 OK, so I'm going to be talking today about a topic which is really intimately connected with turbulent mixing in this sort of theme of this conference. And I'm going to try to show you some very unique experiments we're trying to do in plasmas where we can study thermal mixing, for example, very controlled conditions. And I want to acknowledge my collaborators, a couple of graduate students that are involved in the project from my institution, Scott Karpiszewski and Matt Paulus from UCLA. And we're also working together with Professor Morales and Dr. Juan Campernello. So first of all, I'm just going to try to motivate this by trying to explain what we mean by filaments, plasma filaments. Magnetize, we'll be looking at magnetized plasmas. So the filaments will be in the regime where charged particles in the filaments are basically guided by magnetic fields and involving electric fields. And so I'll try to go through some of the basic physical processes, the transport physics. Essentially, we're trying to understand transition from classical transport to anomalous or transport driven by collective processes. And then we'll try to go into the new regimes we're trying to access. And that's situations where you can have actually filament-filament interaction. And we can build structures that are basically complex, can generate quite complex patterns. And we can study, for example, things like chaotic heat flow in quite a bit of detail. So here's the working definition of a plasma filament, is that it's a coherent mesoscale plasma structure that's strongly aligned to a magnetic field. And so mesoscale here would be plasma, magnetized plasma typically has scales like the gyratius of the electrons, say gyratius of the ions, collisionless skin depth, CO omega p for the species. And so we'll be studying mesoscale structure so that we're talking about plasma filament structures that are sort of on the order of the collisionless skin depth or gyratius of the ions or a little bit larger. So that's what we call mesoscale. Microscale would be subscales below the electron inertial, the CO omega p collisionless skin depth scale. So filamentry structures are typically observed in boundary layers of plasmas. As George McGee was explaining, that you have core turbulence, for example, but you also have boundary turbulence. And some of the boundary turbulence observed in magnetically confined plasmas contain these filamentry structures, which I'm going to be talking about. And of course, filamentry structures are found in many plasma environments, for example, stellar atmospheres. I'll try to show you some example. And so we would like to basically study the transport physics associated with filaments because they can often carry large amounts of heat and particles, and their transport theory is not just simply dictated by a fixed law. So the transport theory for these could be much more involved and involves non-local effects. So just through a couple of examples of the types of filamentry structures we're kind of interested in studying, why are we studying magnetized plasma filaments? So I just pulled out of the literature just a very couple examples that these structures have been imaged in quite a bit of detail with very, very high-speed cameras, for example. So here's an example of a magnetized toroidal plasma more spherical-like plasma that is at the Princeton Plasma Physics Laboratory called NSTX. And there's a similar toroidal device called Mast in the UK, which has very similar characteristics. And both of these are experiments that are studying filamentry structures in particular and the transport associated with them. So that's why I kind of use those as examples. So for example, here's a high-speed video image of one of these sort of spherical-like plasmas which evolves and then there's a formation of so-called ELMS, edge localized mode. So these are kind of occurring near the boundary of the plasma. And you can see this filamentry structure kind of winding around the plasma device sort of toroidally. And here's kind of another picture of it, sort of the time evolution sequence of the development of these filamentry structures. And then here is some modeling of those using fluid models to describe sort of the characteristics, the length scale. As they wind around, many times around the device, these things could be many meters long, for example. And they're fairly highly aligned with the magnetic field as I was trying to explain earlier. And here's another example from the NSTX Plasma Princeton. For example, they can image the formation of the filaments at the boundary layer of the plasma. And so you can see a time sequence here of the development of it. They call it a blob, but it's actually more what we call a blob filament because actually this thing has very elongated structure. But if you look in a plane, it looks like a blob, right? So sometimes they're called blob structures. But basically, they have a lot of dynamics. And you can imagine these blob filaments, they will be carrying heat and particles out as they sort of shed away from the boundary layer of the plasma. So these structures are quite important in terms of the overall transport of heat and particles from the edge region to the outer boundaries of the plasma. And just one more example, this is kind of connected with experiments at General Tomics. For example, you can sort of see the development here of these so-called edge localized modes, which are essentially filamentary structures in both density and temperature. So we're sort of looking at profiles across the boundary, the edge of the plasma. And you can see here sort of structures. So the elevation of the density above the background is quite significant here. So here's a filament density. And then if you look at the temperature, you also see temperature filaments as well. So you could think of this as being a lot of fluctuations that involve filamentary structures of temperature and density. And of course, we can go to the solar surface, for example. And we can see beautiful images from extreme ultraviolet light. We can see that also the plasma structures, the magnetic arches, have very kind of filamentate structures. And these things are very dynamic. And the main thing here is that the luminosity along any of these arches is very non-uniform. So there's a lot of possibilities that the distribution of the heat and the particles is very non-uniform in the magnetic arches, for example. So these are just some examples of filamentary structures. And here in this case, there'll be multiple filaments interacting. OK, so what we want to do is take that sort of background of filamentary structures in the magnetized plasma. And we want to see if we can actually isolate filaments. So we would like to make the filament in a linear device so we don't have the effects of curvature as we have in the toroidal plasma. So we want to make long filaments, linear filaments. And just study, for example, the fluctuations and waves internal to those filaments and then see how we can characterize the transport physics and also the modes. Because you can think of filaments as sort of like an antenna. If you can actually develop modes, internal modes through gradients, for example, this thing will be radiating. We want to know the near field of this filament antenna, if you will, and also some of the far field effects. So to see how much non-locality effects come into play. So here's a cartoon of what we want to set up. And it has been studied in the lab since the late 1990s, early 2000s, in the large plasma device at UCLA, which Walter Guckman gave an excellent introduction to. So what we'd like to do is create a filament that's far away from the boundaries. And also the filament should be isolated. And so it should not be in contact with, say, the anode where you're sort of biasing. If this is your cathode source, we would like to have the filament have a finite length, for example. So what we're attempting to do is basically create a filament which has an elevated temperature above a background. So typically we work with a background plasma which is cold, like a quarter of an electron volt. So that's almost the energies that you'd find in a solid, for example. And then we would have a heated region which is about, say, 20 times larger so we get a very good contrast. And this filament would have the very large gradients if we could localize it in space. And then we could study, for example, the normal modes associated with that filament. So here we study in plasmas which have a density of like 10 to the 12 particles per cubic centimeter. And this is, say, studied helium, but we could study these in hydrogen or other plasmas. And so this is the kind of thing that we set up. So what we first need is a heat source, a good heat source that's very controllable. And so what we're gonna show you now is basically how we actually make the filament from a particular heat source which provides us with a kind of electron beam. And if the plasma is of this density, the mean free path of the particles, electrons, for example, it's rather short. It's about like 10 to 15 centimeters. But this filament is like 10 meters long. So what'll happen is if we have a beam launched into that plasma, the beam will slow down and thermalize and the energy will basically be transported classically along the field line if there's no like a wave processes. And so then we'll get a very nice sort of, very clean sort of temperature filament. This is what we're attempting to do here. So just the pure energy transport. So the gradients of density will be very, very mild and we'll have just everything from, coming from mostly the thermal components. Okay, so Walter has already explained the large plasma device and others in Chris Neiman's talk as well. So we do the experiments here in the, and length here is very important. And this motivates, this work will motivate the reason why we need to do this in a very long chamber. You think, well, why don't we just do this in a very small plasma? Why do we have to have such 18 meter long plasmas? So I'll explain that in a moment why we do need that very, the length and also the very large diameter. So the main plasmas created in this barium oxide cathode and it's pulsed every one second. And plasmas are again magnetized, so one kilo gauss magnetic field, as I told you the density. And the electron, the beta, the ratio of electron, the thermal pressure to the magnetic pressure is rather small, it's like 10 to the minus four. So it's considered low beta plasma. And we work in a particular phase of that discharge. So when the plasma is turned on, that barium oxide source is turned on, you have this background plasma. And then the main plasma source is turned off and then what happens is you get what's called an afterglow. And in this afterglow region is the region we wanna work in because we can then turn on a heat source in this afterglow region when the plasma is very cold. So here's the kind of profile of after the main plasma source is turned off, then there's a slow decay of the density and almost no decay, well very little decay of the temperature, okay? And so we have basically about 15 milliseconds of time to actually put in a heat source and study the waves in which are processes which are occurring on microsecond time scales, for example. So this is the important point. And so we get basically this one quarter of an electron volt. This is electron temperature versus time in units of electron volts, so it's here. And we also have to pay attention that the density is changing over the time scale, but we want to work over time scales where the density changes is rather modest. Okay, so the heat source. So now we've got our background plasma that's really cold and now we have to make it warm, heat it up. And so what we use, so there are a couple of different ways we can introduce the heat source. So the heat source, we could make very small heat sources using these so-called Lantium hexaborate crystals which give us a heat source that's a scale of about a few millimeters. And then we could also use another larger, say 10 centimeter diameter Lantium hexaborate source to actually make filaments that are actually much larger in scale. But we're gonna talk, mostly I'll talk today about the small, very localized heat source. And so what we do is we introduce this crystal into the plasma at some region and then we bias it relative to some anode. Now, Walter described flux rope experiments. Now those biases are like 100 plus 200, or 100 to 200 volt biases. So here we want to make the filament a finite length so we have to do low biasing and the biases on this crystals are typically like about 10 to 20 volts in that range. If you bias above 20 volts, then you start ionizing helium and so then you have to worry about the density changes in the discharge. Okay, so when you actually introduce that Lantium hexaborate crystal as your heat source, again, as this explained the thermalization, then you get this nice kind of thermal filament and I'll explain like how we can control the length. But basically here's the, this looks like a kind of Gaussian looking thermal profile. Here's the, this is taken at a plane, a few meters away from the source. And so in a plane, we kind of use probes to kind of create a map of the saturation current and saturation current is actually proportional to density and square root of electron temperature. But the density here, we can separate these quantities and actually this, when I show these maps, you can think of these as almost purely coming from the temperature, not so significantly varying from, the density's almost a constant over this size of the filament. So here's a nice sort of localized Gaussian filament that we, and these kind of experiments were like as I said done before and we're building upon them. Now the other way we can make larger filaments is to actually use a larger land theme hexabyte source. So this is a disc about 10 centimeters or so in size. And so that can actually be used to make temperature filaments where we're working more in the centimeter scale, not the millimeter scale as I've been explaining. And so the nice thing about this technique with masks, with this larger source is you can put a mask in front of it and then you can actually make controlled sort of thermal structures. So for example, I'm gonna show you here. And the result of you put a mask and you drill three holes and then you can vary the spacing so you can make multiple masks. And here you can see that we can form three filaments. And this kind of idea came from Walter Gecklman's experiments in flux ropes where they use a similar technique to create multiple flux ropes. Walter showed some nice examples of two flux ropes, but he's done three flux ropes. And so here the mask method, we can drill an arbitrary number of holes, and well, not arbitrary, but some finite number of holes and we can actually make dozens of filaments. Of course, depending on the size of the hole, the size of the filament can be like small scale or larger scale. So this is another second method that we can use to create filament structures. And we also built masks where we can actually make a ring shape heat source. So this is some work that we did with Bart and creating a ring shape to hollow profile of the temperature. And so we were studying various kinds. We observed very interesting phenomena associated with intermittent transport and also avalanche type thermal avalanches. So I won't go into those, but I'll just to show you the capability that you can make more or less designer filaments. Okay, so previous work was using these, localized like these Lanty Maxwell crystals. They were used previously for studying fundamental transitions from classical collisional transport in a plasma to anomalous transport where we had from the gradients, they drive the fluctuations in the waves and those would be in addition to the classical effects. And so the other element of this work is that if you have like these steep temperature or pressure gradients in the plasma, you can actually drive alphane, shirofane modes. And just to remind you, shirofane mode is basically a perturbation that is perpendicular to the main magnetic field. So it's driven by a parallel currents, but these induce perturbations in the magnetic field perpendicular to the main field. This is so-called line bending perturbation, which propagates just like when you pluck a guitar string, you can think of that as a shirofane mode equivalent in a plasma. So yeah, so this is the kind of physics we've been investigating with these experiments. So first of all, let me just explain to you a single filament and then we'll just discuss the situation of multiple filaments. So if you have a single filament, how can you sort of characterize it through sort of classical transport theory? So basically filament length control is determined by the ratio of the perpendicular heat diffusivity and to the parallel heat diffusivity. Obviously in a magnetized plasma, the parallel diffusivity along the field is much higher than the perpendicular. Just in classical processes where we have just say electron ion collisions or some collisional transport is dominant. So basically the length of the filament, you can control it mainly by adjusting these perpendicular and parallel transport coefficients, those collisional ones. And so if you just sort of go into any standard textbook description of transport in a magnetized plasma with just collisions, electron ion collisions dominantly, you see how they scale, how these parallel and perpendicular transport coefficients scale. So if you just go down to the bottom here, the parallel thermal diffusivity scales roughly like temperature to the five halves power. However, the perpendicular component because you have electrons are magnetized, they scale like the inverse of the magnetic field squared and the electron temperature square root. So based on these scalings, you can then organize the ratio of this kappa parallel over kappa perp. And then that's how you can then work out what the length of the filament will be, right? By just adjusting the magnetic field and electron temperatures, that's how we actually get the length. That's why the length is 10 meters in the way we designed the parameters for those five electron volts, for example, case. So that's really, this works very, very well. We could predict very accurately from classical theory at least the length of a filament. And so here's kind of, and then that's just a simple heat equation, but if you couple it to the continuity equation, so equation for density and momentum equation, and if you include the heat source here, and so these are called so-called Brighinsky equations, and if you sort of solve those equations, you can actually predict the length based on the properties of your heat source. So for example, early time and then late time, this is your steady state filament. So in this case, this is an eight meter long filament, which goes along with those sort of predictions back to the envelope. Okay, now that's the classical picture. Now the wave picture, you have basically fluctuations that develop from the steep plasma, the temperature gradients. And so here's a kind of early work that characterize this transport from classical when the filament is getting set up, and then when the gradients are sufficiently steep. So here's a plot of like the electron temperature, right? Here's the filament, and if you take the profile, the radial yet, as with the average profile, you see here, it's got a very steep gradient here, and that drives fluctuations, like really large, and these are on the order of 20, 30, 40 kilohertz type fluctuations, and they start, of course, diffusing, they contribute to the perpendicular, so anomalous thermal diffusivity. And so here's kind of now point measurements. So there's actually several modes that are very important here. So this, if you actually are sitting just at the region where the gradient is maximum, you get these M equal one type modes, that's this red here, and these frequencies are on order of 30 kilohertz or so. And then actually at the very center of the filament, you get these lower frequency oscillations, which are about five to eight kilohertz. And those are essentially, the filament acts like a thermal resonator. So basically you have sound waves, which are basically trapped in this thermal filament, and then this creates this low frequency oscillation, which actually we're still trying to understand. We don't have a full picture of how that works, but we, yeah, it's easily observed in experiments. And then outside the filament, you get these spikes, which are actually sort of Lorentzian pulses, which Walter had talked a bit at the end toward the end of his talk. And so that shows that these are kind of far field effects, because we're kind of now as you move out, you can actually see some of these pulses. And so when we put multiple filaments, like film is near each other, then these interaction of Lorentzian pulses is kind of what we're trying to understand. So just a kind of quick physical picture of this electron temperature instability. So essentially when you have a temperature gradient, so here's one filament with a thermal gradient, and here's a magnetic field. So what happens is that you have a perturbation. So this is the shaded part is hot, and this is cold. And so you get basically electric potential fluctuation, which is in phase with the electron temperature fluctuation. And so you get this kind of electron diamagnetic drift effect. But then if the phase relation between potential and temperature is not, if it's finite, then from coupling to the parallel dynamics, you can actually get this perturbation to amplify. This will grow and this will decrease here. So you get basically kind of like an interchange-like type of mode, and then this is the origin of this drift-alphaene fluctuation. So parallel dynamics couples to this perpendicular dynamics. And if you're a fluid theorist or experimentalist, you're probably familiar with the Peckley number. So Peckley numbers for the kind of plasmas, which is the rate of advection to heat diffusion. So if you kind of use the E cross B motion, that's how the plasma really moves in a perpendicular direction to the magnetic field. This Peckley number scales something like the collisionality times the fluctuation. And so typically these are around three to 30. So we're at a greater than one Peckley number, where advection kind of dominates. So I'll just kind of go to, yeah. So basically we've been able to successfully model these temperature gradient instabilities. And I'll just kind of go to sort of the simulations. These are basically simulations based on the so-called gyro-kinetic equations, which gyro-kinetic description of plasma is like an E cross B, each charged particle is sort of E cross B drifting. And we solve this kinetically with millions of particles. And here's a cross section of the electric potential which develops in time. And we see the m equal one mode pattern consistent with what we see in the experiments. But we also see smaller scales, features, and that's connected with convective cells, which are purely, these are essentially k parallel equals zero, or actually average sort of modes that are present. So they relax the temperature profiler simulation showing the relaxation of the temperature from these temperature gradient driven modes. So I'll just go now to the multiple film and experiments that'll be, this is my ending here. So basically what we're now trying to do is do experiments where we have multiple filaments like, and we started with three. And our goal is to study sort of nonlinear convective transport and presence of fluctuating magnetic fields. If they're close enough, the field lines can become stochastic and nonlinear E cross B convection. So what we do is, now we introduce into the plasma not just one heat source, but we introduce three. So we put these crystals on a probe drive, on three probe drives, and here we can make them arbitrary close from this close to arbitrary separation. And each crystal is on a separate electronics. So basically we can bias each crystal independently and so we can say, for example, have one bias to say 15 volts and then these would be biased to say four volts, for example. So this is the view down the machine when you put the crystals on these probe drives and then we can just move them spatially in close together or far apart. And so here's, yeah, that's just a single heat source. But here if we have the three, so we started off with a far separation. So again, this is the electron temperature essentially. Each of the three crystals, they're biased the same amount in this case. So we get the five electron volt peak temperatures here and then we see that there's actually convective tail. So these surprisingly, even though these are very far apart, there are many gyro-radi or skin depth apart, we still get actually interaction between the filaments. So a single filament, if we turn these two off, a single filament would just have a nice sort of Gaussian type shape, whereas now we sort of get interaction. And so here's the fluctuations in the saturation curve essentially, fluctuation in temperature. And we see modes that develop like mode one, M equal one, M equal two modes that develop around each of these filaments. And so there's a symmetry breaking, these offer symmetry breaking perturbations. So now the mode structure around here is not the same as that structure there. And so what we see is that we put the filaments very close together, we form a very complex structure, but then there's an outer gradient that forms because the inside is relaxed very quickly and we build up an outer gradient. And so here's just, we've been modeling this with again, Bruginski transport equations and trying to study the nonlinear convective transport. So just in summary, we've made isolated magnetized electron temperature filaments, various sizes, using crystals from a few millimeters to, and then with masks on larger cathode, we can make a centimeter scale or even larger. And the close-case-case patterns, we tend to get very sort of chaotic cross-field transport enhanced E-Crosby flows and convective tails. And so what we're trying to actually do is characterize these structures as the so-called Lagrangian-coherent structures. And it's kind of interesting because the turbulent mixing, the logo is actually a Lagrangian-coherent structure where you actually see large convective tails. In this case, they're using four filaments and we're using three in a triangular pattern, if you will. So we can try to make sort of fluid analogs in these magnetized plasmas. So thanks very much for listening.