A report on a static ideal MHD equilibrium (no background flow

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Plasma Physics · Fluid and magnetohydrodynamic description

Plasma Physics, Ch Whether it stays held: the energy principle, interchange, kink and tearing

Problem

A report on a static ideal MHD equilibrium (no background flow, no resistivity) contains this claim: "The normal-mode analysis gives

\omega = (2.0 + 1.5i)\times10^{4}\,\mathrm{s^{-1}}

for one eigenmode. This means the displacement oscillates at angular frequency

2.0\times10^{4}\,\mathrm{rad\,s^{-1}}

while simultaneously growing at rate

1.5\times10^{4}\,\mathrm{s^{-1}}

, so the mode is both a wave and an instability at once." ((a)) Compute

\omega^2

for the quoted

\omega

and state whether it is real. ((b)) Explain, from the self-adjointness of the ideal MHD force operator

\mathbf{F}(\boldsymbol{\xi})

, why the claim cannot describe a mode of a static ideal equilibrium, and say what a correct eigenvalue

\omega^2

must look like instead.