A search for Chandler and nearly diurnal free wobbles using Liouville equations

Abstract

The basic equations describing the dynamical effects of the Earth's fluid core (Liouville, Navier-Stokes and elasticity equations) are derived for an ellipsoidal earth model without axial symmetry but with an homogeneous and deformable fluid core and elastic mantle.We develop the balance of moment of momentum up t o the second order and use Love numbers t o describe the inertia tensor's variations. The inertial torque takes into account the ellipticity and the volume change of the liquid core. On the core-mantle boundary we locate dissipative, magnetic and viscous torques. In this way we obtain quite a complete formulation for the Liouville equations.These equations are restricted in order to obtain the usual Chandler and nearly diurnal eigenfrequencies.Then we propose a method for calculating the perturbations of these eigenfrequencies when considering additional terms in the Liouville equations.All these results were obtained by applying the balance of moment of momentum to a fluid-filled ellipsoidal cavity within a rigid shell.This theory, developed for over-simplified earth models, was extended to elastic models with more realistic stratification by Jeffreys & Vicente (1957a, b) and Molodensky (1961); these authors considered a spherical elastic mantle without rotation and very simple core models. Numerical integration has shown the existence of a free period about 3 min shorter than the sidereal day (except for model I1 of Jeffreys & Vicente 1957b), which affects through resonance the amplitude of certain diurnal tidal waves and associated nutations.Shen & Mansinha (1976) have extended Molodensky's theory of harmonic oscillations of any order iz and degree m using a less restricted representation of the core response, and making use of the P(r) stability parameter defined by Pekeris & Accad (1972). Their results are in good agreement with Molodensky's in the case of a neutral stratified core (P(r) = 0).

Lunar laser ranging test of the Nordtvedt parameter and a possible variation in the gravitational constant

“…The previous model for an elastic, homogeneous, and dissipative Moon has been extended by including the influence of a liquid core . Adopting the expressions of Hinderer et al (1982) for the Moon, this leads to Eq. (4) for the whole Moon, Eq.…”

Rotation of the Terrestrial Planets

“…A particular consequence of this approximation is that there is no term in B -A in eqn [25], because that term is of second order in the rotation variations (e.g., Hinderer et al, 1982). Furthermore, the first two equations decouple from the third equation.…”

“…(e.g., Rochester, 1976;Sasao et al, , 1980and Hinderer et al, 1982;Van Hoolst and Dehant, 2002 for the triaxial case).…”

“…The main reason is that for polar motion, changes in the inertia tensor are compared to differences in the moments of inertia (see eqns [23], [24], [32], and [33]), and for rotation rate variations, they are compared to the polar moment of inertia C itself (see eqns [25] and [34]), which is several orders of magnitude larger than the differences in principal moments of inertia for the terrestrial planets. The incremental core inertia terms can be expressed in terms of compliances or generalized Love numbers (Chapter 3.10; Dehant et al, 1993;Hinderer et al, 1982;Mathews et al, 1991;Sasao et al, , 1980. By solving the Liouville equations for the entire planet and the liquid core, wobble can then be expressed as Sasao et al, 1980), where A m is the equatorial moment of inertia of the mantle with respect to the first body-fixed axis.…”

Non-linear equations for the rotation of a viscoelastic planet taking into account the influence of a liquid core

“…(40) and (41) are the well-known Euler equations relative to an Earth's model with an inviscid fluid core and a deformable mantle with a pure linear elastic rheology. After Molodensky (1961), this closed set of equations have been developed for an homogeneous incompressible core by Sasao et al (1977), and extended to the case of a stratified compressible core by Sasao et al (1980) (see also Hinderer et al, 1982).…”