= Magnetic phase mixing under differential rotation
{title2=$\exp[-\eta m^2f'(r)^2t^3/3]$}
A nonuniform angular velocity $f(r)$ winds an azimuthal magnetic mode into phase $m(\phi-f(r)t)$. Its radial phase gradient grows as $-mt f'(r)$, creating small scales even when its ideal amplitude does not decay. The corresponding leading resistive damping rate is $\eta m^2t^2f'(r)^2$, so a local short-wavelength estimate gives damping factor $\exp[-\eta m^2f'(r)^2t^3/3]$. Hence <magnetic diffusion> can become important at time $[\eta m^2f'^2]^{-1/3}$ although it is initially small. This is a local phase-mixing estimate, not a globally exact solution in a finite radial domain: boundary forcing, field stretching and spatial variation of $f'$ must also be retained there.
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