514 The Tides in Relation to Geophysical and Cosmic Problems 



acceleration of the moon's mean motion, which is about 5 to 12 arc sec in a 

 century. This acceleration would then be a direct consequence of the increase 

 of the time of earth's rotation. A value of 9 arc sec per century, for the 

 secular acceleration of the moon's mean motion will require according to 

 Jeffreys (1924, p. 216) a dissipation of energy of 1-39 x 10 19 erg/sec, which 

 would have to be supplied by the frictional effects of tidal currents. According 

 to Adams and Delaunay, the value of 12 arc sec corresponds to an increase 

 of about 1 sec in the time of rotation of the earth. 



The fact that the water height can precede or follow the moon's transit by several hours (establish- 

 ment) was attributed by Airy (1842) to the influence of tidal friction. More recent theories have shown 

 that this is a consequence from dynamical principles which will also occur in the absence of any 

 friction. The differences in phase, which can be explained by equation (XV. 8) remains very small, 

 even in the case of an equatorial canal of 3510 m (1 1 ,250 ft) depth which corresponds to the average 

 depth of the Ocean. From (XV. 8), and using the numerical values of page 292, 



1 1 In 



tan2y = = -0191 — , 



1-311 hjR tn m 



in which t = 2/fi is the modulus of decay of the free oscillations. It seems rational to suppose that 

 this modulus of decay in the present case would be a considerable multiple of the lunar day ln\n, in 

 which event the change produced by friction in the time of high water would be comparable with 

 (2jiIht) x 22 min. Hence we cannot account in this way for phase changes of more than a few 

 minutes. Consequently, the observed differences in phase cannot be explained this way. 



As the required dissipation of energy could not possibly be supplied by 

 a tidal friction in the solid body of the earth, the tides of the ocean should 

 supply it. However, as stated above, it appears that the effect of ordinary 

 friction is insufficient to explain either these shifts in phases, or the secular 

 acceleration of the mean sun and moon's motion. It is easy to evaluate the 

 quantities of energy which are available. The components of the tidal current 

 wand a (except the bottom layers) must satisfy the equations of motion (IX. 4). 

 As resonance hardly comes into question for the vast expanses of the open 

 oceans, the amplitude of the tide will be of the same order of magnitude 

 as that of the equilibrium tide r\. In the open ocean u and v will then have 

 the order of magnitude of 1 cm/sec. The frictional force is k()(u 2 + v 2 ) and 

 is directed against the resultant of the tidal current (see equation (XI. 32) 

 and p. 346). The dissipation of energy per unit area is then 



kQ(u 2 + v 2 yi*. (XV. 9) 



The factor k has a value between 002 and 0001 6. The dissipation of energy 

 per cm 2 will, therefore, be of the order of magnitude 004 erg/sec. The 

 surface of all the oceans is nearly 3-62 x 10 18 cm 2 , so that the total dissipation 

 of energy in the entire ocean will be of the order of magnitude 10 16 erg/sec. 

 This is only a .small fraction of the dissipation of energy which would be 

 required to explain the secular acceleration of the mean sun and moon. 



