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