The investigators are focused on developing a deeper understanding of the behavior of the dynamics of partially molten rock (magma) in the Earth's mantle. This problem has important implications for the dynamics of large scale mantle convection and plate tectonics as well as the geochemical evolution of the planet. Magma migration is also an intrinsically interesting problem in coupled fluid/solid mechanics as it requires understanding the non-linear interactions of low viscosity reactive fluids in a strongly deformable and permeable matrix. For the past 20 years, the predominant theory for magma migration in the mantle has been a system of mathematical models (partial differential equations or PDEs), which describe the evolution of macroscopic quantities, e.g. the porosity or proportion of molten rock at a given position and time. These equations have proved useful for exploring the behavior of magmatic systems through both simplified model problems and more complex geoscience specific applications. However, more recent experimental work suggests that there exist some important quantitative discrepancies between the experiments and predictions of these models. In particular, these models become singular as the porosity becomes small. These and other results suggest that a modeling and mathematical understanding of the behavior of the current equations is necessary to gauge the utility of these models and to develop improved mathematical descriptions (e.g. by introducing physically reasonable regularizations) for partially molten regions in large scale mantle dynamics. The purpose of this proposal is to combine expertise in the analysis of non-linear PDE's with the physics and computation of magma dynamics to develop better insight and better models for complex coupled fluid/solid systems. This work is collaborative effort between Michael I. Weinstein (Columbia Applied Mathematics) and Marc Spiegelman (Columbia Joint appointment between Earth and Env. Sciences and Applied Math) and will be the primary Ph. D. research of Gideon Simpson who is jointly supervised by Weinstein and Spiegelman. This project will attack two separate problems that arise at different scales in partially molten regions. At large scales, variations in porosity can propagate as dispersive non-linear waves with a poorly understood non-linear dispersion term. At smaller scales, laboratory experiments demonstrate that shear deformation of the solid matrix can drive localization of melt-rich bands We will consider a series of analytic and numerical model problems to develop a better understanding of the current models as well as exploring ways to improve the formulation.
This project will form the primary source of funding for a promising graduate student (Gideon Simpson) to work at the intersection of Applied Mathematics and Earth Science. This work should also have general applications in science and engineering to problems involving the flow of fluids in permeable deformable solids such as those arising in petroleum engineering, hydrology and nuclear/toxic waste confinement.
