= Maximum growth rate of the vertical shear instability
{title2=$s_{\max}=\Omega\sqrt{(\sqrt{1+4q^2}-1)/2}$}
For the unstratified local <vertical shear instability>, maximize $s^2/\Omega^2=(2qR-1)/(1+R^2)$ over $R=k_x/k_z$. Its derivative vanishes when $qR^2-R-q=0$. For $q>0$ the maximum uses $R=(1+\sqrt{1+4q^2})/(2q)$, and substitution gives the displayed expression. For small $q$, $R\sim q^{-1}$ and $s_{\max}=\Omega q(1-q^2/2+O(q^4))$. Changing the sign of the shear changes the preferred sign of $R$ but gives the same maximum magnitude; there is no exponential growth at $q=0$.
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