 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 $\beta (\gamma -1)<1$.