= Solution
Let $R=\overline{u'v'}$ be the eddy <momentum flux>, $F=\overline{\rho'v'}$ the eddy <density> flux, and $S=d\rho_s/dz<0$. The PDF's hydrostatic equation is $g\overline\rho=-\overline p_z$; the converted TeX mistakenly replaces the <pressure> derivative by a <density> derivative. With $N_0^2=-gS/\rho_0$, define
$$
A=F/S,\qquad v^*=\overline v_a-A_z,\qquad w^*=\overline w_a+A_y.
$$
The <residual mean circulation> is nondivergent because $v_y^*+w_z^*=\overline v_{ay}+\overline w_{az}$. Substituting in the original mean momentum and <density> equations gives the <transformed Eulerian mean> equations
$$
\boxed{\overline u_t-f_0v^*=-R_y+f_0A_z=\partial_y\mathsf F_y+\partial_z\mathsf F_z,}
\qquad
\boxed{\overline\rho_t+Sw^*=0,\quad v_y^*+w_z^*=0,}
$$
with the unchanged geostrophic and hydrostatic balances. Here the <Eliassen–Palm flux> components are
$$
\mathsf F_y=-R,\qquad \mathsf F_z=f_0A
=\frac{f_0\overline{v'b'}}{N_0^2},\qquad b'=-g\rho'/\rho_0.
$$
The <eddy buoyancy flux> has been absorbed into the residual transport, while both types of wave forcing enter the momentum equation through the <Eliassen–Palm flux> divergence.
For geostrophic disturbances, the <Taylor identity for quasi-geostrophic flux> gives
$$
\nabla\cdot\boldsymbol{\mathsf F}=\overline{v'q'}.
$$
For example, zonal averaging removes $\overline{\psi'_x\psi'_{xx}}$ and converts the remaining horizontal and vertical terms in $\overline{\psi'_xq'}$ into derivatives of $\overline{\psi'_x\psi'_y}$ and $(f_0^2/N_0^2)\overline{\psi'_x\psi'_z}$, giving precisely the displayed fluxes.
Locally, for a basic flow with nonzero meridional PV gradient $Q_y$, the <quasi-geostrophic wave pseudomomentum> is $\mathcal A=\overline{q'^2}/(2Q_y)$. Multiplying the linear perturbation PV equation by $q'/Q_y$ gives
$$
\mathcal A_t+\nabla\cdot\boldsymbol{\mathsf F}=\frac{\overline{q'\mathcal D}}{Q_y},
$$
where $\mathcal D$ is a PV source or sink. In a locally uniform conservative <Rossby wave> packet, $\boldsymbol{\mathsf F}=\mathcal A\mathbf c_g$. Thus the flux tracks <group velocity> rather than the generally different <phase velocity>, with a signed <wave activity> if $Q_y<0$.
The <non-acceleration theorem for quasi-geostrophic waves> states that statistically steady conservative waves, with no wave-activity sources, sinks or independent boundary forcing, produce no balanced mean acceleration. Their flux divergence vanishes. In the present stable, impermeable mean-flow problem the homogeneous residual-circulation equation then has the zero solution, so $v^*=w^*=0$ and $\overline u_t=0$. The <Eulerian mean flow> can still circulate because its eddy-induced correction need not vanish. Dissipation, absorption at a critical layer, time-dependent <wave activity> or boundary forcing produces a nonzero flux divergence and invalidates the non-acceleration conclusion.
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