 Uniform-current toroidal-field curl reduction

For constant-density incompressible ideal magnetohydrodynamics about $\mathbf{B}=Jr{\mathbf{e}}_{\varphi }$, 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 $\tilde{J}=J/\sqrt{{\mu }_0\rho }$. A second curl gives a vector Helmholtz equation and squared frequencies ${\omega }^2={\tilde{J}}^2[m^2\pm 2|mk|/({\alpha }^2+k^2{)}^{1/2}]$. The lower $|m|=1$ branch grows when $3k^2>{\alpha }^2$.