The flute instability of low-β axially symmetric plasma confinement devices is studied. From the set of equations for the low-density case, without an equilibrium space-charge potential, and with a near δ-function distribution of energies appropriate to neutral injection devices, a variational principle is derived for the three-dimensional mirror machine geometry. In general, for small electron temperatures the variational principle furnishes a dispersion relation which for given trial functions is cubic. Two roots of the cubic furnish the stability criterion. The third root describes a density-independent mode of oscillation at a frequency equal to the ion precessional frequency, and is identified with the density-independent mode observed in the ALICE machine. The order of magnitude of the critical plasma density for flute instability is the same as that obtained by previous authors using the drift approximation, but by use of the variational principle the present author is able to obtain a more quantitative stability criterion, in particular an estimate of the effect of the finite plasma length. The critical plasma density is presented as a function of the ratio of the length to the radius of the plasma.
Non-linear theory of convective instability