= Uniform-current rotating magnetohydrodynamic wave dispersion
{title2=$\sigma=i\delta\Omega\pm\sqrt{J^2(2m\delta-m^2)/(4\mu_0\rho)-\delta^2\Omega^2}$}
For an ideal <incompressible flow> rotating about the axis of the field $\mathbf B=(J/2)s\widehat{\boldsymbol\phi}$, use cylindrical <Fourier modes> $e^{\sigma t+im\phi}$. 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 $J=\nabla\times\mathbf B\cdot\widehat{\mathbf z}=\mu_0J_{\rm phys}$ for SI <current density>; the formula is conditional on that boundary eigenrelation.
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