Earth tides

Earth tides The gravitational attraction between the Earth and Moon keeps them in orbit about their common centre of mass, which is a point within the Earth 4670 km from the centre. Both the Earth and the Moon are nearly spherical, which means that the total attractive force between them is nearly the same as if they were point masses at their centres. Thus the centripetal force on each is just that required to keep point masses at their centres in orbit about one another. The side of the Earth nearer to the Moon experiences a stronger gravitational pull than is needed to keep it in orbit with the rest of the Earth. It is therefore pulled towards the Moon. Conversely, on the far side of the Earth the gravitational force of the Moon is weaker than the average. The result is a pattern of tidal force pulling the Earth into a prolate ellipsoidal form aligned with the Earth–Moon axis, as shown in Fig. 1.

The tide due to the Sun is smaller than the lunar tide, by a factor of 0.46: the effect of the Sun's much greater mass is more than compensated by its distance. The total tide is therefore a superposition of the lunar and solar tidal bulges, with a relative alignment that changes progressively through the lunar month. At new and full moon, when the Sun and Moon are in the same direction or opposite one another, the two tidal components are added and maximum (spring) tides are observed. Weaker (neap) tides occur at half-moon, when the sun and Moon are 90° apart.

The observed tide is a result of the rotation of the Earth within the envelope of the two deformations, which remain fixed in orientation relative to the Moon and Sun. In most places the lunar semidiurnal tide is dominant, which means a tidal period of 12.42 hours, with an amplitude that oscillates semi-monthly as a beat with the 12-hour solar tide. Misalignment of the Equator with the orbits gives a prominent diurnal tide at some latitudes, and there are other periods in tidal records arising from the orbital ellipticities and precession of the lunar orbit.

The equilibrium tide is the sum of the two prolate ellipsoidal elongations in the directions of the Moon and Sun. The solid Earth is deformed elastically in this manner, but the motion, about 0.4 m, can be observed only with sensitive instruments. The marine tide is more obvious, but is much more complicated. Even in the deep oceans the natural speed of a tidal wave around the Earth is much less than the speed of the Earth's rotation, and the ocean consequently responds to tidal forces with a phase lag of nearly 90°. This means that high tides appear where lows would be expected for an equilibrium tide. The tidal bulge observed by analysis of satellite orbits is the sum of the solid Earth and marine tides, and so is smaller than would be observed with the solid Earth alone. The term Earth tide sometimes refers to this total tide, but historically it has meant the solid Earth tide.

The marine tide is locally very variable. Large tidal lags and resonant amplifications occur in shallow areas, particularly in bays and estuaries. Tidal movements in shallow water are believed to account for most of the dissipation of tidal energy, which globally amounts to 3 × 10¹² watts. A very small fraction of this is due to imperfect elasticity of the solid Earth.

As seen by satellites, the energy dissipation appears as a 2.9° lag in the orientation of the globally averaged lunar tidal bulge relative to the Earth–Moon axis. Since the bulge is caused by the Moon itself, the misalignment produces a torque in the direction that would pull the bulge back into line. This torque acts as a brake on the Earth's rotation, causing the length of the day to increase gradually. The combined effect of the lunar and solar tidal torques is to increase the length of the day by 24 microseconds each year. Remembering that the Earth is about 4.5 × 10⁹ years old, it is evident that over geological time tidal friction has had a major effect: 650 million years ago there were 400 days per year, and when the Earth was very young the day was probably only 6 to 8 hours long.

The lunar torque on the Earth is balanced by a torque on the Moon's orbit, causing the radius of the orbit to increase by 3.7 cm per year. The angular momentum lost by the Earth's axial rotation appears in the lunar orbit, the total angular momentum being conserved. The Moon was much closer early in the life of the Earth. It may also have been rotating, but now presents a constant face to the Earth because the friction of the large tide raised in the Moon by the Earth has completely stopped its relative rotation.

Frank D. Stacey

Bibliography

Darwin, G. H. (1898) The tides and kindred phenomena in the solar system. (Reprinted 1962). W. H. Freeman, San Francisco.
Stacey, F. D. (1992) Physics of the Earth (3rd edn), pp. 115–33. Brookfield Press, Brisbane.