= Solution
In a steady state the time derivative of quasi-geostrophic potential vorticity vanishes, leaving the local balance
$$
\beta\psi_x=F.
$$
Choose the undisturbed state immediately east of the compact forcing as the integration condition. The solution is then
$$
\boxed{
\psi(x,y,z)=
\begin{cases}
-\displaystyle\frac{1}{\beta}
\int_x^\infty G(x',y)\,dx',
&-D<z<0,\\[1.2em]
0,&-H<z<-D.
\end{cases}
}
$$
The 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.
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