# Quarter Annulus

Test C1-Ex3: Tidal Propagation in a Quarter Annulus

Purpose

The purpose of this verification test is to assess the model performance in simulating long wave propagation. The case is useful for testing the model performance and symmetry for a non-rectangular domain with a tidal forcing specified on one of the curved boundaries. Because there is no bottom friction or mixing, the test case is also useful for looking at numerical dissipation.

Problem

Lynch and Gray (1978) presented the analytical solution for depth-averaged long-wave propagation in an annular domain. The case was for a linearly sloping bed, and without bottom friction, Coriolis force, or horizontal mixing. The offshore boundary consisted of a single tidal constituent (see Figure7). Table7 summarizes the important model settings used for this test.

Table 7. Quarter annulus setup parameters
Parameter Value
Deepwater tidal amplitude 0.3048 m (1 ft)
Tidal period 12.42 hr (M2 tide)
Inner water depth 10.02 m
Outer water depth 25.05 m
Bathymetry profile Linear
Bottom friction None
Mixing terms Off
Coriolis force Off

Figure 7. Computational domain for tidal propagation in a quarter annulus.

Model Setup

The computational grid (Figure 8) consists of a three-level telescoping Cartesian grid with resolution of 4, 2, and 1 km for the three levels. 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 grid has 1,160 active ocean cells. Model settings are shown in Table 8.

Table 8. CMS-Flow setup parameters for the quarter annulus test case.
Parameter Value
Solution scheme Implicit
Time step 10 min
Simulation duration 120 hr
Ramp duration 24 hr
Mixing terms Off
Wall friction Off
Coriolis force Off

Figure 8. Computation grid used for tidal propagation in a quarter annulus.

Results and Discussion

Figure 9 shows a time series of water levels at the inner edge of the simulation domain. The goodness-of-fit statistics are listed in Table 9. The model accurately predicts the wave phase but slightly overestimates the amplitude by approximately 0.01 m. No significant numerical dissipation is observed or numerical instability. The simulation takes about 1 min on a Windows PC on a single 2.67 GHz processor.

Figure 9. Comparison of analytical (solid black) and calculated (red dots) water surface elevations at the center of the inner radius.

Table 9. Water level goodness-of-fit statistics for the quarter annulus test case
Statistic Value
NRMSE, % 3.3
NMAE, % 2.7
${\displaystyle R^{2}}$ 0.999
Bias, m 0.002 m

An example of the simulated water level and current velocity magnitude fields is shown in Figure 10. The water level contours are very smooth and do not show any significant instabilities. However, the current magnitude shows some errors at the offshore boundary. This is due to staircase representation of the curved open boundary. This problem would be fixed by specifying the analytical current velocities at the offshore boundary, but it was not done in this example. Sensitivity tests with smaller time steps showed that the problem persists for smaller time steps. For practical applications all model forcing is specified on straight boundaries and this problem does not occur as demonstrated in subsequent test cases.

Figure 10. Snap shot of water levels at 62 hr (a) and current magnitude at 65.5 hr (b).

Conclusions and Recommendations

The CMS-Flow can accurately simulate linear long-wave propagation in a quarter annulus with a linear bed and zero bottom friction and Coriolis. The water level NRMSE, NMAE, and R2 were 3.3%, 2.7%, and 0.999, respetively. For practical applications, it is recommended to specify water level boundary conditions on straight boundaries. If a curved forcing boundary is necessary, then it is recommended to specify both water levels and current velocities.