 Okay, so good morning, and thank you very much for this opportunity. I'd like to thank the organizers of this conference for the chance here to speak to you today about the dynamics of 2D turbulence in magnetically-confined Tokamak plasmas. So I've noticed that there isn't too much coverage of magnetically-confined plasmas at this conference, so I'm going to take a little step back and review some of the basics of it, kind of put the problem in context, why it is we care about turbulence in fusion plasmas. Just a brief introduction, again, I'm George McKee. I work with the University of Wisconsin, but I work at the D3D National Fusion Facility, which is located at General Atomics in San Diego, California. So why do we care about turbulence? Well, it turns out that it's a critical issue for the development of fusion energy, which has been a goal for many scientists for a long time. It turns out that it's a very fundamental and basic topic to plasma science, fusion energy sciences, and ultimately the development of fusion energy. So the reason we care about it is because turbulence drives radial cross-field transport of particles, energy, and momentum across the magnetically-confining surfaces, and ultimately it helps set a fundamental parameter, which is called the global energy confinement time. This is basically how well the plasma keeps the heat and particles inside of it, and as a result of turbulence, there is a significant reduction in the energy confinement time relative to what would happen from basic neoclassical processes, and ultimately this determines the size and cost of fusion reactors. I show you here a picture of the Iter Tokamak, which is under construction in catarache France. It's a very large international project. It's designed to demonstrate the feasibility of fusion energy, but its size is quite large. There's a canonical person down here, you probably can't even see him because he's too small. The scales of this are tens of meters, and that's basically driven by the fundamentals of the energy confinement, which come down to the turbulence that's inside the plasma. But more fundamentally, and perhaps of more interest to this community, is that the turbulence is a highly complex system. It exhibits strongly nonlinear dynamics across multiple spatial and temporal scales, and the plasma turbulence is largely two-dimensional, not entirely, there are important 3D effects, but it can be described largely as two-dimensional in nature, and therefore has some strong connections to fluid dynamics, planetary, atmospheric, geophysics, and astrophysics. So just to be a little more particular in this, some of the commonalities between plasma turbulence and, say, planetary or fluid turbulence, such as atmospheres, you have a very similar phenomenon of having rotation sources, whether it's Coriolis or the Lorentz force in a magnetized plasma. You have energy sources sort of in the middle of the plasma or at the equator. Common physical equations, Hasagawa Mima in plasmas versus the Charniyokukov in planetary plasmas. Rospy waves are very analogous to drift waves in plasma. Then, of course, you have large-scale flows, zonal flows, we call them in plasmas, or jet streams as you might see in planetary atmospheres. And I'll be talking about measurements of some of these parameters later. So how do we end up with large turbulence in a magnetically confined plasma? Well, in order to achieve the conditions required for fusion energy, you have to have a very hot core, typically the center, you want it to be on the order of 15 kilo-electron volts in order to sustain the nuclear fusion reactions. Correspondingly, there's a relatively high density, and so you end up with a system that at the core has a pressure close to one atmosphere. It turns out it's coincidentally about a million times hotter than the atmosphere, but also about a million times less dense, but around one to ten atmospheres in a reactor at the core. This drops off to near zero at the edge, and we essentially have the vacuum condition. This leads to very large pressure gradients that vary, and when combined with the magnetic geometry, lead to very strong gradients in both separately the density, ion temperature, electron temperature, and flow velocity, which can also be a stabilizing factor. This in turn drives turbulence in multiple fields, density, temperature, electrostatic potential, and magnetic fields, which then drives cross-field transport of the various kinetic properties, density, temperature, etc., which then feeds back on and limits the gradients that are allowed to be sustained in the plasma. There's sort of a fundamental nature to this turbulence. It has both an electrostatic and electromagnetic component, and they both add up and result in a particle flux and a heat flux. And there's some exotic names for some of the specific instabilities that arise, ion temperature, electron temperature gradient modes, trapped electron modes, etc., that arise because of the specific geometry, the kinetic particle populations, etc. The basic nature of this is that it's similar to turbulence in any system, that you have some sort of driving force or driving scale. In this case, there's energy that goes in at around the ion gyro radius, and for a typical magnetically confined plasma, this can be on the order of several millimeters, and then that energy gets dissipated through your standard forward cascade to high k, dissipated regimes, but because it's largely two-dimensional in nature, you also have an inverse cascade to a low wave number and ultimately leading to the development of stabilizing or saturating zonal flow phenomena, and we'll talk more about that later. This has been observed experimentally. Here's a wave number spectrum from the Tor Supra experiment in Catarache, France, and they've seen the wave number does seem to have a break point at sort of near the ion gyro radius, so this makes some sense. And I'll be showing you some more experimental data later that sort of supports that viewpoint. So where are we making these measurements? I realize people may not be so familiar with the facility here. So this is the D3D National Fusion Facility. It's a Tokamak facility operated again at General Atomics in San Diego, and it's designed to ultimately do the science behind developing fusion energy systems, and it has several major elements to the research program. It pursues sort of fundamental fusion science, which includes a study of turbulence, alphanic modes, energetic particle modes, magnetohydrodynamics, et cetera. It also addresses some specific issues towards the technology of developing fusion energy and also optimizing the plasma confinement sort of for a given magnetic field, which tends to be expensive, how much pressure can you contain in there by optimizing the current density, the kinetic profiles, et cetera. One thing I wanted to highlight to this community is that we have started just this past year, a so-called Frontier Science Campaign, and this is basically a campaign that opens up the facility to basic plasma science that may not be, or in fact, in principle, it's not connected to the fusion application, but perhaps can be done in the toroidal geometry with the parameters and the heating systems and the diagnostic systems that we have available. So we actually reached out to a community from universities and national labs at work in related fields, but not necessarily directly in fusion energy to pursue some fundamental topics in, again, turbulence and alphanic instabilities, flux ropes, there's reconnection physics, et cetera. And this will be continuing next year, so if people have ideas and are interested in doing this work, it's sort of a new experimental campaign but it seems to be successful after its first year. So I'm not going to go through this in detail. I just wanted, in a sense, convey how seriously we take turbulence in magnetically confined plasmas. We've developed a very wide range of diagnostics, microwave-based, optically-based systems that measure a range of fields, density, temperature, electron temperature, ion temperature, magnetic fields and electrostatic potential over a range of different scales that are relevant to turbulent transport. So you could spend hours just going through the various systems. I'm not going to do that here, but I want to let you know that these do exist and we work in concert to obtain these measurements in a wide range of experiments to understand the role and impact of turbulence on the magnetically confined plasmas. Again, I'm not going to go into too much detail, but the data I'm going to be showing you for the rest of this talk was primarily obtained with a density fluctuation diagnostic called beam emission spectroscopy. It's an optical diagnostic system that observes emissions from a heating neutral beam that are excited but not ionized by plasma collisions. They emit a photon that's observed spectroscopically and then with high-speed optics detectors, we measure the localized density characteristics because emission density can be related directly to the local density fluctuations in the plasma. So for what I'll be showing today, we've deployed a 2D grid because, again, the turbulence primarily is two-dimensional in nature and that's primarily in the so-called radial direction. That's horizontal in this orientation and the poloidal or azimuthal direction, which is sort of vertical in this direction here. The characteristic scales of the turbulence are, again, a couple centimeters, so we have about one centimeter resolution in this radial poloidal plane. So you can think of it, it's about the size of your hand, the detection zone that we're looking at. And we locate it near the outside edge of the plasma here because that's where there is a particularly strong turbulence effects. It's where particles and energy get transported out of the plasma into the surrounding zone. So what does the turbulence look like? I don't typically show raw data, but I thought since you're a turbulence community you could probably appreciate that. So here's some actual raw traces. This is a 300-microsecond time window from three poloidally adjacent channels showing the turbulence here and how it evolves. So primarily it looks like noise, it is the turbulence. And you can actually see that there's pretty high coherence among these three different spatially separated channels. In fact, if you measure the coherence, it can be up to 0.8 or so between these channels and is being uniformly convicted across the plasma or as a mutally by background E cross B forces. So that leads to a finite convection and a so-called phase shift between these poloidally adjacent channels. So these measurements are fairly high quality. They allow us to look at the dynamics in 1D and two dimensions near the edge of this plasma. So here's a couple of images. There are a set of images that were obtained in a so-called L-mode plasma in D3D. Each of these frames is separated by four microseconds. We use kind of an interpolation method to colorize the pictures and show the advancement, but keep in mind that the original data is obtained on an 8x8 grid. And if you follow some of the individual structures, I'll show you a movie in a moment, but you can kind of see them advecting from frame to frame. And it's very asymmetric radially and vertically, so I want to point that out and just orient you ahead of time. So the horizontal direction is a radial direction sort of going out from the hot core to the cold edge and the vertical direction is the azimuthal direction. That's sort of along a confining magnetic flux surface. There's a background advection in the plasma, again coming from an E cross B flow that you'll be able to see in the visualization. So here I'm going to show you the... Let's see how this works. Okay, this is an actual movie and it's advancing at the rate of 10 microseconds per second of real time, so it's got an inflation factor of 10 to the fifth year. And you can see the individual eddy structures, they're sort of advecting upwards. It's sort of the dominant kinetic that your eye will catch. But you can see that there's a lot of complex interactions going on within the plasma. You see eddies sometimes merging together and sometimes kind of tearing apart. There is a shear flow in the background here which interacts strongly with the eddy structures. And again, you can see visually that the structures have a few centimeter correlation length. This is directly traceable back to the ion gyro radius in the magnetic field as sort of the fundamental length scale, which is again a few millimeters typically in these conditions. But there's a very strong asymmetry between the radial and the poloidal dynamics and also with the correlation lengths as it turns out. So you can watch these for a long time. But the question is what can we determine from the dynamics of the turbulence? So we're going to apply some analysis techniques that actually were developed in fluid mechanics to try to infer some... or quantify some of the detailed dynamics of the turbulent eddy structures and ultimately how they're resulting in the cross-field transport of particles and density. So here this quantifies the images, the spectra in a little more detail. Again, you like looking at spectra. So these are time average spectra at several locations. The parameter, the spatial parameter here is the so-called normalized minor radius where one is the boundary of the plasma, zero will be the core of the plasma. So you can see there's a very strong variation in both the amplitude as well as the spectral shape across the plasma. You get a combination of both local amplitude and e-cross speed Doppler shift effects that change the frequency. The fluctuation amplitude has a large dynamic range. At the very edge of the plasma it can be up to near 10%. That's the n tilde over n, the equilibrium density. So you get this sort of turbulence, virtual turbulence storm near the boundary of the plasma. Then it can drop down below 1% within just a few centimeters spatially or about 20% of the minor radius. And this very strong edge turbulence might in fact be a source of the spreading that might impact core confinement in a stronger way than we might think. You heard about this, Paxu Ham presented this talk on Monday or Tuesday on this topic. And we think that perhaps this very large edge turbulence is a source of that avalanche spreading of turbulence into the core. Some of the other spatial characteristics are shown here. This is a radial correlation function where I've taken a reference channel in the middle of the array, and you can see that there's a radially asymmetric function here. And this can be traced back to the ion gyro- sorry, ion gyro radius being shorter on one side rather than the other side because there is a strong temperature gradient at this region of the plasma. And colloidally you have this wave-like structure. This shows that there's a very strong asymmetry in the radial and colloidal structure of the turbulence. And here this shows a full 2D map of that. So you have this sort of wave-like structure in the azimuthal or colloidal direction and a sort of monotonically decaying radial structure. Again, showing you the strong... And we can invert these and look at the S of K structures and do that routinely. I'm not going to show that right here, but one of the things I claimed earlier was that the ion gyro-radius was sort of a fundamental scaling parameter. And I just wanted to show you some data that backs that up. So we did experiments where we varied the ion gyro-radius by varying the confining magnetic field and other dimensionless variables were held constant. And when we did that, we saw that the measured correlation length of the turbulence scaled just with the ion gyro-radius. So if you look at the normalized ratio of plasma turbulence correlation length to the gyro-radius, you see a fairly flat uniform dependence, about a factor of 5 there. But again, it does change quite a bit numerically. Likewise, the temporal decorrelation time can be measured. Tends to be on the order of about 5 to 10 or 20 microseconds. But it scales with this so-called gyro-kinetic time scale, which comes from the underlying theory of how turbulence is driven in these plasmas. And you can see that the... you can see the parameters quite a bit. When you normalize it to this parameter, you get fairly uniform normalization of the temporal decorrelation time. And this can be thought of as sort of a turnover time or any lifetime, which is important, very important, for how rapidly it convects energy and particles. Here's also some measurements with a much different diagnostic technique on a different machine, but this lineup, when it's normalized properly, gives you some support for the idea that this is a good normalizing factor and you're getting similarly or self-similar turbulence on much different machines with different measurement techniques. So, looking a little more critically and quantitatively at the dynamics, we've applied some villosimetry techniques to the data. These techniques have been developed in fluid mechanics, and they... we particular... oops, sorry about that. We use a particular technique called orthogonal dynamic programming. It results in a velocity field, basically a radial-polloidal field that's a function of space and time, and it shows us a lot of the dynamics of these turbulent eddy structures. Again, we apply this to the data. It gives us a V of r, z, and t. And from these measurements, then we can infer quantities such as zonal flow behavior, the Reynolds stress, particle flux, vorticity, entropy, et cetera. And interpret, again, some of the dynamics of the turbulence here. So, I'm not going to go through this in detail. This is just the technique as it was developed in fluid mechanics, particularly to look at particle imaging villosimetry. It turns out to apply fairly well to the much lower spatial resolution, but sort of the blobby or smoke-like structure of the turbulent eddy structures. So, this is similar background images to what I showed you earlier, but now I'm superimposing on top of that the velocity field, a 2D vector, again, in r and z. And when I show you the movie of this, you can see that there's some very unique dynamics, again, some differences in the radial and colloidal structure that are very important to the particle transport that results from this. So, this is actually a different plasma from the one I showed you earlier. So, actually, the background turbulent eddies are behaving in a somewhat different fashion. One thing you'll note is that there's sort of a natural shear flow that develops in this plasma. You can see a very strong sort of vertically upwards advection at this one radius, and then just a few centimeters over, you actually see a downward advection of the turbulent eddies. So, it's a very strong shear layer that's developing right across this area here. And this turns out to be very important for a phenomenon that's known as the low confinement to high confinement transition. This plasma is about to undergo such a transition. And you'll see the turbulence get very large. The eddy structures are getting increasing in amplitude and getting torn apart by this shear layer. And then very quickly, it gets suppressed. And now you see a much quieter turbulence zone. And at some point, there's a whole story behind the Reynolds stress that's driven by the turbulence and how it drives the flows that ultimately trigger this transition. I'm not going to go into that in detail here other than to just mention that in passing. But now we have this 2D flow field that's derived from these measurements and looking at how the individual eddy structures convect in space and time. And we're going to apply, look at the distribution of these velocities in more detail. So, here's some spectra, long time averaged windows of the velocity. So, this is not now the density turbulence, but the inferred velocity. So, here's a radial velocity spectrum and a poloidal velocity spectrum. And you can see there's much different characteristics between the two. Spatially, this sort of makes sense. If you increase the separation between the point measurements, these are cross-power spectra. You get a monotonically decaying cross-power over the scale of a few centimeters that immediately tells you that you, again, have a velocity correlation length of a few centimeters like you do with the turbulence density fluctuations itself. And it's kind of a broadband spectrum extending up to a few hundred kilohertz. In the poloidal direction, we have a much different structure. Again, there's sort of a more rapidly decaying broadband structure, but then there's this somewhat coherent mode at low frequency. And we didn't know this at the time, but it turns out that this is a feature called a geodesic acoustic mode, which is driven by the turbulence and actually acts to self-stabilize. But again, this shows the strong radial and azimuthal or poloidal asymmetry in the turbulence, because this structure does not show up at all in the radial velocity. So on the time scale, the movie I showed you back here, you wouldn't see this mode. It's about a 15 kilohertz oscillation and so about a 70 microsecond period. So you don't see it by eye, but it's going on in the background, kind of oscillating up and down at relatively low amplitude. But nevertheless, it comes out quite clearly when you do these long-time average spectra. And this feature turns out to be critical to the oscillation mechanism of turbulence. So this turbulence is here, but is it important? Is it actually convecting particles and transport? It's not immediately obvious that it is, but so we interpreted both the density... Sorry. Pardon me. I'm learning how to use this. Both the density and the radial velocity component that's inferred from the velocimetry techniques to get a inferred particle flux. And an important quantity is the phase relationship between the density and the radial velocity. And again, from the density field alone, we don't know what the underlying dynamics are because there's an important but unmeasured quantity here, which is the electrostatic potential, but we assume that there is some E cross B motion, and if it's phased in such a way, then you can get some net outward flux. And when we do this ensemble-averaged measurement here, we do in that indeed see it there is a net radially outward particle flux over the last 10% of the minor radius. So that turbulence I was showing you earlier is in fact moving the particles from the core of the plasma out across the separatrix into the boundary zone. This is the sort of magnetic boundary here at unity minor radius. So this last 10% of the minor radius is a particularly critical zone where the turbulence is basically mixing and conveying the particles and therefore the energy and momentum they carry out into the separatrix. And it gives us, it tells us a lot about the unmeasured electrostatic potential, but that is very important for these measurements. And separate measurements with probes that can both measure the density and electrostatic potential directly give a similar result. So we can see this. Now just to, this is sort of a qualitative impact, but if you look at the raw data signals, you see another factor about how this transport is taking place. It turns out that the skewness of the probability distribution function of the turbulence changes very rapidly across this boundary zone near the edge of the plasma. So the skewness here, again toward the third moment of the probability distribution function is slightly negative inside, just inside the plasma, but it's statistically negative. The error bars are small enough that this is below zero. It then gets very negative right inside the separatrix and then quickly changes sign at the separatrix, the boundary, and then goes positive here. And because of the 2D array and because it was not perfectly aligned with the magnetic geometry, several more data points had effectively higher spatial resolution. These are shown in blue. And they kind of show a very consistent story of a rapidly changing skewness. And you can see this sort of qualitatively by eye in the raw data here. You see this sort of negatively spiking density of plasma sort of developing, and then right at the boundary it becomes more Gaussian-like. And then in the scrape-off or boundary region you actually have these positive structures. So this is sort of qualitative evidence that you do have this mixing and the sort of scooping of the particles from inside to outside as a result of the turbulence there. So, how am I in time? A couple minutes? Okay, that's good. One more topic here. A question comes up as to when you have these strong gradients in this geometry, how does the turbulence stabilize itself? And I mentioned earlier the observation of this geodesic acoustic mode. But for many years the theorists in our community were asking us to try to find this elusive so-called zonal flow. And it was predicted theoretically, basically the underlying mechanism is that the turbulence in the jet stream, and this was predicted to be critical to the stabilization of the turbulence, essentially the saturation mechanism. And the basic nature of this zonal flow is that it's basically a potential of radially localized but as a mutally-anteroidally uniform potential structure that results in a radial electric field that's shown by these black arrows here. And then in a magnetically confined plasma you have an E cross B flow that kind of goes up and down locally. And essentially the concept is that the turbulence drives this flow which then shears the turbulence so it actually has a self-stabilizing mechanism. But did this actually exist in the plasma? So we went to look for these zonal flow phenomena and what we found was in the poloidal velocity spectrum, as I showed you earlier these very coherent modes that had the features of this theoretically predicted geodesic acoustic mode. Then we looked at the frequency of this mode it turned out to match very closely with the predicted gam frequency. There's a slight offset which might come from geometric factors but basically it depends on the sound speed of the plasma. That's the underlying parameter and the geometry. It scales as CS over A and we showed that experimentally. Also there's a uniformity to this structure so it was observed on a different tokamak here. This is a measurement obtained with a Doppler back-scattering system on the Azdex tokamak. A completely different technique different part of the world but fortunately there was a uniformity to the plasma behavior it also showed this very coherent structure which we've since identified as the geodesic acoustic mode. The question then comes, well this is there but is it actually doing something? And it turns out if you look carefully at the amplitude spectrum of the underlying turbulence, what you do see is that there's an amplitude modulation right at the gam frequency which tells you that yes this geodesic acoustic mode or GAM is in fact modulating and helping to stabilize or saturate the underlying turbulence. And if you look more carefully at the dynamics of it you perform a bispectrum between the density fluctuations, the poloidal density gradient of the fluctuations and the poloidal velocity fluctuations there's a lot of math behind this which I won't go through here but this three-way bispectrum is a measure of energy being transferred within the turbulence spectrum. And when this was evaluated you did indeed see these very clear bands and it's not very clear here but the frequency separation between sort of the f1 equals f2 line here is exactly the gam frequency. And there's a positive band at along here and a negative band below that. And what this says is that the geodesic acoustic mode is basically moving energy from lower frequency to higher frequency in steps of the gam frequency so it again provide further assurance that it wasn't just sitting there in the background but it was in fact helping to mediate and saturate the underlying turbulence there. So just one final note before I conclude most of the pretty much everything I've showed you here today was data measurements of the turbulence but there's been a lot of effort in our field and this came up yesterday in the general public discussion about the collaboration and interactions between the experimental community, theoretical and simulation community. And we've been doing some quantitative comparisons between the measured turbulence spectra and the calculated spectra. There's been very advanced numerical codes. You heard earlier as an expert in these. They've been developed. They can predict quantitatively what the turbulence is like. So we did a comparison between the measured turbulence spectrum which is shown in blue here and then the predicted or calculated spectrum from what's called the gyro code. This is a nonlinear turbulence simulation code in red. This is a quantitative graph. There's no free variables here and it shows that you in fact have very good agreement, very good quantitative agreement between the amplitude and the spectrum of the underlying turbulence there. This is at a mid-radius location of the plasma. If you move out to the edge of the plasma there is somewhat of an anomaly. We found that there was almost a factor of 10 difference. And at this point this is still an unresolved issue. It might be a code issue. There might be something to do with the comparisons but we don't understand why there's such a difference. So there are some significant outstanding problems. Some other codes have done this and they show less of a discrepancy between the measured turbulence and the calculated but it seems to be important and we are working to try to understand that. So then let me just summarize and conclude here. So I hope I've demonstrated to you that plasma turbulence and the resulting cross-field transport is a very important phenomena in magnetically confined plasmas. It drives transport of particles, energy and momentum. Generally this is deleterious to plasma confinement although it can have some benefits such as the transport of impurity particles and other things that you don't want in the core plasma. We've developed many measurement techniques to investigate the properties and it's again largely dimensional in nature which is kind of convenient and gives us a lot of mathematical and physical analogs to like atmospheric turbulence. The turbulence obeys the theoretically predicted spatial temporal scaling with respect to the ion gyro radius which is sort of a fundamental driving parameter here and there's a few beneficial impacts like I showed you it does drive a self-stabilizing or saturating normal flows and it actually helps to trigger a phenomenon that actually improves the confinement of the plasma quite a bit which is kind of a subject of a whole different talk and at least our computational simulations are providing reasonable comparisons although not in all cases as I showed you which are helping to reinforce our predictive capability for the performance of burning plasma reactors but there are many outstanding problems how the turbulence saturates how different fields look and so there's a lot more to do in this area. So I appreciate your time and be happy to answer any questions you might have.