= Shearing-coordinate magnetic flux equation
{title2=$\partial_ta=\eta[(\partial_s-imt\Omega')^2+s^{-1}(\partial_s-imt\Omega')-m^2/s^2]a$}
For a planar <magnetic field> written as $\mathbf B=\nabla\times(A\widehat{\mathbf z})$ and a circular <velocity> $\mathbf u=s\Omega(s)\widehat{\boldsymbol\phi}$, the <resistive induction equation> reduces, after choosing the additive <gauge transformation>, to $\partial_tA+\Omega\partial_\phi A=\eta\nabla^2A$. A <Fourier mode> $A=a(s,t)e^{im[\phi-\Omega(s)t]}$ therefore obeys the equation in the title. Indeed, each radial <derivative> acting on the exponential replaces $\partial_s$ by $\partial_s-imt\Omega'$, while the azimuthal <Laplacian> contributes $-m^2/s^2$. At zero <magnetic diffusivity>, $a$ is constant in time, but the physical <magnetic field> can still be amplified algebraically by <differential rotation>.
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