= Toroidal interchange field threshold in a thin Keplerian disk
For $B_\phi\propto r^p$, constant <density> and Keplerian $\kappa^2=\Omega^2$, the <Michael criterion for axisymmetric toroidal-field interchange> becomes $\Omega^2-2(p-1)v_A^2/r^2$. Instability therefore requires $p>1$ and $v_A^2>r^2\Omega^2/[2(p-1)]$. For a field varying on the radial scale $r$, this is an orbital-speed field, much stronger than a gas-pressure-scale field with $v_A\lesssim c_s\sim H\Omega$ in a <thin disk>. Steep radial field gradients require the exact derivative criterion rather than this scale estimate. The instability is powered by the toroidal field distribution and is distinct from weak vertical-field <magnetorotational instability>.
// Target: fluid-mechanics.bigb
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