Tidal dissipation within Earth’s solidifying magma ocean: I. Effects of inertia and lunar orbital eccentricity

When modeling the early tidal evolution of the Earth–Moon system, we need to calculate tidal dissipation within Earth’s solidifying magma ocean for a wide range of tide-raising frequency. This is because some giant impact hypotheses posit an initially rapidly spinning Earth. Also, the evection resonance, which is likely to have been encountered by the early Earth–Moon system, increases lunar orbital eccentricity, which can broaden the frequency range. It is shown that the standard solution methods such as the propagator matrix method and the shooting method are insufficient to handle this wide range of frequency. As tide-raising frequency increases, the effect of inertia becomes significant, but the propagator matrix method is commonly implemented only with the governing equations without inertia. The shooting method can handle high frequencies well, but it becomes unbearably unstable toward the low frequency end. The relaxation method is found to be extremely stable over the entire frequency range considered, being able to handle huge viscosity variations over 20 orders of magnitudes. For an initially rapidly spinning Earth, the effects of inertia and lunar orbital eccentricity are considerable, warranting a careful reevaluation of various scenarios for the early lunar evolution.