is carried out in conjunction with the internal mode computation. 

 Depending on the problem of interest, the internal mode may be computed 

 e'^ery so often with a time step equal to or greater than the external 

 time step. 



Internal Mode 



The internal mode of the flow is described by the vertical flow 

 structures and the density. Defining perturbation velocities as 

 u'=u-U/H and v'ev-V/H, the equations for the internal mode are obtained 

 by subtracting the vertically-averaged momentum equations from the 

 three-dimensional equations: 



1 3Hu' 

 H at 



H 



8t 



1 



and B. 





" 





a 

 aa 



. a /hu'+u\ 



'^v aa V H / 





a 

 aa 



. a (Hv'+v) 

 V ao \ H / 





r 



epresent all 



terms 



(17) 



(18) 



in the transformed 



where B^^ 



three-dimensional mi^mentum equations except the surface slopes and the 

 vertical diffusion terms, and D^ and Dy are defined in Eq. (16). 

 Notice that the above equations retain the three-dimensionality and 

 hence are different from the model of Nihoul and Ronday (1983) which is 

 actually a superposition of a two-dimensional model and a vertical 

 one-dimensional model . 



The above equations do not contain the surface slope terms and 

 hence a large time step may be used in the numerical computation. In 

 the present model , a two-time-level or three-time-level scheme with a 

 vertically implicit scheme is generally used. The bottom friction 

 terms are also treated implicitly to ensure unconditional numerical 

 stability in shallow waters. Care must be taken to ensure that the 

 vertically-integrated perturbation velocities at each horizontal 

 location (i,j) always equal to zero. 



Once the equations for (u',v') are solved, and (u,v) obtained, 

 vertical velocity w and density p may be computed. As mentioned 

 before, the internal mode may be computed as often as the external mode 

 or as desired and as dictated by the problems of interest. The 

 numerical time step for the internal mode is limited by the CFL 

 condition based on the advection speed. In the present study, the drag 

 coefficient C^j in the quadratic bottom stress law-Eq. (9) is generally 

 specified as a function of the bottom roughness (Zq)i the distance 

 above the bottom (z ), at which (u ,v ) is computed, and the stability 

 function of the bottom flow (4)5): 



k Un — + 



^0 



(19) 



where k is the von Karman constant. It can be shown that the stability 

 may increase (unstable case) or decrease (stable case) the drag 



273 



Sheng 



