Magnetic buoyancy instability

A magnetic buoyancy instability is a gravitational instability of a magnetized stratified fluid in which displacements of the gas and magnetic field can release Newtonian gravitational potential energy. Magnetic pressure supports part of the weight. Both interchange displacements and undular displacements can enter this broad category; Parker instability is the undular mechanism involving drainage along bent magnetic field lines.

Parker instability is an undular magnetic buoyancy instability of a gravitationally stratified atmosphere supported partly by magnetic pressure. A bent horizontal magnetic field can permit gas to drain along its magnetic field lines, leaving buoyant rising regions. Magnetic tension resists bending, so wavelengths and boundary conditions affect the instability threshold. For a magnetized isothermal atmosphere, the unrestricted-wavevector threshold is given by the magnetic buoyancy energy criterion.

For the magnetized isothermal atmosphere, set $a=1/(2H)$, $s=v_s^2>0$, $b=v_a^2>0$ and $\Delta =i{k}_x{\xi }_x+i{k}_y{\xi }_y+(i{k}_z+a){\xi }_z$. After weighting the displacement by $e^{z/(2H)}$, the restoring Hermitian matrix $\mathsf{K}$ satisfiesThus $gH\le s$ makes the form nonnegative for every wavevector. If $gH>s$, choose $k_z=0$, $k_x\ne 0$, ${\xi }_z=1$, ${\xi }_x=-a/(i{k}_x)$ and ${\xi }_y=g/(si{k}_y)$. The squares vanish, and sufficiently small nonzero $k_y$ gives a negative form. The Rayleigh quotient proves instability when these wavelengths are admissible. Since $s=\gamma c_s^2$ and $\beta =2c_s^2/b$, the condition is precisely $\beta (\gamma -1)<1$.