Nonlinear Theories — Inertial 



119 



Another relation between ^* and D* can be obtained from the Bernoulli 

 equation (21), because at the outer edge of the boundary layer v*^-^0, 



^^^^"^'" g'D* = B(r). (26) 



Thus the function B{\]r*) is determined by 



Bmo,y)-\ = g'[D((),y)] = g' 



i)2(o,o)+5^ + 



/^. 



g' g'M,{f), 0) 



^*2 



1/2 



(27) 



When this value ofB{rJr) is substituted in equation (21) and D* is eUminated 

 by equation (25), the following equation is obtained: 



m^^' 



2/; 



^ iff* - 



I^kV^M 



1)2(0,0)+-^^ 



g 



x(0. 0) j 

 ,0)^ J 



1/2 



(28) 



9 9 



1/2 



The proper boundary condition is that a: = be the streamline ^*(0, y)=0. 

 It is important to note that if /?£■ = 0, the quantity di/r*jdx = for all x. Thus, 

 no boundary layer forms if the variation of the Coriolis parameter with 

 latitude is neglected. Morgan proceeds to analyze the nonlinear differential 

 equation (28) by introducing dimensionless variables, x, y, x/r*: 



x = sx, y = sy, r/f* = M^{0, 0)sijr*; 

 and the parameters e and 8 are defined as follows: 



s= 



Pk^' 



The differential equation (28) thus may be written 





X {[e + #* + ^*2]i/2 _ [e + 8f* + ^yxjf* - y2]i/2}. 



(29) 



(30) 

 (31) 



(32) 



For numerical integration, the involved numerical coefficient in equation 

 (32) can be absorbed into the independent variable by the transformation 



i-- 



i^l 



'3/2^3 



\3fnO.O)sr 



'1/2I 



1/2 



(33) 



