Threshold gradient for overland flow
In flat areas with ponded water, the head gradient between grid cells will be zero or nearly zero. As the head gradient goes to zero, Dt must also become very small to maintain accuracy. To allow the simulation to run with longer time steps and dampen any numerical instabilities in areas with low lateral gradients, the calculated intercell flows are multiplied by a damping factor when the gradients are close to zero.
Essentially, the damping factor reduces the flow between cells. You can think of the damping function as an increased resistance to flow as the gradient goes to zero. In other words, the flow goes to zero faster than the time steps goes to zero. This makes the solution more stable and allows for larger time steps. However, the resulting gradients will be artificially high in the affected cells and the solution will begin to diverge from the Manning solution. At very low gradients this is normally insignificant, but as the gradient increases the differences may become noticeable. Therefore, the damping function is only applied when the gradient between cells is below a user-defined threshold.
The details of the available functions can be found in the Section Low gradient damping function (V1 p. 467).
For both functions and both the explicit and implicit solution methods, each calculated intercell flow in the current time step is multiplied by the local damping factor, FD, to obtain the actual intercell flow. In the explicit method, the flow used to calculate the Courant criteria are also corrected by FD.
The damping function is controlled by the user-specified threshold gradient (see Common stability parameters (V1 p. 193) for the Overland Flow), below which the damping function becomes active.
The choice of appropriate threshold value depends on the slope of the flow surface. Based on both actual model tests in Florida and synthetic setups, the following conclusions can be reached:
· A Threshold gradient greater than the surface slope can lead to excessive OL storage on the surface that takes a long time to drain away.
· A Threshold gradient equal to the surface slope is often reasonable, but there may still be some excess storage on the surface.
· Threshold values less than the surface slope typically cause rapid drainage and give nearly the same answers.
· Threshold values below 1e-7 do not significantly improve the results even if the topography is perfectly flat.
· In general, you should used the highest value possible. Lower values may increase accuracy but at the expense of run time.
Therefore, we can safely recommend a Threshold gradient in the range of 1e-4 to 1e-5, with a default value of 1e-4. For many floodplains, 1e-4 or 1e-5 should be sufficient. In flood plains with very flat relief, 1e-6 may be used. Lower values are probably never necessary.
Since most discharge happens during and immediately after an event, the Threshold gradient is likely to be most important when there is significant ponding that lasts over several time steps and drains to a boundary or MIKE Hydro River. Ponded water that infiltrates or evaporates and experiences limited lateral flow will not be affected by the Threshold value.
If the topography slope requires a low Threshold, but the solution is unstable at low threshold values, solution stability may be improved with the Explicit solver by reducing the Maximum Courant number until the solution becomes stable. With the Implicit solver, you may need to change the solver parameters.
A low Threshold gradient will increase your simulation time. So, the final value that you use, may be a compromise between simulation length and accuracy of the flow in low gradient conditions.
If you have stagnant ponded water in your model, then the intercell gradient in these areas will be nearly zero. If you lower your Threshold gradient, your simulation performance may be adversely impacted, simply because the OL solver will begin to calculate flow sloshing back and forth in these areas. Not only will the OL solver have to work harder, the OL time step will likely also decrease because of the very low gradients. Thus, the Threshold gradient effectively reduces intercell flow in stagnant areas to zero allowing the Courant criteria to be satisfied at much higher time step lengths.