Abstract
Abstract. Computational models of mantle convection must accurately represent curved boundaries and the associated boundary conditions of a 3-D spherical shell, bounded by Earth's surface and the core–mantle boundary. This is also true for comparable models in a simplified 2-D cylindrical geometry. It is of fundamental importance that the codes underlying these models are carefully verified prior to their application in a geodynamical context, for which comparisons against analytical solutions are an indispensable tool. However, analytical solutions for the Stokes equations in these geometries, based upon simple source terms that adhere to physically realistic boundary conditions, are often complex and difficult to derive. In this paper, we present the analytical solutions for a smooth polynomial source and a delta-function forcing, in combination with free-slip and zero-slip boundary conditions, for both 2-D cylindrical- and 3-D spherical-shell domains. We study the convergence of the Taylor–Hood (P2–P1) discretisation with respect to these solutions, within the finite element computational modelling framework Fluidity, and discuss an issue of suboptimal convergence in the presence of discontinuities. To facilitate the verification of numerical codes across the wider community, we provide a Python package, Assess, that evaluates the analytical solutions at arbitrary points of the domain.
Highlights
Mantle convection transports Earth’s internal heat to its surface: it is the “engine” driving our dynamic Earth (e.g. Davies, 1999)
The number of published analytical Stokes solutions in 2-D cylindrical-shell domains, which are suitable as geodynamical benchmarks and include a complete derivation, is limited. By presenting this extensive set of explicit analytical solutions, we provide a suite of verification cases for use by the wider community of mantle dynamics code developers
We show the convergence of the numerical solutions obtained with Fluidity, using the P2–P1 element pair, towards the analytical solutions
Summary
Mantle convection transports Earth’s internal heat to its surface: it is the “engine” driving our dynamic Earth (e.g. Davies, 1999). Mantle convection transports Earth’s internal heat to its surface: it is the “engine” driving our dynamic Earth The structure, composition, and flow regime within the mantle are reflected in near-surface phenomena such as plate tectonics, mountain building, dynamic topography, sea-level change, volcanism, and the activity of Earth’s magnetic field McKenzie et al, 1974; Gurnis and Davies, 1986; Davies and Stevenson, 1992; Moresi and Solomatov, 1995; van Keken et al, 2002; Hunt et al, 2012; Garel et al, 2014; Davies et al, 2016; Jones et al, 2016, 2019), 3-D spherical geometry is required to simulate global mantle dynamics. Global 3-D spherical mantle convection models, and studies focussing on their application, are in common use As a consequence, simplifying geometries are often used, including the axisymmetric spherical shell (e.g. Solheim and Peltier, 1994; van Keken and Yuen, 1995), the 2-D cylinder
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