= Solution
The <quasi-geostrophic approximation> isolates slowly evolving, nearly balanced motions in a rotating, stably stratified fluid. Ocean eddies and mid-latitude atmospheric disturbances often have this character: a leading <Coriolis force>–pressure balance suppresses rapid horizontal acceleration, while <hydrostatic balance> suppresses rapid vertical acceleration. Small <ageostrophic flow> nevertheless supplies the vertical motion that changes the leading <vorticity>. The approximation filters fast <inertia-gravity waves> while retaining nonlinear transport, vertical shear and <Rossby waves>.
Use a reference density $\rho_0$ and remove the background <hydrostatic pressure>. Let $\pi=p'/\rho_0$ and let the upward <buoyancy perturbation> be $b=-g\rho'/\rho_0$. A vertically varying reference <density stratification> has $N^2(z)>0$. Under the <Boussinesq approximation>, the inviscid and adiabatic equations are
$$
\frac{D\mathbf u_H}{Dt}+f\widehat{\mathbf z}\times\mathbf u_H=-\nabla_H\pi,\qquad
\frac{Dw}{Dt}=-\pi_z+b,\qquad
\frac{Db}{Dt}+N^2w=0,\qquad
\nabla_H\cdot\mathbf u_H+w_z=0.
$$
Here $D/Dt=\partial_t+\mathbf u_H\cdot\nabla_H+w\partial_z$. Density variations are small compared with $\rho_0$ but retained in the vertical buoyancy force. This is a useful local, incompressible description; a compressible atmosphere needs the corresponding weighted or pressure-coordinate version.
Let horizontal and vertical lengths be $L,H$, speed be $U$, and take the slow time $L/U$. Take $f_0>0$ for definiteness; use $|f_0|$ in the scale estimates in the opposite hemisphere. The principal small parameter is the <Rossby number> $\mathrm{Ro}=U/(f_0L)\ll1$ at a latitude where $f_0\ne0$. For the hydrostatic, strongly stratified regime, $H/L\ll1$ and $f_0/N\ll1$. Retaining both horizontal <vorticity> and <vortex stretching> gives the <Burger number>
$$
\mathrm{Bu}=\frac{N^2H^2}{f_0^2L^2}=O(1).
$$
The <geostrophic balance> pressure scale is $f_0UL$ and the <buoyancy perturbation> scale is $B=f_0UL/H$. Hence $B/(N^2H)=\mathrm{Ro}/\mathrm{Bu}\ll1$: <buoyancy perturbations> produce small displacements relative to the background stratification scale. The <ageostrophic flow> is $O(\mathrm{Ro}\,U)$ and <incompressibility> then gives $w=O(\mathrm{Ro}\,UH/L)$. Vertical acceleration is smaller than hydrostatic terms by order $\mathrm{Ro}^2(f_0/N)^2$ in this scaling. Weak forcing or diffusion can be added at the slow order, but neither is present here.
Define the <quasi-geostrophic streamfunction> by $\psi=\pi/f_0$. Leading <geostrophic balance> and <hydrostatic balance> imply
$$
\boxed{u_g=-\psi_y,\qquad v_g=\psi_x,\qquad b=f_0\psi_z.}
$$
Differentiating horizontally and vertically gives the <thermal-wind balance>
$$
f_0u_{g,z}=-b_y,\qquad f_0v_{g,z}=b_x.
$$
Thus vertical shear is determined by horizontal <buoyancy gradients>. The homogeneous <Taylor–Proudman theorem> is recovered only when those gradients vanish. Stable stratification allows sloping density surfaces and different <geostrophic flow> at different levels, without abandoning leading balance.
The <Prandtl ratio of scales> makes that modification quantitative. Horizontal and vertical terms in balanced <potential-vorticity inversion> have sizes $\psi/L^2$ and $(f_0^2/N^2)\psi/H^2$. Equality gives
$$
\boxed{\frac HL\sim\frac{|f_0|}{N}.}
$$
For constant $N$, $Z=(N/|f_0|)z$ makes the inversion operator an ordinary three-dimensional <Laplacian>. Equivalently, an interior balanced disturbance of horizontal size $L$ naturally penetrates a depth of order $|f_0|L/N$. With $N\gg|f_0|$ this is a thin, vertically sheared structure, rather than the arbitrarily tall homogeneous columns suggested by strict <Taylor–Proudman theorem> behaviour. This is an aspect-ratio statement, not the diffusivity <Prandtl number>.
To allow a weak meridional variation of the <Coriolis parameter>, use the <beta plane> $f=f_0+\beta y$, with $\beta L/f_0=O(\mathrm{Ro})$, or equivalently $\beta L^2/U=O(1)$. The small correction to the leading balance is dynamically important on the slow time. Let $\xi=\nabla_H^2\psi$ be vertical relative <vorticity> and let $D_g=\partial_t+\mathbf u_g\cdot\nabla_H$. At the first nontrivial order, the vertical <vorticity equation> and <buoyancy> equation reduce to
$$
D_g\xi+\beta v_g=f_0w_z,\qquad D_gb+N^2w=0.
$$
Relative-vorticity stretching and tilting are higher order here; planetary-vorticity stretching $f_0w_z$ must remain. Eliminate $w$ to expose a materially conserved balanced scalar. Because $\mathbf u_{g,z}=(-b_y/f_0,b_x/f_0)$, the commutator term $\mathbf u_{g,z}\cdot\nabla_H b$ is exactly zero. For $N=N(z)$ this gives
$$
D_g\!\left[\partial_z\left(\frac{f_0b}{N^2}\right)\right]
=\partial_z\left(\frac{f_0}{N^2}D_gb\right)=-f_0w_z.
$$
Adding the two equations proves the <quasi-geostrophic potential-vorticity equation>
$$
\boxed{Q=\nabla_H^2\psi+\partial_z\left(\frac{f_0^2}{N^2}\psi_z\right)+\beta y,\qquad D_gQ=0.}
$$
An additive constant such as $f_0$ changes no dynamics. The horizontal <streamfunction advection bracket> can express this as $Q_t+\psi_xQ_y-\psi_yQ_x=0$. Although small departures from balance generated this evolution equation, the leading <geostrophic flow> itself advects $Q$ nonlinearly. The stretching term couples horizontal motion to vertical displacement of density surfaces.
This gives the practical advection-and-inversion description. Start with interior <three-dimensional quasi-geostrophic potential vorticity>, advect it with $\mathbf u_g$, and recover $\psi$ from
$$
\left[\nabla_H^2+\partial_z\left(\frac{f_0^2}{N^2}\partial_z\right)\right]\psi=Q-\beta y.
$$
The <stratified quasi-geostrophic inversion> is elliptic for positive $N^2$. It must be supplied with lateral and vertical <boundary conditions> and any circulation or mean-flow constraints; interior <potential vorticity> alone is not sufficient. At rigid horizontal lids, $w=0$ makes boundary <buoyancy> satisfy $D_gb=0$, and $b=f_0\psi_z$ supplies the vertical derivative data for inversion. Boundary <buoyancy perturbations> can therefore support flow even when the interior anomaly is zero. Once $\psi$ is found, its horizontal derivatives give velocity, its vertical derivative gives buoyancy, and the thermodynamic equation diagnoses $w$. This balanced evolution excludes the independent fast-wave initial data present in the full equations.
Finally, a gradient in $f$ supplies a restoring mechanism even in an otherwise uniform fluid. Linearizing about rest with constant $N$, the <quasi-geostrophic potential-vorticity equation> becomes
$$
\partial_t\left(\nabla_H^2\psi+\frac{f_0^2}{N^2}\psi_{zz}\right)+\beta\psi_x=0.
$$
Substitution of $\psi\propto e^{i(kx+ly+mz-\omega t)}$ gives the <Rossby wave> dispersion
$$
\boxed{\omega=-\frac{\beta k}{k^2+l^2+f_0^2m^2/N^2}.}
$$
For $\beta>0$, its zonal phase speed is westward relative to a resting background. A meridionally displaced parcel retains <potential vorticity>, so it acquires a relative-vorticity anomaly that induces the velocity tending to return the disturbance. Background shear and stratification modify the full <potential-vorticity gradient>; they can alter propagation or permit instability. On a constant <f-plane> the planetary contribution vanishes, but gradients of relative <vorticity>, boundary buoyancy or topography can still support balanced wave motion. Close enough to the equator that $f_0$ vanishes, this mid-latitude <quasi-geostrophic approximation> and its scaling must be replaced.
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