382 FOFONOFF [sect. 3 



barotropic component is uniform with dejith and equal to the velocity in the 

 lower layer. The baroclinic component is equal to the relative motion between 

 the upper and lower layer. Waves in which the baroclinic component is absent 

 will be termed barotropic waves and those in which the baroclinic component is 

 present, baroclinic waves. i 



We can integrate the time-dependent equations vertically from the bottom 

 of the ocean to the interface and from the interface to the surface. The integra- 

 tion yields the two sets of equations 



du^hi r . 7 S{p-po)2 ,^,„. 



/)2— ^-P2>B^2 = -h2 ^ (216) 



^%^2 , ^ . . . S{p-po)2 

 p2 o. + p2jU^n2 = -fl2 7- l^i') 



8h2 ^ 8uBh2 ^ dvBh2 ^ ^ ,2i8) 



8t 8x 8y 



for the lower layer, and 



pi g^ pif{vB + Vg)hi ^ -hi — (219) 



8{VB + Vg)hl /■/ , M, I, ^{P-P0)l ,oOA\ 



^1 + g/ +pif{uB + Ug)hi = -hi ^^^ (220) 



8hi 8(UB + Ug)hi 8{VB + Vg)hi ^ 

 8t 8x 8y ^ ' 



for the upper layer, where ^i, h2 are the depths and pi, p2 the densities of the 

 upper and lower layers respectively. 



The pressure in the upper layer, from (214), is 



P = pigil-z), po = pigC^-z), (222) 



where -q is the instantaneous and 7; the mean position of the free surface. 

 Similarly, in the lower layer 



p = pig{-n-Zi) + p2g{zi-z), po = pig{^-zi) + p2g{zi-z), (223) 



where Zi is the instantaneous and Zi the mean position of the interface between 

 the two layers. In the absence of steady flow, we can assume that rj is zero and 

 that Zi is constant. The horizontal gradients of pressure can then be expressed as 



^<^ = ..4: (224) 



for the upper layer and 



8{p-po)2 87] 8zi 



—^—=p,g-+(f.,-p,)g- (226) 



1 This coincides with the terminology used by Veronis and Stommel (1956), who con- 

 sidered absolute velocity components in each layer. A barotropic component is present in 

 both wave types. 



