= Solution
Define the residual-circulation <stream function> by
$$
\boxed{
(\overline v_a^*,\overline w_a^*)
=(\overline\chi_{a,z}^*,-\overline\chi_{a,y}^*)}.
$$
This satisfies residual <mass conservation> identically. Let
$$
G=\nabla\mathbin\cdot\overline{\mathbf F},
\qquad
N^2=-\frac g{\rho_0}\frac{d\rho_s}{dz}.
$$
The momentum equation is
$$
\overline u_t-f_0\overline\chi_{a,z}^*=G.
$$
Differentiate <geostrophic balance> vertically and <hydrostatic pressure> meridionally to obtain <thermal-wind balance>
$$
f_0\overline u_z
=\frac g{\rho_0}\overline\rho_y.
$$
After a time derivative, the transformed density equation gives
$$
f_0\overline u_{tz}
=\frac g{\rho_0}\overline\rho_{ty}
=-N^2\overline\chi_{a,yy}^*.
$$
On the other hand, the $z$ derivative of momentum gives
$$
\overline u_{tz}
-f_0\overline\chi_{a,zz}^*=G_z.
$$
Eliminating $\overline u_{tz}$ yields the <Eliassen equation for residual circulation>
$$
\boxed{
f_0^2\overline\chi_{a,zz}^*
+N^2\overline\chi_{a,yy}^*
=-f_0(\nabla\mathbin\cdot\overline{\mathbf F})_z}.
$$
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