= Phase mixing in magnetic flux expulsion
{title2=$t_{\rm exp}\sim(L/U)\mathrm{Rm}^{1/3}$}
Differential circulation of a weak <magnetic field> winds its flux function into progressively smaller radial scales. In a smooth eddy with angular-velocity gradient of order $U/L^2$, its radial wavenumber grows as $k_r\sim Ut/L^2$. Resistive damping becomes substantial when $\eta\int_0^t k_r^2dt\sim1$, giving $t\sim(L^4/(\eta U^2))^{1/3}=(L/U)\mathrm{Rm}^{1/3}$. This estimate presumes differential rotation and negligible magnetic feedback; a rigidly rotating eddy does not create this phase mixing. Finite <magnetic diffusivity> then permits weak interior field and boundary concentration, unlike exact ideal flux freezing.
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