1. A finite volume solution method for thermal convection in a spherical shell with strong temperature- und pressure-dependent viscosity CIG Workshop 2005 Boulder, Colorado K. Stemmer, H. Harder and U. Hansen stemmer@uni-muenster.de Münster University, Germany Department of Geophysics

2. Outline Motivation: Importance of mantle rheology Basic principles of thermal convection with variable viscosity  Mathematical model  Numerical model Simulation results: Thermal convection in a spherical shell  Temperature-dependent viscosity  Temperature- and pressure-dependent viscosity Conclusions Münster University

3. Motivation Importance of mantle rheology Laboratory experiments of mantle material:  viscosity is temperature-, pressure- and stress-dependent Many models have constraints:  Cartesian  isoviscous / depth-dependent viscosity High numerical and computational effort for lateral variable viscosity  mode coupling  sophisticated numerical methods Münster University

4. Thermal convection mathematical model Rayleigh-Bénard convection continuity equation equation of motion heat transport equation Rayleigh number Arrhenius equation. viscosity contrast with pressure viscosity contrast with temperature Münster University

5. Thermal convection with lateral variable viscosity numerical model Implemented methods: Discretization with Finite Volumes (FV) Collocated grid Equations in Cartesian formulation Primitive variables Spherical shell topologically divided in 6 cube surfaces Massive parallel, domain decomposition (MPI) Time stepping: implicit Crank-Nicolson method Solver: conjugate gradients (SSOR) Pressure correction: SIMPLER and PWI Münster University

6. control volume Thermal convection with lateral variable viscosity numerical model grid generationlateral grid Münster University Advantages of this spatial discretization: Efficient parallelization No singularities at the poles Approximately perpendicular grid lines Implicit solver (finite volumes)

7. discretization of the viscous term Problem: required: derivatives of velocities in x-,y- und z-direction available: curved gridlines (not in x-,y- und z-direction) Solution: transformation of the viscous term applying Gauß / Stokes theorem and lokal CV coordinate systems simplification of integrals CV: control volume Thermal convection with lateral variable viscosity numerical model Münster University

8. Gauss integral theorem known Laplacian solution applying Stokes theorem change to local orthonormal basis to simplify notation: local orthonormal basis of the CV surface Thermal convection with lateral variable viscosity discretisation of the viscous term stress tensor viscous term: Münster University

9. solution of integral : normal vector many terms are vanishing due to the use of local coordinates remains the calculation of the curl of velocities on the CV surfaces Thermal convection with lateral variable viscosity discretisation of the viscous term Münster University

10. integration along selected paths Thermal convection with lateral variable viscosity discretisation of the viscous term linear approximation of line integrals Calculation of the curl of velocities on the CV surfaces: applying Stokes theorem a,b,c coupling of velocity components central weight is a vector Münster University

11. Thermal convection with lateral variable viscosity pressure weighted interpolation (PWI) Solution: mathematical principle [Rhie and Chow, 1983] small regularizing terms are added that excludes spurious modes perturb the continuity equation with pressure terms regulating pressure terms do not influence the accuracy of the discretisation pressure is defined to an intermediate time level pressure correction: fluxes are pertubated with pressure terms Problem: insufficient coupling checkerboard oscillations Münster University

12. Thermal convection in a spherical shell temperature-dependent viscosity, basal heating residual temperature dt = +/- 0.1 temperature isosurfaces and slices Münster University

13. T=0.25 T=0.60T=0.83 Thermal convection in a spherical shell temperature-dependent viscosity, basal heating Münster University

14. Three regimes: 1)mobile lid 2)transitional (sluggish) 3)stagnant lid velocitiesminimum with depth of lateral velocities Thermal convection in a spherical shell temperature-dependent viscosity, basal heating Münster University

15. „high viscosity zone“ Thermal convection in a spherical shell temperature- and pressure-dependent viscosity Temperature dependence and pressure dependence of viscosity compete each other! Münster University

16. isosurfaces: slices: red = high viscous blue = low viscous „high viscosity zones“ Thermal convection in a spherical shell temperature- and pressure-dependent viscosity Münster University

17. Conclusions Mantle Convection Importance of spherical shell geometry Importance of mantle rheology... ? …thanks for your attention! High numerical and computational effort for lateral variable viscosity BUT: temperature-dependent viscosity has a strong effect on… …convection pattern …heat flow …temporal evolution Münster University, Germany Department of Geophysics stemmer@uni-muenster.de