= Solution
Put $D(x,y)=H(x,y)-h(x,y)$ and let $\mathbf u=(u,v)$ be the depth-independent interior velocity. The kinematic boundary conditions on the sloping upper and lower surfaces are
$$
w_t=w_E-\mathbf u\mathbin\cdot\nabla h,
\qquad
w_b=-\mathbf u\mathbin\cdot\nabla H.
$$
Integrating <mass conservation> through the layer gives
$$
\frac{DD}{Dt}+D\nabla_h\mathbin\cdot\mathbf u=-w_E.
$$
The inviscid vertical-vorticity equation is
$$
\frac D{Dt}(f+\zeta)
=-(f+\zeta)\nabla_h\mathbin\cdot\mathbf u,
\qquad
\zeta=v_x-u_y.
$$
Consequently the <shallow-water potential vorticity>
$$
q=\frac{f+\zeta}{D}
$$
obeys the forced evolution equation
$$
\boxed{
\frac{Dq}{Dt}=\frac{q}{D}w_E}.
$$
Positive upward <Ekman pumping> removes layer thickness and raises the potential vorticity of the remaining column.
For a steady small-<Rossby number> flow, $q\simeq f/D$, so
$$
\mathbf u\mathbin\cdot\nabla\left(\frac fD\right)
=\frac{f}{D^2}w_E.
$$
Equivalently,
$$
\boxed{
\beta Dv-f\mathbf u\mathbin\cdot\nabla D
=fw_E}.
$$
The same result follows from the integrated vortex-stretching balance
$$
\beta Dv=f(w_t-w_b).
$$
When $\nabla D=(c,0)$ with $c>0$,
$$
\beta Dv-fcu=fw_E.
$$
If the upper pumping is absent or weak and the upper surface is locally level, the impermeable-bottom condition is $w_b=-cu$. The stipulated $w_b<0$ then gives $u>0$, and in the Northern Hemisphere
$$
v\simeq\frac{fc}{\beta D}u>0.
$$
The steady interior flow is therefore directed northeastward, along contours of $f/D$ in the unforced limit. As a parcel moves eastward into deeper water, its vortex column stretches; a poleward displacement increases $f$ and preserves <potential vorticity>. Nonzero <Ekman pumping> drives motion across the $f/D$ contours. This is <topographic potential-vorticity steering> and the associated <topographic Sverdrup balance>.
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