Magnetic phase mixing under differential rotation ($\exp [-\eta m^2{f}^{\prime }(r{)}^2{t}^3/3]$)

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.

For azimuthal flow $u_{\varphi }=s({\Omega }_0+\omega s^2)$, a planar magnetic flux mode $s^{m}g(t)e^{im\varphi -im{\Omega }_{0}t-i{s}^{2}f(t)}$ obeys the resistive induction equation if $\dot{f}=m\omega -4i\eta f^2$ and $\dot{g}/g=-4i\eta (m+1)f$. With $f(0)=0$ and $g(0)=1$, their solution is $f=(m\omega /\mu )\tanh \mu t$ and $g=(\cosh \mu t{)}^{-(m+1)}$. Choose $\mathrm{Re}\mu >0$. For positive $\eta ,\omega$, the fixed-position field decays asymptotically at rate $(m+1)\sqrt{2\eta m\omega }$. Differential rotation creates small spatial scales, accelerating magnetic diffusion.