= Exact quadratic-shear magnetic flux solution
{title2=$\mu^2=4i\eta m\omega$}
For azimuthal flow $u_\phi=s(\Omega_0+\omega s^2)$, a planar <magnetic flux> mode $s^mg(t)e^{im\phi-im\Omega_0t-is^2f(t)}$ 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 $\operatorname{Re}\mu>0$. 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>.
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