A matrix-dependent transfer multigrid method for strongly variable viscosity infinite Prandtl number thermal convection

• Published in: Geophysical and astrophysical fluid dynamics Vol. 92; no. 3-4; pp. 151 - 195
• Main Authors: Yang, Woo-Sun, Baumgardner, John R.
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.01.2000
• Subjects: Algorithms; Finite element method; Galerkin methods; Mantle convection; Matrix algebra; Matrix-dependent transfer; Multigrid; Prandtl number; Problem solving; Rheology; Strain rate; Viscosity; Yield stress
• ISSN: 0309-1929; 1029-0419
• DOI: 10.1080/03091920008203715

We apply a two-dimensional Cartesian finite element treatment to investigate infinite Prandtl number thermal convection with temperature, strain rate and yield stress dependent rheology using parameters in the range estimated for the mantles of the terrestrial planets. To handle the strong viscosity variations that arise from such nonlinear rheology in solving the momentum equation, we exploit a multigrid method based on matrix-dependent intergrid transfer and the Galerkin coarse grid approximation. We observe that the matrix-dependent transfer algorithm provides an exceptionally robust and efficient means for solving convection problems with extreme viscosity gradients. Our algorithm displays a convergence rate per multigrid cycle about five times better than what other published methods (e.g., CITCOM of Moresi and Solomatov, 1995) offer for cases with similar extreme viscosity variation. The algorithm is explained in detail in this paper. When this method is applied to problems with temperature and strain rate dependent rheologies, we obtain strongly time dependent solutions characterized by episodic avalanching of cold material from the upper boundary layer to the bottom of the convecting domain for a significantly broad range of parameter values. In particular, we observe this behavior for the relatively simple case of temperature dependent Newtonian rheology with a plastic yield stress. The intensity and temporal character of the episodic behavior depends sensitively on the yield stress value. The regions most strongly affected by the yield stress are thickened portions of the cold upper boundary layer which can suddenly become unstable and form downgoing diapirs. These computational results suggest that the finite yield properties of silicate rocks must play a vitally important role in planetary mantle dynamics. Although our example calculations were selected mainly to illustrate the power of our multigrid method, they suggest that many possible exotic behaviors in planetary mantles have yet to be discovered.