Set of partial differential equations that describe the flow below a pressure surface in a fluid
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 (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface).[1] The shallow-water equations in unidirectional form are also called (de) Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant (see the related section below).
The equations are derived[2] 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 scale of the fluid is small compared to the horizontal velocity scale. 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 horizontal velocity field is constant throughout the depth of the fluid. Vertically integrating allows the vertical velocity to be removed from the equations. The shallow-water equations are thus derived.
While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. This is an important distinction because, for example, the vertical velocity cannot be zero when the floor changes depth, and thus if it were zero only flat floors would be usable with the shallow-water equations. Once a solution (i.e. the horizontal velocities and free surface displacement) has been found, the vertical velocity can be recovered via the continuity equation.
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 modeling, 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.
^Vreugdenhil, C.B. (1986). Numerical Methods for Shallow-Water Flow. Water Science and Technology Library. Vol. 13. Springer, Dordrecht. p. 262. doi:10.1007/978-94-015-8354-1. ISBN 978-90-481-4472-3.
^"The Shallow Water Equations" (PDF). Archived from the original (PDF) on 2012-03-16. Retrieved 2010-01-22.
and 23 Related for: Shallow water equations information
The shallow-waterequations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the...
water Shallow waterequationsShallowWater (film) Shallow-water blackout (disambiguation) All pages with titles containing Shallowwater This disambiguation...
equation – Physics phenomenon and formula Shallow water equations – Set of partial differential equations that describe the flow below a pressure surface in...
the second equation of the shallowwaterequations are respectively called the zonal and meridional momentum equations, and the third equation is the continuity...
for defining a Sensor Web Shallowwaterequations, a set of equations that describe flow below a pressure surface Snow water equivalent Society of Women...
surface, the shallowwater approximation can be used. Using this approximation, Rossby showed in 1939, by integrating the shallowwaterequations over the...
scheme of the primitive equation model with an icosahedral-hexagonal grid system and its application to the shallow-waterequations". Short- and Medium-Range...
hydrodynamics General equation of heat transfer Geophysical fluid dynamics Potential vorticity Quasi-geostrophic equationsShallowwaterequations Taylor–Goldstein...
Mild-slope equation – Physics phenomenon and formula Rogue wave – Unexpectedly large transient ocean surface wave Shallowwaterequations – Set of partial...
solves the so-called shallowwaterequations, also known as the Saint Venant equations. TELEMAC-2D solves the Saint-Venant equations using the finite-element...
simplest equations that describe the dynamics of Kelvin waves are the linearized shallowwaterequations for homogeneous, in-viscid flows. These equations can...
textbook which made extensive use of ripple tanks to illustrate waves Shallowwaterequations Wave tank Breithaupt, Jim (2000) New Understanding Physics for...
systems, the Conservative Vorticity Equation, the Rayleigh-Bénard Convection Equations, and the ShallowWaterEquations. Moreover, Lorenz can be credited...
plasmas, liquid metals, and salt water. The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism. Relativistic...
later paper in 1940 relaxed this theory from 2D flow to quasi-2D shallowwaterequations on a beta plane. In this system, the atmosphere is separated into...
near the leading edge. Gravity currents may be simulated by the shallowwaterequations, with special dispensation for the leading edge which behaves as...
the well water level: F = A d z d t {\displaystyle F=A{\frac {dz}{dt}}} Combining the above three equations yields a simple differential equation in z: R...
Liao functionals. When calculating a solution to the shallowwaterequations, the solution (water height) might only be calculated for points every few...
since usually the equations are considered in the high Froude limit of negligible external field, leading to homogeneous equations that preserve the mathematical...
referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known...
Global Information