Download - MHD Dynamo Simulation by GeoFEM
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MHD Dynamo Simulation by GeoFEM
Hiroaki Matsui
Research Organization for Informatuion Science & Technology(RIST), JAPAN
3rd ACES WorkshopMay, 5, 2002Maui, Hawai’i
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Introduction-Simple Model for MHD Dynamo-
Crust
Mantle
Outer CoreInner Core
CMB
ICB
Conductive fluid
InsulatorConductive solid or insulator
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∂ω∂t
+∇ × ω×u( ) =Pr∇2ω−PrTa
0.5∇ × ˆ z ×u( )
−PrRa∇ × T −T0( )r{ }+Pr∇ × ∇ ×B( )×B{ }
∇ ⋅u=∇ ⋅ω=0
∂T∂t
+ u⋅∇T( ) =∇2T
∂B∂t
=Pr
Pm
∇2B+∇ × u×B( )
∇ ⋅B=0
Introduction- Basic Equations -
Coriolis term
Lorentz term
Induction equation
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Introduction- Dimensionless Numbers -
Rayleighnumber
Taylor
number
Prandtlnumber
MagneticPrandtlnumber
Estimated values for the Outer core
Ra =αg0ΔTL3
κν=
Buoyancydiffusion
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 6E30
Ta =2ΩL2
ν
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
=Coriolisforce
Viscosity
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
1E30
Pr =νκ
=Viscosity
Thermaldiffusion
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ 0.1
Pm =νη
=Viscosity
Magneticdiffusion
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ 1E −6
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Introduction - Dimensionless Numbers -
To approach such large paramteres…High spatial resolution is required!
Estimated values for the outer core Ra
1E14
1E12
1E10
1E8
1E6
1E4
1E2 1E4 1E6 1E8 1E10
Ta
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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
Octopole field
B=∇ ×∇ × BS10(r)⋅Y1
0(θ,φ) ˆ r ( )
Boundary Condition∂BS1
0
∂r+
1r
BS10 =0 onCMB
B=∇ ×∇ × BS30(r)⋅Y3
0(θ,φ) ˆ r ( )
Boundary Condition
Composition of dipole and octopole
Boundary conditions can not be set locally!!
∂BS30
∂r+
3r
BS30 =0 onCMB
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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€
∇×A =B, ∇ ⋅A = 0
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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
Entire mesh Mesh for the fluid shell Grid pattern for center
rm=14.8r0−ri ()=5.09Re
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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€
∇×A =B, ∇ ⋅A = 0
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∂ω∂t
+∇ × ω×u( ) =Pr∇2ω−PrTa
0.5∇ × ˆ z ×u( )
−PrRa∇ × T −T0( )r{ }+Pr∇ × ∇ ×B( )×B{ }
∇ ⋅u=∇ ⋅ω=0
∂T∂t
+ u⋅∇T( ) =∇2T
∂B∂t
=Pr
Pm
∇2B+∇ × u×B( )
∇ ⋅B=0
Treatment of the Magnetic Field - Basic Equations for Spectral Method-
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∂u∂t
+ u⋅∇u( ) =−∇P +Pr∇2u−PrTa
0.5 ˆ z ×u( )
−PrRa T −T0( )r+Pr ∇ ×B( )×B
∇ ⋅u=0
∂T∂t
+ u⋅∇T( ) =∇2T
∂A∂t
=−∇ϕ +Pr
Pm
∇2A+ u×B( )
∇ ⋅A=0
0=∇2A
∇ ⋅A=0
∇ ×A=B
Treatment of the Magnetic Field - Basic Equations for GeoFEM/MHD -
for conductive fluid
for conductor
for insulator
Coriolis term
Lorentz term
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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 -
Insulator Conductive fluid
Ra=αg0ΔTL3
κν=1.2×104
Ta =2ΩL2
ν
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
=9.0×104
Pr =νκ
=1.0
Pm =νη
=10.0
Dimensionless numbers
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
T =1 atr =ri
T =0 atr =ro
u=0 atr =ri,ro
A=0 atr =rm
• For the northern hemisphere• 81303 nodes• 77760 element
Finite element mesh for the present simulation
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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
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Comparison with Spectral MethodCross Sections at z = 0.35
Spectral method
GeoFEM
3.5E+1
3.5E+1
-9.8E0
-9.8E0
0.0
0.0-1.8E+2
2.3E+2
2.3E+2
-1.8E+2
0.0
0.0
Magnetic field Vorticity
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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, 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.