Exact quadratic-shear magnetic flux solution (${\mu }^2=4i\eta m\omega$)

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.