Diffusive Wave Equation

Diffusive Wave Approximation

The diffusive wave equation (DWE) is a simplification of the Shallow Water Equations valid when the flow is dominated by the balance between gravity and bottom friction. Neglecting the local and advective accelerations, turbulent mixing, Coriolis, and meteorological forcing, the momentum equation reduces to

$\begin{array}{l}\displaystyle g \nabla \eta = - \dfrac{ \boldsymbol{\tau}_b }{ \rho R }\end{array}$

where

$\begin{array}{l}g\end{array}$ : gravitational acceleration [L/T²]
$\begin{array}{l}\eta\end{array}$ : water surface elevation [L]
$\begin{array}{l}\boldsymbol{\tau}_b\end{array}$ : bottom shear stress vector [M/L/T²]
$\begin{array}{l}\rho\end{array}$ : water density [M/L³]
$\begin{array}{l}R\end{array}$ : hydraulic radius [L]

Introducing Manning's friction law (see Bottom Friction) and solving for the velocity yields a flow that is always directed down the water surface slope with a magnitude given by Manning's equation with the friction slope equal to the magnitude of the water surface slope

$\begin{array}{l}\displaystyle \boldsymbol{V} = - \dfrac{ R^{2/3} }{ n } \dfrac{ \nabla z_s }{ \left| \nabla z_s \right|^{1/2} }\end{array}$

where

$\begin{array}{l}\boldsymbol{V}\end{array}$ : depth-averaged velocity vector [L/T]
$\begin{array}{l}n\end{array}$ : Manning roughness coefficient [T/L^1/3]
$\begin{array}{l}\left| \nabla z_s \right|\end{array}$ : magnitude of the water surface slope [-]

Governing Equation

Substituting the velocity into the continuity equation produces a single scalar equation for the water surface elevation referred to as the Diffusive Wave Equation (DWE):

1)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial h}{\partial t} = \nabla \cdot (\beta \nabla z_s) + S\end{array}$

where:

$\begin{array}{l}h\end{array}$ : water depth [L]
$\begin{array}{l}\beta\end{array}$ : non-linear diffusion coefficient
$\begin{array}{l}z_s\end{array}$ : water surface elevation [L]
$\begin{array}{l}S\end{array}$ : water source-sink term [L]

The non-linear diffusion coefficient is given by

2)	$\begin{array}{l}\displaystyle \displaystyle \displaystyle \beta = \frac{R^{2/3}h}{n \left\| \nabla z_s \right\| ^{1/2}}\end{array}$

where

$\begin{array}{l}R\end{array}$ : hydraulic radius [L]
$\begin{array}{l}n\end{array}$ : Manning's roughness coefficient [T/L^1/3]

As the name implies, this is essentially a diffusion equation in which the diffusion coefficient is a function of the slope of the water surface. Since the diffusion coefficient $\begin{array}{l}\beta\end{array}$ includes the water surface slope in the denominator the diffusion coefficient goes to infinity as the water surface approaches flat. The time step limitation of explicit schemes is inversely related to the diffusion coefficient, leading to infinitesimal time steps as the water surface becomes flat. Various approaches such as linearization of $\begin{array}{l}\beta\end{array}$ below some threshold slope are possible solutions, at the expense of a less accurate solution.

Applicability

The DWE is appropriate for gradually varied, friction-dominated flow: floodplain filling and draining, overland flow, and backwater-dominated systems. It is not appropriate where inertia matters: rapid flood waves and dam breaks, flow around structures, high-velocity or supercritical transitions, seiching, and momentum-driven eddies. Since the DWE has no advection or mixing terms, forces from wind, Coriolis, and turbulence are also unavailable. In exchange, the DWE is solved implicitly with the same machinery as the full equations (see Semi-Implicit Shallow Water Solver) and typically tolerates substantially larger time steps.