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Let $\mathbf{u}=(u,v)$ and let the layer thickness be $H=H_0+\eta$. The inviscid shallow water equations areThey follow from a homogeneous incompressible fluid with small aspect ratio, hydrostatic pressure, negligible vertical acceleration, horizontal velocity nearly uniform through the depth, a material free surface, and a rigid stationary bottom. Rotation may be represented by an f-plane or beta plane. The model is useful because many atmospheric and oceanic motions are horizontally much broader than their depth, while the free surface or an internal density interface still supports waves and potential vorticity dynamics.

Let $\zeta =v_x-u_y$. Taking the vertical curl of momentum giveswhere $Df/Dt=\beta v$ supplies the planetary-vorticity term when $f$ varies. Continuity gives $DH/Dt=-H\nabla \cdot \mathbf{u}$. Combining the two equations yields material conservation of shallow-water potential vorticity:

Initially $\zeta =0$, so conservation of the linear potential vorticity givesThe final flow is in geostrophic balance, henceConsequently the adjusted height solves the modified Helmholtz equationwhere $L_R$ is the Rossby deformation radius.

The initial condition is independent of $x$ and odd in $y$. The bounded solution that is continuously differentiable at $y=0$ isIt givesDuring geostrophic adjustment, the part of the initial energy incompatible with the conserved potential-vorticity distribution radiates away as inertia-gravity waves. The remaining current has width $O(L_R)$ and is geostrophically balanced.

Because $v={\psi }_x=(g/f_0){\eta }_x$, the mean boundary-current velocity is exactly related to the height change byFor the e-folding convention used in part d,Its robust narrow-layer scaling isThus weaker drag makes the current proportionally narrower and faster. Their product is independent of $\gamma$ to leading order:Mass conservation requires the narrow return transport to cancel the broad Sverdrup balance transport. The meridional gradient of planetary potential vorticity makes a frictional closure possible on the western side and produces western intensification; bottom drag supplies the vorticity sink that permits fluid parcels to cross potential-vorticity contours there.

In unforced adiabatic inviscid motion, potential vorticity is conserved following a parcel:The linearization about a basic state is $({\partial }_t+\bar{\mathbf{u}}\cdot \nabla )q^{\prime }+{\mathbf{u}}^{\prime }\cdot \nabla \bar{q}=0$. This separates advection of a disturbance from its generation by displacement across a basic potential-vorticity gradient.

A potential vorticity budget has the formThe material derivative follows the fluid motion and $S_q$ collects forcing or dissipation. With $S_q=0$ it becomes potential-vorticity conservation. For a quasi-geostrophic streamfunction, horizontal advection is $J(\psi ,q)={\psi }_x{q}_y-{\psi }_y{q}_x$.

The spatial gradient of potential vorticity controls the anomaly created when a parcel is displaced. The linear potential-vorticity equation contains ${\mathbf{u}}^{\prime }\cdot \nabla \bar{q}$. A gradient of the Coriolis parameter contributes planetary vorticity, but gradients of relative vorticity and layer stretching can be equally important.

Potential-vorticity inversion recovers a balanced velocity field from potential vorticity and specified boundary and global data. In two-dimensional vortex dynamics this is a Poisson equation; shallow-water quasi-geostrophic inversion and stratified quasi-geostrophic inversion use different elliptic boundary value problems. Advance the materially conserved scalar, then invert again to recover the evolving flow. Boundary buoyancy can induce flow with zero interior anomaly. The balance relation and boundary conditions are essential: unrestricted gravity waves are additional degrees of freedom. McIntyre's potential-vorticity account explains the general invertibility principle.

Homogenizing the potential vorticity over a finite constant bottom slope creates the odd anomaly displayed here while leaving the exterior unchanged. Inverting $({\partial }_y^2-L_R^{-2})\psi =\Delta Q$ with decay at both infinities and matching $\psi ,{\psi }_y$ gives a central return current and flanking jets. The deformation radius sets the smoothing length. For $L_R\ll a$, the central surface follows the bottom slope to leading order, but edge corrections and decaying exterior flow are essential.

The quasi-geostrophic approximation describes slowly evolving, nearly geostrophic flow with small Rossby number. Its dynamics are governed by advection and forcing of potential vorticity.

In zonally averaged barotropic quasi-geostrophic approximation flow with zero mean meridional transport, the Taylor identity for quasi-geostrophic flux relates the meridional potential vorticity flux to the convergence of Reynolds stress. A local downgradient closure $\overline{v^{\prime }{q}^{\prime }}=-\kappa {\overline{Q}}_y$, with $\kappa \ge 0$, therefore gives westward mean acceleration where the background gradient is positive. Rossby wave breaking can stir PV into fine filaments; coarse-graining or weak diffusion produces the inferred mixing. Globally the momentum-flux convergence integrates to boundary stresses, so localized westward changes need compensating transport or momentum changes elsewhere in an isolated system.

A lower-boundary Eady edge wave in the semi-infinite Eady model has a smooth exponentially decaying vertical structure and a real frequency. It is supported by the horizontal boundary buoyancy gradient while its interior potential vorticity anomaly vanishes. Its speed equals the basic velocity at one penetration depth.

A uniformly stratified quasi-geostrophic approximation fluid with constant vertical shear occupies $z>0$, has a flat rigid lower wall, and has decaying perturbations aloft. On an f-plane, the basic interior potential vorticity is constant. Its smooth edge-wave modes have zero interior anomaly and propagate along the lower-boundary buoyancy gradient. The absence of an upper wall removes the pair of interacting edge waves that makes the finite-depth Eady model unstable.

The shallow-water quasi-geostrophic potential vorticity relation is a modified Helmholtz equation for the quasi-geostrophic streamfunction. Its inverse with specified boundary conditions recovers horizontal geostrophic flow and surface elevation $\eta =f_0\psi /g$. The Rossby deformation radius $L_D=\sqrt{gH}/|f_0|$ is the range of the balanced response: for a horizontal Fourier component, $\hat{\psi }=-\hat{q}/(|{\mathbf{k}}_h{|}^2+L_D^{-2})$, after subtracting background potential vorticity.

For a semi-infinite stratified fluid with thermal wind $U=\Lambda z$ above a gently sloping boundary $z=\alpha y$, a zero-interior-potential vorticity perturbation decays as $e^{-\mu z}$, where $\mu =N|k|/|f_0|$. The boundary buoyancy perturbation obeys a single oscillatory equation with phase velocity $c=(\Lambda -\alpha N^2/f_0)/\mu$. The topographic and thermal-wind contributions cancel when the boundary follows a background isopycnal.

The three-dimensional quasi-geostrophic potential vorticity relation defines an anisotropic elliptic boundary value problem. Interior potential vorticity and boundary buoyancy perturbation $b=f_0{\psi }_z$, together with lateral and global conditions, determine the quasi-geostrophic streamfunction up to its allowed constant. If two solutions have the same data, their difference has zero elliptic source and homogeneous boundary data. Integration by parts gives zero integral of $|{\nabla }_h\psi {|}^2+(f_0^2/N^2)|{\psi }_z{|}^2$, proving uniqueness modulo a constant in a connected domain when the boundary terms vanish. Eady edge waves show that zero interior source alone is insufficient.

In a slowly varying inviscid layer, parcels tend to follow contours of $f/H$. If depth increases in the direction of motion, conservation of potential vorticity requires a poleward displacement or a compensating change of relative vorticity. Ekman pumping permits streamlines to cross those contours.

For a uniform normalized upper-layer wind stress curl $W$ and no lower-layer forcing, a uniform local Sverdrup balance isThe upper layer flows meridionally and the lower layer rests. Although the relative vorticity vanishes, the potential vorticity has the zonal gradients ${\bar{q}}_{1,x}=-FV$, ${\bar{q}}_{2,x}=FV$. These are essential for the perturbation stability and do not vanish merely because the background velocity is uniform.

The stretching contribution to upper-layer potential vorticity is $-F({\psi }_1-{\psi }_2)$, with the opposite sign below. An interface displacement thickens one layer and thins the other. In a parcel's potential-vorticity equation, its material change balances changes of relative and planetary vorticity and any forcing. The two layer derivatives cannot in general be replaced by one common advection operator.

The vertical curl of surface wind stress is $({\nabla }_h\times \mathbf{\tau }{)}_z={\tau }_{y,x}-{\tau }_{x,y}$. In a layer of density ${\rho }_0$ and depth $h$, its normalized potential vorticity source is $W=({\nabla }_h\times \mathbf{\tau }{)}_z/({\rho }_{0}h)$. It drives Sverdrup balance in the weakly nonlinear steady ocean interior.