Drying and flooding

The main requirement for drying and flooding is that the vertically integrated fluxes $ U$ and $ V$ are controlled such that at no point a negative water depth occurs. It is clear that parts of the physics which play an important role in very shallow water of a few centimetres depth like non-hydrostatic effects are not included in the equations. However, the model is designed in a way that the control of $ U$ and $ V$ in very shallow water is mainly motivated by the physics included in the equations rather than by defining complex drying and flooding algorithms. It is assumed that the major process in this situation is a balance between pressure gradient and bottom friction. Therefore, in the case of very shallow water, all other terms are multiplied with the non-dimensional factor $ \alpha$ which approaches zero when a minimum water depth is reached.

By using formulation (71) for calculating the bottom drag coefficient $ R$, it is guaranteed that $ R$ is exponentially growing if the water depth approaches very small values. This slows the flow down when the water depth in a velocity point is sinking and also allows for flooding without further manipulation.

Figure 10: Sketch explaining the principle of pressure gradient minimisation during drying and flooding over sloping bathymetry.
\begin{figure}\begin{center}
\fbox{
\begin{picture}(23.0,21.0)(-2.5,-0.5)
\thinl...
...17.2,8){\makebox(0,0)[l]{$-H_{i+1,j}$}}
\end{picture}}\end{center}\end{figure}

In this context, one important question is how to calculated the depth in the velocity points, $ H^u$ and $ H^v$, since this determines how shallow the water in the velocity points may become on sloping beaches. In ocean models, usually, the depth in the velocity points is calculated as the mean of depths in adjacent elevation points (T-points):

$\displaystyle H^u_{i,j}=\frac12\left(H_{i,j}+H_{i+1,j}\right), \qquad H^v_{i,j}=\frac12\left(H_{i,j}+H_{i,j+1}\right).$ (54)

Other models which deal with drying and flooding such as the models of Duwe (1988) and Casulli and Cattani (1994) use the minimum of the adjacent depths in the T-points:

$\displaystyle H^u_{i,j}= \min\{H_{i,j},H_{i+1,j}\}, \qquad H^v_{i,j}= \min\{H_{i,j},H_{i,j+1}\}.$ (55)

This guarantees that all depths in the velocity points around a T-point are not deeper than the depth in the T-point. Thus, when the T-point depth is approaching the minimum depth, then all depths in the velocity points are also small and the friction coefficient correspondingly large.

Each of the methods has however drawbacks: When the mean is taken as in (54), the risk of negative water depths is relatively big, and thus higher values of $ D_{min}$ have to be chosen. When the minimum is taken, large mud-flats might need unrealistically long times for drying since all the water volume has to flow through relatively shallow velocity boxes. Also, velocities in these shallow boxes tend to be relatively high in order to provide sufficient transports. This might lead to numerical instabilities.

Therefore, GETM has both options, (54) and (55) and the addition of various other options such as depth depending weighting of the averaging can easily be added. These options are controlled by the GETM variable vel_depth_method, see section 6.1.9 (subroutine uv_depths) documented on page [*].

If a pressure point is dry (i.e. its bathymetry value is higher than a neighbouring sea surface elevation), the pressure gradient would be unnaturally high with the consequence of unwanted flow acceleration. Therefore this pressure gradient will be manipulated such that (only for the pressure gradient calculation) a virtual sea surface elevation $ \tilde \zeta$ is assumed (see figure 10). In the situation shown in figure 10, the left pressure point is dry, and the sea surface elevation there is for numerical reasons even slightly below the critical value $ -H_{i,j}+H_{\min}$. In order not to let more water flow out of the left cell, the pressure gradient between the two boxes shown is calculated with a manipulated sea surface elevation on the right, $ \tilde \zeta_{i+1,j}$.

See also Burchard et al. (2004) for a description of drying and flooding numerics in GETM.

kklingbe 2017-10-02