The moment of inertia of Mars and the existence of a core

Summary. A relation is obtained between the true value of the moment of inertia of a planet and the value calculated from the dynamical form factor, J ₂, on the assumption of hydrostatic equilibrium. The result is applied to Mars and it is shown that the difference between the true and calculated moments of inertia is probably insignificant in considering models of the interior of Mars and in particular does not affect an argument for a core based on models calculated by Lyttleton.     

A dislocation model of microplate boundary ruptures n the presence of a viscoelastic asthenosphere

Summary. A microplate is modelled as an elastic plate with two long strike-slip boundaries, lying over a Maxwell-type viscoelastic asthenosphere. The microplate is subjected to a constant and uniform shear strain rate by the opposite motions of two adjoining larger plates. After the occurrence of an earthquake at one of the microplate boundaries, the time evolution of shear stress at the other boundary is studied. It is found that stress build-up at the second boundary is delayed due to stress diffusion governed by the asthenosphere relaxation. Earthquake occurrence at this latter boundary would be delayed depending upon both the microplate width and the ratio between the Maxwell relaxation time of the asthenosphere and a characteristic time required for tectonic strain to recover rupture conditions. It turns out that the parameters which determine the occurrence of seismic activity along the microplate boundaries are more strictly constrained in the presence of a viscoelastic asthenosphere than in the case of an elastic half-pace model.     

Normal modes of the viscoelastic earth

Summary A uniformly valid linear viscoelastic rheology is described which takes the form of a 'generalized' Burgers' body and which appears capable of reconciling the behaviour of the Earth's mantle across the complete spectrum of geodynamic time-scales. This spectrum is bracketed by the short time-scales of body wave and free oscillation seismology on which anelastic effects are dominant, and the long time-scale of mantle convection on which the Earth behaves viscously. The parameters of the model which control the viscous response are fixed by post-glacial rebound data whereas those which govern the anelasticity are to be determined by fitting the model to observations of seismic Q. The paper is concerned primarily with a discussion of the normal mode spectrum of the Earth as a generalized Burgers' body. Focusing upon the homogeneous model, it includes an initial analysis of the accuracy of first-order perturbation theory as a method of calculating the respective Q s of the elastic gravitational free oscillations. Also considered are the quasi-static modes of relaxation which only exact eigenanalysis can reveal. The importance of these modes is assessed within the context of a discussion of the effect of viscoelasticity upon the efficiency of Chandler wobble excitation.     

Instabilities of fluid conduits in a flowing earth — are plates lubricated by the asthenosphere?

Summary A laboratory and theoretical study of the stability of conduits of buoyant fluid in a viscous shear flow has been conducted. The object of the study is to explain the formation of discrete islands in island chains such as the Hawaiian Emperor seamount chain, and to investigate a new method by which the variation of shear with depth in the mantle may be determined. The conduits were made by injecting oil into a more viscous oil of greater density. Initially a growing chamber of lower viscosity oil formed near the injector, but when the chamber got sufficiently large it rose as a buoyant spheroid. Behind this trailed a vertical cylindrical conduit through which fluid could continue to rise to the surface as long as the source continued. If the more viscous fluid was sheared laterally the conduit was gradually rotated to a more horizontal position. The diameter of the conduit increased with time due to a decreasing component of gravitational force along the axis of the conduit. When the conduit was tilted to more than 60° with the vertical, it began to go unstable by developing bumps which ultimately initiated a new chamber which rose to a new spot. In addition, if the Reynolds number of the conduit was greater than approximately ten, an axisymmetric wavy instability appeared in the walls of the conduit and the conduit had to be tilted less before a new chamber was initiated. If shear under the Pacific plate has to tilt buoyant mantle plumes to as much as 60° to form the relatively regular island chains associated with hot spots, most of the shear would be found in a zone with a vertical extent of less than 200 km.     

Magnetotelluric response of a uniformly stratified earth containing a magnetized layer

Summary The magnetotelluric (MT) response is studied of a uniformly stratified earth which contains a magnetized layer. The impedance as a function of the layer parameters (resistivity, ρ permeability, μ and thickness h ) is discussed. The MT response from a layer (μ, ρ, h ) is equivalent to that from a layer (μ/μ_r=μ₀, μ_rρ, μ_r h ) where μ_r is the relative permeability of the layer. Thus the effect of a magnetized layer is to make it apppear μ_r times more resistive and μ_r times thicker than an unmagnetized equivalent layer. Master curves of apparent resistivity and phase are computed for three-layer models with varying permeability associated with varying resistivity in each layer. An example of MT field data is presented in which the most reasonable interpretation is that a magnetized layer exists beneath the observatory site.     

Effect of 3-D viscoelastic structure on post-seismic relaxation from the 2004 M= 9.2 Sumatra earthquake

The 2004 M = 9.2 Sumatra–Andaman earthquake profoundly altered the state of stress in a large volume surrounding the ∼1400 km long rupture. Induced mantle flow fields and coupled surface deformation are sensitive to the 3-D rheology structure. To predict the post-seismic motions from this earthquake, relaxation of a 3-D spherical viscoelastic earth model is simulated using the theory of coupled normal modes. The quasi-static deformation basis set and solution on the 3-D model is constructed using: a spherically stratified viscoelastic earth model with a linear stress–strain relation; an aspherical perturbation in viscoelastic structure; a 'static' mode basis set consisting of Earth's spheroidal and toroidal free oscillations; a "viscoelastic" mode basis set; and interaction kernels that describe the coupling among viscoelastic and static modes. Application to the 2004 Sumatra–Andaman earthquake illustrates the profound modification of the post-seismic flow field at depth by a slab structure and similarly large effects on the near-field post-seismic deformation field at Earth's surface. Comparison with post-seismic GPS observations illustrates the extent to which viscoelastic relaxation contributes to the regional post-seismic deformation.     

Analytical approach for the toroidal relaxation of viscoelastic earth

This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core–mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes—unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core–mantle boundary and Earth's surface—however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works.