432 General Theory of Ocean Currents in a Homogeneous Sea 



of the stream lines would be reduced, and secondly, the entire wave form would be 

 displaced so that the bottom waves would coincide more with that of the stream lines. 

 Both changes depend on the depth of the water, as well as on the current velocity 

 and wave length of the bottom waves. As long as the expression W/Dt is only a small 

 fraction the deviations from the previous state remain small, but they become con- 

 siderable when it approaches or even exceeds 1. Therein r is the time in pendulum 

 hours (see p. 316) in which the deep current requires to move through the wave length 

 of a single bottom wave. The values found for this expression from observed data are 

 relatively large, so that it is probable that bottom waves and stream lines are therefore 

 closely in phase. 



In general, the effects of the three factors are of the same type as before but they are 

 no longer independent of each other; the topographical and the planetary vortex 

 effects especially are interrelated in a complex way and disturb each other in extended 

 oceanic areas during the generation of a uniform deep current. In general, an irregular 

 bottom topography seems to have a tendency to reduce the velocity of the deep currents. 

 Deep currents do not then play the dominant role ascribed to them earlier. This is 

 probably the reason why many results of the earlier theory based on the most simple 

 assumptions were in good agreement with the observed data, although these assump- 

 tions were only approximately satisfied in nature. If the topography of the sea bottom 

 is very irregular the topographical and planetary vortex effects will disturb and some- 

 times destroy the deep currents, so that essentially there will remain only pure steady 

 drift currents. 



The investigation of the effects of the bottom topography on ocean currents has a 

 direct connection with the discussion on p. 386, where it was stated that a deflection 

 of a current cum sole would occur on top of a rising sea bottom and a deflection 

 contra solem on top of a bottom fall. Without taking friction into account a quantita- 

 tive estimate of this vortex effect can be made. For an extended bottom wave with a 

 triangular shape (Fig. 185 ; x-axis at right angles to its crest, >'-axis along its crest), 

 and assuming a uniform current U in front of the ridge extending throughout the total 

 water mass (depth of water H) and flowing towards the crest, equation (XIII.29) gives: 



?^ dC 



-^n =f^ and ^ =0; V^O. 



dy 8x 



Over this bottom ridge under stationary conditions (duldt = dvjdt = 0) the equations 

 of motion will be 



''fx = -^dy-^''=-^^^-''^- 



If the origin of the co-ordinate system is placed at O vertically underneath the highest 

 point of the ridge, the half-width of which {OA = ^45) is /, and height of which at O 

 is h, then the depth of water will be 



d=cl,^{hll)x, 



where the upper sign applies for the forefront side and the lower sign for the rear of 

 the bottom ridge. The equation of continuity requires the same transport through 

 every cross-section, that is 



UH = u{d^{hll)x]. 



