94
5 2D Shallow-Water Modelling
5.1.5 Stability Criterion
The CFL criterion for the two-dimensional shallow-water equations is given by:
Δt ≤
min (Δx, Δy)
√
2gh max
(5.4)
where h max is the maximum water depth encountered in the model domain.
5.2 Exercise 8: Long Waves in a Shallow Lake
5.2.1 Aim
The aim of this exercise is to simulate the progression of long circular surface gravity waves in a two-dimensional domain.
5.2.2 Task Description
We consider a square lake of 500 m× 500 m in areal extent and 10 m in depth
using equidistant lateral grid spacings of Δx = Δy = 10 m. Lateral boundaries
are closed. The floodin algorithm is included. Lake water is of uniform density.
Forcing consists of an initial sea-level elevation of 1 m in the central grid cell that,
when released, will create a tsunami-type wave spreading out in all directions. Such
waves, created by a point-source disturbance, are called circular waves. The time
step is chosen at Δt = 0.1 s, which satisfie the CFL stability criterion. The simulation is run for 100 s with data outputs at every 0.5 s.
5.2.3 Sample Code and Animation Script
The two-dimensional shallow-water model is a straight-forward extension of the 1D
channel model used in Exercise 6. Model variables are now two-dimensional arrays
such as “eta(j,k)” where “j” and “k” are grid cell pointers. The folder “Exercise 8”
of the CD-ROM contains the computer codes for this exercise. The f le “info.txt”
contains additional information. Note that SciLab animation scripts can be run while
the FORTRAN code is executed in the background. This is useful for long simulations to check whether the results are reasonable. If not, the FORTRAN run can
be stopped by simultaneously pressing and in the Command Prompt
window.