= Neutral quadrupolar perturbation of a uniform-current toroidal field
{title2=$\mathbf b=Cr(\mathbf e_r+i\mathbf e_\phi)e^{2i\phi+ikz}$}
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 $\mathbf B=Jr\mathbf e_\phi$ 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.
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