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Substitute the stated scales and divide the momentum equation by $2\Omega V$. The ratio of inertial to Coriolis acceleration is the Rossby numberwhile the dimensionless buoyancy coefficient isThusThe dimensionless $B$ is the Burgers number for this rotating stratified flow.

Assume a small Rossby number, a beta plane, and small fractional changesAt leading order, geostrophic balance givesExpanding the reciprocal depth in the shallow-water potential vorticity givesThis is the stated shallow-water quasi-geostrophic potential vorticity $P_g$. The term ${\nabla }^2\psi$ is the parcel's relative vorticity, while $-f_0^2\psi /(g{H}_0)=-\psi /L_D^2$ is vortex stretching caused by free-surface displacement, whereis the barotropic deformation radius.

For $h_b=\alpha y$, omit the irrelevant constant factor $f_0/H_0$. The active PV anomaly isLinearizing $Q_t+J(\psi ,Q)=0$ about rest and using $\psi \propto e^{i(kx+ly-\omega t)}$ gives the Topographic Rossby-wave dispersion relationA northward displacement raises planetary PV when $\beta >0$ and raises topographic PV when the bottom rises northward, $\alpha >0$. PV conservation then requires anticyclonic relative vorticity, producing westward phase propagation. A negative slope opposes $\beta$ and reverses propagation when $f_0\alpha /H_0<-\beta$.

Under a rigid lid, $h=H_0-h_b$ is fixed but need not be close to $H_0$. With $q=(f+{\nabla }^2\psi )/h(y)$, linearization of $Dq/Dt=0$ givesFor $h=H_0{e}^{\gamma y}$ and over a region where $f\simeq f_0$, plane waves obeyThe planetary-vorticity gradient dominates when $|\beta |\gg |f_0\gamma |$, while exponential depth variation dominates when $|f_0\gamma |\gg |\beta |$. If the variation of $f=f_0+\beta y$ across the region is retained, this is the corresponding local WKB approximation with effective gradient $\beta -f\gamma$.

Assume a steady, small-Rossby number surface boundary layer, neglect horizontal viscosity and nonlinear acceleration, take the pressure gradient to be independent of depth, impose no normal flow at the surface, and let the viscous stress vanish at $z=-h$. Subtract the depth-independent geostrophic balance from horizontal momentum. For the ageostrophic velocity,Integrating from $-h$ to $0$ and usinggives the Ekman transport

Depth-integrated mass conservation, with $w(0)=0$, givesHence the vertical velocity entering the ocean interior is the Ekman pumping velocityFor constant $f$ this reduces to

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 $u$.

Put $D(x,y)=H(x,y)-h(x,y)$ and let $\mathbf{u}=(u,v)$ be the depth-independent interior velocity. The kinematic boundary conditions on the sloping upper and lower surfaces areIntegrating mass conservation through the layer givesThe inviscid vertical-vorticity equation isConsequently the shallow-water potential vorticityobeys the forced evolution equationPositive upward Ekman pumping removes layer thickness and raises the potential vorticity of the remaining column.

For a steady small-Rossby number flow, $q\simeq f/D$, soEquivalently,The same result follows from the integrated vortex-stretching balance

When $\nabla D=(c,0)$ with $c>0$,If the upper pumping is absent or weak and the upper surface is locally level, the impermeable-bottom condition is $w_b=-cu$. The stipulated $w_b<0$ then gives $u>0$, and in the Northern HemisphereThe steady interior flow is therefore directed northeastward, along contours of $f/D$ in the unforced limit. As a parcel moves eastward into deeper water, its vortex column stretches; a poleward displacement increases $f$ and preserves potential vorticity. Nonzero Ekman pumping drives motion across the $f/D$ contours. This is topographic potential-vorticity steering and the associated topographic Sverdrup balance.

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.

At small Rossby number, $q=(\zeta +f_0)/H\simeq f_0/H$. For steady flow with $H=y^2$, the forced potential-vorticity equation becomessoDefine a transport streamfunction by $Hu=-{\psi }_y$ and $Hv={\psi }_x$. Choosing the eastern wall as $\psi =0$ gives the topographic Sverdrup balance solutionand thereforeIn each half-basin, the transport streamlines are the level curves $(L_X-x)y^3=\text{constant}$. They move westward and toward $y=0$. This interior solution treats the two sides of the degenerate line $H=0$ separately and requires boundary layers to enforce solid-wall conditions.

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