Long wave propagation
Problem
Lynch and Gray (1978) presented the analytical solution for depth-averaged long-wave propagation in am annular domain. Here the case tested with a linearly sloping bed, and without bottom friction, Coriolis, or horizontal mixing. The offshore boundary consists of an single tidal constituent. Table 1 summarizes the important model settings used for this test.
Table 1. General Settings for quarter annulus test
| Parameter | Value |
| Deepwater tidal amplitude | 0.3048 m |
| Tidal period | 12.42 hrs (M2) |
| Inner radius | 60.96 km |
| Outer radius | 152.4 km |
| Depth at inner radius | 10.02 m |
| Depth at outher | 25.05 m |
| Bottom friction | None |
| Horizontal eddy viscosity | 0.0 |
| Corilis | 0.0 |
Model Setup
A Cartesian grid is used with two levels of refinement. The computational grid is shown in Figure 1. The general model settings are shown in Table 2.
Table 2. General Settings for Flow over a Bump
| Parameter | Value |
| Time step | 600 sec (10 min) |
| Ramp period | 24 hrs |
| Simulation time | 120 hrs |
| Grid resolution | 2-8 km |
| Number of active ocean cells | 1160 |
| Wall friction | Off |
Results
The figure below shows a time series of water levels at the inner edge of the model domain. The goodness of fit statistics are listed in the below. The model accurately predicts the wave phase but slightly overestimates the amplitude by approximately 0.01 m.
Table 3. Goodness of fit statistics for the water elevation
| Statistic | Value |
| RMSE | 0.014 m |
| RMAE | 0.028 |
| R^2 | 0.999 |
| Bias | 0.003 m |
References
- Lynch, D.R., and Gray, W.G. (1978). "Analytical solutions for computer flow model testing," J. Hydraulics Division, 104, 1409-28.