Self-similar magnetic mode in a stagnation flow ($B_z=f(t)b(r/g(t))\cos (m\varphi )$)

For magnetic concentration by an incompressible stagnation flow, one possible normalization gives $(g^2{)}^{\prime }=2\eta -2\omega g^2$, $f^{\prime }/f=2\omega -\lambda \eta /g^2$, and $b^{\prime \prime }+(q^{-1}+q)b^{\prime }+(\lambda -m^2/q^2)b=0$. The mode parameter $\lambda$ is not fixed by decay at infinity alone. The regular axisymmetric Gaussian has $\lambda =2$, $b=e^{-q^2/2}$ and conserved axial magnetic flux, with energy proportional to $g^{-2}$. The radial equation also has regular algebraic-tail solutions.