1. MHD Dynamo Simulation by GeoFEM Hiroaki Matsui Research Organization for Informatuion Science & Technology(RIST), JAPAN 3rd ACES Workshop May, 5, 2002 Maui, Hawai’i

2. Introduction -Simple Model for MHD Dynamo- Crust Mantle Outer Core Inner Core CMB ICB Conductive fluid Insulator Conductive solid or insulator

3. Introduction - Basic Equations - Coriolis term Lorentz term Induction equation

4. Introduction - Dimensionless Numbers - Rayleigh number Taylor number Prandtl number Magnetic Prandtl number Estimated values for the Outer core

5. Introduction - Dimensionless Numbers - To approach such large paramteres …High spatial resolution is required! Estimated values for the outer core 1E14 1E12 1E10 1E8 1E6 1E4 1E21E41E61E81E10

6. Introduction - FEM and Spectral Method - SpectralFEM AccuracyHighLow ParallelizationDifficult and complex Easy Boundary Condition for B Easy to applyDifficult Simulation Results ManyFew Application of heteloginity DifficultEasy

7. Purposes Develop a MHD simulation code for a fluid in a Rotating Spherical Shell by parallel FEM Construct a scheme for treatment of the magnetic field in this simulation code

8. Treatment of the Magnetic Field - FEM and Spectral Method - SpectralFEM AccuracyHighLow ParallelizationDifficult and complex Easy Boundary Condition for B Easy to applyDifficult Simulation Results ManyFew Application of heteloginity DifficultEasy

9. Treatment of the Magnetic Field - Boundary Condition on CMB - Dipole field Octopole field Boundary Condition Composition of dipole and octopole Boundary conditions can not be set locally!!

10. Treatment of the Magnetic Field Finite Element Mesh is considered for the outside of the fluid shell Consider the vector potential defined as Vector potential in the fluid and insulator is solved simultaneously

11. Treatment of the Magnetic Field - Finite Element Mesh - Element type –Tri-linear hexahedral element Based on Cubic pattern Requirement –Considering to the outside of the Core –Filled to the Center Entire mesh Mesh for the fluid shell Grid pattern for center

12. Treatment of the Magnetic Field Finite Element Mesh is considered for the outside of the fluid shell Consider the vector potential defined as The vector potential in the fluid and insulator is solved simultaneously

13. Treatment of the Magnetic Field - Basic Equations for Spectral Method-

14. Treatment of the Magnetic Field - Basic Equations for GeoFEM/MHD - for conductive fluid for conductor for insulator Coriolis term Lorentz term

15. Methods of GeoFEM/MHD Valuables –Velocity and pressure –Temperature –Vector potential of the magnetic field and potential Time integration –Fractional step scheme Diffusion terms: Crank-Nicolson scheme Induction, forces, and advection:Adams-Bashforth scheme –Iteration of velocity and vector potential correction –Pressure solving and time integration for diffusion term ICCG method with SSOR preconditioning

16. Model of the Present Simulation - Current Model and Parameters - InsulatorConductive fluid Dimensionless numbers Properties for the simulation box

17. Model of the Present Simulation - Geometry & Boundary Conditions - Boundary Conditions Velocity: Non-Slip Temperature: Constant Vector potential: Symmetry with respect to the equatorial plane Velocity: symmetric Temperature:symmetric Vector potential: symmetric Magnetic field: anti-symmetric For the northern hemisphere 81303 nodes 77760 element Finite element mesh for the present simulation

18. Comparison with Spectral Method Comparison with spectral method (Time evolution of the averaged kinetic and magnetic energies in the shell) Radial magnetic field for t = 20.0

19. Comparison with Spectral Method Cross Sections at z = 0.35 Spectral method GeoFEM 3.5E+1 -9.8E0 0.0 -1.8E+2 2.3E+2 -1.8E+2 0.0 Magnetic fieldVorticity

20. Conclusions We have developed a simulation code for MHD dynamo in a rotating shell using GeoFEM platform Simulation results are compared with results of the same simulation by spherical harmonics expansion Simulation results shows common characteristics of patterns of the convection and magnetic field. To verify more quantitatively, the dynamo benchmark test (Christensen et. Al., 2001) is running.

21. Near Future Challenge The Present Simulation will be performed on Earth Simulator (ES). On ES, E=10 -7 (Ta=10 14 ) is considered to be a target of the present MHD simulation. A simulation with 1x10 8 elements can be performed if 600 nodes of ES can be used. These target are depends on available computation time and performance of the test simulation.