Magnetic helicity simulation Magnetic helicity simulation in solar atmospherein solar atmosphere

Shangbin YangShangbin YangCollaborators: Jean Carlos SantosCollaborators: Jean Carlos Santos

JJöörg Prof. Brg Prof. Büüchnerchner

Solar Group Seminar

Introduction to magnetic helicityIntroduction to magnetic helicity

Motivation to magnetic helicity simulation studyMotivation to magnetic helicity simulation study

MHD simulation modelMHD simulation model

Magnetic helicity calculation processMagnetic helicity calculation process

Results Results

Conclusion and DiscussionConclusion and Discussion

Introduction to Magnetic helicityIntroduction to Magnetic helicity

Magnetic helicity

( ) 3

3

mH A A d x

A Bd x

= ⋅ ∇×

= ⋅

∫∫

Helicity is an integral that measures some topological properties of vector fields,

3H F Fd x= ⋅∇×∫

Examples:

Magnetic helicity

3vH v vd x= ⋅∇×∫Fluid velocity helicity

Current helicity ( ) 3cH B B d x= ⋅ ∇×∫

Example 1Example 2

Magnetic helicity is conserved in ideal magneto hydrodynamics (Woltjer 1958)

It conserves approximately if magnetic Reynolds number is large enough (Taylor 1974)

.... even for any fast reconnection !

2H B nd xψΔ = ⋅∫∫| 0sB n⋅ =

A A ψ′ = +∇

( ) 3m v

H A Bd xψ′ = +∇ ⋅∫

0HΔ =

3mH A Bd x= •∫

Invariance of magnetic helicity despite of different ways to determine the vector potential

There should not be different magnetic helicity for different gauges!

Magnetic helicity is determined for definite boundary conditions

Unfortunately, these boundary conditions are usually not satisfied

Relative magnetic helicity

( )1 VB B x= ∈

B A= ∇×

s sB n P n• = •

2 ( )B P x V= ∈

pP A= ∇×

(Ref. Berger 1984)

E=mgh

h=0

h=z

E=0

E=mgz

V

V V

SS

0A∇• = 0pA∇• =

Gravitational potential energy

( ) ( )( )

a

pV

H H B H P

A B A P dV

Δ = −

= • − •∫

( ) (0)E E z Emgh

Δ = −=

Gravitationalpotential energy

Relative MagneticHelicity

The relative magnetic helicity is gauge invariant

Magnetic helicity change rate

( ) ( ) ( )2 2RP

V S

dH VE B dV A E dS

dt= − • + × •∫ ∫

0=• SP nA

0P SA n• =

0PA∇• = PP A= ∇×

s sB n P n• = •

Magnetic helicity change rate in a volume :V

The first term represents helicity dissipation inside the volume

The second term represents the helicity flux across the boundary

( ) ( ) ( )2 2a

R aP

V S

dH VE B dV A E dS

dt= − • + × •∫ ∫

E V B= − ×Ideal conducting fluid

( ) ( )( )2RP P

S

dH A V B A B V dSdt

= • − • •∫

1. The first term is due to footpoint motion of the flux.

2. The second term is due to emergence of new magnetic flux.

1 2

Deducing V

( )2RP

S

dH A B V dSdt

= − • •∫

Local Correlation Tracking can be used to calculate the first term

( )2RP LCT n

s

dH A U B dSdt

= − •∫

The two terms in the formula can combine together under the ideal magneto hydrodynamics if we use LCT method to deduce V Demoulin et al. (2003)

Chae et al. (2001)

( ) ( )0

t RR

dH tH t dt

dt= ∫

Magnetic helicity at a certain time t

( )2RP LCT n

s

dH A U B dSdt

= − •∫

We use the two formulas to calculate the magnetic helicityin the observation.

V

S

Counterclockwise Vortex

H<0

V

S

V

S

Clockwise Vortex

H>0

Motivation to Magnetic helicity simulation studyMotivation to Magnetic helicity simulation study

Helicity hemisphere Rule:

Southern hemisphere H>0 Northern hemisphere H<0

This result is based on analyzing the current helicity

If the magnetic helicity flow accumulated in active regions (ARs) is calculated then, however, the hemispheric rule is not fulfilled very well as shown for 393 ARs in Labonte et al. (2007) and for 58 newly emerging ARs in Yang et al. (2009).

What is the more important phase of an AR evolution for magnetichelicity injection into the solar corona above an active region

(a) the emerging phase (Nindos and Zhang 2002 ) or (b) the shearing phase related to flows in the photosphere, e.g.

differential rotation (Devore 2000) ?

( )c || ||h B B= ⋅ ∇×

or the force free parameter α ( B= B)α∇×

Moreover, Devore (2000) found that the input of magnetic helicity to corona by differential rotation depends both on time and on the initial orientation of the bipole.

Redistribution problem of magnetic helicity in the Corona

Corona Ideal conducting fluid Magnetic helicity is conserved

How does the magnetic helicity added to an AR evolve ? How does the magnetic helicity redistribute in the Corona ?

Interaction of active regions from the point view of magnetic helicity exchange.

NOAA 9188

NOAA 9192

Yang, S.; Buechner, J. and Zhang, H. 2009, ApJL, 695, L25

From EIT/SOHO 171Å

From TRACE 171Å

Magnetic helicity accumulation in the two active regions

NOAA 9188 NOAA 9192

TT1 T2 T3 T1 T2 T3

-> strongly implies that magnetic helicity exchange had happened.

Based on the three points above, it’s appropriate to investigate the magnetichelicity evolution by using 3D MHD simulation method especially wedon’t have relative accurate magnetic field measurement in corona by now.

MHD simulation model MHD simulation model

Data-Driven MHD model

Buchner et al. 2004ab; Buchner 2006; Santos and Buchner 2007Santos et al. 2008

( )

( ) ( )

0

2

( )

1 1

utu uu p j B u ut

B u B jtp pu p u jt

ρ ρ

ρ ρ νρ

η

γ γ η

∂= −∇⋅

∂∂

= −∇ ⋅ −∇ + × − −∂∂

= −∇× × −∂∂

= −∇ ⋅ − − ∇ ⋅ + −∂

0

2 B

E u B j

B jp nk T

η

μ

= × −

∇× =

=

Ohm Laws:

Ampere equation:

State equation:

Magnetic helicity calculation processMagnetic helicity calculation process

x

y

z

( ) ( )( )

a

pV

H H B H P

A B A P dV

Δ = −

= • − •∫

B A= ∇×

s sB n P n• = •

pP A= ∇×

0A∇• = 0pA∇• =

0p sA n• =

s sB n P n• = • 0p sA n• =

At the six boundary:

px

py

Ay

Ax

ϕ

ϕ

∂= −

∂∂

=∂

0px pyA Ax y

∂ ∂+ =

∂ ∂

pP A= ∇×

2 2

2 2ˆ

sB n

x yϕ ϕ ϕ∂ ∂+ = Δ =

∂ ∂i

Such as Bottom boundary

Boundary Value

Solving of Poisson equation

( ) ( ) 0p p pA A A∇× ∇× = ∇ ∇• −Δ =

( ) ( )A A A J∇× ∇× = ∇ ∇• −Δ =

0P∇× =

JB∇× =

0

0p

p

A

A

∇• =

⎧Δ =⎪⎨⎪⎩ 0

A J

A∇

⎧Δ =

•⎪

⎩ =

−⎨⎪

(a) (b)

We use a fast Poisson solver based on the HODIE finite-difference scheme (Boisvert, R. 1984) to resolve of the two Poisson equation .

We also clean the divergence of and pA A

Magnetic helicity information from the dataMagnetic helicity in the simulation box

Magnetic helicity dissipation in the simulation box

Magnetic helicity flux across the boundary

( ) ( ) ( )a

R pV

H H B H P A B A P dV= − = • − •∫

( )2a

R

disspation V

dH E B dVdt

= − •∫

( )6

1

nR

PiBoundary

dH A E dSdt

=

=

= × •∑

For one data with grid points 131×131×131. It takes about 5 minutes to get one magnetic helicity result.

Test case AR 8210 Test case AR 8210

Results Results

AR 8210AR 8210

Testing of magnetic helicity calculation 1

1 2

1 2

( ) ( ) ( )... ( )...

R R R R n

n

H V H V H V H VV V V V

= + += + +

1 2( ) ( )R RH V H V+( )RH V

Data Courtesy of Dr. Santos

Testing of magnetic helicity calculation 2

( ) ( )( )2RP P

S

dH A V B A B V dSdt

= • − • •∫

Helicity from CalculationHelicity across boundary

(Ideal conducting fluid without resistivity )

Data Courtesy of Dr. Santos

( ) ( ) ( )2 2a

RP

V S

dH VE B dV A E dS

dt= − • + × •∫ ∫

Testing of magnetic helicity calculation 3

time

( )RdH Vdt

Heli

city

flux

(conducting fluid with resistivity )

R

Boundary

dHdt

R R

disspation Boundary

dH dHdt dt

+

R

disspation

dHdt

Case of the interacting ARs Case of the interacting ARs (AR 9188 and AR 9192)(AR 9188 and AR 9192)

MDI data

NOAA 9188

NOAA 9192

Spatial Fourier filter to select the first 8 modes for the data

L (AR9192) R (AR9188)

Adding Vortex at the boundary (step 1)

L RClockwise

Adding Vortex at the boundary (step 2)

L R

Counterclockwise

Helicity evolution for L and R in step 1

Left region H>0

Right region H>0

Helicity evolution for L and R in step 2

Left region from H>0 to H<0

Right region H>0 while decreasingNegative magnetic helicity Accumulated in this region

Conclusion and Discussion Conclusion and Discussion

We have developed a scheme for calculating magnetic helicity of We have developed a scheme for calculating magnetic helicity of active regions from simulation data and verified for the case ofactive regions from simulation data and verified for the case ofAR 8210.AR 8210.

The helicity transport calculations also verified the physical The helicity transport calculations also verified the physical correctness of the 3D MHD simulation.correctness of the 3D MHD simulation.

Initial results for two interacting ARs (9188 and 9192) show thaInitial results for two interacting ARs (9188 and 9192) show that t the magnetic helicity exchange process between active regions the magnetic helicity exchange process between active regions can be confirmed and appropriately described by the simulation.can be confirmed and appropriately described by the simulation.

So far the imposed boundary conditions (velocity vortex) are So far the imposed boundary conditions (velocity vortex) are simple (clock or counterclockwise). A real velocity pattern wilsimple (clock or counterclockwise). A real velocity pattern will be l be applied in the future.applied in the future.

Further detailed analysis of helicity accumulation and exchange Further detailed analysis of helicity accumulation and exchange processes in ARs will be carried out using the MHD simulation .processes in ARs will be carried out using the MHD simulation .

Thank you !Thank you !

3 25mH A Bd x Twist Writhe= ⋅ = Φ = +∫

Φ

Ref. Berger (1999)

Magnetic fluxReturn

2kH = +Φ

2kH = −Φ

Φ Magnetic flux

(a)

(b)

Ref. Berger (1984)

Return

1V 2V

Simulation box V