4.4 Effects of Bottom Boundary Condition on Wind-Driven Currents 

 in a Shal low Sea 



Next we shall illustrate the effect of bottom boundary condition on the 



response of a lake under an impulsive wind stress. Consider a 50 Km square 



lake with the bottom varying linearly from 2.5 m at y=0 to 7.5 m at y=50 Km. 



2 

 At t=0, a wind stress of 1 dyne/cm is applied in the x direction. Assuming a 



2 



constant vertical eddy viscosity of 25 cm /sec, the vertical Ekman number 

 ranges between 0.44 and 4.0. Based on the analytical results discussed above, 

 for the present case T =2.5 hrs, T =3 to 5.6 hrs, T =17.5 hrs, and T^ is 

 irrelevant since H/d « 1 and e is between 1 and 3. Numerical computations 

 were performed with two types of bottom boundary conditions: (1) quadratic 

 str'^«;s law with 0^^^=0.004, and (2) no-slip condition, i.e., (tjjj^, 

 "^by^ " P^v (^i» ^i)/^^ where u and v are the horizontal velocities at Lz 

 above the bottom. The transient results (Figure 4.5) obtained with the 

 no-slip condition clearly exhibit a decay time on the order of T but T is 

 not apparent in the results. With the quadratic stress law, T„ is apparent in 

 the results and the decay time is more than doubled than the no-slip case, due 

 primarily to the smaller bottom dissipation. At the steady-state, the 

 vertical profiles of horizontal velocities due to the two different boundary 

 conditions are quite different. As shown in Figure 4.6, the quadratic stress 

 law yields a flatter velocity profile above the bottom much like a turbulent 

 boundary layer over a flat bottom. The no-slip condition, on the other hand, 

 yields a parabolic profile above the bottom resembling a laminar boundary 

 layer. The large difference in the near-bottom currents computed by the two 

 boundary conditions is of particular significance when considering the 

 transport of sediments or other materials near the bottom. 



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