Earth tides in MacDonald's model

TL;DR

This paper extends MacDonald's 1964 tidal theory, analyzes tidal variations and potential at a point on Earth, and discusses the relationship between tide component lags and frequencies, contrasting different rheological models.

Contribution

It provides a Fourier analysis of tidal variations based on expanded equations and clarifies the rheological assumptions in different tidal models.

Findings

No intrinsic law relates tide lag to frequency in MacDonald's model.

Singer's modification also lacks a fixed rheology.

Williams and Efroimsky's model aligns with Darwin rheology, fixing lag proportional to frequency.

Abstract

We expand the equations used in MacDonald's 1964 theory and Fourier analyze the tidal variations of the height at one point on the Earth surface, and also the tidal potential at such point. It is shown that no intrinsic law is relating the lag of the tide components to their frequencies. In other words, no simple rheology is being intrinsically fixed by MacDonald's equations. The same is true of the modification proposed by Singer(1968). At variance with these two cases, the modification proposed by Williams and Efroimsky (2012) fix the standard Darwin rheology in which the lags are proportional to the frequencies and their model is, in this sense, equivalent to Mignard's 1979 formulation of Darwin's theory.