SECT. 3] DYNAMICS OF OCEAN CUKRENTS 355 



simple for a relatively complete analytical examination of its features to 

 be made. In the two-layer approximation, the ocean is assumed to consist of 

 two homogeneous layers of water of slightly different density with the lighter 

 water overlying the more dense layer. Velocities within each layer are constant 

 with depth and change abruptly across the interface between the two layers. 

 Frictional stresses along the interface are neglected. 



In the general case, motion can exist in both layers, i.e. the flow can be a 

 combination of both barotropic and baroclinic modes. However, if there is 

 motion in both layers, it does not appear possible to find an interface such that 

 the normal component of velocity is zero everywhere except in the special case 

 of non-divergent zonal flow.i The equations are considerably simplified if 

 either the barotropic or baroclinic mode is absent, or if the baroclinic velocities 

 are equal and opposite to the barotropic velocities (motion in the lower layer 

 only). The barotropic flow has already been examined for the homogeneous 

 ocean and can be applied with minor changes to the two-layer ocean. The 

 two remaining simple flows are basically similar to each other. If the 

 barotropic mode is absent, the flow is entirely in the upper layer and 

 pressure gradients in the lower layer are zero. Conversely, if the baroclinic flow 

 is equal and opposite to the barotropic flow, the motion is confined entirely to 

 the bottom layer. Provided we assume the bottom of the ocean to be level, we 

 can consider these two types of flow to be "mirror images" of each other. 

 Hence, we shall consider only the baroclinic mode. 



In the absence of the barotropic mode, the baroclinic velocities become 

 absolute velocities relative to the co-ordinate system. We may write the 

 momentum equations in the form 



i^Vi^V/M = -I (.07) 



^ + ^ = 0, (108) 



OX oy 



where h = ^ — r]i, rn denoting the position of the interface between the two 

 layers, and 



X = r%8 dp = lApgh"^ = \pg'h\ g' = {Aplp)g, 



(109) 



where Jp is the difference of density between the lower and upper layer. 



1 Unfortunately, a proof that no steady interface can exist for motion in both layers 

 has not been constructed for the general inertial system of equations with friction neg- 

 lected. It can be given if the non-linear acceleration terms are neglected. Such a proof 

 would enable us to assert that steady flow in both layers could exist only in the presence 

 of processes, such as internal mixing, that alter density along a stream-line of the flow. 

 If implicit mechanisms are postulated to allow flow across the interface, it is possible to 

 construct a simplified theory of thermohaline circulation (Stommel, Arons and Faller, 

 1958; Stommel and Arons, 1960a). 



