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.