= Solution
Initially $P=-f\phi_0\operatorname{sgn}(x)/c^2$. In the final steady state,
$$
u=-\frac1f\Phi_y,
\qquad v=\frac1f\Phi_x.
$$
Equating its potential vorticity to the initial value gives
$$
\boxed{\Phi_{xx}+\Phi_{yy}-\frac{f^2}{c^2}\Phi
=-\frac{f^2}{c^2}\phi_0\operatorname{sgn}(x).}
$$
This is <geostrophic adjustment>: inertia-gravity waves remove the unbalanced part. Their group speed is at most $c$, so at any finite time only a region whose distance from the initial jump is $O(ct)$ can have reached the steady state.
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