 Solution

In a steady state the time derivative of quasi-geostrophic potential vorticity vanishes, leaving the local balanceChoose the undisturbed state immediately east of the compact forcing as the integration condition. The solution is thenThe discontinuity in this ideal expression reflects the discontinuous prescribed forcing at $z=-D$; a vertically smooth forcing gives the corresponding smooth vertical profile.

The adjustment begins with the rapidly propagating barotropic mode and is therefore initially almost depth-independent. Successively slower baroclinic modes then arrive from the forcing region. Their Fourier series in the vertical normal modes progressively reconstructs the forcing's upper-layer profile, tending to the stated steady solution while points to the east remain unaltered.