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MHD Dynamo Simulation by GeoFEM

3rd ACES Workshop May, 5, 2002 Maui, Hawai’i. MHD Dynamo Simulation by GeoFEM. Hiroaki Matsui Research Organization for Informatuion Science & Technology(RIST), JAPAN. Outer Core. Inner Core. CMB. ICB. Mantle. Crust. Introduction -Simple Model for MHD Dynamo-. Conductive fluid.

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MHD Dynamo Simulation by GeoFEM

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

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

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

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

  5. 1E14 1E12 1E10 1E8 1E6 1E4 1E2 1E4 1E6 1E8 1E10 Introduction- Dimensionless Numbers - Estimated values for the outer core To approach such large paramteres …High spatial resolution is required!

  6. Introduction- FEM and Spectral Method -

  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 -

  9. Treatment of the Magnetic Field- Boundary Condition on CMB - Dipole field Composition of dipole and octopole Boundary Condition Octopole field Boundary Condition 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 Mesh for the fluid shell Entire mesh 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 - Coriolis term Lorentz term for conductive fluid for conductor for insulator

  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 - Dimensionless numbers Insulator Conductive fluid 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 respectto 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 Radial magnetic field for t = 20.0 Comparison with spectral method (Time evolution of the averaged kinetic and magnetic energies in the shell)

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

  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=1014) is considered to be a target of the present MHD simulation. • A simulation with 1x108 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.

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