Forced nutations of a rigid mantle-liquid core earth model in canonical formulation

Abstract

S U M M A R YThe canonical formalism for the rotation of an earth model composed of a rigid mantle and a liquid core, established in Getino (1995b), starting from the model of Sevilla & Romero (1987), is applied here to the study of the forced nutations caused by the perturbing lunisolar potential. The main purpose of this paper is not to get accurate values for the nutation terms, since the earth model considered does not include the effect of the elasticity. This approach should be understood as a generalization of Kinoshita's (1977) theory for a rigid earth. Under the canonical formulation, using Andoyer-like variables and following a procedure very similar to that of Kinoshita, we obtain similar expressions (to those of Kinoshita) for the nutations of the fundamental planes, the only difference being a corrector factor fc in the amplitudes of the nutation to account for the effect of the liquid core. This corrector factor is equal to 1 when particularizing for the rigid case; the results are then the same as those of Kinoshita.In this way, we develop a complete canonical theory for a non-rigid earth (not as yet including deformations) which includes the effect of the liquid core from the beginning. and which enables the amplitudes of the forced nutations to be obtained directly, unlike the usual theories for a non-rigid earth, which require the support of results for the rigid case.

Canonical treatment of dissipative forces between Earth mantle and core

“…If we neglect the dissipation (Αχ = 0), the out-of-phase terms disappears, and we get similar expressions as those obtained in Getino (1995b) for the conservative case. Even more, if we particularize to the rigid case (A c = 0), then F a = 1, and we obtain the same expressions as in Kinoshita's theory.…”

“…The study of the Earth rotation is not an exception, and the theory of Kinoshita (1977) is the most complete and accurate among those for a rigid Earth model. Recently, the Hamiltonian approach has been followed by the authors in order to extend Kinoshita's theory to general models of deformable Earth through a series of papers, first considering an elastic mantle (Getino and Ferrândiz, 1995), then a liquid core (Getino, 1995a(Getino, , 1995b. We only point out here that the results obtained are completely analytical, and depend on parameters that are provided by the different Earth models available.…”

“…We can write the components of M and M c in the Oxyz frame in terms of the canonical variables as follows Then, the canonical expression of the kinetic energy is (Getino, 1995b):…”

“…As a first step, we have chosen a fairly well known simplified model, in which the dissipation appears through a torque proportional to the difference of angular velocities (Kubo,1979). Using canonical Andoyer-like variables as in Getino (1995b), it is only necessary to introduce the generalized forces in a straightforward way. By integrating the unperturbed Hamiltonian equations, solutions for the free wobbles are obtained, which in our model represent the CW and the NDFW, using the standard terminology for conservative free oscillations.…”

Hamiltonian theory for the non-rigid Earth: Semidiurnal terms

“…(37) is of the second order in σ. Therefore, due to the smallness of variables σ, σ c ∼ 10 −6 rad (Nikitina 1990), neglecting this term we obtain the first order the relationship (Getino 1995b;Getino & Ferrándiz 1997) µ +ν = Ω .…”