360 FOFONOFF [sect. 3 



predominantly poleward in the boundary region, the component normal to 

 the coast will be small. Thus, we obtain the approximate equation 



or, solving for the derivative, 



^Ah\2 Of 2 



t) =f (^,-A.). (135) 



Equation (135) is an ordinary non-linear differential equation for hb as hi is 

 given in terms of hb by (134). 



We can convert (135) to a non-dimensional form by introducing h'b = hblho, 

 h'i = hilho and x' = xlW, where Pf is a characteristic width. The non-dimensional 

 equations are 



and 



h'i-^ = l-a{h'bo^-h'bn + ^^^^^^{h'bo^-h'b^)^. (137) 



All the variables in (136) will be of unit magnitude if we choose W = 

 {g'hol2fo^y'^K In general, the solutions to (136) will have to be obtained numeri- 

 cally. However, as x' does not enter the equation explicitly, the solutions are 

 readily obtainable in inverse form by numerical evaluation of the integral 



2x' = f* -^;*iL^. (138) 



An explicit solution of (136) can be obtained in the special case amin = 

 /o//max = i. For this case, the expression for the interior depth (137) can be 

 reduced to 



h'b^ + {i+yr 

 ^'- 2(1+2/') ' ^^^^^ 



where y' = ylyma\, by substituting 1/(1 -(-?/') for a and l—y'^ for h'bo^ from 

 (126). Substituting (139) into (138) and evaluating the integral, we obtain 



h'b = l+2/'-[l+l/'-(l-2/'2)'/^]e-^'[2/(l+l/')]'/^ (140) 



Having obtained h'b, we can derive the transport function from (121) and the 

 velocity from (135). Hence, the boundary -layer solution is complete. 

 In the limiting case y -^ ymax, (140) reduces to 



h'b = 2(1 -e-^') (141) 



