distribution is hydrostatic in the vertical direction, (2) Boussinesq 

 approximation is valid, and (3) non-constant eddy viscosities and 

 diffusivities are used to describe the turbulence. The resulting 

 equations are as follows: 



iii + i^ + i^=o (1) 



3x 3y 32 



!£= . fii^^iH^^ iH^U fv .±1L^±(a^^JlV V, . (AuV) (2) 



3t \3X 3y 32/ Pq 3X 32 \ V 32^ H H H 



2 

 3v /3uv . 3v . 3vw 



3t \ 3X 3y 32 / p- 3y 32 



+ ^^^ - f u 



-liP.AfA 



(avIi)^^h- (Vhv) (3) 



iP=-pg (4) 



32 



ia = . 3ii^ + 1)^ + 1^ + A (viA^^ (K.Vup) (5) 



3t 3X 3y 32 32 \ V 32/ " " " 



where x and y are the hori2ontal coordinates; z is the vertical 

 coordinate pointing vertically upward to form a right-handed coordinate 

 system with x and y; u, v, and w are the three-dimensional velocities 

 in the x, y, and z directions; t is time; f is the Coriolis 

 parameter; g is the gravitational acceleration; p is the pressure; p 

 is the density; Au, and K^ are the hori2ontal eddy coefficients; Ay 

 and Ky are the vertical eddy coefficients; and 



'H- IVhE)=^ (A„||)*^(AH|i) (6) 



At the free surface, the appropriate boundary conditions 

 are: (a) the wind stress is specified, 



I ^^ 3v\ , , _ , 2^ 2.1/2 , . ,,. 



[jl' Jl)^ ^^sx.^sy^ = Pa^da ^V^^ ("w^w^ <7) 



where x-^^ and t^ are the wind stresses in the x and y directions 

 respectively, p^ is the air density, C^^, is the drag coefficient and 

 (u ,v ) are the wind velocities at a certain height above the surface; 

 (b) the kinematic condition is satisfied, 



w = |^+ u |^+ v|^ (8) 



3t 3x 3y 



where c is the elevation of the free surface; (c) the dynamic 

 condition is satisfied, i.e., p=P3, where Pg is the atmospheric 

 pressure; and (d) the density flux, i.e., the heat flux and the salt 

 flux, is specified. 



At the bottom, the boundary conditions are: (a) a quadratic 

 stress law is val id: 



'0 \ 



269 



Sheng 



