= Solution
Let the wave-forcing layer have vertical scale $D$ and meridional scale $L$. In the <Eliassen equation for residual circulation>, the two restoring terms scale as
$$
f_0^2\frac{\chi^*}{D^2},
\qquad
N^2\frac{\chi^*}{L^2}.
$$
Their ratio is controlled by
$$
\Lambda=\frac{ND}{f_0L}.
$$
In the shallow-forcing limit $\Lambda\ll1$, the vertical derivative term dominates:
$$
f_0^2\chi_{zz}^*
\simeq-f_0(\nabla\mathbin\cdot\mathbf F)_z.
$$
After one vertical integration,
$$
f_0\overline v_a^*
\simeq-\nabla\mathbin\cdot\mathbf F.
$$
The Coriolis force on the <residual mean circulation> therefore balances most of the wave forcing, and
$$
\overline u_t
=f_0\overline v_a^*
+\nabla\mathbin\cdot\mathbf F
$$
is small at leading order. The response is primarily an overturning circulation with an associated density tendency.
In the deep-forcing limit $\Lambda\gg1$, the meridional term dominates:
$$
N^2\chi_{yy}^*
\simeq-f_0(\nabla\mathbin\cdot\mathbf F)_z.
$$
Now
$$
\frac{f_0\overline v_a^*}
{\nabla\mathbin\cdot\mathbf F}
=O(\Lambda^{-2}),
$$
so the residual circulation is too weak to balance the forcing. The dominant momentum response is direct zonal acceleration,
$$
\boxed{
\overline u_t\simeq
\nabla\mathbin\cdot\mathbf F},
$$
with a comparatively weak overturning and density response.
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