 Neutral quadrupolar perturbation of a uniform-current toroidal field

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.