Rotationally Corrected Equilibrium Tide Theory: Discussion

Abstract

Michelson (1973) has suggested modifications to the classical equilibrium tide theory, and his paper has received an enthusiastic review (Anon. 1974). I feel, however, that it is his modifications that are in error rather than the original equilibrium theory. Michelson claims that proponents of the classical theory, 'Newton, Bernoulli, Laplace, Darwin, Bassett, and Lamb, among others,' do not properly treat the rotation of the earth and the of its centre. In fact, the classical development of the equilibrium tide theory correctly allows for of the earth's centre by requiring it to be exactly that demanded by the gravitational attraction of sun and moon acting at the centre of the earth. Since the coordinate system being employed is fixed in the rigid body of the earth, this of the centre must enter the equation of motion as a component of the of the coordinate system. Thus far there seems to be no difference of opinion between Michelson and the earlier workers. The difference arises in consideration of the component due to the rotation of the coordinate system, that is, the rotation of the rigid body of the earth about its axis. The term expressing this component of does not commonly appear explicitly in the equilibrium theory because it is simply the centripetal term usually incorporated as part of the earth's 'gravity.' This development is fairly accurately described by Michelson in his Case I, but it does not require the earth's motion to be irrotational as he implies. In his Case 11, Michelson derives an expression that would be appropriate for an earth on which tidal friction had constrained the rotation to one revolution per lunar month about an normal to the lunar orbit; however, until such a condition is achieved this seems to have little pertinence to the real situation. In his Case 111, Michelson attempts to derive the terms appropriate to the orbiting and rotating coordinate system fixed to the rigid body of the earth. He does this by a rather indirect route which should, however, arrive at the same result as the classical equilibrium theory. That it does not is due to an error in the tortuous path of logic. This error seems to enter when Michelson considers rotation of the earth about an instantaneous of rotation displaced from but parallel to the polar axis. He states that acceleration of any point of the Earth is then directed toward the new displaced axis (p. 1756) and seems to overlook the fact that the instantaneous of rotation has an which must be considered. Correction for this error will bring Michelson's equation [lo] and what follows into agreement with the classical theory. I fear, however, that Michelson's readers may have been made unduly mistrustful of tidal predictions as a result of the article. It needs to be emphasized that the equilibrium tide theory is used primarily to describe the character of the tide-raising forces, particularly their frequencies and relative magnitudes. Predictions are made by harmonically analysing actual records of water level at the frequencies prescribed by the theory and by projecting the contributions of these harmonic constituents into the future. It is well realized that such a theory as the equilibrium theory that denies the presence of continental barriers, irregular bottom topography, and friction, and that neglects the response times of the system cannot be expected to predict to any accuracy the actual heights or times of local tides. Tidal predictions based on harmonic analysis at the frequencies indicated by the equilibrium tide theory, however, have been eminently successful.