WEBVTT 00:00:00.000 --> 00:00:02.420 align:middle line:90% [SQUEAKING] 00:00:02.420 --> 00:00:04.356 align:middle line:90% [RUSTLING] 00:00:04.356 --> 00:00:06.776 align:middle line:90% [CLICKING] 00:00:06.776 --> 00:00:09.680 align:middle line:90% 00:00:09.680 --> 00:00:11.430 align:middle line:84% JACK HARE: So I'm going to begin by giving 00:00:11.430 --> 00:00:14.010 align:middle line:84% a recap of the last lecture, just very briefly. 00:00:14.010 --> 00:00:16.830 align:middle line:84% And then we'll keep going with where we left off before. 00:00:16.830 --> 00:00:19.650 align:middle line:84% So you remember we talked about B-dot probes. 00:00:19.650 --> 00:00:22.440 align:middle line:84% These were little loops of wire, like this. 00:00:22.440 --> 00:00:24.870 align:middle line:84% They've got some area, A. They've 00:00:24.870 --> 00:00:28.080 align:middle line:84% got some magnetic field pointing through them. 00:00:28.080 --> 00:00:29.700 align:middle line:84% We're only sensitive to the component 00:00:29.700 --> 00:00:32.130 align:middle line:84% of the magnetic field that is parallel 00:00:32.130 --> 00:00:33.570 align:middle line:90% to the normal of this loop. 00:00:33.570 --> 00:00:36.750 align:middle line:84% And what we want to do is measure some voltage 00:00:36.750 --> 00:00:38.370 align:middle line:90% across this loop. 00:00:38.370 --> 00:00:43.140 align:middle line:84% And we showed that the flux through this loop, phi, 00:00:43.140 --> 00:00:47.358 align:middle line:84% was going to be equal to the integral of B dot 00:00:47.358 --> 00:00:49.800 align:middle line:90% ds over this area. 00:00:49.800 --> 00:00:51.690 align:middle line:84% And the voltage that we were going to get out 00:00:51.690 --> 00:00:54.270 align:middle line:84% was going to be equal to the time 00:00:54.270 --> 00:00:59.790 align:middle line:84% rate of change of the flux, which is equal to B dot A, 00:00:59.790 --> 00:01:00.700 align:middle line:90% like that. 00:01:00.700 --> 00:01:03.810 align:middle line:84% So by measuring the voltage on this loop 00:01:03.810 --> 00:01:06.240 align:middle line:84% and then integrating it up, we can work out 00:01:06.240 --> 00:01:10.020 align:middle line:84% what the magnetic field is doing as a function of time here. 00:01:10.020 --> 00:01:11.780 align:middle line:90% We also talked about-- 00:01:11.780 --> 00:01:15.050 align:middle line:90% these are B-dots. 00:01:15.050 --> 00:01:22.240 align:middle line:84% We also talked about Rogowski coils, 00:01:22.240 --> 00:01:25.870 align:middle line:84% which is effectively a set of B-dots 00:01:25.870 --> 00:01:27.770 align:middle line:90% arranged around the circle. 00:01:27.770 --> 00:01:31.370 align:middle line:90% So we have a circle like this. 00:01:31.370 --> 00:01:34.580 align:middle line:84% But now, we have a load of B-dots 00:01:34.580 --> 00:01:36.240 align:middle line:90% arranged all the way around. 00:01:36.240 --> 00:01:37.490 align:middle line:90% And we join them all together. 00:01:37.490 --> 00:01:43.340 align:middle line:90% 00:01:43.340 --> 00:01:46.380 align:middle line:84% And we measure the voltage again here. 00:01:46.380 --> 00:01:49.340 align:middle line:84% And this tells us something about the current 00:01:49.340 --> 00:01:51.890 align:middle line:84% which is flowing through this loop here. 00:01:51.890 --> 00:01:54.920 align:middle line:84% And we found that the voltage that we get out here 00:01:54.920 --> 00:02:00.110 align:middle line:84% is equal to the number of turns per unit length. 00:02:00.110 --> 00:02:04.590 align:middle line:84% So for example, this might be three turns per millimeter. 00:02:04.590 --> 00:02:10.660 align:middle line:84% The area of each of these little loops here times by I dot-- 00:02:10.660 --> 00:02:14.020 align:middle line:84% and there was an optional factor of mu r here. 00:02:14.020 --> 00:02:15.100 align:middle line:90% This is different from 1. 00:02:15.100 --> 00:02:16.360 align:middle line:84% Or if it's different from 1 because you've 00:02:16.360 --> 00:02:18.280 align:middle line:84% chosen a material that saturates, like steel 00:02:18.280 --> 00:02:19.520 align:middle line:90% or something like that. 00:02:19.520 --> 00:02:21.850 align:middle line:84% So these Rogowski coils are used for measuring 00:02:21.850 --> 00:02:23.980 align:middle line:84% current in a very similar way to B-dots. 00:02:23.980 --> 00:02:27.610 align:middle line:84% We get out of voltage, which we can then integrate up and find 00:02:27.610 --> 00:02:29.500 align:middle line:90% the current. 00:02:29.500 --> 00:02:33.120 align:middle line:84% So then we talked about how we could use these simple devices 00:02:33.120 --> 00:02:40.290 align:middle line:84% to measure the plasma conductivity, 00:02:40.290 --> 00:02:43.020 align:middle line:84% which we represented by this symbol, sigma. 00:02:43.020 --> 00:02:46.230 align:middle line:84% And the bar here is just some sort of average conductivity 00:02:46.230 --> 00:02:46.740 align:middle line:90% here. 00:02:46.740 --> 00:02:49.620 align:middle line:84% We made lots of assumptions, such as the fact 00:02:49.620 --> 00:02:55.440 align:middle line:84% that we're in steady state, which allows 00:02:55.440 --> 00:02:57.300 align:middle line:90% us to ditch a load of terms. 00:02:57.300 --> 00:02:59.760 align:middle line:84% And we then ended up with a power balance 00:02:59.760 --> 00:03:03.180 align:middle line:84% where we balanced the ohmic dissipation 00:03:03.180 --> 00:03:06.420 align:middle line:84% within the plasma, which is the integral of the current density 00:03:06.420 --> 00:03:09.240 align:middle line:84% squared over this plasma conductivity 00:03:09.240 --> 00:03:12.450 align:middle line:84% with the Poynting flux, which is the energy we're pushing 00:03:12.450 --> 00:03:15.960 align:middle line:84% into the plasma in the form of a loop voltage driven 00:03:15.960 --> 00:03:17.100 align:middle line:90% by our transformer. 00:03:17.100 --> 00:03:20.190 align:middle line:84% In the case of something like a tokamak, 00:03:20.190 --> 00:03:22.140 align:middle line:84% this could be the voltage inside a Z-pinch 00:03:22.140 --> 00:03:26.100 align:middle line:84% as well, and times by the current flowing 00:03:26.100 --> 00:03:27.790 align:middle line:90% in that same direction here. 00:03:27.790 --> 00:03:34.070 align:middle line:84% And so this was a balance between the ohmic heating 00:03:34.070 --> 00:03:39.533 align:middle line:84% and the external applied power here. 00:03:39.533 --> 00:03:41.575 align:middle line:84% And we found that, for something like a tokamak-- 00:03:41.575 --> 00:03:46.660 align:middle line:90% 00:03:46.660 --> 00:03:49.360 align:middle line:84% I'm just giving you half a donut here-- 00:03:49.360 --> 00:03:54.530 align:middle line:84% if we put a Rogowski around the cross-section here, 00:03:54.530 --> 00:03:56.810 align:middle line:90% we could measure I phi. 00:03:56.810 --> 00:04:01.440 align:middle line:84% And if we put a voltage loop around the tokamak 00:04:01.440 --> 00:04:02.970 align:middle line:84% like this, which will effectively 00:04:02.970 --> 00:04:06.210 align:middle line:84% sense the same voltage that's being driven inside the tokamak 00:04:06.210 --> 00:04:09.720 align:middle line:84% by our transformer, we measure V phi. 00:04:09.720 --> 00:04:11.250 align:middle line:84% So we can measure those two things. 00:04:11.250 --> 00:04:13.830 align:middle line:84% And with a little bit of rearrangement here, 00:04:13.830 --> 00:04:17.850 align:middle line:84% we ended up with a formula for our conductivity, 00:04:17.850 --> 00:04:27.740 align:middle line:84% which was equal to I phi over V phi with some geometric terms 2R 00:04:27.740 --> 00:04:32.840 align:middle line:84% over A squared, where R is the major radius and A is the minor 00:04:32.840 --> 00:04:34.530 align:middle line:90% radius of our tokamak. 00:04:34.530 --> 00:04:36.170 align:middle line:84% And this was particularly interesting 00:04:36.170 --> 00:04:39.590 align:middle line:84% because we have good models for plasma resistivity. 00:04:39.590 --> 00:04:42.380 align:middle line:84% And we know that this conductivity is proportional 00:04:42.380 --> 00:04:47.490 align:middle line:84% to a load of constants times Te to the 3/2, like that. 00:04:47.490 --> 00:04:49.610 align:middle line:84% And so by measuring the plasma conductivity, 00:04:49.610 --> 00:04:51.470 align:middle line:84% we could effectively measure the temperature 00:04:51.470 --> 00:04:55.220 align:middle line:84% inside our plasma just using two loops of wire, 00:04:55.220 --> 00:04:58.370 align:middle line:90% which is pretty cool. 00:04:58.370 --> 00:05:03.450 align:middle line:84% Finally, we started talking about pressure balance. 00:05:03.450 --> 00:05:06.630 align:middle line:84% So we looked at our MHD equation-- 00:05:06.630 --> 00:05:11.070 align:middle line:84% J cross B minus gradient P equals 0. 00:05:11.070 --> 00:05:14.940 align:middle line:84% This is steady state MHD in completely general 00:05:14.940 --> 00:05:16.020 align:middle line:90% three dimensions. 00:05:16.020 --> 00:05:23.850 align:middle line:84% And the idea is we want to be able to measure B and use that 00:05:23.850 --> 00:05:26.310 align:middle line:90% to infer the pressure. 00:05:26.310 --> 00:05:28.368 align:middle line:84% And so then the pressure is an important quantity 00:05:28.368 --> 00:05:30.660 align:middle line:84% because the density and the temperature of the tokamak, 00:05:30.660 --> 00:05:32.730 align:middle line:84% or whatever device you're trying to measure, 00:05:32.730 --> 00:05:35.320 align:middle line:84% gives you the fusion power output here. 00:05:35.320 --> 00:05:37.890 align:middle line:84% And we decided that trying to solve this 00:05:37.890 --> 00:05:40.170 align:middle line:84% for like a three dimensional tokamak or a stellarator 00:05:40.170 --> 00:05:41.080 align:middle line:90% was very hard. 00:05:41.080 --> 00:05:42.747 align:middle line:84% And so we were just going to focus, just 00:05:42.747 --> 00:05:46.830 align:middle line:84% to start with, on a so-called straight tokamak, which 00:05:46.830 --> 00:05:50.280 align:middle line:84% is just where we take the torus and unfold it into a cylinder, 00:05:50.280 --> 00:05:51.190 align:middle line:90% like this. 00:05:51.190 --> 00:05:53.310 align:middle line:84% So we went for cylindrical geometry. 00:05:53.310 --> 00:06:02.040 align:middle line:90% 00:06:02.040 --> 00:06:06.720 align:middle line:84% And we also said we would focus on just the first two modes, 00:06:06.720 --> 00:06:10.470 align:middle line:84% the m equals 0 and the m equals 1 modes. 00:06:10.470 --> 00:06:15.720 align:middle line:84% And this was the idea that if we put some probe out here 00:06:15.720 --> 00:06:19.080 align:middle line:84% measuring the theta, the magnetic field going 00:06:19.080 --> 00:06:21.180 align:middle line:84% around our little cylinder here, we 00:06:21.180 --> 00:06:27.360 align:middle line:84% could decompose that B theta, as we can always do, 00:06:27.360 --> 00:06:29.500 align:middle line:90% using a Fourier decomposition. 00:06:29.500 --> 00:06:31.950 align:middle line:84% So would have some Bc component-- 00:06:31.950 --> 00:06:32.940 align:middle line:90% C0. 00:06:32.940 --> 00:06:37.500 align:middle line:84% And then we'd have a sum over these series of m modes from 1 00:06:37.500 --> 00:06:44.750 align:middle line:84% to infinity with coefficients like Cm, [? pulse, ?] m theta-- 00:06:44.750 --> 00:06:47.120 align:middle line:90% I'm running out of space-- 00:06:47.120 --> 00:06:53.620 align:middle line:84% plus Sm phi m theta, just about made it. 00:06:53.620 --> 00:06:54.640 align:middle line:90% OK, cool. 00:06:54.640 --> 00:06:56.540 align:middle line:84% And the idea is really, most of the time, 00:06:56.540 --> 00:06:57.910 align:middle line:84% we're just going to be interested in these lower order 00:06:57.910 --> 00:06:58.310 align:middle line:90% modes. 00:06:58.310 --> 00:07:00.040 align:middle line:84% But of course, if you have lots and lots 00:07:00.040 --> 00:07:01.720 align:middle line:84% of B-dots and digitizers, you might 00:07:01.720 --> 00:07:03.280 align:middle line:84% want to go after some of these high order modes. 00:07:03.280 --> 00:07:05.260 align:middle line:84% But what we're going to show in today's class 00:07:05.260 --> 00:07:08.470 align:middle line:84% is that the size of the m equals 0 mode tells you something 00:07:08.470 --> 00:07:09.700 align:middle line:90% about energy confinement. 00:07:09.700 --> 00:07:12.100 align:middle line:84% And the size of the m equals 1 mode tells you 00:07:12.100 --> 00:07:15.130 align:middle line:84% something about the displacement of your plasma, both of which 00:07:15.130 --> 00:07:17.480 align:middle line:84% are useful things to be able to measure. 00:07:17.480 --> 00:07:19.168 align:middle line:84% So I'm going to pause there and let 00:07:19.168 --> 00:07:20.710 align:middle line:84% you ask any questions on the material 00:07:20.710 --> 00:07:24.384 align:middle line:90% that we covered last lecture. 00:07:24.384 --> 00:07:25.866 align:middle line:90% Mm-hmm? 00:07:25.866 --> 00:07:28.336 align:middle line:90% STUDENT: [INAUDIBLE] 00:07:28.336 --> 00:07:33.280 align:middle line:90% 00:07:33.280 --> 00:07:34.050 align:middle line:90% JACK HARE: Yes. 00:07:34.050 --> 00:07:36.515 align:middle line:90% STUDENT: [INAUDIBLE] 00:07:36.515 --> 00:07:40.470 align:middle line:90% 00:07:40.470 --> 00:07:42.750 align:middle line:84% JACK HARE: I haven't drawn the return loop there. 00:07:42.750 --> 00:07:45.310 align:middle line:84% If you want to put the return loop on, 00:07:45.310 --> 00:07:47.850 align:middle line:84% which is a good idea for canceling out 00:07:47.850 --> 00:07:50.430 align:middle line:84% some of the stray fields, when you get to this end, 00:07:50.430 --> 00:07:54.870 align:middle line:84% you simply wind it back here and put it out this side. 00:07:54.870 --> 00:07:56.280 align:middle line:84% So one way you can do this simply 00:07:56.280 --> 00:07:59.520 align:middle line:84% is you take coaxial cable, like a BNC cable, 00:07:59.520 --> 00:08:02.080 align:middle line:84% you strip off the outer conductor. 00:08:02.080 --> 00:08:03.560 align:middle line:84% And so you just have the dielectric 00:08:03.560 --> 00:08:04.560 align:middle line:90% and the inner conductor. 00:08:04.560 --> 00:08:05.820 align:middle line:90% You take some magnet wire. 00:08:05.820 --> 00:08:08.610 align:middle line:84% And you back-wind it along the section that you've stripped, 00:08:08.610 --> 00:08:11.190 align:middle line:84% and then solder it back onto the outer sheet. 00:08:11.190 --> 00:08:15.423 align:middle line:84% And so that makes this configuration very easily. 00:08:15.423 --> 00:08:16.090 align:middle line:90% Other questions? 00:08:16.090 --> 00:08:16.965 align:middle line:90% Anyone from Colombia? 00:08:16.965 --> 00:08:21.490 align:middle line:90% 00:08:21.490 --> 00:08:22.874 align:middle line:90% Yeah? 00:08:22.874 --> 00:08:25.359 align:middle line:90% STUDENT: [INAUDIBLE] 00:08:25.359 --> 00:08:36.390 align:middle line:90% 00:08:36.390 --> 00:08:38.640 align:middle line:84% JACK HARE: Yeah, so the question is, does the Rogowski 00:08:38.640 --> 00:08:40.679 align:middle line:90% have to form a complete loop? 00:08:40.679 --> 00:08:41.988 align:middle line:90% Or can you go halfway? 00:08:41.988 --> 00:08:43.530 align:middle line:84% And the answer is, no, you definitely 00:08:43.530 --> 00:08:45.725 align:middle line:90% have to form a complete loop. 00:08:45.725 --> 00:08:47.100 align:middle line:84% When you're deriving this, you're 00:08:47.100 --> 00:08:49.440 align:middle line:84% using Ampere's law, where you have a loop integral. 00:08:49.440 --> 00:08:52.500 align:middle line:84% And that requires you to do a full circuit around the surface 00:08:52.500 --> 00:08:55.990 align:middle line:84% that you're integrating around, but yeah. 00:08:55.990 --> 00:08:57.252 align:middle line:90% Matthew? 00:08:57.252 --> 00:08:59.460 align:middle line:84% STUDENT: Yeah, I think you mentioned this last class, 00:08:59.460 --> 00:09:01.175 align:middle line:90% but I just want to make sure-- 00:09:01.175 --> 00:09:01.610 align:middle line:90% JACK HARE: I can't hear you. 00:09:01.610 --> 00:09:03.902 align:middle line:84% And that's probably my fault. So just give me a second. 00:09:03.902 --> 00:09:09.350 align:middle line:90% 00:09:09.350 --> 00:09:11.090 align:middle line:84% Can you try to say something now? 00:09:11.090 --> 00:09:12.050 align:middle line:90% STUDENT: Test, test. 00:09:12.050 --> 00:09:12.650 align:middle line:90% JACK HARE: Yeah, OK. 00:09:12.650 --> 00:09:13.275 align:middle line:90% I can hear you. 00:09:13.275 --> 00:09:15.320 align:middle line:84% Can you say your question again, please? 00:09:15.320 --> 00:09:17.970 align:middle line:84% STUDENT: Yeah, so I think you mentioned this last class, 00:09:17.970 --> 00:09:19.178 align:middle line:90% but I just want to make sure. 00:09:19.178 --> 00:09:23.600 align:middle line:84% So another assumption in your computation of the conductivity 00:09:23.600 --> 00:09:25.760 align:middle line:84% is that all the current in the plasma 00:09:25.760 --> 00:09:27.230 align:middle line:90% is chemically driven, right? 00:09:27.230 --> 00:09:31.910 align:middle line:84% You can't have ECCD or neutral beam-driven current 00:09:31.910 --> 00:09:33.170 align:middle line:90% or anything like that. 00:09:33.170 --> 00:09:35.450 align:middle line:84% Otherwise, that V phi measurement 00:09:35.450 --> 00:09:37.325 align:middle line:90% wouldn't be accurate, right? 00:09:37.325 --> 00:09:38.700 align:middle line:84% JACK HARE: Yeah, absolutely true. 00:09:38.700 --> 00:09:40.880 align:middle line:84% So in case anyone didn't hear properly, 00:09:40.880 --> 00:09:43.070 align:middle line:84% here, we're assuming that all the things that's 00:09:43.070 --> 00:09:47.150 align:middle line:84% driving the current here is to do with a transformer that's 00:09:47.150 --> 00:09:48.140 align:middle line:90% driving the current. 00:09:48.140 --> 00:09:48.530 align:middle line:90% STUDENT: Right. 00:09:48.530 --> 00:09:50.840 align:middle line:84% JACK HARE: It's not to do with waves or something else 00:09:50.840 --> 00:09:51.648 align:middle line:90% like that. 00:09:51.648 --> 00:09:53.690 align:middle line:84% If you have a tokamak where you're driving waves, 00:09:53.690 --> 00:09:56.000 align:middle line:84% this very simple treatment won't work anymore. 00:09:56.000 --> 00:09:57.830 align:middle line:84% But many tokamaks, for a long time, 00:09:57.830 --> 00:09:59.412 align:middle line:84% they were just inductively driven. 00:09:59.412 --> 00:10:00.620 align:middle line:90% STUDENT: Right, right, right. 00:10:00.620 --> 00:10:01.507 align:middle line:90% All right, thank you. 00:10:01.507 --> 00:10:02.090 align:middle line:90% JACK HARE: OK. 00:10:02.090 --> 00:10:10.410 align:middle line:90% 00:10:10.410 --> 00:10:10.910 align:middle line:90% Right. 00:10:10.910 --> 00:10:12.620 align:middle line:84% So yeah, we started with this derivation 00:10:12.620 --> 00:10:13.460 align:middle line:90% with the Poynting flux. 00:10:13.460 --> 00:10:15.543 align:middle line:84% But I quickly simplified it to this circuit model. 00:10:15.543 --> 00:10:17.915 align:middle line:84% But you're right, if I kept the Poynting flux in there, 00:10:17.915 --> 00:10:19.790 align:middle line:84% then obviously I'm capturing all of the power 00:10:19.790 --> 00:10:20.957 align:middle line:90% I'm putting into the plasma. 00:10:20.957 --> 00:10:23.720 align:middle line:84% But we wouldn't be able to do it in this very simple way with two 00:10:23.720 --> 00:10:26.430 align:middle line:84% loops anymore because these two loops are only 00:10:26.430 --> 00:10:28.430 align:middle line:84% measuring the contributions to the Poynting flux 00:10:28.430 --> 00:10:29.480 align:middle line:90% from the transformer. 00:10:29.480 --> 00:10:32.730 align:middle line:84% These loops won't capture your lower hybrid current. 00:10:32.730 --> 00:10:35.030 align:middle line:90% STUDENT: [INAUDIBLE] 00:10:35.030 --> 00:10:35.893 align:middle line:90% 00:10:35.893 --> 00:10:36.560 align:middle line:90% JACK HARE: Yeah. 00:10:36.560 --> 00:10:39.060 align:middle line:84% So if you know that power, if you know how much power you're 00:10:39.060 --> 00:10:39.710 align:middle line:90% coupling-- 00:10:39.710 --> 00:10:40.907 align:middle line:90% STUDENT: [INAUDIBLE] 00:10:40.907 --> 00:10:41.990 align:middle line:90% JACK HARE: Right, exactly. 00:10:41.990 --> 00:10:44.282 align:middle line:84% Yeah, so you might know you're putting in 10 megawatts. 00:10:44.282 --> 00:10:46.350 align:middle line:84% But you may not be coupling 10 megawatts. 00:10:46.350 --> 00:10:49.856 align:middle line:90% So yeah, any other questions? 00:10:49.856 --> 00:10:56.530 align:middle line:90% 00:10:56.530 --> 00:10:58.885 align:middle line:90% All right, let's keep going. 00:10:58.885 --> 00:11:05.010 align:middle line:90% 00:11:05.010 --> 00:11:08.463 align:middle line:90% So I'm just going to write that. 00:11:08.463 --> 00:11:09.880 align:middle line:84% Actually, I don't think I need it. 00:11:09.880 --> 00:11:11.370 align:middle line:84% I'm not going to write the decomposition again. 00:11:11.370 --> 00:11:12.787 align:middle line:84% Hopefully, you've remembered this. 00:11:12.787 --> 00:11:14.920 align:middle line:84% I'll write it again if I need it sometime later. 00:11:14.920 --> 00:11:16.740 align:middle line:84% So we're going to focus, first of all, 00:11:16.740 --> 00:11:20.890 align:middle line:84% on the m equals 0 term from that decomposition. 00:11:20.890 --> 00:11:24.135 align:middle line:84% This is often called the diamagnetic term. 00:11:24.135 --> 00:11:31.240 align:middle line:90% 00:11:31.240 --> 00:11:33.400 align:middle line:84% We'll see, in fact, that the plasma does not 00:11:33.400 --> 00:11:34.540 align:middle line:90% have to be diamagnetic. 00:11:34.540 --> 00:11:36.280 align:middle line:84% It just turns out, for many plasmas 00:11:36.280 --> 00:11:38.440 align:middle line:84% we study, like tokamaks and stellarators, 00:11:38.440 --> 00:11:39.460 align:middle line:90% they are diamagnetic. 00:11:39.460 --> 00:11:41.570 align:middle line:84% And so this is a reasonable thing to call them. 00:11:41.570 --> 00:11:44.410 align:middle line:84% But we'll talk about paramagnetism briefly as well. 00:11:44.410 --> 00:11:46.960 align:middle line:84% And the point of studying this, just so 00:11:46.960 --> 00:11:48.437 align:middle line:84% you don't lose all hope as we plow 00:11:48.437 --> 00:11:50.770 align:middle line:84% through all this mathematics-- the thing we're going for 00:11:50.770 --> 00:11:53.650 align:middle line:84% is it allows us to measure the energy confinement 00:11:53.650 --> 00:11:56.680 align:middle line:90% time inside our plasma, tau e. 00:11:56.680 --> 00:12:00.730 align:middle line:84% And you probably recall that the Lawson criterion 00:12:00.730 --> 00:12:05.422 align:middle line:84% depends on the density and on this tau e quantity. 00:12:05.422 --> 00:12:06.880 align:middle line:84% And if you want to put a T on there 00:12:06.880 --> 00:12:09.213 align:middle line:84% and call it the triple product, you can do that as well. 00:12:09.213 --> 00:12:11.860 align:middle line:84% And pretty clearly, right now, we're 00:12:11.860 --> 00:12:13.997 align:middle line:84% able to measure T very roughly using two loops. 00:12:13.997 --> 00:12:16.330 align:middle line:84% And we're going to see it only takes three loops of wire 00:12:16.330 --> 00:12:17.608 align:middle line:90% to measure tau e. 00:12:17.608 --> 00:12:19.150 align:middle line:84% It takes significantly more than that 00:12:19.150 --> 00:12:20.410 align:middle line:90% to measure n, unfortunately. 00:12:20.410 --> 00:12:21.910 align:middle line:84% But we can get a pretty decent way 00:12:21.910 --> 00:12:25.060 align:middle line:84% to being able to measure the Lawson parameter for our plasma 00:12:25.060 --> 00:12:29.240 align:middle line:84% with just some little loops of wire, which is pretty cool. 00:12:29.240 --> 00:12:31.400 align:middle line:90% So let's have a look at this. 00:12:31.400 --> 00:12:34.570 align:middle line:84% So the m equals 0 term, as we discussed last week, 00:12:34.570 --> 00:12:39.070 align:middle line:84% is a term that says something about the size of the plasma. 00:12:39.070 --> 00:12:41.500 align:middle line:84% This is something that tells us-- 00:12:41.500 --> 00:12:44.980 align:middle line:84% has no information whatsoever about the azimuthal profile 00:12:44.980 --> 00:12:45.660 align:middle line:90% of our plasma. 00:12:45.660 --> 00:12:46.910 align:middle line:90% We've thrown all of that away. 00:12:46.910 --> 00:12:49.860 align:middle line:84% So we're only interested in things that 00:12:49.860 --> 00:12:51.270 align:middle line:90% move in the radial coordinate. 00:12:51.270 --> 00:12:59.320 align:middle line:84% So can, for example, take my MHD equation 00:12:59.320 --> 00:13:03.780 align:middle line:84% and dot it with the radial unit vector and say, 00:13:03.780 --> 00:13:05.640 align:middle line:90% this is equal to 0. 00:13:05.640 --> 00:13:07.620 align:middle line:84% Previously, the whole equation was equal to 0. 00:13:07.620 --> 00:13:10.050 align:middle line:84% But now, I don't care about all the other terms. 00:13:10.050 --> 00:13:11.370 align:middle line:90% They'll be 0 by symmetry. 00:13:11.370 --> 00:13:12.870 align:middle line:84% I'm just going to focus on this one. 00:13:12.870 --> 00:13:16.800 align:middle line:84% And that allows us to write down a simplified form of this MHD 00:13:16.800 --> 00:13:28.100 align:middle line:84% equation, which is minus dP/dR plus dBphi/dR 00:13:28.100 --> 00:13:43.140 align:middle line:84% times B phi upon mu 0 plus 1 upon RdR/d theta dR times B 00:13:43.140 --> 00:13:47.583 align:middle line:90% theta over mu 0. 00:13:47.583 --> 00:13:50.000 align:middle line:84% And you'll recall that all of these mu 0's are popping out 00:13:50.000 --> 00:13:55.478 align:middle line:84% because we've replaced J with the curl of B over mu 0. 00:13:55.478 --> 00:13:57.770 align:middle line:84% And you'll also remember that, although we don't really 00:13:57.770 --> 00:13:59.900 align:middle line:84% have a theta in our analysis because we're 00:13:59.900 --> 00:14:02.510 align:middle line:84% dealing with our straight cylinder, 00:14:02.510 --> 00:14:04.370 align:middle line:84% we agreed that we would still continue 00:14:04.370 --> 00:14:09.410 align:middle line:84% to call this direction along the cylinder phi, 00:14:09.410 --> 00:14:12.980 align:middle line:84% like that, because, if we do bend this back around 00:14:12.980 --> 00:14:14.630 align:middle line:84% into a torus, that's what it'll be. 00:14:14.630 --> 00:14:17.210 align:middle line:84% And of course, the other direction that's important 00:14:17.210 --> 00:14:20.610 align:middle line:84% here is the theta direction, which wraps around 00:14:20.610 --> 00:14:22.095 align:middle line:90% [? but ?] your poloidal angle. 00:14:22.095 --> 00:14:25.010 align:middle line:90% 00:14:25.010 --> 00:14:26.030 align:middle line:90% OK. 00:14:26.030 --> 00:14:30.220 align:middle line:84% If you take this equation 1 and you spend a little bit 00:14:30.220 --> 00:14:31.670 align:middle line:90% of time fiddling with it-- 00:14:31.670 --> 00:14:33.310 align:middle line:84% and this is where I'm not going to do 00:14:33.310 --> 00:14:34.930 align:middle line:90% the whole derivation in class-- 00:14:34.930 --> 00:14:37.840 align:middle line:84% you can take this equation 1 and you 00:14:37.840 --> 00:14:40.610 align:middle line:90% can multiply it by R squared. 00:14:40.610 --> 00:14:42.970 align:middle line:84% And then you can integrate it up-- 00:14:42.970 --> 00:14:47.330 align:middle line:84% equation 1 times R squared dR. And you can integrate it 00:14:47.330 --> 00:14:52.890 align:middle line:84% between 0 and A, where A is the edge of your plasma, here. 00:14:52.890 --> 00:14:57.650 align:middle line:84% And if you do all of that plus some rearrangement-- 00:14:57.650 --> 00:15:00.960 align:middle line:90% 00:15:00.960 --> 00:15:03.030 align:middle line:84% and I suggest you go away in your own time 00:15:03.030 --> 00:15:04.920 align:middle line:84% and give this a go, especially if you're 00:15:04.920 --> 00:15:06.630 align:middle line:84% studying for quals at MSE, because this 00:15:06.630 --> 00:15:08.460 align:middle line:84% is the sort of question we love to ask-- 00:15:08.460 --> 00:15:11.640 align:middle line:84% you will end up being able to rearrange it 00:15:11.640 --> 00:15:16.090 align:middle line:84% into a dimensionless parameter called beta theta. 00:15:16.090 --> 00:15:21.950 align:middle line:84% And beta theta is defined as 2 times mu 0 00:15:21.950 --> 00:15:27.090 align:middle line:90% over B theta A squared. 00:15:27.090 --> 00:15:31.660 align:middle line:84% So this is the magnetic field at the edge of the plasma. 00:15:31.660 --> 00:15:35.590 align:middle line:84% And this is multiplied by the average pressure. 00:15:35.590 --> 00:15:39.450 align:middle line:84% So this is not necessarily the standard definition of beta 00:15:39.450 --> 00:15:40.500 align:middle line:90% that you've seen. 00:15:40.500 --> 00:15:42.750 align:middle line:84% It's a definition of beta that uses 00:15:42.750 --> 00:15:45.667 align:middle line:84% an average of the magnetic pressure-- sorry, 00:15:45.667 --> 00:15:48.000 align:middle line:84% an average of the thermal pressure over the whole plasma 00:15:48.000 --> 00:15:49.870 align:middle line:90% cross-section. 00:15:49.870 --> 00:15:53.150 align:middle line:84% And we're comparing that to the magnetic pressure at the edge. 00:15:53.150 --> 00:15:55.630 align:middle line:84% So it's a little bit subtle because it's not actually 00:15:55.630 --> 00:15:57.260 align:middle line:90% a local dimensionless parameter. 00:15:57.260 --> 00:15:59.860 align:middle line:84% This is an average dimensionless parameter here. 00:15:59.860 --> 00:16:03.110 align:middle line:90% 00:16:03.110 --> 00:16:03.860 align:middle line:90% You've taken this. 00:16:03.860 --> 00:16:04.490 align:middle line:90% You've integrated it up. 00:16:04.490 --> 00:16:06.140 align:middle line:84% You've rearranged it so that you have-- 00:16:06.140 --> 00:16:07.610 align:middle line:84% on your left-hand side of your equation, 00:16:07.610 --> 00:16:09.723 align:middle line:84% you have this, which is what you're going after. 00:16:09.723 --> 00:16:11.390 align:middle line:84% On the right-hand side of your equation, 00:16:11.390 --> 00:16:18.930 align:middle line:84% you find out you have 1 minus B phi at the edge-- so B phi 00:16:18.930 --> 00:16:27.040 align:middle line:84% A squared minus the average of B phi squared across the plasma 00:16:27.040 --> 00:16:34.530 align:middle line:84% cross-section over B theta A squared. 00:16:34.530 --> 00:16:39.380 align:middle line:90% 00:16:39.380 --> 00:16:45.210 align:middle line:84% So again, these two are at the edge. 00:16:45.210 --> 00:16:47.643 align:middle line:90% 00:16:47.643 --> 00:16:49.060 align:middle line:84% And so you could measure them just 00:16:49.060 --> 00:16:51.580 align:middle line:84% using B-dot probes, which are sitting at the edge, 00:16:51.580 --> 00:16:53.410 align:middle line:90% not inside your plasma. 00:16:53.410 --> 00:16:58.370 align:middle line:84% This one is going to give you more trouble measuring it. 00:16:58.370 --> 00:17:05.030 align:middle line:84% Now, if I asked you to measure B phi, you'd say, 00:17:05.030 --> 00:17:06.480 align:middle line:90% that's not a big problem. 00:17:06.480 --> 00:17:11.190 align:middle line:84% I'm just going to stick a B-dot all the way around my plasma. 00:17:11.190 --> 00:17:14.435 align:middle line:84% And you see how this is going to average the average B 00:17:14.435 --> 00:17:18.010 align:middle line:90% phi all the way over the area. 00:17:18.010 --> 00:17:19.180 align:middle line:90% And you say, job done. 00:17:19.180 --> 00:17:21.520 align:middle line:90% But I didn't ask you for B phi. 00:17:21.520 --> 00:17:23.390 align:middle line:90% I asked you for B squared phi. 00:17:23.390 --> 00:17:29.670 align:middle line:84% I also didn't ask you for B phi squared, which you could do just 00:17:29.670 --> 00:17:31.080 align:middle line:90% by squaring that. 00:17:31.080 --> 00:17:33.630 align:middle line:84% So clearly, this is going to cause us some problems. 00:17:33.630 --> 00:17:37.080 align:middle line:84% And we don't have a diagnostic that 00:17:37.080 --> 00:17:38.940 align:middle line:90% just measures B squared phi. 00:17:38.940 --> 00:17:41.280 align:middle line:84% So we can't just get this out easily. 00:17:41.280 --> 00:17:42.825 align:middle line:84% And so to make some progress on this, 00:17:42.825 --> 00:17:44.700 align:middle line:84% we're going to have to make some assumptions. 00:17:44.700 --> 00:17:47.940 align:middle line:84% And this is where we're making even more assumptions in order 00:17:47.940 --> 00:17:49.115 align:middle line:90% to be able to make progress. 00:17:49.115 --> 00:17:50.490 align:middle line:84% Before we keep going, does anyone 00:17:50.490 --> 00:17:52.260 align:middle line:90% have any questions about this? 00:17:52.260 --> 00:17:53.448 align:middle line:90% Yes? 00:17:53.448 --> 00:17:55.938 align:middle line:90% STUDENT: [INAUDIBLE] 00:17:55.938 --> 00:18:09.930 align:middle line:90% 00:18:09.930 --> 00:18:11.475 align:middle line:90% JACK HARE: All I need is for-- 00:18:11.475 --> 00:18:13.995 align:middle line:84% the question was, can I put a B-dot probe at A 00:18:13.995 --> 00:18:15.100 align:middle line:90% and have it survive? 00:18:15.100 --> 00:18:17.190 align:middle line:84% In fact, all I need is for the plasma pressure 00:18:17.190 --> 00:18:19.260 align:middle line:90% to be 0 where I place my probe. 00:18:19.260 --> 00:18:22.470 align:middle line:84% And it could have been 0 at A over 2, much further in. 00:18:22.470 --> 00:18:24.510 align:middle line:84% As long as it's still 0 where I put my probe, 00:18:24.510 --> 00:18:25.500 align:middle line:90% this will all work. 00:18:25.500 --> 00:18:27.420 align:middle line:84% So it needs to be a vacuum measurement. 00:18:27.420 --> 00:18:31.530 align:middle line:84% There cannot be any plasma at a larger radius than my probe. 00:18:31.530 --> 00:18:33.720 align:middle line:90% But yeah, OK, other questions? 00:18:33.720 --> 00:18:35.124 align:middle line:90% Yes? 00:18:35.124 --> 00:18:37.278 align:middle line:90% STUDENT: [INAUDIBLE] 00:18:37.278 --> 00:18:43.645 align:middle line:90% 00:18:43.645 --> 00:18:44.520 align:middle line:90% JACK HARE: It's free. 00:18:44.520 --> 00:18:45.570 align:middle line:90% It's a free agent. 00:18:45.570 --> 00:18:46.700 align:middle line:90% This is a fraction. 00:18:46.700 --> 00:18:47.930 align:middle line:90% The 1 is by itself. 00:18:47.930 --> 00:18:50.042 align:middle line:90% Yes, cool. 00:18:50.042 --> 00:18:52.250 align:middle line:84% I can't remember what equation this is in Hutchinson, 00:18:52.250 --> 00:18:54.083 align:middle line:84% but this is definitely one of his equations. 00:18:54.083 --> 00:18:55.340 align:middle line:90% Yeah, other questions? 00:18:55.340 --> 00:18:56.640 align:middle line:90% Yeah? 00:18:56.640 --> 00:18:58.748 align:middle line:90% STUDENT: [INAUDIBLE] 00:18:58.748 --> 00:19:00.872 align:middle line:90% 00:19:00.872 --> 00:19:02.330 align:middle line:84% JACK HARE: Yeah, so we're bothering 00:19:02.330 --> 00:19:04.227 align:middle line:90% with this definition of beta. 00:19:04.227 --> 00:19:05.810 align:middle line:84% Actually, the beta, it doesn't matter. 00:19:05.810 --> 00:19:08.520 align:middle line:84% What we've done is-- just ignore that if you don't like it. 00:19:08.520 --> 00:19:10.910 align:middle line:84% I've shown that I can write the average pressure 00:19:10.910 --> 00:19:13.280 align:middle line:90% in terms of some quantities. 00:19:13.280 --> 00:19:15.838 align:middle line:84% And we can certainly measure these quantities. 00:19:15.838 --> 00:19:17.630 align:middle line:84% I could multiply this up by the way, right? 00:19:17.630 --> 00:19:19.520 align:middle line:84% I could put this over on the right-hand side. 00:19:19.520 --> 00:19:22.340 align:middle line:84% Then I'd have a B theta A squared minus this blah, blah, 00:19:22.340 --> 00:19:23.060 align:middle line:90% blah. 00:19:23.060 --> 00:19:25.370 align:middle line:84% So far, it's looking good apart from we can't measure this one. 00:19:25.370 --> 00:19:26.850 align:middle line:84% And we're going to explain how to measure that. 00:19:26.850 --> 00:19:28.310 align:middle line:84% So if you don't like thinking about B theta, 00:19:28.310 --> 00:19:30.410 align:middle line:84% just think, hey, you've measured the average thermal pressure 00:19:30.410 --> 00:19:31.070 align:middle line:90% inside the plasma. 00:19:31.070 --> 00:19:32.210 align:middle line:90% That's already pretty good. 00:19:32.210 --> 00:19:33.470 align:middle line:90% So questions from Columbia? 00:19:33.470 --> 00:19:36.960 align:middle line:90% 00:19:36.960 --> 00:19:37.950 align:middle line:90% All right. 00:19:37.950 --> 00:19:39.840 align:middle line:90% Yeah, another question here? 00:19:39.840 --> 00:19:42.090 align:middle line:90% STUDENT: [INAUDIBLE] 00:19:42.090 --> 00:19:44.610 align:middle line:84% JACK HARE: Apart from the last one, the squared term, yes, 00:19:44.610 --> 00:19:46.780 align:middle line:90% yeah. 00:19:46.780 --> 00:19:47.410 align:middle line:90% OK, good. 00:19:47.410 --> 00:19:50.598 align:middle line:84% So to make progress, we are going to make some assumptions. 00:19:50.598 --> 00:19:51.640 align:middle line:90% This happens quite a lot. 00:19:51.640 --> 00:19:53.630 align:middle line:84% Some people are like, is that assumption good? 00:19:53.630 --> 00:19:54.730 align:middle line:90% Well, it doesn't matter. 00:19:54.730 --> 00:19:56.570 align:middle line:84% It lets me keep going with the mathematics. 00:19:56.570 --> 00:19:57.760 align:middle line:90% So it's good enough for now. 00:19:57.760 --> 00:19:59.920 align:middle line:84% If you want to do it properly, you'll 00:19:59.920 --> 00:20:02.290 align:middle line:84% often have to resort to numerics as opposed 00:20:02.290 --> 00:20:03.470 align:middle line:90% to analytical results. 00:20:03.470 --> 00:20:05.480 align:middle line:84% So we want to make analytical progress here. 00:20:05.480 --> 00:20:08.140 align:middle line:84% So we're going to assume three things-- 00:20:08.140 --> 00:20:13.930 align:middle line:84% first of all, that our toroidal magnetic field, B phi, 00:20:13.930 --> 00:20:18.250 align:middle line:84% as a function of R is roughly constant, 00:20:18.250 --> 00:20:22.910 align:middle line:84% as in constant spatially, it doesn't change much in R. We're 00:20:22.910 --> 00:20:31.660 align:middle line:84% going to assume that B phi is much, much larger than B theta. 00:20:31.660 --> 00:20:35.652 align:middle line:84% And we're also going to assume that beta phi-- 00:20:35.652 --> 00:20:37.360 align:middle line:84% and we haven't actually defined that yet. 00:20:37.360 --> 00:20:39.610 align:middle line:84% But if you squint over here, you can probably work out 00:20:39.610 --> 00:20:41.370 align:middle line:90% what it's going to look like. 00:20:41.370 --> 00:20:42.730 align:middle line:90% Oh no, I'll define it here. 00:20:42.730 --> 00:20:51.510 align:middle line:84% OK, 2 mu 0 B phi at the edge, B phi A squared times 00:20:51.510 --> 00:20:57.020 align:middle line:84% by the average total pressure is much, much less than 1. 00:20:57.020 --> 00:20:58.820 align:middle line:84% Can anyone tell me a system for which 00:20:58.820 --> 00:21:02.350 align:middle line:84% this is a reasonable set of assumptions? 00:21:02.350 --> 00:21:02.860 align:middle line:90% Yeah? 00:21:02.860 --> 00:21:03.830 align:middle line:90% STUDENT: [INAUDIBLE] 00:21:03.830 --> 00:21:07.120 align:middle line:84% JACK HARE: Yeah, high aspect ratio tokamak, high aspect ratio 00:21:07.120 --> 00:21:08.360 align:middle line:90% stellarator as well. 00:21:08.360 --> 00:21:16.830 align:middle line:84% So some sort of tokamak or stellarator-- 00:21:16.830 --> 00:21:20.280 align:middle line:84% these are machines-- stellarator, OK. 00:21:20.280 --> 00:21:22.680 align:middle line:84% These are machines where we are deliberately 00:21:22.680 --> 00:21:26.700 align:middle line:84% operating at low beta to avoid MHD instabilities, where we have 00:21:26.700 --> 00:21:30.870 align:middle line:84% a very strong toroidal magnetic field to prevent MHD 00:21:30.870 --> 00:21:32.310 align:middle line:90% instabilities, like the kink. 00:21:32.310 --> 00:21:34.680 align:middle line:84% And if you're at high aspect ratio, 00:21:34.680 --> 00:21:39.860 align:middle line:84% then your magnetic field as a function of R 00:21:39.860 --> 00:21:43.670 align:middle line:84% is going to be dropping as roughly 1 over R. 00:21:43.670 --> 00:21:48.460 align:middle line:84% So I just put my circular cross-section 00:21:48.460 --> 00:21:50.988 align:middle line:84% far enough out that this is roughly 00:21:50.988 --> 00:21:52.780 align:middle line:84% constant across the circular cross-section. 00:21:52.780 --> 00:21:54.697 align:middle line:84% So these are actually not terrible assumptions 00:21:54.697 --> 00:21:56.068 align:middle line:90% to make for certain plasmas. 00:21:56.068 --> 00:21:57.610 align:middle line:84% But they are violated for other ones. 00:21:57.610 --> 00:21:59.560 align:middle line:84% This won't work for reversed field pinch 00:21:59.560 --> 00:22:01.180 align:middle line:90% where this is not true. 00:22:01.180 --> 00:22:03.490 align:middle line:84% But it does work quite well for these. 00:22:03.490 --> 00:22:04.810 align:middle line:90% OK. 00:22:04.810 --> 00:22:07.370 align:middle line:84% So using these assumptions, what can we do? 00:22:07.370 --> 00:22:13.370 align:middle line:84% We can say, again, that B phi is roughly constant, which 00:22:13.370 --> 00:22:20.720 align:middle line:84% means it's also roughly equal to the average value of B phi. 00:22:20.720 --> 00:22:22.970 align:middle line:84% But to take into account the fact that B phi probably 00:22:22.970 --> 00:22:28.220 align:middle line:84% does vary just a little bit, I'm going to write this as B phi 00:22:28.220 --> 00:22:33.800 align:middle line:84% at the edge, B phi A, plus some small quantity delta B 00:22:33.800 --> 00:22:37.690 align:middle line:84% phi, which will vary as a function of R. 00:22:37.690 --> 00:22:41.030 align:middle line:84% If I take the square of this then, 00:22:41.030 --> 00:22:44.360 align:middle line:84% B phi squared, I'm going to get something that's 00:22:44.360 --> 00:22:55.560 align:middle line:84% roughly equal to B phi A squared plus 2 delta B E phi A 00:22:55.560 --> 00:23:00.270 align:middle line:84% plus a term, which is quadratic in this small quantity, delta B 00:23:00.270 --> 00:23:01.870 align:middle line:90% phi. 00:23:01.870 --> 00:23:04.030 align:middle line:84% And we will say that we are going 00:23:04.030 --> 00:23:06.530 align:middle line:90% to drop the quadratic terms. 00:23:06.530 --> 00:23:08.660 align:middle line:84% And we're just going to keep the linear terms here. 00:23:08.660 --> 00:23:11.368 align:middle line:84% So this is a standard perturbation theory trick. 00:23:11.368 --> 00:23:13.160 align:middle line:84% And if you squint at this for a little bit, 00:23:13.160 --> 00:23:20.030 align:middle line:84% you'll realize you can then write B phi A squared 00:23:20.030 --> 00:23:30.280 align:middle line:84% minus the average of B phi squared is equal to 2 00:23:30.280 --> 00:23:41.460 align:middle line:84% B phi A B phi A minus the average of B phi. 00:23:41.460 --> 00:23:48.870 align:middle line:84% Conveniently, this term here is the same as this term here. 00:23:48.870 --> 00:23:50.670 align:middle line:84% But we've now rewritten it in terms 00:23:50.670 --> 00:23:52.840 align:middle line:90% of things which we can measure. 00:23:52.840 --> 00:23:55.450 align:middle line:84% So we can now finally make some progress. 00:23:55.450 --> 00:23:59.460 align:middle line:84% I will say, I find this derivation slightly hand-wavy. 00:23:59.460 --> 00:24:01.890 align:middle line:84% And it's not in Hutchinson's book. 00:24:01.890 --> 00:24:04.380 align:middle line:84% He skips straight to the result. So you 00:24:04.380 --> 00:24:06.150 align:middle line:84% might want to sit down and work it out 00:24:06.150 --> 00:24:08.610 align:middle line:84% yourself and convince yourself that it's really true. 00:24:08.610 --> 00:24:11.160 align:middle line:84% We're definitely playing fast and loose with our averages 00:24:11.160 --> 00:24:11.920 align:middle line:90% as we do this. 00:24:11.920 --> 00:24:14.340 align:middle line:84% So this suddenly randomly including 00:24:14.340 --> 00:24:16.950 align:middle line:84% these angle bracket signs is maybe a little bit dodgy. 00:24:16.950 --> 00:24:18.600 align:middle line:90% So give it a go. 00:24:18.600 --> 00:24:21.605 align:middle line:90% You might find it interesting. 00:24:21.605 --> 00:24:22.105 align:middle line:90% OK. 00:24:22.105 --> 00:24:24.650 align:middle line:90% 00:24:24.650 --> 00:24:33.580 align:middle line:84% So that means we can now write our colloidal beta 00:24:33.580 --> 00:24:42.738 align:middle line:84% as roughly equal to 1 plus 2 B phi A 00:24:42.738 --> 00:24:59.100 align:middle line:84% by A minus the average of B phi over B theta A squared. 00:24:59.100 --> 00:25:00.800 align:middle line:84% And if I go back and I have a look 00:25:00.800 --> 00:25:05.570 align:middle line:84% at my little straight tokamak, I can put in a B-dot here 00:25:05.570 --> 00:25:08.405 align:middle line:90% to measure B theta A. 00:25:08.405 --> 00:25:13.945 align:middle line:84% I can put it in a loop that's aligned with phi. 00:25:13.945 --> 00:25:18.750 align:middle line:84% And I can measure B phi A. And as I've already said, 00:25:18.750 --> 00:25:22.200 align:middle line:84% I can put a big loop that goes around the entire plasma 00:25:22.200 --> 00:25:27.150 align:middle line:84% and measure the average value across the plasma of B phi. 00:25:27.150 --> 00:25:33.100 align:middle line:84% If I number those 1, 2, and 3, you 00:25:33.100 --> 00:25:35.470 align:middle line:84% can see that we have now measured 00:25:35.470 --> 00:25:37.900 align:middle line:90% all the things that we wanted. 00:25:37.900 --> 00:25:38.950 align:middle line:90% Where's 2? 00:25:38.950 --> 00:25:43.580 align:middle line:90% This one, this one, this one. 00:25:43.580 --> 00:25:45.950 align:middle line:84% Great, so it's pretty clear that we can actually 00:25:45.950 --> 00:25:49.160 align:middle line:84% measure the average pressure inside our plasma using 00:25:49.160 --> 00:25:54.680 align:middle line:84% just three loops and a lot of assumption. 00:25:54.680 --> 00:25:57.952 align:middle line:84% But these assumptions are reasonable for the sorts 00:25:57.952 --> 00:25:59.410 align:middle line:84% of plasmas that many of us work on. 00:25:59.410 --> 00:26:01.160 align:middle line:84% Any questions on that before we keep going 00:26:01.160 --> 00:26:03.125 align:middle line:84% and I show you how to get the [INAUDIBLE]? 00:26:03.125 --> 00:26:24.420 align:middle line:90% 00:26:24.420 --> 00:26:29.457 align:middle line:84% Let's briefly return to our definition for beta phi 00:26:29.457 --> 00:26:32.040 align:middle line:84% because this tells us something about the [? damn ?] magnetism 00:26:32.040 --> 00:26:33.340 align:middle line:90% that I hinted at earlier. 00:26:33.340 --> 00:26:36.930 align:middle line:84% So beta phi we can write very simply as beta theta. 00:26:36.930 --> 00:26:40.470 align:middle line:84% But we need to swap out the magnetic field 00:26:40.470 --> 00:26:42.940 align:middle line:84% theta in beta theta for a beta phi field. 00:26:42.940 --> 00:26:47.640 align:middle line:84% And so we just multiply this by beta theta A squared-- 00:26:47.640 --> 00:26:54.280 align:middle line:84% sorry, not beta, B theta A squared over B phi A squared. 00:26:54.280 --> 00:26:56.940 align:middle line:84% And if you remember what that looked like, 00:26:56.940 --> 00:26:59.940 align:middle line:84% you're just swapping out the B thetas for the B phis. 00:26:59.940 --> 00:27:02.790 align:middle line:84% And so then you'll have your definition of B phi. 00:27:02.790 --> 00:27:06.630 align:middle line:84% And if we pop in this B theta definition, 00:27:06.630 --> 00:27:14.860 align:middle line:84% we'd end up with B theta squared A over B phi squared 00:27:14.860 --> 00:27:23.780 align:middle line:84% at the edge plus 2 times 1 minus the average value of B 00:27:23.780 --> 00:27:27.815 align:middle line:90% phi over B phi at the edge. 00:27:27.815 --> 00:27:32.810 align:middle line:90% 00:27:32.810 --> 00:27:34.000 align:middle line:90% I can't go down low enough. 00:27:34.000 --> 00:27:37.240 align:middle line:84% But if you look at this, you multiply it 00:27:37.240 --> 00:27:41.500 align:middle line:84% by that ratio of B theta over B phi. 00:27:41.500 --> 00:27:43.315 align:middle line:90% The 1 picks up that term. 00:27:43.315 --> 00:27:47.300 align:middle line:84% The B theta conveniently cancels with the denominator on this. 00:27:47.300 --> 00:27:49.860 align:middle line:84% We end up with something like this. 00:27:49.860 --> 00:27:51.953 align:middle line:84% We can go further than this for the assumptions 00:27:51.953 --> 00:27:53.370 align:middle line:84% that we've made because we already 00:27:53.370 --> 00:27:57.190 align:middle line:84% assumed that the poloidal magnetic field is weak compared 00:27:57.190 --> 00:27:58.850 align:middle line:90% to the toroidal magnetic field. 00:27:58.850 --> 00:28:01.290 align:middle line:90% So we'll drop this. 00:28:01.290 --> 00:28:06.330 align:middle line:84% Now, can anyone tell me any bounds on beta? 00:28:06.330 --> 00:28:08.650 align:middle line:90% STUDENT: [INAUDIBLE] 00:28:08.650 --> 00:28:10.128 align:middle line:90% 00:28:10.128 --> 00:28:11.170 align:middle line:90% JACK HARE: It's positive. 00:28:11.170 --> 00:28:12.250 align:middle line:90% Why is it positive? 00:28:12.250 --> 00:28:14.390 align:middle line:90% STUDENT: [INAUDIBLE] 00:28:14.390 --> 00:28:21.240 align:middle line:84% JACK HARE: So the definition is 2 mu 0 over B phi A squared 00:28:21.240 --> 00:28:22.440 align:middle line:90% average pressure. 00:28:22.440 --> 00:28:24.690 align:middle line:84% Every single one of those terms is positive. 00:28:24.690 --> 00:28:26.220 align:middle line:90% So beta phi is positive. 00:28:26.220 --> 00:28:28.298 align:middle line:90% It has to be. 00:28:28.298 --> 00:28:30.340 align:middle line:84% There might be other limits to MHD instabilities. 00:28:30.340 --> 00:28:32.930 align:middle line:84% But this is the most basic of them. 00:28:32.930 --> 00:28:39.050 align:middle line:84% That also implies that this term is positive because that's 00:28:39.050 --> 00:28:42.130 align:middle line:90% just the same as beta phi. 00:28:42.130 --> 00:28:47.650 align:middle line:84% So from that, we can see that B phi at the edge 00:28:47.650 --> 00:28:49.360 align:middle line:84% here is always going to be larger 00:28:49.360 --> 00:28:53.550 align:middle line:90% than the average value of B phi. 00:28:53.550 --> 00:28:57.460 align:middle line:84% If that were not true, then this could be a negative number, 00:28:57.460 --> 00:28:59.873 align:middle line:90% which would be bad. 00:28:59.873 --> 00:29:02.040 align:middle line:84% And what this is effectively telling us is something 00:29:02.040 --> 00:29:04.990 align:middle line:84% that you already know, but is nice to show just 00:29:04.990 --> 00:29:09.850 align:middle line:84% using some MHD, which is-- if you think about your tokamak 00:29:09.850 --> 00:29:13.000 align:middle line:84% as a function of major radius, you've 00:29:13.000 --> 00:29:17.730 align:middle line:84% got your magnetic field, which is falling off as 1 upon R, 00:29:17.730 --> 00:29:18.780 align:middle line:90% like that. 00:29:18.780 --> 00:29:23.780 align:middle line:84% The cross-section of your tokamak is something like this. 00:29:23.780 --> 00:29:29.230 align:middle line:84% We have decreed that, in fact, B phi at the edge 00:29:29.230 --> 00:29:30.370 align:middle line:90% is the same on either side. 00:29:30.370 --> 00:29:32.680 align:middle line:84% So maybe I need to go a little bit further out 00:29:32.680 --> 00:29:34.300 align:middle line:90% until this line flattens off. 00:29:34.300 --> 00:29:37.510 align:middle line:84% But what this is saying is that the average magnetic field 00:29:37.510 --> 00:29:40.060 align:middle line:84% has to be less than the magnet field of the edges, which 00:29:40.060 --> 00:29:42.010 align:middle line:84% means that this magnetic field has 00:29:42.010 --> 00:29:44.403 align:middle line:90% a lovely little dip inside it. 00:29:44.403 --> 00:29:46.570 align:middle line:84% So the magnetic field is less than you would predict 00:29:46.570 --> 00:29:49.000 align:middle line:84% from the 1 over R. And it is, indeed, 00:29:49.000 --> 00:29:55.610 align:middle line:84% that dip that the thermal pressure sits inside, there. 00:29:55.610 --> 00:29:58.550 align:middle line:84% So the plasma reduces the magnetic field. 00:29:58.550 --> 00:30:07.300 align:middle line:90% 00:30:07.300 --> 00:30:10.440 align:middle line:84% And that is the definition of a diamagnetic substance. 00:30:10.440 --> 00:30:17.030 align:middle line:90% 00:30:17.030 --> 00:30:18.780 align:middle line:90% Any questions on that? 00:30:18.780 --> 00:30:19.290 align:middle line:90% Yes? 00:30:19.290 --> 00:30:21.705 align:middle line:84% STUDENT: [INAUDIBLE] magnetic [INAUDIBLE]? 00:30:21.705 --> 00:30:28.535 align:middle line:90% 00:30:28.535 --> 00:30:30.410 align:middle line:84% JACK HARE: If there was no pressure gradient, 00:30:30.410 --> 00:30:31.460 align:middle line:90% there'd be no plasma. 00:30:31.460 --> 00:30:33.112 align:middle line:90% And so it wouldn't happen. 00:30:33.112 --> 00:30:34.320 align:middle line:90% There would just be a vacuum. 00:30:34.320 --> 00:30:35.990 align:middle line:84% You can't have a plasma with pressure gradient 00:30:35.990 --> 00:30:37.790 align:middle line:84% because you need to have a plasma which 00:30:37.790 --> 00:30:41.200 align:middle line:84% has 0 pressure A because otherwise you don't 00:30:41.200 --> 00:30:42.293 align:middle line:90% have the edge of a plasma. 00:30:42.293 --> 00:30:44.710 align:middle line:84% And if you have no gradients, then the only other pressure 00:30:44.710 --> 00:30:46.877 align:middle line:84% available to the plasma throughout the entire volume 00:30:46.877 --> 00:30:48.241 align:middle line:90% is 0, yeah. 00:30:48.241 --> 00:30:50.596 align:middle line:90% STUDENT: [INAUDIBLE] 00:30:50.596 --> 00:30:55.495 align:middle line:90% 00:30:55.495 --> 00:30:57.620 align:middle line:84% JACK HARE: It would, under these strong assumptions 00:30:57.620 --> 00:30:58.827 align:middle line:90% that we've made. 00:30:58.827 --> 00:31:01.160 align:middle line:84% And we're going to talk about relaxing these assumptions 00:31:01.160 --> 00:31:02.952 align:middle line:84% and finding a plasma which is paramagnetic, 00:31:02.952 --> 00:31:05.520 align:middle line:84% which actually enhances the magnetic field, in a moment. 00:31:05.520 --> 00:31:07.790 align:middle line:84% But for something like a tokamak, 00:31:07.790 --> 00:31:11.090 align:middle line:84% then this is always paramagnetic-- 00:31:11.090 --> 00:31:14.430 align:middle line:90% diamagnetic, sorry, yeah. 00:31:14.430 --> 00:31:16.790 align:middle line:90% Other questions? 00:31:16.790 --> 00:31:18.380 align:middle line:90% Hm? 00:31:18.380 --> 00:31:20.632 align:middle line:90% STUDENT: [INAUDIBLE] 00:31:20.632 --> 00:31:36.940 align:middle line:90% 00:31:36.940 --> 00:31:40.830 align:middle line:84% JACK HARE: I mean, this is a pretty good model. 00:31:40.830 --> 00:31:43.230 align:middle line:84% You can go to Hutchinson's textbook 00:31:43.230 --> 00:31:45.540 align:middle line:84% and see measurements he made in 1973 when 00:31:45.540 --> 00:31:48.750 align:middle line:84% I assume his mustache was absolutely brilliant. 00:31:48.750 --> 00:31:50.167 align:middle line:84% And he does a very good job there. 00:31:50.167 --> 00:31:52.417 align:middle line:84% The error bars are quite tight on measuring the energy 00:31:52.417 --> 00:31:53.200 align:middle line:90% confinement time. 00:31:53.200 --> 00:31:55.710 align:middle line:84% So this was possible in the '70s. 00:31:55.710 --> 00:31:57.547 align:middle line:84% So I imagine it is still possible now. 00:31:57.547 --> 00:31:59.880 align:middle line:84% We're going to go on a little bit about how difficult it 00:31:59.880 --> 00:32:02.530 align:middle line:84% is because effectively you're trying to measure-- 00:32:02.530 --> 00:32:03.630 align:middle line:90% it's a very small dip. 00:32:03.630 --> 00:32:05.070 align:middle line:90% I've made it look very big here. 00:32:05.070 --> 00:32:06.570 align:middle line:84% But of course, the actual difference 00:32:06.570 --> 00:32:08.795 align:middle line:84% between the toroidal magnetic field 00:32:08.795 --> 00:32:10.920 align:middle line:84% at the edge and the average toroidal magnetic field 00:32:10.920 --> 00:32:13.450 align:middle line:84% is very small because the beta is so low. 00:32:13.450 --> 00:32:15.450 align:middle line:84% So this is a very difficult measurement to make. 00:32:15.450 --> 00:32:17.820 align:middle line:84% But as Hutchinson said in his book, 00:32:17.820 --> 00:32:21.210 align:middle line:84% if you do it very carefully, you can get reasonable results 00:32:21.210 --> 00:32:21.760 align:middle line:90% from it. 00:32:21.760 --> 00:32:25.410 align:middle line:84% So yeah, I think it is a real technique that one can use. 00:32:25.410 --> 00:32:26.985 align:middle line:90% Yeah? 00:32:26.985 --> 00:32:29.235 align:middle line:84% STUDENT: [INAUDIBLE] so this A [INAUDIBLE] first where 00:32:29.235 --> 00:32:31.900 align:middle line:90% [INAUDIBLE]? 00:32:31.900 --> 00:32:33.940 align:middle line:84% JACK HARE: Yes, the A refers to a point 00:32:33.940 --> 00:32:35.080 align:middle line:90% where pressure goes to 0. 00:32:35.080 --> 00:32:37.510 align:middle line:84% I'm saying at the edge of the vacuum chamber 00:32:37.510 --> 00:32:39.410 align:middle line:84% because it definitely should go to 0 there. 00:32:39.410 --> 00:32:42.610 align:middle line:84% But if you want to put A further out, you can also do that, yeah. 00:32:42.610 --> 00:32:45.310 align:middle line:84% STUDENT: Does it matter [INAUDIBLE]? 00:32:45.310 --> 00:32:46.842 align:middle line:84% JACK HARE: Oh, it absolutely matters 00:32:46.842 --> 00:32:48.800 align:middle line:84% that it's outside the last closed flux surface. 00:32:48.800 --> 00:32:51.500 align:middle line:84% You cannot have any plasma further out than this probe 00:32:51.500 --> 00:32:53.300 align:middle line:84% because that plasma could carry current. 00:32:53.300 --> 00:32:55.175 align:middle line:84% And that would violate all of the assumptions 00:32:55.175 --> 00:32:56.170 align:middle line:90% we've made so far. 00:32:56.170 --> 00:32:57.628 align:middle line:84% STUDENT: So if there's [INAUDIBLE]? 00:32:57.628 --> 00:33:00.730 align:middle line:90% 00:33:00.730 --> 00:33:02.088 align:middle line:90% JACK HARE: I don't understand. 00:33:02.088 --> 00:33:04.630 align:middle line:84% I would never put a B-dot probe inside [? the ?] [? last-- ?] 00:33:04.630 --> 00:33:06.940 align:middle line:84% even anywhere close to the last closed vector. 00:33:06.940 --> 00:33:09.460 align:middle line:84% You'd have them very far away because of reasons mentioned 00:33:09.460 --> 00:33:11.390 align:middle line:90% earlier, that they would melt. 00:33:11.390 --> 00:33:12.470 align:middle line:90% STUDENT: Right, OK. 00:33:12.470 --> 00:33:14.178 align:middle line:84% JACK HARE: So let me give you an example. 00:33:14.178 --> 00:33:17.620 align:middle line:84% If this is your tokamak with a divertor, 00:33:17.620 --> 00:33:21.040 align:middle line:84% we've got our core plasma and we've got all of this stuff 00:33:21.040 --> 00:33:22.630 align:middle line:90% here. 00:33:22.630 --> 00:33:25.480 align:middle line:84% You can just put your B-dots out here, somewhere where 00:33:25.480 --> 00:33:27.250 align:middle line:90% the pressure is equal to 0. 00:33:27.250 --> 00:33:30.070 align:middle line:84% The pressure is not equal to 0 in the last closed flux service. 00:33:30.070 --> 00:33:32.380 align:middle line:84% And it's not equal to 0 in the core. 00:33:32.380 --> 00:33:35.050 align:middle line:84% And that effectively reflects that there is some plasma there. 00:33:35.050 --> 00:33:40.060 align:middle line:84% That plasma could carry some toroidal current density. 00:33:40.060 --> 00:33:42.737 align:middle line:84% And that would mess with of the magnetic field calculations 00:33:42.737 --> 00:33:44.320 align:middle line:84% we've been doing where we've carefully 00:33:44.320 --> 00:33:48.340 align:middle line:84% assumed that there is no further plasma carrying current outside. 00:33:48.340 --> 00:33:50.590 align:middle line:84% So we certainly couldn't suddenly 00:33:50.590 --> 00:33:53.880 align:middle line:84% introduce a blob of plasma out here. 00:33:53.880 --> 00:33:56.402 align:middle line:90% That would ruin our day, yeah. 00:33:56.402 --> 00:33:58.807 align:middle line:90% STUDENT: [INAUDIBLE] 00:33:58.807 --> 00:34:05.070 align:middle line:90% 00:34:05.070 --> 00:34:07.300 align:middle line:84% JACK HARE: I mean, the pressure of the plasma 00:34:07.300 --> 00:34:08.409 align:middle line:90% has to go to 0 somewhere. 00:34:08.409 --> 00:34:10.449 align:middle line:84% Otherwise, it has no edge, right? 00:34:10.449 --> 00:34:12.670 align:middle line:84% And you can define where that goes, like you say, 00:34:12.670 --> 00:34:15.260 align:middle line:84% by sticking a limiter in, like that. 00:34:15.260 --> 00:34:18.260 align:middle line:84% And so you can put it anywhere that is behind the limiter, 00:34:18.260 --> 00:34:18.760 align:middle line:90% yeah. 00:34:18.760 --> 00:34:19.659 align:middle line:90% STUDENT: OK, cool. 00:34:19.659 --> 00:34:20.949 align:middle line:90% Anywhere? 00:34:20.949 --> 00:34:23.605 align:middle line:84% JACK HARE: Anywhere where the plasma pressure is 0, yes. 00:34:23.605 --> 00:34:26.235 align:middle line:90% 00:34:26.235 --> 00:34:29.600 align:middle line:84% Yeah, I see a question from online. 00:34:29.600 --> 00:34:30.929 align:middle line:90% STUDENT: Hi there. 00:34:30.929 --> 00:34:37.014 align:middle line:84% So my question is about the B phi at the edge on both sides. 00:34:37.014 --> 00:34:38.389 align:middle line:84% You said at one point that we are 00:34:38.389 --> 00:34:40.940 align:middle line:84% like confining them to be the same on either edge. 00:34:40.940 --> 00:34:43.159 align:middle line:90% And so it's the dip? 00:34:43.159 --> 00:34:46.409 align:middle line:84% JACK HARE: Yeah, so we made an assumption here 00:34:46.409 --> 00:34:48.699 align:middle line:90% that B phi was roughly constant. 00:34:48.699 --> 00:34:50.598 align:middle line:90% So all I'm saying-- 00:34:50.598 --> 00:34:54.150 align:middle line:84% probably I shouldn't have drawn the diagram exactly like this. 00:34:54.150 --> 00:35:00.210 align:middle line:84% In the model that we're doing at the moment, what we have is 00:35:00.210 --> 00:35:03.780 align:middle line:90% a plasma like this. 00:35:03.780 --> 00:35:06.150 align:middle line:84% And we're saying that the magnetic field 00:35:06.150 --> 00:35:15.792 align:middle line:84% up to the edge of our plasma is some value B phi A. 00:35:15.792 --> 00:35:18.720 align:middle line:84% And inside that, it has some dip. 00:35:18.720 --> 00:35:20.490 align:middle line:84% We don't know the exact shape of the dip. 00:35:20.490 --> 00:35:24.060 align:middle line:84% That's going to be due to the current profile. 00:35:24.060 --> 00:35:28.270 align:middle line:84% So the dip could look like this, for example. 00:35:28.270 --> 00:35:31.515 align:middle line:90% But I'm just-- some sort of dip. 00:35:31.515 --> 00:35:32.390 align:middle line:90% Does that make sense? 00:35:32.390 --> 00:35:33.723 align:middle line:90% STUDENT: Yeah, that makes sense. 00:35:33.723 --> 00:35:36.810 align:middle line:90% And then the thermal energy is-- 00:35:36.810 --> 00:35:39.000 align:middle line:84% the thermal-- what did you say about thermal? 00:35:39.000 --> 00:35:39.870 align:middle line:90% I missed that part. 00:35:39.870 --> 00:35:41.100 align:middle line:84% You said there was a thermal energy-- 00:35:41.100 --> 00:35:42.000 align:middle line:84% JACK HARE: Well, we'll go through it in a little bit 00:35:42.000 --> 00:35:43.000 align:middle line:90% more detail in a moment. 00:35:43.000 --> 00:35:45.918 align:middle line:84% But the thermal pressure here is just the thermal energy density. 00:35:45.918 --> 00:35:48.210 align:middle line:84% So you can measure the thermal energy inside the plasma 00:35:48.210 --> 00:35:51.220 align:middle line:84% now that you know the average thermal energy density. 00:35:51.220 --> 00:35:51.880 align:middle line:90% STUDENT: OK. 00:35:51.880 --> 00:35:53.740 align:middle line:84% JACK HARE: Yeah, we'll get to that in a second, yeah. 00:35:53.740 --> 00:35:54.740 align:middle line:90% OK, more questions here? 00:35:54.740 --> 00:35:55.463 align:middle line:90% Yeah? 00:35:55.463 --> 00:35:57.140 align:middle line:90% STUDENT: [INAUDIBLE] 00:35:57.140 --> 00:35:58.940 align:middle line:90% 00:35:58.940 --> 00:36:00.470 align:middle line:84% JACK HARE: We haven't got there yet. 00:36:00.470 --> 00:36:03.797 align:middle line:84% If you'll wait five minutes, I will get there. 00:36:03.797 --> 00:36:04.880 align:middle line:90% The big reveal at the end. 00:36:04.880 --> 00:36:06.812 align:middle line:90% OK, other questions? 00:36:06.812 --> 00:36:09.242 align:middle line:90% STUDENT: [INAUDIBLE] 00:36:09.242 --> 00:36:17.040 align:middle line:90% 00:36:17.040 --> 00:36:19.680 align:middle line:84% JACK HARE: Well, the way we've done this is we 00:36:19.680 --> 00:36:23.130 align:middle line:84% have taken such a large aspect ratio 00:36:23.130 --> 00:36:27.640 align:middle line:84% that our 1 over R is effectively constant 00:36:27.640 --> 00:36:29.215 align:middle line:90% where we've chosen this. 00:36:29.215 --> 00:36:32.620 align:middle line:84% This is our unwrapping of our torus into a cylinder. 00:36:32.620 --> 00:36:33.647 align:middle line:90% It's not true. 00:36:33.647 --> 00:36:35.980 align:middle line:84% And we will talk a little bit about the not truthfulness 00:36:35.980 --> 00:36:38.850 align:middle line:90% of it in a moment as well. 00:36:38.850 --> 00:36:42.350 align:middle line:90% Any other questions? 00:36:42.350 --> 00:36:45.730 align:middle line:84% Cool, I'm just going to raise these answers to questions. 00:36:45.730 --> 00:36:46.870 align:middle line:90% And then I'll keep going. 00:36:46.870 --> 00:36:51.310 align:middle line:90% 00:36:51.310 --> 00:36:54.150 align:middle line:84% So I want to briefly touch on times when 00:36:54.150 --> 00:36:56.445 align:middle line:90% we have paramagnetism instead. 00:36:56.445 --> 00:37:07.720 align:middle line:90% 00:37:07.720 --> 00:37:12.570 align:middle line:84% So in the case where our poloidal magnetic field is not 00:37:12.570 --> 00:37:19.393 align:middle line:84% necessarily much smaller than our toroidal magnetic field, 00:37:19.393 --> 00:37:20.810 align:middle line:84% that doesn't necessarily mean it's 00:37:20.810 --> 00:37:22.550 align:middle line:84% bigger than the toroidal magnetic field. 00:37:22.550 --> 00:37:25.920 align:middle line:84% It might just be that it's not negligible. 00:37:25.920 --> 00:37:29.400 align:middle line:84% If you go and you work through this equation and stare at it 00:37:29.400 --> 00:37:32.220 align:middle line:84% for a little bit, it's actually pretty obvious 00:37:32.220 --> 00:37:38.150 align:middle line:84% that you can end up with a situation in which B phi is 00:37:38.150 --> 00:37:43.490 align:middle line:84% greater than B phi A, where the average B phi is greater 00:37:43.490 --> 00:37:54.510 align:middle line:84% than B phi A. So that would be a situation which looks like that. 00:37:54.510 --> 00:37:57.450 align:middle line:84% Anyone got any idea what's going on here, why that could happen? 00:37:57.450 --> 00:38:03.180 align:middle line:90% 00:38:03.180 --> 00:38:03.750 align:middle line:90% Yeah? 00:38:03.750 --> 00:38:05.714 align:middle line:90% STUDENT: [INAUDIBLE] 00:38:05.714 --> 00:38:07.747 align:middle line:90% 00:38:07.747 --> 00:38:08.830 align:middle line:90% JACK HARE: That'd be cool. 00:38:08.830 --> 00:38:11.690 align:middle line:90% 00:38:11.690 --> 00:38:14.085 align:middle line:84% Dynamo is forbidden in axisymmetric systems 00:38:14.085 --> 00:38:14.960 align:middle line:90% by Cowling's theorem. 00:38:14.960 --> 00:38:16.070 align:middle line:90% So I think we can't have that. 00:38:16.070 --> 00:38:17.300 align:middle line:90% But OK, it'd be a good idea. 00:38:17.300 --> 00:38:18.664 align:middle line:90% Yeah? 00:38:18.664 --> 00:38:21.024 align:middle line:90% STUDENT: [INAUDIBLE] 00:38:21.024 --> 00:38:22.153 align:middle line:90% 00:38:22.153 --> 00:38:22.820 align:middle line:90% JACK HARE: Good. 00:38:22.820 --> 00:38:24.960 align:middle line:90% OK, so let's draw that out. 00:38:24.960 --> 00:38:28.300 align:middle line:84% So let's start with a big plasma that 00:38:28.300 --> 00:38:32.320 align:middle line:84% has got a uniform magnetic field across it. 00:38:32.320 --> 00:38:38.800 align:middle line:84% But this plasma also has some current going in this direction. 00:38:38.800 --> 00:38:46.740 align:middle line:90% 00:38:46.740 --> 00:38:53.730 align:middle line:84% And that current is generating a poloidal magnetic field, 00:38:53.730 --> 00:38:54.450 align:middle line:90% B theta. 00:38:54.450 --> 00:38:57.460 align:middle line:84% And this poloidal magnetic field is allowed inside the plasma 00:38:57.460 --> 00:38:58.290 align:middle line:90% as well, of course. 00:38:58.290 --> 00:39:00.840 align:middle line:90% 00:39:00.840 --> 00:39:03.245 align:middle line:84% This configuration gives rise to J 00:39:03.245 --> 00:39:07.430 align:middle line:84% cross B forces, which will always pinch inwards. 00:39:07.430 --> 00:39:08.990 align:middle line:90% These are pinching forces. 00:39:08.990 --> 00:39:12.290 align:middle line:84% This is like two wires attracting that you know about 00:39:12.290 --> 00:39:13.440 align:middle line:90% from electromagnetism. 00:39:13.440 --> 00:39:16.270 align:middle line:90% So this is your J cross B force. 00:39:16.270 --> 00:39:20.110 align:middle line:84% Now, the magnetic field inside a sufficiently conducting plasma 00:39:20.110 --> 00:39:21.680 align:middle line:90% is frozen to it. 00:39:21.680 --> 00:39:24.880 align:middle line:84% So as this plasma compresses, the magnetic field 00:39:24.880 --> 00:39:26.930 align:middle line:90% inside will also be compressed. 00:39:26.930 --> 00:39:31.540 align:middle line:84% So we'll end up with a situation where the plasma is now smaller, 00:39:31.540 --> 00:39:34.630 align:middle line:84% the magnetic field on the outside is still pinned, 00:39:34.630 --> 00:39:38.090 align:middle line:84% but the magnetic field on the inside 00:39:38.090 --> 00:39:40.827 align:middle line:84% can have some pattern which, again, 00:39:40.827 --> 00:39:43.160 align:middle line:84% is going to depend on some details of how the current is 00:39:43.160 --> 00:39:48.260 align:middle line:84% distributed such that this condition is fulfilled. 00:39:48.260 --> 00:39:51.180 align:middle line:90% 00:39:51.180 --> 00:39:55.140 align:middle line:84% And it turns out, if you go through mathematically, 00:39:55.140 --> 00:39:58.230 align:middle line:84% whether you get ferromagnetism or diamagnetism depends 00:39:58.230 --> 00:40:05.520 align:middle line:84% not on beta phi, but it depends on beta theta. 00:40:05.520 --> 00:40:10.260 align:middle line:84% And if you have beta theta less than 1, it's diamagnetic. 00:40:10.260 --> 00:40:13.190 align:middle line:84% And if you have beta theta greater than 1, 00:40:13.190 --> 00:40:15.210 align:middle line:90% it's paramagnetic. 00:40:15.210 --> 00:40:17.160 align:middle line:84% Don't ask me what happens if it's equal to 1, 00:40:17.160 --> 00:40:18.240 align:middle line:90% probably nothing. 00:40:18.240 --> 00:40:22.720 align:middle line:84% So cool, and so that situation, can anyone 00:40:22.720 --> 00:40:24.890 align:middle line:84% tell me some sort of device where this could happen? 00:40:24.890 --> 00:40:27.280 align:middle line:84% So this was relevant for stellarators and tokamaks. 00:40:27.280 --> 00:40:30.610 align:middle line:84% Any sorts of devices where this could occur? 00:40:30.610 --> 00:40:33.962 align:middle line:90% STUDENT: [INAUDIBLE] 00:40:33.962 --> 00:40:35.670 align:middle line:84% JACK HARE: Yeah, Z-pinches don't actually 00:40:35.670 --> 00:40:38.710 align:middle line:90% have a toroidal magnetic field. 00:40:38.710 --> 00:40:41.160 align:middle line:84% So it wouldn't work, but a screw pinch would, a generic 00:40:41.160 --> 00:40:42.858 align:middle line:90% screw pinch, Yeah. 00:40:42.858 --> 00:40:44.400 align:middle line:84% Something like a reversed field pinch 00:40:44.400 --> 00:40:46.120 align:middle line:90% is like the classic example. 00:40:46.120 --> 00:40:48.960 align:middle line:84% So reversed field pinches, which look topologically 00:40:48.960 --> 00:40:51.845 align:middle line:84% like tokamaks, fulfill this condition here. 00:40:51.845 --> 00:40:53.220 align:middle line:84% In fact, they have a region where 00:40:53.220 --> 00:40:55.290 align:middle line:84% the toroidal magnetic field goes to 0 00:40:55.290 --> 00:40:57.870 align:middle line:84% and reverses, hence reversed field pinch. 00:40:57.870 --> 00:41:00.780 align:middle line:84% And they definitely see this paramagnetic effect. 00:41:00.780 --> 00:41:03.940 align:middle line:90% 00:41:03.940 --> 00:41:05.310 align:middle line:90% Yeah? 00:41:05.310 --> 00:41:07.540 align:middle line:90% STUDENT: [INAUDIBLE] 00:41:07.540 --> 00:41:09.640 align:middle line:84% JACK HARE: Yeah, I think you probably would, yes. 00:41:09.640 --> 00:41:13.440 align:middle line:90% 00:41:13.440 --> 00:41:15.090 align:middle line:84% Sorry, for Columbia people, I think 00:41:15.090 --> 00:41:18.950 align:middle line:84% you would see this in a [? sparmac, ?] but I don't know. 00:41:18.950 --> 00:41:21.260 align:middle line:84% Any questions from Colombia or anyone else in the room? 00:41:21.260 --> 00:41:24.670 align:middle line:90% 00:41:24.670 --> 00:41:26.300 align:middle line:90% Yeah? 00:41:26.300 --> 00:41:28.460 align:middle line:90% STUDENT: [INAUDIBLE] 00:41:28.460 --> 00:41:35.697 align:middle line:90% 00:41:35.697 --> 00:41:37.573 align:middle line:84% JACK HARE: Ah, that's a good question. 00:41:37.573 --> 00:41:38.990 align:middle line:84% I guess the way we're coming at it 00:41:38.990 --> 00:41:40.760 align:middle line:84% here is what we can measure and so what 00:41:40.760 --> 00:41:43.250 align:middle line:84% we can learn about our plasma rather than 00:41:43.250 --> 00:41:45.320 align:middle line:90% what we can do to our plasma. 00:41:45.320 --> 00:41:48.870 align:middle line:84% So I'm not sure Hutchinson goes into much detail on this. 00:41:48.870 --> 00:41:52.860 align:middle line:84% I'm not sure the answer to your question there. 00:41:52.860 --> 00:41:54.690 align:middle line:84% Yeah, I don't have a good answer. 00:41:54.690 --> 00:41:57.620 align:middle line:90% 00:41:57.620 --> 00:41:58.890 align:middle line:90% STUDENT: I have a question. 00:41:58.890 --> 00:42:00.760 align:middle line:90% JACK HARE: Yes, please. 00:42:00.760 --> 00:42:03.560 align:middle line:84% STUDENT: So how does this relate to the diamagnetic drift? 00:42:03.560 --> 00:42:06.080 align:middle line:84% Often in a tokamak, if it's diamagnetic, 00:42:06.080 --> 00:42:08.750 align:middle line:84% there will be some poloidal drift associated with it. 00:42:08.750 --> 00:42:11.870 align:middle line:84% But does that drift [? switch ?] direction 00:42:11.870 --> 00:42:14.082 align:middle line:90% in the paramagnetic case? 00:42:14.082 --> 00:42:15.790 align:middle line:84% JACK HARE: Yeah, that's a great question. 00:42:15.790 --> 00:42:17.415 align:middle line:84% So if people didn't hear, how does this 00:42:17.415 --> 00:42:19.390 align:middle line:84% relate to diamagnetic drifts that we talk about 00:42:19.390 --> 00:42:20.980 align:middle line:90% from single particle motion? 00:42:20.980 --> 00:42:22.710 align:middle line:84% The answer is this is equivalent. 00:42:22.710 --> 00:42:24.270 align:middle line:84% So you can get these same results 00:42:24.270 --> 00:42:26.790 align:middle line:84% by following single particles as they orbit. 00:42:26.790 --> 00:42:28.472 align:middle line:84% And you can also get it from MHD, 00:42:28.472 --> 00:42:30.930 align:middle line:84% which is quite nice because you can't always get everything 00:42:30.930 --> 00:42:32.710 align:middle line:90% single particle from MHD. 00:42:32.710 --> 00:42:36.100 align:middle line:84% But in this case, MHD retains that information. 00:42:36.100 --> 00:42:38.520 align:middle line:84% And so think, in the paramagnetic case, you're right. 00:42:38.520 --> 00:42:40.020 align:middle line:84% The drifts in the opposite direction 00:42:40.020 --> 00:42:43.250 align:middle line:84% would cancel the magnetic field, yeah. 00:42:43.250 --> 00:42:50.090 align:middle line:84% STUDENT: [INAUDIBLE] theta theta that [INAUDIBLE] definition? 00:42:50.090 --> 00:42:52.130 align:middle line:84% JACK HARE: No, it's still this one. 00:42:52.130 --> 00:42:56.870 align:middle line:90% 00:42:56.870 --> 00:42:58.520 align:middle line:84% So it's at that beta theta, yeah. 00:42:58.520 --> 00:43:03.250 align:middle line:90% 00:43:03.250 --> 00:43:07.060 align:middle line:84% All right, let's measure the energy confinement time. 00:43:07.060 --> 00:43:30.930 align:middle line:90% 00:43:30.930 --> 00:43:33.540 align:middle line:84% So all of this has led up to us being 00:43:33.540 --> 00:43:43.650 align:middle line:84% able to measure the volume averaged pressure. 00:43:43.650 --> 00:43:48.810 align:middle line:84% And we found that we could do that using B theta at the edge, 00:43:48.810 --> 00:43:55.770 align:middle line:84% B phi at the edge, and the average of B phi. 00:43:55.770 --> 00:43:57.390 align:middle line:84% The reason we might want to do that 00:43:57.390 --> 00:44:00.980 align:middle line:84% is that this average thermal pressure 00:44:00.980 --> 00:44:04.400 align:middle line:84% is just equal to the density times the temperature. 00:44:04.400 --> 00:44:06.140 align:middle line:84% Using temperature in energy units, 00:44:06.140 --> 00:44:07.970 align:middle line:84% we swallow the Boltzmann constant. 00:44:07.970 --> 00:44:17.250 align:middle line:84% And the stored thermal energy is just equal to this average 00:44:17.250 --> 00:44:22.020 align:middle line:84% energy density times the volume of our plasma times a factor 00:44:22.020 --> 00:44:24.522 align:middle line:84% of 3/2, which comes from equipartition in three 00:44:24.522 --> 00:44:26.730 align:middle line:84% dimensions And you've probably seen from ideal gases. 00:44:26.730 --> 00:44:28.188 align:middle line:84% And I'm not going to go through it. 00:44:28.188 --> 00:44:29.850 align:middle line:90% But just, it's there. 00:44:29.850 --> 00:44:31.460 align:middle line:90% Cool. 00:44:31.460 --> 00:44:37.190 align:middle line:84% So the trouble is measuring this is still difficult because-- 00:44:37.190 --> 00:44:38.540 align:middle line:90% where's my board with it on-- 00:44:38.540 --> 00:44:43.810 align:middle line:90% 00:44:43.810 --> 00:44:46.840 align:middle line:84% the thing that we're measuring in order to get this pressure 00:44:46.840 --> 00:44:49.190 align:middle line:84% is the ratio of these two things. 00:44:49.190 --> 00:44:52.977 align:middle line:84% And it turns out that this dip is going to be very, very small. 00:44:52.977 --> 00:44:55.310 align:middle line:84% And so it's going to be very, very hard to measure this. 00:44:55.310 --> 00:44:57.810 align:middle line:84% So I'm going to put that back now. 00:44:57.810 --> 00:45:00.560 align:middle line:90% 00:45:00.560 --> 00:45:04.820 align:middle line:84% The thing we're trying to measure, all of the errors 00:45:04.820 --> 00:45:07.985 align:middle line:84% are going to come from the average value of B 00:45:07.985 --> 00:45:12.150 align:middle line:90% phi over B phi at the edge. 00:45:12.150 --> 00:45:15.080 align:middle line:90% And this is going to be-- 00:45:15.080 --> 00:45:17.730 align:middle line:84% it's going to vary within about 10%. 00:45:17.730 --> 00:45:21.330 align:middle line:84% So it's not really the right use of approximately equal sign, 00:45:21.330 --> 00:45:24.510 align:middle line:90% so 10% variation. 00:45:24.510 --> 00:45:28.210 align:middle line:90% 00:45:28.210 --> 00:45:30.030 align:middle line:84% So this is difficult to measure accurately. 00:45:30.030 --> 00:45:31.650 align:middle line:90% But you can still do it. 00:45:31.650 --> 00:45:35.020 align:middle line:90% It is hard, but possible. 00:45:35.020 --> 00:45:38.630 align:middle line:84% And as I said, you should have a look in the textbook. 00:45:38.630 --> 00:45:42.810 align:middle line:84% There's a very nice figure from a tokamak, I think in Australia. 00:45:42.810 --> 00:45:46.353 align:middle line:84% And they still had tokamaks where they did this. 00:45:46.353 --> 00:45:48.020 align:middle line:84% And the reason you might want to do that 00:45:48.020 --> 00:45:51.080 align:middle line:84% is because you might want to calculate the energy confinement 00:45:51.080 --> 00:45:52.610 align:middle line:90% time. 00:45:52.610 --> 00:45:56.990 align:middle line:84% So can anyone tell me a useful definition for the energy 00:45:56.990 --> 00:45:58.340 align:middle line:90% confinement time? 00:45:58.340 --> 00:46:00.200 align:middle line:84% And you can do it in words, if you 00:46:00.200 --> 00:46:01.760 align:middle line:90% don't want to do it in formula. 00:46:01.760 --> 00:46:06.242 align:middle line:90% 00:46:06.242 --> 00:46:07.238 align:middle line:90% Mm-hmm? 00:46:07.238 --> 00:46:10.226 align:middle line:90% STUDENT: [INAUDIBLE] 00:46:10.226 --> 00:46:11.298 align:middle line:90% 00:46:11.298 --> 00:46:12.090 align:middle line:90% JACK HARE: Perfect. 00:46:12.090 --> 00:46:13.060 align:middle line:90% Thank you very much. 00:46:13.060 --> 00:46:16.765 align:middle line:84% So it is defined as the total stored energy 00:46:16.765 --> 00:46:20.500 align:middle line:90% W over the input power. 00:46:20.500 --> 00:46:22.510 align:middle line:84% And I'm writing that as a big P because it's 00:46:22.510 --> 00:46:23.890 align:middle line:90% not the same as our little p. 00:46:23.890 --> 00:46:26.170 align:middle line:90% It's the power going in. 00:46:26.170 --> 00:46:32.240 align:middle line:90% So as I said, this is simply-- 00:46:32.240 --> 00:46:33.990 align:middle line:84% well, I'm not going to write it out again. 00:46:33.990 --> 00:46:36.070 align:middle line:90% It's there. 00:46:36.070 --> 00:46:40.660 align:middle line:84% And this, or an ohmically driven, inductively driven 00:46:40.660 --> 00:46:42.730 align:middle line:84% tokamak, doesn't include all the other heating 00:46:42.730 --> 00:46:44.770 align:middle line:84% terms you might want to do is going 00:46:44.770 --> 00:46:52.580 align:middle line:84% to be I phi squared RP where this is the plasma resistivity 00:46:52.580 --> 00:46:53.900 align:middle line:90% or plasma resistance. 00:46:53.900 --> 00:46:56.570 align:middle line:84% So we're treating the entire plasma, the entire donut 00:46:56.570 --> 00:47:07.560 align:middle line:84% here, as some sort of resistor with resistance RP, like that. 00:47:07.560 --> 00:47:08.160 align:middle line:90% OK. 00:47:08.160 --> 00:47:11.640 align:middle line:84% And if you crank the handle on all of this, 00:47:11.640 --> 00:47:15.120 align:middle line:84% you remember the volume of your torus 00:47:15.120 --> 00:47:21.660 align:middle line:84% is pi A squared times 2 pi r, at least 00:47:21.660 --> 00:47:26.610 align:middle line:84% for large aspect ratio torus so that we have a simple formula. 00:47:26.610 --> 00:47:32.160 align:middle line:84% And you remember that your toroidal make-- 00:47:32.160 --> 00:47:34.320 align:middle line:84% your toroidal current from Ampere's law 00:47:34.320 --> 00:47:45.640 align:middle line:84% is going to be 2 pi A B theta at the edge over mu 0 squared. 00:47:45.640 --> 00:47:56.850 align:middle line:84% You can write all of this triumphantly as 3/8 mu 0 beta 00:47:56.850 --> 00:48:03.730 align:middle line:90% theta R over RP. 00:48:03.730 --> 00:48:06.572 align:middle line:84% I personally dislike the way that we use R here because these 00:48:06.572 --> 00:48:07.780 align:middle line:90% are completely different R's. 00:48:07.780 --> 00:48:09.800 align:middle line:84% It makes it look like it's a ratio of things 00:48:09.800 --> 00:48:10.842 align:middle line:90% with the same quantities. 00:48:10.842 --> 00:48:18.450 align:middle line:84% But this is the R for your torus, the major radius. 00:48:18.450 --> 00:48:22.830 align:middle line:84% And this, again as we said, is the plasma resistive, 00:48:22.830 --> 00:48:25.570 align:middle line:90% which is going to be-- 00:48:25.570 --> 00:48:29.320 align:middle line:90% let's see-- da, da, da-- 00:48:29.320 --> 00:48:31.860 align:middle line:90% it should go up with length. 00:48:31.860 --> 00:48:40.950 align:middle line:84% So 2 pi r dotted with the resistivity over pi A squared-- 00:48:40.950 --> 00:48:44.310 align:middle line:84% and that resistive we can just replace with a 1 00:48:44.310 --> 00:48:47.630 align:middle line:90% over the conductivity. 00:48:47.630 --> 00:48:48.350 align:middle line:90% Huzzah. 00:48:48.350 --> 00:48:49.700 align:middle line:90% So we've measured that already. 00:48:49.700 --> 00:48:52.320 align:middle line:84% Remember, we used two loops to measure that. 00:48:52.320 --> 00:48:57.480 align:middle line:84% We used a Rogowski and we measured B phi. 00:48:57.480 --> 00:49:00.820 align:middle line:84% And then we also need to know this. 00:49:00.820 --> 00:49:04.170 align:middle line:84% And again, we can measure that using 00:49:04.170 --> 00:49:17.262 align:middle line:84% three loops, which are B theta A B phi A and average of phi. 00:49:17.262 --> 00:49:22.400 align:middle line:84% Three loops-- so the conclusion of all this 00:49:22.400 --> 00:49:29.720 align:middle line:84% is 5 loops gets you tau E with a fair few assumptions 00:49:29.720 --> 00:49:30.870 align:middle line:90% along the way. 00:49:30.870 --> 00:49:31.730 align:middle line:90% But that's not bad. 00:49:31.730 --> 00:49:34.470 align:middle line:90% 00:49:34.470 --> 00:49:35.130 align:middle line:90% Questions? 00:49:35.130 --> 00:49:38.250 align:middle line:90% 00:49:38.250 --> 00:49:39.656 align:middle line:90% Yeah? 00:49:39.656 --> 00:49:42.046 align:middle line:90% STUDENT: [INAUDIBLE] 00:49:42.046 --> 00:49:45.400 align:middle line:90% 00:49:45.400 --> 00:49:47.950 align:middle line:84% JACK HARE: Yeah, I guess you can also 00:49:47.950 --> 00:49:52.600 align:middle line:84% extract rather more noisily the density at this point. 00:49:52.600 --> 00:49:59.110 align:middle line:84% So your argument is that from the conductivity 00:49:59.110 --> 00:50:00.850 align:middle line:90% we've also got temperature. 00:50:00.850 --> 00:50:03.460 align:middle line:84% And because we've got the average pressure, which 00:50:03.460 --> 00:50:06.640 align:middle line:84% is presumably going to be the average of n times t, 00:50:06.640 --> 00:50:10.360 align:middle line:84% we're going to be able to get out an estimate for n 00:50:10.360 --> 00:50:13.070 align:middle line:84% with lots of exciting averaging going on. 00:50:13.070 --> 00:50:15.460 align:middle line:84% But again, for five loops, that's not bad. 00:50:15.460 --> 00:50:17.470 align:middle line:84% So you're going to argue that we can also get n. 00:50:17.470 --> 00:50:19.600 align:middle line:84% And so we can get the Lawson criterion as well. 00:50:19.600 --> 00:50:20.475 align:middle line:90% I'll talk about that. 00:50:20.475 --> 00:50:21.970 align:middle line:90% That sounds like fun. 00:50:21.970 --> 00:50:24.290 align:middle line:90% Someone should try that. 00:50:24.290 --> 00:50:24.790 align:middle line:90% OK. 00:50:24.790 --> 00:50:27.730 align:middle line:84% Yeah, I see there's a question online. 00:50:27.730 --> 00:50:28.530 align:middle line:90% STUDENT: Oh, hi. 00:50:28.530 --> 00:50:31.410 align:middle line:84% Would you just quickly mind showing where 00:50:31.410 --> 00:50:33.093 align:middle line:90% you would place all of these-- 00:50:33.093 --> 00:50:35.010 align:middle line:84% JACK HARE: Thank you, that's a great question. 00:50:35.010 --> 00:50:44.600 align:middle line:90% 00:50:44.600 --> 00:50:46.100 align:middle line:84% I'm just going to erase some of this 00:50:46.100 --> 00:50:47.683 align:middle line:84% because I think you guys have seen it. 00:50:47.683 --> 00:50:51.200 align:middle line:84% And I want to have this space, so I can draw nice and big 00:50:51.200 --> 00:50:54.040 align:middle line:90% what's going on here. 00:50:54.040 --> 00:51:03.090 align:middle line:84% So I'm going to draw my tokamak or my toroidal device 00:51:03.090 --> 00:51:06.470 align:middle line:90% in cross-section, like this. 00:51:06.470 --> 00:51:08.830 align:middle line:90% So I need to have a Rogowski. 00:51:08.830 --> 00:51:14.760 align:middle line:84% I'm going to put that around some poloidal cross-section. 00:51:14.760 --> 00:51:16.930 align:middle line:84% I'm going to back-wind it to keep everyone happy. 00:51:16.930 --> 00:51:20.160 align:middle line:84% But that's I. I am going to stick 00:51:20.160 --> 00:51:27.670 align:middle line:84% in a probe which measures the toroidal magnetic field 00:51:27.670 --> 00:51:29.270 align:middle line:90% at the edge. 00:51:29.270 --> 00:51:31.330 align:middle line:84% I'm going to stick in another probe. 00:51:31.330 --> 00:51:34.595 align:middle line:90% 00:51:34.595 --> 00:51:36.720 align:middle line:84% I hope you can appreciate the beautiful perspective 00:51:36.720 --> 00:51:38.910 align:middle line:90% I've put into this circle here. 00:51:38.910 --> 00:51:42.540 align:middle line:84% And this is going to be measuring B theta at the edge. 00:51:42.540 --> 00:51:45.660 align:middle line:84% So remember, that probe should be lined up 00:51:45.660 --> 00:51:47.800 align:middle line:84% with the long axis of the torus here. 00:51:47.800 --> 00:51:52.380 align:middle line:84% So if you want me to, I could just draw it like this, I guess. 00:51:52.380 --> 00:51:53.050 align:middle line:90% There we go. 00:51:53.050 --> 00:51:55.140 align:middle line:90% Cool. 00:51:55.140 --> 00:51:56.850 align:middle line:84% I also want to have something that 00:51:56.850 --> 00:52:04.060 align:middle line:84% measures the average magnetic field B phi. 00:52:04.060 --> 00:52:06.370 align:middle line:84% Note how this looks a little bit like the Rogowski, 00:52:06.370 --> 00:52:09.070 align:middle line:84% but it measures something very, very different. 00:52:09.070 --> 00:52:18.050 align:middle line:84% And then finally, I need to have my flux loop, which 00:52:18.050 --> 00:52:21.325 align:middle line:90% was measuring B phi like this. 00:52:21.325 --> 00:52:22.700 align:middle line:84% Now, remember what we were saying 00:52:22.700 --> 00:52:26.090 align:middle line:84% is we're talking about here an ohmically driven, inductively 00:52:26.090 --> 00:52:27.960 align:middle line:90% driven device. 00:52:27.960 --> 00:52:32.910 align:middle line:84% And so that transformer, which will 00:52:32.910 --> 00:52:37.000 align:middle line:84% fit around the cross-section here, 00:52:37.000 --> 00:52:39.280 align:middle line:84% is driving some voltage that drives 00:52:39.280 --> 00:52:41.620 align:middle line:90% the voltage in the plasma. 00:52:41.620 --> 00:52:45.770 align:middle line:84% But exactly the same voltage is being driven across this loop. 00:52:45.770 --> 00:52:47.350 align:middle line:84% And so we can measure the voltage 00:52:47.350 --> 00:52:50.570 align:middle line:84% that's in the plasma using this loop here. 00:52:50.570 --> 00:52:51.230 align:middle line:90% OK. 00:52:51.230 --> 00:52:53.030 align:middle line:90% Hopefully, that is clear. 00:52:53.030 --> 00:52:53.930 align:middle line:90% Any more questions? 00:52:53.930 --> 00:52:55.160 align:middle line:90% STUDENT: Yeah, thank you. 00:52:55.160 --> 00:52:56.243 align:middle line:90% JACK HARE: You're welcome. 00:52:56.243 --> 00:53:06.670 align:middle line:90% 00:53:06.670 --> 00:53:09.820 align:middle line:84% OK, let's do m equals 1 [? after ?] m equals 0. 00:53:09.820 --> 00:53:49.150 align:middle line:90% 00:53:49.150 --> 00:53:54.205 align:middle line:84% So as we discussed last week, m equals 1 is a displacement mode. 00:53:54.205 --> 00:54:00.160 align:middle line:90% 00:54:00.160 --> 00:54:01.960 align:middle line:84% And so what we're doing is we're talking 00:54:01.960 --> 00:54:04.660 align:middle line:84% about a cylindrical vacuum vessel 00:54:04.660 --> 00:54:08.830 align:middle line:84% here that has inside it some sort of cylindrical plasma. 00:54:08.830 --> 00:54:12.940 align:middle line:84% And the center of that plasma is offset from the geometric center 00:54:12.940 --> 00:54:16.090 align:middle line:84% of our vacuum vessel by a distance 00:54:16.090 --> 00:54:21.660 align:middle line:84% that we're going to call delta, capital delta. 00:54:21.660 --> 00:54:27.520 align:middle line:84% And we'll say that the vacuum vessel has a radius A here. 00:54:27.520 --> 00:54:30.160 align:middle line:84% And around the edge of this vacuum vessel, 00:54:30.160 --> 00:54:34.890 align:middle line:84% we're going to place a series of B-dot probes. 00:54:34.890 --> 00:54:45.090 align:middle line:84% And these B-dot probes are going to be lined up 00:54:45.090 --> 00:54:47.115 align:middle line:90% to measure the B theta. 00:54:47.115 --> 00:54:50.490 align:middle line:90% 00:54:50.490 --> 00:54:55.150 align:middle line:84% So if we look at what B theta is going 00:54:55.150 --> 00:54:56.643 align:middle line:90% to be as a function of theta-- 00:54:56.643 --> 00:54:58.060 align:middle line:84% and remember, each of these probes 00:54:58.060 --> 00:55:00.910 align:middle line:84% is at a different angle-- theta 1, theta 2, so on. 00:55:00.910 --> 00:55:04.390 align:middle line:84% So effectively, we are measuring this function sparsely 00:55:04.390 --> 00:55:05.950 align:middle line:90% at each B-dot probe here. 00:55:05.950 --> 00:55:12.520 align:middle line:84% We're going to see that this is equal to mu 0 over 2 pi A times 00:55:12.520 --> 00:55:14.335 align:middle line:90% the current inside this plasma. 00:55:14.335 --> 00:55:18.082 align:middle line:90% 00:55:18.082 --> 00:55:20.175 align:middle line:84% For Ampere's law, we don't care about how 00:55:20.175 --> 00:55:21.300 align:middle line:90% the current is distributed. 00:55:21.300 --> 00:55:23.400 align:middle line:84% It's just the total current here. 00:55:23.400 --> 00:55:26.640 align:middle line:84% And this is going to be modified by a term that 00:55:26.640 --> 00:55:40.480 align:middle line:84% looks like sine squared theta plus cos theta minus delta 00:55:40.480 --> 00:55:46.480 align:middle line:84% upon A [INAUDIBLE] vacuum brackets squared square rooted. 00:55:46.480 --> 00:55:49.140 align:middle line:90% 00:55:49.140 --> 00:55:51.000 align:middle line:90% This is just geometry. 00:55:51.000 --> 00:55:56.910 align:middle line:84% This term here is just R, where R 00:55:56.910 --> 00:56:01.320 align:middle line:84% is the distance from the center of the plasma to some probe, 00:56:01.320 --> 00:56:04.170 align:middle line:84% which is now different depending on your angle 00:56:04.170 --> 00:56:05.910 align:middle line:84% that you're going at, which is why 00:56:05.910 --> 00:56:09.210 align:middle line:90% I'm just going to clarify it. 00:56:09.210 --> 00:56:11.430 align:middle line:84% This now has a theta dependence to it. 00:56:11.430 --> 00:56:15.630 align:middle line:90% 00:56:15.630 --> 00:56:18.270 align:middle line:84% And we're going to make an assumption again 00:56:18.270 --> 00:56:22.920 align:middle line:90% that delta upon A is small. 00:56:22.920 --> 00:56:25.940 align:middle line:90% Is this a good assumption? 00:56:25.940 --> 00:56:27.440 align:middle line:84% What would happen if it wasn't true? 00:56:27.440 --> 00:56:30.110 align:middle line:90% 00:56:30.110 --> 00:56:30.942 align:middle line:90% Yeah? 00:56:30.942 --> 00:56:32.360 align:middle line:90% STUDENT: [INAUDIBLE] 00:56:32.360 --> 00:56:33.410 align:middle line:90% JACK HARE: Plasma's gone. 00:56:33.410 --> 00:56:34.997 align:middle line:84% It's hit the wall already, exactly. 00:56:34.997 --> 00:56:36.080 align:middle line:90% So we're going to make it. 00:56:36.080 --> 00:56:38.497 align:middle line:84% And it's a very good assumption because otherwise you have 00:56:38.497 --> 00:56:40.580 align:middle line:90% already failed the [INAUDIBLE]. 00:56:40.580 --> 00:56:45.290 align:middle line:84% So having made that assumption, we can approximate this as mu 00:56:45.290 --> 00:56:52.217 align:middle line:90% 0 I upon 2 pi A. 00:56:52.217 --> 00:57:04.210 align:middle line:84% And we get out 1 plus delta upon A cos theta, the Taylor 00:57:04.210 --> 00:57:05.260 align:middle line:90% expansion here. 00:57:05.260 --> 00:57:08.110 align:middle line:84% And you can see that there's going to be a second order term 00:57:08.110 --> 00:57:09.460 align:middle line:90% [INAUDIBLE] A that disappears. 00:57:09.460 --> 00:57:11.200 align:middle line:84% And there's going to be a sine squared and a cos squared. 00:57:11.200 --> 00:57:13.158 align:middle line:84% And they're going to come together to make a 1. 00:57:13.158 --> 00:57:16.910 align:middle line:84% So this all looks like it's going to work. 00:57:16.910 --> 00:57:18.410 align:middle line:84% Remember here, in this case, we have 00:57:18.410 --> 00:57:21.330 align:middle line:84% specified that delta is in the x direction. 00:57:21.330 --> 00:57:25.470 align:middle line:90% So this is delta x, like that. 00:57:25.470 --> 00:57:30.510 align:middle line:84% If it was in the y direction, this would be 1 plus delta y 00:57:30.510 --> 00:57:34.060 align:middle line:90% upon A sine theta. 00:57:34.060 --> 00:57:35.970 align:middle line:84% And if you remember the Fourier decomposition 00:57:35.970 --> 00:57:37.887 align:middle line:84% that we said we were going to do at the start, 00:57:37.887 --> 00:57:40.810 align:middle line:84% for this magnetic field pattern there are only two terms. 00:57:40.810 --> 00:57:46.560 align:middle line:84% There's the B-- well, this is the C0 coefficient. 00:57:46.560 --> 00:57:49.950 align:middle line:84% And these two are the C1 coefficients. 00:57:49.950 --> 00:57:54.360 align:middle line:84% And so we can say that we can directly measure 00:57:54.360 --> 00:58:00.870 align:middle line:84% the displacement x as 2A C1 over C0 or the displacement in y 00:58:00.870 --> 00:58:04.620 align:middle line:90% as 2A S1 over C0. 00:58:04.620 --> 00:58:06.040 align:middle line:90% So this is C1. 00:58:06.040 --> 00:58:07.575 align:middle line:90% This is S1. 00:58:07.575 --> 00:58:12.730 align:middle line:90% 00:58:12.730 --> 00:58:15.040 align:middle line:84% So by doing a Fourier decomposition of our signal 00:58:15.040 --> 00:58:18.520 align:middle line:84% and therefore finding the coefficients C1 and S1, 00:58:18.520 --> 00:58:21.850 align:middle line:84% we can immediately find the displacement of our plasma. 00:58:21.850 --> 00:58:23.810 align:middle line:90% And that's pretty fast to do. 00:58:23.810 --> 00:58:26.350 align:middle line:84% And so we can see if our plasma is starting to move. 00:58:26.350 --> 00:58:28.903 align:middle line:84% And we can do something about it, like active feedback, 00:58:28.903 --> 00:58:30.070 align:middle line:90% to stop it hitting the wall. 00:58:30.070 --> 00:58:32.240 align:middle line:90% So that's pretty useful. 00:58:32.240 --> 00:58:34.490 align:middle line:90% Any questions on this? 00:58:34.490 --> 00:58:35.914 align:middle line:90% Yeah? 00:58:35.914 --> 00:58:41.022 align:middle line:84% STUDENT: [INAUDIBLE] as you're varying theta [INAUDIBLE]? 00:58:41.022 --> 00:58:47.340 align:middle line:90% 00:58:47.340 --> 00:58:48.270 align:middle line:90% JACK HARE: Oh, yes. 00:58:48.270 --> 00:58:49.790 align:middle line:90% STUDENT: [INAUDIBLE] 00:58:49.790 --> 00:58:57.093 align:middle line:90% 00:58:57.093 --> 00:58:59.260 align:middle line:84% JACK HARE: Yeah, so if your plasma is in the center, 00:58:59.260 --> 00:59:00.640 align:middle line:90% you would have-- 00:59:00.640 --> 00:59:05.810 align:middle line:90% B theta would be like this. 00:59:05.810 --> 00:59:08.170 align:middle line:84% And you might reconstruct a flat profile. 00:59:08.170 --> 00:59:11.470 align:middle line:84% If your plasma moves towards theta 1, 00:59:11.470 --> 00:59:12.992 align:middle line:90% you might have a profile-- 00:59:12.992 --> 00:59:14.200 align:middle line:90% let me use a different color. 00:59:14.200 --> 00:59:20.420 align:middle line:90% 00:59:20.420 --> 00:59:22.890 align:middle line:90% Yeah, that goes back to there. 00:59:22.890 --> 00:59:26.160 align:middle line:84% And you'll notice that, in this case, I've got four sensors. 00:59:26.160 --> 00:59:33.310 align:middle line:84% And I can just about reconstruct the cosinusoidal variation 00:59:33.310 --> 00:59:33.810 align:middle line:90% there. 00:59:33.810 --> 00:59:35.352 align:middle line:84% And you'll find that very interesting 00:59:35.352 --> 00:59:38.930 align:middle line:84% when you come to the P set because this sort of question 00:59:38.930 --> 00:59:41.540 align:middle line:84% is fundamental to how many B-dot probes do you need 00:59:41.540 --> 00:59:42.780 align:middle line:90% to measure some displacement. 00:59:42.780 --> 00:59:45.500 align:middle line:84% So thinking in terms of Nyquist-Shannon sampling 00:59:45.500 --> 00:59:49.160 align:middle line:84% theorem, things like that might get you a long way. 00:59:49.160 --> 00:59:51.443 align:middle line:90% Cool. 00:59:51.443 --> 00:59:52.110 align:middle line:90% Other questions? 00:59:52.110 --> 00:59:53.220 align:middle line:90% Yeah? 00:59:53.220 --> 00:59:56.070 align:middle line:84% STUDENT: [INAUDIBLE] for measuring 00:59:56.070 --> 00:59:57.575 align:middle line:90% [INAUDIBLE] 0 [INAUDIBLE]? 00:59:57.575 --> 01:00:03.160 align:middle line:90% 01:00:03.160 --> 01:00:05.880 align:middle line:84% JACK HARE: How many did we need to measure? 01:00:05.880 --> 01:00:07.486 align:middle line:90% We just needed 1. 01:00:07.486 --> 01:00:09.310 align:middle line:84% STUDENT: But now we're [INAUDIBLE]? 01:00:09.310 --> 01:00:16.890 align:middle line:90% 01:00:16.890 --> 01:00:19.380 align:middle line:90% JACK HARE: No, it's still here. 01:00:19.380 --> 01:00:23.220 align:middle line:84% You see the C0 term when you do your Fourier decomposition. 01:00:23.220 --> 01:00:27.190 align:middle line:84% No matter how many probes you have, 01:00:27.190 --> 01:00:31.330 align:middle line:84% you can always Fourier decompose it to C0 upon 2, 01:00:31.330 --> 01:00:43.850 align:middle line:84% and then the sum to infinity of Cm cos m theta Sm sine m theta. 01:00:43.850 --> 01:00:45.550 align:middle line:84% And what we're saying is if you want 01:00:45.550 --> 01:00:48.190 align:middle line:84% to measure, for example, Te and all 01:00:48.190 --> 01:00:50.980 align:middle line:84% that beta stuff we did before, you just take this one. 01:00:50.980 --> 01:00:52.810 align:middle line:90% This is the m equals 0 term. 01:00:52.810 --> 01:00:56.110 align:middle line:84% If you want to measure displacements of the plasma, 01:00:56.110 --> 01:01:01.160 align:middle line:84% you take these first two terms, C1. 01:01:01.160 --> 01:01:04.580 align:middle line:84% If you want to measure more exciting higher order 01:01:04.580 --> 01:01:07.250 align:middle line:84% modes where your plasma gets pancaked 01:01:07.250 --> 01:01:11.360 align:middle line:84% or it comes into increasingly unlikely shapes, 01:01:11.360 --> 01:01:13.520 align:middle line:84% then you want the higher order ones of these. 01:01:13.520 --> 01:01:15.770 align:middle line:84% But John made a very good point, which 01:01:15.770 --> 01:01:18.770 align:middle line:84% is that you'll have to think about how many-- 01:01:18.770 --> 01:01:21.650 align:middle line:84% what m you need to get to will set a requirement 01:01:21.650 --> 01:01:24.695 align:middle line:84% on how many probes that you have in order to properly do 01:01:24.695 --> 01:01:25.820 align:middle line:90% this Fourier decomposition. 01:01:25.820 --> 01:01:28.460 align:middle line:84% And this, again, is related to Nyquist theorem. 01:01:28.460 --> 01:01:32.300 align:middle line:90% So there's a bit to think about. 01:01:32.300 --> 01:01:33.560 align:middle line:90% Any other questions on this? 01:01:33.560 --> 01:01:36.520 align:middle line:90% 01:01:36.520 --> 01:01:37.884 align:middle line:90% Yeah? 01:01:37.884 --> 01:01:39.255 align:middle line:90% STUDENT: What [INAUDIBLE]? 01:01:39.255 --> 01:01:41.285 align:middle line:90% 01:01:41.285 --> 01:01:43.410 align:middle line:84% JACK HARE: Yeah, so the question was, at what point 01:01:43.410 --> 01:01:44.850 align:middle line:84% do you need to start worrying about eddy currents 01:01:44.850 --> 01:01:46.892 align:middle line:84% in the vacuum vessel messing up this measurement. 01:01:46.892 --> 01:01:49.920 align:middle line:84% These would definitely be affected by eddy currents. 01:01:49.920 --> 01:01:54.567 align:middle line:84% So that's a higher order correction 01:01:54.567 --> 01:01:56.150 align:middle line:84% that you need to do to understand what 01:01:56.150 --> 01:01:59.110 align:middle line:90% the eddy current does, yeah. 01:01:59.110 --> 01:02:00.925 align:middle line:84% So in this assumption, we have assumed 01:02:00.925 --> 01:02:02.050 align:middle line:90% there are no eddy currents. 01:02:02.050 --> 01:02:05.200 align:middle line:84% We have a nonconducting vacuum vessel or something else. 01:02:05.200 --> 01:02:06.100 align:middle line:90% Yeah, exactly. 01:02:06.100 --> 01:02:09.340 align:middle line:90% 01:02:09.340 --> 01:02:10.594 align:middle line:90% Any other questions? 01:02:10.594 --> 01:02:15.340 align:middle line:90% 01:02:15.340 --> 01:02:21.767 align:middle line:84% So just very briefly, Hutchinson spends a long time 01:02:21.767 --> 01:02:22.600 align:middle line:90% on this in the book. 01:02:22.600 --> 01:02:24.212 align:middle line:84% And I think it's very interesting. 01:02:24.212 --> 01:02:25.920 align:middle line:84% But it's very hard to go through in class 01:02:25.920 --> 01:02:27.240 align:middle line:90% because it's very mathematical. 01:02:27.240 --> 01:02:29.940 align:middle line:84% So I just want to give you a little taster of what would 01:02:29.940 --> 01:02:32.350 align:middle line:90% happen for a toroidal system. 01:02:32.350 --> 01:02:34.980 align:middle line:84% So remember, we've really been assuming all along here 01:02:34.980 --> 01:02:38.460 align:middle line:84% that we've got a cylinder, a very high aspect ratio tokamak, 01:02:38.460 --> 01:02:42.090 align:middle line:84% so that we don't have to worry about toroidal curvature. 01:02:42.090 --> 01:02:43.980 align:middle line:84% But what happens in a toroidal system 01:02:43.980 --> 01:02:47.670 align:middle line:84% is that we have a very complicated force balance. 01:02:47.670 --> 01:02:51.660 align:middle line:84% Remember, our MHD equations, which we simplified massively, 01:02:51.660 --> 01:02:53.220 align:middle line:84% we're going to have some more terms. 01:02:53.220 --> 01:02:54.720 align:middle line:84% But there's a complex force balance. 01:02:54.720 --> 01:03:00.130 align:middle line:90% 01:03:00.130 --> 01:03:02.920 align:middle line:90% And that shifts. 01:03:02.920 --> 01:03:05.080 align:middle line:84% If this is our axis of symmetry of our tokamak 01:03:05.080 --> 01:03:07.660 align:middle line:84% and this is the vacuum vessel, that 01:03:07.660 --> 01:03:10.720 align:middle line:84% shifts the magnetic flux surfaces so that they're 01:03:10.720 --> 01:03:11.950 align:middle line:90% no longer concentric. 01:03:11.950 --> 01:03:14.140 align:middle line:84% So we may have a flux surface like this, 01:03:14.140 --> 01:03:16.570 align:middle line:84% and a flux surface like this, a flux surface like this, 01:03:16.570 --> 01:03:19.950 align:middle line:90% and a flux surface like this. 01:03:19.950 --> 01:03:22.430 align:middle line:84% And again, I'm not explaining where you get to from this. 01:03:22.430 --> 01:03:24.620 align:middle line:84% This is like three or four pages of algebra. 01:03:24.620 --> 01:03:26.428 align:middle line:84% But you start ending up with systems 01:03:26.428 --> 01:03:27.720 align:middle line:90% that are much more complicated. 01:03:27.720 --> 01:03:30.020 align:middle line:84% We have kind of been assuming a lot of the time 01:03:30.020 --> 01:03:32.107 align:middle line:84% here that we have concentric flux surfaces. 01:03:32.107 --> 01:03:34.190 align:middle line:84% It's actually not important for most of the things 01:03:34.190 --> 01:03:35.010 align:middle line:90% we're driving. 01:03:35.010 --> 01:03:36.830 align:middle line:84% But we certainly would have no reason 01:03:36.830 --> 01:03:39.260 align:middle line:84% to believe the flux surfaces are not concentric 01:03:39.260 --> 01:03:41.430 align:middle line:84% because our system is azimuthally symmetric. 01:03:41.430 --> 01:03:45.010 align:middle line:84% There'd be nothing pushing them from one side to the other. 01:03:45.010 --> 01:03:47.640 align:middle line:84% So if you start doing this analysis properly, 01:03:47.640 --> 01:03:55.070 align:middle line:84% you get to equation 2.2.25 in Hutchinson's book, 01:03:55.070 --> 01:03:57.890 align:middle line:84% just to let you know where I've pulled this away from. 01:03:57.890 --> 01:04:03.070 align:middle line:84% And you find out that your magnetic field, the m equals 1 01:04:03.070 --> 01:04:10.260 align:middle line:84% term, is going to be equal to the m equals 0 term times 01:04:10.260 --> 01:04:16.040 align:middle line:84% a correction for where you are inside the plasma. 01:04:16.040 --> 01:04:20.990 align:middle line:84% So this coordinate R measures radially outwards 01:04:20.990 --> 01:04:23.480 align:middle line:84% from the center of our poloidal cross-section. 01:04:23.480 --> 01:04:28.130 align:middle line:84% And this R0 is the distance from the axis the center 01:04:28.130 --> 01:04:30.540 align:middle line:90% of our poloidal cross-section. 01:04:30.540 --> 01:04:33.650 align:middle line:84% And we find-- sorry, this isn't very clear. 01:04:33.650 --> 01:04:38.800 align:middle line:90% 01:04:38.800 --> 01:04:40.780 align:middle line:84% And all of this is going to be multiplied 01:04:40.780 --> 01:04:48.220 align:middle line:84% by beta theta plus a term, Li upon 2, which I'll 01:04:48.220 --> 01:04:51.250 align:middle line:90% explain in a moment, minus 1. 01:04:51.250 --> 01:04:53.050 align:middle line:90% Again, none of this is derived. 01:04:53.050 --> 01:04:54.970 align:middle line:84% I'm just throwing this equation at you 01:04:54.970 --> 01:04:56.762 align:middle line:84% to show you some of the things that you can 01:04:56.762 --> 01:04:58.270 align:middle line:90% do in the toroidal geometry. 01:04:58.270 --> 01:05:02.440 align:middle line:84% Now, beta theta is something that we 01:05:02.440 --> 01:05:05.057 align:middle line:84% can measure from our m equals 0 diagnostics. 01:05:05.057 --> 01:05:06.640 align:middle line:84% They're still going to work just fine. 01:05:06.640 --> 01:05:09.370 align:middle line:84% What's interesting is that this shift here 01:05:09.370 --> 01:05:14.610 align:middle line:84% results in a term that is to do with the inductance-- 01:05:14.610 --> 01:05:17.250 align:middle line:84% not quite the inductance, but it's similar to the inductance. 01:05:17.250 --> 01:05:19.920 align:middle line:90% 01:05:19.920 --> 01:05:22.080 align:middle line:84% And we talked before about how the inductance 01:05:22.080 --> 01:05:24.780 align:middle line:84% is a measure of the geometry of your current, 01:05:24.780 --> 01:05:26.920 align:middle line:84% so how the current is flowing in the system. 01:05:26.920 --> 01:05:28.860 align:middle line:84% And so by measuring this very precisely, 01:05:28.860 --> 01:05:32.670 align:middle line:84% you may be able to determine whether your system has 01:05:32.670 --> 01:05:34.800 align:middle line:84% a current profile that is nicely peaked 01:05:34.800 --> 01:05:36.780 align:middle line:84% or whether it has a profile which 01:05:36.780 --> 01:05:39.910 align:middle line:84% has a hollow in the center, something like that. 01:05:39.910 --> 01:05:43.290 align:middle line:84% And so you actually can end up getting more information out 01:05:43.290 --> 01:05:46.140 align:middle line:84% of your system when you have a toroidal geometry. 01:05:46.140 --> 01:05:49.650 align:middle line:84% But the price you pay for that is significantly more complexity 01:05:49.650 --> 01:05:51.490 align:middle line:90% with all the equations. 01:05:51.490 --> 01:05:54.240 align:middle line:84% So again, that's all I'm going to go into in terms of this. 01:05:54.240 --> 01:05:56.607 align:middle line:84% If you work with magnetic diagnostics for tokamaks, 01:05:56.607 --> 01:05:58.440 align:middle line:84% or stellarators, or other devices like that, 01:05:58.440 --> 01:06:00.270 align:middle line:84% you will definitely come across some 01:06:00.270 --> 01:06:02.560 align:middle line:90% of this very complicated stuff. 01:06:02.560 --> 01:06:04.164 align:middle line:90% So any questions? 01:06:04.164 --> 01:06:12.700 align:middle line:90% 01:06:12.700 --> 01:06:13.960 align:middle line:90% OK. 01:06:13.960 --> 01:06:17.857 align:middle line:84% Now, we're going to stick the probe in the plasma, 01:06:17.857 --> 01:06:18.565 align:middle line:90% see what happens. 01:06:18.565 --> 01:06:49.850 align:middle line:90% 01:06:49.850 --> 01:06:52.595 align:middle line:84% So now, we're discussing internal probes. 01:06:52.595 --> 01:07:03.770 align:middle line:90% 01:07:03.770 --> 01:07:08.090 align:middle line:84% So again, we'll have some vacuum chamber with some sort 01:07:08.090 --> 01:07:10.010 align:middle line:90% of clever re-entrance port. 01:07:10.010 --> 01:07:14.855 align:middle line:84% And we can stick into it our B-dot probe. 01:07:14.855 --> 01:07:18.510 align:middle line:90% 01:07:18.510 --> 01:07:22.600 align:middle line:84% We'll probably put a bit of shielding insulation 01:07:22.600 --> 01:07:24.588 align:middle line:90% on the B-dot probe, like that. 01:07:24.588 --> 01:07:26.380 align:middle line:84% And we'll talk a little bit more about that 01:07:26.380 --> 01:07:29.290 align:middle line:84% in the context of Langmuir probes next lecture. 01:07:29.290 --> 01:07:32.800 align:middle line:90% And there is some plasma here. 01:07:32.800 --> 01:07:36.580 align:middle line:84% And there will be some magnetic fields, 01:07:36.580 --> 01:07:39.500 align:middle line:84% which may be three dimensional in nature. 01:07:39.500 --> 01:07:43.600 align:middle line:84% Now, despite what it looks like, this is actually not 01:07:43.600 --> 01:07:46.390 align:middle line:90% very perturbative. 01:07:46.390 --> 01:07:49.240 align:middle line:84% Despite the fact you would stick a probe right in your plasma, 01:07:49.240 --> 01:07:52.420 align:middle line:84% this is not as bad as it might seem. 01:07:52.420 --> 01:07:53.890 align:middle line:90% What do we mean by perturbative? 01:07:53.890 --> 01:07:56.290 align:middle line:84% We mean that it changes the measurement 01:07:56.290 --> 01:07:59.108 align:middle line:90% that we're trying to make. 01:07:59.108 --> 01:08:00.900 align:middle line:84% This turns out not to be very perturbative. 01:08:00.900 --> 01:08:03.960 align:middle line:84% The main reason is that the magnetic fields at a given point 01:08:03.960 --> 01:08:07.230 align:middle line:84% here are not just set by the currents locally. 01:08:07.230 --> 01:08:08.760 align:middle line:84% They're set by currents over here, 01:08:08.760 --> 01:08:10.900 align:middle line:90% and over here, and over here. 01:08:10.900 --> 01:08:16.890 align:middle line:90% So B is set by global currents. 01:08:16.890 --> 01:08:20.609 align:middle line:84% And so disturbing just a little bit of the plasma 01:08:20.609 --> 01:08:23.580 align:middle line:84% doesn't actually change the magnetic field at a point 01:08:23.580 --> 01:08:25.170 align:middle line:90% that much. 01:08:25.170 --> 01:08:28.260 align:middle line:84% Again, we're going to put an insulator around it. 01:08:28.260 --> 01:08:31.490 align:middle line:90% 01:08:31.490 --> 01:08:38.210 align:middle line:84% So we would set I into the probe equal to 0. 01:08:38.210 --> 01:08:40.609 align:middle line:84% And we're going to make the probe small. 01:08:40.609 --> 01:08:45.350 align:middle line:84% We want it to be small to reduce the perturbation on the plasma, 01:08:45.350 --> 01:08:54.510 align:middle line:84% but also to make delta B over the probe small, 01:08:54.510 --> 01:08:57.210 align:middle line:84% as we discussed before, which makes our analysis easier 01:08:57.210 --> 01:09:00.180 align:middle line:84% because it means the magnetic field is constant over the probe 01:09:00.180 --> 01:09:01.080 align:middle line:90% here. 01:09:01.080 --> 01:09:02.460 align:middle line:84% So you wouldn't want to stick one 01:09:02.460 --> 01:09:06.240 align:middle line:84% of these inside a very hot and dense plasma. 01:09:06.240 --> 01:09:07.472 align:middle line:90% Well, you might. 01:09:07.472 --> 01:09:09.180 align:middle line:84% If went on to stick one of these inside a 01:09:09.180 --> 01:09:10.380 align:middle line:84% tokamak because it would just melt 01:09:10.380 --> 01:09:12.120 align:middle line:84% and it would fill your plasma with impurities. 01:09:12.120 --> 01:09:13.109 align:middle line:90% And your plasma would crash. 01:09:13.109 --> 01:09:13.920 align:middle line:90% That would be bad. 01:09:13.920 --> 01:09:16.439 align:middle line:84% If you're doing, for example, a pulse power-driven experiment, 01:09:16.439 --> 01:09:18.356 align:middle line:84% you might want to stick these inside a plasma. 01:09:18.356 --> 01:09:21.149 align:middle line:84% But you might not expect it to last more than one experiment 01:09:21.149 --> 01:09:23.490 align:middle line:90% because of the heat flux on it. 01:09:23.490 --> 01:09:26.340 align:middle line:84% Now, what people do on some machines is 01:09:26.340 --> 01:09:33.270 align:middle line:84% they have sticks of probes that they can position. 01:09:33.270 --> 01:09:37.790 align:middle line:84% The stick has a load of probes wound like this to measure, 01:09:37.790 --> 01:09:39.520 align:middle line:90% for example, Bx. 01:09:39.520 --> 01:09:42.220 align:middle line:84% It also has a load of probes wound 01:09:42.220 --> 01:09:45.430 align:middle line:84% like this in the same location to measure By. 01:09:45.430 --> 01:09:51.450 align:middle line:84% And it'll have a load of probes wound like this to measure Bz. 01:09:51.450 --> 01:09:54.820 align:middle line:84% So you'll be able to get all three of the axes here. 01:09:54.820 --> 01:09:57.430 align:middle line:90% And the nice thing about that-- 01:09:57.430 --> 01:09:58.600 align:middle line:90% you put these all together. 01:09:58.600 --> 01:10:01.240 align:middle line:84% You can find out what the current is locally 01:10:01.240 --> 01:10:04.360 align:middle line:84% inside your plasma because the current is simply 1 01:10:04.360 --> 01:10:09.355 align:middle line:84% over mu 0 curl of B. Now, you're taking 01:10:09.355 --> 01:10:10.480 align:middle line:90% the gradient of the signal. 01:10:10.480 --> 01:10:11.890 align:middle line:90% And the signal might be noisy. 01:10:11.890 --> 01:10:15.070 align:middle line:84% So you've got to be a little bit careful here 01:10:15.070 --> 01:10:18.100 align:middle line:84% because this is going to be a noisy measurement itself. 01:10:18.100 --> 01:10:19.922 align:middle line:84% But the fact is you can still have 01:10:19.922 --> 01:10:22.630 align:middle line:84% some go at measuring the current internally to a plasma, which is 01:10:22.630 --> 01:10:24.610 align:middle line:90% a very, very hard thing to do. 01:10:24.610 --> 01:10:27.650 align:middle line:90% 01:10:27.650 --> 01:10:32.070 align:middle line:84% I'll just make a note here that this is noisy because we 01:10:32.070 --> 01:10:36.720 align:middle line:90% have delta B over delta x. 01:10:36.720 --> 01:10:40.350 align:middle line:84% Delta x is your spacing of your probes here. 01:10:40.350 --> 01:10:42.810 align:middle line:84% So the smaller you can make delta x, 01:10:42.810 --> 01:10:46.020 align:middle line:84% the better you're going to be measuring this gradient. 01:10:46.020 --> 01:10:48.420 align:middle line:84% And this looks very Nyquisty again all of a sudden. 01:10:48.420 --> 01:10:50.345 align:middle line:84% Sorry, we can't escape from them. 01:10:50.345 --> 01:10:53.080 align:middle line:90% 01:10:53.080 --> 01:10:56.170 align:middle line:84% The nice thing about measuring J is 01:10:56.170 --> 01:11:03.480 align:middle line:84% that then we would have J and B. And we can use 01:11:03.480 --> 01:11:05.740 align:middle line:90% that to measure the pressure. 01:11:05.740 --> 01:11:08.610 align:middle line:84% So we can actually measure the pressure inside our plasma 01:11:08.610 --> 01:11:10.915 align:middle line:84% using these closely spaced probes. 01:11:10.915 --> 01:11:12.540 align:middle line:84% Or at least we can measure the pressure 01:11:12.540 --> 01:11:16.110 align:middle line:90% along the line of the probe. 01:11:16.110 --> 01:11:19.580 align:middle line:84% This turns out to be hard for low beta plasmas 01:11:19.580 --> 01:11:21.600 align:middle line:90% because grad P is small. 01:11:21.600 --> 01:11:25.420 align:middle line:84% So you won't see much change in J cross B. 01:11:25.420 --> 01:11:29.941 align:middle line:84% But when beta is reasonable, you might be able to do this. 01:11:29.941 --> 01:11:34.260 align:middle line:84% A very nice trick is if you have a steady plasma or a plasma 01:11:34.260 --> 01:11:37.040 align:middle line:84% which is highly reproducible, so you can do this experiment 01:11:37.040 --> 01:11:37.925 align:middle line:90% over and over again. 01:11:37.925 --> 01:11:43.950 align:middle line:90% 01:11:43.950 --> 01:11:48.410 align:middle line:84% You can actually take this probe and say 01:11:48.410 --> 01:11:51.910 align:middle line:84% you've got a plasma that looks like this. 01:11:51.910 --> 01:11:59.170 align:middle line:84% And you can step the probe across the plasma. 01:11:59.170 --> 01:12:00.880 align:middle line:84% And either the plasma is steady or you're 01:12:00.880 --> 01:12:03.047 align:middle line:84% doing the same experiment again and again and again. 01:12:03.047 --> 01:12:06.010 align:middle line:84% And you can build up a complete map of B 01:12:06.010 --> 01:12:10.840 align:middle line:84% as a function of space and time and therefore 01:12:10.840 --> 01:12:14.650 align:middle line:84% J as a function of space and time as well. 01:12:14.650 --> 01:12:15.970 align:middle line:90% And people do this. 01:12:15.970 --> 01:12:20.560 align:middle line:84% One great experiment is the large area plasma device 01:12:20.560 --> 01:12:22.438 align:middle line:90% at UCLA, LAPD. 01:12:22.438 --> 01:12:24.730 align:middle line:84% And on this device where they do astrophysics relevance 01:12:24.730 --> 01:12:27.550 align:middle line:84% experiments, they can map out all of this. 01:12:27.550 --> 01:12:29.560 align:middle line:84% They will do a shot every second. 01:12:29.560 --> 01:12:32.530 align:middle line:84% There'll be a servo stepper motor that moves this across. 01:12:32.530 --> 01:12:34.540 align:middle line:84% And they just press go after programming it. 01:12:34.540 --> 01:12:36.610 align:middle line:84% And it will run for 24 hours a day for a week. 01:12:36.610 --> 01:12:38.710 align:middle line:84% And you'll have a huge amount of data 01:12:38.710 --> 01:12:41.740 align:middle line:84% with a full three-dimensional map of the magnetic fields. 01:12:41.740 --> 01:12:43.240 align:middle line:84% So this is the sort of thing you can 01:12:43.240 --> 01:12:45.220 align:middle line:90% do with high rep rate devices. 01:12:45.220 --> 01:12:47.850 align:middle line:90% It's very, very cool. 01:12:47.850 --> 01:12:49.700 align:middle line:90% OK, any questions so far? 01:12:49.700 --> 01:12:50.640 align:middle line:90% Yes? 01:12:50.640 --> 01:12:51.990 align:middle line:90% STUDENT: [INAUDIBLE]. 01:12:51.990 --> 01:12:57.926 align:middle line:90% 01:12:57.926 --> 01:13:00.300 align:middle line:84% But when you're entering [INAUDIBLE]? 01:13:00.300 --> 01:13:06.613 align:middle line:90% 01:13:06.613 --> 01:13:08.030 align:middle line:84% JACK HARE: It's the local current. 01:13:08.030 --> 01:13:09.197 align:middle line:90% So this is a local equation. 01:13:09.197 --> 01:13:12.210 align:middle line:90% 01:13:12.210 --> 01:13:14.200 align:middle line:90% This is B of x. 01:13:14.200 --> 01:13:18.040 align:middle line:84% So you get J of x wherever your measurement is being taken. 01:13:18.040 --> 01:13:21.070 align:middle line:84% So you would get J at the locations of each of these coils 01:13:21.070 --> 01:13:22.870 align:middle line:84% or, depending on your differencing scheme, 01:13:22.870 --> 01:13:25.810 align:middle line:84% halfway between the locations of each coil, yeah. 01:13:25.810 --> 01:13:27.420 align:middle line:90% STUDENT: [INAUDIBLE] 01:13:27.420 --> 01:13:31.830 align:middle line:90% 01:13:31.830 --> 01:13:36.270 align:middle line:84% JACK HARE: Yeah, so the thing is that this equation here 01:13:36.270 --> 01:13:37.560 align:middle line:90% does not imply causality. 01:13:37.560 --> 01:13:40.158 align:middle line:84% You don't have magnetic fields telling currents where to go 01:13:40.158 --> 01:13:42.450 align:middle line:84% or currents going where-- telling magnetic fields where 01:13:42.450 --> 01:13:42.990 align:middle line:90% to go. 01:13:42.990 --> 01:13:46.050 align:middle line:84% It's just true simultaneously that a global magnetic field 01:13:46.050 --> 01:13:48.240 align:middle line:84% also constrains the global current. 01:13:48.240 --> 01:13:51.797 align:middle line:84% And so you don't end up having too much perturbation here. 01:13:51.797 --> 01:13:53.130 align:middle line:90% There will be some perturbation. 01:13:53.130 --> 01:13:55.090 align:middle line:84% You'll be cooling down the plasma and things like that. 01:13:55.090 --> 01:13:56.730 align:middle line:84% But if you're on small length scales 01:13:56.730 --> 01:13:58.740 align:middle line:84% compared to the plasma dynamics, the current 01:13:58.740 --> 01:14:00.550 align:middle line:90% will still be roughly the same. 01:14:00.550 --> 01:14:01.050 align:middle line:90% Yeah? 01:14:01.050 --> 01:14:05.513 align:middle line:84% STUDENT: Can you repeat why [INAUDIBLE] theta [INAUDIBLE]? 01:14:05.513 --> 01:14:07.180 align:middle line:84% JACK HARE: It's just that, in that case, 01:14:07.180 --> 01:14:10.230 align:middle line:84% you have P much less than B squared 01:14:10.230 --> 01:14:14.520 align:middle line:84% over 2 mu 0, which means that this J cross term is just 01:14:14.520 --> 01:14:18.970 align:middle line:90% going to look like this. 01:14:18.970 --> 01:14:25.720 align:middle line:84% And so this is x and this is B squared over 2 mu 0. 01:14:25.720 --> 01:14:30.470 align:middle line:84% So this is only going to vary by, say, 1% or so. 01:14:30.470 --> 01:14:32.470 align:middle line:84% But you're trying to measure the pressure, which 01:14:32.470 --> 01:14:36.250 align:middle line:84% is going to be the opposite of this, down at about 01:14:36.250 --> 01:14:39.280 align:middle line:84% a 1% level, which means you need to have 01:14:39.280 --> 01:14:42.187 align:middle line:84% 1% accuracy on this reconstruction. 01:14:42.187 --> 01:14:44.020 align:middle line:84% So you need to have very low signal to noise 01:14:44.020 --> 01:14:48.670 align:middle line:84% because if your signal was noisy, 01:14:48.670 --> 01:14:52.100 align:middle line:84% you would just reconstruct a completely noisy pressure 01:14:52.100 --> 01:14:52.600 align:middle line:90% signal. 01:14:52.600 --> 01:14:53.892 align:middle line:90% And it wouldn't make any sense. 01:14:53.892 --> 01:14:59.545 align:middle line:90% 01:14:59.545 --> 01:15:00.045 align:middle line:90% Okey doke. 01:15:00.045 --> 01:15:06.820 align:middle line:90% 01:15:06.820 --> 01:15:10.150 align:middle line:84% Some other cool tricks you can do with probes-- 01:15:10.150 --> 01:15:12.850 align:middle line:84% if you can't move the probe inside the plasma 01:15:12.850 --> 01:15:15.490 align:middle line:84% because the plasma is moving too quickly 01:15:15.490 --> 01:15:17.650 align:middle line:84% or it's not steady enough, you can just 01:15:17.650 --> 01:15:20.450 align:middle line:84% set your probe up and let the plasma flow over it. 01:15:20.450 --> 01:15:25.340 align:middle line:84% So for example, we could have a B-dot probe like this. 01:15:25.340 --> 01:15:29.800 align:middle line:84% And we can have a plasma that is moving 01:15:29.800 --> 01:15:34.970 align:middle line:84% towards the probe with some velocity, v. 01:15:34.970 --> 01:15:37.610 align:middle line:84% And we're assuming here that this plasma is taking 01:15:37.610 --> 01:15:40.400 align:middle line:90% with it some magnetic field. 01:15:40.400 --> 01:15:42.920 align:middle line:84% And maybe you know which direction the magnetic field 01:15:42.920 --> 01:15:44.840 align:middle line:84% is pointing in prior to the experiment. 01:15:44.840 --> 01:15:46.880 align:middle line:84% And you set up your B-dot probe so that it's 01:15:46.880 --> 01:15:48.990 align:middle line:90% aligned with this field. 01:15:48.990 --> 01:15:51.050 align:middle line:84% And if you have a set of these B-dot probes-- 01:15:51.050 --> 01:15:53.420 align:middle line:84% for example, if you have one here and then you 01:15:53.420 --> 01:15:59.910 align:middle line:84% have one up here and they've got some separation delta 01:15:59.910 --> 01:16:04.590 align:middle line:84% x, like this, the signal that you'll measure on the two probes 01:16:04.590 --> 01:16:05.350 align:middle line:90% will be delayed. 01:16:05.350 --> 01:16:10.010 align:middle line:84% So if one probe goes like this and gives you 01:16:10.010 --> 01:16:13.880 align:middle line:84% this magnetic field as a function of time, 01:16:13.880 --> 01:16:19.090 align:middle line:90% well, that one is 2, 1. 01:16:19.090 --> 01:16:25.140 align:middle line:90% And the next one goes like this. 01:16:25.140 --> 01:16:26.910 align:middle line:84% You can do some rather neat tricks 01:16:26.910 --> 01:16:31.500 align:middle line:84% and identify similar features in these two signals-- so the start 01:16:31.500 --> 01:16:35.520 align:middle line:84% of the magnetic field, the start of this foot, 01:16:35.520 --> 01:16:39.550 align:middle line:84% the peak of the magnetic field here, and the fall of this foot 01:16:39.550 --> 01:16:40.050 align:middle line:90% here. 01:16:40.050 --> 01:16:46.180 align:middle line:90% 01:16:46.180 --> 01:16:48.320 align:middle line:84% And you can calculate the time differences here. 01:16:48.320 --> 01:16:49.987 align:middle line:84% So you'll have a set of time differences 01:16:49.987 --> 01:16:55.790 align:middle line:84% like delta T1, delta T2, delta T3, and so on like that. 01:16:55.790 --> 01:16:57.700 align:middle line:84% And from this, you can put it all together. 01:16:57.700 --> 01:17:00.640 align:middle line:84% And you can get out an estimate for the flow velocity 01:17:00.640 --> 01:17:05.470 align:middle line:84% of your plasma as a function of time at a specific point, 01:17:05.470 --> 01:17:08.740 align:middle line:84% so at the location of your probe, just 01:17:08.740 --> 01:17:17.000 align:middle line:84% from doing very simple time of flight differencing like this. 01:17:17.000 --> 01:17:19.210 align:middle line:84% It turns out this formula is a bit simplified. 01:17:19.210 --> 01:17:22.000 align:middle line:84% You actually have to think in a Lagrangian sense 01:17:22.000 --> 01:17:23.898 align:middle line:84% rather than this is an Eulerian sense. 01:17:23.898 --> 01:17:26.440 align:middle line:84% And one of my grad students, [? Rachaad ?] [? Datta, ?] wrote 01:17:26.440 --> 01:17:29.420 align:middle line:84% a nice paper on how to do this properly last year, 01:17:29.420 --> 01:17:30.910 align:middle line:90% which is in RSI. 01:17:30.910 --> 01:17:35.000 align:middle line:84% So if you want to know any more details, [? Datta ?] et al. 01:17:35.000 --> 01:17:37.480 align:middle line:90% RSI 2022. 01:17:37.480 --> 01:17:39.230 align:middle line:84% Another thing that [? Rachaad ?] then did, 01:17:39.230 --> 01:17:41.860 align:middle line:84% which is the final thing I want to talk about today, 01:17:41.860 --> 01:17:48.170 align:middle line:84% is he asked what happens if the plasma, which is flowing here, 01:17:48.170 --> 01:17:58.203 align:middle line:84% is flowing supersonically, has a Mach number greater than 1. 01:17:58.203 --> 01:17:59.620 align:middle line:84% Anyone know what will happen then? 01:17:59.620 --> 01:18:02.760 align:middle line:90% 01:18:02.760 --> 01:18:03.777 align:middle line:90% Yeah? 01:18:03.777 --> 01:18:05.030 align:middle line:90% STUDENT: [INAUDIBLE] 01:18:05.030 --> 01:18:07.880 align:middle line:84% JACK HARE: Yeah, so you'll get a bow shock 01:18:07.880 --> 01:18:10.390 align:middle line:90% forming around your probe. 01:18:10.390 --> 01:18:13.660 align:middle line:84% Anyone taking a course in shock physics and 01:18:13.660 --> 01:18:18.870 align:middle line:84% know anything interesting about the shape of the bow shock? 01:18:18.870 --> 01:18:21.260 align:middle line:90% STUDENT: [INAUDIBLE] 01:18:21.260 --> 01:18:23.180 align:middle line:90% 01:18:23.180 --> 01:18:24.110 align:middle line:90% JACK HARE: Indeed. 01:18:24.110 --> 01:18:27.300 align:middle line:84% So the opening angle of this shock, 01:18:27.300 --> 01:18:32.510 align:middle line:84% which we call mu, from simple shock theory, 01:18:32.510 --> 01:18:35.390 align:middle line:84% we get that the sine of this opening angle 01:18:35.390 --> 01:18:38.550 align:middle line:84% is equal to 1 over the Mach number. 01:18:38.550 --> 01:18:41.510 align:middle line:84% So if we have a probe and a camera taking 01:18:41.510 --> 01:18:43.610 align:middle line:84% a picture of this bow shock and we 01:18:43.610 --> 01:18:45.620 align:middle line:84% can measure the opening angle of it, 01:18:45.620 --> 01:18:49.480 align:middle line:90% we can get out the Mach number. 01:18:49.480 --> 01:18:51.430 align:middle line:90% What's the Mach number equal to? 01:18:51.430 --> 01:18:55.006 align:middle line:90% The dimensionless parameter? 01:18:55.006 --> 01:18:57.112 align:middle line:90% STUDENT: [INAUDIBLE] 01:18:57.112 --> 01:18:58.520 align:middle line:90% 01:18:58.520 --> 01:19:00.470 align:middle line:84% JACK HARE: Speed over the sound speed. 01:19:00.470 --> 01:19:01.070 align:middle line:90% Great. 01:19:01.070 --> 01:19:07.510 align:middle line:84% Well, we've already measured the speed using this technique. 01:19:07.510 --> 01:19:11.000 align:middle line:84% We've just measured the Mach number using this technique. 01:19:11.000 --> 01:19:15.670 align:middle line:84% So now, we have an estimate of the sound speed. 01:19:15.670 --> 01:19:18.520 align:middle line:84% And in a plasma, the sound speed is 01:19:18.520 --> 01:19:22.060 align:middle line:84% going to be something like gamma average ionization 01:19:22.060 --> 01:19:26.710 align:middle line:84% times the electron temperature over the ion mass all 01:19:26.710 --> 01:19:28.720 align:middle line:90% [? halved, ?] like this. 01:19:28.720 --> 01:19:31.360 align:middle line:84% Now, some of these coefficients we don't know very well. 01:19:31.360 --> 01:19:33.370 align:middle line:84% We should know what our plasma is made out of. 01:19:33.370 --> 01:19:34.990 align:middle line:90% We should know the ion mass. 01:19:34.990 --> 01:19:39.100 align:middle line:84% It turns out that this gamma for high density 01:19:39.100 --> 01:19:43.290 align:middle line:84% plasma where you're going to see shocks is about 1.1. 01:19:43.290 --> 01:19:45.000 align:middle line:90% E would be 5/3 in an ideal gas. 01:19:45.000 --> 01:19:46.980 align:middle line:84% But it's reduced by the effect of ionization. 01:19:46.980 --> 01:19:49.200 align:middle line:84% And so all of this together gives you 01:19:49.200 --> 01:19:55.360 align:middle line:84% a measurement of the ionization times the electron temperature, 01:19:55.360 --> 01:19:57.870 align:middle line:84% which is kind of cool because it means that, just by using 01:19:57.870 --> 01:20:02.060 align:middle line:84% a single B-dot and a camera, you can get out the electron 01:20:02.060 --> 01:20:04.550 align:middle line:84% temperature of your plasma, which is the sort of thing 01:20:04.550 --> 01:20:07.340 align:middle line:84% that you normally need to do optical Thomson scattering for. 01:20:07.340 --> 01:20:09.200 align:middle line:84% We're going to spend four lectures on optical Thomson 01:20:09.200 --> 01:20:09.700 align:middle line:90% scattering. 01:20:09.700 --> 01:20:11.630 align:middle line:84% It's not an easy thing to understand, 01:20:11.630 --> 01:20:14.540 align:middle line:84% whereas this is a relatively simple diagnostic and relatively 01:20:14.540 --> 01:20:15.540 align:middle line:90% cheap as well. 01:20:15.540 --> 01:20:17.180 align:middle line:84% So this is kind of a nice way of being 01:20:17.180 --> 01:20:19.250 align:middle line:84% able to measure the temperature of your plasma 01:20:19.250 --> 01:20:21.020 align:middle line:84% just by taking pictures of bow shocks 01:20:21.020 --> 01:20:23.730 align:middle line:84% and measuring the magnetic field flowing over them. 01:20:23.730 --> 01:20:27.580 align:middle line:84% So I see there were some questions. 01:20:27.580 --> 01:20:30.345 align:middle line:90% STUDENT: [INAUDIBLE] 01:20:30.345 --> 01:20:45.270 align:middle line:90% 01:20:45.270 --> 01:20:47.770 align:middle line:84% JACK HARE: So the question is, does the heating of the probe 01:20:47.770 --> 01:20:48.880 align:middle line:90% cause problems? 01:20:48.880 --> 01:20:50.530 align:middle line:84% Definitely, these probes get blown up. 01:20:50.530 --> 01:20:52.510 align:middle line:84% And normally, you see the voltage signal go haywire. 01:20:52.510 --> 01:20:54.468 align:middle line:84% And then you stop trusting it after that point. 01:20:54.468 --> 01:20:56.650 align:middle line:84% So you integrate up until time of death. 01:20:56.650 --> 01:20:58.704 align:middle line:90% And then you leave it, yeah. 01:20:58.704 --> 01:21:01.024 align:middle line:90% STUDENT: [INAUDIBLE] 01:21:01.024 --> 01:21:01.868 align:middle line:90% 01:21:01.868 --> 01:21:04.160 align:middle line:84% JACK HARE: The thing is that heat transport is actually 01:21:04.160 --> 01:21:06.110 align:middle line:90% very slow on these timescales. 01:21:06.110 --> 01:21:08.630 align:middle line:84% These are like nanosecond timescale experiments here. 01:21:08.630 --> 01:21:11.390 align:middle line:84% So you're not too worried about heat transport. 01:21:11.390 --> 01:21:13.813 align:middle line:84% So yeah, ablation is actually like photoionization 01:21:13.813 --> 01:21:15.230 align:middle line:84% on the surface and heating through 01:21:15.230 --> 01:21:18.500 align:middle line:84% as an ablation [INAUDIBLE] problem, yeah. 01:21:18.500 --> 01:21:20.888 align:middle line:90% STUDENT: [INAUDIBLE] 01:21:20.888 --> 01:21:29.663 align:middle line:90% 01:21:29.663 --> 01:21:32.025 align:middle line:84% JACK HARE: So the question is, are you 01:21:32.025 --> 01:21:33.650 align:middle line:84% looking at optical wavelengths, are you 01:21:33.650 --> 01:21:34.860 align:middle line:90% looking at X-ray wavelengths? 01:21:34.860 --> 01:21:36.527 align:middle line:84% Yes, all of those wavelengths will work. 01:21:36.527 --> 01:21:38.210 align:middle line:84% In general, the brightness will be 01:21:38.210 --> 01:21:39.872 align:middle line:84% to do with the density of the plasma. 01:21:39.872 --> 01:21:42.330 align:middle line:84% It has a stronger dependence on density than anything else. 01:21:42.330 --> 01:21:45.200 align:middle line:84% So if the plasma is hot enough to be emitting, 01:21:45.200 --> 01:21:47.240 align:middle line:84% then you will see the density jump. 01:21:47.240 --> 01:21:49.007 align:middle line:84% And you'll see that as a dark region, 01:21:49.007 --> 01:21:50.840 align:middle line:84% and then a bright region where the shock is. 01:21:50.840 --> 01:21:53.900 align:middle line:90% So measuring this is uncertain. 01:21:53.900 --> 01:21:57.205 align:middle line:84% You want to measure mu as far back as possible, 01:21:57.205 --> 01:21:58.580 align:middle line:84% where you've got to a weak shock. 01:21:58.580 --> 01:21:59.780 align:middle line:84% But where you've got the weak shock, 01:21:59.780 --> 01:22:01.860 align:middle line:84% you have a small density jump, so you can't see it. 01:22:01.860 --> 01:22:03.818 align:middle line:84% So what you actually do is you measure up here, 01:22:03.818 --> 01:22:05.060 align:middle line:90% and then eyeball it. 01:22:05.060 --> 01:22:08.217 align:middle line:90% And you go uh, maybe like that. 01:22:08.217 --> 01:22:09.300 align:middle line:90% And then you measure that. 01:22:09.300 --> 01:22:11.300 align:middle line:84% So that contributes a lot to the errors. 01:22:11.300 --> 01:22:12.967 align:middle line:84% And then, of course, because you're only 01:22:12.967 --> 01:22:15.020 align:middle line:84% measuring the sound speed with temperature 01:22:15.020 --> 01:22:17.630 align:middle line:84% inside a square root, when you square the sound speed 01:22:17.630 --> 01:22:20.160 align:middle line:84% to get the temperature, that amplifies those errors. 01:22:20.160 --> 01:22:22.730 align:middle line:84% So we measured like 22 plus or minus 01:22:22.730 --> 01:22:25.640 align:middle line:84% 9 electron volts, which is not a very precise measurement. 01:22:25.640 --> 01:22:29.690 align:middle line:84% But it's still better than you can do with [INAUDIBLE]. 01:22:29.690 --> 01:22:30.890 align:middle line:90% Yeah, any other questions? 01:22:30.890 --> 01:22:31.560 align:middle line:90% Yes? 01:22:31.560 --> 01:22:33.960 align:middle line:90% STUDENT: [INAUDIBLE] 01:22:33.960 --> 01:22:35.070 align:middle line:90% 01:22:35.070 --> 01:22:38.280 align:middle line:84% JACK HARE: Z is the average ionization of the plasma. 01:22:38.280 --> 01:22:40.350 align:middle line:84% So for hydrogen, that would just be 1. 01:22:40.350 --> 01:22:42.720 align:middle line:84% For fully stripped carbon, that would be 6. 01:22:42.720 --> 01:22:46.020 align:middle line:84% But in the plasmas I work with, your atom can have 01:22:46.020 --> 01:22:47.410 align:middle line:90% some electrons attached to it. 01:22:47.410 --> 01:22:48.690 align:middle line:90% So it's still an ion. 01:22:48.690 --> 01:22:51.180 align:middle line:84% But it doesn't have all of the electrons removed. 01:22:51.180 --> 01:22:53.880 align:middle line:84% And we'll talk a lot more about ionization states later on. 01:22:53.880 --> 01:22:56.340 align:middle line:84% But just throwing that in there, if you're a tokamak person 01:22:56.340 --> 01:23:00.590 align:middle line:84% you don't like thinking about z, you can just make that 1. 01:23:00.590 --> 01:23:02.840 align:middle line:84% But then you probably won't get any radiation 01:23:02.840 --> 01:23:04.160 align:middle line:90% because it's fully stripped. 01:23:04.160 --> 01:23:06.410 align:middle line:84% So you probably won't be able to see this very easily. 01:23:06.410 --> 01:23:09.827 align:middle line:84% So you need to have something going on, yeah. 01:23:09.827 --> 01:23:10.660 align:middle line:90% Any other questions? 01:23:10.660 --> 01:23:11.618 align:middle line:90% Anything from Colombia? 01:23:11.618 --> 01:23:15.660 align:middle line:90% 01:23:15.660 --> 01:23:17.700 align:middle line:84% All right, well, we're at time or past time. 01:23:17.700 --> 01:23:18.550 align:middle line:90% Thank you very much. 01:23:18.550 --> 01:23:20.480 align:middle line:90% See you next week. 01:23:20.480 --> 01:23:23.000 align:middle line:90%