= Solution
Work with the complex representation and take its real part to obtain the physical <magnetic field>. For constant <magnetic diffusivity> and the prescribed circular <velocity> $\mathbf u=s\Omega(s)\widehat{\boldsymbol\phi}$, the <resistive induction equation> and $\mathbf B=\nabla\times(A\widehat{\mathbf z})$ give
$$
B_s=\frac1s\partial_\phi A,\qquad B_\phi=-\partial_s A,\qquad
\partial_tA+\Omega\partial_\phi A=\eta\left(\partial_s^2A+\frac1s\partial_sA+\frac1{s^2}\partial_\phi^2A\right).
$$
To see the last equation directly, $\mathbf u\times\mathbf B=-s\Omega B_s\widehat{\mathbf z}=-\Omega A_\phi\widehat{\mathbf z}$. Taking its <curl> reproduces the induction equation if $A_t=-\Omega A_\phi+\eta\nabla^2A$, up to a time-dependent additive constant removable by the <gauge transformation>. This is the scalar form of advection and <magnetic diffusion> in <cylindrical coordinates>.
Put $\psi=\phi-\Omega(s)t$. The <material derivative> of $ae^{i\psi}$ is $a_t e^{i\psi}$, because $\psi_t+\Omega\psi_\phi=0$. A radial <derivative> instead acts on both amplitude and phase:
$$
\partial_sA=(a_s-it\Omega'a)e^{i\psi},\qquad
\partial_s^2A=(a_{ss}-2it\Omega'a_s-it\Omega''a-t^2(\Omega')^2a)e^{i\psi}.
$$
Also $A_{\phi\phi}=-ae^{i\psi}$. Substitution gives the <shearing-coordinate magnetic flux equation>, here for azimuthal mode number one:
$$
\boxed{a_t=\eta\left[a_{ss}+\frac{a_s}{s}-\frac a{s^2}-(\Omega')^2t^2a-2it\Omega'a_s-\frac{it\Omega'}s a-it\Omega''a\right].}
$$
Writing it with $a_t$ on the left is important when considering zero <magnetic diffusivity>: one must not literally divide by $\eta=0$. \b[In the ideal limit, $a_t=0$], so $A(s,\phi,t)=A(s,\phi-\Omega(s)t,0)$. This is <magnetic flux freezing>: the scalar flux pattern is carried with each rotating fluid ring. It does not make the <magnetic field> stationary. Indeed,
$$
B_s=\frac{ia(s,0)}s e^{i\psi},\qquad B_\phi=[-a_s(s,0)+it\Omega'a(s,0)]e^{i\psi},
$$
so <differential rotation> winds the field and can make its azimuthal component grow linearly in time. Rigid rotation, for which $\Omega'=0$, only rotates the pattern.
For the specified interior <angular velocity>, $\Omega'=-\Omega_0/d$ and $\Omega''=0$ away from the circulation edge. Let $T=\Omega_0t$, $x=s/d$ and $\mathrm{Rm}=\Omega_0d^2/\eta$. The amplitude equation becomes
$$
\partial_Ta=\frac1{\mathrm{Rm}}\left[a_{xx}+\frac{a_x}{x}-\frac a{x^2}-T^2a+2iTa_x+\frac{iT}{x}a\right].
$$
For a smooth initial amplitude on the cell scale, at $x$ bounded away from zero and the edge, the $-T^2a$ term dominates when $T\gg1$. Its accumulated damping is order one at $T=O(\mathrm{Rm}^{1/3})$, whereas the integrated terms proportional to $T$ are only $O(\mathrm{Rm}^{-1/3})$ and the terms independent of $T$ are $O(\mathrm{Rm}^{-2/3})$. The resulting <cubic-time resistive damping in a differentially rotating cell> is
$$
a(s,t)\simeq a(s,0)\exp\left[-\frac{\eta\Omega_0^2t^3}{3d^2}\right],\qquad
\boxed{\tau_c=\left(\frac{3d^2}{\eta\Omega_0^2}\right)^{1/3},\quad\Omega_0\tau_c=(3\mathrm{Rm})^{1/3}.}
$$
The <magnetic field> has this leading <exponential decay> envelope. Its components also contain the algebraic winding factor visible in $B_\phi$, so the claim concerns the cubic exponent, not an exact amplitude with no prefactor. Equivalently the radial <wavenumber> grows as $|k_s|\simeq|\Omega'|t$ and the integrated resistive damping is $\int_0^t\eta k_s^2dt=\eta(\Omega')^2t^3/3$.
This is <phase mixing in magnetic flux expulsion>. The circulation creates ever finer gradients, allowing weak <magnetic diffusion> to act much sooner than its unsheared time $t_\eta=d^2/\eta$:
$$
\Omega_0^{-1}\ll\tau_c\ll t_\eta,\qquad \frac{\tau_c}{t_\eta}=3^{1/3}\mathrm{Rm}^{-2/3}.
$$
\b[Rapid winding followed by resistive smoothing expels the imposed field from the cell interior.] The maintained exterior <magnetic field> is redistributed into a circulation-edge <boundary layer>. The result is an interior high-<magnetic Reynolds number> transient, not a uniform assertion at the axis and the nonsmooth edge, nor the exact $t\to\infty$ solution with a continuously imposed far field. The diffusion length at $\tau_c$ is $\sqrt{\eta\tau_c}/d=3^{1/6}\mathrm{Rm}^{-1/3}$, identifying where a local bulk approximation can fail. The linear profile also needs smoothing at a regular rotation axis and at $s=d$ if used as a fully smooth global flow.
In photospheric <magnetoconvection>, circulating and diverging <convection> transports weak <magnetic flux> towards converging downflow regions. <Solar granulation> concentrates field in <intergranular lanes>, while larger-scale <supergranulation> helps form the <solar magnetic network>. <Flux expulsion> explains why a highly conducting <photosphere> can have relatively weak field in cell interiors and intermittent concentrations at their edges: high <electrical conductivity> delays diffusion on the original scale, but does not prevent diffusion on the fine scales created by advection. A long-lived circulation must last long enough for the expulsion time; rapidly changing cells need not reach this idealized state. Once the field becomes strong, the <Lorentz force> modifies the <convection>, so this prescribed-flow calculation alone neither determines the final field strength nor describes strong-field <sunspots>.
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