= Fastest discrete mode of a stratified magnetorotational instability
{title2=$s^2/\Omega^2=[-(1+2a)+\sqrt{1+16a}]/2$}
The growing root of the <ideal magnetorotational dispersion relation> is maximized continuously at $a=v_A^2k^2/\Omega^2=15/16$, with growth $3\Omega/4$. Discrete vertical modes instead have $a_n=2n(n+1)/\beta$. The fastest one is found by comparing admissible integers on either side of the continuous maximum, rather than rounding a vertical wavenumber blindly. At $\beta=24$, $n=3$ gives $s=\Omega\sqrt{(\sqrt{17}-3)/2}$.
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