| Case 6.Rossby Equatorial Soliton
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| 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.
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Fig
1:
Initial Surface Elevation
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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.
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Fig
2: Subregion of Mesh Showing Triangulation
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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. |
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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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