= Convective instability of a square-root exponential atmospheric profile
For a homogeneous <ideal gas> of constant <specific-heat ratio> $\gamma>1$, the <Schwarzschild criterion> applied to the <square-root exponential atmospheric profile> gives
$$
\nabla=\frac{d\log T}{d\log P}=\frac{2\sqrt{T-T_0}}{\alpha T}>\nabla_{\rm ad}=\frac{\gamma-1}{\gamma}.
$$
For $T_0>0$ and $\alpha>0$, put $s=\sqrt{T-T_0}$ and $A=\nabla_{\rm ad}$. The <quadratic inequality> $A\alpha s^2-2s+A\alpha T_0<0$ has solutions precisely when $A\alpha\sqrt{T_0}<1$, between $s_\pm=[1\pm\sqrt{1-(A\alpha)^2T_0}]/(A\alpha)$. This is a local instability of the imposed <temperature gradient>; actual <convection> adjusts that gradient and the prescribed profile need not survive.
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