= Solution
A field in a circulation plane has components crossing circular streamlines. <Magnetic flux freezing> carries those lines with <fluid elements> at different <angular speeds>: the lines become spirals, their azimuthal components are stretched, and neighboring turns acquire increasingly small separations. This is a nonaxisymmetric winding and phase-mixing problem.
For illustration, in the ideal limit take the initial horizontal field along $x$ and fix a horizontal plane. With $\chi=\phi-\omega(s,z)t$, the induction equations give
$$
B_s=B_0\cos\chi,\qquad B_\phi=-B_0\sin\chi+s\,t\,\partial_s\omega\,B_0\cos\chi,\qquad B_z=0.
$$
The first term describes advection around a circle; the term proportional to $t\partial_s\omega$ is the stretching by <differential rotation>. Spatial variation of $\omega$ also produces phase gradients proportional to $t\nabla\omega$. A rigidly rotating region merely turns the horizontal field, whereas a shearing region progressively winds it up.
Even small <magnetic diffusivity> becomes important once these gradients are large. Locally a phase mode of azimuthal order $m$ has damping rate of order $\eta m^2|\nabla\omega|^2t^2$, leading to attenuation of order $\exp[-\eta m^2|\nabla\omega|^2t^3/3]$. The associated <magnetic phase mixing under differential rotation> time is
$$
t_{\mathrm{mix}}\sim[\eta m^2|\nabla\omega|^2]^{-1/3}.
$$
For a horizontal uniform seed, $m=1$; if $|\nabla\omega|\sim\Omega/\ell$, this is $\Omega^{-1}R_m^{1/3}$, much shorter than the ordinary diffusion time at large $R_m$. Resistive smoothing cancels the closely spaced oppositely directed field and, in persistent circulation regions with an imposed surrounding field, drives <flux expulsion> from the interiors toward their boundaries. The detailed final distribution depends on the geometry and boundary forcing. Winding alone does not supply a self-sustaining poloidal–toroidal regeneration loop, and magnetic feedback can again limit the kinematic process.
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