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The shallow-water approximation requires depth much less than horizontal length, a homogeneous incompressible fluid, gentle surface/bottom slopes, and frequencies slow enough that vertical acceleration is negligible compared with gravity. Continuity gives $w\sim (H/L)U$, making vertical inertial acceleration smaller than the horizontal inertial scale by the aspect ratio. The vertical momentum equation therefore reduces to $p_z=-{\rho }_{0}g$. With constant atmospheric pressure,The horizontal pressure gradient is independent of depth. Linearizing about rest and retaining the leading depth-uniform horizontal motion givesTheir horizontal curl gives ${\zeta }_t=-f(u_x+v_y)$, soHere $\zeta =v_x-u_y$ is relative vorticity; the PDF sentence identifying $\eta$ as relative vorticity is a symbol error. This is the linear shallow-water potential vorticity anomaly; advection of that perturbation is second order.

In the adjusted state take alongshore independence, decay offshore, and an impermeable coast. Steady continuity gives $u=0$, and geostrophic balance gives $v=g{\eta }_x/f$. Assume $f>0$ so the printed $a=\sqrt{gH}/f$ is a positive barotropic deformation radius. For either hemisphere the decay length is $\sqrt{gH}/|f|$, with the corresponding change in current direction. Initial rest makes the conserved anomaly $-{\eta }_{\mathrm{initial}}/H$, and thereforeThe coastal adjustment of an elevated strip uses a decaying offshore solution and a particular solution in the elevated strip are matched with continuous $\eta$ and $v$, hence continuous ${\eta }_x$, at $x=-L$. Integrating this equation over $x<0$ and conserving the initial volume per alongshore length, ${\int }_{-\infty }^0\eta dx={\eta }_{0}L$, gives $a^2{\eta }_x(0)=0$. Solving those matching conditions yieldsExpanding the hyperbolic functions gives exactly the alternative summed form in the PDF. Both matching values at $-L$ are ${\eta }_0(1-e^{-2L/a})/2$, and the wall elevation is ${\eta }_0(1-e^{-L/a})$.

The adjusted flow isThus both velocity components vanish at the coast. In this inviscid problem tangential no-slip is not generally an independently imposed wall condition; here it follows from initial rest and alongshore uniformity. Indeed $v_t+fu=0$ at the impermeable wall keeps $v(0,t)=0$. The mass constraint recovers the same adjusted-state condition.

For $L\gg a$, the surface retains nearly its initial elevation through most of the strip. The offshore edge is smoothed over width of order $a$, with half-height at the original edge, while the coast remains at nearly ${\eta }_0$. The current is concentrated near the smoothed edge and vanishes at the coast.

For $L\ll a$, the elevation spreads over a much larger offshore scale $a$, with wall height approximately ${\eta }_{0}L/a$. Away from the thin original strip, $\eta \simeq {\eta }_0(L/a)e^{x/a}$. Within it the leading elevation is nearly that reduced constant, with a small curvature required to bring $v$ to zero at the wall. Wide strips retain a broad high plateau; narrow strips spread into a low deformation-scale coastal bulge. The total anomalous volume remains ${\eta }_{0}L$ in both limits.

The vertical component $q_z=\zeta -f\eta /H$ is $H$ times the first-order anomaly in shallow-water potential vorticity about rest:Thus $\mathbf{q}=q_z\hat{\mathbf{z}}$ records the combination of relative vorticity and column stretching that is conserved by the unforced linearized shallow water equations. Indeed their curl and continuity equation give $(q_z{)}_t=0$. It is the persistent forcing of the balanced component during geostrophic adjustment, while inertia-gravity waves carry away the unbalanced component.

Initially $q_z=-f{\eta }_0\mathrm{sgn}(x)/H$. In the final geostrophic flow, $u=0$, $v=g{\eta }_x/f$, and $\zeta =g{\eta }_{xx}/f$. Conservation of the linearized shallow-water potential vorticity therefore givesBoundedness on both sides and continuity of $\eta ,{\eta }_x$ at the origin determine the constants:The solution is the locally attained balanced state after outgoing inertia-gravity waves have departed; the full inviscid system need not lose its globally conserved wave energy. The signed volume flux in the $y$ direction is

For double Kelvin-wave adjustment at a depth step, put $c_{\pm }=\sqrt{g{H}^{\pm }}$ and use $R_D^{\pm }=c_{\pm }/f$ on the corresponding side of the step. The PDF's far-field wording must mean $\eta \to {\eta }_0\mathrm{sgn}(x)$, rather than ${\eta }_0$ for every $x$. In the exponentially trapped outer response, cross-step geostrophic balance gives, at $y=0$,To retain the propagation of this slowly varying trapped disturbance, the along-step acceleration in $u_t-fv=-g{\eta }_x$ must also be retained. It supplies the small cross-step ageostrophic flow. Thus the two cross-step volume fluxes areHere $d\mathrm{sgn}(x)/dx=2\delta (x)$ as a distributional derivative. Equating the fluxes and dividing by $c_{+}+c_{-}$ givesThe printed forcing is missing a factor of two. It cannot follow from the printed sign-function surface profile with the usual Dirac delta distribution normalization. Pure geostrophic balance in both horizontal directions would also omit $A_t$ and could not determine the wave evolution.

The method of characteristics with zero initial amplitude gives the corrected weak solutionBehind the eastward front, $0<x<\Delta ct$, the surface elevation at the step becomes $-{\eta }_0$, matching the upstream side. All the original cross-step transport is diverted into flow along the step once the wave has passed a fixed location: $v=0$ there, with opposing along-step velocities above and below the step. This is the complete barrier in the adjusted region, and in the long-time limit at every fixed $x$. At finite time a front still carries cross-step flux: the formulas above give $H^{\pm }{v}^{\pm }=(2{\eta }_0{c}_{+}{c}_{-}/f)\delta (x-\Delta ct)$. The discontinuous outer representation does not resolve the short-scale fast adjustment at either front.

For completeness, solving the equation literally as printed instead gives $A={\eta }_0[\Theta (x)-\Theta (x-\Delta ct)]$. Its amplitude is half that required; $\eta (x,0,t)$ retains a jump ${\eta }_0$ at the origin after passage, so that literal equation does not imply complete blocking. The corrected equation and the literal printed equation are distinguished rather than asserting an inconsistent barrier result.

Assume inviscid, unforced evolution, so shallow-water potential vorticity is conserved. If a column has initial depth $H_1$ and relative vorticity ${\zeta }_1$, moving to $H_2>H_1$ at fixed Coriolis parameter givesIts absolute vorticity is amplified by vortex stretching. A column initially without relative vorticity develops cyclonic spin: positive in the Northern Hemisphere and negative in the Southern Hemisphere. More generally the increase is in the direction of its original absolute vorticity, not necessarily the direction of $f$ if the initial spin is unusually strong.

Conservation of shallow-water potential vorticity with fixed relative vorticity givesFor ordinary planetary-vorticity-dominated columns, moving towards the equator decreases the magnitude of absolute vorticity, so the column must become shallower and spread horizontally. This is the compression counterpart of vortex stretching. For $\zeta =0$, the depth ratio is $f_2/f_1$, in either hemisphere before reaching the equator. The displayed general formula is essential if the fixed relative vorticity is large: the sign of the depth change then depends on the original absolute vorticity, and conservation cannot continue through zero absolute vorticity with positive finite depth and nonzero conserved potential vorticity.

Use a one-layer, hydrostatic, homogeneous fluid with reference depth $H_0$, no forcing or friction, small Rossby number $ϵ=U/(|f_0|L)\ll 1$, slow evolution, and leading geostrophic balance. For the beta plane, take $\beta L/|f_0|=O(ϵ)$; the relative free-surface displacement is also small, $\eta /H_0=O(ϵ)$, with the Rossby deformation radius retained at the chosen horizontal scale. This excludes the equatorial limit. If a bottom elevation $h_b$ is included, additionally require $h_b/H_0=O(ϵ)$ and stationary topography. Set $h_b=0$ for a flat bottom.

The quasi-geostrophic streamfunction satisfies $\psi =g\eta /f_0$, $u=-{\psi }_y$, $v={\psi }_x$, so relative vorticity is $\zeta ={\nabla }_h^2\psi$. Expanding the shallow-water potential vorticity using actual depth $H_0+\eta -h_b$ givesConsequently the depth-scaled anomaly, the shallow-water quasi-geostrophic potential vorticity, isThe leading material derivative uses the geostrophic flow. Define $J(a,b)=a_x{b}_y-a_y{b}_x$; then conservation yields $Q_t+J(\psi ,Q)=0$. The term $-\psi /R_D^2$ represents free-surface stretching, $\beta y$ the variation of planetary rotation, and $f_0{h}_b/H_0$ topographic stretching. Under a rigid lid or at scales much smaller than the Rossby deformation radius, the free-surface term is absent or negligible and the flat-bottom Barotropic Rossby wave model has $Q={\nabla }_h^2\psi +\beta y$.

Let $Q_b=f/H$. At sufficiently small Rossby number, retain advection of this full background shallow-water potential vorticity while dropping the higher-order advection of $\zeta /H$. For uniform-depth Sverdrup balance, this also requires relative-vorticity advection to be small against planetary-vorticity advection; small Rossby number alone does not guarantee this when $\beta$ is very small. Multiplying the resulting steady potential vorticity equation by $H$ givesDefine the topographic potential-vorticity pseudovelocityIts sign is important: it is opposite to the coefficient vector on the left of the preceding equation. Substituting the ocean transport streamfunction expression for relative vorticity yieldsThe pseudovelocity follows contours of $f/H$ and is not the fluid velocity; its units are inverse length squared per time because $\psi$ is a transport streamfunction. If $r=F=0$, the ocean transport streamfunction is constant along each connected background potential vorticity contour. Locally, at regular points,with arbitrary differentiable $\Phi$. Different disconnected components of a level set may carry different functions until boundary conditions identify them. If $f/H$ is constant over an open region, the leading equation gives no restriction there; critical points similarly require regularity and global matching rather than division by a vanishing gradient.

Write $\zeta =(\nabla \times \mathbf{u})\cdot {\mathbf{e}}_z$ and $q=\zeta -f\eta /h_0$, so the PDF's vector potential vorticity is $\mathbf{Q}=q{\mathbf{e}}_z$. The local TeX loses some of this boldface. Taking the vertical curl of the momentum equation gives ${\zeta }_t=-f\nabla \cdot \mathbf{u}$, while mass conservation gives ${\eta }_t=-h_0\nabla \cdot \mathbf{u}$. ThusThis is the linear perturbation of shallow-water potential vorticity, scaled by $h_0$; it should not be confused with the full nonlinear conserved ratio $(f+\zeta )/h$.

Since $\nabla \cdot ({\mathbf{e}}_z\times \mathbf{u})=-\zeta$, taking the divergence of momentum gives $(\nabla \cdot \mathbf{u}{)}_t=f\zeta -g{\nabla }^2\eta$. Differentiate mass conservation and use $\zeta =q+f\eta /h_0$ to obtainFor $\mathbf{Q}=0$, substitution of the stated plane wave gives the linear rotating shallow-water dispersion relation, with $c=\sqrt{g{h}_0}$:The two even branches approach $\pm c|k|$ at large $|k|$ and have values $\pm |f|$ at $k=0$. The original dispersion diagram below also compares the speed magnitudes.

For $k\ne 0$, the phase velocity and group velocity magnitudes areThe printed convention $e^{i(kx+\omega t)}$ means the signed phase velocity is $-\omega /k$ and the signed group velocity is $-d\omega /dk$. The speed magnitudes above are independent of that sign convention.

For $f\ne 0$, longer wavelengths have greater phase velocity but smaller group velocity: individual crests and wave packets therefore answer “faster or slower” differently. At long wavelength, $|k|\ll |f|/c$, the frequency is nearly the inertial frequency $|f|$, $|c_p|\sim |f|/|k|$, and $|c_g|\sim c^2|k|/|f|$. Rotation has its largest relative effect here, and the waves are strongly dispersive. At short wavelength, $|k|\gg |f|/c$, both speeds tend to $c$ and rotation gives only a small, weakly dispersive correction. If $f=0$, the gravity waves are nondispersive, with speed $c$; at $k=0$ the phase velocity formula is undefined.

The vertical relative vorticity isDifferentiate the $v$-momentum equation with respect to $x$ and the $u$-momentum equation with respect to $y$, then subtract. The mixed pressure derivatives cancel, givingThe mass-conservation equation says ${\eta }_t=-h(u_x+v_y)$. Therefore the linearized shallow-water potential vorticitysatisfiesThus

Differentiate the second momentum equation in $x$, the first in $y$, subtract, and use continuity:Hence the linear shallow-water potential vorticitysatisfies $P_t=0$.

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 $\beta -f\gamma$.

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.

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:

The horizontal velocities $u$ and $v$ point east and north, $\eta$ is the displacement of the free surface from its mean level, $H_0$ is the undisturbed depth, $g$ is gravitational acceleration, and $f$ is the constant Coriolis parameter on an f-plane. The three linearized shallow water equations are horizontal momentum balance and mass conservation:They follow from the rotating Navier-Stokes equation by assuming an inviscid homogeneous layer, hydrostatic pressure, horizontal scales much larger than $H_0$, depth-independent horizontal velocity, a flat impermeable bottom, constant $f$, and small surface displacement and velocity so that nonlinear products are neglected.

In a steady state, geostrophic balance givesThe relative vorticity is $\zeta =v_x-u_y$. Expanding the shallow-water potential vorticity $(f+\zeta )/(H_0+\eta )$ to first order givesThus one convenient normalization of its disturbance isTaking the curl of momentum and using continuity shows $q_t=0$.

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.

For a steady, slowly varying rotating channel, negligible transverse velocity and zero shallow-water potential vorticity give $u_y=f$. Cross-channel geostrophic balance gives $g{h}_y=-fu$, producing the displayed profiles. The specific energy $E=h+u^2/(2g)$ is constant across the channel, and $E+H$ is constant along it. Zero potential vorticity is not the value in an ordinary finite-depth reservoir at rest in the rotating frame; it requires vanishing absolute vorticity or a deep-reservoir approximation.