Shallow water equations

The shallow water equations are a set of equations that describe the flow below a horizontal pressure surface in a fluid. The flow that these equations describe is the horizontal flow that has been forced by changes in the height of the pressure surface of the fluid. Shallow water equations can be used in atmospheric and ocean modelling but are much simpler then the primitive equations. Shallow water equation models only have one vertical level so can not encompass anything that varies with height.

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.

Definitions

• $u$ 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.
• $\eta$ is the deviation of the horizontal pressure surface from its mean.
• $g$ is gravity
• $f$ is the term corresponding to the Coriolis force, and is equal to $2\Omega sin(\phi )$, where $\Omega$ is the angular rotation rate of the Earth ($2\pi /24$ radians/hour), and $\phi$ is the latitude.
• $b$ is the viscous drag.

The equations

${\frac {Du}{Dt}}-fv=-g{\frac {\partial \eta }{\partial x}}-bu$

${\frac {Dv}{Dt}}+fu=-g{\frac {\partial \eta }{\partial y}}-bv$

${\frac {\partial \eta }{\partial t}}-{\frac {\partial (u(H+\eta ))}{\partial x}}-{\frac {\partial (v(H+\eta ))}{\partial y}}=0$

Wave modelling by shallow water equations

Shallow water equations can be used to model Rossby and Kelvin waves in the atmosphere and in the oceans as well as gravity waves in a smaller domain (e.g. surface waves in a bath).

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 propogate away from the splash locations and reflect off of the bathtub walls.