shallowwater

Shallow water refers to a set of fluid-dynamics models that describe the flow of a thin layer of fluid where horizontal scales greatly exceed vertical depth. This regime assumes hydrostatic pressure and negligible vertical accelerations, providing a depth-averaged description of the flow. The primary variables are the water depth h(x,y,t) and the depth-averaged horizontal velocity U(x,y,t).

The governing equations express conservation of mass and momentum. Let η be the free-surface elevation and b the bottom topography, with h = η − b. The equations in conservative form are: ∂h/∂t + ∇·(h U) = 0; ∂(h U)/∂t + ∇·(h U ⊗ U) + g h ∇η = 0. These capture advection of depth and momentum and the pressure force due to surface slope; bed slope and friction can be added as source terms.

Applications include dam-break analysis, river and coastal hydraulics, tsunami and storm-surge propagation, and some atmospheric flows. The model is computationally efficient and forms the basis of many numerical schemes, especially finite-volume methods.

Limitations include neglect of vertical density stratification, nonhydrostatic effects, and complex vertical structure; may be inaccurate for steep slopes, rapidly varying depth, or strongly three-dimensional flows. Extensions such as nonhydrostatic shallow-water equations increase fidelity at higher computational cost.