= Solution
The ansatz gives $B_r=A_\phi/r$, $B_\phi=-A_r$, and $B_z=B$. Initially $A=B_0r\sin\phi$ and $B=0$, since $B_r=B_0\cos\phi$ and $B_\phi=-B_0\sin\phi$ are a uniform Cartesian field. In <ideal magnetohydrodynamics>, the transverse flux function is advected, while the axial field is stretched by the axial <velocity> <gradient>. More explicitly, the transverse induction equations are equivalent to $(\partial_t+f\partial_\phi)A=0$; the axial component of $\partial_t\mathbf B+\mathbf u\cdot\nabla\mathbf B=\mathbf B\cdot\nabla\mathbf u$ is $(\partial_t+f\partial_\phi)B=B_rg'$. Substituting the azimuthal mode therefore gives
$$
\partial_t\widehat A+if\widehat A=0,\qquad\partial_t\widehat B+if\widehat B=i\frac{g'}r\widehat A.
$$
The initial coefficients are $\widehat A(r,0)=B_0r$ and $\widehat B(r,0)=0$. Multiplication by $e^{ift}$ integrates both equations, giving $\widehat A=B_0r e^{-ift}$ and $\widehat B=iB_0tg'e^{-ift}$. Thus the <ideal magnetic response to a cylindrical cyclonic event> at its end is
$$
\boxed{A(r,\phi,T)=B_0r\sin(\phi-f(r)T),\qquad B(r,\phi,T)=B_0Tg'(r)\cos(\phi-f(r)T).}
$$
With ideal induction these fields remain frozen after the flow is switched off. The EMF calculated below is the active-flow value at $T^-$; at $T^+$ the <velocity> itself is zero.
In cylindrical components, $\mathbf u\times\mathbf B=(rfB+gA_r)\widehat{\mathbf r}+(gA_\phi/r)\widehat{\boldsymbol\phi}-fA_\phi\widehat{\mathbf z}$. Put $\psi=\phi-fT$. Since $A_r=B_0(\sin\psi-rTf'\cos\psi)$, its Cartesian $x$-component reduces to
$$
(\mathbf u\times\mathbf B)_x=B_0\left[rT(fg'-gf')\cos\phi\cos\psi-g\sin(fT)\right].
$$
Use $\int_0^{2\pi}\cos\phi\cos(\phi-fT)d\phi=\pi\cos(fT)$ and the area element $r\,dr\,d\phi$. The <integrated electromotive response of a finite cyclonic event> is
$$
\boxed{\mathcal E(T)=\pi B_0\int_0^\infty\left[Tr^2(fg'-gf')\cos(fT)-2rg\sin(fT)\right]dr.}
$$
When the boundary term vanishes, the last two terms combine as $-g\,\partial_r[r^2\sin(fT)]$, giving the useful equivalent <integral>
$$
\mathcal E(T)=\pi B_0\int_0^\infty r^2g'[\sin(fT)+Tf\cos(fT)]dr.
$$
For the compactly supported profile, $fg'-gf'=0$ on $0<r<a$. Substitution of $v=a^2-r^2$ into the first <integral> gives
$$
\mathcal E(T)=-\pi B_0\int_0^{a^2}v\sin(vT)dv=\boxed{\pi B_0\left[\frac{a^2\cos(a^2T)}T-\frac{\sin(a^2T)}{T^2}\right].}
$$
At zero the continuous limiting value is zero, and $\mathcal E(T)=-\pi B_0a^6T/3+O(T^3)$. For $B_0>0$ it begins negative; the nonzero zeros satisfy $\tan(a^2T)=a^2T$, the first at $a^2T\simeq4.4934$. It then alternates sign, with an asymptotic $1/T$ envelope. The requested sketch uses $x=a^2T$ and scales the EMF by $\pi B_0a^4$:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-74-cyclonic-emf.png]
{title=Sign changes of the ideal cyclonic-event electromotive response}
This sign alternation is possible even though the flow has fixed-sign <kinetic helicity>. Indeed, its <vorticity> components are $(0,-g',2f+rf')$, so $\mathbf u\cdot\boldsymbol\omega=2fg+r(gf'-fg')$, which reduces to $2f^2\geq0$ for this profile. The finite-duration response retains winding and phase information; it cannot be assigned a universal sign from the instantaneous helicity alone.
A small <magnetic diffusivity> replaces the ideal mode equations by
$$
\partial_t\widehat A+if\widehat A=\eta D_1\widehat A,\qquad\partial_t\widehat B+if\widehat B=i\frac{g'}r\widehat A+\eta D_1\widehat B,\quad D_1=\partial_r^2+r^{-1}\partial_r-r^{-2}.
$$
<Differential rotation> creates radial phase <gradients> of order $Tf'$, so the resistive damping rate grows like $\eta T^2f'^2$. Locally, <magnetic phase mixing under differential rotation> gives an accumulated damping factor of order $e^{-\eta f'^2T^3/3}$. Thus the ideal result is a good early-time approximation but <diffusion> eventually limits winding and stretching, smooths the sharp field gradients at $r=a$, and modifies the later EMF oscillations. Small <diffusion> does not immediately remove all finite-time sign reversals, and a long-time resistive response requires the diffusive boundary-value problem rather than the ideal formula. After the event, the remaining distorted field relaxes by <diffusion> toward the imposed uniform background.
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