This dissertation presents new theoretical methods that could contribute to constraining the deep Earth’s long-wavelength density structure.
In Chapter 2 we present a numerically exact method for calculating the internal and external gravitational potential of aspherical and heterogeneous planets. Such calculations are crucial in computing Earth’s long-period deformation. Our approach is based on the transformation of Poisson’s equation into an equivalent equation posed on a spherical computational domain. This new problem is solved in an efficient iterative manner based on a hybrid pseudo-spectral/spectral- element discretisation. The main advantage of our method is that its computational cost reflects the planet’s geometric and structural complexity, being in many situations only marginally more expensive than boundary perturbation theory. Several numerical examples are presented to illustrate the method’s efficacy and potential range of applications.
In Chapter 3 we investigate theoretically the dependence of the elastic tensor on the equilibrium stress, our aim being to understand the effect of nonzero stress on seismic wave propagation and Earth’s long-period motion. Using ideas from finite elasticity, it is first shown that both the equilibrium stress and elastic tensor are given uniquely in terms of the equilibrium deformation gradient relative to a fixed choice of reference body. Inversion of the relation between the deformation gradient and stress might, therefore, be expected to lead neatly to the desired expression for the elastic tensor. Unfortunately, the deformation gradient can only be recovered from the stress up to a choice of rotation matrix. Hence it is not possible in general to express the elastic tensor as a unique function of the equilibrium stress. By considering material symmetries, though, it is shown that the degree of non-uniqueness can sometimes be reduced, and in some cases even removed entirely. These results are illustrated through a range numerical calculations, and we also obtain linearised relations applicable to small perturbations in equilibrium stress. Finally, we make a comparison with previous studies before considering implications for geophysical forward- and inverse-modelling.
Finally, in Chapter 4 we present a theoretical framework for modelling the rotational dynamics of solid elastic bodies. It takes full account of: Earth’s variable rotation, aspherical topography, and lateral variations in density and wave-speeds. It is based on an exact decomposition of the body’s motion that separates out the motion’s elastic and rotational components, in a way that we make precise in the main text. As a prelude to the elastic problem, we show how Hamilton’s principle provides an elegant means of deriving the exact and linearised equations of rigid body motion, then study the normal modes of a rigid body in uniform rotation. We subsequently build on these ideas to write down the exact equations of motion of a variably rotating elastic body. We linearise the equations and discuss their numerical solution, before showing how to extend these ideas to analyse N elastic bodies interacting through gravity.
We conclude in Section 4.7 by discussing the extensions that would be necessary in order to describe layered fluid-solid bodies. Importantly, with those extensions the framework will bypass the numerical difficulties commonly associated with the Earth’s fluid layers, and could therefore readily be used to model diurnal tides. Thus, it would allow the abundant data provided by diurnal tides to be used to constrain lateral variations in mantle density. Furthermore, it would allow for a systematic investigation of the effect of lateral density heterogeneities on the Earth’s rotation; this has never yet been undertaken, and will provide additional constraints on mantle density and other parameters of interest
