On the Design, Implementation, and Use of a Volume-of-fluid Interface Tracking Algorithm for Modeling Convection and Other Processes in the Earth’s Mantle

Abstract

In this dissertation we describe the design, implementation, and use of a two-dimensional, second-order accurate volume-of-fluid interface tracking algorithm in the open source finite element software package ASPECT, which is designed to model convection and other processes in the Earth's mantle. This involves the solution of the incompressible Stokes equations coupled to an advection-diffusion equation for the temperature, a Boussinesq approximation that governs the dependence of the density on the temperature, and an advection equation for composition or some other quantity, such as volume fraction, which is passively advected in the underlying flow field. Our volume-of-fluid method is is fully parallelized and integrated with ASPECT's adaptive mesh refinement algorithm.

To the best of our knowledge, the volume-of-fluid method has not yet been implemented in any other software, which is designed to model convection or other processes in the Earth's mantle. Furthermore we are not aware of any interface tracking methods that have been implemented in a finite element method code for use by the computational mantle convection community. In fact, we are only aware of one other interface tracking algorithm designed to model convection and other processes in the Earth's mantle and this interface tracking algorithm is implemented in a finite difference method rather than a finite element method.

We review the history of the volume-of-fluid method and then describe in detail the the design and implementation of our volume-of-fluid algorithm in ASPECT. After introducing the underlying partial differential equations we use to model mantle convection, we present the results of several interface tracking benchmarks designed to demonstrate numerically that our volume-of-fluid methodology is indeed second-order accurate on smooth flows, as it was designed to be. In addition we demonstrate that our methodology accurately reproduces two benchmarks that are commonly used in the computational mantle convection community.

We also present the results of two more realistic computations in geodynamics. The first of these problems is a survey of the behavior of a computationally stratified fluid for varying values of a nondimensional buoyancy parameter. This model problem is intended to provide insight into how thermal plumes, which eventually reach the Earth's surface where they melt to form ocean island basalts separate from structures near the core-mantle boundary, which are denser than the surrounding mantle. These structures, known as Large Low Shear wave Velocity Provinces or ``LLSVPs''. LLSVPs lie in parts of the lowermost portion of the Earth's mantle and are characterized by slow shear wave velocities and higher density than the surrounding mantle. They were first discovered by seismic tomography of the deep Earth.

The second problem in computational mantle convection we present is that of a subducting slab. This computation is a basic model of, for example, the subduction of the Pacific tectonic plate beneath the South American tectonic plate. This problem involves a more complex material model than the other problems presented in this dissertation. The slab consists of an overriding crustal layer on top of a layer consisting largely of harzburgite thereby demonstrating how, with the aid of adaptive mesh refinement, one can use our volume-of-fluid methodology to track more than two materials in a single problem.