Z-Pinch Fusion and Magnetic Pinch Relatives

Fusion needs a plasma at over a hundred million kelvin, and a plasma that hot pushes outward hard. Here is the cheapest way anyone has proposed to hold one still. Drive a current straight through it, and let the current's own field do the squeezing. So take a straight column of plasma and pass a current along its axis. That axial current is the z in Z-pinch, and notice that nothing outside the plasma carries it. Look at the column itself for a moment. It is a cylinder of ionized gas hotter than the core of the sun, and the only thing between it and the vessel wall is the field we are about to build. Now that field. A current is wrapped by a magnetic field, and the right hand rule puts those loops in the azimuthal direction, closing around the axis at every height. The field is azimuthal and the current is axial, so the force per unit volume, J cross B, points inward. Every part of the plasma is pushed toward the axis at once. How hard? Ampere's law gives the azimuthal field at radius r, and a field of strength B always carries a magnetic pressure equal to B squared over two mu nought. Put the two together and the pinch pressure at the surface of the column goes as the current squared over the radius squared. Squeeze the column inward and the field that squeezes it only grows. Balance that magnetic pressure against the plasma's own pressure, and one relation falls out. Mu nought times the current squared equals eight pi N k T, with N the number of particles per unit length. Read it as a design rule. The current alone fixes the pressure you can confine, so a few mega-amps reaches fusion conditions with no external magnet anywhere. That is the whole appeal of the pinch, and in a moment it will also be the whole problem.

The pinch has a fatal flaw, and it is written in the formula we just derived. The inward pressure goes as one over the radius squared, so it cannot be the same everywhere unless the radius is. Give that column a ripple. A slight narrowing here, a slight bulge there: exactly the kind of perturbation that thermal noise supplies for nothing. Now look at the narrow place. Its radius is smaller, so the field there is stronger, and the inward push is stronger. The neck squeezes down harder than its neighbors do. At the bulge the opposite happens. Larger radius, weaker field, weaker squeeze. So the perturbation feeds itself, and the neck runs away. That is the m equals zero mode, the sausage instability. It grows in tens of nanoseconds, and it does not stop until the column has been cut into separate blobs. The second mode is worse, because nothing about the radius has to change at all. Suppose the whole column simply bends sideways. On the inside of the bend the azimuthal field lines are crowded closer together, so the field there is stronger than on the outside. The net push is sideways, the same way the column has already moved. So this one feeds itself too. The more it bends, the harder it is pushed, and that is the m equals one kink mode. Both modes come straight out of ideal magnetohydrodynamics, both are driven by the confining current itself, and both are fast. Everything that follows is an answer to these two pictures.

The first fix is the obvious one. Add a magnetic field along the axis, from external coils or driven by the plasma itself, and watch what it does to the field line geometry. The azimuthal field wraps around the column and the axial field runs along it, so the total field is a helix. A field line now has tension along the column, and bending the column means stretching that line. How much do we need? Compare the pitch of the helix with the length of the column. The safety factor q is two pi a B z over L B theta, and the kink is held off when q is greater than one. Written out, that says a field line must not close on itself before it reaches the end of the column. Bend the column and the tension pulls it straight again. The trapped axial field resists the sausage mode too. Squeezing the column compresses the flux it encloses, and compressed flux pushes back. Now bend the whole pinch into a torus, so the current has no ends to leak from and the plasma can organize its own field. This is the geometry of every steady magnetic confinement machine, and the pinch version of it is called the reversed field pinch. The plasma current runs the long way around the ring. The name comes from its field profile. Measure the axial field from the center of the plasma out to the wall, with r over a the fraction of the way out. It starts strong on the axis, falls, and reverses sign near the edge. The azimuthal field does the opposite: zero on the axis and largest at the edge. That helical, self-organized profile is what a resistive plasma relaxes into, and it is stable against the kink with no large external field. The other route keeps the column straight and gives it a velocity shear. Let the plasma flow along the axis, faster in the middle than at the edge. A growing sausage or kink is a wave that has to stay coherent across the radius. If neighboring layers are sliding past each other fast enough, the wave is pulled apart before it can grow. The requirement is a shear rate of order a tenth of the growth rate. Sheared flow experiments have then held a quiet pinch for microseconds, thousands of times the ideal growth time.

The pinch and the torus both use a field that is open at the ends or held in place by coils. There is a third option: let the plasma make a closed field of its own. Here is a cross-section taken along the axis. The current in this plasma is azimuthal, running around the axis, and the field it makes is poloidal, lying in the plane of the picture. Inside the separatrix the field lines are closed loops around a magnetic axis, one above the machine axis in this section and one below. The separatrix meets the axis at two X points, and outside it the field runs the other way. That reversal is the name of the thing: a field-reversed configuration. Because the field closes, it can hold a plasma whose pressure is comparable to the magnetic pressure, beta of order one, rather than a few percent. High beta is the economic argument. For a given magnetic field you confine far more plasma, and there is no coil threading the plasma at all. An FRC is usually made in one place and used in another. Form the plasma at one end of the machine, and it can be pushed along the axis as a self-contained object. Then squeeze it. Coils at the far end compress the plasma inward and along the axis at once, and because beta is high, that compression heats it efficiently. Formation, translation, compression. That separation is the appeal: the difficult plasma physics happens in one section of the machine and the heating happens in another.

The last idea gives up on holding the plasma steady, and crushes it instead. This is magnetized liner inertial fusion, MagLIF, and it runs on the Z machine at Sandia. Start with a centimeter-scale metal cylinder full of deuterium fuel. Step one: put an axial magnetic field through the fuel before anything else happens, about ten tesla from a pair of coils. The fuel is now magnetized. Step two: a laser pulse of about a kilojoule enters through a window at the top and preheats the fuel to a couple of hundred electron volts, so the implosion does not have to do that work. Step three: twenty million amps flows axially through the liner in a hundred nanoseconds. It is the same J cross B force as the classic pinch, but now it acts on solid metal rather than on plasma. The magnetic pressure driving the liner is the expression we started with: mu nought I squared over eight pi squared r squared. At twenty mega-amps on a centimeter radius, that is megabars. The liner implodes at seventy kilometers a second. The fuel column is compressed by a factor of about twenty five in radius, and the axial field goes with it, because flux is conserved: B times r squared stays put. Ten tesla becomes several thousand tesla, hundreds of times stronger than any steady magnet. That trapped field is the point of the scheme: it holds back electron heat conduction and it keeps the alpha particles inside the fuel. So put the family side by side. Each row is confined by something different, and each one pays a different price. The classic Z-pinch is confined by its own current and destroyed by the sausage and the kink. Sheared flow keeps that same confinement and buys stability with a velocity profile that has to be maintained. The reversed field pinch lets the plasma organize a helical field for itself, and pays for it in resistive transport. The FRC closes the field lines and runs at high beta, and fights global stability and confinement time. MagLIF does not confine in the usual sense at all: an imploding liner holds the fuel together for a hundred nanoseconds, and everything depends on that liner staying smooth. A tokamak is the opposite extreme, with big external coils, low beta, and a steady state. All of them chase the same product: density, times confinement time, times temperature. A pinch pushes the density up and the time down, and a tokamak does the reverse. Which of those trades wins is still, sixty years on, an open question.