 Magnetic phase mixing under differential rotation

A nonuniform angular velocity $f(r)$ winds an azimuthal magnetic mode into phase $m(\varphi -f(r)t)$. Its radial phase gradient grows as $-mt{f}^{\prime }(r)$, creating small scales even when its ideal amplitude does not decay. The corresponding leading resistive damping rate is $\eta m^2{t}^2{f}^{\prime }(r{)}^2$, so a local short-wavelength estimate gives damping factor $\exp [-\eta m^2{f}^{\prime }(r{)}^2{t}^3/3]$. Hence magnetic diffusion can become important at time $[\eta m^2{f}^{\prime 2}{]}^{-1/3}$ 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 $f^{\prime }$ must also be retained there.