M. G. Rochester

Compressibility, core dynamics and the subseismic wave equation

Abstract It is shown that in the dynamics of a deep fluid of planetary scale such as the Earths core, compressibility, stratification and self-gravitation are all important as well as rotation. The existing proof of Cowlings theorem prohibiting non-stationary axisymmetric dynamos, and the application of the Proudman-Taylor theorem to core flows, both based on the assumption of solenoidal flow, need to be reconsidered. For sufficiently small (subacoustic) frequencies or reciprocal time scales, an approximation which neglects the effect of flow pressure on the density is valid. We call this the “subseismic approximation” and show that it leads to a new second-order partial differential equation in a single scalar variable describing the low frequency dynamical behaviour. The new “subseismic wave equation” allows a direct connection to be made between the various possible physical regimes of core structure and its dynamics.

Slichter modes and Love numbers

The recent claim by Smylie [1992] to have detected the Slichter modes of the Earths inner core rests on what appears to be a remarkable agreement between theory and observation. However the theoretical eigenperiods which are used by Smylie conflict significantly with all previously published periods for the Slichter modes, both for non-rotating and rotating Earth models. A closer examination of the theory used in these calculations reveals the use of static Love numbers to represent the response of the inner core and mantle to dynamics in the liquid core. We here show that the use of dynamic Love numbers restores the eigenperiods to those obtained using standard seismological theory and consequently destroys agreement between these periods and the claimed observations.

CORE-MANTLE INTERACTIONS: GEOPHYSICAL AND ASTRONOMICAL CONSEQUENCES

Abstract The existence of the liquid core, and its coupling to the solid mantle, give variety and complexity to the possible ways in which the Earths rotation can change. The interaction may be inertial (including what Hide has called topographic coupling), viscous (laminar or turbulent boundary layer friction), or electromagnetic. This paper compares the roles of the several coupling mechanisms in causing detectable fluctuations in the axial rate of rotation and in the position of the pole both in inertial space and with respect to a geographical frame of reference.

Computing core oscillation eigenperiods for the rotating Earth: a test of the subseismic approximation

Abstract The liquid outer core (OC) density profile of the preliminary reference Earth model (PREM) is modified to have various values of the Brunt-Vaisala frequency N in order to study long-period gravity modes modified by rotation. The eigenperiods of gravity modes of both non-rotating and rotating, self-gravitating, stably and spherically stratified, compressible OC models with rigid-fixed boundaries are obtained by solving the exact two-potential description (TPD) of core dynamics and the subseismic wave equation (SSWE) respectively, so as to test the subseismic approximation (SSA). Comparison of the resulting eigenperiods shows the high accuracy of the SSA: (1) the eigenperiods computed from the SSWE differ from those calculated from the TPD by at most 0.8% for a non-rotating core with 2π N = 6 h ; (2) the weaker the stability of the OC, the smaller the difference; (3) the more complex the spatial variation of the eigenfunction, the smaller the difference; (4) the SSA gives the eigenperiods more accurately for a rotating Earth than for a non-rotating model. The results also show that the SSA always shortens the eigenperiods. Convergence of the truncated series of trial functions in the Galerkin method used in the calculations is carefully checked to guarantee reliability of the numerical results.

Gravity and Slichter modes of the rotating Earth

Abstract The exact two-potential description of the dynamics of the liquid outer core (OC) is solved numerically, using a Galerkin method, for a rotating spherical earth model (original and modified PREM). Coriolis effects are taken fully into account in the OC. Deformation of the inner core and mantle, incorporating Coriolis self-coupling, is included via internal load Love numbers. Special attention is given to conditions at the geocentre in computing Love numbers at the inner core boundary (ICB). Internal undertone oscillation eigenperiods are computed for PREM modified to have a uniformly stable OC. For those modes involving no significant degree-1 spheroidal field, the assumption of rigid OC boundaries leads to smaller errors than adopting the subseismic approximation. Slichter mode eigenperiods are computed for PREM and for PREM modified to have a uniformly stable or unstable OC. The stability structure of the OC has little effect on Slichter eigenperiods, which are almost inversely proportional to the square root of the ICB density jump for earth models with the OC so modified. The effects of ellipticity and rotational potential on the Slichter eigenperiods are estimated using perturbation theory, and the effects of melting/freezing at the ICB during a Slichter oscillation are shown to be negligible.

Long Period Core Dynamics

Fundamental to the determination of the dynamical behaviour of the whole Earth is the computation of the dynamical response of the fluid core. It is a rotating, stratified, self-gravitating, contained, compressible fluid exhibiting a rich array of possible behaviours. At periods beyond a substantial fraction of a day, Coriolis coupling between terms in traditional spherical harmonic representations of the motions is very strong and the series are poorly convergent. To overcome this problem a single scalar second order governing equation has been developed, called the subseismic equation, which is valid at frequencies below the elasto-gravitational mode spectrum. This equation has been separated and new eigenfunctions which are uncoupled in the body of the fluid core are being investigated as representations of the motions. In general, the boundary conditions are not separable and superpositions of the eigenfunctions of the subseismic equation are required for their satisfaction. Successful computation of the normal modes of the fluid core will allow the complete determination of its contribution to the body tide response, free and forced nutations, changes in the length of day and polar motion as well as its long period free mode spectrum.

Zenomagnetic core-mantle coupling and the rotation of Jupiter's Great Red Spot

Abstract Fluctuations of the order of several seconds in the period of rotation of Jupiters Great Red Spot have been observed by Reese and Solberg (1966) to occur within intervals of not more than a few months. On Hides (1961) hypothesis that the Red Spot is a persistent feature attached to the solid surface of the planet, these fluctuations may indicate an internal redistribution of Jupiters angular momentum. Evidence for a strong zenomagnetic field and for a transition from the molecular to the metallic phase of hydrogen at pressures of a few Mb have led Gallet (1961) and Runcorn (1967) to suggest that angular momentum may be transferred between the solid molecular hydrogen mantle and a liquid metallic core. Adopting their model for Jupiters interior and using the theory of electromagnetic coupling which has successfully explained the irregular changes in the Earths rotation period, the electrical conductivity of the Jovian mantle can be estimated. The result is not inconsistent with such slight evidence as exists for changes in the rotation period of the decametre radio sources.