Case 01. Windinduced Surface Gravity Waves in a Circular Lake
Case 1. Windinduced 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. 

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 nondimensional variables , and are derived as 

and ; J_{1}and I_{1} are the original and modified firstkind 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 windinduced 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




(Note: ECOMsi 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 semiimplicit scheme provides flexibility for larger time step).


3. Results  


FVCOMcomputed 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, ECOMsi shows the exact same results as POM. 

