Shallow water equations

An incoming tide can form a moving surge that propagates upstream against a river's current. A discontinuous idealization is a shock wave of the shallow water equations, while weaker or dispersive bores can have an undular wave train. A boundary overtaking upstream-propagating characteristic curves warns that a smoothly attached quasi-steady description may fail; the appropriate discontinuity must satisfy the conservative jump conditions of the chosen model.

The one-dimensional open-channel shallow water equations express conservation of water volume and depth-averaged momentum. Here $h$ is water depth, $u$ the mean velocity, $\eta$ the free-surface elevation and ${\tau }_b$ the resisting bed shear stress. The elevation includes the bed profile, so its gradient already includes gravitational driving by bed slope. A constant volume flux in space and time forces $h_t=0$ through the first equation; an unsteady reduction that imposes that flux while retaining momentum relaxation is consequently an approximation to this full system.

This scalar quasilinear partial differential equation results by imposing $h=Q/u$ in a frictional river momentum equation and measuring $X$ upstream. It retains a relaxation toward $u_0=Q^{1/3}$ and carries information at speed $c(u)=Q/(F^2{u}^2)-u$. Near equilibrium the relaxation time is $\epsilon F^2{u}_0/3$. A slowly moving downstream boundary can generate an attached boundary layer when $c(u)$ exceeds its speed throughout the relaxation. The model retains momentum dynamics only: exactly constant flux in both original Saint-Venant equations would instead force time-independent depth.

In a quasi-one-dimensional inviscid channel with depth $H$ and width $b(x)=b_0{e}^{\mu x}$, the linear shallow water equations are ${\eta }_t+(H/b)(bu{)}_x=0$ and $u_t+g{\eta }_x=0$. Combining them gives ${\eta }_{tt}=gH({\eta }_{xx}+\mu {\eta }_x)$.

For a time dependence $e^{i\omega t}$, write the elevation amplitude as $N(x)=e^{-\mu x/2}V(x)$. The spatial equation becomesThere is a travelling spatial phase only when the bracket is positive, so the cutoff frequency is ${\omega }_c=\mu \sqrt{gH}/2$. Above it, an outgoing wave has a phase proportional to $\omega t-kx$ and amplitude proportional to $e^{-\mu x/2}$; below it the spatial response is evanescent. Geometric widening introduces the cutoff even though a constant-width channel has the nondispersive speed $\sqrt{gH}$.

A flat pool of depth $h_0$ between a wall at $-a$ and a removed dam at zero initially develops the shallow water equations rarefaction $u=2(c_0+x/t)/3$, $c=(2c_0-x/t)/3$, where $c_0=\sqrt{g{h}_0}$. It has $Q_{+}=u+2c=2c_0$ and front $x=2c_{0}t$. Its right-running characteristics satisfy $x_{+}=2c_{0}t+C{t}^{1/3}$, obtained by integrating $x_{+}^{\prime }-x_{+}/(3t)=4c_0/3$.

The first rarefaction signal reaches the wall at $(-a,a/c_0)$. The right-running characteristic through that event is the displayed curve. To its right the original fan remains valid; to its left reflected wall data alter the invariant. Continuing the original fan to the wall after that time would give nonzero wall velocity, contradicting impermeability. This is a concrete domain of dependence calculation for a nonlinear hyperbolic system.

For frictional wide-channel shallow water equations, a smooth increasing wave connecting $(h,u)=(1,1)$ to water depth $H>1$ and moving upstream has $c=(H^{3/2}+1)/(H-1)$ and $h^{\prime }=[h^3-(1+c-ch{)}^2]/[h^3-(1+c{)}^2{F}^2]$. Positivity of the denominator at the upstream endpoint requires $F<1/(1+c)$. For the numerator put $r=\sqrt{h}$, $R=\sqrt{H}$: its nontrivial factor is $(r+1)(r-R)(r-R/(R-1))$. The endpoint at $H$ is the first positive downstream zero exactly when $R\le 2$, giving the amplitude restriction. At $H=4$ that zero is double; otherwise the profile cannot cross an intervening equilibrium.

For the dimensionless shallow water equations with gravity coefficient one, let the left state be $(h,u)$, $h>0$, and the right state $(1,0)$. The Rankine-Hugoniot conditions are $s(h-1)=hu$ and $shu=h{u}^2+(h^2-1)/2$. For $h\ne 1$ and $s>0$, eliminating $u$ gives the displayed formulas. The characteristic speeds are $u\pm \sqrt{h}$ and $\pm 1$. Because $u-\sqrt{h}<s$ always, the positive branch can only be an admissible family-plus shock. Its remaining Lax shock admissibility conditions, $1<s<u+\sqrt{h}$, both reduce to $h>1$. At $h=1$ there is no jump and the labelled interface has no determined shock speed.

For a rectangular channel with stationary width $b(x)$ and constant reduced gravity $g^{\prime }$, volume conservation and momentum conservation give $h_t+u{h}_x+h{u}_x=-hu{b}^{\prime }/b$ and $u_t+u{u}_x+g^{\prime }{h}_x=0$. With $c=\sqrt{g^{\prime }h}$ the characteristic curves have speeds $u\pm c$. The combinations $u\pm 2c$ obey the displayed compatibility equations; they are Riemann invariants only when the width source term vanishes.

For an outgoing simple wave into water initially at rest at depth $H$, the incoming Riemann invariant is $u-2\sqrt{gh}=-2\sqrt{gH}$. Thus $\sqrt{gh}=\sqrt{gH}+u/2$ and the outgoing characteristic speed is $v=\sqrt{gH}+3u/2$. A boundary characteristic emitted at time $s$ is $x=v(s)(t-s)$. Its envelope satisfies $t=s+v/v^{\prime }$ and $x=v^2/v^{\prime }$. At a zero-velocity onset with positive acceleration, the leading-edge intersection is the displayed distance. Later characteristic curves can form a shock earlier elsewhere in a sufficiently strong waveform.

In a sectionally uniform flow, a volume source $Q_x$ per downstream length brings axial velocity $u_a$. Momentum conservation gives the displayed source correction, with $A$ the wetted area. The head derivative is $[h+H+u^2/(2g){]}_x=Q_x(u_a-u)/(gA)$. Removing fluid at the local velocity preserves the head in this approximation. Adding fluid at rest does not: its acceleration reduces that head. The volume source alone cannot specify the momentum exchange without a velocity assumption.

For a steady rectangular-channel flow with discharge $Q=bhu$, the shallow-water specific energy relative to the bed is $E=h+Q^2/(2g{b}^2{h}^2)$. At fixed discharge and width, $E_h=1-F^2$, where the Froude number satisfies $F^2=Q^2/(g{b}^2{h}^3)$. The minimum is $E_c=3h_c/2$, with $h_c=(Q^2/(g{b}^2){)}^{1/3}$. Without dissipation, $E+H$ is the conserved Bernoulli equation head for bed elevation $H$.

For cross-sectional area $S(x,h)$ and cross-sectionally uniform velocity, volume conservation gives $S_h(h_t+u{h}_x)+S{u}_x+u{S}_x=0$, while hydrostatic inviscid momentum gives $u_t+u{u}_x+g{h}_x=0$. Their characteristic speeds are $u\pm c$. With $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$, the robust compatibility equation is $(g/c)D_{\pm }h\pm D_{\pm }u=-uc{S}_x/S$. Replacing $D_{\pm }h$ by $D_{\pm }c/c_h$ requires $c_x=0$ at fixed height; otherwise an explicit geometry term remains.

For constant long-wave speed and $S_0\propto e^{\beta x}$, a standing free-surface amplitude obeys ${\eta }^{\prime \prime }+\beta {\eta }^{\prime }+k^2\eta =0$. The transformation $\eta =e^{-\beta x/2}Y$ gives $Y^{\prime \prime }+(k^2-{\beta }^2/4)Y=0$. Impermeable ends at $0,L$ impose ${\eta }^{\prime }=0$, yielding the displayed nonzero-frequency eigenvalues and ${\eta }_n\propto e^{-\beta x/2}[\cos (n\pi x/L)+\beta L\sin (n\pi x/L)/(2n\pi )]$. A separate zero-frequency mode changes the uniform water level.

A hydraulic jump connects shallow supercritical flow to deeper subcritical flow while conserving discharge and momentum and dissipating mechanical energy. Its Rankine-Hugoniot condition uses the channel's actual area and integrated hydrostatic pressure, which depend on cross-sectional geometry.

In a stationary rectangular-channel hydraulic jump, orient the discharge $q=hu>0$ from side 1 to side 2. Momentum conservation gives $q^2=g{h}_1{h}_2(h_1+h_2)/2$ for unequal depths. The specific mechanical energy is $B=u^2/2+gh$, soDividing by $g$ gives the displayed loss of head, measured as a length. The mechanical energy flux lost is $\rho q(B_1-B_2)$, so a dissipative jump must have $h_2>h_1$. With $R=h_2/h_1>1$, the squared Froude numbers are $R(1+R)/2>1$ upstream and $(1+R)/(2R^2)<1$ downstream. Thus the shallow water equations connect supercritical flow to subcritical flow through the admissible jump.

A hydraulic bore is a propagating abrupt depth change described as a moving hydraulic jump. In the shock frame, an admissible rightward through-flow goes from supercritical flow to subcritical flow, with the appropriate characteristic curves entering the jump.

A wall moving at constant speed $U>0$ into stationary water of depth $h_0$ produces a hydraulic bore of speed $V>U$ in the uniform-state shallow water equations model. Behind the bore the fluid velocity equals $U$ and its depth $H$ obeys $H/h_0=V/(V-U)$. Put $F=U/\sqrt{g{h}_0}$ and $v=V/U$. The moving Rankine-Hugoniot conditions give $F^{2}v=[(H/h_0{)}^2-1]/2$, hence the displayed equation for $v>1$. Its left side is a strictly increasing function from $F^2$ to infinity, while its right side is a strictly decreasing function from infinity to one. The intermediate value theorem therefore gives exactly one admissible $V$ for every $F>0$.

In a channel whose transverse width $b(z)$ does not depend on downstream position, the wetted area is $A(h)={\int }_0^{h}b(z)dz$ and the integrated hydrostatic pressure force divided by density is $P(h)=g{\int }_0^h(h-z)b(z)dz$. With cross-sectionally uniform velocity, volume conservation and momentum conservation give the displayed shallow water equations. Since $P^{\prime }(h)=gA(h)$, the two characteristic speeds are $u\pm c$ with $c^2=gA/A^{\prime }$. For $b(z)=C{z}^p$, $c^2=gh/(p+1)$ and the Riemann invariants are $u\pm 2(p+1)c$.

A channel with width $b(z)=\alpha z^2$ has wetted area $A=\alpha h^3/3$. Its inviscid shallow water equations have characteristic speeds $u\pm c$ and the displayed Riemann invariants. Releasing a reservoir of depth $H$ at rest into a dry channel gives a left-family rarefaction wave: $c=(6c_0-x/t)/7$, $u=6(c_0+x/t)/7$, $h/H=[(6-x/(c_{0}t))/7{]}^2$ between $x=-c_{0}t$ and $x=6c_{0}t$, where $c_0=\sqrt{gH/3}$. This width law differs from a geometrically parabolic trough, whose width is proportional to $\sqrt{z}$.

An impermeable wall at $x=-\ell$ first receives the upstream-moving head at $t_1$. The leading downstream characteristic curve of the reflected adjustment obeys $\dot{x}=u+c=(12c_0+5x/t)/7$ in the unmodified rarefaction wave, giving the displayed curve. It stays behind the inviscid dry tip $x_f=6c_{0}t$ at every finite time. The finite reservoir drains and its upstream depth decreases, while only the region behind $x_r$ has yet received the wall signal. Bottom drag or other tip physics can change the front law.

In a prismatic triangular channel with wetted area proportional to $h^2$, the Riemann invariants are $u\pm 4c$, with $c^2=gh/2$. Releasing water initially at rest into a dry channel produces a rarefaction wave with $u+4c=4c_0$ and $x/t=u-c$. Hence $c=(4c_0-x/t)/5$, $u=4(c_0+x/t)/5$ and the displayed depth profile for $-c_{0}t<x<4c_{0}t$. The dry front advances at $4c_0$ before finite-reservoir or varying-channel effects become important.

For water initially at rest at depth $H$ on one side of a dam and a dry bed on the other, a channel with $b(z)=C\sqrt{z}$ has $c_0=\sqrt{2gH/3}$. The centered rarefaction wave has $u+3c=3c_0$ and $x/t=u-c$. Thus $c=(3c_0-x/t)/4$ and $u=3(c_0+x/t)/4$ within $-c_{0}t<x<3c_{0}t$. The reservoir remains undisturbed to the left and the bed dry to the right. This is an outer shallow-water approximation before finite-reservoir boundary signals return.

For linear surface gravity waves with wavelength much larger than the depth, ${\omega }^2=gh{\kappa }^2$. It follows from ${\omega }^2=g\kappa \tanh (\kappa h)$ when $\kappa h\ll 1$, and the wave speed is $\sqrt{gh}$.

The rigid-lid approximation suppresses free surface displacement in a fluid layer while retaining a variable pressure that enforces the mass constraint. At constant depth it gives horizontally nondivergent motion. In a one-layer quasi-geostrophic approximation, removing free-surface stretching eliminates the $-\psi /R_D^2$ term from shallow-water quasi-geostrophic potential vorticity. This is also the small-horizontal-scale limit compared with the Rossby deformation radius, when that term is negligible rather than exactly absent.

The rigid-lid approximation removes external surface displacement from volume conservation while retaining its pressure as a constraint multiplier. It does not mean zero horizontal pressure gradient. In two-layer geostrophic flow, $f_0{\psi }_1=\Pi$ and $f_0{\psi }_2=\Pi -g\prime \chi$, so the upper-layer depth anomaly is $\chi =f_0({\psi }_1-{\psi }_2)/g\prime$. This pressure difference supplies internal potential-vorticity coupling.

A vertically mixed gravity current with horizontally varying reduced gravity $b$ obeys $h_t+(hu{)}_x=0$ and $(hu{)}_t+(h{u}^2+b{h}^2/2{)}_x=0$. Consequently the material acceleration is $Du/Dt=-b{h}_x-h{b}_x/2$. Advected density variables add repeated contact characteristic curves of speed $u$ to the gravity-wave characteristic curves of speeds $u\pm \sqrt{bh}$.

The shallow-water approximation applies when a fluid layer's vertical scale is much smaller than its horizontal scale. Vertical acceleration is then negligible to leading order, pressure is hydrostatic pressure, and the horizontal velocity can be represented by a depth average.

A seiche is a standing oscillation of a body of water. Its natural frequencies depend on the basin geometry, water depth and the boundary conditions at its closed walls or open connections.

An ideal uniform shallow basin, closed at one end and held at zero displacement at its open end, has its lowest standing wave with a displacement node at the opening and an antinode at the closed end. Its wavelength is four times its length. Real mouths need not satisfy this perfectly reflecting idealization.

For a homogeneous layer of mean depth $H_0$ on an f-plane, linearization about rest gives

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 $\beta >0$.

For a depth-independent horizontal velocity in a layer of reference depth $H$, the linear transports are $U=Hu$, $V=Hv$. They have dimensions of area per time and satisfy the surface continuity equation ${\eta }_t+U_x+V_y=0$. Their momentum equations are obtained by multiplying the depth-independent velocity equations by $H$.

In a homogeneous layer under the hydrostatic approximation, the horizontal pressure gradient is independent of depth. The depth derivatives of velocity obey $(u_z{)}_t-f{v}_z=0$, $(v_z{)}_t+f{u}_z=0$. If the motion starts without shear, these derivatives remain zero. Hydrostatic pressure alone does not eliminate a pre-existing freely oscillating shear.

For linear rotating shallow water equations with fixed depth $H(y)$ and constant Coriolis parameter $f$, elimination of the horizontal velocity givesAt a depth step this is interpreted distributionally, with continuity of surface elevation and normal volume flux. Because velocity elimination raises the time order, initial height alone is insufficient; the original velocity and continuity equations determine the remaining initial derivatives.

In a rotating shallow-water layer,is the horizontal scale on which gravity and Coriolis effects balance.

A steady horizontal pressure gradient can balance the Coriolis force:

The Rossby number compares advective acceleration with Coriolis acceleration, and the second ratio compares local time dependence with it. Both should be small when pressure and Coriolis terms dominate horizontal momentum. Surface stress, sharp curvature, small scales, rapid evolution and the near-equatorial region can invalidate the approximation. The vertical hydrostatic equilibrium assumption is a separate scale approximation.

An atmospheric isobar is a line of equal pressure on a specified atmospheric map. Its spacing indicates the horizontal pressure gradient at that map's reference level; a closer spacing means a larger gradient for the same contour interval.

The gradient wind retains the centripetal acceleration of a curved flow as well as the Coriolis force and horizontal pressure gradient force. For cyclonic circular flow in the northern hemisphere, $v^2/r+fv={\rho }^{-1}dp/dr$, with $r$ increasing away from the low-pressure centre.

For a conventional Northern-Hemisphere anticyclone let $G=-p_r/\rho >0$. Gradient wind balance gives $V^2/r-fV+G=0$, with $V_{\pm }=fr[1\pm \sqrt{1-4G/(f^{2}r)}]/2$. The minus branch approaches the geostrophic wind at weak forcing; the plus branch is anomalous. Reality requires $G\le f^{2}r/4$, so a smooth central pressure maximum must have a radial pressure force vanishing at least linearly toward its centre.

The geostrophic wind is the horizontal flow for which the pressure gradient force balances the Coriolis force. It runs along atmospheric isobars and has speed proportional to their pressure gradient, at fixed density and nonzero Coriolis parameter. Curvature, friction and small spatial scales can invalidate that approximation.

A steady horizontal parcel with drag acceleration $-r\mathbf{u}$ obeys $0=-{\nabla }_{h}p/\rho -f\hat{\mathbf{z}}\times \mathbf{u}-r\mathbf{u}$. Inverting the horizontal rotation-plus-drag matrix gives the title. For $r>0$, the velocity has a component toward lower pressure and a component along atmospheric isobars. For $p_x=0$, $f>0$, $p_y>0$, both $u,v$ are negative. The cross-isobar angle is $\arctan (r/f)$, and the speed is smaller than the corresponding frictionless wind. A vertically resolved turbulent stress is not generally equivalent to one constant $r$.

Dot the steady linear-drag momentum equation with the velocity. The Coriolis force does no work, so pressure-gradient work balances frictional dissipation. In a strictly geostrophic flow the wind is along isobars and pressure work vanishes. Cross-isobar motion supplies the positive pressure work needed to maintain a dissipative equilibrium.

With $x$ eastward and $y$ northward, a purely meridional pressure gradient gives the velocity in the title. For $f>0$, pressure increasing northward gives westward wind, leaving lower pressure to the wind's left. Reversing the pressure gradient reverses the velocity. At a fixed gradient and density the speed scales as $1/|\sin \varphi |$; the formal equatorial divergence signals breakdown of geostrophic balance, rather than a real infinite wind.