ALBERT

Notes: Whereas the present-day true polar wander and the secular non-tidal acceleration of the Earth have usually been attributed to postglacial rebound, it has recently been suggested that non-glacially induced vertical tectonic movements taking place under non-isostatic conditions can also be effective in changing the Earth's rotation. We present a case study in which we analyse the effects of some simple uplift histories of the Himalayas and the Tibetan Plateau on the rotational axis and on the second-degree zonal harmonic of the geoid, for time-scales of up to a few million years. We first assume a permanent amount of overcompensation, which is consistent with observed geoid anomalies over the Himalayas, and then we model by means of the normal-mode techniques, the viscous relaxation in the mantle, with the elastic lithosphere supporting elastically 2 per cent of isostatic disequilibrium. In our normal-mode analysis, the Earth is divided into five layers: an effectively elastic lithosphere, a viscoelastic shallow upper mantle, transition zone and lower mantle characterized by the Maxwell rheology and an inviscid core. The readjustment of the equatorial bulge due to viscous flow in the mantle is taken into account in our studies by solving the linearized Liouville equations for the conservation of angular momentum, via the Love numbers formalism.Polar wander is sensitive to the rate of relaxation of the modes M1 and M2 due to the discontinuities between the three mantle layers, positioned at 420 and 670 kilometres depth. The rate of readjustment is sensitive to the viscosity of the transition zone whenever the lower mantle/shallow upper mantle viscosity ratio is small. The highest present-day velocity of polar wander due to Himalayan and Tibetan Plateau uplift is estimated to be 1° Myr−1 for an isoviscous mantle that has the same magnitude of the observed value, reduced to 0.1° Myr−1 for a factor 50 viscosity increase in the lower mantle. These numbers are about the same as those found from postglacial rebound that occurs on the short time-scale of a thousand years instead of the million years of our analysis, but represent upper bounds for mountain building, obtained only in the case in which a permanent deviation from isostasy of at least 2 per cent is assumed. In general, the proposed mechanism is less efficient in driving long-term rotation instabilities than deep-seated processes characterized by the same time-scale of a million years such as subduction; polar-wander velocity is extremely sensitive to the depth of the uncompensated anomalous root of the topography for the models in which full mantle relaxation is allowed.

Notes: The notion that the self-gravitation problem of a perfectly elastic solid body involves small strains from a reference state with arbitrary large initial stresses is reconfirmed and extended. It is shown that the Lagrangian, from which the equations of motion, the boundary conditions and Poisson's equation can be derived in Eulerian or Lagrangian coordinates by the Variational Principle, can be written in a general form which incorporates the various stress measures (Piola-Kirchhoff, Cauchy). Especially for an Eulerian description it is shown that the derived equations of motion and boundary conditions lead to a complete set of mutually orthogonal seismic normal modes.The Lagrangian which Geller (1988) gives is re-evaluated. It appears that in his considerations Geller neglected finite pre-stresses and only accounted for gravitational contributions. This, in general, is not correct. Though the static equilibrium equation has no unique solution for the initial stress components, this does not imply that in physical situations there would not be a specified initial pre-stress. Geller's statement that the asymmetric stress tensor Woodhouse & Dahlen (1978) employ in their Lagrangian necessarily leads to non-conservation of angular momentum is not valid. This stress tensor is a first Piola-Kirchhoff stress tensor, which is a so-called two-point tensor, associating two vector fields defined in different coordinate systems.

Notes: In some studies on glacially induced true polar wander (TPW). the tidal-effective relaxation of the fundamental mantle mode (M0) is lacking. We show that this is caused by the deletion of the Chandler wobble in an early stage of the theory development to facilitate the retrieval of the rotational relaxation modes. We derive an analytical approximation formula for the M0 rotational relaxation mode (including the Chandler wobble), which can be of practical value for TPW simulations with realistically stratified earth models. However, we point out that the contribution of the M0 rotational relaxation mode has, to a high approximation, the same effect on secular TPW as the contribution from an elastic term in models that do not have the M0 rotational relaxation mode. The two model approaches lead to the same polar wander results whenever the Chandler wobble is filtered from models in which the M0 rotational relaxation mode is retained.

Notes: In recent years a number of studies have investigated the influence of compressibility on geophysical observables such as postglacial rebound deformation rates and the geoid. Some of these studies indicate that long-term signatures such as the geoid might be sensitive to compressibility. As both load relaxation and tidal-effective relaxation of the equatorial bulge are operative in a dependent way, polar wander can potentially be more sensitive to compressible rheologies if the interference between the two relaxation mechanisms is constructive. This has motivated us to study the influence of compressibility on true polar wander by means of spherical, laterally homogeneous, self-gravitating analytical earth models. As we wish to study both short-term rotational changes and polar wander on geological time-scales, we employ a Maxwell viscoelastic model instead of a Newtonian viscous model. The latter is commonly used in geoid modelling. The purpose of this paper is to concentrate on the basic physical aspects of the differences between compressible and incompressible rotational deformation, rather than applying the procedures to fine-graded multi-layered PREM models with realistic forcing functions. An important issue of our method concerns the analytical instead of numerical way of solving the differential equations by the propagator matrix method. Compressible viscoelastic relaxation has usually been treated numerically until now.The results show that homogeneous earth models do not have significant differences on long time-scales between compressible and the corresponding incompressible cases. Compressibility introduces a denumerably infinite set of short-time relaxation modes. The relaxation times of these dilatation modes can be approximated analytically. Two-layer core-mantle models show relatively large differences between incompressible and compressible Maxwell rheologies. Simplified models of true polar wander triggered by Heaviside loads show that differences of several tens of per cent between incompressible and compressible Maxwell rheologies are possible. True polar wander is decreased in the compressible case on both short and long time-scales, which means that smaller viscosities are required to explain polar-wander measurements than in the incompressible case.