Uniform-current rotating magnetohydrodynamic wave dispersion ($\sigma =i\delta \Omega \pm \sqrt{J^2(2m\delta -m^2)/(4{\mu }_0\rho )-{\delta }^2{\Omega }^2}$)

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.

In the uniform-current rotating magnetohydrodynamic wave dispersion, exponential growth requires $J^2(2m\delta -m^2)/(4{\mu }_0\rho )>{\delta }^2{\Omega }^2$. For integer $m$ and $|\delta |<1$ this is possible only for $|m|=1$ and $m\delta >1/2$. Positive and negative $m$ branches are related by complex conjugation. The equality case is a repeated frequency and requires separate analysis of possible algebraic growth.