Vg(I,J) = V|^(I,J) - jAt f(J){Ug(I,J) + U^(I,J)} 



-7y(I,J)6y\(/^(I,J) + Xi + AtFg. (46) 



The even time step counterpart of (46) is 



Ue(I,J) = U^(I,J) - jAt f(J){Vg(I,J) + V^(I,J)} 



-7^(1,3) 8^x1^(1, J) + Xi + At Fg. (47) 



Likewise, the explicit coding, at odd and even time steps, 

 respectively, for the internal mode computation is obtained by 

 replacing the total depth by the equivalent depth, interchanging 

 modes of all field variables and coupling terms, and employing the 

 proper forcing terms in (46) and (47). 



Eqs. (42) and (43) or (44) and (45) form a system of linear 

 algebraic equations in the collective I or J, depending upon time 

 step, of i// cind either U or V at time level n+1. The coefficient 

 matrix is tridiagonal for which there exists a double sweep solution 

 algorithm for inversion, provided that boundary conditions on U or V 

 or some combination of conditions on U or V and 4/ are supplied at 

 each end of the array of variables. 



c) Surface, interface, and bottom stress 



The forcing term F in (42) and (44) consists of the surface 

 stress, the bottom stress and either the atmospheric pressure force 

 due to a surface pressure deficit for the external mode forcing or 

 the interface stress for the internal mode forcing, respectively. 

 That is 



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