= Michael criterion for axisymmetric toroidal-field interchange
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{title2=$\mathcal D=\kappa^2-\frac r{\mu_0\rho}\frac{d}{dr}\left(\frac{B_\phi}{r}\right)^2$}
For constant-density cylindrical rotation $r\Omega(r)\mathbf e_\phi$ and a purely toroidal <magnetic field>, an axisymmetric radial displacement gives $b_\phi=(B_\phi/r-B_\phi')\xi_r$. Combining its curvature force with conservation of angular momentum gives the restoring coefficient $\mathcal D$ above. A local incompressible radial-vertical mode has $\omega^2=(k_z^2/k^2)\mathcal D$. Positive $\mathcal D$ gives stability in this ideal interchange setting; negative $\mathcal D$ permits growth. This is distinct from weak-field <magnetorotational instability>: a steeply increasing toroidal field can release current energy even without differential rotation.
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