2.3 Time Scales and Dimensionless Parameters 



It is convenient to rewrite the above system of equations and boundary 

 conditions in dimensionless form such that the relative orders of magnitude of 

 the various terms become more apparent. However, it is instructive to first 

 examine the time scales associated with the large scale motions in a lake. 

 Assuming H and L as the reference lengths in lateral and vertical directions, 

 Up as the reference velocity, and Ap as the reference density difference in 

 stratified flow, (A^)p, (Ay)p, (KH)r' ^"^ ^^\K ^^ ^^^ reference eddy 

 coefficients, several time scales can be defined as in Table 2.3. 



The list in Table 2.3 is not meant to be exhaustive. For example, two 

 time scales for the turbulent thermal diffusion, T^.^ and T^^, can be defined 

 similar to T^j^ and T^j^ by replacing the eddy viscosities with the eddy 

 diffusivities. The inverse of the Brunt-Vaisala frequency, 

 N = (1/pp 3p/3z) * , the time scale of the lowest mode of free oscillation 

 in a stably stratified fluid, can be used instead of T^. Other time scales 

 will be mentioned at appropriate places later. A number of dimensionless 

 parameters can be defined from the ratio of these time scales and are listed 

 in Table 2.4. 



The time scales and dimensionless numbers so defined in Tables 2.3 and 



2.4 are wery useful in comparing the relative importance of various terms in 

 the equations of motion. If the Coriolis terms in the momentum equation is of 

 order 1, then orders of the unsteady, nonlinear, lateral diffusion, and 

 vertical diffusion terms are Un, Ro, E^, and E^, respectively. 



2.4 Vertical Grid Resolution 



Two types of vertical grid are generally used in the finite-difference 

 models of coastal currents. In the first type, the vertical domain of the 

 water body is separated by layers of constant depth as was done by Leendertse 

 and Liu (1975). Although the overall features of the flow field may be well 

 represented, this type of grid has two potential problems: (1) unless a large 

 number of layers is used, there is generally insufficient resolution in the 

 shallow nearshore region and hence the nearshore dynamics is poorly 

 represented, (2) continuous variation in the bottom topography cannot be 



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