A Pointwise Conservative Method for Thermochemical Convection Under the Compressible Anelastic Liquid Approximation

In prior work we found that precise approximation of the continuity constraint is crucial for accurate propagation of tracer data when advected through a background incompressible velocity field (Sime et al., 2021, https://doi.org/10.1029/2020gc009349). Here we extend this investigation to compressi...

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Published in:Geochemistry, geophysics, geosystems : G3 geophysics, geosystems : G3, 2022-02, Vol.23 (2), p.n/a
Main Authors: Sime, Nathan, Wilson, Cian R., Keken, Peter E.
Format: Article
Language:English
Subjects:
Advection
Approximation
Compressibility
compressible flow
Conservation
Convection
Diffusion
finite element analysis
geodynamics
mantle convection
Momentum
Numerical experiments
tracer methods
Tracers
Velocity
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Summary:In prior work we found that precise approximation of the continuity constraint is crucial for accurate propagation of tracer data when advected through a background incompressible velocity field (Sime et al., 2021, https://doi.org/10.1029/2020gc009349). Here we extend this investigation to compressible flows using the anelastic liquid approximation (ALA) and address four related issues: (a) Exact conservation of tracer discretized fields through a background compressible velocity; (b) Exact mass conservation; (c) Addition and removal of tracers without affecting (exact) conservation to preserve a consistent number of tracers per cell; and (d) the diffusion of tracer data, for example, as induced by thermal or chemical effects. In this process we also present an formulation of the interior penalty hybrid discontinuous Galerkin (HDG) finite element formulation for diffusion problems and apply it to the advection‐diffusion and compressible Stokes systems. Finally we present numerical experiments exhibiting the HDG compressible Stokes momentum formulation's superconvergent compressibility approximation and reproduce examples of a community benchmark for the ALA. Key Points Pointwise divergence free momentum fields offer an accurate approximation of tracer advection in compressible velocity fields Partial differential equation (PDE)‐constrained l2 projection of tracer data advected with an appropriately discretized momentum field exactly conserves mass in compressible flows Tracers that discretize fields may be added and removed with no impact on conservation when using a PDE‐constrained l2 projection
ISSN:1525-2027
1525-2027
DOI:10.1029/2021GC009922