Tidal Friction.--The effect of dissipation of the ocean tide by friction has so far been left out of the discussion, but it is known to be an important factor. Harold Jeffreys has calculated that if the tide-generating forces of the moon and sun could be suddenly turned off, half the tidal energy in the earth's oceans would be dissipated in approximately a day. This dissipation is known to occur practically entirely by the friction of tidal currents against the bottom of a few shallow seas of the earth, and against the long shorelines of the continents and islands. It is a curious fact that probably half of the entire tidal dissipation of energy in the ocean takes place in the shallow Bering sea. This rapid rate of energy dissipation should have a considerable effect on the tides of the world.

The matter of tidal friction is of considerable importance in the motions of the planets and their satellites because it provides a mechanism whereby angular momentum is transferred from one type of motion to another. The tidal protuberance created on the primary by the tide-generating force of one of its satellites is not symmetrical with respect to the line joining their centres, and the satellite's own gravitational attraction on this protuberance exerts a torque on the primary. It can be shown in general that if there were no tidal dissipation, there would be no such torque, and that if the angular speed of the planet's rotation is faster than that of

the revolution of its satellite about it, then this torque always tends to decelerate the planet's rotation. In such a case, the gravitational force of the primary on its satellite deviates slightly from the direction toward the primary's centre of gravity, and tends to accelerate the revolution of the satellite about its primary, provided this revolution has the same sense as the rotation of the primary about its axis. Then, the distance between the primary and its satellite will increase as the satellite's speed of revolution is increased, according to Kepler's laws. The above-stated conditions apply, of course, in the case of the earth-moon system.

The same deductions can be arrived at by an alternative point of view: the frictional force of the tidal currents on the sea bottom and coasts must itself exert a decelerating torque on the earth, If the tidal currents were known accurately enough over all critical regions where the bulk of the tidal dissipation takes place, the decelerating torque could be calculated, since it is known that by far the greater part of the tidal dissipation occurs in the oceans, and only a negligible amount in the lithosphere. Actually, the form of the tide over the vast regions of the oceans is so poorly known that neither the gravitational torque method nor the frictional torque method can be applied quantitatively, and it is through records of ancient eclipses and modern observations on the moon's position that the average rate of lengthening of the day and lunar period of revolution have been estimated. It should be remarked that the lunar effect on the earth's rotation should be about three to five times as large as the solar effect.

There are four principal effects: (1) tides raised in the satellites by their primaries will tend to make each keep the same face toward its primary (as the moon presently faces the earth); (2) tides raised in the primaries by the satellites will alter the rates of rotation of the primaries; (3) tides raised in the primaries by the satellites will alter the distances between them; and (4) solar tides will affect all the rotations.

On the basis of the above considerations it has been deduced that during previous ages the earth had a much shorter day and much shorter month (period of lunar revolution), and that the moon was much closer to earth. The day has probably lengthened by a second in the last 120,000 years.

Internal Tides in the Ocean.--It has been shown that the tidal current would be uniform from top to bottom at any given locality and instant provided that the sea water had uniform density throughout the oceans. In the actual case, however, variations in the salinity and temperature lead to both time and space variations in the density, and it is possible that this condition would lead to a vertical shear in the tidal current. In any case, there is no reason to believe that there should not exist tidai variations in the water density at any given point in the ocean, and there is evidence on several occasions that such tidal variations may have been actually observed.

The outstanding feature of the density structure of the ocean is that the lighter water is found in the upper layers of the ocean, and in a large number of dynamical problems the ocean can be treated successfully as though the water were stratified: i.e., as though the water in any horizontal plane were homogeneous. In such a situation there can exist an important class of wave motions in which the maximum amplitude of the vertical displacent of the water particles occurs at some depth beneath the surface. These are called internal waves to contrast them with the familiar surface waves, for which the vertical displacements the water particles have their maximum amplitude at the surface.  Internal waves are similar in many respects to surface waves, have some important differences: their speed of propagation is much less than that of a surface wave having the same wave length; and the vertical movement of the water particles (at the level of maximum movement) is a great deal larger than in the case of a surface wave having the same energy. Such waves been observed by making periodical determinations of the vertical density profile from an anchored ship, and the results of such observations indicate that such internal wave motion is common, and that amplitudes of the order of tens of metres are to be expected. The oscillations are irregular, but frequently there appear to be dominant waves of diurnal or semidiurnal period having all the characteristics of a tidal variation. In order to prove conclusively the tidal origin of such waves, a considerable series of observations would be necessary, and the difficulties and expense of having a ship remain at the same location for a protracted period of time have prevented this study from being carried out.

As air, like all other matter, is subject to gravitational influence, there will be tides in the atmosphere possessing many features of similarity with those in the ocean. One of the characteristics of these tides will be a very small oscillatory variation in the atmospheric pressure at any place, and this may be regarded as the superposition of harmonic constituents with the ordinary tidal periods. By the systematic analysis of long series of regular barometric records, the principal lunar semidiurnal constituent M₂ Of the barometric variation has been determined for a number of places, and found to have an amplitude of the order of 0.001 in. The dynamical theory of these tides has been the subject of considerable study. The derivation of the equations is not so simple as for ocean tides, one difficulty being that of taking account of the physical conditions in the upper regions. The equations which have been proposed are of the same general form as those for ocean tides, but the theory is still imperfect. The results of analysis for the S₂ constituent show a much larger oscillation with an amplitude of barometric variation of the order of 0.03 in. For this constituent, however, it is certain that thermal factors play a larger part than gravitational factors, and the theory is far from complete.

The solid body of the earth, or lithosphere, suffers periodic deformation because of the tide-generating forces just as do the oceans and atmosphere. These bodily tides manifest themselves in the following ways: (1) a variation of the vertical, or plumb line, with respect to any solid structure imbedded in the earth's crust; and (2) a variation in the acceleration of gravity at any point due to the potential of the displaced matter in the earth's body, quite aside from the variation of gravity directly associated with the tide-generating forces.

The gravest free elastic oscillations of the earth's body as a whole have periods of the order of an hour, which is much less than those of the principal tide-generating forces. Therefore, it is probably correct to assume that the bodily tides will approximate closely to their equilibrium forms, or in other words, that they may be calculated on the principles of statics. Owing to the earth's rigidity, the vertical rise and fall of the earth's solid surface at any position on it is somewhat less than that based on the equilibrium tide. The fractional height of the equilibrium tide attained by this surface vertical displacement is practically independent of geographical location, and is designated by the symbol h. The potential of the displaced matter is additive to the tidegenerating potential, its fractional contribution to the latter at any point on the earth's surface being designated by the symbol k. As in the case of h, k also is practically independent of geographical location. The numbers h and k were introduced by the theoretician in the field of elasticity, A. E. H. Love, and henceforth have been called the Love numbers.

The values of h and k depend on the geographical form of the disturbing potential, but it is common usage to consider only those values corresponding to a disturbing potential having the form of a spherical harmonic of the second degree, of which the semidiurnal equilibrium form is an example.

Various types of geophysical observations have been used to evaluate the Love numbers. For instance, it can be shown that the acceleration of the earth's gravity at the surface is altered by the amount

Thus it is seen that precise measurements of the acceleration of the earth's gravity at a point would give a determination of the quantity  while measurements of the water level in a long sheltered tube, free from winds and other disturbances, would give a determination of -the quantity 1 + k - h. This quantity also determines the position of the vertical as can be measured by a plumb line or a special instrument called the horizontal pendulum.

It should also be possible to determine the quantity 1 + k - h  by means of observations of the ordinary ocean tide, provided its dynamics were well understood. The long-period tides, especially the lunar fortnightly tide Mf, have usually been utilized for this purpose. It is generally assumed that the periods of these constituents are very much longer than those of all natural oscillations in the ocean basins, and that the equilibrium law would therefore hold. This assumption is certainly valid in the case of ordinary inertio-gravitational standing wave motion in the oceans, but serious objections have been raised on the basis of the possible existence of geostrophic or planetary wave modes characterized by steady or nearly steady ocean currents. There is a further serious difficulty in isolating the purely astronomical long-period constituents in analysis, and it may well be doubted whether reliable constants for the Mf constituent have ever yet been obtained from the records of observation. The values of h and k determined vary considerably depending on the type of observation used, and for this reason are subject to appreciable error. The value of h is somewhere in the vicinity of 0.6, and that of k is near 0.3.

The above discussion of the bodily tide has omitted any reference to the complications introduced by the tides in the ocean. There are two effects: the warping of the earth's crust by the varying load on the ocean floor; and the contribution to the total gravitational potential by the displaced water mass. Generally, these effects are more serious closer to the coast and the greater the tidal range at the coast nearby. For this reason measurements of the bodily tide are usually attempted near the centre of large continents.

If the rigidity and density at all points in the earth's body were exactly known, it should be possible to calculate the Love numbers theoretically without need for recourse to observations. The internal constitution of the earth is not precisely known, but certain hypotheses concerning this internal constitution can be rejected on the basis of results of bodily tide observations. Lord Kelvin used the observed height of the fortnightly tide to show that the earth's rigidity, on the average, is greater than that of steel. See also TIDAL POWER and references under "Tide" in the Index.

BIBLIOGRAPHY.-P. Schureman, "Manual of Harmonic Analysis and Prediction of Tides," U.S. Coast and Geodetic Survey, Special Publication 98 (1941); A. T. Doodson and H. D. Warburg, Admiralty Manual of Tides (1941); Sir Horace Lamb, Hydrodynamics (1945); Harald U. Sverdrup, et at., The Oceans (1942) ; Sir Harold Jeffreys, The Earth (1952) ; J. Proudman, Dynamical Oceanography (1953) ; G. H. Darwin, The Tides and Kindred Phenomena in the Solar System (1962).
(J. PR.; G. W. Gs.)