Fundamentals of Plasma Physics 2025, 35 - Stability of Static MHD
Presupposing knowledge of Rayleigh-Taylor Stable systems, a magnetofluid (as in satisfies MHD) when a magnetofluid is supported by atmospheric pressure, and a vertical magnetic field gradient balances the downwards gravitational force has an equilibrium magnetic field of the general form B0 = Bx0(y) + Bz0(y). If assumed incompressible, one can adapt a linearized equation of motion.
To identify B1y get the full vector B1 through the curl of Ohm's law. Solving for γ2
At kB0 = 0, then density gradient is positive, leading to instability. At finite kB0 the effect of the destabilizing density gradient reducing the growth rate.
A system initially in equilibrium, but subject to small perturbations by thermal noise will feature the perturbation affect all the various dependent variables in a way consistent with the relevant equations and boundary conditions. The perturbations can be read as a low-level excitation of some allowed normal mode of the system. Each mode has its own pattern for displacing the magnetofluid volume elements from their equilibrium positions. The displacement is noted as a vector function of position, applied to a fluid volume element. The energy content of a magnetofluid can be obtained from the ideal MHD equation, assuming the motions happen sufficiently fast to qualify as adiabatic, and slow enough to retain isotropic pressure. Pressure and density may vanish at the magnetofluid surface, even though the Poynting flux E × B can be finite. If the tangential electric field vanishes at the surface, the normal Poynting flux vanishes. Energy in this setup may flow between the magnetofluid and vacuum region. The internal energy for static equilibrium and T0 = 0 requires all internal energy to be in the form of stored potential energy. The displacement due to noise is ξ = ∫0tU1dt' and kinetic energy associated with the mode is of order ϵ2
