Shallow water equations

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The shallow water equations (also called Saint Venant equations after Adhémar Jean Claude Barré de Saint-Venant) are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface).

The equations are derived from depth-integrating the Navier-Stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity of the fluid is small. It can be shown from the momentum equation that vertical pressure gradients are nearly hydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the velocity field is nearly constant throughout the depth of the fluid. Taking the vertical velocity and variations throughout the depth of the fluid to be exactly zero in the Navier-Stokes equations, the shallow water equations are derived.

Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow water equations are widely applicable. They are used with Coriolis forces in atmospheric and oceanic modelling, as a simplification of the primitive equations of atmospheric flow.

Shallow water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the mean state is sufficiently simple, the vertical variations can be separated from the horizontal and several sets of shallow water equations can describe the state.

Shallow water waves
Output from a shallow water equation model of water in a bathtub. The water experiences five splashes which generate surface gravity waves that propagate away from the splash locations and reflect off the bathtub walls.

1 Equations
- 1.1 Conservative form
- 1.2 Non-conservative form
2 Wave modelling by shallow water equations

[edit] Equations

[edit] Conservative form

The shallow water equations are derived from equations of conservation of mass and conservation of momentum (the Navier-Stokes equations), which hold even when the assumptions of shallow water break down, such as across a hydraulic jump. In the case of no Coriolis, frictional or viscous forces, the shallow-water equations are:

$\begin{align} \frac{\partial \eta }{\partial t} + \frac{\partial (\eta u)}{\partial x} + \frac{\partial (\eta v)}{\partial y} = 0\\[3pt] \frac{\partial (\eta u)}{\partial t}+ \frac{\partial}{\partial x}\left( \eta u^2 + \frac{1}{2}g \eta^2 \right) + \frac{\partial (\eta u v)}{\partial y} = 0\\[3pt] \frac{\partial (\eta v)}{\partial t} + \frac{\partial (\eta uv)}{\partial x} + \frac{\partial}{\partial y}\left(\eta v^2 + \frac{1}{2}g \eta ^2\right) = 0. \end{align}$

[edit] Non-conservative form

The equations can also be written in terms of velocities instead of momenta. Since velocities are not subject to a fundamental conservation equation, in this form the equations do not hold across a shock or hydraulic jump.

$\begin{align} \frac{Du}{Dt} - f v& = -g \frac{\partial \eta}{\partial x} - b u\\[3pt] \frac{Dv}{Dt} + f u& = -g \frac{\partial \eta}{\partial y} - b v\\[3pt] \frac{\partial \eta}{\partial t}& = - \frac{\partial (u(H + \eta))}{\partial x} - \frac{\partial (v(H + \eta))}{\partial y} \end{align}$

$u$ is the velocity in the x dimension, or zonal velocity
$v$ is the velocity in the y dimension, or meridional velocity
$H$ is the mean height of the horizontal pressure surface.
$η$ is the deviation of the horizontal pressure surface from its mean
$g$ is the acceleration due to gravity.
$f$ is the coefficient of the Coriolis force, on Earth equal to 2Ω sin(φ), where Ω is the angular rotation rate of the Earth (π/12 radians/hour), and φ is the latitude.
$b$ is the viscous drag.

[edit] Wave modelling by shallow water equations

Shallow water equations can be used to model Rossby and Kelvin waves in the atmosphere, rivers, lakes and oceans as well as gravity waves in a smaller domain (e.g. surface waves in a bath). In order for shallow water equations to be valid, the wave length of the phenomenon they are supposed to model has to be much higher than the depth of the basin where the phenomenon takes place. Shallow water equations are especially suitable to model tides which have very large length scales (over hundred of kilometers). For tidal motion, even a very deep ocean may be considered as shallow as its depth will always be much smaller than the tidal wave length.

The image on the right is output from a shallow water equation model of water in a bathtub. The water experiences 5 splashes which generate surface gravity waves that propagate away from the splash locations and reflect off of the bathtub walls.