Long period free-oscillations of the Earth

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Free oscillations provide one of the most important constraints on the Earth's internal structure at long wavelengths. Indeed, free-oscillation periods were the main data used in determining the Earth's spherically symmetric density structure, while free-oscillation spectra can be used to image lateral density variations in the mantle. This latter point is of particular relevance to the question of whether convection within the lowermost mantle is driven by thermal or compositional buoyancy. The Earth's free-oscillation spectrum is very diverse, being comprised both of familiar "acoustic modes" associated with seismic wave propagation, and more exotic long period modes such as the free core nutations, the Chandler wobble, and undertone modes that are largely confined to the outer core. At these long periods, however, the dynamics of the Earth is greatly complicated by the Earth's rotation, the nature of the density stratification within the outer core, viscous or viscoelasticity dissipation, and possible interactions with the Earth's magnetic field.This project will extend our understanding of the dynamics of the solid Earth at periods between about one year and one hour, through both theoretical and numerical studies of the Earth's free-oscillations. Were the Earth entirely solid and perfectly elastic, then free-oscillations theory is completely understood. However, the Earth's outer core and oceans are fluid, and within seismology they are often regarded as being inviscid. The inclusion of such fluid regions fundamentally changes the nature of the eigenvalue problem associated with free-oscillations, and the precise nature of the spectrum at low frequencies is not yet understood. Furthermore, the effects of a finite viscosity or viscoelasticity at these long periods has received little attention, and is of particular relevance for dynamic interactions of the core and mantle. All these questions could have significant implications for numerical studies of the longest period free-oscillations, and hence for applications to the study of Earth structure.The student will work collaboratively to develop new theoretical and computational approaches to study the longest period free oscillations of the Earth. A major question is the precise nature of the free-oscillation spectrum at low frequencies. Due to the inclusion of inviscid fluid regions, the operator occurring in the eigenvalue problem lacks the usual Fredholm property, and this leads to the existence of an essential spectrum near to zero frequency. In previous work it has been argued - without full proof - that this part of the spectrum is continuous, and that observable long period free-oscillations such as the free core nutations are embedded within region. If this is true, then there are significant complications related to computation and excitation of these oscillations. A further question is how a finite viscosity within the fluid core, or more generally of viscoelasticity throughout the Earth, modifies the long period oscillations of the Earth, and in particular the undertone modes within the fluid outer core.