Book
The mathematics of magma migration
01 Jan 2008
Abstract
Magma, molten rock in the Earth's interior, surfaces at many sites of geologic interest. Perhaps its most vivid manifestation is the volcanic eruption. Magmatism is also essential to global mantle convection and plate tectonics; it is universally present at plate boundaries, including mid-ocean ridges and subduction zones. However, the coupling between convection and magmatism is not well understood, spurring us to develop better models. We expand on one particular model for transport in the upper mantle, percolation through viscously deformable permeable rock. Historically, this type of magma migration has been modeled with macroscopic conservation equations for two coupled fluids. In this work, we present a new systematic derivation for this mechanism based on homogenization, which begins at the microscopic scale. Our use of homogenization clarifies the assumptions needed to derive macroscopic equations. With this approach, we derive a system of equations, similar to the earlier models. For these equations, constitutive relationships for permeability, shear viscosity, and bulk viscosity are consistently calculated with respect to the microscopic structures. In particular, we deduce an inverse relationship between the bulk viscosity of the porous rock and the volume fraction of molten rock. Under certain assumptions, these systems of equations simplify to a scalar, nonlinear, dispersive, degenerate wave equation. While this equation has been previously studied, there are few formal results. We undertake rigorous analysis of the equation and prove several theorems for it: well-posedness in a natural function space, existence of a Hamiltonian case of the equation and the nonlinear stability of its solitary waves, and asymptotic stability of small amplitude solitary waves in the general case. The degeneracy of this equation is also common to the full macroscopic system; understanding this simplified problem may help treat the larger one.
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Details
- Title
- The mathematics of magma migration
- Creators
- Gideon Simpson
- Resource Type
- Book
- Language
- English
- Academic Unit
- Mathematics
- Identifiers
- 9780549858294; 0549858296; 991019296801604721