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

Figure 1. Quarter Annulus

The computational domain (Figure 1) consists of a three-level telescoping Cartesian grid with a grid resolution of 4, 2, and 1 km for each level. Higher resolution is specified near the inner and outer boundaries in order to reduce errors associated with the representation of the curved boundaries with squares. The computational grid is shown in Figure 1. The grid has 1,160 active ocean cells. The important model settings are shown in Table 2.

Table 2. General settings for the quarter annulus test

Parameter	Value
Solution scheme	Implicit
Time step	600 sec (10 min)
Simulation duration	120 hrs
Ramp period	24 hrs
Advection terms	Off
Mixing terms	Off
Wall friction	Off
Coriolis	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.

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Test_Cases