# Z-Pinch Fusion and Magnetic Pinch Relatives > 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. - Canonical watch page: [Z-Pinch Fusion and Magnetic Pinch Relatives](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Physics - Published: 2026-08-30T23:51:44.701Z - Updated: 2026-08-30T23:51:44.701Z - Duration: PT633S (10 minutes 33 seconds) - Chapters: 5 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M1AFM0CEW468WY026NSSG2ZK/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M1AFM0CEW468WY026NSSG2ZK/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M1AFM0CEW468WY026NSSG2ZK/0/dark/poster.jpg) ## Description How a plasma's own current confines it, why the Z-pinch is unstable, and how RFPs, sheared flow, FRCs and MagLIF answer that. ## Chapters - [00:00–02:0.403 · The Pinch Effect](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=0) - [02:0.403–03:37.321 · Sausages and Kinks](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997) - [03:37.321–05:59.925 · Stabilizing the Pinch](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=217.32129166666664) - [05:59.925–07:40.658 · Field-Reversed Configurations](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=359.92479166666664) - [07:40.658–10:33 · MagLIF and the Trade-offs](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=460.65806249999997) ## Transcript ### [00:00 · The Pinch Effect](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=0) 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. ### [02:0.403 · Sausages and Kinks](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997) 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. ### [03:37.321 · Stabilizing the Pinch](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=217.32129166666664) 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. ### [05:59.925 · Field-Reversed Configurations](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=359.92479166666664) 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. ### [07:40.658 · MagLIF and the Trade-offs](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=460.65806249999997) 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. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M1AFM0CEW468WY026NSSG2ZK/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [The Pinch Effect](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=0) Span: 00:00–02:0.403 (0s–120.40268749999997s). #### 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=\, 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 $p$. Can a current driven through the plasma itself hold it together?" - radius: a VariableNumber (initial\_value=1.5) - squeeze: a Vector \[blue\] labelled "arrow(J) times arrow(B)" drawn in axes (start=(2.0, 0.0, 0.0), end=(0.75, 0.0, 0.0)) - squeeze\_2: a Vector \[blue\] drawn in axes (start=(-2.0, 0.0, 0.0), end=(-0.75, 0.0, 0.0)) - squeeze\_3: a Vector \[blue\] drawn in axes (start=(0.0, 2.0, 0.0), end=(0.0, 0.75, 0.0)) - squeeze\_4: a Vector \[blue\] drawn in axes (start=(0.0, -2.0, 0.0), end=(0.0, -0.75, 0.0)) - work: a Derivation \[text\] that says "$B\_theta (r) &= frac(mu\_0 I, 2 pi r) \\ p\_B &= frac(B\_theta^2, 2 mu\_0) \\ &= frac(mu\_0 I^2, 8 pi^2 r^2) \\ mu\_0 I^2 &= 8 pi N k\_B T$" #### Beats ##### [00:00](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=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. Board: Empty. Actions: - [00:00](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=0): question is shown on the screen, written out. - [00:15.011](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=15.0115): question moves to a new place on the board. ##### [00:16.212](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=16.2115) Narration: 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. Board: question — a Panel that says "A fusion plasma pushes outward with pressure $p$. Can a current driven through the plasma itself hold it together?" Actions: - [00:16.212](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=16.2115): axes is shown on the screen, written out. - [00:17.512](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=17.512): column is shown on the screen, written out. - [00:20.217](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=20.217): current is shown on the screen, written out. ##### [00:28.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=28.305) Narration: 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. Board: question — a Panel that says "A fusion plasma pushes outward with pressure $p$. Can a current driven through the plasma itself hold it together?"; 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=\, 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)) Actions: - [00:28.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=28.305): axes turns in its own slot. ##### [00:39.877](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=39.8765) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [00:42.106](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=42.106): loop\_low is shown on the screen, written out. - [00:45.577](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=45.577): loop\_sense is shown on the screen, written out. - [00:49.559](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=49.559): loop\_high is shown on the screen, written out. ##### [00:50.96](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=50.960499999999996) Narration: 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. Board: question — a Panel that says "A fusion plasma pushes outward with pressure $p$. Can a current driven through the plasma itself hold it together?"; 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=\, 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\_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); 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)) Actions: - [00:57.996](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=57.99599999999999): squeeze is shown on the screen, written out. - [00:58.379](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=58.37907885379626): squeeze\_2 is shown on the screen, written out. - [00:58.762](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=58.762157707592536): squeeze\_3 is shown on the screen, written out. - [00:59.145](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=59.145236561388806): squeeze\_4 is shown on the screen, written out. ##### [01:3.809](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=63.809) Narration: 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. Board: question — a Panel that says "A fusion plasma pushes outward with pressure $p$. Can a current driven through the plasma itself hold it together?"; 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=\, 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\_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); 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)); squeeze — a Vector \[blue\] labelled "arrow(J) times arrow(B)" drawn in axes (start=(2.0, 0.0, 0.0), end=(0.75, 0.0, 0.0)); squeeze\_2 — a Vector \[blue\] drawn in axes (start=(-2.0, 0.0, 0.0), end=(-0.75, 0.0, 0.0)); squeeze\_3 — a Vector \[blue\] drawn in axes (start=(0.0, 2.0, 0.0), end=(0.0, 0.75, 0.0)); squeeze\_4 — a Vector \[blue\] drawn in axes (start=(0.0, -2.0, 0.0), end=(0.0, -0.75, 0.0)) Actions: - [01:6.723](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=66.723): axes moves to a new place on the board. - [01:6.723](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=66.723): work is shown on the screen, written out. - [01:12.923](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=72.923): work is shown on the screen, written out. ##### [01:16.797](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=76.797) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [01:17.691](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=77.691): work is shown on the screen, written out. - [01:21.731](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=81.73100000000001): work (the "I^2" part) is emphasized. - [01:24.727](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=84.727): column is redrawn as the numbers it depends on change. - [01:24.727](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=84.727): work (the "I^2" part) is no longer emphasized. - [01:24.727](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=84.727): radius ticks to 0.8. ##### [01:29.843](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=89.84299999999999) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [01:34.394](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=94.39399999999999): work is shown on the screen, written out. ##### [01:43.701](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=103.70149999999998) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [01:45.2](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=105.19999999999999): A box is drawn around work. - [01:46.453](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=106.45299999999997): work (the "I^2" part) is emphasized. - [01:55.335](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=115.33499999999998): work (the "I^2" part) is no longer emphasized. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): axes is hidden from the screen — left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): column is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): current is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): loop\_low is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): loop\_sense is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): loop\_high is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): squeeze is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): squeeze\_2 is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): squeeze\_3 is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): squeeze\_4 is hidden from the screen — axes left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): question is hidden from the screen — left the board. - [01:59.361](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=119.36102083333331): work is hidden from the screen — left the board. ### Scene 2: [Sausages and Kinks](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997) Span: 02:0.403–03:37.321 (120.40268749999997s–217.32129166666664s). #### Objects - bend: a VariableNumber - board\_kink: a Heading that says "The $m = 1$ Kink Mode" - board\_sausage: a Heading that says "The $m = 0$ Sausage Mode" - hard: a Vector \[blue\] drawn in sausage (start=(0.0, 1.92), end=(0.0, 0.65)) - hard\_2: a Vector \[blue\] drawn in sausage (start=(0.0, -1.92), end=(0.0, -0.65)) - kink: a Figure (x\_range=(-3.0, 3.0), y\_range=(-2.0, 2.0), aspect=(6.0, 4.0)) - kink\_axis: a Line \[gray\] drawn in kink (start=(-2.7, 0.0), end=(2.7, 0.0), dashed=True) - kink\_law: a Math \[text\] that says "$B\_(upright("in")) \> B\_(upright("out"))$" - kink\_note: a Panel that says "Crowded field lines on the inside of a bend push the bend further over, so a sideways displacement grows as well." - line: a Line \[red\] drawn in sausage (start=(-2.7, (0.0 + (1.0 \* (0.9 - (neck \* 0.8090169943749473))))), end=(-2.4000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…) - line\_10: a Line \[red\] drawn in sausage (start=(-1.5000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0))))), end=(-1.2000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…) - line\_11: a Line \[red\] drawn in sausage (start=(-1.2000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…, end=(-0.9000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* -0.309016994…) - line\_12: a Line \[red\] drawn in sausage (start=(-1.2000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…, end=(-0.9000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699…) - line\_13: a Line \[red\] drawn in sausage (start=(-0.9000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* -0.309016994…, end=(-0.6000000000000001, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…) - line\_14: a Line \[red\] drawn in sausage (start=(-0.9000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699…, end=(-0.6000000000000001, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…) - line\_15: a Line \[red\] drawn in sausage (start=(-0.6000000000000001, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…, end=(-0.30000000000000027, (0.0 + (1.0 \* (0.9 - (neck \* 0.809016994…) - line\_16: a Line \[red\] drawn in sausage (start=(-0.6000000000000001, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…, end=(-0.30000000000000027, (0.0 + (-1.0 \* (0.9 - (neck \* 0.80901699…) - line\_17: a Line \[red\] drawn in sausage (start=(-0.30000000000000027, (0.0 + (1.0 \* (0.9 - (neck \* 0.809016994…, end=(-4.440892098500626e-16, (0.0 + (1.0 \* (0.9 - (neck \* 1.0)))))) - line\_18: a Line \[red\] drawn in sausage (start=(-0.30000000000000027, (0.0 + (-1.0 \* (0.9 - (neck \* 0.80901699…, end=(-4.440892098500626e-16, (0.0 + (-1.0 \* (0.9 - (neck \* 1.0)))))) - line\_19: a Line \[red\] drawn in sausage (start=(-4.440892098500626e-16, (0.0 + (1.0 \* (0.9 - (neck \* 1.0))))), end=(0.2999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…) - line\_2: a Line \[red\] drawn in sausage (start=(-2.7, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943749473))))), end=(-2.4000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…) - line\_20: a Line \[red\] drawn in sausage (start=(-4.440892098500626e-16, (0.0 + (-1.0 \* (0.9 - (neck \* 1.0))))), end=(0.2999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…) - line\_21: a Line \[red\] drawn in sausage (start=(0.2999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…, end=(0.5999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…) - line\_22: a Line \[red\] drawn in sausage (start=(0.2999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…, end=(0.5999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…) - line\_23: a Line \[red\] drawn in sausage (start=(0.5999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…, end=(0.8999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…) - line\_24: a Line \[red\] drawn in sausage (start=(0.5999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…, end=(0.8999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…) - line\_25: a Line \[red\] drawn in sausage (start=(0.8999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…, end=(1.1999999999999997, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…) - line\_26: a Line \[red\] drawn in sausage (start=(0.8999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…, end=(1.1999999999999997, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…) - line\_27: a Line \[red\] drawn in sausage (start=(1.1999999999999997, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…, end=(1.5, (0.0 + (1.0 \* (0.9 - (neck \* -1.0)))))) - line\_28: a Line \[red\] drawn in sausage (start=(1.1999999999999997, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…, end=(1.5, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0)))))) - line\_29: a Line \[red\] drawn in sausage (start=(1.5, (0.0 + (1.0 \* (0.9 - (neck \* -1.0))))), end=(1.7999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…) - line\_3: a Line \[red\] drawn in sausage (start=(-2.4000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…, end=(-2.1, (0.0 + (1.0 \* (0.9 - (neck \* -0.30901699437494756)))))) - line\_30: a Line \[red\] drawn in sausage (start=(1.5, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0))))), end=(1.7999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…) - line\_31: a Line \[red\] drawn in sausage (start=(1.7999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…, end=(2.0999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…) - line\_32: a Line \[red\] drawn in sausage (start=(1.7999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…, end=(2.0999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…) - line\_33: a Line \[red\] drawn in sausage (start=(2.0999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…, end=(2.3999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…) - line\_34: a Line \[red\] drawn in sausage (start=(2.0999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…, end=(2.3999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…) - line\_35: a Line \[red\] drawn in sausage (start=(2.3999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…, end=(2.6999999999999993, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…) - line\_36: a Line \[red\] drawn in sausage (start=(2.3999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…, end=(2.6999999999999993, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…) - line\_37: a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + 0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6))) - line\_38: a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + -0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6))) - line\_39: a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + 0.6))) - line\_4: a Line \[red\] drawn in sausage (start=(-2.4000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…, end=(-2.1, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699437494756)))))) - line\_40: a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + -0.6))) - line\_41: a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + 0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6))) - line\_42: a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + -0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6))) - line\_43: a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6))) - line\_44: a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6))) - line\_45: a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6))) - line\_46: a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6))) - line\_47: a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6))) - line\_48: a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6))) - line\_49: a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6))) - line\_5: a Line \[red\] drawn in sausage (start=(-2.1, (0.0 + (1.0 \* (0.9 - (neck \* -0.30901699437494756))))), end=(-1.8000000000000003, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…) - line\_50: a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6))) - line\_51: a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6))) - line\_52: a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6))) - line\_53: a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…) - line\_54: a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…) - line\_55: a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6))) - line\_56: a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6))) - line\_57: a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6))) - line\_58: a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6))) - line\_59: a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6))) - line\_6: a Line \[red\] drawn in sausage (start=(-2.1, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699437494756))))), end=(-1.8000000000000003, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…) - line\_60: a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6))) - line\_61: a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6))) - line\_62: a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6))) - line\_63: a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6)), end=(1.5, ((bend \* 0.9848077530122081) + 0.6))) - line\_64: a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6)), end=(1.5, ((bend \* 0.9848077530122081) + -0.6))) - line\_65: a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + 0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6))) - line\_66: a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + -0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6))) - line\_67: a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6))) - line\_68: a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6))) - line\_69: a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6))) - line\_7: a Line \[red\] drawn in sausage (start=(-1.8000000000000003, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…, end=(-1.5000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -1.0)))))) - line\_70: a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6))) - line\_71: a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + 0.6))) - line\_72: a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + -0.6))) - line\_8: a Line \[red\] drawn in sausage (start=(-1.8000000000000003, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…, end=(-1.5000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0)))))) - line\_9: a Line \[red\] drawn in sausage (start=(-1.5000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -1.0))))), end=(-1.2000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…) - neck: a VariableNumber - pinch\_law: a Math \[text\] that says "$p\_B = frac(mu\_0 I^2, 8 pi^2 r^2)$" - push: a Vector \[blue\] drawn in kink (start=(1.35, -0.05), end=(1.35, 0.35)) - push\_2: a Vector \[blue\] drawn in kink (start=(-1.35, 0.05), end=(-1.35, -0.35)) - sausage: a Figure (x\_range=(-3.0, 3.0), y\_range=(-2.0, 2.0), aspect=(6.0, 4.0)) - sausage\_note: a Panel that says "A narrowing of the column squeezes itself harder than its neighbors, so any ripple grows without limit." - soft: a Vector \[blue\] drawn in sausage (start=(1.5, 1.92), end=(1.5, 1.45)) - soft\_2: a Vector \[blue\] drawn in sausage (start=(1.5, -1.92), end=(1.5, -1.45)) #### Beats ##### [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=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. Board: Empty. Actions: - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): sausage is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_2 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_3 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_4 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_5 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_6 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_7 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_8 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_9 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_10 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_11 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_12 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_13 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_14 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_15 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_16 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_17 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_18 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_19 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_20 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_21 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_22 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_23 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_24 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_25 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_26 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_27 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_28 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_29 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_30 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_31 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_32 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_33 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_34 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_35 is shown on the screen, written out. - [02:0.403](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=120.40268749999997): line\_36 is shown on the screen, written out. - [02:3.143](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=123.14268749999997): sausage moves to a new place on the board. - [02:3.143](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=123.14268749999997): pinch\_law is shown on the screen, written out. ##### [02:12.137](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=132.13668749999997) Narration: Give that column a ripple. A slight narrowing here, a slight bulge there: exactly the kind of perturbation that thermal noise supplies for nothing. Board: pinch\_law — a Math \[text\] that says "$p\_B = frac(mu\_0 I^2, 8 pi^2 r^2)$"; sausage — a Figure (x\_range=(-3.0, 3.0), y\_range=(-2.0, 2.0), aspect=(6.0, 4.0)); line — a Line \[red\] drawn in sausage (start=(-2.7, (0.0 + (1.0 \* (0.9 - (neck \* 0.8090169943749473))))), end=(-2.4000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…); line\_2 — a Line \[red\] drawn in sausage (start=(-2.7, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943749473))))), end=(-2.4000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…); line\_3 — a Line \[red\] drawn in sausage (start=(-2.4000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…, end=(-2.1, (0.0 + (1.0 \* (0.9 - (neck \* -0.30901699437494756)))))); line\_4 — a Line \[red\] drawn in sausage (start=(-2.4000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…, end=(-2.1, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699437494756)))))); line\_5 — a Line \[red\] drawn in sausage (start=(-2.1, (0.0 + (1.0 \* (0.9 - (neck \* -0.30901699437494756))))), end=(-1.8000000000000003, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…); line\_6 — a Line \[red\] drawn in sausage (start=(-2.1, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699437494756))))), end=(-1.8000000000000003, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…); line\_7 — a Line \[red\] drawn in sausage (start=(-1.8000000000000003, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…, end=(-1.5000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -1.0)))))); line\_8 — a Line \[red\] drawn in sausage (start=(-1.8000000000000003, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…, end=(-1.5000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0)))))); line\_9 — a Line \[red\] drawn in sausage (start=(-1.5000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -1.0))))), end=(-1.2000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…); line\_10 — a Line \[red\] drawn in sausage (start=(-1.5000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0))))), end=(-1.2000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…); line\_11 — a Line \[red\] drawn in sausage (start=(-1.2000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…, end=(-0.9000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* -0.309016994…); line\_12 — a Line \[red\] drawn in sausage (start=(-1.2000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…, end=(-0.9000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699…); line\_13 — a Line \[red\] drawn in sausage (start=(-0.9000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* -0.309016994…, end=(-0.6000000000000001, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…); line\_14 — a Line \[red\] drawn in sausage (start=(-0.9000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699…, end=(-0.6000000000000001, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…); line\_15 — a Line \[red\] drawn in sausage (start=(-0.6000000000000001, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…, end=(-0.30000000000000027, (0.0 + (1.0 \* (0.9 - (neck \* 0.809016994…); line\_16 — a Line \[red\] drawn in sausage (start=(-0.6000000000000001, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…, end=(-0.30000000000000027, (0.0 + (-1.0 \* (0.9 - (neck \* 0.80901699…); line\_17 — a Line \[red\] drawn in sausage (start=(-0.30000000000000027, (0.0 + (1.0 \* (0.9 - (neck \* 0.809016994…, end=(-4.440892098500626e-16, (0.0 + (1.0 \* (0.9 - (neck \* 1.0)))))); line\_18 — a Line \[red\] drawn in sausage (start=(-0.30000000000000027, (0.0 + (-1.0 \* (0.9 - (neck \* 0.80901699…, end=(-4.440892098500626e-16, (0.0 + (-1.0 \* (0.9 - (neck \* 1.0)))))); line\_19 — a Line \[red\] drawn in sausage (start=(-4.440892098500626e-16, (0.0 + (1.0 \* (0.9 - (neck \* 1.0))))), end=(0.2999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…); line\_20 — a Line \[red\] drawn in sausage (start=(-4.440892098500626e-16, (0.0 + (-1.0 \* (0.9 - (neck \* 1.0))))), end=(0.2999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…); line\_21 — a Line \[red\] drawn in sausage (start=(0.2999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…, end=(0.5999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…); line\_22 — a Line \[red\] drawn in sausage (start=(0.2999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…, end=(0.5999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…); line\_23 — a Line \[red\] drawn in sausage (start=(0.5999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…, end=(0.8999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…); line\_24 — a Line \[red\] drawn in sausage (start=(0.5999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…, end=(0.8999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…); line\_25 — a Line \[red\] drawn in sausage (start=(0.8999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…, end=(1.1999999999999997, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…); line\_26 — a Line \[red\] drawn in sausage (start=(0.8999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…, end=(1.1999999999999997, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…); line\_27 — a Line \[red\] drawn in sausage (start=(1.1999999999999997, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…, end=(1.5, (0.0 + (1.0 \* (0.9 - (neck \* -1.0)))))); line\_28 — a Line \[red\] drawn in sausage (start=(1.1999999999999997, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…, end=(1.5, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0)))))); line\_29 — a Line \[red\] drawn in sausage (start=(1.5, (0.0 + (1.0 \* (0.9 - (neck \* -1.0))))), end=(1.7999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…); line\_30 — a Line \[red\] drawn in sausage (start=(1.5, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0))))), end=(1.7999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…); line\_31 — a Line \[red\] drawn in sausage (start=(1.7999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…, end=(2.0999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…); line\_32 — a Line \[red\] drawn in sausage (start=(1.7999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…, end=(2.0999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…); line\_33 — a Line \[red\] drawn in sausage (start=(2.0999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…, end=(2.3999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…); line\_34 — a Line \[red\] drawn in sausage (start=(2.0999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…, end=(2.3999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…); line\_35 — a Line \[red\] drawn in sausage (start=(2.3999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…, end=(2.6999999999999993, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…); line\_36 — a Line \[red\] drawn in sausage (start=(2.3999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…, end=(2.6999999999999993, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…) Actions: - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_2 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_3 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_4 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_5 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_6 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_7 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_8 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_9 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_10 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_11 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_12 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_13 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_14 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_15 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_16 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_17 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_18 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_19 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_20 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_21 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_22 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_23 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_24 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_25 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_26 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_27 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_28 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_29 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_30 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_31 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_32 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_33 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_34 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_35 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): line\_36 is redrawn as the numbers it depends on change. - [02:13.286](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=133.28568749999997): neck ticks to 0.16. ##### [02:22.071](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=142.07068749999996) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [02:23.035](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=143.03468749999996): hard is shown on the screen, written out. - [02:23.035](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=143.03468749999996): hard\_2 is shown on the screen, written out. - [02:30.674](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=150.67368749999997): soft is shown on the screen, written out. - [02:30.674](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=150.67368749999997): soft\_2 is shown on the screen, written out. ##### [02:32.423](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=152.42318749999998) Narration: At the bulge the opposite happens. Larger radius, weaker field, weaker squeeze. So the perturbation feeds itself, and the neck runs away. Board: pinch\_law — a Math \[text\] that says "$p\_B = frac(mu\_0 I^2, 8 pi^2 r^2)$"; sausage — a Figure (x\_range=(-3.0, 3.0), y\_range=(-2.0, 2.0), aspect=(6.0, 4.0)); line — a Line \[red\] drawn in sausage (start=(-2.7, (0.0 + (1.0 \* (0.9 - (neck \* 0.8090169943749473))))), end=(-2.4000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…); line\_2 — a Line \[red\] drawn in sausage (start=(-2.7, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943749473))))), end=(-2.4000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…); line\_3 — a Line \[red\] drawn in sausage (start=(-2.4000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…, end=(-2.1, (0.0 + (1.0 \* (0.9 - (neck \* -0.30901699437494756)))))); line\_4 — a Line \[red\] drawn in sausage (start=(-2.4000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…, end=(-2.1, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699437494756)))))); line\_5 — a Line \[red\] drawn in sausage (start=(-2.1, (0.0 + (1.0 \* (0.9 - (neck \* -0.30901699437494756))))), end=(-1.8000000000000003, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…); line\_6 — a Line \[red\] drawn in sausage (start=(-2.1, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699437494756))))), end=(-1.8000000000000003, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…); line\_7 — a Line \[red\] drawn in sausage (start=(-1.8000000000000003, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…, end=(-1.5000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -1.0)))))); line\_8 — a Line \[red\] drawn in sausage (start=(-1.8000000000000003, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…, end=(-1.5000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0)))))); line\_9 — a Line \[red\] drawn in sausage (start=(-1.5000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -1.0))))), end=(-1.2000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…); line\_10 — a Line \[red\] drawn in sausage (start=(-1.5000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0))))), end=(-1.2000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…); line\_11 — a Line \[red\] drawn in sausage (start=(-1.2000000000000002, (0.0 + (1.0 \* (0.9 - (neck \* -0.809016994…, end=(-0.9000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* -0.309016994…); line\_12 — a Line \[red\] drawn in sausage (start=(-1.2000000000000002, (0.0 + (-1.0 \* (0.9 - (neck \* -0.80901699…, end=(-0.9000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699…); line\_13 — a Line \[red\] drawn in sausage (start=(-0.9000000000000004, (0.0 + (1.0 \* (0.9 - (neck \* -0.309016994…, end=(-0.6000000000000001, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…); line\_14 — a Line \[red\] drawn in sausage (start=(-0.9000000000000004, (0.0 + (-1.0 \* (0.9 - (neck \* -0.30901699…, end=(-0.6000000000000001, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…); line\_15 — a Line \[red\] drawn in sausage (start=(-0.6000000000000001, (0.0 + (1.0 \* (0.9 - (neck \* 0.3090169943…, end=(-0.30000000000000027, (0.0 + (1.0 \* (0.9 - (neck \* 0.809016994…); line\_16 — a Line \[red\] drawn in sausage (start=(-0.6000000000000001, (0.0 + (-1.0 \* (0.9 - (neck \* 0.309016994…, end=(-0.30000000000000027, (0.0 + (-1.0 \* (0.9 - (neck \* 0.80901699…); line\_17 — a Line \[red\] drawn in sausage (start=(-0.30000000000000027, (0.0 + (1.0 \* (0.9 - (neck \* 0.809016994…, end=(-4.440892098500626e-16, (0.0 + (1.0 \* (0.9 - (neck \* 1.0)))))); line\_18 — a Line \[red\] drawn in sausage (start=(-0.30000000000000027, (0.0 + (-1.0 \* (0.9 - (neck \* 0.80901699…, end=(-4.440892098500626e-16, (0.0 + (-1.0 \* (0.9 - (neck \* 1.0)))))); line\_19 — a Line \[red\] drawn in sausage (start=(-4.440892098500626e-16, (0.0 + (1.0 \* (0.9 - (neck \* 1.0))))), end=(0.2999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…); line\_20 — a Line \[red\] drawn in sausage (start=(-4.440892098500626e-16, (0.0 + (-1.0 \* (0.9 - (neck \* 1.0))))), end=(0.2999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…); line\_21 — a Line \[red\] drawn in sausage (start=(0.2999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…, end=(0.5999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…); line\_22 — a Line \[red\] drawn in sausage (start=(0.2999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…, end=(0.5999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…); line\_23 — a Line \[red\] drawn in sausage (start=(0.5999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…, end=(0.8999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…); line\_24 — a Line \[red\] drawn in sausage (start=(0.5999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…, end=(0.8999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…); line\_25 — a Line \[red\] drawn in sausage (start=(0.8999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…, end=(1.1999999999999997, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…); line\_26 — a Line \[red\] drawn in sausage (start=(0.8999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…, end=(1.1999999999999997, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…); line\_27 — a Line \[red\] drawn in sausage (start=(1.1999999999999997, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…, end=(1.5, (0.0 + (1.0 \* (0.9 - (neck \* -1.0)))))); line\_28 — a Line \[red\] drawn in sausage (start=(1.1999999999999997, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…, end=(1.5, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0)))))); line\_29 — a Line \[red\] drawn in sausage (start=(1.5, (0.0 + (1.0 \* (0.9 - (neck \* -1.0))))), end=(1.7999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…); line\_30 — a Line \[red\] drawn in sausage (start=(1.5, (0.0 + (-1.0 \* (0.9 - (neck \* -1.0))))), end=(1.7999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…); line\_31 — a Line \[red\] drawn in sausage (start=(1.7999999999999998, (0.0 + (1.0 \* (0.9 - (neck \* -0.8090169943…, end=(2.0999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…); line\_32 — a Line \[red\] drawn in sausage (start=(1.7999999999999998, (0.0 + (-1.0 \* (0.9 - (neck \* -0.809016994…, end=(2.0999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…); line\_33 — a Line \[red\] drawn in sausage (start=(2.0999999999999996, (0.0 + (1.0 \* (0.9 - (neck \* -0.3090169943…, end=(2.3999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…); line\_34 — a Line \[red\] drawn in sausage (start=(2.0999999999999996, (0.0 + (-1.0 \* (0.9 - (neck \* -0.309016994…, end=(2.3999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…); line\_35 — a Line \[red\] drawn in sausage (start=(2.3999999999999995, (0.0 + (1.0 \* (0.9 - (neck \* 0.30901699437…, end=(2.6999999999999993, (0.0 + (1.0 \* (0.9 - (neck \* 0.80901699437…); line\_36 — a Line \[red\] drawn in sausage (start=(2.3999999999999995, (0.0 + (-1.0 \* (0.9 - (neck \* 0.3090169943…, end=(2.6999999999999993, (0.0 + (-1.0 \* (0.9 - (neck \* 0.8090169943…); hard — a Vector \[blue\] drawn in sausage (start=(0.0, 1.92), end=(0.0, 0.65)); hard\_2 — a Vector \[blue\] drawn in sausage (start=(0.0, -1.92), end=(0.0, -0.65)); soft — a Vector \[blue\] drawn in sausage (start=(1.5, 1.92), end=(1.5, 1.45)); soft\_2 — a Vector \[blue\] drawn in sausage (start=(1.5, -1.92), end=(1.5, -1.45)) Actions: - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_2 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_3 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_4 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_5 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_6 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_7 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_8 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_9 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_10 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_11 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_12 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_13 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_14 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_15 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_16 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_17 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_18 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_19 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_20 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_21 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_22 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_23 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_24 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_25 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_26 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_27 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_28 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_29 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_30 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_31 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_32 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_33 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_34 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_35 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): line\_36 is redrawn as the numbers it depends on change. - [02:39.854](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=159.85368749999998): neck ticks to 0.45. ##### [02:43.333](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=163.33268749999996) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [02:46.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=166.30468749999997): sausage\_note is shown on the screen, written out. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): pinch\_law is hidden from the screen — left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): sausage is hidden from the screen — left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_2 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_3 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_4 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_5 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_6 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_7 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_8 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_9 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_10 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_11 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_12 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_13 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_14 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_15 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_16 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_17 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_18 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_19 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_20 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_21 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_22 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_23 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_24 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_25 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_26 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_27 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_28 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_29 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_30 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_31 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_32 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_33 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_34 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_35 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): line\_36 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): hard is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): hard\_2 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): soft is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): soft\_2 is hidden from the screen — sausage left the board. - [02:54.211](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=174.21068749999998): sausage\_note is hidden from the screen — left the board. ##### [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997) Narration: The second mode is worse, because nothing about the radius has to change at all. Suppose the whole column simply bends sideways. Board: Empty. Actions: - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): kink is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): kink\_axis is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_37 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_38 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_39 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_40 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_41 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_42 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_43 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_44 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_45 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_46 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_47 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_48 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_49 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_50 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_51 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_52 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_53 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_54 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_55 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_56 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_57 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_58 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_59 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_60 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_61 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_62 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_63 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_64 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_65 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_66 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_67 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_68 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_69 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_70 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_71 is shown on the screen, written out. - [02:55.411](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=175.41068749999997): line\_72 is shown on the screen, written out. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_37 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_38 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_39 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_40 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_41 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_42 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_43 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_44 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_45 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_46 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_47 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_48 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_49 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_50 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_51 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_52 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_53 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_54 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_55 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_56 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_57 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_58 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_59 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_60 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_61 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_62 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_63 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_64 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_65 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_66 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_67 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_68 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_69 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_70 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_71 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): line\_72 is redrawn as the numbers it depends on change. - [03:1.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=181.66868749999998): bend ticks to 0.35. ##### [03:3.72](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=183.72018749999995) Narration: 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. Board: kink — a Figure (x\_range=(-3.0, 3.0), y\_range=(-2.0, 2.0), aspect=(6.0, 4.0)); kink\_axis — a Line \[gray\] drawn in kink (start=(-2.7, 0.0), end=(2.7, 0.0), dashed=True); line\_37 — a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + 0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6))); line\_38 — a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + -0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6))); line\_39 — a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + 0.6))); line\_40 — a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + -0.6))); line\_41 — a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + 0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6))); line\_42 — a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + -0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6))); line\_43 — a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6))); line\_44 — a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6))); line\_45 — a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6))); line\_46 — a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6))); line\_47 — a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6))); line\_48 — a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6))); line\_49 — a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6))); line\_50 — a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6))); line\_51 — a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6))); line\_52 — a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6))); line\_53 — a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…); line\_54 — a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…); line\_55 — a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6))); line\_56 — a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6))); line\_57 — a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6))); line\_58 — a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6))); line\_59 — a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6))); line\_60 — a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6))); line\_61 — a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6))); line\_62 — a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6))); line\_63 — a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6)), end=(1.5, ((bend \* 0.9848077530122081) + 0.6))); line\_64 — a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6)), end=(1.5, ((bend \* 0.9848077530122081) + -0.6))); line\_65 — a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + 0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6))); line\_66 — a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + -0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6))); line\_67 — a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6))); line\_68 — a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6))); line\_69 — a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6))); line\_70 — a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6))); line\_71 — a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + 0.6))); line\_72 — a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + -0.6))) Actions: - [03:6.762](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=186.76168749999997): kink moves to a new place on the board. - [03:6.762](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=186.76168749999997): kink\_law is shown on the screen, written out. - [03:11.975](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=191.97468749999996): push is shown on the screen, written out. - [03:11.975](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=191.97468749999996): push\_2 is shown on the screen, written out. ##### [03:15.675](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=195.67468749999995) Narration: So this one feeds itself too. The more it bends, the harder it is pushed, and that is the m equals one kink mode. Board: kink\_law — a Math \[text\] that says "$B\_(upright("in")) \> B\_(upright("out"))$"; kink — a Figure (x\_range=(-3.0, 3.0), y\_range=(-2.0, 2.0), aspect=(6.0, 4.0)); kink\_axis — a Line \[gray\] drawn in kink (start=(-2.7, 0.0), end=(2.7, 0.0), dashed=True); line\_37 — a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + 0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6))); line\_38 — a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + -0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6))); line\_39 — a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + 0.6))); line\_40 — a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + -0.6))); line\_41 — a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + 0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6))); line\_42 — a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + -0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6))); line\_43 — a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6))); line\_44 — a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6))); line\_45 — a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6))); line\_46 — a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6))); line\_47 — a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6))); line\_48 — a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6))); line\_49 — a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6))); line\_50 — a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6))); line\_51 — a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6))); line\_52 — a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6))); line\_53 — a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…); line\_54 — a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…); line\_55 — a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6))); line\_56 — a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6))); line\_57 — a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6))); line\_58 — a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6))); line\_59 — a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6))); line\_60 — a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6))); line\_61 — a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6))); line\_62 — a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6))); line\_63 — a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6)), end=(1.5, ((bend \* 0.9848077530122081) + 0.6))); line\_64 — a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6)), end=(1.5, ((bend \* 0.9848077530122081) + -0.6))); line\_65 — a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + 0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6))); line\_66 — a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + -0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6))); line\_67 — a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6))); line\_68 — a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6))); line\_69 — a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6))); line\_70 — a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6))); line\_71 — a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + 0.6))); line\_72 — a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + -0.6))); push — a Vector \[blue\] drawn in kink (start=(1.35, -0.05), end=(1.35, 0.35)); push\_2 — a Vector \[blue\] drawn in kink (start=(-1.35, 0.05), end=(-1.35, -0.35)) Actions: - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_37 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_38 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_39 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_40 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_41 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_42 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_43 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_44 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_45 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_46 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_47 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_48 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_49 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_50 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_51 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_52 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_53 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_54 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_55 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_56 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_57 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_58 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_59 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_60 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_61 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_62 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_63 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_64 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_65 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_66 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_67 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_68 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_69 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_70 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_71 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): line\_72 is redrawn as the numbers it depends on change. - [03:18.531](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=198.53068749999994): bend ticks to 0.98. - [03:22.757](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=202.75668749999994): kink\_note is shown on the screen, written out. ##### [03:24.437](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=204.43668749999995) Narration: 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. Board: kink\_law — a Math \[text\] that says "$B\_(upright("in")) \> B\_(upright("out"))$"; kink\_note — a Panel that says "Crowded field lines on the inside of a bend push the bend further over, so a sideways displacement grows as well."; kink — a Figure (x\_range=(-3.0, 3.0), y\_range=(-2.0, 2.0), aspect=(6.0, 4.0)); kink\_axis — a Line \[gray\] drawn in kink (start=(-2.7, 0.0), end=(2.7, 0.0), dashed=True); line\_37 — a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + 0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6))); line\_38 — a Line \[red\] drawn in kink (start=(-2.7, ((bend \* -1.2246467991473532e-16) + -0.6)), end=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6))); line\_39 — a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + 0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + 0.6))); line\_40 — a Line \[red\] drawn in kink (start=(-2.4000000000000004, ((bend \* -0.3420201433256689) + -0.6)), end=(-2.1, ((bend \* -0.6427876096865395) + -0.6))); line\_41 — a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + 0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6))); line\_42 — a Line \[red\] drawn in kink (start=(-2.1, ((bend \* -0.6427876096865395) + -0.6)), end=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6))); line\_43 — a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + 0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6))); line\_44 — a Line \[red\] drawn in kink (start=(-1.8000000000000003, ((bend \* -0.8660254037844385) + -0.6)), end=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6))); line\_45 — a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6))); line\_46 — a Line \[red\] drawn in kink (start=(-1.5000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6))); line\_47 — a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + 0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6))); line\_48 — a Line \[red\] drawn in kink (start=(-1.2000000000000002, ((bend \* -0.984807753012208) + -0.6)), end=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6))); line\_49 — a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + 0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6))); line\_50 — a Line \[red\] drawn in kink (start=(-0.9000000000000004, ((bend \* -0.8660254037844388) + -0.6)), end=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6))); line\_51 — a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + 0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6))); line\_52 — a Line \[red\] drawn in kink (start=(-0.6000000000000001, ((bend \* -0.6427876096865393) + -0.6)), end=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6))); line\_53 — a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + 0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…); line\_54 — a Line \[red\] drawn in kink (start=(-0.30000000000000027, ((bend \* -0.34202014332566905) + -0.6)), end=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…); line\_55 — a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + 0.6…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6))); line\_56 — a Line \[red\] drawn in kink (start=(-4.440892098500626e-16, ((bend \* -5.167212589642417e-16) + -0.…, end=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6))); line\_57 — a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + 0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6))); line\_58 — a Line \[red\] drawn in kink (start=(0.2999999999999998, ((bend \* 0.3420201433256685) + -0.6)), end=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6))); line\_59 — a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + 0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6))); line\_60 — a Line \[red\] drawn in kink (start=(0.5999999999999996, ((bend \* 0.6427876096865389) + -0.6)), end=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6))); line\_61 — a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + 0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6))); line\_62 — a Line \[red\] drawn in kink (start=(0.8999999999999995, ((bend \* 0.8660254037844383) + -0.6)), end=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6))); line\_63 — a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + 0.6)), end=(1.5, ((bend \* 0.9848077530122081) + 0.6))); line\_64 — a Line \[red\] drawn in kink (start=(1.1999999999999997, ((bend \* 0.984807753012208) + -0.6)), end=(1.5, ((bend \* 0.9848077530122081) + -0.6))); line\_65 — a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + 0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6))); line\_66 — a Line \[red\] drawn in kink (start=(1.5, ((bend \* 0.9848077530122081) + -0.6)), end=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6))); line\_67 — a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + 0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6))); line\_68 — a Line \[red\] drawn in kink (start=(1.7999999999999998, ((bend \* 0.8660254037844389) + -0.6)), end=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6))); line\_69 — a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + 0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6))); line\_70 — a Line \[red\] drawn in kink (start=(2.0999999999999996, ((bend \* 0.6427876096865398) + -0.6)), end=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6))); line\_71 — a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + 0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + 0.6))); line\_72 — a Line \[red\] drawn in kink (start=(2.3999999999999995, ((bend \* 0.3420201433256697) + -0.6)), end=(2.6999999999999993, ((bend \* 1.0106430996148606e-15) + -0.6))); push — a Vector \[blue\] drawn in kink (start=(1.35, -0.05), end=(1.35, 0.35)); push\_2 — a Vector \[blue\] drawn in kink (start=(-1.35, 0.05), end=(-1.35, -0.35)) Actions: - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): kink is hidden from the screen — left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): kink\_axis is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_37 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_38 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_39 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_40 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_41 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_42 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_43 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_44 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_45 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_46 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_47 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_48 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_49 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_50 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_51 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_52 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_53 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_54 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_55 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_56 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_57 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_58 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_59 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_60 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_61 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_62 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_63 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_64 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_65 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_66 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_67 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_68 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_69 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_70 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_71 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): line\_72 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): push is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): push\_2 is hidden from the screen — kink left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): kink\_law is hidden from the screen — left the board. - [03:36.28](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=216.27962499999998): kink\_note is hidden from the screen — left the board. ### Scene 3: [Stabilizing the Pinch](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=217.32129166666664) Span: 03:37.321–05:59.925 (217.32129166666664s–359.92479166666664s). #### 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=\) - bz\_curve: a FunctionPlot \[blue\] labelled "B\_z" drawn in profile (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=\, 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=\, 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=\) - tube\_radius: a VariableNumber (initial\_value=1.05) #### Beats ##### [03:37.321](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=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. Board: Empty. Actions: - [03:37.321](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=217.32129166666664): depth is shown on the screen, written out. - [03:37.321](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=217.32129166666664): tube is shown on the screen, written out. - [03:41.512](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=221.51229166666664): axial\_1 is shown on the screen, written out. - [03:41.512](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=221.51229166666664): axial\_2 is shown on the screen, written out. ##### [03:48.846](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=228.84629166666664) Narration: 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. Board: depth — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), z\_range=(-2.6, 2.6)); tube — a Cylinder \[red\] drawn in depth (start=(0.0, 0.0, -1.9), end=(0.0, 0.0, 1.9), radius=\); 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)) Actions: - [03:54.941](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=234.94129166666664): helix is shown on the screen, drawn. ##### [04:2.635](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=242.63529166666663) Narration: 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. Board: depth — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), z\_range=(-2.6, 2.6)); tube — a Cylinder \[red\] drawn in depth (start=(0.0, 0.0, -1.9), end=(0.0, 0.0, 1.9), radius=\); 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)); helix — a ParametricCurve \[green\] labelled "arrow(B)" drawn in depth (function=\, t\_range=(0.0, 18.84955592153876)) Actions: - [04:4.795](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=244.79529166666663): depth moves to a new place on the board. - [04:4.795](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=244.79529166666663): q\_work is shown on the screen, written out. - [04:11.342](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=251.34229166666665): q\_work (the "B\_z" part) is emphasized. - [04:14.639](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=254.63929166666662): q\_work is shown on the screen, written out. - [04:14.639](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=254.63929166666662): q\_work (the "B\_z" part) is no longer emphasized. ##### [04:16.517](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=256.5167916666666) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [04:16.865](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=256.8652916666666): axial\_note is shown on the screen, written out. ##### [04:26.37](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=266.36979166666663) Narration: The trapped axial field resists the sausage mode too. Squeezing the column compresses the flux it encloses, and compressed flux pushes back. Board: 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."; depth — an Axes3D (x\_range=(-2.0, 2.0), y\_range=(-2.0, 2.0), z\_range=(-2.6, 2.6)); tube — a Cylinder \[red\] drawn in depth (start=(0.0, 0.0, -1.9), end=(0.0, 0.0, 1.9), radius=\); 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)); helix — a ParametricCurve \[green\] labelled "arrow(B)" drawn in depth (function=\, t\_range=(0.0, 18.84955592153876)) Actions: - [04:30.387](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=270.38729166666667): tube is redrawn as the numbers it depends on change. - [04:30.387](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=270.38729166666667): tube\_radius ticks to 0.78. - [04:34.369](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=274.36929166666664): tube is redrawn as the numbers it depends on change. - [04:34.369](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=274.36929166666664): tube\_radius ticks to 1.05. - [04:35.46](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=275.46029166666665): axial\_note is hidden from the screen — left the board. - [04:35.46](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=275.46029166666665): depth is hidden from the screen — left the board. - [04:35.46](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=275.46029166666665): tube is hidden from the screen — depth left the board. - [04:35.46](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=275.46029166666665): axial\_1 is hidden from the screen — depth left the board. - [04:35.46](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=275.46029166666665): axial\_2 is hidden from the screen — depth left the board. - [04:35.46](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=275.46029166666665): helix is hidden from the screen — depth left the board. - [04:35.46](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=275.46029166666665): q\_work is hidden from the screen — left the board. ##### [04:36.66](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=276.66029166666664) Narration: 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. Board: Empty. Actions: - [04:36.66](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=276.66029166666664): ring is shown on the screen, written out. - [04:38.669](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=278.66929166666665): torus is shown on the screen, written out. - [04:39.644](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=279.6442916666666): ring\_current is shown on the screen, written out. - [04:39.644](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=279.6442916666666): ring\_sense is shown on the screen, written out. ##### [04:44.783](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=284.7832916666666) Narration: 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. Board: ring — an Axes3D (x\_range=(-3.2, 3.2), y\_range=(-3.2, 3.2), z\_range=(-1.8, 1.8)); torus — a Surface \[red\] drawn in ring (function=\, u\_range=(0.0, 6.283185307179586), v\_range=(0.0, 6.283185307179586)); 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) Actions: - [04:44.783](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=284.7832916666666): ring turns in its own slot. ##### [04:56.018](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=296.01779166666665) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [04:58.885](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=298.88529166666666): ring moves to a new place on the board. - [04:58.885](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=298.88529166666666): profile is shown on the screen, written out. - [05:5.677](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=305.67729166666663): bz\_curve is shown on the screen, written out. - [05:8.475](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=308.47529166666664): point is shown on the screen, grown. - [05:10.962](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=310.961832025411): point is hidden from the screen. ##### [05:11.293](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=311.29279166666663) Narration: 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. Board: ring — an Axes3D (x\_range=(-3.2, 3.2), y\_range=(-3.2, 3.2), z\_range=(-1.8, 1.8)); profile — an Axes (y\_range=(-0.5, 1.2), x\_ticks\_every=0.5, y\_ticks\_every=0.5); torus — a Surface \[red\] drawn in ring (function=\, u\_range=(0.0, 6.283185307179586), v\_range=(0.0, 6.283185307179586)); 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); bz\_curve — a FunctionPlot \[blue\] labelled "B\_z" drawn in profile (function=\) Actions: - [05:13.371](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=313.37129166666665): btheta\_curve is shown on the screen, written out. - [05:26.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=326.30479166666663): profile is hidden from the screen — left the board. - [05:26.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=326.30479166666663): bz\_curve is hidden from the screen — profile left the board. - [05:26.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=326.30479166666663): btheta\_curve is hidden from the screen — profile left the board. - [05:26.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=326.30479166666663): ring is hidden from the screen — left the board. - [05:26.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=326.30479166666663): torus is hidden from the screen — ring left the board. - [05:26.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=326.30479166666663): ring\_current is hidden from the screen — ring left the board. - [05:26.305](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=326.30479166666663): ring\_sense is hidden from the screen — ring left the board. ##### [05:27.505](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=327.5047916666666) Narration: 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. Board: Empty. Actions: - [05:27.505](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=327.5047916666666): flow is shown on the screen, written out. - [05:27.505](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=327.5047916666666): flow\_top is shown on the screen, written out. - [05:27.505](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=327.5047916666666): flow\_bottom is shown on the screen, written out. - [05:34.075](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=334.07529166666666): jets is shown on the screen, written out. - [05:34.199](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=334.19895371357495): jets\_2 is shown on the screen, written out. - [05:34.323](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=334.3226157604833): jets\_3 is shown on the screen, written out. - [05:34.446](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=334.44627780739165): jets\_4 is shown on the screen, written out. - [05:34.528](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=334.5283762437811): jets\_5 is shown on the screen, written out. - [05:34.585](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=334.5845080845771): jets\_6 is shown on the screen, written out. - [05:34.638](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=334.6383513681592): jets\_7 is shown on the screen, written out. - [05:35.329](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=335.3292916666667): speed\_label is shown on the screen, written out. ##### [05:36.638](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=336.63779166666666) Narration: 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. Board: flow — a Figure (x\_range=(-0.5, 3.7), y\_range=(-1.8, 1.8), aspect=(4.2, 3.6)); flow\_top — a Line \[red\] drawn in flow (start=(-0.2, 1.25), end=(3.5, 1.25)); flow\_bottom — a Line \[red\] drawn in flow (start=(-0.2, -1.25), end=(3.5, -1.25)); 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)); speed\_label — a Math \[text\] that says "$v\_z (r)$" drawn in flow Actions: - [05:43.267](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=343.26729166666667): jets is indicated — a transient flash. - [05:43.267](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=343.26729166666667): jets\_6 is indicated — a transient flash. ##### [05:48.349](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=348.34879166666667) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [05:48.79](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=348.79029166666663): flow moves to a new place on the board. - [05:48.79](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=348.79029166666663): shear\_law is shown on the screen, written out. - [05:53.12](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=353.1202916666666): shear\_note is shown on the screen, written out. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): flow is hidden from the screen — left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): flow\_top is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): flow\_bottom is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): jets is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): jets\_2 is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): jets\_3 is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): jets\_4 is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): jets\_5 is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): jets\_6 is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): jets\_7 is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): speed\_label is hidden from the screen — flow left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): shear\_law is hidden from the screen — left the board. - [05:58.883](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=358.88312499999995): shear\_note is hidden from the screen — left the board. ### Scene 4: [Field-Reversed Configurations](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=359.92479166666664) Span: 05:59.925–07:40.658 (359.92479166666664s–460.65806249999997s). #### Objects - blob: a Point \[red\] labelled "upright("FRC")" drawn in machine (location=(\, 0.0), marker\_radius=\) - 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=\, t\_range=(0.0, 6.283185307179586)) - inner\_upper: a ParametricCurve \[green\] drawn in cross (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=(\ 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=\, 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 ##### [05:59.925](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=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. Board: Empty. Actions: - [05:59.925](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=359.92479166666664): cross is shown on the screen, written out. - [05:59.925](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=359.92479166666664): plasma is shown on the screen, written out. - [05:59.925](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=359.92479166666664): machine\_axis is shown on the screen, written out. - [06:8.656](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=368.65579166666663): separatrix is shown on the screen, written out. - [06:8.656](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=368.65579166666663): sep\_label is shown on the screen, written out. ##### [06:11.194](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=371.19429166666663) Narration: 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. Board: cross — a Figure (x\_range=(-3.4, 3.4), y\_range=(-2.1, 2.1), aspect=(6.8, 4.2)); plasma — a Region \[red\] drawn in cross (predicates=(\ at 0x2a7c587a7ec0\>,), opacity=0.16); machine\_axis — a Line \[gray\] drawn in cross (start=(-3.2, 0.0), end=(3.2, 0.0), dashed=True); separatrix — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); sep\_label — a Math \[text\] that says "$upright("separatrix")$" drawn in cross Actions: - [06:16.314](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=376.31379166666665): cross moves to a new place on the board. - [06:16.314](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=376.31379166666665): frc\_work is shown on the screen, written out. - [06:20.064](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=380.06379166666665): frc\_work is shown on the screen, written out. - [06:21.399](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=381.3987916666666): separatrix\_sense is shown on the screen, written out. ##### [06:23.357](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=383.35729166666664) Narration: 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. Board: cross — a Figure (x\_range=(-3.4, 3.4), y\_range=(-2.1, 2.1), aspect=(6.8, 4.2)); plasma — a Region \[red\] drawn in cross (predicates=(\ at 0x2a7c587a7ec0\>,), opacity=0.16); machine\_axis — a Line \[gray\] drawn in cross (start=(-3.2, 0.0), end=(3.2, 0.0), dashed=True); separatrix — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); sep\_label — a Math \[text\] that says "$upright("separatrix")$" drawn in cross; 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) Actions: - [06:26.156](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=386.15579166666663): inner\_upper is shown on the screen, written out. - [06:26.156](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=386.15579166666663): inner\_lower is shown on the screen, written out. - [06:26.864](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=386.86379166666666): o\_upper is shown on the screen, written out. - [06:26.864](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=386.86379166666666): o\_lower is shown on the screen, written out. - [06:34.352](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=394.35179166666666): x\_left is shown on the screen, written out. - [06:34.352](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=394.35179166666666): x\_right is shown on the screen, written out. - [06:36.976](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=396.9757916666666): ext\_top is shown on the screen, written out. - [06:36.976](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=396.9757916666666): ext\_bottom is shown on the screen, written out. ##### [06:38.528](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=398.52779166666664) Narration: 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. Board: cross — a Figure (x\_range=(-3.4, 3.4), y\_range=(-2.1, 2.1), aspect=(6.8, 4.2)); plasma — a Region \[red\] drawn in cross (predicates=(\ at 0x2a7c587a7ec0\>,), opacity=0.16); machine\_axis — a Line \[gray\] drawn in cross (start=(-3.2, 0.0), end=(3.2, 0.0), dashed=True); separatrix — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); sep\_label — a Math \[text\] that says "$upright("separatrix")$" drawn in cross; 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); inner\_upper — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); inner\_lower — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); o\_upper — a Point \[yellow\] labelled "O" drawn in cross (location=(0.0, 0.95)); o\_lower — a Point \[yellow\] labelled "O" drawn in cross (location=(0.0, -0.95)); 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)); ext\_top — a Vector \[blue\] labelled "arrow(B)\_(upright("ext"))" drawn in cross (start=(-3.2, 1.75), end=(3.2, 1.75)); ext\_bottom — a Vector \[blue\] drawn in cross (start=(-3.2, -1.75), end=(3.2, -1.75)) Actions: - [06:50.15](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=410.14979166666666): frc\_work is shown on the screen, written out. - [06:52.855](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=412.85479166666664): frc\_note is shown on the screen, written out. ##### [06:54.465](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=414.46529166666664) Narration: 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. Board: 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."; cross — a Figure (x\_range=(-3.4, 3.4), y\_range=(-2.1, 2.1), aspect=(6.8, 4.2)); plasma — a Region \[red\] drawn in cross (predicates=(\ at 0x2a7c587a7ec0\>,), opacity=0.16); machine\_axis — a Line \[gray\] drawn in cross (start=(-3.2, 0.0), end=(3.2, 0.0), dashed=True); separatrix — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); sep\_label — a Math \[text\] that says "$upright("separatrix")$" drawn in cross; 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); inner\_upper — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); inner\_lower — a ParametricCurve \[green\] drawn in cross (function=\, t\_range=(0.0, 6.283185307179586)); o\_upper — a Point \[yellow\] labelled "O" drawn in cross (location=(0.0, 0.95)); o\_lower — a Point \[yellow\] labelled "O" drawn in cross (location=(0.0, -0.95)); 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)); ext\_top — a Vector \[blue\] labelled "arrow(B)\_(upright("ext"))" drawn in cross (start=(-3.2, 1.75), end=(3.2, 1.75)); ext\_bottom — a Vector \[blue\] drawn in cross (start=(-3.2, -1.75), end=(3.2, -1.75)) Actions: - [07:2.406](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=422.40579166666663): separatrix is indicated — a transient flash. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): cross is hidden from the screen — left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): plasma is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): machine\_axis is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): separatrix is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): sep\_label is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): separatrix\_sense is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): inner\_upper is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): inner\_lower is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): o\_upper is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): o\_lower is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): x\_left is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): x\_right is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): ext\_top is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): ext\_bottom is hidden from the screen — cross left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): frc\_note is hidden from the screen — left the board. - [07:4.6](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=424.60029166666664): frc\_work is hidden from the screen — left the board. ##### [07:5.8](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=425.8002916666666) Narration: 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. Board: Empty. Actions: - [07:5.8](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=425.8002916666666): machine is shown on the screen, written out. - [07:5.8](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=425.8002916666666): vessel\_top is shown on the screen, written out. - [07:5.8](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=425.8002916666666): vessel\_bottom is shown on the screen, written out. - [07:10.166](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=430.1657916666666): blob is shown on the screen, written out. - [07:10.166](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=430.1657916666666): form\_label is shown on the screen, written out. - [07:13.115](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=433.11479166666663): blob is redrawn as the numbers it depends on change. - [07:13.115](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=433.11479166666663): place ticks to 4.9. ##### [07:16.803](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=436.8027916666666) Narration: 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. Board: machine — a Figure (x\_range=(-0.4, 6.4), y\_range=(-2.0, 2.0), aspect=(6.8, 4.0)); vessel\_top — a Line \[gray\] drawn in machine (start=(0.2, 1.15), end=(6.0, 1.15)); vessel\_bottom — a Line \[gray\] drawn in machine (start=(0.2, -1.15), end=(6.0, -1.15)); blob — a Point \[red\] labelled "upright("FRC")" drawn in machine (location=(\, 0.0), marker\_radius=\); form\_label — a Math \[text\] that says "$upright("formation")$" drawn in machine Actions: - [07:18.777](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=438.7767916666666): coils is shown on the screen, written out. - [07:18.777](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=438.7767916666666): coils\_2 is shown on the screen, written out. - [07:18.777](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=438.7767916666666): coils\_3 is shown on the screen, written out. - [07:18.777](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=438.7767916666666): coils\_4 is shown on the screen, written out. - [07:18.777](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=438.7767916666666): coils\_5 is shown on the screen, written out. - [07:18.777](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=438.7767916666666): coils\_6 is shown on the screen, written out. - [07:18.777](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=438.7767916666666): squeeze\_label is shown on the screen, written out. - [07:19.984](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=439.98379166666666): blob is redrawn as the numbers it depends on change. - [07:19.984](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=439.98379166666666): size ticks to 0.16. ##### [07:28.269](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=448.2692916666666) Narration: 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. Board: machine — a Figure (x\_range=(-0.4, 6.4), y\_range=(-2.0, 2.0), aspect=(6.8, 4.0)); vessel\_top — a Line \[gray\] drawn in machine (start=(0.2, 1.15), end=(6.0, 1.15)); vessel\_bottom — a Line \[gray\] drawn in machine (start=(0.2, -1.15), end=(6.0, -1.15)); blob — a Point \[red\] labelled "upright("FRC")" drawn in machine (location=(\, 0.0), marker\_radius=\); form\_label — a Math \[text\] that says "$upright("formation")$" drawn in machine; 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)); squeeze\_label — a Math \[text\] that says "$upright("compression")$" drawn in machine Actions: - [07:30.383](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=450.3827916666666): blob is indicated — a transient flash. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): machine is hidden from the screen — left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): vessel\_top is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): vessel\_bottom is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): blob is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): form\_label is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): coils is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): coils\_2 is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): coils\_3 is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): coils\_4 is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): coils\_5 is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): coils\_6 is hidden from the screen — machine left the board. - [07:39.616](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=459.6163958333333): squeeze\_label is hidden from the screen — machine left the board. ### Scene 5: [MagLIF and the Trade-offs](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=460.65806249999997) Span: 07:40.658–10:32.832 (460.65806249999997s–632.8321874999999s). #### 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=\) - 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=\) - 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 ##### [07:40.658](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=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. Board: Empty. Actions: - [07:40.658](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=460.65806249999997): stack is shown on the screen, written out. - [07:54.694](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=474.6940625): liner is shown on the screen, written out. - [07:56.215](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=476.2150625): fuel is shown on the screen, written out. ##### [07:57.605](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=477.6045625) Narration: 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. Board: stack — an Axes3D (x\_range=(-2.2, 2.2), y\_range=(-2.2, 2.2), z\_range=(-2.9, 2.9)); liner — a Cylinder \[gray\] drawn in stack (start=(0.0, 0.0, -1.8), end=(0.0, 0.0, 1.8), radius=\); fuel — a Cylinder \[red\] drawn in stack (start=(0.0, 0.0, -1.75), end=(0.0, 0.0, 1.75), radius=\) Actions: - [07:57.953](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=477.9530625): step\_field is shown on the screen, written out. - [07:59.474](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=479.47406249999995): bz\_1 is shown on the screen, written out. - [07:59.474](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=479.47406249999995): bz\_2 is shown on the screen, written out. - [08:4.118](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=484.11806249999995): readout is shown on the screen, written out. ##### [08:9.443](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=489.4430625) Narration: 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. Board: step\_field — a Text \[text\] that says "Pre-magnetize the fuel with an axial field of about 10 T."; stack — an Axes3D (x\_range=(-2.2, 2.2), y\_range=(-2.2, 2.2), z\_range=(-2.9, 2.9)); liner — a Cylinder \[gray\] drawn in stack (start=(0.0, 0.0, -1.8), end=(0.0, 0.0, 1.8), radius=\); fuel — a Cylinder \[red\] drawn in stack (start=(0.0, 0.0, -1.75), end=(0.0, 0.0, 1.75), radius=\); 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)); readout — a Point \[green\] labelled "B\_z = 10 thin upright("T")" drawn in stack (location=(1.75, 0.0, 2.6), show\_marker=False) Actions: - [08:9.791](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=489.79106249999995): step\_heat is shown on the screen, written out. - [08:11.312](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=491.31206249999997): laser is shown on the screen, written out. ##### [08:21.549](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=501.5485625) Narration: 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. Board: 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."; stack — an Axes3D (x\_range=(-2.2, 2.2), y\_range=(-2.2, 2.2), z\_range=(-2.9, 2.9)); liner — a Cylinder \[gray\] drawn in stack (start=(0.0, 0.0, -1.8), end=(0.0, 0.0, 1.8), radius=\); fuel — a Cylinder \[red\] drawn in stack (start=(0.0, 0.0, -1.75), end=(0.0, 0.0, 1.75), radius=\); 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)); readout — a Point \[green\] labelled "B\_z = 10 thin upright("T")" drawn in stack (location=(1.75, 0.0, 2.6), show\_marker=False); laser — a Vector \[magenta\] labelled "upright("laser")" drawn in stack (start=(0.0, 0.0, 2.75), end=(0.0, 0.0, 0.9)) Actions: - [08:21.897](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=501.89706249999995): step\_push is shown on the screen, written out. - [08:23.696](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=503.6960625): drive is shown on the screen, written out. - [08:29.733](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=509.73306249999996): crush is shown on the screen, written out. - [08:29.863](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=509.8633261188474): crush\_2 is shown on the screen, written out. - [08:29.994](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=509.99358973769483): crush\_3 is shown on the screen, written out. - [08:30.117](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=510.1167971295323): crush\_4 is shown on the screen, written out. - [08:35.271](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=515.2710625): step\_field is hidden from the screen — left the board. - [08:35.271](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=515.2710625): step\_heat is hidden from the screen — left the board. - [08:35.271](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=515.2710625): step\_push is hidden from the screen — left the board. ##### [08:36.471](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=516.4710625) Narration: 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. Board: stack — an Axes3D (x\_range=(-2.2, 2.2), y\_range=(-2.2, 2.2), z\_range=(-2.9, 2.9)); liner — a Cylinder \[gray\] drawn in stack (start=(0.0, 0.0, -1.8), end=(0.0, 0.0, 1.8), radius=\); fuel — a Cylinder \[red\] drawn in stack (start=(0.0, 0.0, -1.75), end=(0.0, 0.0, 1.75), radius=\); 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)); readout — a Point \[green\] labelled "B\_z = 10 thin upright("T")" drawn in stack (location=(1.75, 0.0, 2.6), show\_marker=False); laser — a Vector \[magenta\] labelled "upright("laser")" drawn in stack (start=(0.0, 0.0, 2.75), end=(0.0, 0.0, 0.9)); drive — a Vector \[yellow\] labelled "I" drawn in stack (start=(1.6, 0.0, 1.9), end=(1.6, 0.0, -1.9)); 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)) Actions: - [08:39.118](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=519.1180625): mag\_work is shown on the screen, written out. - [08:47.187](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=527.1870624999999): mag\_work (the "I^2" part) is emphasized. - [08:50.856](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=530.8560625): mag\_work (the "I^2" part) is no longer emphasized. ##### [08:51.456](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=531.4560624999999) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [08:56.448](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=536.4480625): liner is redrawn as the numbers it depends on change. - [08:56.448](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=536.4480625): fuel is redrawn as the numbers it depends on change. - [08:56.448](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=536.4480625): readout is redrawn as the numbers it depends on change. - [08:56.448](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=536.4480625): r\_liner ticks to 0.22. - [08:56.448](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=536.4480625): r\_fuel ticks to 0.16. - [08:56.448](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=536.4480625): b\_field ticks to 6000.0. - [09:1.951](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=541.9510625): mag\_work is shown on the screen, written out. ##### [09:6.604](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=546.6035625) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [09:8.426](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=548.4260625): mag\_work is shown on the screen, written out. - [09:14.603](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=554.6030625): A box is drawn around mag\_work. - [09:16.959](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=556.9590625): mag\_note is shown on the screen, written out. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): mag\_note is hidden from the screen — left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): mag\_work is hidden from the screen — left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): stack is hidden from the screen — left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): liner is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): fuel is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): bz\_1 is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): bz\_2 is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): readout is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): laser is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): drive is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): crush is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): crush\_2 is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): crush\_3 is hidden from the screen — stack left the board. - [09:20.477](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=560.4770625): crush\_4 is hidden from the screen — stack left the board. ##### [09:21.677](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=561.6770624999999) Narration: So put the family side by side. Each row is confined by something different, and each one pays a different price. Board: Empty. Actions: - [09:24.858](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=564.8580625): table is shown on the screen, written out. ##### [09:29.754](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=569.7540624999999) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [09:30.265](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=570.2650625): table is shown on the screen, written out. - [09:35.443](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=575.4430625): table is shown on the screen, written out. ##### [09:42.191](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=582.1905624999999) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [09:42.568](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=582.5680625): table is shown on the screen, written out. - [09:49.603](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=589.6030625): table is shown on the screen, written out. ##### [09:56.948](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=596.9480625) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [10:0.687](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=600.6870624999999): table is shown on the screen, written out. - [10:8.175](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=608.1750625): table is shown on the screen, written out. ##### [10:15.486](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=615.4860625) Narration: 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. Board: Unchanged from the preceding beat in this scene. Actions: - [10:17.007](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=617.0070625): trade is shown on the screen, written out. - [10:17.936](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=617.9360624999999): table is indicated — a transient flash. - [10:26.388](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=626.3880624999999): table is indicated — a transient flash. - [10:31.791](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=631.7905208333334): table is hidden from the screen — left the board. - [10:31.791](https://academa.ai/@adamierymenko/lectures/z-pinch-fusion-and-magnetic-pinch-relatives?t=631.7905208333334): trade is hidden from the screen — left the board.