Marine Ecosystem Dynamics Modeling Laboratory

Case 06. Rossby Equatorial Soliton

Case 6.Rossby Equatorial Soliton

Test Case Summary: In this test case the flowfield is initialized with a zeroth order solution of an equatorial Rossby soliton of small amplitude.  In the analytical solution, the wave progresses unchanged to the west at a constant celerity c.   A full description of the problem setup can be found here.  The computational domain and initial conditions for the free surface elevation are shown below in Figure 1.    This test case is beneficial for analyzing the accuracy of the advective terms of a model.  Computational results for this test case can indicate the spatial resolution required to achieve a desired accuracy in the modeling of geophysical waveforms.   

Fig 1: Initial Surface Elevation

As implemented, the spatial order of accuracy of the advective fluxes in FVCOM is second order accurate.  A log-log plot of the r.m.s. error versus spatial resolution for this test case shown in Figure 3 confirms this.   Several error metrics are shown in Table 1 for various grid spacing.  The values are comparable to the results obtained using 4th order elements in the SEOM code found here.  At low spatial resolutions on the order of 10 elements/wavelength, the temporal energy loss is unacceptable.  However, as the grid is refined to 20 and 40 elements/wavelength, this discrepancy can be quickly reduced to acceptable levels.  Calculations for an updated test case incorporating a longer channel with periodic boundary conditions are currently underway.

Computations Description and Results: Solitons were computed for grid spacings (dx,dy) of 1., .5,.25, and .125. The mesh was constructed using a triangulation of a regular grid as shown in Figure 2.  The inviscid nonlinear shallow water equations were utilized (FVCOM external mode with zero horizontal diffusion).  Due to the inconsistency between the computed equations and the analytical solution, the computed wave goes through an initial adjustment during which a small amplitude wave propagates to the East.   Results for several error metrics are shown in Table 1. A graph depicting the dependency on spatial resolution is provided in Figure 3.  A description of the error metrics is located at the bottom of this document.  These results can be compared with results computed using the SEOM code located here.

Fig 2: Subregion of Mesh Showing Triangulation

Discussion:  As expected, the coarsest mesh was unable to resolve the wave form for more than one wave period (T=25).  The decay of the maximum surface elevation was quite rapid, falling to less than 50% of the initial amplitude after only 40 time units.  As the mesh density is increased, one can see that both the r.m.s. error and the defect in maximum surface amplitude decrease due to the increase in spatial resolution.  The wave speed also converges towards the analytical solution as the mesh spacing approaches .125.  Note that some error will always remain regardless of the mesh resolution due to the inconsistency between the zeroth order solution for the initial conditions and the governing equations used in the simulation.  In Figure 3, a log-log plot of r.m.s. error vs. grid spacing is shown. The average slope of this line (1.83) can be correlated with the spatial order of accuracy.  The animations provided compare the computed and analytical waveforms as they propagate to the west over a period of 40 time units.  In the finer meshes, a spurious wave can be seen propagating to the east following the initial adjustment.  This wave is also present in the SEOM results and may be eliminated by initializing with a higher order analytical solution.  The error metrics from the FVCOM computations shown in Table 1 compare well with results computed using the high-order spectral element model SEOM.

dx rms error (el) c/c_theory el_max/el0_max
1. 4.64e-4 .89 .43
.5 1.33e-4 .98 .79
.25 2.62e-5 .98 .84
.125 1.06e-5 .99 .92

Description of Metrics

dx = grid spacing in meters

rms error (el)  = root mean square error of computed wave elevation (el) computed with:
        sqrt   [ sum( (el(computed) – el(analytical))**2 ) ]/num_nodes ; sum taken over num_nodes

c/c_theory = ratio of computed wave speed to wave speed from analytical solution.  computed wave speed is calculated using the progression of the maximum wave elevation vs. time.  

el_max/el0_max = ratio of the maximum of the computed wave elevation to the maximum of the initial wave elevation from the analytical solution.

Geoff Cowles  26/08/2003

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