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Let $D=H-h$ be the constant interior depth. Assume a homogeneous hydrostatic interior with depth-independent horizontal velocity, negligible friction, steady small-Rossby number flow, and no normal flow through the flat bottom. The vertical velocity varies from $w(-H)=0$ to $w(-h)=w_E$, so incompressible flow givesThe leading horizontal momentum balance is geostrophic balance:Taking its vertical curl on a beta plane givesCombining the last two equations yields Sverdrup balanceThe zonal velocity is then fixed, up to its value on one side boundary, byEquivalently, substituting part a gives the depth-integrated formA lateral boundary condition, normally supplied by matching to a boundary current, determines the remaining zonally uniform part of $u$.

Begin with the Boussinesq approximation primitive equations on a beta plane, decompose every field into a zonal mean and a disturbance, and average over longitude. The zonal momentum equation then contains the divergence of the eddy momentum flux $\overline{u^{\prime }{v}^{\prime }}$, while the mean density equation contains the divergence of the eddy density flux $\overline{{\rho }^{\prime }{v}^{\prime }}$. At small Rossby number, use geostrophic balance, hydrostatic pressure, thermal-wind balance, and the leading eddy equations to combine those fluxes.

Defineand introduce the residual mean circulationThe added eddy-induced velocity is nondivergent, soIt absorbs the eddy density-flux divergence into advection by the transformed circulation, giving

The mean zonal momentum equation becomeswhere the zonally averaged Eliassen–Palm flux in the meridional-vertical plane isThus the transformed Eulerian mean gathers the wave forcing into one flux divergence and makes density evolve under one residual circulation.

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:

Sverdrup balance applies to a steady, large-scale, small-Rossby number, hydrostatic and nearly geostrophic ocean interior on a beta plane. The flow is depth-integrated, relative-vorticity advection and interior friction are negligible, density is treated as constant for the barotropic balance, and the principal vorticity source is the curl of wind stress. The balancesays that wind input of vertical vorticity is balanced by meridional advection of planetary vorticity, or equivalently by the stretching needed to conserve potential vorticity as parcels move across latitude circles.

The quasi-geostrophic approximation requires small Rossby number, nearly horizontal geostrophic balance, hydrostatic vertical balance, small interface or density displacements, stable background stratification, and horizontal scales much larger than the vertical scale. The Boussinesq approximation and a beta plane are used, ageostrophic motion enters only at the order needed to evolve potential vorticity, and here the buoyancy frequency $N$ is constant. Under these assumptions the materially conserved three-dimensional quasi-geostrophic potential vorticity is

Assume a steady, small-Rossby-number interior in which drag is negligible and the nonlinear advection of relative vorticity,is small compared with advection of planetary vorticity. The stretching term proportional to $\psi$ has zero Jacobian with $\psi$. On the beta plane,Hence the forced quasi-geostrophic equation reduces to Sverdrup balanceFor the zonal wind stress $\mathbf{\tau }=\tau (y)\hat{\mathbf{x}}$,so

The planetary contribution to absolute vorticity is supplied by the rotating frame. Its local vertical component is the Coriolis parameter $f=2\Omega \sin \varphi$. On a beta plane, motion with northward velocity $v$ changes this contribution at the rate $\beta v$.

The quasi-geostrophic potential-vorticity equation states that three-dimensional quasi-geostrophic potential vorticity is materially conserved in inviscid, adiabatic flow. On a beta plane with constant buoyancy frequency,

For a uniform eastward current $U>0$ on a beta plane with $\beta >0$, stationary nondivergent Rossby waves have $k^2+l^2=\beta /U$. Their ground-frame group velocity is $2U(k^2,kl)/(k^2+l^2)$, tracing a circle centered at $(U,0)$ with radius $U$. Packets emitted by a localized source throughout $0<s<t$ occupy the disk shown in geometrical ray theory. This envelope is distinct from the full balanced near field. A westward current has no nontrivial stationary propagating branch in this model.

In the steady small-Rossby number transport equation, this coefficient field advects the ocean transport streamfunction along contours of background shallow-water potential vorticity $f/H$. It is not a physical velocity: when $\psi$ has units of volume transport, its units are inverse length squared per time. For uniform depth on a beta plane it points westward, $\tilde{\mathbf{u}}=(-\beta /H,0)$. With no forcing or drag, $\psi$ is constant on each connected characteristic; locally $\psi =\Phi (f/H)$ at regular points.

On a beta plane with $f=f_0+\beta y$, set ${\mathcal{F}}_y={\partial }_t^2+f(y{)}^2$. Exact elimination of the depth-integrated shallow-water transports givesThe essential commutator is ${\partial }_y({\mathcal{F}}_{y}V)={\mathcal{F}}_y{V}_y+2f\beta V$. Freezing undifferentiated $f$ factors to $f_0$ is a local approximation. The slow-wave limit gives the shallow-water Rossby-wave dispersion relation with a negative zonal phase velocity for $\beta >0$.