Time-Dependent Analysis 



The time-dependent, wind-driven currents in an infinitely long shallow 

 basin with a free surface has been studied in detail by Haq, Lick, and 

 Sheng (1974). Adding time-dependence into Welander's steady-state analysis, 

 they analytically computed the unit drift current and the unit slope current 

 and analyzed the various time scales of large scale motion by examining the 

 asymptotic behavior and singularities of the integro-differential equation for 

 the free surface displacement ?. The results depend strongly on the following 

 parameter: 



(Ti/Tg) 



^fj- (Ti/T„) (4.9) 



where D and L are the representative length scales in the vertical and lateral 

 directions. The square root of the parameter can be looked upon as the ratio 

 between the time scales associated with inertia! oscillation and the surface 

 gravity wave. Another parameter of importance is D/d, where d is the depth of 



0. 5 



frictional influence defined as it/(2Ay/f) 



In the limit of B -»• » (rigid-lid approximation), ? is a linear 

 combination of two solutions, one of which decays with a time scale T^, a 

 viscous diffusion time An^D^/Sfd^ when D/d«l or a spin-up time ZiiO/fd when 

 D/d>>l, while another solution decays with a time scale of T but oscillates 



0. 5 



with a period of T2=2L/m(gD) (m=l ,2,3,. .. ), the period of seiches in a 

 lake. In the limit of g + 0, however, it exhibits a decaying oscillatory 

 solution with a period T =2ii/f corresponding to inertial oscillation. In 

 addition, there is a decaying solution with a decay time of 



2 2 2 2 



T =T L f /m Ti gD (m=l,2,3...) corresponding to the movement of fluid from one 

 side of the basin to another under the action of a wind stress. For 

 intermediate values of B corresponding to real basins, the general solution 

 consists of: (1) a decaying part that has a decay time varying from T^ to T^^ 

 as B goes from » to 0, and (2) an imposed decaying oscillatory part with a 

 decaying time of the order of T. and a period of oscillation varying from T 

 to T as B goes from "o to 0. The rigid-lid model approaches the steady state 

 much quicker than the free-surface model and is only valid in the limit of 



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