A Hamiltonian approach to dissipative phenomena between the Earth's mantle and core, and effects on free nutations

Abstract

S U M M A R YThe Hamiltonian formalism was recently applied by Getino (1995a,b) for the study of the rotation of a non-rigid earth with a heterogeneous and stratified liquid core. That earth model is generalized here by including the effect of the dissipation arising from the mantle-core interaction, using a model similar to that of Sasao, Okubo 8z Saito (1980), which includes both viscous and electromagnetic coupling. First, a solution for the free nutations is obtained following a classical approach, which in our opinion is more familiar to most of the readers than the Hamiltonian treatment. This solution provides a theoretical basis clear enough to study both the qualitative and quantitative effects of the dissipations considered in the hypotheses. The main qualitative features are, besides the delays, that the free core nutation (FCN) suffers an exponential damping, while the chandler wobble (CW) is not damped at first order, by the dissipation considered. The numerical values obtained for the complex compliances agree with the most recent experimental computations.Next, the problem is studied under a Hamiltonian formalism, and a solution equivalent to the above is obtained. Besides its interest from a theoretical point of view, this formalism is necessary in order to apply canonical perturbation methods in order to obtain analytical nutation series.

Hamiltonian theory for the non-rigid Earth: Semidiurnal terms

“…Further information, as well as a detailed study of this problem, can be found in Getino & Ferrándiz (1997, 2000a.…”

“…In the Hamiltonian framework the dissipative effects are formulated by means of the corresponding generalized forces Q p , Q q , whose expression, developed in Getino and Ferrándiz (1997), will be given below.…”

“…The generalized forces appearing in (23) are obtained in Getino & Ferrándiz (1997). The expressions of Q i which will be used in the following are…”

“…(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 .…”

“…At present, we have performed a comprehensive treatment of an Earth model composed of an elastic, axis-symmetric mantle, and a liquid core, including the dissipative effects in the core-mantle boundary (CMB) and the delay in the response of the deformation due to the inelasticity of the mantle. The corresponding free problem, with the determination of the free frequencies, is developed in Getino & Ferrándiz (1997), while the nutation series concerning the forced motion can be found in Getino & Ferrándiz (1999, 2000a.…”