Theory of the flute instability in mirror machines

Abstract

The flute instability of a low β and neutral plasma contained in a mirror machine is investigated. The particle motion in a real mirror field is fully taken into account by using a Hamiltonian method for the solution of the Vlasov equation. An eigenvalue equation is obtained for the potential of the flute-like perturbation assuming an arbitrary radial dependence of plasma density and of the electron and ion energies. Using a variational method, the eigenvalue equation has been discussed in detail for a plasma with no energy gradients. The stability condition has been calculated taking into account terms up to second order in Larmor radius. Finite Larmor radius stabilization is effective for the higher-order modes and cold electron plasmas. The lowest mode, k = 1, is practically not stabilized by finite Larmor radius effects. The discussion of the eigenvalue equation with finite energy gradients has been limited to the case of a hot electron plasma surrounded by a sheath of cold plasma. If the density of the cold plasma is an order of magnitude higher than the density of the hot plasma, the system is stable at all densities. The stabilization is due to the high dielectric constant of the cold plasma.