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Geoscientific Model Development An interactive open-access journal of the European Geosciences Union
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https://doi.org/10.5194/gmd-2020-194
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-2020-194
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.

Submitted as: methods for assessment of models 24 Aug 2020

Submitted as: methods for assessment of models | 24 Aug 2020

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This preprint is currently under review for the journal GMD.

Analytical solutions for mantle flow in cylindrical and spherical shells

Stephan C. Kramer1, D. Rhodri Davies2, and Cian R. Wilson3 Stephan C. Kramer et al.
  • 1Department of Earth Science and Engineering, Imperial College London, UK
  • 2Research School of Earth Sciences, The Australian National University, Canberra, Australia
  • 3Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC, USA

Abstract. Computational models of mantle convection must accurately represent curved boundaries and the associated boundary conditions within 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 natural 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.

Stephan C. Kramer et al.

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Stephan C. Kramer et al.

Model code and software

Fluidity - an open-source computational fluid dynamics code with adaptive unstructured mesh capabilities Stephan C. Kramer, Cian R. Wilson, D. Rhodri Davies, Timothy Greaves, Alexandros Avdis, Michael Lange, James Percival, Christopher Matthews, Angus Gibson, Simon Mouradian, Thomas Duvernay, Gheorghe-Teodor Bercia, Lucy Bricheno, Andrew Buchan, Adam Candy, Gareth Clay, Gareth Collins, Colin Cotter, Angus Creech, Rhodri Davies, Jason Dunstall, Juan Du, Matthew Eaton, Fangxin Fang, Patrick Farrell, Simon Funke, David Ham, Jon Hill, Ana Garcia-Sagrado, Jefferson Gomes, Gerard Gorman, Daryl Harrison, Christian Jacobs, Shan Jiang, Stephan Kramer, Hedong Liu, Nick Lutsko, James Maddison, Ben Martin, Bryan Miles, Frank Milthaler, Julian Mindel, Andrew Mitchell, Lawrence Mitchel, Rhodri Nelson, Dimitrios Pavlidis, Ralph Perpeet, Matthew Piggott, Hubert Plociniczak, Philip Power, Mariano Sallito, Jon Saunders, Sabu Shahdatullah, Brendan Tollit, Axelle Vire, Hongbin Wang, Michele Weiland, Martin Wells, Matthew Whitworth, Adrian Umpleby, Pablo R. Brito-Parada, Gaurav Bhutani, Peter Allison, Elsa Aristodemou, Xiahu Guo, Florian Rathgeber, Mark Goffin, Dave Robinson, Steven Dargaville, Alistair Everett, Satoshi Kimura, Juan Nunez Rattia, Giannis Nikiteas, and Adam Cavendish https://doi.org/10.5281/zenodo.3988620

Assess - Analytical Solutions for the Stokes Equations in Spherical Shells Stephan C. Kramer https://doi.org/10.5281/zenodo.3891545.

Stephan C. Kramer et al.

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Short summary
Computational models of Earth's mantle require rigorous verification and validation. Analytical solutions of the underlying Stokes equations provide a method to verify that these equations are accurately solved for. However their derivation in spherical and cylindrical shell domains with physically relevant boundary conditions is involved. This paper works out a number of solutions. They are provided in a python package Assess and their use is demonstrated in a convergence study with Fluidity.
Computational models of Earth's mantle require rigorous verification and validation. Analytical...
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