118 Nonlinear Theories — Inertial 



In this way we obtain the following expression for use in the two-layer 



system : 



2M / 1 \ 



DHO,y) = -^[f^y + -^^f-j+D^O,0). (19) 



The quantity D (0, 0), the interior depth of the interface at the outer edge 



of the boundary layer, and at ?/ = 0, is a constant which we can choose 



arbitrarily so far as this theory is concerned. In this way we can form a 



variety of models. 



We now turn to the boundary layer itself. Morgan (1956) uses the 



following dynamical equations: 



8D* 

 f''*=9~. (20) 



^+j'I>* = B(f). (21) 



The asterisks are used to denote quantities defined -nithin the boundary 

 layer. Equation (20) states that the dynamical equation for force com- 

 ponents normal to the coast is essentially geostrophic. Equation (21) is 

 Bernoulli's (nonhnear) equation for the upper layer. Next to the coast, u*^ 

 is insignificant compared to v*^. In the boundary layer the kinetic plus 

 potential energies are a function B{\Jf*) of the transport function only. 

 Outside the boundary laj^er this is not so, because the wind stress has a long 

 time to do work on the water there. It is convenient to transform these two 

 equations into a form containing only i/r* and D*, by making use of the 

 definition v*D* = di/f*]8x: 



^* = ^D*^ + C{y), (22) 



?^ = V2 D* [Biir*) - g'D*Yl^ . (23) 



ox 



The quantity C{y) is easily evaluated along the outer edge of the boundary 

 layer by matching ijr* and D* with interior solutions ijr and D at a; = 

 [equations (16) and (19)]. In this way Morgan obtains the result: 



where we write M^{0, 0)=Mx{0, y). When he substitutes this in equation 

 (22) and solves for D*^, Morgan obtains 



D*^ = D%0, 0)+g^^*- ^^^'^f^^' ^\ (25) 



g 9 



