Earth is a chemically heterogeneous body. While mantle convection can be approximated as a thermally driven process, recent studies indicate that chemical heterogeneities may play an important role in governing the form and planform of mantle dynamics. Additionally, a growing body of geochemical observations argue for distinct geochemical "reservoirs" within the mantle. Geochemical observations indicate that these reservoirs exist, but additional constraints are required to ascertain their scale and location. If mantle dynamics is modelled as a purely thermal process, the chemical properties of the system are effectively ignored, which can potentially limit the accuracy of resulting simulations. It is therefore desirable to simulate thermos-chemical mantle convection.

When simulating incompressible thermally-driven mantle convection, there are three fundamental equations governing the physics of fluid flow. These are the equations for conservation of momentum, conservation of mass, and conservation of energy. When simulating thermochemical mantle convection, two amendments are made to these governing equations. First, the chemical advection equation is added to the system, accounting for the advection of chemical materials during convection. Second, an additional term is added to the conservation of momentum equation, to account for the influence of chemical density contrasts on buoyancy.

The chemical advection equation can be challenging to implement. Unlike the temperature advection-diffusion equation, chemical advection equation contains zero diffusion on the right-hand side. As a result, using a standard finite element (grid based) approach to solve for the chemical field can lead to numerical problems, as a degree of numerical diffusion will inherently be introduced into the system. This can smear the interface between distinct chemical bodies, causing a loss of accuracy as the simulation proceeds.

In this project, the Fluidity computational modelling framework will be modified to accurately model thermochemical convection processes. Fluidity utilizes an unstructured, anisotropic, adaptive simplex mesh, and employs a continuous Galerkin FEM approach to mesh discretization. While Fluidity has proven to be accurate when modelling both 2D and 3D thermal convection problems, the available schemes used to model thermochemical convection are limited in their applicability. To alleviate the challenges associated with simulating thermochemical convection, it has been proposed to incorporate a particle-in-cell scheme into the Fluidity computational framework.

In the particle method, distinct chemical bodies are represented by many particles that each have an independent mass. Here, the model domain is effectively discretised into small cells, with many particles then placed within each cell. Each particle is given a physical mass value depending on the chemical properties of the material at which it is located, where the combined sum of individual particle masses in a region is equal to the total mass of the chemical body located in that same region. The particles are then advected and tracked throughout the simulation.

The particle method is extremely flexible in its application, and is widely used in the geodynamics community.  A major benet of this method is that particle attributes can evolve over time. This allows the modelling of phenomena such as chemical reactions and melting, enabling more complex thermochemical models to be simulated.