SECT. 3] DYNAMICS OF OCEAN CURRENTS 383 



for the lower layer. Similar expressions obtain for the ^/-component. The 

 presence of steady flow will not affect the difference between the instantaneous 

 and mean pressure, p—po. However, in the presence of a steady baroclinic 

 component, the interface zt can no longer be assumed to be level. The variation 

 in depth of the upper and lower layer will enter the equations for the time- 

 dependent motion through the continuity equations, (218) and (221). The 

 influence on time-dependent motion of the variation of thickness of the layers, 

 or, in the related problem, of the variation of bottom topography, has not been 

 studied adequately as yet. 



We assume that the variations in the depths of both layers is sufficiently 

 small so that we can replace the instantaneous depths, hi, hz, by their means. 

 Hi and H2, except where the depths are multiplied by gravity, g. Introducing 

 -qi for Zi — Zi to represent the deviation of the interface from its mean position 

 and substituting (224) and (225) into the integrated equations, we obtain 



«'" + /,, (^ + ^) = (228) 



dt \ dx . By 



for the barotropic velocity, and 



t-/"--^'^^ (229) 



^-/"--^'^ ,230) 



drj /8ub dvB\ rj /^% ^^9 



for the baroclinic velocity, where go={pilp2)g, g' — {Aplp)g and H = Hi + H2. 

 The equations for a homogeneous ocean are obtained by setting g' and Hi to 

 zero. 



We can study the wave modes allowed by the two-layer system of equations 

 by assuming that the deviations of the free surface and the interface from their 

 mean positions and the velocity components are periodic functions of time and 

 horizontal distance. As the equations governing the motion are linear, more 

 complex motions can be considered as linear combinations of simple plane 

 waves. In addition to wave motion, the linear equations admit aperiodic 

 solutions that are of importance in interpreting simple interactions with steady 

 flow. We assume that all of the variables can be represented by functions of the 

 general form ^ei("'<+*a;+i2/)^ where ^ is a complex dimensional coefficient that 

 is independent of time but may vary with latitude. The coefficients a», k and I 

 will be complex in general. For wave solutions, oj is real and is interpreted as 



