Maximum growth rate of the vertical shear instability ($s_{\max }=\Omega \sqrt{(\sqrt{1+4q^2}-1)/2}$)

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 $q{R}^2-R-q=0$. 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 $q=0$.