Physics of the Earth - II The Figure of the Earth Bulletin of the National Research Council (1931) / Chapter Skim
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Chapter VI. Tidal Friction
Chapter VI. Tidal Friction
Pages 81-100
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From page 81... ...
This its, however, only a makeshift, as the coefficient of' turbulence depends on the velocities attained To what extent friction a.fI:'ects the amplitudes and phases of the tide is not easy to say. At one time it was common to attribute many familiar tidal phenomena mainly to I'riction; for instance: the interval between the transit of the moon and the occurrence of high water; the interval between syzygy and the occurrence of spring tides or between quadrature and Heap tides, and the interval between the time of the moon's maximun~ declination and the occurrence of tropic tides; the fact that in a river the duration of fall exceeds that of rise, a difference that in general increases with the distance from the mouth of the river and which when extremely exaggerated becomes the tidal bore (eager, a.gger or eyt,re)
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From page 82... ...
A very natural but entirely incorrect mental picture of the phenomenon is that of an ocean enveloping the rotating earth, the ocean tendin, to remain stationary in space and so acting directly as a brake '1 ~1 ~· ~1 1 1 V V upon tne earths axial rotation. This mental picture is commonly associated with the idea of tides as equilibrium phenomena and the inadequacy, already pointed out, of this simple conception of the tides should put us on our guard against any such over-simplified idea.
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From page 83... ...
. Perhaps the most striking result of these investigations is that frictional dissipation occurs almost entirely in the shallow seas, the part contributed by the oceans, although they vastly exceed the shallow seas in area, being almost negligible.
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From page 84... ...
r= radius of the moon's orbit, assumed to be circular. o.=angular velocity of the earth's axial rotation.
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From page 85... ...
TVe know that it =0 and that 2~ has a value estimated from oceanic tidal friction; :t'or purposes of' numerical illustration we shall adopt Jeff'rey's value of' 1.l x ~ Oi9 eras per second. Alter some rather easy manipulation we find: dT d(,)
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From page 86... ...
Part of this is provided for in the ordinary lunar theory, quite apart from any considerations of tidal friction. The remainder is plausibly attributed to tidal friction and mainly, as has been said, to tidal friction in the shallow seas, for Jeffreys' estimate of a frictional dissipation of 1.!
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From page 87... ...
Moreover; the theory of friction in the body tides of Me ebb ma be used as ~ basis on which to develop ~notber method of estimating friction in the ocean tides ~ metbo] tat bus the double advantage of enabling ~n estimate to he made from data of ~ Ji~ereDt nature from those Just mentioned, and of lending itself better to the formation of ~ mental idea of the process.
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From page 88... ...
88 FIGURE OF THE EARTH appears whether we consider the total acceleration acting on the bulges ~ and B in Figure 1, or whether we consider merely the tide-producing acceleration, which is the resultant of the total acceleration due to the tide-producing I'orce of' the moon and the centri:f'ugal acceleration of the center of the earth due to the orbital motion. (See Figure 2.)
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From page 89... ...
TIDAL FRICTION 89 perpendicular to AB than the line BM, and hence gives a greater component perpendicular to AB, which is the component tending to retard rotation. We must get the same result if we consider tidal forces only, as in Figure 2, for the tidal acceleration is merely the total acceleration combined with the centr;:tugal acceleration of the center of the earth; this latter component is the same for A and B and since these are on opposite sides of the center its effect on the rotation is null.
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From page 90... ...
90 FIGURE OF THE EARTH observed acceleration of the moon. The quantities E, p, and ~ depend on the physical properties of the earth, that is, on how the density, elasticity and viscosity vary from center to surface.
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From page 91... ...
As far as the earth tides alone ale collcerneJ, Mere is at present r~ reason Baby we sold not be gibing to accept this result Ibe tangle between the axis of tiff protuter~Dces ~d the meridian plane of the moon implies ~ lag in the obese of the observed ebb tides Rebind the corresponding please of the tidy forces. But this lag in pose Stolid in the present i~st~nce'5 be only 25~, ~]
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From page 92... ...
The conception may be adapted to the case of the ocean tides and enables us to base an estimate of tidal friction on observational data different from those used in deducing the estimate so far used. The semidiurnal tides in the solid earth conform approximately to the equilibrium theory, that is, the two high tides lie in the line of the meridian plane of the moon.
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From page 93... ...
Noxious sets of cotidal lines have been drawn which differ considerably in cletai.1 According to one of the fundamental properties of spherical harmonics only the second-degree component of the tidal height that is, the ellipsoid, contributes to the moment. All other spherical-harmonic components, when thus multiplied and integrated, give a zero contribution.
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From page 94... ...
The formula may be derived from quite a different method, that of considering the work done against the tidal forces by the vertical rise and fall of the water. This is the method followed by Taylor and Heiskanen.5 ~ Heiskanen made the quadratures, as stated, but treated the result not as ~ different method of estimating the same quantity as had been derived from a study of the currents; but as a correction to the latter quantity.
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From page 95... ...
~ith the r~tcs ~t ~bicb thc lc~gtb oI thc sidore~1 d~> ~d siicre~1 ~ontb ~re cb~nging. B~ either oT thc two ~il~Blc mctbods tbo r~te ~t ~hicb energy js Deing iissip~te]
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From page 96... ...
How the rate of frictional dissipation of tidal energy during the era of extensive shallow seas compared with the corresponding rate today we are, therefore, unable to say. But some tidal friction must have been present always; accordingly as in imagination we go further and further back into the past we find the day and the month getting shorter and the moon getting nearer the earth.
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From page 97... ...
He supposes that the rotation of the earth was originally so rapid that the solar sem;diurnal tides had a period approximating to the natural period of vibration of the earth for a disturbance of the tidal type, that is, a period o-t' about two hours. The near coincidence in period of the free vibration of the earth with the forced tidal vibration raised such enormous solar tides that the earth was disruptecl.
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From page 98... ...
~., more than counterbalances the contrary small effect of NVe now return to the postponed question as to why the moment due to I'riction which obviously retards the axial rotation of the earth should also slow clown the Notion of' the moon and tend to push the moon farther away from the earth. At -first ,la~-~ce it should appear that the moment would make the snooty move -['aster and would have no effect on the distance.
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From page 99... ...
Uber den Einfluss der Gezeiten auf die sakulare Acceleration des Mondes. Annales Academiae Scientiarum Fennicae, 18A: 1 (1921)
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