 Ideal magnetic response to a cylindrical cyclonic event

For an axisymmetric velocity $\mathbf{u}=(0,rf(r),g(r))$ independent of $z$, write $\mathbf{B}=B_z\hat{\mathbf{z}}+\nabla \times (A\hat{\mathbf{z}})$. Ideal induction becomesThe first equation expresses magnetic flux freezing of the transverse flux function, while the second is axial stretching $B_r{g}^{\prime }$. Starting from a uniform field $B_0\hat{\mathbf{x}}$ gives $A(r,\varphi ,t)=B_{0}r\sin (\varphi -f(r)t)$ and $B_z(r,\varphi ,t)=B_{0}t{g}^{\prime }(r)\cos (\varphi -f(r)t)$. In complex azimuthal-mode notation, $\hat{A}=B_{0}r{e}^{-ift}$ and ${\hat{B}}_z=i{B}_{0}t{g}^{\prime }{e}^{-ift}$, obtained by elementary integrating factors. The transverse field has $B_r=r^{-1}{A}_{\varphi }$ and $B_{\varphi }=-A_r$, preserving zero divergence.