Implementation of a Volume-of-Fluid method in a finite element code with applications to thermochemical convection in a density stratified fluid in the Earth’s mantle

Abstract

We describe the implementation of a second-order accurate volume-of-fluid interface tracking algorithm in the open source finite element code ASPECT, which is designed to model convection 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 a marker indicating the two initial density states. The volume-of-fluid method is fully parallelized and is integrated with the adaptive mesh refinement algorithm in ASPECT. We present the results of several standard interface tracking benchmarks in order to demonstrate the accuracy of the method as well as the results of several benchmarks commonly used in the computational mantle convection community. Finally, we present the results of computations with and without adaptive mesh refinement of a model problem involving thermochemical convection in a computationally stratified fluid designed to provide insight into how thermal plumes, that eventually reach the Earth's surface as ocean island basalts, originate at structures on the core-mantle boundary known as Large Low Shear wave Velocity Provinces.

101 Geodynamic modelling: How to design, interpret, and communicate numerical studies of the solid Earth

“…This method is often found in finite element (Tackley and King, 2003;Deubelbeiss and Kaus, 2008;Thielmann et al, 2014;Gassmöller et al, 2018Gassmöller et al, , 2019 and finite difference codes alike (Gerya and Yuen, 2007;Pusok et al, 2016). Other employed techniques are the level-set method (Samuel and Evonuk, 2010;Hillebrand et al, 2014;Braun et al, 2008), the grid-based advection method (sometimes referred to as compositional fields method, Figure 4) (Gerya, 2019), the marker-chain method (van Keken et al, 1997), and the volume-of-fluid method (Robey and Puckett, 2019;Louis-Napoléon et al, 2020).…”

An Exactly Mass Conserving and Pointwise Divergence Free Velocity Method: Application to Compositional Buoyancy Driven Flow Problems in Geodynamics

“…In the past decade or so, there has been significant progress improving methods for the accurate solution of thermochemical convection which are based on, among others, application of discontinuous Galerkin (DG) finite element (FE) methods (He et al., 2017), further use of level‐set methods (Hillebrand et al., 2014; Samuel & Evonuk, 2010), and adaptive remeshing (Davies et al., 2007; Leng & Zhong, 2011). Modeling of thermochemical convection has also become more readily available to a larger group of researchers through community codes such as Advanced Solver for Problems in Earth's ConvecTion (as demonstrated by Gassmöller et al., 2018; and Robey & Puckett, 2019). The reliability of new methods has been demonstrated by careful benchmarking and error testing (e.g., Thielmann et al., 2014; Vynnytska et al., 2013).…”

“…In geodynamical applications several distinct methods have been used to represent the chemical buoyancy. These include (i) methods that delineate the boundary between two volumes of distinct chemical composition such as the marker chain method (Christensen & Yuen, 1984; Lin & van Keken, 2006; Schmeling, 1987) and volume or moment of fluid methods (Pilliod & Puckett, 2004; Robey & Puckett, 2019; Zalesak, 1979); (ii) tracer methods where individual tracers carry a relative proportion of the chemical buoyancy (Brandenburg et al., 2008; Christensen & Hofmann, 1994; Gerya & Yuen, 2003; O'Neill et al., 2006; Tackley & King, 2003); and (iii) representing the chemical density as a composition field and solving the advection‐diffusion equation with low chemical diffusivity (e.g., Hansen & Yuen, 2000; Kellogg & King, 1993), which in the presence of thermal buoyancy leads to double‐diffusive convection (Turner, 1974). A comprehensive, if now slightly dated, comparison of these three methods is provided in van Keken et al.…”

Towards the application of Stokes flow equations to structural restoration simulations

“…First, rocks can behave as viscous fluids when subjected to deformation that includes large stress and strain over periods of millions of years (Massimi et al, 2006;Cornet, 2015). This principle has been used for more than three decades to simulate the flow of lithospheric scale to deep mantle material (Poliakov et al, 1993;Fullsack, 1995;Hassani et al, 1997;Schubert et al, 2001;Morra and Regenauer-Lieb, 2006;Thieulot, 2011;Gerya, 2019;Robey and Puckett, 2019;Louis-Napoléon et al, 2020), so there is extensive litterature on how to solve the associated equations. Secondly, considering the rocks as having a creeping flow behavior allows the simulation to reproduce the flow of salt layers (Nalpas and Brun, 1993), and possibly also other rock rheologies through relevant effective viscosities (Moresi et al, 2003;Glerum et al, 2018).…”