WEBVTT 00:00:00.046 --> 00:00:03.040 Fusion needs a plasma at over a hundred million kelvin, 00:00:03.123 --> 00:00:06.084 and a plasma that hot pushes outward hard. 00:00:06.281 --> 00:00:10.194 Here is the cheapest way anyone has proposed to hold one still. 00:00:10.507 --> 00:00:12.142 Drive a current straight through it, 00:00:12.225 --> 00:00:15.163 and let the current's own field do the squeezing. 00:00:16.560 --> 00:00:21.344 So take a straight column of plasma and pass a current along its axis. 00:00:21.540 --> 00:00:24.499 That axial current is the z in Z-pinch, 00:00:24.582 --> 00:00:27.857 and notice that nothing outside the plasma carries it. 00:00:28.653 --> 00:00:30.976 Look at the column itself for a moment. 00:00:31.173 --> 00:00:35.654 It is a cylinder of ionized gas hotter than the core of the sun, and the only 00:00:35.654 --> 00:00:39.428 thing between it and the vessel wall is the field we are about to build. 00:00:40.225 --> 00:00:41.430 Now that field. 00:00:41.513 --> 00:00:45.879 A current is wrapped by a magnetic field, and the right hand rule puts those loops 00:00:45.879 --> 00:00:50.512 in the azimuthal direction, closing around the axis at every height. 00:00:51.309 --> 00:00:56.452 The field is azimuthal and the current is axial, so the force per unit volume, 00:00:56.452 --> 00:00:58.984 J cross B, points inward. 00:00:59.180 --> 00:01:03.303 Every part of the plasma is pushed toward the axis at once. 00:01:04.215 --> 00:01:05.493 How hard? 00:01:06.108 --> 00:01:11.319 Ampere's law gives the azimuthal field at radius r, and a field of strength B 00:01:11.402 --> 00:01:16.349 always carries a magnetic pressure equal to B squared over two mu nought. 00:01:17.145 --> 00:01:19.351 Put the two together and the pinch pressure 00:01:19.351 --> 00:01:24.275 at the surface of the column goes as the current squared over the radius squared. 00:01:24.727 --> 00:01:29.267 Squeeze the column inward and the field that squeezes it only grows. 00:01:30.319 --> 00:01:33.812 Balance that magnetic pressure against the plasma's own pressure, 00:01:33.895 --> 00:01:36.079 and one relation falls out. 00:01:36.275 --> 00:01:41.140 Mu nought times the current squared equals eight pi N k T, with N the number 00:01:41.140 --> 00:01:43.195 of particles per unit length. 00:01:44.108 --> 00:01:46.094 Read it as a design rule. 00:01:46.291 --> 00:01:50.876 The current alone fixes the pressure you can confine, so a few mega-amps reaches 00:01:50.876 --> 00:01:54.349 fusion conditions with no external magnet anywhere. 00:01:54.545 --> 00:01:56.517 That is the whole appeal of the pinch, 00:01:56.600 --> 00:01:59.341 and in a moment it will also be the whole problem. 00:02:00.461 --> 00:02:05.059 The pinch has a fatal flaw, and it is written in the formula we just derived. 00:02:05.256 --> 00:02:06.986 The inward pressure goes as one 00:02:06.986 --> 00:02:11.689 over the radius squared, so it cannot be the same everywhere unless the radius is. 00:02:12.485 --> 00:02:14.239 Give that column a ripple. 00:02:14.436 --> 00:02:17.625 A slight narrowing here, a slight bulge there: 00:02:17.709 --> 00:02:21.623 exactly the kind of perturbation that thermal noise supplies for nothing. 00:02:22.419 --> 00:02:23.984 Now look at the narrow place. 00:02:24.068 --> 00:02:25.241 Its radius is smaller, 00:02:25.241 --> 00:02:28.930 so the field there is stronger, and the inward push is stronger. 00:02:29.014 --> 00:02:31.975 The neck squeezes down harder than its neighbors do. 00:02:32.772 --> 00:02:35.118 At the bulge the opposite happens. 00:02:35.221 --> 00:02:38.647 Larger radius, weaker field, weaker squeeze. 00:02:38.844 --> 00:02:42.885 So the perturbation feeds itself, and the neck runs away. 00:02:43.681 --> 00:02:48.117 That is the m equals zero mode, the sausage instability. 00:02:48.313 --> 00:02:50.354 It grows in tens of nanoseconds, 00:02:50.438 --> 00:02:54.409 and it does not stop until the column has been cut into separate blobs. 00:02:55.713 --> 00:02:59.963 The second mode is worse, because nothing about the radius has to change at all. 00:03:00.160 --> 00:03:03.272 Suppose the whole column simply bends sideways. 00:03:04.069 --> 00:03:08.164 On the inside of the bend the azimuthal field lines are crowded closer together, 00:03:08.248 --> 00:03:10.989 so the field there is stronger than on the outside. 00:03:11.186 --> 00:03:15.227 The net push is sideways, the same way the column has already moved. 00:03:16.023 --> 00:03:18.172 So this one feeds itself too. 00:03:18.369 --> 00:03:20.897 The more it bends, the harder it is pushed, 00:03:20.981 --> 00:03:23.989 and that is the m equals one kink mode. 00:03:24.785 --> 00:03:28.463 Both modes come straight out of ideal magnetohydrodynamics, 00:03:28.547 --> 00:03:32.971 both are driven by the confining current itself, and both are fast. 00:03:33.168 --> 00:03:36.338 Everything that follows is an answer to these two pictures. 00:03:37.391 --> 00:03:39.853 The first fix is the obvious one. 00:03:39.956 --> 00:03:43.474 Add a magnetic field along the axis, from external coils 00:03:43.474 --> 00:03:48.444 or driven by the plasma itself, and watch what it does to the field line geometry. 00:03:49.148 --> 00:03:53.790 The azimuthal field wraps around the column and the axial field runs along it, 00:03:53.873 --> 00:03:56.057 so the total field is a helix. 00:03:56.509 --> 00:03:59.154 A field line now has tension along the column, 00:03:59.237 --> 00:04:02.233 and bending the column means stretching that line. 00:04:02.937 --> 00:04:04.482 How much do we need? 00:04:04.795 --> 00:04:07.941 Compare the pitch of the helix with the length of the column. 00:04:08.045 --> 00:04:12.803 The safety factor q is two pi a B z over L B theta, 00:04:12.886 --> 00:04:16.068 and the kink is held off when q is greater than one. 00:04:16.865 --> 00:04:20.150 Written out, that says a field line must not close on itself 00:04:20.150 --> 00:04:22.473 before it reaches the end of the column. 00:04:22.577 --> 00:04:25.863 Bend the column and the tension pulls it straight again. 00:04:26.776 --> 00:04:30.190 The trapped axial field resists the sausage mode too. 00:04:30.387 --> 00:04:33.218 Squeezing the column compresses the flux it encloses, 00:04:33.301 --> 00:04:35.658 and compressed flux pushes back. 00:04:36.962 --> 00:04:39.259 Now bend the whole pinch into a torus, 00:04:39.342 --> 00:04:44.381 so the current has no ends to leak from and the plasma can organize its own field. 00:04:45.085 --> 00:04:48.694 This is the geometry of every steady magnetic confinement machine, 00:04:48.777 --> 00:04:52.203 and the pinch version of it is called the reversed field pinch. 00:04:52.399 --> 00:04:55.569 The plasma current runs the long way around the ring. 00:04:56.366 --> 00:04:58.689 The name comes from its field profile. 00:04:58.885 --> 00:05:03.030 Measure the axial field from the center of the plasma out to the wall, with r 00:05:03.030 --> 00:05:05.306 over a the fraction of the way out. 00:05:05.503 --> 00:05:10.635 It starts strong on the axis, falls, and reverses sign near the edge. 00:05:11.850 --> 00:05:17.180 The azimuthal field does the opposite: zero on the axis and largest at the edge. 00:05:17.283 --> 00:05:22.437 That helical, self-organized profile is what a resistive plasma relaxes into, 00:05:22.520 --> 00:05:26.456 and it is stable against the kink with no large external field. 00:05:27.853 --> 00:05:31.708 The other route keeps the column straight and gives it a velocity shear. 00:05:31.904 --> 00:05:36.236 Let the plasma flow along the axis, faster in the middle than at the edge. 00:05:36.939 --> 00:05:41.967 A growing sausage or kink is a wave that has to stay coherent across the radius. 00:05:42.164 --> 00:05:46.135 If neighboring layers are sliding past each other fast enough, the wave is pulled 00:05:46.135 --> 00:05:47.993 apart before it can grow. 00:05:48.604 --> 00:05:52.297 The requirement is a shear rate of order a tenth of the growth rate. 00:05:52.493 --> 00:05:56.276 Sheared flow experiments have then held a quiet pinch for microseconds, 00:05:56.359 --> 00:05:58.833 thousands of times the ideal growth time. 00:05:59.983 --> 00:06:03.672 The pinch and the torus both use a field that is open at the ends 00:06:03.756 --> 00:06:05.800 or held in place by coils. 00:06:05.904 --> 00:06:10.688 There is a third option: let the plasma make a closed field of its own. 00:06:11.601 --> 00:06:14.539 Here is a cross-section taken along the axis. 00:06:14.736 --> 00:06:18.901 The current in this plasma is azimuthal, running around the axis, 00:06:18.985 --> 00:06:22.909 and the field it makes is poloidal, lying in the plane of the picture. 00:06:23.706 --> 00:06:28.347 Inside the separatrix the field lines are closed loops around a magnetic axis, 00:06:28.431 --> 00:06:32.147 one above the machine axis in this section and one below. 00:06:32.344 --> 00:06:33.957 The separatrix meets the axis 00:06:33.957 --> 00:06:38.126 at two X points, and outside it the field runs the other way. 00:06:38.830 --> 00:06:41.269 That reversal is the name of the thing: 00:06:41.373 --> 00:06:44.032 a field-reversed configuration. 00:06:44.229 --> 00:06:48.664 Because the field closes, it can hold a plasma whose pressure is comparable 00:06:48.664 --> 00:06:53.959 to the magnetic pressure, beta of order one, rather than a few percent. 00:06:54.872 --> 00:06:57.601 High beta is the economic argument. 00:06:57.797 --> 00:07:01.521 For a given magnetic field you confine far more plasma, 00:07:01.605 --> 00:07:04.752 and there is no coil threading the plasma at all. 00:07:06.149 --> 00:07:09.969 An FRC is usually made in one place and used in another. 00:07:10.166 --> 00:07:12.497 Form the plasma at one end of the machine, 00:07:12.581 --> 00:07:16.401 and it can be pushed along the axis as a self-contained object. 00:07:17.105 --> 00:07:18.580 Then squeeze it. 00:07:18.777 --> 00:07:22.689 Coils at the far end compress the plasma inward and along the axis 00:07:22.689 --> 00:07:27.821 at once, and because beta is high, that compression heats it efficiently. 00:07:28.618 --> 00:07:31.413 Formation, translation, compression. 00:07:31.497 --> 00:07:33.785 That separation is the appeal: 00:07:33.889 --> 00:07:35.735 the difficult plasma physics happens 00:07:35.735 --> 00:07:39.637 in one section of the machine and the heating happens in another. 00:07:40.704 --> 00:07:45.523 The last idea gives up on holding the plasma steady, and crushes it instead. 00:07:45.836 --> 00:07:49.840 This is magnetized liner inertial fusion, MagLIF, 00:07:49.923 --> 00:07:52.803 and it runs on the Z machine at Sandia. 00:07:52.907 --> 00:07:57.156 Start with a centimeter-scale metal cylinder full of deuterium fuel. 00:07:57.953 --> 00:08:01.643 Step one: put an axial magnetic field through the fuel 00:08:01.726 --> 00:08:06.684 before anything else happens, about ten tesla from a pair of coils. 00:08:06.881 --> 00:08:08.995 The fuel is now magnetized. 00:08:09.791 --> 00:08:11.044 Step two: 00:08:11.127 --> 00:08:15.085 a laser pulse of about a kilojoule enters through a window at the top 00:08:15.085 --> 00:08:16.699 and preheats the fuel to a couple 00:08:16.699 --> 00:08:21.100 of hundred electron volts, so the implosion does not have to do that work. 00:08:21.897 --> 00:08:25.645 Step three: twenty million amps flows axially through the liner 00:08:25.728 --> 00:08:27.865 in a hundred nanoseconds. 00:08:28.178 --> 00:08:31.719 It is the same J cross B force as the classic pinch, 00:08:31.916 --> 00:08:35.423 but now it acts on solid metal rather than on plasma. 00:08:36.819 --> 00:08:41.034 The magnetic pressure driving the liner is the expression we started with: 00:08:41.138 --> 00:08:42.346 mu nought 00:08:43.147 --> 00:08:46.468 I squared over eight pi squared r squared. 00:08:46.572 --> 00:08:51.008 At twenty mega-amps on a centimeter radius, that is megabars. 00:08:51.804 --> 00:08:55.242 The liner implodes at seventy kilometers a second. 00:08:55.438 --> 00:08:58.492 The fuel column is compressed by a factor of about twenty five 00:08:58.492 --> 00:09:03.366 in radius, and the axial field goes with it, because flux is conserved: 00:09:03.449 --> 00:09:06.155 B times r squared stays put. 00:09:06.952 --> 00:09:09.643 Ten tesla becomes several thousand tesla, 00:09:09.726 --> 00:09:12.897 hundreds of times stronger than any steady magnet. 00:09:13.093 --> 00:09:15.694 That trapped field is the point of the scheme: it 00:09:15.694 --> 00:09:20.629 holds back electron heat conduction and it keeps the alpha particles inside the fuel. 00:09:22.025 --> 00:09:24.395 So put the family side by side. 00:09:24.591 --> 00:09:26.923 Each row is confined by something different, 00:09:27.006 --> 00:09:29.306 and each one pays a different price. 00:09:30.102 --> 00:09:33.933 The classic Z-pinch is confined by its own current and destroyed 00:09:33.933 --> 00:09:35.360 by the sausage and the kink. 00:09:35.443 --> 00:09:40.180 Sheared flow keeps that same confinement and buys stability with a velocity profile 00:09:40.180 --> 00:09:41.899 that has to be maintained. 00:09:42.382 --> 00:09:46.885 The reversed field pinch lets the plasma organize a helical field for itself, 00:09:46.968 --> 00:09:52.296 and pays for it in resistive transport. The FRC closes the field lines and runs 00:09:52.296 --> 00:09:56.500 at high beta, and fights global stability and confinement time. 00:09:57.296 --> 00:10:00.429 MagLIF does not confine in the usual sense at all: 00:10:00.512 --> 00:10:04.644 an imploding liner holds the fuel together for a hundred nanoseconds, 00:10:04.727 --> 00:10:07.816 and everything depends on that liner staying smooth. 00:10:08.013 --> 00:10:13.398 A tokamak is the opposite extreme, with big external coils, low beta, 00:10:13.481 --> 00:10:15.038 and a steady state. 00:10:15.834 --> 00:10:17.853 All of them chase the same product: 00:10:17.936 --> 00:10:22.058 density, times confinement time, times temperature. 00:10:22.255 --> 00:10:27.408 A pinch pushes the density up and the time down, and a tokamak does the reverse. 00:10:27.491 --> 00:10:31.799 Which of those trades wins is still, sixty years on, an open question.