MHD Dynamo Simulation by GeoFEM

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MHD Dynamo Simulation by GeoFEM
3rd ACES Workshop May, 5, 2002 Maui, Hawaii

• Hiroaki Matsui
• Research Organization for Informatuion Science
  Technology(RIST), JAPAN

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Introduction-Simple Model for MHD Dynamo-
Conductive fluid
Conductive solid or insulator
Insulator
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Introduction- Basic Equations -
Coriolis term
Lorentz term
Induction equation
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Introduction- Dimensionless Numbers -
Estimated values for the Outer core
Rayleigh number Taylor number Prandtl number Magn
etic Prandtl number
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Introduction - Dimensionless Numbers -
Estimated values for the outer core
To approach such large paramteres High spatial
resolution is required!
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Introduction- FEM and Spectral Method -
Spectral FEM
Accuracy High Low
Parallelization Difficult and complex Easy
Boundary Condition for B Easy to apply Difficult
Simulation Results Many Few
Application of heteloginity Difficult Easy
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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

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Treatment of the Magnetic Field- FEM and
Spectral Method -
Spectral FEM
Accuracy High Low
Parallelization Difficult and complex Easy
Boundary Condition for B Easy to apply Difficult
Simulation Results Many Few
Application of heteloginity Difficult Easy
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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!!
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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

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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
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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

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

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Model of the Present Simulation - Current Model
and Parameters -
Dimensionless numbers
Insulator
Conductive fluid
Properties for the simulation box
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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
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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)
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Comparison with Spectral MethodCross Sections at
z 0.35
GeoFEM
Spectral method
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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.

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Near Future Challenge

• The Present Simulation will be performed on Earth
  Simulator (ES).
• On ES, E10-7 (Ta1014) 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.