 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. 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. Image File history File links No higher resolution available. ...
Adhémar Jean Claude Barré de Saint-Venant (August 23, 1797 - January 6, 1886) was a mechanician who contributed to early stress analysis and also developed the one-dimensional unsteady open channel flow shallow water equations or Saint-Venant equations that are a fundamental set of equations used in modern...
The primitive equations are a version of the Navier-Stokes equations that describe hydrodynamical flow on the sphere under the assumptions that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere. ...
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}](http://upload.wikimedia.org/math/6/3/6/636ee84a1c5c8a185a5c13f3c2376909.png) - 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.
In geography, geophysics, and meteorology, zonal usually means along a latitude circle, i. ...
Meridional is a geographic term that means along a north-south direction, or relative to a meridian (opposite: zonal, east-west). ...
Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...
Gravity redirects here. ...
In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ...
An object falling through a gas or liquid experiences a force in direction opposite to its motion. ...
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. Rossby (or planetary) waves are large-scale motions in the ocean or atmosphere whose restoring force is the variation in Coriolis effect with latitude. ...
A Kelvin wave is a wave in the ocean or atmosphere that balances the Earths Coriolis force against a topographic boundary such as a coastline. ...
Ocean wave Wave clouds over Theresa, Wisconsin, USA Atmospheric gravity waves as seen from space. ...
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. Ocean wave Wave clouds over Theresa, Wisconsin, USA Atmospheric gravity waves as seen from space. ...
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