122 Nonlinear Theories — Inertial 



with increasing x. In this way the streams showTi by curves 1 b and 2 a are 

 of about the same width, but the latter is displaced somewhat away from 

 the coast. 



Although Morgan's curves indicate that stratification does not change the 

 form of the stream in an inertial boundary layer, there is an important 

 point that, as he has pointed out, needs emphasis. First, let us write equa- 

 tion (25) in terms of the dimensionless variables: 



fl« = i>=(0. 0) + 2(4±^ J^.(0, 0) r - ^'^M^^ . (34) 



Thus MJ^, 0) ^^s^lg' is a measure of the change of depth along a streamhne, 

 and e is the ratio of the characteristic depth Z)*(0, 0) to the characteristic 

 change in depth. For fixed y the depth is smallest at the coast. It vanishes 

 when _ -D"(0,0)g' ,„., 



Thus Morgan shows that if e < 1, the solution cannot be vahd for y>€, and 

 the value y = e might be regarded as a latitude north of which a new regime 

 of flow must occur. The region of decay, of meanders and eddies beyond 

 Cape Hatteras, comes to mind. 



There are two further remarks that we can make concerning what, in 

 nature, this hmiting condition (35) may imply. First, it should be remem- 

 bered that the warm water of the upper layer is produced in the interior by 

 climatic influences acting upon the sea surface in subtropical and equatorial 

 regions. Therefore it seems reasonable to suppose that the interface (or 

 thermocUne) ought to come to the surface in a latitude between the latitude 

 of maximum curl of the wind stress and that of the northern hmit of the 

 subtropical gyre — between 35 and 50° N. latitude. In other words, the 

 choice of Z)(0, 0) is arbitrary only so long as we do not include chmatic 

 influence in our models. Furthermore, if D{x, y) = does not coincide fairly 

 well with x/r{x, y) = in the region of maximum westerly wind stress, water 

 of the upper layer is not conserved in the upper layer. Of course, the real 

 ocean is not a simple two-layer system, and we cannot demand complete 

 conservation in reaUty, but the climatic influences acting upon the sea do 

 not appear to be powerful enough to ofi'set an excessive loss of surface water. 

 Much of it must recirculate without major density modification. 



Secondly, the quantity D{0, 0) may be determined, to some extent, by the 

 control action (in the hydrauhc engineering sense) of the Gulf Stream itself. 

 As the upper layer is made thinner and thinner, the velocities within it 

 become higher and higher, until finally a minimum depth of that layer is 

 reached beyond which the transport required by the interior wind-stress 

 solution cannot be sustained [equation (35)]. It is at this very stage that 

 the crude meander theory outlined in Chapter IX indicates the onset of 



