magnitude than the analytical results. Notice that the numerical computation 

 was made with a time step of 7200 sec which is much greater than that allowed 

 by an explicit numerical scheme {^ 1000 sec). This is apparently responsible 

 for the damping in the numerical solutions. 



4.3 Dominant Time Scales for Wind-Driven Currents in a Shallow Sea 



The general three-dimensional, time-dependent motion in a large shallow 

 sea, with complex geometry and topography, as described the basic equations 

 and boundary conditions in the previous section, is too complicated for 

 analytical solution and can only be computed numerically. 



Under limiting conditions, however, the system of equations can become 

 linearized and sufficiently simplified such that analytical solutions are 

 available. Although idealized, these analytical solutions can provide much 

 insight into the dynamics of the basin. In addition, they can be used as 

 reference solutions for checking the results from complicated 

 multi-dimensional numerical models. 



Steady-State Analysis 



Welander (1957) analyzed the steady-state currents in a homogeneous 

 shallow sea, where non-linear inertia and lateral diffusion are negligible 

 compared to Coriolis acceleration and vertical turbulent diffusion, and 

 obtained the following analytical expression (in dimensional form) for 

 horizontal velocities as a function of depth with wind stress and surface 

 slope as parameters: 



u ^ iv = sinhx(h-Hz) (^sx-^^Jsy) . j^ /coshxz .A /li + i li) (4.8) 

 coshxh ^ ^3/2 f \coshxh / \ax ay/ 



0.5 0.5 

 where i=(-l) , A=(if/Av) , z=-h(x,y) is the bottom, and z=c(x,y) is the 



free surface. This solution represents the sum of the drift current, which is 



proportional to the wind stress, and the slope current, which is proportional 



to the surface slope. Despite its limitations, this solution has been the 



basis of many numerical models. 



56 



