= Maximum growth rate of radially stratified magnetorotational instability
{title2=$\sigma_{\max}=(3\Omega^2-N_r^2)/(4\Omega)$}
For $-5\Omega^2<N_r^2<3\Omega^2$ and a continuum of vertical <wavenumbers>, the growing branch of the <radially stratified magnetorotational dispersion relation> is maximized at $k^2v_A^2=\Omega^2-(N_r^2+\Omega^2)^2/(16\Omega^2)$, with $\sigma_{\max}=(3\Omega^2-N_r^2)/(4\Omega)$. For $N_r^2\leq-5\Omega^2$ the optimum moves to the $k\to0$ boundary and the limiting rate is $\sqrt{-N_r^2-\Omega^2}$. For $N_r^2\geq3\Omega^2$ there is no exponential instability. Boundaries that discretize the allowed <wavenumbers> can alter the actual maximum.
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