344 roFONOFF [sect. 3 



D. Ocean Boundaries 



The reduced transport equations contain only first derivatives of the trans- 

 port components so that only one boundary condition can be satisfied. In order 

 to be able to consider oceans that are bounded meridionally or fully enclosed 

 by land, we must retain friction or accelerations or both in the vicinity of the 

 boundaries. 



Munk (1950) discussed the effects of meridional boundaries on wind-driven 

 baroclinic transport by retaining frictional terms in the equations and neglect- 

 ing accelerations. We shall consider his analysis in simplified form to illustrate 

 the boundary-layer method of treating the more complete equations and to 

 examine the special features of the solutions obtained. 



We assume that the fiow in the interior of the ocean is of class (iv) with the 

 barotropic mode absent. Adding (62) and (63), we obtain 



^(F,+ F.) = ^F = ^»-^' (68) 



for the total meridional transport F. Equation (59) can be satisfied by intro- 

 ducing a mass transport function i/f defined by 



[/ = |A, y^J± (69) 



dy ox 



The general solution to (68) is 



where the subscript is introduced to denote the interior solution, and C is a 

 function of y only. The interior solution (70) contains only one arbitrary con- 

 stant of integration. Therefore, only one condition at one of the boundaries 

 can be satisfied. However, if we include friction in the equations, both the 

 normal and the tangential components of transport must vanish at the lateral 

 boundaries. From (69), we see that these conditions can be satisfied by requiring 

 i/f and its derivative normal to the boundary to be zero. Thus, for an ocean with 

 meridional boundaries, there are four conditions to be satisfied, two at each 

 boundary. 



The equation for the transport function in the boundary region can be 

 obtained from (47) and (48) by neglecting the non-linear terms and cross- 

 differentiating to eliminate the potential energy anomaly. The equation 

 reduces to 



^«VV-^|^ = ^-^^ (71) 



ox dx oy 



where W^= d'^ldx^ + 2d^ldx^ dy^ + d'^jdy'^ is the biharmonic differential operator. 

 We anticipate that the transport function ip will change rapidly with distance 

 from the boundary in the boundary region but will not change as rapidly along 

 the boundary. We can introduce a characteristic width W of the boundary 

 region and a length L to denote the meridional extent of the circulation. If we 



