= Uniform-current toroidal-field curl reduction
{title2=$(\omega^2-m^2\widetilde J^2)\nabla\times\mathbf b=-2mk\widetilde J^2\mathbf b$}
For constant-density incompressible <ideal magnetohydrodynamics> about $\mathbf B=Jr\mathbf e_\phi$, the cylindrical basis contributions make the linearized <Lorentz force> equal to $imJ\mathbf b+2J\mathbf e_z\times\mathbf b$, apart from a <magnetic pressure> gradient. The induction equation reduces to $i\omega\mathbf b=imJ\mathbf u$. Taking the curl after eliminating velocity gives the displayed relation, with $\widetilde J=J/\sqrt{\mu_0\rho}$. A second curl gives a vector Helmholtz equation and squared frequencies $\omega^2=\widetilde J^2[m^2\pm2|mk|/(\alpha^2+k^2)^{1/2}]$. The lower $|m|=1$ branch grows when $3k^2>\alpha^2$.
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