= Finite-thickness magnetorotational instability criterion
If the horizontal <magnetic field> perturbations vanish at $z=\pm H$, their allowed <vertical wavenumbers> are $k_m=m\pi/(2H)$, $m=1,2,\ldots$. The lowest <wavenumber> is unstable exactly when
$$
0<\frac{\pi^2B_z^2}{8q\mu_0\rho H^2\Omega^2}<1.
$$
If that magnetic mode is stable, all higher magnetic modes are stable as well. A weak field admits longer unstable wavelengths that fit inside the slab; sufficiently strong <magnetic tension> stabilizes every allowed wavelength.
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