 Uniform-current rotating magnetohydrodynamic wave dispersion

For an ideal incompressible flow rotating about the axis of the field $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, use cylindrical Fourier modes $e^{\sigma t+im\varphi }$. The azimuthal derivative of a cylindrical vector Fourier mode reduces induction to $\sigma \mathbf{b}=imJ\mathbf{u}/2$. If the boundary eigenproblem gives $\lambda =i\delta \nu$, with real $|\delta |<1$, eliminating the perturbation gives the displayed dispersion relation. Here $J=\nabla \times \mathbf{B}\cdot \hat{\mathbf{z}}={\mu }_0{J}_{\mathrm{phys}}$ for SI current density; the formula is conditional on that boundary eigenrelation.