APPENDIX A 



Vertical Stretching of the Coordinates 



The vertical stretching of the coordinates for the three-dimensional, 

 free-surface model is obtained by letting 



^ ^' y y' h(x.y)+C(x.y.t) H ^'^ '' 



By chain rule, first-order derivatives in the original coordinates (x,y,z) are 

 related to those in the transformed coordinates (x',y',°) as: 



9 3 1 8c 3 a 3h 9 



3x"9x^"H8x9a""H379a' 



3 3 111 JL g 8H 3 

 3y ~ 3y' " H^ 3y 3a ~ H 3y 3o 



i_ - i A. 



9z ~ H 9a 



(A-2) 



(A-3) 



(A-4) 



The vertical velocity w in the (x,y,z) coordinates is related to that in the 

 transformed coordinates {^=do/dt) as: 



fl^-l^) 



Ho. + (u^+ v^l + (1+0) ^ (A-5; 



Utilizing (A-2) through (A-5), the continuity equation and the advective terms 

 of the x-momentum equation in the original coordinates can be transformed as: 



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