Shallow water equations

The shallow water equations (also called Saint Venant equations after Adhémar Jean Claude Barré de Saint-Venant) are a set of equations that describe the flow below a horizontal pressure surface in a fluid. The flow these equations describe is the horizontal flow caused by changes in the height of the pressure surface of the fluid. Shallow water equations can be used in atmospheric and oceanic modelling, but are much simpler than the primitive equations. Shallow water equation models have only one vertical level, so they cannot encompass any factor that varies with height.

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In general, nearly all forms of the shallow water equations relate to the three variables (u, v, η), and their evolution over space and time.

Equations

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

$u$ is the zonal velocity (or velocity in the x dimension).
$v$ is the meridional velocity (or velocity in the y dimension).
$H$ 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 of gravity.
$f$ is the term corresponding to the Coriolis force, and is 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.

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.

Category: Equations of fluid dynamics