| Case 1. Wind-induced Surface
Gravity Waves in a Circular Lake |
1. Analytical Solution Considering that a constant wind stress
imposes on the surface in the x direction in a flat bottom
circular lake shown in Fig. 1, the inviscous linear transport
process in a polar coordinate system satisfies the following
governing equations:
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Fig. 1: Schematic of an idealized circular
lake |
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 |
Eqs. (1.1)-(1.3), which satisfy
conditions (1.4) and (1.5), could be solved analytically
(Csanady, 1968; Birchfield, 1969), and the exact solution
of non-dimensional variables  , and  are derived as |
 |
| and  ;
J1and I1 are the original
and modified first-kind Bessel’s functions, respectively.
The kth mode frequency 
is determined by solving the following equation:
The solutions (1.6)-(1.8) consist of two parts:
one is a wind-induced steady state motion, and another is
the Kelvin/Poincare waves. Amplitudes of the surface elevation
and velocity decrease rapidly as mode number increases;
the exact solutions of ,and can be
accurately expressed by a sum of the first 7 modes with
frequencies of 
=7.0; -7.84; 21.41; -21.48; 34.3; -34.33; and -47.03.
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| 2. Design of the Numerical Experiment |
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| (Note: ECOM-si shows a
significant decay in the amplitudes of the surface elevation
and transport for a given time step as that used in FVCOM
and POM. It requires much shorter time step to reach the
same result as POM, even the semi-implicit scheme provides
flexibility for larger time step). |
| |
| 3. Results |
| |
 Fig.3: Comparison of the time series of
surface elevation (z), x-component (U) transport between
analytical solution (heavy solid line), FVCOM (thin solid
line), and POM/ECOM-si (dashed line) at a location shown in
Fig. 2. | |
FVCOM-computed surface elevation and transport
accurately match the analytical solutions regarding both amplitudes
and phases, while POM shows a phase delay after one model hour. The
time delay in phase increases with model hours: 17.5 minutes at the
end of the first model day and then up to 68.4 minutes at the end of
the fourth model day. With a time step of 1 sec, ECOM-si shows the
exact same results as POM. |
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The time delay in phases of model-computed
elevation and transport found in POM and ECOM-si is clearly
related to inaccurate fitting of the coastal boundary of the
circular lake. This phase delay decreases with an increase of
the horizontal resolution. For example, as the cell number for
POM or ECOM-si is coupled (i.e, km), the time delay in the
phases of elevation and transport decreases to 8.3 minutes at
the end of the first model day and to 36.3 minutes at the end
of the fourth model day, about one time as small as those
reported in the case with a horizontal resolution of about
1.78 km.A much better agreement in amplitude and phase between
POM or ECOM-si and analytical solution is found as the cell
number for these two finite-difference models is tripled.
|
 Fig. 4: Comparison of the spatial
distributions of the surface elevation at the end
of 1st hour, 1st day, and 5th day between
analytical, FVCOM and POM/ECOM-si.
Click
here to see animation |
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Theoretically speaking, the phase delay caused
by POM or ECOM-si would approach zero as horizontal resolution
increases to a certain level. | |
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and 
are the radius and angle axes of the polar coordinate; 
and 
are the Coriolis parameter and gravity acceleration; H is
the mean water depth, and 
is
the x (eastward) component of the surface wind stress.
s-1 ,
km, 
m , 
m s-1 and 
=
0.4016
for FVCOM and POM 
for ECOM-si
= 15
sec and time step used for ECOM-si 
