-^ + u ^ + V IJ + fu + g -^ (n - n^) + ^ (u2 + v^)'^' 



3t 9x 3y Jy a ^Zj 



_ , ^ + ^ + F = (3) 



3x 3y" ^ 



The dependent variables in the problem are n , u , and v , which represent 



the water-surfa^3 elevation above datum and the vertically averaged water 



velocities in the x- nd y-direct ions, respectively.* Tlie other variables in 



the equations are: h , the local ground (cell) elevation above datum; 



d = n - h , the total water depth; t , time; f , the Coriolis parameter; 



C , the Chezy coefficient for bocto.ii friction; £ , tiie eddy viscosity coer- 



ficient; g , the acceleration due to gravity; and R , the rate of water 



volume change in tlie system due to rainfall or evaporation. The coefficient 



n accounts for hydrostatic water elevations due to atmospheric pressure 



a 

 differences, and F and F are terms representing external forces such as 

 X y 



wind stress. 



9. The computational grid used for the finite difference approximations 

 in UTFM empl&vs a stretching transformation for each of the two coordinate 

 directions (x and y) . This transformation from physical space to computational 

 space offers the major advantage of allowing increased grid resolution in 

 areas of interest, by controlling the arbitrary constants a , b , and c in 

 the equation: 



X = a + ba*^ W 



Physical distances are defined by x , and the computational grid lines in 

 each direction are defined by positive integer values of a . The valu3s of 



a , b , and c are determined not only from desired grid resolutions, but 



3x 

 also from the requirement that the derivative — be continuous everywhere. 



Many stability problems commonly associT>ted with variable grid schemes are 



eliminated due to this continuity constraint. The transformation is applied 



* For convenience, symbols and unusual abbreviations are listed and defined 

 in the Notation (Appendix A). 



13 



