Diapiric ascent : asymptotics and numerics of slow flow with strongly temperature-dependent viscosity.

In the field of geophysics, it has long been accepted that the bodies of igneous rock, known as intrusions, that are often found at shallow levels of the Earth’s continental crust are a result of the solidification of granitic magma that was generated in the deep lithosphere. This raises the question of how the magma is transported a distance of tens of kilometres through the lithosphere before being emplaced. One possibility is that large volumes of hot buoyant magma - or diapirs - are transported en masse from their point of origin to the shallow crust. In this thesis we investigate, through mathematical modelling, the viability of magmatic diapirism as an ascent mechanism. Whilst the problem has been tackled by earlier authors, a literature review indicates a litany of algebraic errors and unwarranted assumptions in earlier work, which requires us to begin from scratch. A moving boundary problem for an ascending diapir, modelled as a hot, buoyant sphere rising through a lithosphere that behaves as a thermoviscous, power-law fluid, is formulated. Numerically, this turns out to be very difficult to solve, and an alternative asymptotics-based approach is adopted. This centres on the non-isothermal, thermoviscous, analogue of the Hadamard-Rybczinski problem for a light and relatively inviscid fluid rising in a denser, more viscous fluid, and is governed by two dimensionless parameters: the P´eclet number, Pe, and a viscosity variation parameter, ǫ. Significant analytical progress is found to be possible in four asymptotic regimes; furthermore, it is possible to recover all of these numerically. The asymptotic analysis is then extended to the case of a power-law fluid. Again, significant progress is possible in four regimes. These asymptotic results are then used to construct a zero-dimensional model for a rising diapir. The results of this model are compared to those of earlier formulations of the problem. It is concluded that the diapir rises through a considerably smaller distance than was predicted previously, and that the role of thermal softening, whereby the diapir’s heat is able to decrease the local crust viscosity allowing it to rise further, has been overstated.