CIG Workshop 2005 Boulder, Colorado K. Stemmer, H. Harder and U. Hansen stemmer@uni-muenster.de

1. Münster University, Germany Department of Geophysics 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

2. Münster University 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

3. Münster University MotivationImportance 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

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

5. Münster University Thermal convection with lateral variable viscositynumerical 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

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

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

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

9. Münster University Thermal convection with lateral variable viscositydiscretisation of the viscous term 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

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

11. Münster University Problem: insufficient coupling checkerboard oscillations pressure is defined to an intermediate time level pressure correction: fluxes are pertubated with pressure terms Thermal convection with lateral variable viscositypressure 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

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

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

14. Münster University Thermal convection in a spherical shelltemperature-dependent viscosity, basal heating • Three regimes: • mobile lid • transitional (sluggish) • stagnant lid velocities minimum with depth of lateral velocities

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

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

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