Andrew Wright - Observations & Simulations of 3D Field Lines Resonances in the Earth's Magnetosphere

Okay, thank you Jasmine and thanks to the seminar organizers for this invitation. I'm going to talk about observations and simulations of 3D field resonances in the Earth's magnetosphere. This is work that I've done in conjunction with an ISSI team. Here's a picture of the ISSI team members. This is our first meeting back in August 2019, first of three meetings that were scheduled, but of course COVID hit. So for the next two years we actually had brief virtual meetings once a year, difficult with the time differences of course. Then we finally after the lockdown finished managed to have our final two in-person meetings. The list of people here are people who've been involved with these activities. The ISSI team ran actually for four years in all because of the interruption with COVID. This actually meant we had a lot more time to complete our research and it turned out to be the most productive ISSI team I've been involved with. I really appreciate the ISSI organizers who were flexible and made the best they could out of the COVID situation and allowed us to run the team to get the best science out. So big thanks to ISSI. For the material I'm going to present today, there were several things that came out of that team, but the material I'll talk about today has big contributions I should recognize from Tom Elphic and our own Jasmine Sandu. So to orient you as to what I'll be talking about, here's a cartoon of the Earth's magnetosphere. Something that will be important is the dark red region, the plasmasphere, and the boundary of that, the plasmapause. I'll bear that in mind and come back to that in a few slides.

The waves I'm going to be talking about are the lowest frequency MHD waves of the system. We can excite these in two important ways: fast modes, which propagate across field lines and transport energy from the magnetopause into the magnetosphere. The artist's impression here is trying to show a standing fast wave. The other wave that's important is the Alfven wave. A good analogy for this is to think of the field line here as being like a string that a wave can stand on or propagate up and down. When you have a non-uniform medium, these two waves are coupled together. What you can do is imagine having an impulsive solar wind nudge. It's a bit like if you nudge a jelly on a plate, it'll reverberate with its natural frequencies. Then these natural oscillations of the fast mode of the system through cavity or waveguide modes can couple into Alfven waves where the conditions are right for that. The title of the talk talks about coupled field line resonances. If you're from a different community outside the magnetosphere, you may refer to these as resonant Alfven waves, so I use these two terms interchangeably. Perhaps I'll just pause there and ask if there's anything in that global setup that isn't clear.

No, okay. So when you have a 2D system, for example if you take the magnetosphere near the Earth to be axisymmetric — you can see it's not when you go far away from Earth, but a simple model would take the magnetosphere to be axisymmetric, so it's independent of the azimuthal coordinate. We refer to it as being two-dimensional, just depends on two coordinates: the L shell and latitudinal coordinate. The coupling of fast waves to Alfven waves in that context has been studied in great detail. It's well known that the Alfven wave you excite where you drive it with a fast frequency that matches the frequency of the Alfven wave will generate an Alfven wave that has a displacement in the azimuthal direction. The frequency of the Alfven wave depends on the field line you're on. Just as with a string, you could crudely think: if I'm close into the Earth here, I've got a short string, and as I move further out like here, I'll have a longer string, so the frequencies are going to change. Typically in most of the magnetosphere, the frequencies decrease as you move away from the Earth. The coupling I'll be talking about is a sort of resonant coupling where the frequency of the fast wave finds the field line where it matches the frequency of the Alfven wave, and you get strong resonant coupling. I've got a movie here to orient you as to some of the ways we'll find useful to think about resonant oscillations. This is taken from YouTube and it's an animation. I'll just run it and you'll see some of the properties that evolve.

Basically we've got a system of pendula of different lengths so they can have different frequencies to oscillate with. This is a view looking from the top where the short one here will oscillate quickly and the long one here will have a lower frequency. Let's have a look at this movie. In particular, notice when they start, they're all just pulled down to one side here and released, so it's not really continually driven, it's more like giving a nudge at the start and then letting it oscillate freely. Pretty patterns start to emerge. If you Google pendulum waves on YouTube, you'll see that some people have actually constructed physical systems like this out of billiard balls and you get a really nice pattern. I haven't run this too far, but they actually all come back into phase and fall back into the initial starting state if you run it long enough. The key thing to notice are one frequency here and a lower frequency here. You'll notice that if you look at a peak in this pattern that emerges, it runs from right left to right, and that phase motion is indicative of the gradient in the frequency of the different oscillators. Another way I could drive this system would be rather than give it that impulsive nudge, which we regard as a broadband driver exciting a broad range of frequencies in the pendula, another way of exciting would be to take this support point here and just oscillate it back and forth at a single frequency. What would happen then is that if I go along these different pendula and I find the one that matches the driving frequency — perhaps it was this one — then this would respond with a really large amplitude oscillation, and the ones here and here either side would be non-resonantly driven so they wouldn't have a large response. So depending how you drive the system, you can get a variety of behavior: you can get all the oscillators with all the different frequencies excited here, or you could drive it so that you just bring out a single preferred frequency determined by your driver.

Armed with those ideas, we'll sort of start to figure out the equilibrium that we're going to be looking at to see these effects of 3D. As I mentioned, if you go close into the Earth you can imagine a simple model would treat the plasmasphere and plasmapause as being axisymmetric. In that case it's 2D, but if it was non-axisymmetric then we'd have to think about wave coupling in 3D, and that's really at the heart of this talk. Here we're looking in the equatorial plane. This is image EUV observations showing the plasmapause boundary. We've got a notional dashed line here which would be a typical plasmapause boundary in quiet times, but when you have a storm things change quite dramatically. This is 8 hours of data and what happens is you can get the plasmaspheric material coming out here in what's called a plasmaspheric plume. If you imagine you were looking at the sort of field line resonance coupling that I'm going to show down here, then the system is pretty axisymmetric there, so you could get away with 2D theory, that would be a good approximation in this region. But if you're up here, well, things change in azimuth quite dramatically as you cross the plume boundaries. So if you want to study wave coupling in these sort of regions, then you're really going to need 3D theory.

To model this we've got three panels here. The first one is the Alfven speed. It's a view in the equatorial plane. Basically we put an axisymmetric plasma density on here but we've added a lot of density in the afternoon sector where you expect these plumes to form. So this is showing the Alfven speed. There's a strong depression in Alfven speed here as you're in the plume. The other panels show B parallel, so this is a compression or magnetic field or the magnetic pressure. This will be indicative of what the fast mode is doing. To identify where the Alfven waves are, we're looking at the field line vorticity. You can also use the field-aligned current to identify Alfven waves. You can note here that I've really got a very narrow range of L shells excited in this region. The equilibrium is 2D, so I'll get the familiar toroidal or azimuthally polarized Alfven waves with plasma displacement in the azimuthal direction. Things change dramatically here; we'll come to that. But just for the moment, note that this is quite narrow in L, so I'm really exciting essentially a single frequency or just a very small narrow bandwidth of Alfven waves.

This simulation result is a final snapshot from a long time evolution where we drive it monochromatically, and it feels like it's driven at a single frequency. Now of course that won't be the case for the first few cycles. If you take the Fourier transform of a sinusoidal wave at a single frequency but you only have a couple of cycles, the Fourier transform of that is not a delta function at the frequency you're driving with; it's quite a broad band centered on the driving frequency. The delta function will eventually emerge at long times. I'll play the movie. What you can see is if you look down in this region to start with, for the first few cycles you'll see a whole range of Alfven waves are excited over a broad range of L shells because the driver is broadband during the early phase, and then it becomes narrower band as you continue to drive it. Let's have a look at the movie and just pay attention down to this section here to start with. Down here you see a broad range of Alfven waves. You can see that phase motion that we saw in the pendula moving from high frequency to low frequency. It's becoming narrower and narrower in time, and eventually it's limited by dissipation just how narrow it gets. The fast mode over here is being driven — I should have said — at around local noon. We just apply a sinusoidal in time magnetic pressure perturbation. It's localized around noon and tapers off as you go towards the flank. So over this section here, it's just pushed with a magnetic pressure to cause the magnetopause to just oscillate in and out. I'll just run that again in case any features you want to see again in more detail. You got the outward phase motion, fast mode sort of standing here at local noon and propagating onto the waveguide flanks here at the sides, and it becomes narrower in time as the driver becomes more and more like a single frequency. I'm not going to say too much about the simulation details, but it's based on the cold plasma MHD equations using a field-aligned coordinate system. So we can get very fine resolution in the perpendicular directions. You don't need fine resolution along the field line, so it's useful to separate the grid into these different coordinates.

Okay, so 2D versus 3D. Over here it's locally 2D. In fact, right in the middle of the plume is going to be locally 2D. If you look in here, this is the Alfven speed again. If we're in a 2D region, then the azimuthal plasma displacement — that's what that is, the direction of the plasma displacement with the Alfven wave — and it will oscillate at the toroidal Alfven frequency. For example here, it would oscillate along tangential to these ridges, would be the plasma displacement, and it would go at the Alfven frequency. If we had a field line here, the plasma displacement would be again in the azimuthal direction at the toroidal Alfven frequency. By toroidal I just mean it's kind of another word for azimuthal, as opposed to poloidal which would be where you have plasma displacement in more of a radial-like direction.

We can see when we cross the plume boundaries here and here, then things become very different. We don't have the Alfven wave sitting on a constant L shell; it actually crosses L shells, and here it actually turns out to being almost poloidal. So if we look at this field line here and we ask what Alfven frequency does that have, well, it actually depends on the polarization or the orientation of the plasma displacement. We've mentioned a lot the toroidal direction or azimuthal direction. In this figure down here I'm plotting the Alfven frequency as a function of polarization angle. We take the convention that theta equals zero corresponds to the toroidal direction. So the frequency here just below 2.2 for field line A: if I have plasma displacement of the Alfven wave in the azimuthal direction, it will have a frequency of just under 2.2. Whereas if I take the same field line and we ask what happens if the Alfven wave oscillates in the radial or poloidal direction, well, Jim Dungey solved this back in the 60s. He wrote down the toroidal wave equation and the poloidal Alfven wave equation, and they have different values. So for a single field line you'll get two different frequencies for the poloidal and the toroidal. That's shown here for field line A: here's the toroidal frequency, then you go to pi by 2 — this is now plasma displacement in the radial or poloidal direction — and you get a frequency of like 1.7. What's only been appreciated relatively recently is we've known about toroidal and poloidal Alfven frequencies for a long time thanks to Jim, but it's now realized you can have actually any intermediate polarization, and you can work out the frequency. We've figured out a way of getting the Alfven frequency for an arbitrary polarization. So for example on the flanks, on the edges of the plume here, we'd have plasma displacement along the ridges. That would be an angle midway between toroidal and poloidal, so you'd have a polarization angle here, and it would correspond to a field line whose trace isn't shown here that would come down like this. This horizontal red line represents the driving frequency. So field line A, if I want to match this driving frequency, I need to choose the toroidal polarization. If I'm up at B where the plasma displacement is going to be like this, I need — this is the Alfven frequency as a function of polarization curve — and I match the poloidal value of the Alfven frequency. Intermediate field lines along this ridge would sit in between here and would have intermediate polarization angles. There's a lot of ideas. I'll just pause and if anyone wants to ask for clarification on any of those concepts I've mentioned, I'll just give you an opportunity to ask.

Okay, well let's move on then. In these simulations we were trying to figure out what would be a good way of identifying when we have a 3D field line resonance as opposed to a 2D one. We experimented and the most reliable way we found was to look at hodograms. Here, let's take this field line labeled with the red dots. What I'm showing here is the velocity hodogram for it. Here I've got the radial plasma velocity and the azimuthal, and you see we get this nice ellipse here, and it's tilted over at some angle — it's not toroidal or poloidal, it's midway between. I've tried to indicate that here. In contrast, this one is azimuthally polarized or toroidal, so the hodogram for that would be aligned with the azimuthal direction, that would be the axis of the ellipse. Then as you cross the boundary it rotates one way, when you're in the middle it's azimuthal again, and then it would adopt this orientation which the hodogram would be over like this. So it seemed that ways of trying to identify observations of 3D field line resonances would be to look at hodograms on a satellite of the perpendicular fields of the Alfven wave. You should see something like this. Another way would be, we took the simulation results here and we just worked out for each point the polarization angle — that is to say, the angle of the major axis of the ellipse here to the toroidal direction — we binned it in local time and we got these results here. So as you move from noon through the afternoon, you cross the plume and the polarization ellipse tilts one way, in the middle it's back to zero, and then it tilts the other way as you cross the other boundary of the plume. This might be more suitable for example for a chain of ground magnetometers rotating under this under the plume. You might expect to see this kind of signature, and on a satellite you'd expect to see these sort of features.

This ellipse here you can see is very clean. It's from a monochromatically driven system that's been driven for a long time, so it's all nice and well established. But of course nature isn't like that. You're going to get much more broadband, impulsive or randomly driven magnetopause pressure pulses. So what we did here is we took the worst case scenario where we drive it by giving the whole system a nudge. Again, around local noon we just apply a magnetic pressure here. It just ramps up in time and switches off, so it's equivalent to just giving this magnetopause boundary here around noon a shove in the radial direction or shove normal to it. We ask what we get. We've got three key positions labeled 1, 2, and 3 that we'll focus on. The panels here, the top two correspond to position 1, middle two for position 2, and the bottom two to position 3. These traces here are a little bit busy, but what we can see is the blue line is B parallel, so this is telling you what the fast mode is doing. If you see here the blue line, this is the impulsive drive coming in, and then after that it just sort of bounces back and forth a bit and disperses away, so the amplitude of the blue curve decreases in time. The main response here in the Alfven wave is going to be in the azimuthal direction, so that's the black trace. What you see with the black trace is it kind of grows linearly with time, but that's representing a kind of resonant response. Then the driver disappearing in these later stages and there's a little bit of dissipation, so it then just decays gradually in time. There's also a radial plasma velocity shown in red. This early on is associated partly with the fast mode, and then a little bit associated with the Alfven wave later on. But you can see the red, that's the radial velocity, is much smaller than the azimuthal one.

If we look at the hodograms for this, we've taken selected periods throughout to just plot a little section so it's not too busy. The hodogram over this section indicated by the blue line gives us this blue hodogram. You can see it's just over one cycle. Then we take another sample from the green interval here that gives us the green line, and finally the red. Really the only difference is that the amplitude is getting smaller as the waves are decaying. The orientation of the ellipse, given that this is a broadband push — we're not really driving a monochromatic wave in any sense — but we've got a very well-defined orientation to the ellipse. If we go to position 2, this is when we're on the plume boundary, so we're going to expect a different orientation. You can see similar features here. I won't labor them too much, but the main thing to note is that the ellipse is tilted over as we predicted from the simulation results. We're getting this sort of behavior. It's tilted over again, three time periods, three little intervals are taken giving a nice consistent orientation to the hodogram. Interestingly, the black line, the azimuthal plasma velocity, and the red line, the radial velocity, are out of phase. That's what you can see here — it's like the line Y is minus X, they're out of phase. That's what you get on this boundary here. That's here. When you cross the other boundary, you'd expect them to be in phase and like this. I haven't shown that in this panel, but it's important that the radial and azimuthal plasma velocity components of the Alfven wave will either be in phase or out of phase. It'll be a straight line like this or like that, depending which boundary you're on. Position 3 is back in a locally 2D medium, so it should be azimuthally polarized, and indeed that's what we see here. It's also interesting to note that there's a little delay here before the wave field starts to be nonzero. That's because when you push at noon, it takes a little while for the wave to propagate down to this region. That's why there's a little delay here which isn't really evident in position 1 and position 2. The take-home message is that when you cross the plume you should see a rotation at the plume boundaries either in this sense or this sense, depending which boundary you're on.

This is results from a Geophysical Research Letters paper from last year led by Jasmine Sandu. Her task was to find these sort of signatures in data. So we found some Van Allen Probes data. This is the trajectory of it. I'll just focus on the equatorial plane, and I've indicated with these two blue dots where you cross the plume boundary. So you'll see it's in exactly the right place in the afternoon, that's where we expect these strong variations of azimuth to occur, forcing us to use 3D ideas for field line resonances. Mixture of data here. The top panel is ACE. The main thing to note is that we got the dynamic pressure variations here. It's pretty quiet — it's the blue line we're looking at — and then it steps up here and there's some variation there, so there's some dynamic pressure fluctuations upstream in the solar wind coming down. SYM-H is in this panel. There's not much happening over this time interval, but if you look over a broader time interval, SYM-H indicates that we're in the main phase of a substorm. Moving on to the Van Allen Probes data here, we got number density, and this steps up here. So this is where we've entered the plume, and then it's reasonably constant, and then we exit the plume here. So it's at this boundary here or here that we'd expect to see the 3D signatures. At the first transit, we've got the radial, azimuthal, and parallel magnetic field perturbations from Van Allen Probes A. There's not much in the way of wave activity during entering the plume, but when we exit the plume, well there's plenty of activity. We think that these are driven by fast mode waves. The reason being that there's dynamic pressure fluctuations in the solar wind, there's compressional wave activity here. The other way of driving Alfven waves would be with a bump-on-tail distribution. These tend to excite very high-end poloidal waves. There was no evidence of a bump-on-tail, and also toroidal waves wouldn't have a significant azimuthal magnetic field, which these waves do. So this is a good candidate for a 3D field line resonance. In particular, note that the radial and azimuthal components are in phase. If you sort of just by eye you can see peak here, here, here, here, so it looks like they're in phase, which is what we expect. They're either going to be out of phase like this or in phase, and as we're crossing the boundary here we'd expect the radial and azimuthal to be in phase, which is exactly what we're seeing here. The other thing to note is the electric field data wasn't too good for this event, but we did manage to salvage the radial electric field. If you compare the radial electric field with the azimuthal magnetic field, you can see here — using the vertical lines to guide your eye — you can see that the two fields are in quadrature, that's a quarter of a cycle out of phase in time, and that's exactly what you expect for a standing Alfven wave. That's further evidence.

So the question is, can we use the hodograms to sort of improve our interpretation of this magnetometer data? That's exactly what Jasmine did. We're focusing now on just exiting the plume boundary, so 1740 to 1910, just focusing on the latter part of this event here. We're just looking at where we exit the plume boundary. We've got little arrows here showing the radial and azimuthal directions here, and the direction of the magnetic field as you go across here. But perhaps the easiest way to see it is in the hodograms. If we take centered on 1740, take half an hour of data from here to here, produce the hodogram, we get this here. I've indicated with this dash line the orientation of the ellipse that we got from the data. Then if we go to the actual crossing the boundary 1810, again we take half an hour of data from here to here, plot the hodogram, and it's tilted over towards the radial direction. This is radial field and azimuthal field, so it's got the radial field dominating the azimuthal now. Then once we've exited the plume, we take half an hour data from here to here, plot the hodogram, and the line through the orientation of that — and this is exactly what we were hoping for. As you go through the plume you rotate towards the radial direction, and then you come out and it rotates back. This was, we felt, convincing enough and the referees agreed that we can show we've got an observation of a field line resonance in 3D.

So just in summary then, what we've got is MHD theory: we've managed to derive the wave equation for Alfven waves with an arbitrary polarization, and it agrees with the orientation you would deduce from the hodogram for the polarization. We've shown how hodograms can be used to sort of show the importance of 3D nature of the equilibrium in coupling of fast waves to Alfven waves. Satellite observations reveal the existence of field line resonances with exactly these properties on exiting a plume, which is where there's no way you can use 2D theory — it's got to be a 3D analysis. Interesting comments to make on this study is that it's maybe not surprising, but it really does bring home how important the cold plasma density is for controlling MHD wave dynamics, particularly when you've got structures like plumes. The fact that we can rotate the wave away from the traditional 2D azimuthal polarization is important for considering how these waves are going to interact with particles, in particular radiation belt particles. In a let's see if I can get a figure to help explain this. If you think about this as an azimuthal 2D one, the velocity of the plasma elements are in the azimuthal direction, and the electric field is perpendicular to that, so it's in the radial direction. So if you have particles drifting along azimuth — radiation belt particles — they're not going to interact strongly with an Alfven wave polarized like this because the electric field is perpendicular to its drift direction. But when you come to this sort of field line, when they drift through here they're going to see an azimuthal electric field which doesn't exist in this case. So these 3D field line resonances driven by fast modes, traditionally they've not been considered as important in interacting with particles. But when you consider their 3D counterpart, then I think it's a different story. This is the subject of ongoing research from ISSI team members. Finally, I'll just finish with an advert. If any of you are interested in learning more about 3D field line resonance theory, there's a review we wrote a couple of years ago outlining the state of our understanding at that time. That's a good resource if you want to find out more. Okay, I'll stop there. Thanks for your attention.