 The first speaker is Professor Gekelman. Yeah. Yes. He's going to present Ohm's Law and the Collision of Magnetic Flux Robes. OK, thank you. This is experimental paper. And so everything that you're going to see is data, experimental data. It's not a simulation. These are my colleagues. I won't read all the names that worked on this with me. And first, I might mention what's a magnetic flux rope? Many may know. But essentially, magnetic flux ropes are bundles of helical twisted magnetic fields. And in the center of the rope, how you have this blue line, the field's nearly straight. It becomes more twisted as you go out. And then, of course, that magnetic field has to be accompanied by a current. And so when the current ends, then the field goes away. There are many, many instances of flux ropes in nature. I'm not even going to begin to tell you what they all are. But people are fairly certain that these things which are coronal ropes, twisted coronal loops, are magnetic flux ropes. They have currents. They're huge. The sun is absolutely covered with them. And there's a photo of the sun in UV light. And there's a picture, a little dot, that would be the size of the Earth compared to these flux ropes. So what I'm going to talk about is the ropes themselves. Because they're currents, and they're in a background field. There are forces on these currents. So they twist, they ride, and they can be kink unstable. And in fact, in this experiment, the ropes are kink unstable. I'll tell you what I mean by that. We're talking about two magnetic ropes. And in this experiment, we can force them to collide periodically when they do. Some magnetic energy is destroyed in a process that's called reconnection. I'll mention something called the quasi separatrix layer briefly, which essentially tells you where the reconnection is happening. The main thrust of this is we've measured every single term in Ohm's law to calculate the plasma resistivity. And what we found is that you can't use Ohm's law. It's non-local. Then we used something called the Kubo resistivity, which comes from the fluctuation dissipation theorem. And that actually allowed us to arrive at what the resistivity of the plasma is in space. And then I'm also going to talk about, I threw this in at the end based on some of the talks I listened to, the ropes are chaotic. So I'm going to talk about something called the entropy and complexity of these ropes. So this is a large plasma device. We've seen a couple pictures of it before. The one things I'll point out that maybe haven't been mentioned is it has 450 access ports, many of them with valves. So there's an enormous amount of locations where you can put probes and spectrometers and so on on the machine. It's also a user facility. It's funded by the Department of Energy mostly and National Science Foundation. And half of the time on the machine is given away to users. And anybody can become a user, including everybody in the room here. All you have to do is come up with a great idea and we'll do the experiment. If you're a theoretician, that's OK. As long as you're actively involved with the experiment, we'll work with you and do it. Use of the machine is free. So how does a machine work? Well, there are two plasma sources. On the left there, and then there's a picture of it down here, is a barium oxide cathode at 60 centimeters across. And it makes a plasma 60 centimeters in diameter. There are magnets, 100 of them, that go the length of the machine. And what I did was foreshorten this. Looks like there's only a few of those purple magnets. There's actually like 66 of them to show you just the geometry. So on one end you have this anode and cathode. Now how do we make the flux ropes? You need very strong currents that have their own azimuthal field. So on the left over here, shown schematically, we have a cathode, 20 centimeters across, made of a material called lanthanum hexaboride, which has an enormous emissivity. And so we can get very large current densities out of it. We then put a mask in front of it with two holes, one for each flux rope. And then 11 meters away we have an anode. So the background plasma has no current flowing through it. The cathode and anode are very close to each other. So it's very quiescent background plasma. But then we make currents of these two ropes done with a transistor switch. This is all experiment that I'm going to show you. So we have a whole variety of probes. These B dot probes are actually three-axis differentially wound magnetic field probes. The whole thing's just a few millimeters in size. And they move by stepping motors on probe drives. So we actually measure the magnetic field, the flow, with a Mach probe, the plasma potential with an emissive probe at about 50,000 locations in the machine versus time. So we were running the machine pulses once a second, 24 hours a day. This took about six weeks to do. And so how do the experiment work? We turn on the background plasma. That's the current versus time from that barium oxide cathode once a second. And then sometime after the plasma settled down, it's in steady state, we switch the rope currents on for a few milliseconds. So during that purple line over there, these are just some of the parameters of the plasma. Solunculus number S can be as high as 10 to the sixth. And for this experiment, the background field was 330 gauss. You can go to 2 kilo gauss in the machine, but it doesn't work well for flux ropes. So how do we take data? This is one component of magnetic field versus time. Now the flux ropes, I'll show you in another slide, are kink unstable. And so they're going to start kinking. And that's when they smash into each other to give you reconnection. But the time that the kink starts varies by 100 microseconds or so every shot. You can't control when it starts. At least we didn't for these two ropes. So if you just took 10 or 15 shots and averaged them, they'd average to a very small number, which wouldn't be the field. So we used something called a conditional trigger for all of this. Essentially correlated. We took one magnetic probe, which was sitting at the edge of one of the flux ropes throughout the entire experiment for six weeks. And it recorded the magnetic field versus time. When the thing starts kinking, you get the sinusoidal signal in the magnetic field. So we use t equals 0, not at the time when we turn on the ropes, but at the time when the ropes start moving. When you do that, you can do a beautiful average. So this is the average over the same number of shots. So this is data. This shows you what the flux ropes look like. On the right, the current density from the two spots in front of the cathode, current is about 5 amps per square centimeter carried by electrons. The blue are magnetic field lines. So this is all measured. So the field lines are derived from the measured magnetic field. I just colored them blue and red to make it easier to see. If you stare at this, you could see the ropes twist about each other, about themselves, like if you wrap up a towel. They twist around each other from the J-course B force and then on the left over here is a place where they collide. So this is shown again and again. This is data acquired at 50,000 locations. If you look at the currents, which we get the curl of B to get J, it's a little easier to see how the ropes kink. So I'm going to show you these currents versus time. Now notice the up-down direction is about 20 centimeters, 25 centimeters. But if you go from right to left, we're going out in this picture to seven meters nearly. So this thing's highly compressed. So these are the rope currents as a function of time. You can see them twisting up on the right. Then the plasma kinks, the ropes collide, the currents stretch out and unravel. And in fact, this happens again and again, this exact same thing for every oscillation you saw in the B field. So what we did was we made the ropes kink unstable. I think that's the next slide. If you have two ropes, there is a threshold current called a kink current. Which depends on the magnetic field, the length of the rope. R is the radius of one of the ropes. And so if you exceed this kink current, which we do by a factor of three, then the ropes are kink unstable. Now if we can go way higher than this, if you go to very large currents, the ropes are kink unstable, but they become so chaotic and horrible that you really can't measure anything. So we adjusted them so that the collisions between the ropes would occur again and again and again and again. And each collision was more or less the same. The ropes have a dispersion relation, derived by Rutov. And this dispersion relation, if you calculate the frequency, predicted frequency of a ropes, it's something between five and eight kilohertz and the experimentally observed frequency of the ropes. And you'll see that everything is oscillating at about five kilohertz. So this is a magnetic field. This is all data. But only the perpendicular magnetic field. At one plane, very, very close to where the ropes are born. So you can clearly see them. And what happens is when they kink, the two ropes smash into each other. And I just drew two little arrows in to show you that when those two ropes collide, you have magnetic fields in opposite direction being forced together. And that's when you get free connection. This is, since we can follow the field lines, we can also follow them in space and in time so we can just look at two field lines and their separation as a function of distance going away. So when you're really close to where the ropes are born, on the bottom, they're not moving very much. The further away you get, these are all offset for clarity, the ropes eventually start moving so much back and forth across the magnetic field that they smash into each other. Okay? You can talk about something called the squashing factor. That is, if you have reconnection and you follow two field lines, there's no reconnection, the field lines will go like this and that. The separation between them will more or less be the same. But if they do reconnect, there'll be a point at which they suddenly may jump apart. So the field lines will rapidly diverge and this q is a measure of that. So q, that q there, the smallest value you can get is four. So if q is much, much greater than that, then one suspects there's reconnection. So this is data, again, this is about nine meters from the top to the bottom but only 25 centimeters across. This is the current at what instant of time those candy stripe things to guide your eye and that blue magenta curve is the quasi-separatrix layer, q 100. So this is really large. So you know the field lines are diverging and that quasi-separatrix layer itself is a flux surface. So it's B times A and it doesn't change and what I've drawn in it and it's hard to see because the lighting is pretty bad in here but the field lines which are colored gray get further and further apart as you go from the bottom to the top of the picture. This is what this looks like if you're viewing this quasi-separatrix. Same picture, just looking at it at different angles. So this quasi-separatrix layer isn't just a flat surface but it's a twisted warp thing and you can see the field lines get further and further apart as you go up. The reconnection rate, this is the definition of it essentially it's an induced voltage from Faraday's law. When you annihilate B field, you get an electric field and then E dot dl is the electric field integrated along a field line. I'll tell you in a minute how we get the electric field but if you do this integration along field lines you can get the reconnection rate. So that's shown on the left there and the reconnection rate is five volts and you could see just in the center that bright line where you have a large reconnection rate and then shown on the right it's just completely superimposed this is a plane of the log of Q, the QSL so 10 to the fifth is really big and the two are identical the geometries of these two things. So Ohm's law. For a long time I've been interested in what the plasma resistivity is is the plasma resistivity not classical and why if it isn't and so I thought the best way to do this is use Ohm's law, this is the MHD Ohm's law so we have to measure all the terms in this except for the resistivity which we don't know and so it turns out that the turn on the left hand side dj dt M over any squared if you put the numbers in that term is so small you don't have to worry about it it's orders of magnitude small than everybody else but all the other terms have to be measured you can't neglect any of them so this is how we measure it we measure the electron temperature everywhere and this is three meters away from where the ropes are born so Z equals zero is where the ropes are born this is at a certain time later and these are contours of electron temperature so it goes up to 12 EV in the middle of the ropes the background plasma from the barium oxide cathode is has about a four EV temperature it's much cooler that's the blue over there and the density which it goes to three times ten to the 12th in the middle of the ropes and turns out that the temperature and the density are contours of constant current look exactly the same right which makes sense because the current in the rope is large the temperature is going to be higher the ropes ionized so the ropes are fully ionized the background plasma is about 50% ionized and so if you have we need NKT that's one of the terms in Ohm's law so the ion temperature is very low about one EV and so this is essentially the pressure NKT so on the bottom is where the ropes are born this goes out to eight meters in this diagram and these are contours of constant pressure you could see the pressure is higher closer to where the ropes are born and get smaller as you go away the ropes actually eventually overlap and mix with each other and very very far away the plasma is cooler so you have a pressure gradient along and across the magnetic field this is the three-dimensional flow which is ion flow at one instant of time and you have this very I just started following a few flow lines so in magenta if you follow them all you won't see a thing but the flow is very very complicated the arrows over there are the flow arrows of flow and you could see if you just look at the arrows they're tiny on the right and they get bigger and bigger as you go down the machine that's because of the pressure gradient the pressure is higher on the right so it's pushing the ions away and so they're picking up energy and they're going faster when you get to the left but their actual motion is really complicated the ions actually meander back and forth from rope to rope so they don't gyrate around the ropes but if you look at the Mach number which is what the Mach probe measures which is the ratio of the velocity to the sound speed and since, by the way, we know the sound speed at every location because we know TE so this is whenever we calculate M it's based on the local sound speed but anyway you could see that the flow or the Mach number oscillates at five kilohertz exactly the rope oscillation frequency this is the electric field which we measured it has two turns, grad phi and the ADT the grad phi we get from an emissive probe which is a hot probe, plasma is dense so we have to make a probe out of cerium hexaboride that's a whole other story but we can make it hot enough to emit enough to measure the potential and the ADT we can calculate the vector potential from the 3D fields and currents so that's a derived quantity it turns out that the ADT is about 1 tenth of grad phi that the potential gradients dominate everything and these arrows show you they're pointing inward the electric field points inward so the ropes are negatively charged you might expect that they have tons of electron current and so close by you can actually see the two ropes in the electric field and you get further and further away the ropes get closer together and merge, twist around each other and you can't see the separation between them and it's too bright in here so you can't see what's happening at six meters this is the J cross B force so we have the current from the curly B the background field dominates though we can add the rope field into this as well and so very close to where the rope are born J cross B points inward just like the electric field further away it looks like the picture of the electric field it points inward toward the ropes but grad P, one over any grad P these are all voltages points outward so the grad P in the perpendicular direction has the opposite sign of J cross B and in fact they nearly cancel so what's left, oh I guess I didn't show it when you take all those terms and ohms a lot and add them up the thing that really dominates everything is grad phi gradient in the plasma potential the space charge, the electrostatic field the biggest thing of all all right so next what I did was put it into ohms law and calculated, I did this 10 different ways and I totally don't have the time to explain but I'd be happy to speak to anyone who wants to know all the various ways I tried to do it but one way I thought was to calculate the resistivity along field lines so I integrated E dot D L this is the total field with all those terms you saw along field lines divided by J E over J is the resistivity and then divided that by the local spitzer resistivity so you get the resistivity 50 or whatever means 50 times spitzer so here at this particular time resistivity is all positive and large and seems to be biggest between the ropes and on the gradients of the current but then at another time the parts of the resistivity go negative right so how can that be either there's some exotic dynamo which we've never seen and or large currents flowing the opposite direction which we never saw or there's something wrong with our measurement but we checked and double-checked for a year before doing this so I began to think maybe we can't use Ohm's law so instead we found this paper by Jacobson and Moses who talked about Ohm's law has the possibility of being non-local essentially that means when this is non-local you either have to derive a new form of Ohm's law but you can't use the one I just used verboten all right so that's the criteria if alpha is greater than one it's non-local that means that you're getting contributions to the local resistivity from large distances away in the plasma as these field lines flap around so we have all the terms in that equation so we calculate alpha and this is the calculation to guide your eye these are the two ropes red and blue and then these are surfaces of constant alpha so the red surface in the middle there is 12 that's bigger than one there's a surface alpha is three another one alpha is equal eight three nested surfaces if you average it over the entire volume it's one and a half so Jacobson's law seems to hold true alpha is greater than one you have to give up and you can't use Ohm's law so what do you do all right well found a paper saying that you can calculate from the fluctuation dissipation theorem it's a famous paper by Kubo 1966 using velocity correlations you can calculate a conductivity ac conductivity but to us that's perfect because the flux ropes are oscillating at five kilohertz so all we have to do is omega there corresponds to five kilohertz okay and we can calculate these terms uh... and I can show you later if you're interested how I did it but it's a correlation in velocity at two different positions uh... and uh... we used v from the current this is how we got the velocity essentially took the current divided by the local density times the charge and use that in that expression and then uh... for the z component of the resistivity I just took this is a tensor right because new and you go from one to three so uh... this resistivity along the field is one over sigma uh... just to show you these are all the components of that conductivity temp sir but the dominant one is the sigma two two and if you take the one over the sigma two two uh... this is not time-dependent remember it's five kilohertz but we can see what it looks like in space so what it looks like in space this is three meters away from the birth of the ropes as you move away while these uh... components change but you can then uh... get the resistivity and these are uh... three surfaces of constant resistivity and that quasi-separatrix layer where reconnections going on is drawn in and so first of all you see it turns out that on the right and left you see these big mountains of high resistivity twenty or fourteen times the spitzer value and in the middle same thing the middle spike in resistivity we associate with the reconnection because it's right on top of the quasi-separatrix layer the other anomalous resistivity regions are just where the currents are there's three regions the resistivity is anomalous in the currents uh... and this is just shown in contour maps uh... it can be up to thirty eight forty times spitzer in uh... right in the middle of the quasi-separatrix layer and then uh... you could see the currents on top and bottom eventually the re... the reconnection layer fades out because the reconnection is only really happening around four meters away three to four meters away and eventually all that's left of the high resistivity due to the currents so from the magnetic helicity here's an expression for relative helicity uh... this was just published uh... you can calculate the dissipation and transport actually the expression we use is a little different than this uh... we included flow density temperature and electrostatic field and what we found is uh... use this expression the arrows are the transport of helicity into the quasi-separatrix layer see that q equals one hundred is one of the contours of q and the hours are going in and then the dissipation is shown in color and you could see inside the quasi-separatrix layer uh... is where all the dissipation is happening and if you plot the dissipation and helicity flux into the quasi-separatrix layer they balance perfectly from this uh... we predicted and then looked in and at the magnetic fields themselves that uh... only point one gauss of the magnetic field is destroyed the flux rope have magnetic fields of up to thirty gauss so a small fraction is destroyed but that was enough to raise the electron temperature by one of the in the quasi-separatrix layer uh... and so uh... i'm not going to read this out but it turns out that when the ropes are colliding from this helicity uh... measurement then all of uh... all of uh... the helicity goes into raising the temperature as i just mentioned that when the ropes move apart there's also change in helicity and it turns out all of that goes into augmenting the flows around the rope so i'm going to talk about this which is uh... the other side of the coin complexity all right uh... one of the speakers mentioned the permutation entropy yesterday uh... i don't think he's here but uh... you can you can calculate the entropy of the time series and there's there's an enormous number of papers on this and how do you entropy is a measure of disorder and so uh... how do you do that you take the time series and break it into bins so there could be typically five seven ten numbers in a bin so for example suppose you would have been with five numbers what you do is just count of those five uh... data points which is the biggest which is the next biggest which is the next biggest right and you get an ordering so for these five points i randomly chose point one is the largest point two is the next to largest point uh... five is the third largest and so on uh... but since there are five bins there's five factorial uh... possible combinations so then you go through the entire data stream and you ask how many times is this p one p two b five p four happen right so turns out just for this one example happens thirty times we're a thousand data points uh... so that's a probability thirty over a thousand and it's a thousand minus the five uh... and this goes into this quantity called permutation entropy so it's a measure of disorder there's another thing called the jensen shannon complexity which you may or may not have heard of i never heard of it until a couple years ago one of my colleagues discovered it the complexity is a funk is this complicated function of uh... the entropy s of p which we just talked about p e is the maximum entropy state it turns out that maximum entropy states of no-brainer to calculate if you have a thousand uh... numbers then if uh... every single data value is one over a thousand then that's maximum entropy total disorder and this is a combination of them when you do this you get something called a complexity entropy diagram you say what the hell is this good for uh... entropy by the way is normalized from zero to one the complexity uh... is always less than one but it turns out that uh... it was proven but by these mathematical people in statistics there are two curves called the minimum and maximum complexity curve and anything any anything at all has to lie between these two curves your data never be outside them and so i just show a few things you could take all these maps logistic map and map any of them they'll be uh... somewhere in there a sine wave is a point all the way on the left white noise if you just put it random numbers random numbers have very high entropy but no complexity so they're all the way on the right of that thing uh... but it turns out that on the top of the arch oh and fb is fractional baronian motion okay so on the toward the top of the curve uh... if you have data there that means uh... that you it's very chaotic because the complexities high so very chaotic stuff means complexity very high and entropy sort of in the middle of the road like point five normalized entropy also what these guys discovered is that uh... if your data is like the hennan map or whatever and it's up there in the region uh... we have high complexity the process that led to it being there is not run of the mill random stuff but it's deterministic chaos that means somewhere there's a set of equations which you probably never can find lead to this chaotic structure so a lot of these maps that you that you can uh... that lead to chaos you put them on this diagram they're all up there so what is our that's deterministic chaos region the chaos on the bottom ran is just random to completely random non-deterministic stuff like white noise this is our data this is uh... flux rope in the present of the background alpha and white and uh... there we are highly highly chaotic it has this uh... beautiful little hook there uh... there's several maps that i drew in the hennan logical map lorenz 3d map you can put in anything you like all right but if you're up there where it's read it's highly chaotic but the chaotic chaos is deterministic even though you may not know why and what we did to plot this there's actually fifty forty eight four hundred thousand dots each one of them is a magnetic field versus time and the thing is here uh... we did not conditionally average the data so when you conditionally average it you get this beautiful uh... sine wave kind of thing i'm just about done uh... but we didn't conditionally average it so essentially what we're seeing is this shows the chaos but all the field line pictures i showed you before show you the recurrent part the part you can average this stuff up there we cannot see this is what this looks like this develops in space so these are the minimum and maxity complexity curves and this is uh... how the complexity varies as you move away from the birth of the ropes the rope motion gets more and more complex as they go whipping around and uh... you get this large uh... hook in the complexity diagram this is what it looks like in three days the last thing i want to say i know uh... he's pointing at me everybody shows a spectrum okay well here is the spectrum for the uh... this is uh... power spectrum for the flux ropes magnetic field it's not it does not have a power law dependence it's exponential in fact we see exponential spectrum in plasmas all the time they're more common than power law spectrum the spikes you see are out there waves on the right over there and uh... the magnetic field those are the flux ropes all those spikes are due to the motion of the ropes on the left so of course they they're not they don't fall on the exponential spectrum but as if you take those things out perfectly exponential because what uh... there are if you look carefully at the magnetic field data uh... before it's conditionally averaged there are larency and spikes structures in the plasma and you can fit them to larency and function and once you have larency and functions uh... you get an exponential spectrum they go together okay i'll stop i'm over