Precession and kink motion of long astron layers

A study is made of the stability of the low-frequency precession and kink modes of a long astron particle layer embedded in a dense low-temperature plasma. The precessional motion corresponds to a toroidal mode number n= 1 and the kink motion to n> or approx. = 2. Two types of precession of nonrelativistic ion layers are analyzed in detail. For frequencies ω such that α2Ω2very-much-less-thanvertical-barωvertical-bar2very-much-less-thanΩ2, there is a precession mode with the real part of the frequency ω/sub r/approx. =-(eta/sub e//2) Ω. Here, Ω is the average circulation frequency for the layer ions; αequivalent v/sub A//v-bar/sub theta/very-much-less-than 1 with v/sub A/ the plasma Alfven wave speed, and v-bar/sub theta/ the average azimuthal velocity of the layer ions; and eta/sub e/ is the effective external magnetic field index. This mode shows the well known negative energy type of instability due to dissipation for eta/sub e/< 0, which corresponds to a mirror field. For frequencies vertical-barωvertical-bar2very-much-less-thanα2Ω2, there is a magnetohydrodynamic precession mode with a frequency ωapprox. = +- (eta/sub e//zeta)/sup 1/2/αΩ, where zeta is the loading factor, which is a measure of the layer strength. This mode has a magnetohydrodynamic type of instability for eta/sub e/< 0. For the kink modes, a necessary and sufficient condition for stability is shown to be eta/sub s/< 3, where eta/sub s/ is the self-magnetic field index