624 GROEN AND GROVES [CHAP. 17 



because of relatively small depth and sufficiently developed turbulence, is little 

 influenced by the Coriolis force. So, in equations (3) and (4), we have 



Ta-Tb = (l+w)Ta- ^^^ • (7) 



The non-linearity of the equations makes it very difficult to solve them 

 adequately and makes the separation of an astronomical effect and an atmos- 

 pheric effect problematic. 



There are three ways of approaching the problem of integrating a system of 

 non-linear equations. First, one may use a procedure of direct integration 

 such as the method of integration along characteristics (see Schonfeld, 1951 ; 

 Freeman, 1954, 1957). Secondly, we may have recourse to numerical integra- 

 tion, approximating the differential equations by suitable difference equations. 

 (This method has been applied by Doodson, 1956.) Thirdly, we may first 

 linearize the equations and then set up an iterative process by which the non- 

 linear equations are solved in successive approximations, as follows. In the 

 dynamical equations we split all the terms which are non-linear into a linear 

 part and a non-linear residue. For instance : 



dtyh+c) h dt hdt\h+C 



The most difficult to linearize are the bottom stress terms. We may, however, 

 introduce a suitable average value Um of U, which may be a function of x and 

 y, and then write 



sUV sUmV , ^. _./ 1 1\ s{U-U„,)V 



If we now write all non-linear residues on the right-hand side of equations (3) 

 and (4), together with all known terms, we obtain : 



= {l+m)p-^Tax-hp-^^-^-h^ + X{^, U,, Uy), (8) 



= {l+m)p-Way-hp-^^-h^+Y{C, U,, Uy), (9) 



where X{i„ Ux, Uy) and Y{t„ Ux, Uy) stand for the sums of all non-linear 

 residues, as defined above. The solving procedure is now as follows. We first 

 neglect X and Y in the above equations and, by solving the equations thus 

 simplified, find first-order approximations ^<i), Ux^^K Uy^^K Next, these are 

 substituted for ^, Ux, Uy in X and Y. By solving the equations with X<i) = 

 ^(^<^), Ux^i\ Uy(^)) and 7(i)= 7(^<i), Ux^^^ Uy(^)) on the right-hand side, we 



