Magnetic compression of ion rings

The ion ring is described by the total number of ions, N; its major and minor radii, R and r; and the ion and electron ring currents Ii and Ie. It is assumed that the toroidal structure maintains itself and that the aspect ratio R/r is large. The evolution of these variables for a prescribed external field is then governed by a set of ordinary differential equations. If the ions alone produce the ring current and if the external field varies slowly, the equations can be integrated analytically to yield simple transcendental relations which determine the radius R, the aspect ratio R/r, the field reversal factor, and the kinetic and thermal energies as functions of the external field B. For a few cases these relations are given in graphic form for normalized, dimensionless variables. Both the field reversal factor and the aspect ratio increase during compression but, it is shown that these quantities approach constant values even when B increases without limit.