Gyrokinetic theory of the screw pinch

Abstract

The gyrokinetic differential equation for waves propagating in a hot collisionless current-carrying plasma is derived in cylindrical geometry. It is shown that the averaging of the wave electric field over the ion Larmor circle leads to a transcendental differential equation (d.e.) of infinite order in the radial derivative. This reduces to a d.e. of sixth order when the scale length of the plasma inhomogeneity Ln is greater than the ion gyroradius ρi by a factor (M/m)1/2, where M and m are, respectively, the ion and electron mass. This sixth-order d.e. describes the properties of the two (compressional and torsional) Alfvén modes and the ion acoustic mode. When Ln <(M/m)1/2ρi, the plasma can only support modes of the magnetokinetic (short wavelength) type. In the absence of a finite Larmor radius (FLR) effect, shear, and equilibrium current, we find that the correct equation to start with is the Hain and Lust [Z. Naturforsch. A 13, 936 (1958)] d.e. of second order that is singular at the Alfvén resonance layer (ARL). The ARL behaves in that case like a Budden absorption layer that traps the global Alfvén eigenmodes (GAEM) inside the plasma cavity where they are damped by transit time magnetic pumping (TTMP). The logarithmic singularity does not disappear with the introduction of the FLR effect in the Hain and Lust d.e., but only with the TTMP damping term. There is no mode conversion between the fast magnetosonic mode and the shear or magnetokinetic mode at the ARL or anywhere in the plasma. In the presence of shear and equilibrium current, the correct equation to use is a d.e. of fourth (or greater) order whose solutions descibe the shear Alfvén mode in the long wavelength limit or the magnetokinetic mode at shorter wavelengths. In the true magnetohydrodynamic (MHD) limit, both modes become degenerate. It is shown that the slow Alfvén eigenmodes are (almost) completely decoupled from the fast magnetosonic wave and therefore the growth rates show no dependence on the poloidal number m. The quicker the current density drops from the cylinder axis, the more unstable the modes are. This fourth-order d.e. is singular at the hybrid resonances, not at the ARL. It is therefore found that no normal mode solution can exist for linear shear Alfvén perturbations in the full FLR limit.