Forced nutations of the Earth: Influence of inner core dynamics: 1. Theory

Abstract

Gravitational and pressure couplings between the solid inner core and the rest of the Earth give rise to torques through which the inner core influences the nutational motions of the Earth. In view of the very small magnitude of the moment of inertia of the inner core relative to that of the the Earth as a whole, one expects from physical considerations that inclusion of the inner core in the dynamics should lead to a new nutational normal mode with a natural frequency not too far from that of the free core nutation, and to an associated weak resonance in the amplitude of forced nutations. We present here a treatment of the nutation problem for an oceanless, elastic, spheroidally stratified Earth, with the dynamical role of the inner core explicitly included in the formulation. As a preliminary to the setting‐up of dynamical equations, we devote some attention to a careful definition of a suitable coordinate system and of certain basic dynamical variables. We use the approach of Sasao et al. (1980), with their system of dynamical equations enlarged by the inclusion of two additional equations which are needed to describe the rotational motion of the inner core. An extension and sharpening of a line of reasoning employed by them enables us to derive expressions for the torques which couple the mantle and the fluid outer core to the solid inner core. Solving the enlarged system of equations, we show that a new nearly diurnal eigenfrequency does emerge; a rough estimate places it not very far from the prograde annual tidal excitation frequency, so that possible resonance effects on nutation amplitudes need careful consideration. Another eigenfrequency, attributable to a wobble of the inner core, is also found; its value is estimated to be a few times smaller than the wobble frequency that the inner core would have in the absence of couplings to the rest of the Earth. Considering an expansion, in terms of resonance contributions, of the amplitude of forced nutations normalized relative to that for a corresponding rigid Earth model, we indicate how the coefficients in the expansion are related to those in expansions of the type used by Wahr (1981b). Finally, we discuss the problem of comparing observed nutation amplitudes with those computed on the basis of Earth models generated from seismological data, with special reference to the fact that the dynamical ellipticity of the Earth, as computed from published Earth models which assume the condition of hydrostatic equilibrium, differs significantly from that determined from the precession constant. Numerical results, corrections for unmodeled effects, and comparison with observational results will be dealt with in accompanying papers.

Nonlinear Core-Mantle Coupling

“…Anelasticity of the mantle is also mentioned as a possibility. Recent VLBI results suggest that magnetic coupling is the dominant dissipative mechanism (Mathews et al 1991).…”

“…For dissipation in a laminar boundary layer with viscosity 0.01 cm2 s~1 and a rotation period of a day, the Q is estimated to be 12,000 (Toomre 1974). For a turbulent boundary layer or magnetic damping, the Q can be up to 2 orders of magnitude smaller (Mathews et al 1991). Defraigne, Dehant, & Hinderer (1994) give an estimate of 75^25 for the e †ective core Q of Earth.…”

“…where b is estimated to be b \ 0.00061 for Earth (Mathews et al 1991). This is to be compared with the core precession frequency without elasticity (eq.…”

Earth’s Rotation: A Challenging Problem in Mathematics and Physics

“…New investigations aimed at explaining the new results, among them DEHANT et al (1999). MATHEWS et al (1991a, b) introduced a solid inner core in the basic Earth structure. An empirical nutation model was adopted in the IERS Conventions 1996(MCCARTHY 1996.…”

Improved Models for Precession and Nutation

“…Of importance to this e ort is the availability of theoretical formulations of the e ects of ocean tides (Sasao and Wahr, 1981) and anelasticity ( W ahr and Bergen, 1986) on nutations, and the awareness of the potential for important c o n tributions to nutation from electromagnetic couplings of the uid outer core (FOC) to the mantle (Bu ett, 1992) and to the solid inner core (SIC). Enhancement of the existing framework of the analytical theory of nutation and wobble (Mathews et al, 1991) by the inclusion of all the above e ects into its basic structure (Mathews et al, 2 0 0 0 ) has been a signi cant step in the e ort to construct an accurate nutation series.…”