General Theory of Ocean Currents in a Homogeneous Sea 429 



place but also with the velocity of the deep current. In that way the theory becomes 

 very complete indeed, but then in most cases the results do not allow a clear insight. 

 It is therefore necessary to investigate the effect of each factor separately. 



The condition for a constant sea-level is that the total transport M, which is made 

 up of M' and M", the transport for the drift and the gradient current should satisfy 

 the equation: 



div M' + div M" = 0. (XIII.55) 



To this must be added the boundary condition along the coast (vertical coast down to 

 the sea bottom at depth d) 



M'n + Ml = 0, (XIII.56) 



where the index n indicates the transport components at right angles to the coast. 

 Disregarding differences in latitude and in the two frictional depths, then the equa- 

 tions (XIII.55, 56 and 57) after some calculation give the differential equation 



8^C , s^C , g [dd ec . 8d en i /er er 



dx^ + 8y^ + B [dx cy + dy 8x) ~ gB \ 8x 8y ) (Xni.57) 



The effect of the difference in depth can be investigated more closely using this equa- 

 tion in special cases. A simple case is shown in Fig. 183 which represents a vertical 

 section in the sea directed along the .v-axis and parallel to the coast. The sea bottom 

 slopes downwards in the direction of the coast by D over a distance /, so that the 

 gradient is 8dl8x = Djl. It is necessary to investigate whether a deep current parallel 

 to the coast is at all possible. If the wind is assumed to be constant over the area 

 {cTyj8x = 8Txl8y = 0), then since 81,1 ex = and since for p = 1 , DjB = In, 

 (XIII. 57) gives the differential equation 



8^C 277 8C 



ey^ + T8y = ^ (XIII.58) 



the solution of which is given by 



8C 



— = /^e-<2-')^ and U = Uoe-<^-'^)\ (XIII. 59) 



cy 



where /q is the slope of the sea surface and Uq is the velocity of the deep current at the 

 coast. The latter decreases rapidly with distance from the coast, so that at a distance 

 hi Uq has fallen to ^'23 Uq. The deep current is limited to a narrow strip off the coast, 

 the individual current filaments perform a shearing motion relative to each other and 

 and observer on the sea would notice a vortex motion contra solem. Figure 183 shows 

 the assumed wind direction off the coast. The thin dotted line shows the decrease in 

 velocity for a frictional depth proportional to the velocity of the deep current. 



For constant D and for a locally constant wind it is also easy to investigate how the 

 deep current is transformed when flowing over a sea bottom shaped like corrugated 

 sheet-iron. The outline of the coast and the wind direction are assumed to be at right 

 angles to the ridges of the bottom waves. The depth of the sea is then a function only 

 of X and with —8CI8y = i^ = const, and if the sea depth d = d^ + 8 cos (2ttII)x, 

 one obtains from (XIII. 57) 



8^ 2tt8 Itt 



T- = ~ ^ /'o COS -J- X. 



ox D ^ l 



