AlJ-lCl 18 



(V • A.V)q represents the eddy viscosity term (discussed 

 ""■ belov/). 



Let us consider the expression for the eddy viscosity 

 term In a rectangular coordinate system, this being the system 

 In which we shall later write our equations. 



V/e define the operator (V • A-V) as follows; 



(V • A.7) = -?.(A-, 9 ) +-i(Ap -1) + A(A^ i-), 



lax-Lox ay 2 ay azSoi' 



where A-, , Ap, A^, may depend on the space coordinates. These 

 three quantities (the coefficients of the lateral and vertical 

 eddy viscosity) have been measured and are knov/n to vary through- 

 out the ocean. The definition of the viscous coefficients and 

 our knowledge of their magnitudes, hov/ever, are rather vague. 

 In view of this, and because of subsequent analytical simplifi- 

 cations, we assume that the lateral kinematic eddy viscosity 

 coefficients are constant and equal, so that 



where A is now a kinematic eddy viscosity and is constant. No 

 simplification will be made concerning A^. 



Our continuity equation is valid for an Incompressible 

 fluid. In the steady problem the density may be more general 

 and we have simply V * (pq) = 0. In the non- steady problem, the 

 assumption of incompressibility is imposed but the fluid may be 

 homogeneous. 



We shall want to make use of [7] regarding the effect 

 of the non-linear terras. Because the results in [7] are discussed 

 in terms of rectangular coordinates and because the use of 



