Tides and the evolution of the Earth—Moon system

Summary. A model of the tides in a hemispherical ocean is used to investigate the effect of changes in the Earth's rotation rate on the power dissipated by the ocean tides. The results obtained are then used in an idealized astronomical model to investigate how they affect the history of the Earth—Moon system.
Using the tidal model it is found that at rotation rates higher than that of the present Earth, the power dissipated by the semi-diurnal tides in the ocean drops off rapidly as a result of the increased tidal frequency. Thus if the Earth's rotation rate is doubled from its present value, then the rate of energy dissipation in the ocean is reduced to approximately one-third of its present value and the tidal torque is reduced by a factor of about 6.
The present value for secular acceleration of the Moon, calculated from the results of the tidal model is -30.5 arcsec century^-2. Using this value in the astronomical model, which has the Moon and Sun in circular orbits above the equator, and assuming that the tidal torque is independent of the tidal frequency, the Gerstenkorn event is predicted to have occurred 1.3 × 10⁹ yr ago.
When the astronomical model is run with a torque determined at all times from the tidal model, the reduction in the energy dissipated early in the history of the system, leads to a Gerstenkorn date of 5.3 × 10⁹ yr ago. However, dissipation within the solid earth is found to be important early in the history of the system and when this effect is included it gives a date for the Gerstenkorn event of 3.9 × 10⁹ yr ago.