= Oblique axisymmetric magnetorotational dispersion relation
{title2=$\frac{k^2}{k_z^2}(s^2+\omega_A^2)^2+\kappa^2(s^2+\omega_A^2)-4\Omega^2\omega_A^2=0$}
In the local incompressible weak-field limit, take an axisymmetric <wavevector> $(k_r,0,k_z)$ and $\omega_A=\mathbf k\cdot\mathbf B/\sqrt{\mu_0\rho}$. Eliminating the pressure projects the radial force by $k_z^2/k^2$. This yields the displayed dispersion relation, where $\kappa$ is the <radial epicyclic frequency>. For a hydrodynamically stable rotation profile with outward-decreasing angular velocity, instability requires $0<\omega_A^2<-(k_z^2/k^2)d\Omega^2/d\ln r$. A purely toroidal field has no axisymmetric local field-line-bending frequency and therefore no ordinary weak-field MRI in this approximation.
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