 Michael criterion for axisymmetric toroidal-field interchange

For constant-density cylindrical rotation $r\Omega (r){\mathbf{e}}_{\varphi }$ and a purely toroidal magnetic field, an axisymmetric radial displacement gives $b_{\varphi }=(B_{\varphi }/r-B_{\varphi }^{\prime }){\xi }_r$. 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.