{"version":1,"lectureId":"01M1AFM0CEW468WY026NSSG2ZK","attempt":0,"publication":{"slug":"z-pinch-fusion-and-magnetic-pinch-relatives","title":"Z-Pinch Fusion and Magnetic Pinch Relatives","subject":"physics","summary":"A current through a plasma makes its own magnetic field, and that field squeezes the plasma carrying it. This lecture starts from that idea, derives the pinch pressure and the Bennett relation, and then shows why the classic Z-pinch tears itself apart through the sausage and kink modes. From there it follows the fixes: an axial field and the Kruskal and Shafranov limit, the reversed-field pinch bent into a torus, sheared-flow stabilization, the closed high-beta topology of a field-reversed configuration, and MagLIF, where a pre-magnetized, laser-preheated fuel column is crushed by an imploding metal liner. It closes by comparing what each scheme is confined by and what each one pays for that confinement. For viewers who already have the Lorentz force, magnetic pressure and basic ideal MHD.","metaDescription":"How a plasma's own current confines it, why the Z-pinch is unstable, and how RFPs, sheared flow, FRCs and MagLIF answer that.","transcript":"Fusion needs a plasma at over a hundred million kelvin, and a plasma that hot pushes outward hard. Here is the cheapest way anyone has proposed to hold one still. Drive a current straight through it, and let the current's own field do the squeezing. So take a straight column of plasma and pass a current along its axis. That axial current is the z in Z-pinch, and notice that nothing outside the plasma carries it. Look at the column itself for a moment. It is a cylinder of ionized gas hotter than the core of the sun, and the only thing between it and the vessel wall is the field we are about to build. Now that field. A current is wrapped by a magnetic field, and the right hand rule puts those loops in the azimuthal direction, closing around the axis at every height. The field is azimuthal and the current is axial, so the force per unit volume, J cross B, points inward. Every part of the plasma is pushed toward the axis at once. How hard? Ampere's law gives the azimuthal field at radius r, and a field of strength B always carries a magnetic pressure equal to B squared over two mu nought. Put the two together and the pinch pressure at the surface of the column goes as the current squared over the radius squared. Squeeze the column inward and the field that squeezes it only grows. Balance that magnetic pressure against the plasma's own pressure, and one relation falls out. Mu nought times the current squared equals eight pi N k T, with N the number of particles per unit length. Read it as a design rule. The current alone fixes the pressure you can confine, so a few mega-amps reaches fusion conditions with no external magnet anywhere. That is the whole appeal of the pinch, and in a moment it will also be the whole problem. The pinch has a fatal flaw, and it is written in the formula we just derived. The inward pressure goes as one over the radius squared, so it cannot be the same everywhere unless the radius is. Give that column a ripple. A slight narrowing here, a slight bulge there: exactly the kind of perturbation that thermal noise supplies for nothing. Now look at the narrow place. Its radius is smaller, so the field there is stronger, and the inward push is stronger. The neck squeezes down harder than its neighbors do. At the bulge the opposite happens. Larger radius, weaker field, weaker squeeze. So the perturbation feeds itself, and the neck runs away. That is the m equals zero mode, the sausage instability. It grows in tens of nanoseconds, and it does not stop until the column has been cut into separate blobs. The second mode is worse, because nothing about the radius has to change at all. Suppose the whole column simply bends sideways. On the inside of the bend the azimuthal field lines are crowded closer together, so the field there is stronger than on the outside. The net push is sideways, the same way the column has already moved. So this one feeds itself too. The more it bends, the harder it is pushed, and that is the m equals one kink mode. Both modes come straight out of ideal magnetohydrodynamics, both are driven by the confining current itself, and both are fast. Everything that follows is an answer to these two pictures. The first fix is the obvious one. Add a magnetic field along the axis, from external coils or driven by the plasma itself, and watch what it does to the field line geometry. The azimuthal field wraps around the column and the axial field runs along it, so the total field is a helix. A field line now has tension along the column, and bending the column means stretching that line. How much do we need? Compare the pitch of the helix with the length of the column. The safety factor q is two pi a B z over L B theta, and the kink is held off when q is greater than one. Written out, that says a field line must not close on itself before it reaches the end of the column. Bend the column and the tension pulls it straight again. The trapped axial field resists the sausage mode too. Squeezing the column compresses the flux it encloses, and compressed flux pushes back. Now bend the whole pinch into a torus, so the current has no ends to leak from and the plasma can organize its own field. This is the geometry of every steady magnetic confinement machine, and the pinch version of it is called the reversed field pinch. The plasma current runs the long way around the ring. The name comes from its field profile. Measure the axial field from the center of the plasma out to the wall, with r over a the fraction of the way out. It starts strong on the axis, falls, and reverses sign near the edge. The azimuthal field does the opposite: zero on the axis and largest at the edge. That helical, self-organized profile is what a resistive plasma relaxes into, and it is stable against the kink with no large external field. The other route keeps the column straight and gives it a velocity shear. Let the plasma flow along the axis, faster in the middle than at the edge. A growing sausage or kink is a wave that has to stay coherent across the radius. If neighboring layers are sliding past each other fast enough, the wave is pulled apart before it can grow. The requirement is a shear rate of order a tenth of the growth rate. Sheared flow experiments have then held a quiet pinch for microseconds, thousands of times the ideal growth time. The pinch and the torus both use a field that is open at the ends or held in place by coils. There is a third option: let the plasma make a closed field of its own. Here is a cross-section taken along the axis. The current in this plasma is azimuthal, running around the axis, and the field it makes is poloidal, lying in the plane of the picture. Inside the separatrix the field lines are closed loops around a magnetic axis, one above the machine axis in this section and one below. The separatrix meets the axis at two X points, and outside it the field runs the other way. That reversal is the name of the thing: a field-reversed configuration. Because the field closes, it can hold a plasma whose pressure is comparable to the magnetic pressure, beta of order one, rather than a few percent. High beta is the economic argument. For a given magnetic field you confine far more plasma, and there is no coil threading the plasma at all. An FRC is usually made in one place and used in another. Form the plasma at one end of the machine, and it can be pushed along the axis as a self-contained object. Then squeeze it. Coils at the far end compress the plasma inward and along the axis at once, and because beta is high, that compression heats it efficiently. Formation, translation, compression. That separation is the appeal: the difficult plasma physics happens in one section of the machine and the heating happens in another. The last idea gives up on holding the plasma steady, and crushes it instead. This is magnetized liner inertial fusion, MagLIF, and it runs on the Z machine at Sandia. Start with a centimeter-scale metal cylinder full of deuterium fuel. Step one: put an axial magnetic field through the fuel before anything else happens, about ten tesla from a pair of coils. The fuel is now magnetized. Step two: a laser pulse of about a kilojoule enters through a window at the top and preheats the fuel to a couple of hundred electron volts, so the implosion does not have to do that work. Step three: twenty million amps flows axially through the liner in a hundred nanoseconds. It is the same J cross B force as the classic pinch, but now it acts on solid metal rather than on plasma. The magnetic pressure driving the liner is the expression we started with: mu nought I squared over eight pi squared r squared. At twenty mega-amps on a centimeter radius, that is megabars. The liner implodes at seventy kilometers a second. The fuel column is compressed by a factor of about twenty five in radius, and the axial field goes with it, because flux is conserved: B times r squared stays put. Ten tesla becomes several thousand tesla, hundreds of times stronger than any steady magnet. That trapped field is the point of the scheme: it holds back electron heat conduction and it keeps the alpha particles inside the fuel. So put the family side by side. Each row is confined by something different, and each one pays a different price. The classic Z-pinch is confined by its own current and destroyed by the sausage and the kink. Sheared flow keeps that same confinement and buys stability with a velocity profile that has to be maintained. The reversed field pinch lets the plasma organize a helical field for itself, and pays for it in resistive transport. The FRC closes the field lines and runs at high beta, and fights global stability and confinement time. MagLIF does not confine in the usual sense at all: an imploding liner holds the fuel together for a hundred nanoseconds, and everything depends on that liner staying smooth. A tokamak is the opposite extreme, with big external coils, low beta, and a steady state. All of them chase the same product: density, times confinement time, times temperature. A pinch pushes the density up and the time down, and a tokamak does the reverse. Which of those trades wins is still, sixty years on, an open question.","watch":{"version":1,"scenes":[{"title":"The Pinch Effect","start":0,"end":120.40268749999997,"objects":{"axes":"an Axes3D (x_range=(-2.4, 2.4), y_range=(-2.4, 2.4), z_range=(-2.8, 2.8))","column":"a Cylinder [red] drawn in axes (start=(0.0, 0.0, -2.0), radius=<VariableNumber radius = 0.8>, fill_end_circles=False)","current":"a Vector [yellow] labelled \"arrow(J)\" drawn in axes (start=(0.0, 0.0, -2.4), end=(0.0, 0.0, 2.5))","loop_high":"a Circle [green] drawn in axes (center=(0.0, 0.0, 1.1), radius=2.0, normal_vector=(0.0, 0.0, 1.0))","loop_low":"a Circle [green] labelled \"arrow(B)_theta\" drawn in axes (center=(0.0, 0.0, -1.1), radius=2.0, normal_vector=(0.0, 0.0, 1.0))","loop_sense":"an Orientation [green] drawn in axes (path=((2.0, 0.0), (1.9903694533443939, 0.1960342806591212), (1.96157…, closed=True, arrows=4)","question":"a Panel that says \"A fusion plasma pushes outward with pressure $p$. 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The inward pressure goes as one over the radius squared, so it cannot be the same everywhere unless the radius is.","live":[],"does":[[120.40268749999997,"sausage is shown on the screen, written out."],[120.40268749999997,"line is shown on the screen, written out."],[120.40268749999997,"line_2 is shown on the screen, written out."],[120.40268749999997,"line_3 is shown on the screen, written out."],[120.40268749999997,"line_4 is shown on the screen, written out."],[120.40268749999997,"line_5 is shown on the screen, written out."],[120.40268749999997,"line_6 is shown on the screen, written out."],[120.40268749999997,"line_7 is shown on the screen, written out."],[120.40268749999997,"line_8 is shown on the screen, written out."],[120.40268749999997,"line_9 is shown on the screen, written out."],[120.40268749999997,"line_10 is shown on the screen, written out."],[120.40268749999997,"line_11 is shown on the screen, written out."],[120.40268749999997,"line_12 is shown on the screen, written out."],[120.40268749999997,"line_13 is shown on the screen, written out."],[120.40268749999997,"line_14 is shown on the screen, written out."],[120.40268749999997,"line_15 is shown on the screen, written out."],[120.40268749999997,"line_16 is shown on the screen, written out."],[120.40268749999997,"line_17 is shown on the screen, written out."],[120.40268749999997,"line_18 is shown on the screen, written out."],[120.40268749999997,"line_19 is shown on the screen, written out."],[120.40268749999997,"line_20 is shown on the screen, written out."],[120.40268749999997,"line_21 is shown on the screen, written out."],[120.40268749999997,"line_22 is shown on the screen, written out."],[120.40268749999997,"line_23 is shown on the screen, written out."],[120.40268749999997,"line_24 is shown on the screen, written out."],[120.40268749999997,"line_25 is shown on the screen, written out."],[120.40268749999997,"line_26 is shown on the screen, written out."],[120.40268749999997,"line_27 is shown on the screen, written out."],[120.40268749999997,"line_28 is shown on the screen, written out."],[120.40268749999997,"line_29 is shown on the screen, written out."],[120.40268749999997,"line_30 is shown on the screen, written out."],[120.40268749999997,"line_31 is shown on the screen, written out."],[120.40268749999997,"line_32 is shown on the screen, written out."],[120.40268749999997,"line_33 is shown on the screen, written out."],[120.40268749999997,"line_34 is shown on the screen, written out."],[120.40268749999997,"line_35 is shown on the screen, written out."],[120.40268749999997,"line_36 is shown on the screen, written out."],[123.14268749999997,"sausage moves to a new place on the board."],[123.14268749999997,"pinch_law is shown on the screen, written out."]]},{"start":132.13668749999997,"say":"Give that column a ripple. A slight narrowing here, a slight bulge there: exactly the kind of perturbation that thermal noise supplies for nothing.","live":["pinch_law","sausage","line","line_2","line_3","line_4","line_5","line_6","line_7","line_8","line_9","line_10","line_11","line_12","line_13","line_14","line_15","line_16","line_17","line_18","line_19","line_20","line_21","line_22","line_23","line_24","line_25","line_26","line_27","line_28","line_29","line_30","line_31","line_32","line_33","line_34","line_35","line_36"],"does":[[133.28568749999997,"line is redrawn as the numbers it depends on change."],[133.28568749999997,"line_2 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_3 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_4 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_5 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_6 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_7 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_8 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_9 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_10 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_11 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_12 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_13 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_14 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_15 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_16 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_17 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_18 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_19 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_20 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_21 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_22 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_23 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_24 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_25 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_26 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_27 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_28 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_29 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_30 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_31 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_32 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_33 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_34 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_35 is redrawn as the numbers it depends on change."],[133.28568749999997,"line_36 is redrawn as the numbers it depends on change."],[133.28568749999997,"neck ticks to 0.16."]]},{"start":142.07068749999996,"say":"Now look at the narrow place. Its radius is smaller, so the field there is stronger, and the inward push is stronger. The neck squeezes down harder than its neighbors do.","live":null,"does":[[143.03468749999996,"hard is shown on the screen, written out."],[143.03468749999996,"hard_2 is shown on the screen, written out."],[150.67368749999997,"soft is shown on the screen, written out."],[150.67368749999997,"soft_2 is shown on the screen, written out."]]},{"start":152.42318749999998,"say":"At the bulge the opposite happens. Larger radius, weaker field, weaker squeeze. So the perturbation feeds itself, and the neck runs away.","live":["pinch_law","sausage","line","line_2","line_3","line_4","line_5","line_6","line_7","line_8","line_9","line_10","line_11","line_12","line_13","line_14","line_15","line_16","line_17","line_18","line_19","line_20","line_21","line_22","line_23","line_24","line_25","line_26","line_27","line_28","line_29","line_30","line_31","line_32","line_33","line_34","line_35","line_36","hard","hard_2","soft","soft_2"],"does":[[159.85368749999998,"line is redrawn as the numbers it depends on change."],[159.85368749999998,"line_2 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_3 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_4 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_5 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_6 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_7 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_8 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_9 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_10 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_11 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_12 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_13 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_14 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_15 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_16 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_17 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_18 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_19 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_20 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_21 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_22 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_23 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_24 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_25 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_26 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_27 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_28 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_29 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_30 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_31 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_32 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_33 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_34 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_35 is redrawn as the numbers it depends on change."],[159.85368749999998,"line_36 is redrawn as the numbers it depends on change."],[159.85368749999998,"neck ticks to 0.45."]]},{"start":163.33268749999996,"say":"That is the m equals zero mode, the sausage instability. It grows in tens of nanoseconds, and it does not stop until the column has been cut into separate blobs.","live":null,"does":[[166.30468749999997,"sausage_note is shown on the screen, written out."],[174.21068749999998,"pinch_law is hidden from the screen — left the board."],[174.21068749999998,"sausage is hidden from the screen — left the board."],[174.21068749999998,"line is hidden from the screen — sausage left the board."],[174.21068749999998,"line_2 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_3 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_4 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_5 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_6 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_7 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_8 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_9 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_10 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_11 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_12 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_13 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_14 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_15 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_16 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_17 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_18 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_19 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_20 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_21 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_22 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_23 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_24 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_25 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_26 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_27 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_28 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_29 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_30 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_31 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_32 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_33 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_34 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_35 is hidden from the screen — sausage left the board."],[174.21068749999998,"line_36 is hidden from the screen — sausage left the board."],[174.21068749999998,"hard is hidden from the screen — sausage left the board."],[174.21068749999998,"hard_2 is hidden from the screen — sausage left the board."],[174.21068749999998,"soft is hidden from the screen — sausage left the board."],[174.21068749999998,"soft_2 is hidden from the screen — sausage left the board."],[174.21068749999998,"sausage_note is hidden from the screen — left the board."]]},{"start":175.41068749999997,"say":"The second mode is worse, because nothing about the radius has to change at all. Suppose the whole column simply bends sideways.","live":[],"does":[[175.41068749999997,"kink is shown on the screen, written out."],[175.41068749999997,"kink_axis is shown on the screen, written out."],[175.41068749999997,"line_37 is shown on the screen, written out."],[175.41068749999997,"line_38 is shown on the screen, written out."],[175.41068749999997,"line_39 is shown on the screen, written out."],[175.41068749999997,"line_40 is shown on the screen, written out."],[175.41068749999997,"line_41 is shown on the screen, written out."],[175.41068749999997,"line_42 is shown on the screen, written out."],[175.41068749999997,"line_43 is shown on the screen, written out."],[175.41068749999997,"line_44 is shown on the screen, written out."],[175.41068749999997,"line_45 is shown on the screen, written out."],[175.41068749999997,"line_46 is shown on the screen, written out."],[175.41068749999997,"line_47 is shown on the screen, written out."],[175.41068749999997,"line_48 is shown on the screen, written out."],[175.41068749999997,"line_49 is shown on the screen, written out."],[175.41068749999997,"line_50 is shown on the screen, written out."],[175.41068749999997,"line_51 is shown on the screen, written out."],[175.41068749999997,"line_52 is shown on the screen, written out."],[175.41068749999997,"line_53 is shown on the screen, written out."],[175.41068749999997,"line_54 is shown on the screen, written out."],[175.41068749999997,"line_55 is shown on the screen, written out."],[175.41068749999997,"line_56 is shown on the screen, written out."],[175.41068749999997,"line_57 is shown on the screen, written out."],[175.41068749999997,"line_58 is shown on the screen, written out."],[175.41068749999997,"line_59 is shown on the screen, written out."],[175.41068749999997,"line_60 is shown on the screen, written out."],[175.41068749999997,"line_61 is shown on the screen, written out."],[175.41068749999997,"line_62 is shown on the screen, written 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change."],[181.66868749999998,"line_41 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_42 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_43 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_44 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_45 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_46 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_47 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_48 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_49 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_50 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_51 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_52 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_53 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_54 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_55 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_56 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_57 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_58 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_59 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_60 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_61 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_62 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_63 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_64 is redrawn as the numbers it depends on change."],[181.66868749999998,"line_65 is redrawn as the numbers it depends on 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The net push is sideways, the same way the column has already moved.","live":["kink","kink_axis","line_37","line_38","line_39","line_40","line_41","line_42","line_43","line_44","line_45","line_46","line_47","line_48","line_49","line_50","line_51","line_52","line_53","line_54","line_55","line_56","line_57","line_58","line_59","line_60","line_61","line_62","line_63","line_64","line_65","line_66","line_67","line_68","line_69","line_70","line_71","line_72"],"does":[[186.76168749999997,"kink moves to a new place on the board."],[186.76168749999997,"kink_law is shown on the screen, written out."],[191.97468749999996,"push is shown on the screen, written out."],[191.97468749999996,"push_2 is shown on the screen, written out."]]},{"start":195.67468749999995,"say":"So this one feeds itself too. The more it bends, the harder it is pushed, and that is the m equals one kink mode.","live":["kink_law","kink","kink_axis","line_37","line_38","line_39","line_40","line_41","line_42","line_43","line_44","line_45","line_46","line_47","line_48","line_49","line_50","line_51","line_52","line_53","line_54","line_55","line_56","line_57","line_58","line_59","line_60","line_61","line_62","line_63","line_64","line_65","line_66","line_67","line_68","line_69","line_70","line_71","line_72","push","push_2"],"does":[[198.53068749999994,"line_37 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_38 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_39 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_40 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_41 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_42 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_43 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_44 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_45 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_46 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_47 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_48 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_49 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_50 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_51 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_52 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_53 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_54 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_55 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_56 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_57 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_58 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_59 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_60 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_61 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_62 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_63 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_64 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_65 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_66 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_67 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_68 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_69 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_70 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_71 is redrawn as the numbers it depends on change."],[198.53068749999994,"line_72 is redrawn as the numbers it depends on change."],[198.53068749999994,"bend ticks to 0.98."],[202.75668749999994,"kink_note is shown on the screen, written out."]]},{"start":204.43668749999995,"say":"Both modes come straight out of ideal magnetohydrodynamics, both are driven by the confining current itself, and both are fast. Everything that follows is an answer to these two pictures.","live":["kink_law","kink_note","kink","kink_axis","line_37","line_38","line_39","line_40","line_41","line_42","line_43","line_44","line_45","line_46","line_47","line_48","line_49","line_50","line_51","line_52","line_53","line_54","line_55","line_56","line_57","line_58","line_59","line_60","line_61","line_62","line_63","line_64","line_65","line_66","line_67","line_68","line_69","line_70","line_71","line_72","push","push_2"],"does":[[216.27962499999998,"kink is hidden from the screen — left the board."],[216.27962499999998,"kink_axis is hidden from the screen — kink left the board."],[216.27962499999998,"line_37 is hidden from the screen — kink left the board."],[216.27962499999998,"line_38 is hidden from the screen — kink left the board."],[216.27962499999998,"line_39 is hidden from the screen — kink left the board."],[216.27962499999998,"line_40 is hidden from the screen — kink left the board."],[216.27962499999998,"line_41 is hidden from the screen — kink left the board."],[216.27962499999998,"line_42 is hidden from the screen — kink left the board."],[216.27962499999998,"line_43 is hidden from the screen — kink left the board."],[216.27962499999998,"line_44 is hidden from the screen — kink left the board."],[216.27962499999998,"line_45 is hidden from the screen — kink left the board."],[216.27962499999998,"line_46 is hidden from the screen — kink left the board."],[216.27962499999998,"line_47 is hidden from the screen — kink left the board."],[216.27962499999998,"line_48 is hidden from the screen — kink left the board."],[216.27962499999998,"line_49 is hidden from the screen — kink left the board."],[216.27962499999998,"line_50 is hidden from the screen — kink left the board."],[216.27962499999998,"line_51 is hidden from the screen — kink left the board."],[216.27962499999998,"line_52 is hidden from the screen — kink left the board."],[216.27962499999998,"line_53 is hidden from the screen — kink left the board."],[216.27962499999998,"line_54 is hidden from the screen — kink left the board."],[216.27962499999998,"line_55 is hidden from the screen — kink left the board."],[216.27962499999998,"line_56 is hidden from the screen — kink left the board."],[216.27962499999998,"line_57 is hidden from the screen — kink left the board."],[216.27962499999998,"line_58 is hidden from the screen — kink left the board."],[216.27962499999998,"line_59 is hidden from the screen — kink left the board."],[216.27962499999998,"line_60 is hidden from the screen — kink left the board."],[216.27962499999998,"line_61 is hidden from the screen — kink left the board."],[216.27962499999998,"line_62 is hidden from the screen — kink left the board."],[216.27962499999998,"line_63 is hidden from the screen — kink left the board."],[216.27962499999998,"line_64 is hidden from the screen — kink left the board."],[216.27962499999998,"line_65 is hidden from the screen — kink left the board."],[216.27962499999998,"line_66 is hidden from the screen — kink left the board."],[216.27962499999998,"line_67 is hidden from the screen — kink left the board."],[216.27962499999998,"line_68 is hidden from the screen — kink left the board."],[216.27962499999998,"line_69 is hidden from the screen — kink left the board."],[216.27962499999998,"line_70 is hidden from the screen — kink left the board."],[216.27962499999998,"line_71 is hidden from the screen — kink left the board."],[216.27962499999998,"line_72 is hidden from the screen — kink left the board."],[216.27962499999998,"push is hidden from the screen — kink left the board."],[216.27962499999998,"push_2 is hidden from the screen — kink left the board."],[216.27962499999998,"kink_law is hidden from the screen — left the board."],[216.27962499999998,"kink_note is hidden from the screen — left the board."]]}]},{"title":"Stabilizing the Pinch","start":217.32129166666664,"end":359.92479166666664,"objects":{"axial_1":"a Vector [blue] labelled \"arrow(B)_z\" drawn in depth (start=(0.55, 0.0, -2.2), end=(0.55, 0.0, 2.2))","axial_2":"a Vector [blue] drawn in depth (start=(-0.55, 0.0, -2.2), end=(-0.55, 0.0, 2.2))","axial_note":"a Panel that says \"If the field line does not close on itself over the length of the column, its tension resists the kink.\"","board_axial":"a Heading that says \"Adding an Axial Field\"","board_flow":"a Heading that says \"Sheared Flow\"","board_ring":"a Heading that says \"The Toroidal Pinch\"","btheta_curve":"a FunctionPlot [green] labelled \"B_theta\" drawn in profile (function=<function>)","bz_curve":"a FunctionPlot [blue] labelled \"B_z\" drawn in profile (function=<function>)","depth":"an Axes3D (x_range=(-2.0, 2.0), y_range=(-2.0, 2.0), z_range=(-2.6, 2.6))","flow":"a Figure (x_range=(-0.5, 3.7), y_range=(-1.8, 1.8), aspect=(4.2, 3.6))","flow_bottom":"a Line [red] drawn in flow (start=(-0.2, -1.25), end=(3.5, -1.25))","flow_top":"a Line [red] drawn in flow (start=(-0.2, 1.25), end=(3.5, 1.25))","helix":"a ParametricCurve [green] labelled \"arrow(B)\" drawn in depth (function=<function>, t_range=(0.0, 18.84955592153876))","jets":"a Vector [yellow] drawn in flow (end=(3.0, 0.0))","jets_2":"a Vector [yellow] drawn in flow (start=(0.0, 0.45), end=(2.711058263971463, 0.45))","jets_3":"a Vector [yellow] drawn in flow (start=(0.0, -0.45), end=(2.711058263971463, -0.45))","jets_4":"a Vector [yellow] drawn in flow (start=(0.0, 0.9), end=(1.84423305588585, 0.9))","jets_5":"a Vector [yellow] drawn in flow (start=(0.0, -0.9), end=(1.84423305588585, -0.9))","jets_6":"a Vector [yellow] drawn in flow (start=(0.0, 1.2), end=(0.9453032104637337, 1.2))","jets_7":"a Vector [yellow] drawn in flow (start=(0.0, -1.2), end=(0.9453032104637337, -1.2))","point":"a Point [yellow] drawn in profile (location=(0.87, 0.0))","profile":"an Axes (y_range=(-0.5, 1.2), x_ticks_every=0.5, y_ticks_every=0.5)","q_work":"a Derivation [text] that says \"$q(a) &= frac(2 pi a B_z, L B_theta) \\ q(a) &> 1$\"","ring":"an Axes3D (x_range=(-3.2, 3.2), y_range=(-3.2, 3.2), z_range=(-1.8, 1.8))","ring_current":"a Circle [yellow] labelled \"arrow(J)\" drawn in ring (center=(0.0, 0.0, 0.0), radius=2.1, normal_vector=(0.0, 0.0, 1.0))","ring_sense":"an Orientation [yellow] drawn in ring (path=((2.1, 0.0), (2.0898879260116137, 0.20583599469207728), (2.0596…, closed=True, arrows=5)","shear_law":"a Math [text] that says \"$frac(dif v_z, dif r) > 0.1 gamma$\"","shear_note":"a Panel that says \"Sheared axial flow pulls an unstable mode apart before it can grow. Experiments have held a quiet pinch for microseconds.\"","speed_label":"a Math [text] that says \"$v_z (r)$\" drawn in flow","torus":"a Surface [red] drawn in ring (function=<function>, u_range=(0.0, 6.283185307179586), v_range=(0.0, 6.283185307179586))","tube":"a Cylinder [red] drawn in depth (start=(0.0, 0.0, -1.9), end=(0.0, 0.0, 1.9), radius=<VariableNumber tube_radius = 1.05>)","tube_radius":"a VariableNumber (initial_value=1.05)"},"beats":[{"start":217.32129166666664,"say":"The first fix is the obvious one. Add a magnetic field along the axis, from external coils or driven by the plasma itself, and watch what it does to the field line geometry.","live":[],"does":[[217.32129166666664,"depth is shown on the screen, written out."],[217.32129166666664,"tube is shown on the screen, written out."],[221.51229166666664,"axial_1 is shown on the screen, written out."],[221.51229166666664,"axial_2 is shown on the screen, written out."]]},{"start":228.84629166666664,"say":"The azimuthal field wraps around the column and the axial field runs along it, so the total field is a helix. A field line now has tension along the column, and bending the column means stretching that line.","live":["depth","tube","axial_1","axial_2"],"does":[[234.94129166666664,"helix is shown on the screen, drawn."]]},{"start":242.63529166666663,"say":"How much do we need? Compare the pitch of the helix with the length of the column. The safety factor q is two pi a B z over L B theta, and the kink is held off when q is greater than one.","live":["depth","tube","axial_1","axial_2","helix"],"does":[[244.79529166666663,"depth moves to a new place on the board."],[244.79529166666663,"q_work is shown on the screen, written out."],[251.34229166666665,"q_work (the \"B_z\" part) is emphasized."],[254.63929166666662,"q_work is shown on the screen, written out."],[254.63929166666662,"q_work (the \"B_z\" part) is no longer emphasized."]]},{"start":256.5167916666666,"say":"Written out, that says a field line must not close on itself before it reaches the end of the column. Bend the column and the tension pulls it straight again.","live":null,"does":[[256.8652916666666,"axial_note is shown on the screen, written out."]]},{"start":266.36979166666663,"say":"The trapped axial field resists the sausage mode too. Squeezing the column compresses the flux it encloses, and compressed flux pushes back.","live":["axial_note","depth","tube","axial_1","axial_2","helix"],"does":[[270.38729166666667,"tube is redrawn as the numbers it depends on change."],[270.38729166666667,"tube_radius ticks to 0.78."],[274.36929166666664,"tube is redrawn as the numbers it depends on change."],[274.36929166666664,"tube_radius ticks to 1.05."],[275.46029166666665,"axial_note is hidden from the screen — left the board."],[275.46029166666665,"depth is hidden from the screen — left the board."],[275.46029166666665,"tube is hidden from the screen — depth left the board."],[275.46029166666665,"axial_1 is hidden from the screen — depth left the board."],[275.46029166666665,"axial_2 is hidden from the screen — depth left the board."],[275.46029166666665,"helix is hidden from the screen — depth left the board."],[275.46029166666665,"q_work is hidden from the screen — left the board."]]},{"start":276.66029166666664,"say":"Now bend the whole pinch into a torus, so the current has no ends to leak from and the plasma can organize its own field.","live":[],"does":[[276.66029166666664,"ring is shown on the screen, written out."],[278.66929166666665,"torus is shown on the screen, written out."],[279.6442916666666,"ring_current is shown on the screen, written out."],[279.6442916666666,"ring_sense is shown on the screen, written out."]]},{"start":284.7832916666666,"say":"This is the geometry of every steady magnetic confinement machine, and the pinch version of it is called the reversed field pinch. The plasma current runs the long way around the ring.","live":["ring","torus","ring_current","ring_sense"],"does":[[284.7832916666666,"ring turns in its own slot."]]},{"start":296.01779166666665,"say":"The name comes from its field profile. Measure the axial field from the center of the plasma out to the wall, with r over a the fraction of the way out. It starts strong on the axis, falls, and reverses sign near the edge.","live":null,"does":[[298.88529166666666,"ring moves to a new place on the board."],[298.88529166666666,"profile is shown on the screen, written out."],[305.67729166666663,"bz_curve is shown on the screen, written out."],[308.47529166666664,"point is shown on the screen, grown."],[310.961832025411,"point is hidden from the screen."]]},{"start":311.29279166666663,"say":"The azimuthal field does the opposite: zero on the axis and largest at the edge. That helical, self-organized profile is what a resistive plasma relaxes into, and it is stable against the kink with no large external field.","live":["ring","profile","torus","ring_current","ring_sense","bz_curve"],"does":[[313.37129166666665,"btheta_curve is shown on the screen, written out."],[326.30479166666663,"profile is hidden from the screen — left the board."],[326.30479166666663,"bz_curve is hidden from the screen — profile left the board."],[326.30479166666663,"btheta_curve is hidden from the screen — profile left the board."],[326.30479166666663,"ring is hidden from the screen — left the board."],[326.30479166666663,"torus is hidden from the screen — ring left the board."],[326.30479166666663,"ring_current is hidden from the screen — ring left the board."],[326.30479166666663,"ring_sense is hidden from the screen — ring left the board."]]},{"start":327.5047916666666,"say":"The other route keeps the column straight and gives it a velocity shear. Let the plasma flow along the axis, faster in the middle than at the edge.","live":[],"does":[[327.5047916666666,"flow is shown on the screen, written out."],[327.5047916666666,"flow_top is shown on the screen, written out."],[327.5047916666666,"flow_bottom is shown on the screen, written out."],[334.07529166666666,"jets is shown on the screen, written out."],[334.19895371357495,"jets_2 is shown on the screen, written out."],[334.3226157604833,"jets_3 is shown on the screen, written out."],[334.44627780739165,"jets_4 is shown on the screen, written out."],[334.5283762437811,"jets_5 is shown on the screen, written out."],[334.5845080845771,"jets_6 is shown on the screen, written out."],[334.6383513681592,"jets_7 is shown on the screen, written out."],[335.3292916666667,"speed_label is shown on the screen, written out."]]},{"start":336.63779166666666,"say":"A growing sausage or kink is a wave that has to stay coherent across the radius. If neighboring layers are sliding past each other fast enough, the wave is pulled apart before it can grow.","live":["flow","flow_top","flow_bottom","jets","jets_2","jets_3","jets_4","jets_5","jets_6","jets_7","speed_label"],"does":[[343.26729166666667,"jets is indicated — a transient flash."],[343.26729166666667,"jets_6 is indicated — a transient flash."]]},{"start":348.34879166666667,"say":"The requirement is a shear rate of order a tenth of the growth rate. Sheared flow experiments have then held a quiet pinch for microseconds, thousands of times the ideal growth time.","live":null,"does":[[348.79029166666663,"flow moves to a new place on the board."],[348.79029166666663,"shear_law is shown on the screen, written out."],[353.1202916666666,"shear_note is shown on the screen, written out."],[358.88312499999995,"flow is hidden from the screen — left the board."],[358.88312499999995,"flow_top is hidden from the screen — flow left the board."],[358.88312499999995,"flow_bottom is hidden from the screen — flow left the board."],[358.88312499999995,"jets is hidden from the screen — flow left the board."],[358.88312499999995,"jets_2 is hidden from the screen — flow left the board."],[358.88312499999995,"jets_3 is hidden from the screen — flow left the board."],[358.88312499999995,"jets_4 is hidden from the screen — flow left the board."],[358.88312499999995,"jets_5 is hidden from the screen — flow left the board."],[358.88312499999995,"jets_6 is hidden from the screen — flow left the board."],[358.88312499999995,"jets_7 is hidden from the screen — flow left the board."],[358.88312499999995,"speed_label is hidden from the screen — flow left the board."],[358.88312499999995,"shear_law is hidden from the screen — left the board."],[358.88312499999995,"shear_note is hidden from the screen — left the board."]]}]},{"title":"Field-Reversed Configurations","start":359.92479166666664,"end":460.65806249999997,"objects":{"blob":"a Point [red] labelled \"upright(\"FRC\")\" drawn in machine (location=(<VariableNumber place = 4.9>, 0.0), marker_radius=<VariableNumber size = 0.16>)","board_machine":"a Heading that says \"Form, Translate, Compress\"","board_topology":"a Heading that says \"A Field That Closes on Itself\"","coils":"a Line [blue] drawn in machine (start=(4.2, 1.0), end=(4.2, 1.5))","coils_2":"a Line [blue] drawn in machine (start=(4.9, 1.0), end=(4.9, 1.5))","coils_3":"a Line [blue] drawn in machine (start=(5.6, 1.0), end=(5.6, 1.5))","coils_4":"a Line [blue] drawn in machine (start=(4.2, -1.0), end=(4.2, -1.5))","coils_5":"a Line [blue] drawn in machine (start=(4.9, -1.0), end=(4.9, -1.5))","coils_6":"a Line [blue] drawn in machine (start=(5.6, -1.0), end=(5.6, -1.5))","cross":"a Figure (x_range=(-3.4, 3.4), y_range=(-2.1, 2.1), aspect=(6.8, 4.2))","ext_bottom":"a Vector [blue] drawn in cross (start=(-3.2, -1.75), end=(3.2, -1.75))","ext_top":"a Vector [blue] labelled \"arrow(B)_(upright(\"ext\"))\" drawn in cross (start=(-3.2, 1.75), end=(3.2, 1.75))","form_label":"a Math [text] that says \"$upright(\"formation\")$\" drawn in machine","frc_note":"a Panel that says \"A compact torus with closed poloidal field lines, no toroidal field, and a plasma pressure comparable to the magnetic pressure.\"","frc_work":"a Derivation [text] that says \"$arrow(J) &= J_theta hat(theta) \\ arrow(B) &= B_r hat(r) + B_z hat(z) \\ beta &= frac(2 mu_0 p, B^2) approx 1$\"","inner_lower":"a ParametricCurve [green] drawn in cross (function=<function>, t_range=(0.0, 6.283185307179586))","inner_upper":"a ParametricCurve [green] drawn in cross (function=<function>, t_range=(0.0, 6.283185307179586))","machine":"a Figure (x_range=(-0.4, 6.4), y_range=(-2.0, 2.0), aspect=(6.8, 4.0))","machine_axis":"a Line [gray] drawn in cross (start=(-3.2, 0.0), end=(3.2, 0.0), dashed=True)","o_lower":"a Point [yellow] labelled \"O\" drawn in cross (location=(0.0, -0.95))","o_upper":"a Point [yellow] labelled \"O\" drawn in cross (location=(0.0, 0.95))","place":"a VariableNumber (initial_value=1.2)","plasma":"a Region [red] drawn in cross (predicates=(<function <lambda> at 0x2a7c587a7ec0>,), opacity=0.16)","sep_label":"a Math [text] that says \"$upright(\"separatrix\")$\" drawn in cross","separatrix":"a ParametricCurve [green] drawn in cross (function=<function>, t_range=(0.0, 6.283185307179586))","separatrix_sense":"an Orientation [green] drawn in cross (path=((2.4, 0.0, 0.0), (2.3884433440132726, 0.12252142541195075, 0.0…, closed=True, arrows=4)","size":"a VariableNumber (initial_value=0.45)","squeeze_label":"a Math [text] that says \"$upright(\"compression\")$\" drawn in machine","vessel_bottom":"a Line [gray] drawn in machine (start=(0.2, -1.15), end=(6.0, -1.15))","vessel_top":"a Line [gray] drawn in machine (start=(0.2, 1.15), end=(6.0, 1.15))","x_left":"a Point [yellow] labelled \"X\" drawn in cross (location=(-2.4, 0.0))","x_right":"a Point [yellow] labelled \"X\" drawn in cross (location=(2.4, 0.0))"},"beats":[{"start":359.92479166666664,"say":"The pinch and the torus both use a field that is open at the ends or held in place by coils. There is a third option: let the plasma make a closed field of its own.","live":[],"does":[[359.92479166666664,"cross is shown on the screen, written out."],[359.92479166666664,"plasma is shown on the screen, written out."],[359.92479166666664,"machine_axis is shown on the screen, written out."],[368.65579166666663,"separatrix is shown on the screen, written out."],[368.65579166666663,"sep_label is shown on the screen, written out."]]},{"start":371.19429166666663,"say":"Here is a cross-section taken along the axis. The current in this plasma is azimuthal, running around the axis, and the field it makes is poloidal, lying in the plane of the picture.","live":["cross","plasma","machine_axis","separatrix","sep_label"],"does":[[376.31379166666665,"cross moves to a new place on the board."],[376.31379166666665,"frc_work is shown on the screen, written out."],[380.06379166666665,"frc_work is shown on the screen, written out."],[381.3987916666666,"separatrix_sense is shown on the screen, written out."]]},{"start":383.35729166666664,"say":"Inside the separatrix the field lines are closed loops around a magnetic axis, one above the machine axis in this section and one below. The separatrix meets the axis at two X points, and outside it the field runs the other way.","live":["cross","plasma","machine_axis","separatrix","sep_label","separatrix_sense"],"does":[[386.15579166666663,"inner_upper is shown on the screen, written out."],[386.15579166666663,"inner_lower is shown on the screen, written out."],[386.86379166666666,"o_upper is shown on the screen, written out."],[386.86379166666666,"o_lower is shown on the screen, written out."],[394.35179166666666,"x_left is shown on the screen, written out."],[394.35179166666666,"x_right is shown on the screen, written out."],[396.9757916666666,"ext_top is shown on the screen, written out."],[396.9757916666666,"ext_bottom is shown on the screen, written out."]]},{"start":398.52779166666664,"say":"That reversal is the name of the thing: a field-reversed configuration. Because the field closes, it can hold a plasma whose pressure is comparable to the magnetic pressure, beta of order one, rather than a few percent.","live":["cross","plasma","machine_axis","separatrix","sep_label","separatrix_sense","inner_upper","inner_lower","o_upper","o_lower","x_left","x_right","ext_top","ext_bottom"],"does":[[410.14979166666666,"frc_work is shown on the screen, written out."],[412.85479166666664,"frc_note is shown on the screen, written out."]]},{"start":414.46529166666664,"say":"High beta is the economic argument. For a given magnetic field you confine far more plasma, and there is no coil threading the plasma at all.","live":["frc_note","cross","plasma","machine_axis","separatrix","sep_label","separatrix_sense","inner_upper","inner_lower","o_upper","o_lower","x_left","x_right","ext_top","ext_bottom"],"does":[[422.40579166666663,"separatrix is indicated — a transient flash."],[424.60029166666664,"cross is hidden from the screen — left the board."],[424.60029166666664,"plasma is hidden from the screen — cross left the board."],[424.60029166666664,"machine_axis is hidden from the screen — cross left the board."],[424.60029166666664,"separatrix is hidden from the screen — cross left the board."],[424.60029166666664,"sep_label is hidden from the screen — cross left the board."],[424.60029166666664,"separatrix_sense is hidden from the screen — cross left the board."],[424.60029166666664,"inner_upper is hidden from the screen — cross left the board."],[424.60029166666664,"inner_lower is hidden from the screen — cross left the board."],[424.60029166666664,"o_upper is hidden from the screen — cross left the board."],[424.60029166666664,"o_lower is hidden from the screen — cross left the board."],[424.60029166666664,"x_left is hidden from the screen — cross left the board."],[424.60029166666664,"x_right is hidden from the screen — cross left the board."],[424.60029166666664,"ext_top is hidden from the screen — cross left the board."],[424.60029166666664,"ext_bottom is hidden from the screen — cross left the board."],[424.60029166666664,"frc_note is hidden from the screen — left the board."],[424.60029166666664,"frc_work is hidden from the screen — left the board."]]},{"start":425.8002916666666,"say":"An FRC is usually made in one place and used in another. Form the plasma at one end of the machine, and it can be pushed along the axis as a self-contained object.","live":[],"does":[[425.8002916666666,"machine is shown on the screen, written out."],[425.8002916666666,"vessel_top is shown on the screen, written out."],[425.8002916666666,"vessel_bottom is shown on the screen, written out."],[430.1657916666666,"blob is shown on the screen, written out."],[430.1657916666666,"form_label is shown on the screen, written out."],[433.11479166666663,"blob is redrawn as the numbers it depends on change."],[433.11479166666663,"place ticks to 4.9."]]},{"start":436.8027916666666,"say":"Then squeeze it. Coils at the far end compress the plasma inward and along the axis at once, and because beta is high, that compression heats it efficiently.","live":["machine","vessel_top","vessel_bottom","blob","form_label"],"does":[[438.7767916666666,"coils is shown on the screen, written out."],[438.7767916666666,"coils_2 is shown on the screen, written out."],[438.7767916666666,"coils_3 is shown on the screen, written out."],[438.7767916666666,"coils_4 is shown on the screen, written out."],[438.7767916666666,"coils_5 is shown on the screen, written out."],[438.7767916666666,"coils_6 is shown on the screen, written out."],[438.7767916666666,"squeeze_label is shown on the screen, written out."],[439.98379166666666,"blob is redrawn as the numbers it depends on change."],[439.98379166666666,"size ticks to 0.16."]]},{"start":448.2692916666666,"say":"Formation, translation, compression. That separation is the appeal: the difficult plasma physics happens in one section of the machine and the heating happens in another.","live":["machine","vessel_top","vessel_bottom","blob","form_label","coils","coils_2","coils_3","coils_4","coils_5","coils_6","squeeze_label"],"does":[[450.3827916666666,"blob is indicated — a transient flash."],[459.6163958333333,"machine is hidden from the screen — left the board."],[459.6163958333333,"vessel_top is hidden from the screen — machine left the board."],[459.6163958333333,"vessel_bottom is hidden from the screen — machine left the board."],[459.6163958333333,"blob is hidden from the screen — machine left the board."],[459.6163958333333,"form_label is hidden from the screen — machine left the board."],[459.6163958333333,"coils is hidden from the screen — machine left the board."],[459.6163958333333,"coils_2 is hidden from the screen — machine left the board."],[459.6163958333333,"coils_3 is hidden from the screen — machine left the board."],[459.6163958333333,"coils_4 is hidden from the screen — machine left the board."],[459.6163958333333,"coils_5 is hidden from the screen — machine left the board."],[459.6163958333333,"coils_6 is hidden from the screen — machine left the board."],[459.6163958333333,"squeeze_label is hidden from the screen — machine left the board."]]}]},{"title":"MagLIF and the Trade-offs","start":460.65806249999997,"end":632.8321874999999,"objects":{"b_field":"a VariableNumber (initial_value=10.0, format_spec='.0f')","board_compare":"a Heading that says \"Two Ways to Reach the Same Product\"","board_flux":"a Heading that says \"Crushing the Field With the Fuel\"","board_steps":"a Heading that says \"Magnetized Liner Inertial Fusion\"","bz_1":"a Vector [green] labelled \"arrow(B)_z\" drawn in stack (start=(0.55, 0.0, -2.4), end=(0.55, 0.0, 2.4))","bz_2":"a Vector [green] drawn in stack (start=(-0.55, 0.0, -2.4), end=(-0.55, 0.0, 2.4))","crush":"a Vector [blue] labelled \"arrow(J) times arrow(B)\" drawn in stack (start=(2.05, 0.0, 0.0), end=(0.9, 0.0, 0.0))","crush_2":"a Vector [blue] drawn in stack (start=(-2.05, 0.0, 0.0), end=(-0.9, 0.0, 0.0))","crush_3":"a Vector [blue] drawn in stack (start=(0.0, 2.05, 0.0), end=(0.0, 0.9, 0.0))","crush_4":"a Vector [blue] drawn in stack (start=(0.0, -2.05, 0.0), end=(0.0, -0.9, 0.0))","drive":"a Vector [yellow] labelled \"I\" drawn in stack (start=(1.6, 0.0, 1.9), end=(1.6, 0.0, -1.9))","fuel":"a Cylinder [red] drawn in stack (start=(0.0, 0.0, -1.75), end=(0.0, 0.0, 1.75), radius=<VariableNumber r_fuel = 0.16>)","laser":"a Vector [magenta] labelled \"upright(\"laser\")\" drawn in stack (start=(0.0, 0.0, 2.75), end=(0.0, 0.0, 0.9))","liner":"a Cylinder [gray] drawn in stack (start=(0.0, 0.0, -1.8), end=(0.0, 0.0, 1.8), radius=<VariableNumber r_liner = 0.22>)","mag_note":"a Panel that says \"The compressed axial field suppresses electron heat loss and holds the alpha particles inside the fuel.\"","mag_work":"a Derivation [text] that says \"$p_B &= frac(mu_0 I^2, 8 pi^2 r^2) \\ B_z r^2 &= upright(\"const\") \\ B_z &arrow.r 10^4 thin upright(\"T\")$\"","r_fuel":"a VariableNumber (initial_value=1.25)","r_liner":"a VariableNumber (initial_value=1.55)","readout":"a Point [green] labelled \"B_z = 10 thin upright(\"T\")\" drawn in stack (location=(1.75, 0.0, 2.6), show_marker=False)","stack":"an Axes3D (x_range=(-2.2, 2.2), y_range=(-2.2, 2.2), z_range=(-2.9, 2.9))","step_field":"a Text [text] that says \"Pre-magnetize the fuel with an axial field of about 10 T.\"","step_heat":"a Text [text] that says \"Preheat it with a laser pulse of about a kilojoule.\"","step_push":"a Text [text] that says \"Implode the metal liner with about 20 MA in 100 ns.\"","table":"a Table [text] that says \"Scheme Confined by The catch Z-pinch its own current sausage and kink Sheared-flow pinch current plus flow shear holding the shear Reversed-field pinch a self-organized field resistive transport FRC closed field, high beta tilt and transpo…\" (rows=(('Scheme', 'Confined by', 'The catch'), ('Z-pinch', 'its own c…, header=True)","trade":"a Tex [text] that says \"Every scheme must reach the same product $n thin tau_E thin T$. A pinch buys it with density, a tokamak with time.\""},"beats":[{"start":460.65806249999997,"say":"The last idea gives up on holding the plasma steady, and crushes it instead. This is magnetized liner inertial fusion, MagLIF, and it runs on the Z machine at Sandia. Start with a centimeter-scale metal cylinder full of deuterium fuel.","live":[],"does":[[460.65806249999997,"stack is shown on the screen, written out."],[474.6940625,"liner is shown on the screen, written out."],[476.2150625,"fuel is shown on the screen, written out."]]},{"start":477.6045625,"say":"Step one: put an axial magnetic field through the fuel before anything else happens, about ten tesla from a pair of coils. The fuel is now magnetized.","live":["stack","liner","fuel"],"does":[[477.9530625,"step_field is shown on the screen, written out."],[479.47406249999995,"bz_1 is shown on the screen, written out."],[479.47406249999995,"bz_2 is shown on the screen, written out."],[484.11806249999995,"readout is shown on the screen, written out."]]},{"start":489.4430625,"say":"Step two: a laser pulse of about a kilojoule enters through a window at the top and preheats the fuel to a couple of hundred electron volts, so the implosion does not have to do that work.","live":["step_field","stack","liner","fuel","bz_1","bz_2","readout"],"does":[[489.79106249999995,"step_heat is shown on the screen, written out."],[491.31206249999997,"laser is shown on the screen, written out."]]},{"start":501.5485625,"say":"Step three: twenty million amps flows axially through the liner in a hundred nanoseconds. It is the same J cross B force as the classic pinch, but now it acts on solid metal rather than on plasma.","live":["step_field","step_heat","stack","liner","fuel","bz_1","bz_2","readout","laser"],"does":[[501.89706249999995,"step_push is shown on the screen, written out."],[503.6960625,"drive is shown on the screen, written out."],[509.73306249999996,"crush is shown on the screen, written out."],[509.8633261188474,"crush_2 is shown on the screen, written out."],[509.99358973769483,"crush_3 is shown on the screen, written out."],[510.1167971295323,"crush_4 is shown on the screen, written out."],[515.2710625,"step_field is hidden from the screen — left the board."],[515.2710625,"step_heat is hidden from the screen — left the board."],[515.2710625,"step_push is hidden from the screen — left the board."]]},{"start":516.4710625,"say":"The magnetic pressure driving the liner is the expression we started with: mu nought I squared over eight pi squared r squared. At twenty mega-amps on a centimeter radius, that is megabars.","live":["stack","liner","fuel","bz_1","bz_2","readout","laser","drive","crush","crush_2","crush_3","crush_4"],"does":[[519.1180625,"mag_work is shown on the screen, written out."],[527.1870624999999,"mag_work (the \"I^2\" part) is emphasized."],[530.8560625,"mag_work (the \"I^2\" part) is no longer emphasized."]]},{"start":531.4560624999999,"say":"The liner implodes at seventy kilometers a second. The fuel column is compressed by a factor of about twenty five in radius, and the axial field goes with it, because flux is conserved: B times r squared stays put.","live":null,"does":[[536.4480625,"liner is redrawn as the numbers it depends on change."],[536.4480625,"fuel is redrawn as the numbers it depends on change."],[536.4480625,"readout is redrawn as the numbers it depends on change."],[536.4480625,"r_liner ticks to 0.22."],[536.4480625,"r_fuel ticks to 0.16."],[536.4480625,"b_field ticks to 6000.0."],[541.9510625,"mag_work is shown on the screen, written out."]]},{"start":546.6035625,"say":"Ten tesla becomes several thousand tesla, hundreds of times stronger than any steady magnet. That trapped field is the point of the scheme: it holds back electron heat conduction and it keeps the alpha particles inside the fuel.","live":null,"does":[[548.4260625,"mag_work is shown on the screen, written out."],[554.6030625,"A box is drawn around mag_work."],[556.9590625,"mag_note is shown on the screen, written out."],[560.4770625,"mag_note is hidden from the screen — left the board."],[560.4770625,"mag_work is hidden from the screen — left the board."],[560.4770625,"stack is hidden from the screen — left the board."],[560.4770625,"liner is hidden from the screen — stack left the board."],[560.4770625,"fuel is hidden from the screen — stack left the board."],[560.4770625,"bz_1 is hidden from the screen — stack left the board."],[560.4770625,"bz_2 is hidden from the screen — stack left the board."],[560.4770625,"readout is hidden from the screen — stack left the board."],[560.4770625,"laser is hidden from the screen — stack left the board."],[560.4770625,"drive is hidden from the screen — stack left the board."],[560.4770625,"crush is hidden from the screen — stack left the board."],[560.4770625,"crush_2 is hidden from the screen — stack left the board."],[560.4770625,"crush_3 is hidden from the screen — stack left the board."],[560.4770625,"crush_4 is hidden from the screen — stack left the board."]]},{"start":561.6770624999999,"say":"So put the family side by side. Each row is confined by something different, and each one pays a different price.","live":[],"does":[[564.8580625,"table is shown on the screen, written out."]]},{"start":569.7540624999999,"say":"The classic Z-pinch is confined by its own current and destroyed by the sausage and the kink. Sheared flow keeps that same confinement and buys stability with a velocity profile that has to be maintained.","live":null,"does":[[570.2650625,"table is shown on the screen, written out."],[575.4430625,"table is shown on the screen, written out."]]},{"start":582.1905624999999,"say":"The reversed field pinch lets the plasma organize a helical field for itself, and pays for it in resistive transport. The FRC closes the field lines and runs at high beta, and fights global stability and confinement time.","live":null,"does":[[582.5680625,"table is shown on the screen, written out."],[589.6030625,"table is shown on the screen, written out."]]},{"start":596.9480625,"say":"MagLIF does not confine in the usual sense at all: an imploding liner holds the fuel together for a hundred nanoseconds, and everything depends on that liner staying smooth. A tokamak is the opposite extreme, with big external coils, low beta, and a steady state.","live":null,"does":[[600.6870624999999,"table is shown on the screen, written out."],[608.1750625,"table is shown on the screen, written out."]]},{"start":615.4860625,"say":"All of them chase the same product: density, times confinement time, times temperature. A pinch pushes the density up and the time down, and a tokamak does the reverse. Which of those trades wins is still, sixty years on, an open question.","live":null,"does":[[617.0070625,"trade is shown on the screen, written out."],[617.9360624999999,"table is indicated — a transient flash."],[626.3880624999999,"table is indicated — a transient flash."],[631.7905208333334,"table is hidden from the screen — left the board."],[631.7905208333334,"trade is hidden from the screen — left the board."]]}]}]},"durationSeconds":633,"chapters":[{"title":"The Pinch Effect","startSeconds":0,"narration":"Fusion needs a plasma at over a hundred million kelvin, and a plasma that hot pushes outward hard. Here is the cheapest way anyone has proposed to hold one still. Drive a current straight through it, and let the current's own field do the squeezing. So take a straight column of plasma and pass a current along its axis. That axial current is the z in Z-pinch, and notice that nothing outside the plasma carries it. Look at the column itself for a moment. It is a cylinder of ionized gas hotter than the core of the sun, and the only thing between it and the vessel wall is the field we are about to build. Now that field. A current is wrapped by a magnetic field, and the right hand rule puts those loops in the azimuthal direction, closing around the axis at every height. The field is azimuthal and the current is axial, so the force per unit volume, J cross B, points inward. Every part of the plasma is pushed toward the axis at once. How hard? Ampere's law gives the azimuthal field at radius r, and a field of strength B always carries a magnetic pressure equal to B squared over two mu nought. Put the two together and the pinch pressure at the surface of the column goes as the current squared over the radius squared. Squeeze the column inward and the field that squeezes it only grows. Balance that magnetic pressure against the plasma's own pressure, and one relation falls out. Mu nought times the current squared equals eight pi N k T, with N the number of particles per unit length. Read it as a design rule. The current alone fixes the pressure you can confine, so a few mega-amps reaches fusion conditions with no external magnet anywhere. That is the whole appeal of the pinch, and in a moment it will also be the whole problem."},{"title":"Sausages and Kinks","startSeconds":120.40268749999997,"narration":"The pinch has a fatal flaw, and it is written in the formula we just derived. The inward pressure goes as one over the radius squared, so it cannot be the same everywhere unless the radius is. Give that column a ripple. A slight narrowing here, a slight bulge there: exactly the kind of perturbation that thermal noise supplies for nothing. Now look at the narrow place. Its radius is smaller, so the field there is stronger, and the inward push is stronger. The neck squeezes down harder than its neighbors do. At the bulge the opposite happens. Larger radius, weaker field, weaker squeeze. So the perturbation feeds itself, and the neck runs away. That is the m equals zero mode, the sausage instability. It grows in tens of nanoseconds, and it does not stop until the column has been cut into separate blobs. The second mode is worse, because nothing about the radius has to change at all. Suppose the whole column simply bends sideways. On the inside of the bend the azimuthal field lines are crowded closer together, so the field there is stronger than on the outside. The net push is sideways, the same way the column has already moved. So this one feeds itself too. The more it bends, the harder it is pushed, and that is the m equals one kink mode. Both modes come straight out of ideal magnetohydrodynamics, both are driven by the confining current itself, and both are fast. Everything that follows is an answer to these two pictures."},{"title":"Stabilizing the Pinch","startSeconds":217.32129166666664,"narration":"The first fix is the obvious one. Add a magnetic field along the axis, from external coils or driven by the plasma itself, and watch what it does to the field line geometry. The azimuthal field wraps around the column and the axial field runs along it, so the total field is a helix. A field line now has tension along the column, and bending the column means stretching that line. How much do we need? Compare the pitch of the helix with the length of the column. The safety factor q is two pi a B z over L B theta, and the kink is held off when q is greater than one. Written out, that says a field line must not close on itself before it reaches the end of the column. Bend the column and the tension pulls it straight again. The trapped axial field resists the sausage mode too. Squeezing the column compresses the flux it encloses, and compressed flux pushes back. Now bend the whole pinch into a torus, so the current has no ends to leak from and the plasma can organize its own field. This is the geometry of every steady magnetic confinement machine, and the pinch version of it is called the reversed field pinch. The plasma current runs the long way around the ring. The name comes from its field profile. Measure the axial field from the center of the plasma out to the wall, with r over a the fraction of the way out. It starts strong on the axis, falls, and reverses sign near the edge. The azimuthal field does the opposite: zero on the axis and largest at the edge. That helical, self-organized profile is what a resistive plasma relaxes into, and it is stable against the kink with no large external field. The other route keeps the column straight and gives it a velocity shear. Let the plasma flow along the axis, faster in the middle than at the edge. A growing sausage or kink is a wave that has to stay coherent across the radius. If neighboring layers are sliding past each other fast enough, the wave is pulled apart before it can grow. The requirement is a shear rate of order a tenth of the growth rate. Sheared flow experiments have then held a quiet pinch for microseconds, thousands of times the ideal growth time."},{"title":"Field-Reversed Configurations","startSeconds":359.92479166666664,"narration":"The pinch and the torus both use a field that is open at the ends or held in place by coils. There is a third option: let the plasma make a closed field of its own. Here is a cross-section taken along the axis. The current in this plasma is azimuthal, running around the axis, and the field it makes is poloidal, lying in the plane of the picture. Inside the separatrix the field lines are closed loops around a magnetic axis, one above the machine axis in this section and one below. The separatrix meets the axis at two X points, and outside it the field runs the other way. That reversal is the name of the thing: a field-reversed configuration. Because the field closes, it can hold a plasma whose pressure is comparable to the magnetic pressure, beta of order one, rather than a few percent. High beta is the economic argument. For a given magnetic field you confine far more plasma, and there is no coil threading the plasma at all. An FRC is usually made in one place and used in another. Form the plasma at one end of the machine, and it can be pushed along the axis as a self-contained object. Then squeeze it. Coils at the far end compress the plasma inward and along the axis at once, and because beta is high, that compression heats it efficiently. Formation, translation, compression. That separation is the appeal: the difficult plasma physics happens in one section of the machine and the heating happens in another."},{"title":"MagLIF and the Trade-offs","startSeconds":460.65806249999997,"narration":"The last idea gives up on holding the plasma steady, and crushes it instead. This is magnetized liner inertial fusion, MagLIF, and it runs on the Z machine at Sandia. Start with a centimeter-scale metal cylinder full of deuterium fuel. Step one: put an axial magnetic field through the fuel before anything else happens, about ten tesla from a pair of coils. The fuel is now magnetized. Step two: a laser pulse of about a kilojoule enters through a window at the top and preheats the fuel to a couple of hundred electron volts, so the implosion does not have to do that work. Step three: twenty million amps flows axially through the liner in a hundred nanoseconds. It is the same J cross B force as the classic pinch, but now it acts on solid metal rather than on plasma. The magnetic pressure driving the liner is the expression we started with: mu nought I squared over eight pi squared r squared. At twenty mega-amps on a centimeter radius, that is megabars. The liner implodes at seventy kilometers a second. The fuel column is compressed by a factor of about twenty five in radius, and the axial field goes with it, because flux is conserved: B times r squared stays put. Ten tesla becomes several thousand tesla, hundreds of times stronger than any steady magnet. That trapped field is the point of the scheme: it holds back electron heat conduction and it keeps the alpha particles inside the fuel. So put the family side by side. Each row is confined by something different, and each one pays a different price. The classic Z-pinch is confined by its own current and destroyed by the sausage and the kink. Sheared flow keeps that same confinement and buys stability with a velocity profile that has to be maintained. The reversed field pinch lets the plasma organize a helical field for itself, and pays for it in resistive transport. The FRC closes the field lines and runs at high beta, and fights global stability and confinement time. MagLIF does not confine in the usual sense at all: an imploding liner holds the fuel together for a hundred nanoseconds, and everything depends on that liner staying smooth. A tokamak is the opposite extreme, with big external coils, low beta, and a steady state. All of them chase the same product: density, times confinement time, times temperature. A pinch pushes the density up and the time down, and a tokamak does the reverse. Which of those trades wins is still, sixty years on, an open question."}]}}