Shallow Water Equations

Overview

The hydraulics model solves the two-dimensional depth-averaged shallow water equations (SWE): a statement of conservation of water volume together with a balance of momentum for the depth-averaged velocity. The SWE are obtained by integrating the three-dimensional incompressible Navier–Stokes equations over the water column between the bed and the free surface, under the following assumptions:

Hydrostatic pressure. Vertical accelerations are negligible compared to gravity, so the pressure at any depth is determined by the weight of the water above it. This is accurate when horizontal length scales are much larger than the water depth.
Depth-uniform velocity. The horizontal velocity is represented by its depth average. The momentum carried by deviations from the depth average is either neglected or represented through closure terms (turbulent mixing, secondary flow).
Incompressible water of constant density.
Mild bed slopes. The bed elevation varies gradually so the vertical component of the bed-normal direction is close to one.

The water surface elevation, bed elevation, and water depth are related by

$\begin{array}{l}\displaystyle z_s = z_b + h\end{array}$

where

$\begin{array}{l}z_s\end{array}$ : water surface elevation [L]
$\begin{array}{l}z_b\end{array}$ : bed elevation [L]
$\begin{array}{l}h\end{array}$ : water depth [L]

Continuity Equation

Conservation of water volume is expressed as

$\begin{array}{l}\displaystyle \dfrac{\partial h}{\partial t} + \nabla \cdot ( h \boldsymbol{V} ) = q\end{array}$

where

$\begin{array}{l}t\end{array}$ : time [T]
$\begin{array}{l}h\end{array}$ : water depth [L]
$\begin{array}{l}\boldsymbol{V} = (u, v)\end{array}$ : depth-averaged velocity vector [L/T]
$\begin{array}{l}q\end{array}$ : net water source/sink rate per unit plan area (e.g. precipitation excess, internal inflows) [L/T]
$\begin{array}{l}\nabla = ( \partial / \partial x, \partial / \partial y )\end{array}$ : horizontal gradient operator [1/L]

An important property of the model is that continuity is actually enforced on the water volume of each computational cell rather than on a cell-average depth, so that the fine-scale terrain within a cell is fully accounted for. This is described in Subgrid Bathymetry and Hydraulic Property Tables.

Momentum Equation

The depth-averaged momentum balance is written in the non-conservative (velocity) form

$\begin{array}{l}\displaystyle \dfrac{\partial \boldsymbol{V}}{\partial t} + ( \boldsymbol{V} \cdot \nabla ) \boldsymbol{V} + f_c \, \textbf{k} \times \boldsymbol{V} = - g \nabla z_s + \dfrac{1}{h} \nabla \cdot ( \boldsymbol{\nu}_t \, h \nabla \boldsymbol{V} ) - \dfrac{ \boldsymbol{\tau}_b }{ \rho R } + \dfrac{ \boldsymbol{\tau}_s }{ \rho h } - \dfrac{1}{\rho} \nabla p_a\end{array}$

where

$\begin{array}{l}\boldsymbol{V}\end{array}$ : depth-averaged velocity vector [L/T]
$\begin{array}{l}g\end{array}$ : gravitational acceleration [L/T²]
$\begin{array}{l}z_s\end{array}$ : water surface elevation [L]
$\begin{array}{l}f_c\end{array}$ : Coriolis parameter [1/T]
$\begin{array}{l}\textbf{k}\end{array}$ : vertical unit vector [-]
$\begin{array}{l}\boldsymbol{\nu}_t\end{array}$ : turbulent (eddy) viscosity tensor [L²/T]
$\begin{array}{l}\boldsymbol{\tau}_b\end{array}$ : bottom shear stress vector [M/L/T²]
$\begin{array}{l}R\end{array}$ : hydraulic radius [L]
$\begin{array}{l}\boldsymbol{\tau}_s\end{array}$ : wind surface stress vector [M/L/T²]
$\begin{array}{l}\rho\end{array}$ : water density [M/L³]
$\begin{array}{l}p_a\end{array}$ : atmospheric pressure at the water surface [M/L/T²]

The terms on the left-hand side are the local acceleration, the advective (convective) acceleration, and the Coriolis acceleration due to the rotation of the Earth. The Coriolis parameter is computed from the latitude as

$\begin{array}{l}\displaystyle f_c = 2 \omega \sin \varphi\end{array}$

$\begin{array}{l}\omega = 2\pi / 86400 \; \text{s}^{-1} \approx 7.2722 \times 10^{-5} \; \text{s}^{-1}\end{array}$ : angular speed of the Earth's rotation, taken as one rotation per day [1/T]
$\begin{array}{l}\varphi\end{array}$ : latitude [-]

The Coriolis effect is negligible for small domains and short reaches but can matter for very large waterbodies; it is disabled by default. The terms on the right-hand side are, in order:

Pressure gradient. Under the hydrostatic assumption the depth-averaged pressure gradient reduces to the gravity-driven slope of the water surface. Because the gradient acts on the water surface elevation rather than on the depth, the bed slope term is included automatically and the formulation is well balanced (a flat water surface over an irregular bed produces no spurious flow).
Turbulent mixing (horizontal momentum diffusion), representing the combined effect of turbulent momentum exchange and vertical shear dispersion. The eddy viscosity may be anisotropic, with different values along and across the flow direction. The closures used to compute it are described in Turbulence Closures. The conservative form, with the eddy viscosity and depth inside the divergence, ensures that mixing redistributes momentum without creating or destroying it.
Bottom friction, parameterized with Manning's roughness formulation acting through the hydraulic radius, as described in RAS Bottom Friction.
Wind surface stress and atmospheric pressure gradient, described in Wind and Atmospheric Pressure Forcing.

An optional secondary (helical) flow correction can additionally be included for curved channels; it adds the dispersive momentum transport produced by the vertical structure of flow in bends, which the depth-averaged equations otherwise cannot represent.

Simplified Forms

When the flow is friction dominated and the inertial terms are negligible, the momentum equation can be reduced to a balance between the water surface slope and bottom friction. This leads to the diffusion wave form of the equations, described in Diffusion Wave Approximation, which is cheaper and more robust but excludes inertial effects.