516 The Tides in Relation to Geophysical and Cosmic Problems 



The amplitude of the current at spring tide is A(I+ v). The dissipation of energy is proportional 



to the third power of the amplitude so that the ratio between the average dissipation of energy 



and that at spring tide will be average value of (l+v 2 + 2i'Cosro/) 3/2 : (1+v) 3 during an entire tidal 



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period. If we neglect v 6 , the numerator of it will be 1 + - v 2 -\ v* J. The mean ratio of the tidal 



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ranges of the lunar and solar tide for coastal localities is approximately v = 1:2-73 and, if we as- 

 sume that the mean ratio of the current velocities is about as large, we have for the above ratio 0-51. 



This reduction gives for the average dissipation of energy 11 x 10 19 erg/sec, 

 i.e. 80% of the required amount. It seems that this dissipation of energy- 

 is sufficient to explain the observed secular acceleration of the moon. This 

 is the more true, as certainly quite a number of adjacent seas and shelf areas 

 have still been neglected. The Patagonian Shelf, the North American Archi- 

 pelago, Barents Sea, the vast North Siberian Shelf, the numerous fjords of 

 Norway, Greenland and other coastal regions will certainly be able to make 

 up for the 20% still lacking. 



Furthermore we have to consider the frictional losses suffered by the tidal 

 energy on the vast ice covered areas of the Arctic and Antarctic. Sverdrup's 

 (1926 B) investigations showed how large they can be. Thus there is hardly 

 any doubt that the tidal friction could account for the dissipation of energy 

 required by the astronomers for explaining the secular acceleration of the sun 

 and moon if the eddy viscosity of the tides on the vast shelf areas of the seas 

 is taken into account. However, the frictional effects of the tidal currents 

 in the open oceans are unimportant. 



[A rough estimate can be made in the following manner. The total shelf 

 area of all the oceans (depth up till 200 m) is about 27-5 x 10 6 km 2 = 27-5 x 

 x 10 16 cm 2 . The average velocity of the tidal current during a period can 

 be estimated for the shelf as a whole, at about | knot, i.e. around 30 cm/sec. 

 Equation (XV. 9) then gives as the dissipation of energy on the shelf 1-53 x 

 x 10 19 erg/sec, which comes very close to the amount required.] 



