms. If we estimate this terms, we can do this purely dimensionally.
We can ask, well, how much power are you injecting into turbulence? Well, just dimensionally.
Right. So if things move at a certain rate, which I call you are a mass, the mean square velocity,
if they have a certain density, if, if my books has a certain size, just the.
Mention that you could convince yourself that if you wanted to concoct a quantity that has the dimensions of power,
it'll look like really you cube over l this l is sort of the size of the system, or rather the size of the spoon.
That's, that's during the turbulence. If you calculate the dissipation just from here, it's density times, viscosity times,
the square of velocity divided by the square of the size because there's a greater power.
And if you compare these two terms, divide one by another, the ratio will be velocity times scale divided by viscosity.
This is this is a dimensionless number known as the Reynolds number.
And it turns out that in most of these systems, it's very, very large.
So there is an imbalance seemingly between the injection of this equation, and nature has to find a way to correct that imbalance.
How does it do that? Well, it does this the following way, the way to make this.
So it's the imbalance. The imbalance is is in the direction of making dissipation smaller.
So how do you make dissipation large? Well, dissipation has gradients in it.
See its viscosity times of the question. Right.
So I can make dissipation large by making scale smaller if I have smaller scales that have larger gradients.
And so even the small viscosity coefficient will go a long way to dissipate all my motions.
And indeed I can even dimensionally estimate how small that scale has to be in order for this to be equal to that.
And that again is by a split of dimensional analysis, its density times, cube of viscosity divided by the injection rate.
All of the one quarter power in that sense turns out to be the size of the system.
Times the Reynolds number to the minus three quarters. This is called the cumulative scale.
In that scale is much smaller than the scale of the system because the Reynolds number is much greater than the scale than one.
And so we have a scale of injection at which we're putting the energy in.
We have a scale of dissipation in which this energy is dissipate, and then somehow that energy has to get from large scales to small scales.
These intermediate scales is something called the initial range. Now, how does it do that is the next question, right?
How do you you set up a large motion. How do you convert that large motion into tiny little things?
Well, it turns out that in most cases, in nature,
the way you do this isn't by somehow directly transferring energy from an enormous swirl into tiny little blips of fluctuations.
What you do is you you make a large motion. Then that motion breaks up into motions that are half the size,
and then it breaks up into motions that are half the size of them, and then half again and again and again.
So this is called the turbulent cascade. And this is an idea that was probably first due to Louis Fry Richardson.
This is this is an illustration of what I've just said. You start with large with large motions, you break into small motions of breaking.
It's more of the motions and small motions and so on.
In terms of, you know, these spectral power laws, you could think of energy going from large scale to smaller scale, smaller scale, smaller, smaller.
So this is key. The wave numbers are large, came in small scale.
He actually didn't put it that way. He put it in inverse.
He said big whirls have little whirls that feed on their velocity and little will also have lesser rules.
And so on. The viscosity easy to remember he this is this this is actually a piece of plagiarism.
Well, maybe not quite. It's a piece of being inspired, cross-disciplinary.
So he was he was inspired. Cross-disciplinary by Jonathan Swift, who was inspired.
Cross-disciplinary by by Hook and Levin Hook, who in those days won one in Britain, one in Holland invented the microscope.
And we're looking at all kinds of thing. And once the intensity microscope, they put all kinds of things under it.
And what they found is this amazing, microscopic world that you couldn't see.
And in particular, you could see, you know, two things. They looked at the fly and then the fly had little flies on top of it and so on.
And so Jonathan Swift said,
So naturalist observe a flea had smaller fleas that only prey and these have smaller yet to point them and so proceed on the ad infinitum.
It's kind of a cascade that's every point that he's quite this bit by him that comes behind.
That's not true about the theoretical physics, so it's only the points that have that property.
Theoretical physicists are collegiate, but no backstabbing whatsoever.
No. The last bit that that that I want to talk to you about is how do you actually make this all more more quantitative right.
So we'll go from verse to mathematics and nobody was better it going from poetry to mathematics.
And then then Komarov, a great Russian mathematician who died in 1987.
And that what he said is this is actually one of the greatest contributions to physics by a mathematician.
I think this this this bit and the and and it all and it all fits on one slope.
So so what did he do?
He basically said, well, let's let's think of the system in the following way.
First of all, let's let's. Let's think.
I mean, if you notice if you noticed what I showed you about this picture, they all had power loss and they all had the same power.
So there was a suggestion there that there's some universal physics going on in in the process.
Right. So. So McConnell said, well, let's assume that there are no special systems.
That all systems are sort of roughly the same. That everything is homogeneous.
There are no special locations in the system. So so it's the turbines in one place.
So the system is no different from turbulence in another place statistically that everything is isotropic, meaning there are no special directions.
We'll consider a system where that way is the same as this way, in the same exact way, in the same exact way.
And everything is local in scale space in the sense that as you break up these motions,
there is nothing special about any scales in between the injection scale and the dissipation scale that that is sort of a natural assumption to make.
And then what we wish to predict is, you know, if you if you thought about how you would measure turbulence, well,
what you would probably do is you would stick two probes into it, registers the velocity at one probe rich to the velocity and another probe.
Right. And then ask how the difference between these velocities scales with the distance between the probes.
Right. That's a that's a very natural thing to ask about this.
And and so and so what we want to predict is this build that you at scale out as a velocity difference.
And so then at each scale you could say because there is look, it's each scale can only depend on things that happen at the same scale.
And so you could see the energy associated with that scale,
which is zero times the square of that quantity divided by some time that it takes for
that structure to break up must be equal to the power that's flowing through your system.
Because you've injected the certain amount of power there, you're going to dissipate it at a much smaller scale, at all the intermediate scales.
You just have to transfer it from one structure to another to it to have a small scale so dimensionally.
This should be the case. And then again, dimensionally the only thing that this time can be.
And I've already introduced that notion sort of surreptitiously, you know, when I talked about typical diffusion dimensionally,
what that time has to be is the size of the structure divided by the velocity at which it which it turns over.
And so if you plug that into here, what you find is rho delta, you cube overall equals to the injected power, which is a constant.
And so they'll tell you the velocity difference across a scale is proportional to the one third power of the scale.
So it's almost a miracle, right? Something out of nothing. We started with a complete mess.
We've just predicted a scaling law for, you know, based on on all these assumptions.
This this corresponds to the incidentally, to the 5/3 spectrum.
I can if anybody wants to know how exactly that works out, explain it.
But the point is that we now know that the typical amplitudes depend on scales in the following way.
Well, recall, you know, going back to the transport recall.
Well, I told you two of them diffusivity is u times l, right.
Very basic on a very basic level.
I mean this estimate that means that so and we now know that you is proportional to L and so the whole thing is proportional to L to the fourth.
Third. So so it's bigger when the scales are bigger.
What this tells us is that the largest scale eddies will make the largest contribution to the twin.
Transport is perhaps intuitively obvious, but.
But with the nice conclusion, the interesting practical question about the answering of which I think you'll hear from Steve is,
is, is what that scale is and how fast these edges are and what you can do to reduce both of those things.
Well, before I conclude, let me tell you that things are not quite as simple as I have described.
Uh, there are complications. First of all, to consider the tokamak is not homogeneous.
Conditions vary with radius, right?
So if this is a tokamak, this is a picture of a typical tokamak took two months would actually be different depending on where you are in this area.
So what we do is we theorise of simulate locally on the magnetic surfaces we have,
we take these magnetic fields that wrap around tourists and that in fact,
what we do is, is, is we take something called the flux tube that moves, that wraps around the door.
And then we look at the turbulence in that curvilinear flux to improve linear coordinates.
There is an absolutely spectacular movie that I have to show you, which was also made by Edmund Haycock.
It's what theoretical physics does to this to this flux tube.
It's so. So how do you reduce it to something we can understand?
Well, it gets transformed into a box and there we go.
Right. And so that when we. All these boxes.
They were not actually boxes in the context of the document. They were those flexed and done in accordance.
Secondly, it's not isotropic. Things are actually very stretched out along the magnetic field.
So all these structures I talked to you about, they're not just little girls and squirrels.
They are actually sort of worms. Right.
So they're whirls and swirls in this direction. So.
But but the law, you see, they have they have a very long coherence length along the magnetic field.
So that brings in lots of interesting new physics. And finally, this is not even happening in three dimensional space,
because if you remember the end of Felix's talk, plasma is actually a six dimensional object.
It's it's described by the probability distribution function of particles which depend on positions and velocities.
This lives in the six dimensional phase space.
And so the small scales that I talk to you about are created not as I described, for water or air in space, but in three space.
Right. These are six dimensional small structures. And and you go through large gradients in both velocities and in space.
You don't want to know what the complication of before is. Anyway, story so far.
Let me summarise. So we want to build a machine to tap the energy that fuels the stones.
And inside that machine, we have a plasma that's locked in a cage and kept out of equilibrium,
but inside called on the outside it rattles its cage, it breaks into whirls and swirls, and it's trying to regain equilibrium this way.
It's rather good at doing that. And so to keep it and to keep it hot, we must stay in the nonlinear beast.
And so that, you know, it's a beast, let's see if this is going to work. So this is a this is what I'm going to show you.
I hope it's loud enough. So so this is this is what the transmission sounds like.
This is Anthony Field, who is a experimentalist, have called them put a piece of plasma to different observations of sound.
And this is this is what it is. Let's see. Um.
So it is a beast.