Misc

# Case 01. Wind-induced Surface Gravity Waves in a Circular Lake

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:  Fig. 1: Schematic of an idealized circular lake

where and are the radius and angle axes of the polar coordinate; and are the and components of water transport; is the surface elevation; 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.

The boundary and initial conditions for (1.1)-(1.3) are given as 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.

2. Design of the Numerical Experiment

 Parameters: s-1 , km, m , m s-1 and = 0.4016 Criterion for numerical instability: for FVCOM and POM for ECOM-si Time step used for FVCOM and POM: = 15 sec and time step used for ECOM-si is 1 sec Model Grid Fig. 2: Unstructured triangular and structured rectangular grids   used for FVCOM and POM/ECOM-si, respectively. Horizontal   resolution is 1.78 km for both models.
(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.

 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. Theoretically speaking, the phase delay caused by POM or ECOM-si would approach zero as horizontal resolution increases to a certain level.