Uniform-current toroidal-field curl reduction ($({\omega }^2-m^2{\tilde{J}}^2)\nabla \times \mathbf{b}=-2mk{\tilde{J}}^2\mathbf{b}$)

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 $3k^2>{\alpha }^2$.

The displayed solenoidal vector field has $\nabla \times \mathbf{b}=k\mathbf{b}$, ${\nabla }^2\mathbf{b}=-k^2\mathbf{b}$ and ${\mathbf{e}}_z\times \mathbf{b}=-i\mathbf{b}$. Thus the non-gradient part of its linearized Lorentz force about $\mathbf{B}=Jr{\mathbf{e}}_{\varphi }$ vanishes. Choosing $p_1=-\mathbf{B}\cdot \mathbf{b}/{\mu }_0$ cancels the magnetic-pressure gradient, and $\mathbf{u}=0$, $\omega =0$ is a neutral perturbation. It is regular at the axis and realizes the endpoint $m=2$, $\alpha =0$. A boundary condition can exclude this growing-in-radius field, but without such a condition strict positivity at the endpoint cannot be asserted.