= Magnetic buoyancy energy criterion
For the <magnetized isothermal atmosphere>, set $a=1/(2H)$, $s=v_s^2>0$, $b=v_a^2>0$ and $\Delta=ik_x\xi_x+ik_y\xi_y+(ik_z+a)\xi_z$. After weighting the displacement by $e^{z/(2H)}$, the restoring <Hermitian matrix> $\mathsf K$ satisfies
$$
\boldsymbol\xi^\dagger\mathsf K\boldsymbol\xi=
s\left|\Delta-\frac g{s}\xi_z\right|^2+
b\left|ik_x\xi_x+(ik_z+a)\xi_z\right|^2+
bk_y^2(|\xi_x|^2+|\xi_z|^2)+
\left(\frac gH-\frac{g^2}{s}\right)|\xi_z|^2.
$$
Thus $gH\leq s$ makes the form nonnegative for every <wavevector>. If $gH>s$, choose $k_z=0$, $k_x\ne0$, $\xi_z=1$, $\xi_x=-a/(ik_x)$ and $\xi_y=g/(sik_y)$. 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 $\beta(\gamma-1)<1$.
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