1 f k+1 g . f k+1 k-1. , jj. 



2^ (V - V*) + 2f^ 6y (n - r, ) = (18) 



17. Noting that v* in Equation 15 is only a function of previously 

 computed variables at the k-1 time-level, its substitution into Equation 18 

 and the substitution of u* (Equation 17) into Equations 13 and 14 yield the 

 simplified forms 



1^ (n* - n^"^) + ^ 6 (u^'^^d + u^^'^d) + -r^ 6 (v^ ^d) = (19) 

 2At 2Ax X Ay y 



1 , k+1 k-1. , g r / ju . k-1, „ ,„-. 



21^ (^''"'' - ^*) + zb ^ (-''^''^ - -''"''^) = (21) 



1 , k+1 k-1. , g r / k+1 , k-1. _ ,--. 



IaT (^ - ^ ^ + 2a7 ^ (^ + '^ ^ ' ° (22) 



18. The details of applying the SC scheme to Equations 7-9 can be found 

 in a report by Butler (in preparation) . The diffusion terms of Equations 8 

 and 9 are also represented with time-centered approximations. The inclusion 

 of diffusion terms contributes to the numerical stability of the scheme 

 (Vreugdenhil 1973) and serves to somewhat account for turbulent momentum 

 dissipation at the larger scales. While the resulting finite difference forms 

 of Equations 7-9 appear cumbersome, they are efficient to solve. Application 

 of the appropriate equation to one row or column of the grid (the "sweeping" 

 process) results in a system of linear algebraic equations whose coefficient 

 matrix is tridiagonal. Tridiagonal matrix problems can be solved directly, 

 without the cost and effort of matrix inversion. 



19. Apart from Courant number considerations, the computational time- 

 step for the SC scheme in WIFM is largely governed by simple mass and momentum 

 conservation principles. The maximum time-step for a problem is characterized 

 by 



19 



