Theemohaline Features 157 



THE VORTICITY EQUATION WITH A 

 THERMOHALINE PROCESS 



Let us now attempt to introduce a simple thermohaline process into this 

 model: For example, we can suppose that, in subtropical latitudes, there is 

 a net vertical flux of heat through the surface of the sea tending to increase 

 the temperature, and hence to decrease the density of the surface waters, 

 and that the density field is actually maintained constant, in spite of this 

 net heat flux, by virtue of a slow vertical mass flux pw bringing deep water 

 from below into the surface layer. The vertical velocity w is certainly very 

 small at the surface and at the bottom. At some intermediate depth z = Zi 

 it reaches an extreme value (maximum in the subtropics). Since, by mass 

 continuity, d d 8 



gj(p«)+3^W=-^(P-), (U) 



the level of extreme vertical velocity, z = 2j, also corresponds to the level of 

 no horizontal divergence. In the layers above and below this level there 

 are net divergences, and we shall suppose that they are balanced by 

 geostrophic divergences in these layers. We now divide the ocean into two 

 such layers, indicating the layer above the level of no horizontal divergence 

 by the subscript 1 and the layer below by the subscript 2. We then proceed 

 to form the integrated vorticity equations for each layer. First, the follow- 

 ing definitions are introduced: 



^xi=\ pudz; Myi=\ pvdz 



J m J zi 



M^2= pudz; My2= pvdz 



J D J D 



(12) 



where D is the depth of the bottom of the ocean. We cross-differentiate the 

 equations of motion (1), obtaining 



fipv+f 





~dz{dx~e^}- <^^' 



If we now assume that vertical shearing stresses are not important below 

 the depth of the Ekman spiral we may take the mid-depth flow as being 

 essentially frictionless at z = Zi. Furthermore, for simphcity, we shall take 

 the depth of the bottom, z — D, as a constant, in order to avoid, at this 

 stage, the rather superfluous complications of carrying terms of the form 

 {pu) 1 2^2) {SD/dx), and so forth. The integrated forms of the vorticity equa- 

 tions of the two layers are, then : 



fiMy, = -fipw), + curlT , (14) 



/]My,=^fipw),. (15) 



