Michael criterion for axisymmetric toroidal-field interchange ($\mathcal{D}={\kappa }^2-\frac{r}{{\mu }_0\rho }\frac{d}{dr}{\left(\frac{B_{\varphi }}{r}\right)}^2$)

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.

For $B_{\varphi }\propto r^p$, 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

For rigid walls $a<r<b$, a nonzero vertical wavenumber $k$ and radial Lagrangian fluid displacement $\xi$, set $y=r\xi$ and $\mathcal{D}={\kappa }^2-r(d/dr)(v_A^2/r^2)$. Axisymmetric modes satisfyThe operator on the left is a positive self-adjoint operator. Its Rayleigh quotient is ${\omega }^2=k^2\int \mathcal{D}|y{|}^2/rdr/\int (|y^{\prime }{|}^2+k^2|y{|}^2)/rdr$. All mode squares are real; a growing mode exists exactly when the continuous coefficient $\mathcal{D}$ is negative somewhere. Sufficiency follows by a test function supported in that negative interval and the compact operator and self-adjoint operator eigenproblem obtained by conjugating the multiplication operator with the inverse square root of the positive differential operator.

// Target: astrophysical-fluid-dynamics.bigb