Precession and kink motion of long astron layers

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Abstract
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≳ 2. Two types of precession of nonrelativistic ion layers are analyzed in detail. For frequencies ω such that α2Ω2≪‖ω‖2≪Ω2, there is a precession mode with the real part of the frequency ωr≈−(ηe/2) Ω. Here, Ω is the average circulation frequency for the layer ions; α≡ vA/v̄ϑ≪ 1 with vA the plasma Alfvén wave speed, and v̄ϑ the average azimuthal velocity of the layer ions; and ηe is the effective external magnetic field index. This mode shows the well known negative energy type of instability due to dissipation for ηe< 0, which corresponds to a mirror field. For frequencies ‖ω‖2≪α2Ω2, there is a magnetohydrodynamic precession mode with a frequency ω≈± (ηe/ζ)1/2αΩ, where ζ is the loading factor, which is a measure of the layer strength. This mode has a magnetohydrodynamic type of instability for ηe< 0. For the kink modes, a necessary and sufficient condition for stability is shown to be ηs< 3, where ηs is the self‐magnetic field index.