Non-rotating Earth Model Research Articles

Postglacial sea-level change on a rotating Earth: first results from a gravitationally self-consistent sea-level equation

We present and solve a gravitationally self-consistent sea-level equation which governs postglacial sea-level variations on a spherically symmetric, self-gravitating, viscoelastic and rotating Earth. We find that the inclusion of a glacio-isostatically induced rotational excitation can significantly affect previous predictions of both present-day sea-level rates and postglacial sea-level histories which were based on a theory that assumed a non-rotating Earth model. To illustrate, we consider present-day sea-level rates (and tide-gauge corrections) along the US east coast, and relative sea-level curves in the far field of the late Pleistocene ice sheets.

The subseismic approximation in core dynamics-II. Love numbers and surface gravity

SUMMARY This paper extends our earlier examinations of the utility of various approximations for treating the dynamics of the Earth’s liquid core on time-scales of the order of lo4 to lo8 s. We discuss the effects of representing the response of the mantle and inner core by static (versus dynamic) Love numbers, and of invoking the subseismic approximation for treating core flow, used either only in the interior of the liquid core (SSA-1) or also at the boundaries (SSA-2). The success of each approximation (or combinations thereof) is measured by comparing the resulting surface gravity effects (computed for a given earthquake excitation), and (for the Slichter mode) the distribution of translational momentum, with reference calculations in which none of these approximations is made. We conclude that for calculations of the Slichter triplet, none of the approximations is satisfactory, i.e. a full solution (using dynamic Love numbers at elastic boundaries and no core flow approximation) is required in order to avoid spurious eigenfrequencies and to yield correct eigenfunctions (e.g. conserving translational momentum) and surface gravity. For core undertones, the use of static Love numbers at rigid boundaries is acceptable, along with SSA-1 (i.e. provided the subseismic approximation is not invoked at the core boundaries). Although the calculations presented here are for a non-rotating earth model, we argue that the principal conclusions should be applicable to the rotating Earth. Shortcomings of the subseismic approximation appear to arise because both SSA-1 and SSA-2 lower the order of the governing system of differential equations (giving rise to a singular perturbation problem), and because SSA-2 overdetermines the boundary conditions (making it impossible for solutions to satisfy all continuity requirements at core boundaries).

The Gravity Effect of Core Modes for a Rotating Earth

Previous work has shown that the surface gravity effect associated with core modes (undertones) for a non-rotating Earth model is at or below the nanogal level of detectability for a large earthquake excitation. The assumption in the non-rotating case is that each undertone consists of a single spherical harmonic. In this study we add the Coriolis force to the equations governing the oscillations in a rigid shell with the Boussinesq approximation. By including up to 20 additional harmonics, we find that there are many solutions that are still dominated by low-degree spherical harmonics when judged by the distribution of kinetic energy among the harmonics. For these modes the contribution of higher degrees to the eigenfunctions is not substantial, which suggests that the addition of rotation will probably not significantly change the gravity effects computed without rotation. This conclusion remains tentative until verified in a fuller treatment that explicitly evaluates the excitation for a rotating Earth model.

On the excitation, detection and damping of core modes

We continue work on the possibility that internal oscillations in the Earth's core might be detectable by a gravity meter at the surface. As a possible excitation mechanism we use earthquakes and for the damping we consider the contribution of anelasticity in both solid and liquid parts of the Earth. The calculation of the eigenfunctions is restricted in this paper to a non-rotating Earth model. We first review the spectrum of modes for degrees l up to 20 using a stably stratified fluid core based on the Preliminary Reference Earth Model (PREM) and compute the earthquake excitation and Q values using standard seismic theory. We then show that the effects of boundary rigidity and both Boussinesq and subseismic approximations in the core have little effect on the eigenfrequencies of the core modes. The rigid boundary core solutions are then extended into both the mantle and inner core by making use of a generalised internal load Love number approach. This method yields good approximations to the elastic eigenfunctions throughout the Earth, the agreement with full theory improving as the degree l increases. Thus we have confidence that this method can be extended successfully to models with rotation.

The influence of rotation on the free oscillations of the Earth

We pursue an abstract investigation of the theory of the infinitesimal free elasticgravitational oscillations of a fairly general rotating Earth model. By considering in some detail the transition to the non-rotating case, we are able to delineate certain of the intrinsic effects of rotation on the normal mode eigensolutions, and to show how profoundly rotation alters the fundamental mathematical and physical properties of these eigensolutions. In particular, we show that the displacement eigenfunctions of a rotating Earth model are not mutually orthogonal, and that the corresponding normal modes of oscillation cannot in general be represented by pure standing waves. We consider the excitation of the normal modes of oscillation of a rotating Earth model by a transient imposed body force distribution, and we show that the complex dynamical amplitude of each normal mode may, in many geophysical applications, be determined separately, in spite of the lack of orthogonality among the displacement eigenfunctions. The calculation of the associated static response after the decay of the normal modes of oscillation is, on the other hand, complicated considerably by the absence of orthogonality. We specifically examine the influence of rotation on the zero-frequency rigid body translational and rotational modes of any non-rotating Earth model, and show how to account for the corresponding rigid body modes of any rotating Earth model in excitation calculations.

NONEQUATORIAL LAUNCHING TO EQUATORIAL ORBITS AND GENERAL NONPLANAR LAUNCHING

The general problem of launching out of the plane of an orbit is investigated by considering first the problem of launching from an arbitrary point on earth to a given equatorial orbit. A simplified analysis using a spherical, nonrotating earth model with no atmosphere is discussed. Generalized curves showing the interrelationship of ideal velocity to burnout conditions and the true heading to burnout conditions are presented for several values of launch latitude and several orbit altitudes. From these, a curve for each orbit altitude showing the minimum ideal velocity as a function of launch latitude is derived. Curves for time and angular range of the ascent trajectory for several orbit altitudes also are included. A method is given for using these curves when the desired circular orbit is inclined.