Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 1 / iv / Solution

Initially $P=-f{\varphi }_0\mathrm{sgn}(x)/c^2$. In the final steady state,Equating its potential vorticity to the initial value givesThis 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.