1

On the Calculation of Magnetic Helicity of a Solar Active Region

and a Cylindrical Flux Rope

Qiang Hu, G. M. Webb, B. Dasgupta

CSPAR, University of Alabama in Huntsville, USA

J. QiuMontana State University, USA

qh0001@uah.edu

2

Acknowledgement

NASA grants NNG04GF47G, NNG06GD41G, NNX07AO73G, and NNX08AH46G

(data provided by various NASA/ESA missions, and ground facilities; images credit: mostly NASA/ESA unless where indicated)

Debi P. ChoudharyB. DasguptaCharlie FarrugiaG.A. GaryYang Liu Dana LongcopeJiong Qiu

D. Shaikh

R. Skoug

C. W. Smith/N.F. Ness

W.-L. TehBengt U. Ö. Sonnerup

Vasyl YurchyshynGary Zank

Collaborators:

3

Overview

• Helical Structures: Interplanetary Coronal Mass Ejections– In-situ detection and magnetic flux rope model

– GS technique and its applications

• MDR-based coronal magnetic field extrapolation• Homotopy formula for magnetic vector potential • Summary and Outlook

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Coronal Mass Ejection (CME)

(Moore et al. 2007)

Simultaneous multi-point in-situ measurements of an Interplanetary CME

(ICME) structure(Adapted from STEREO website,

http://sprg.ssl.berkeley.edu/impact/instruments_boom.html)

5in-situ spacecraft data

Cylindrical flux-rope model fit (Burlaga, 1995; Lepping et al., 1990, etc.)

In-situ Detection and Modeling

6x: projected s/c path

x: projected s/c path

GS Reconstruction method: derive the axis orientation (z) and the cross section of locally 2 ½ D structure from in-situ single spacecraft measurements (e.g., Hu and Sonnerup 2002).

•Main features:

- 2 ½ D

- self-consistent

- non-force free

- flux rope boundary definition

- multispacecraft

actual result:actual result:

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• Output:1. Field configuration2. Spatial config.3. Electric Current.4. Plasma pressure p(A).5. Magnetic Flux ：

- axial (toroidal) flux t= Bzxy- poloidal flux p=|Ab - Am|*L

6. Relative Helicity:Krel=2L A’· Bt dxdy

A’=Bzz

^

Reconstruction of ICME Flux Ropes (1D2D)

• Ab

Am

ACE Halloween event (Hu et al. 2005)

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z=(0.057, 0.98, -0.18) ± (0.08, 0.01, 0.03) RTN

July 11, 1998 [Hu et al, 2004]

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Field line twist, Flux rope 1 Flux rope 2

Pink 4.22/AU 4.72/AU

Blue 1.92/AU 1.92/AU

View towards Sun:

2 ~ 1025-1026 Wb2z

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CME ICME

Sun at Earth

propagation

Sun-Earth Connection

1. Orientation of flux rope CME/ICME (Yurchyshyn et al. 2007)2. Quantitative comparison of magnetic flux (Qiu et al. 2007)

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poloidal or azimuthal magnetic flux P: the amount of twist

along the field lines

The helical structure, in-situ formed flux rope, results from magnetic reconnection.

toroidal or axial magnetic flux t

Longcope et al (2007)

ribbons

poloidal flux P

reconnection flux r

reconnection

3D view3D view3D view3D view

(Gosling et al. 1995)

(Moore et al. 2007)

Credit: ESA

reconnection

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• Comparison of CME and ICME fluxes (independently measured for 9 events; Qiu et al., 2007):

- flare-associated CMEs and flux-rope ICMEs with one-to-one correspondence; - reasonable flux-rope solutions satisfying diagnostic measures; - an effective length L=1 AU (uncertainty range 0.5-2 AU) .

GS method

Leamon et al. 04

Lynch et al. 05

P ~ r

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• One existing simple model, variational principle of minimum energy (e.g., Taylor, 1974; Freidberg, 1987):

, // B B j B

However, Amari and Luciani (2000), among others, showed by 3D numerical simulation that in certain solar physics situation, …, the final “relaxed state is far from the constant- linear force-free field that would be predicted by Taylor’s conjecture” …, and suggested to derive alternative variational problem.

Linear force-free field (LFFF, const)Or, Nonlinear FFF ( varies)

, j B 0

Coronal Magnetic Field Extrapolation (2D3D)

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An An alternative...alternative...

Simple Examples:Simple Examples:

• Current distribution in a circuit

Total ohmic dissipation is minimum

• Velocity profile of a viscous liquid flowing through a duct

Total viscous dissipation is minimum

• Principle of Minimum Dissipation Rate (MDR): the energy dissipation rate is minimum. (Montgomery and Phillips,1988; Dasgupta et al. 1998; Bhattacharyya and Janaki, 2004))

(Several extended variational principles of minimum (Several extended variational principles of minimum energy (energy (Mahajan 2008; Turner 1986Mahajan 2008; Turner 1986) yield solutions that ) yield solutions that are subsets to the above)are subsets to the above)

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• For an open system with flow, the MDR theory yields (Bhattacharyya et al. 2007; Hu et al., 2007; Hu and Dasgupta, 2008, Sol. Phys.)

Take an extra curl to eliminate the undetermined potential field , one obtains

New ApproachNew Approach

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(5)

(7)

(8)

• Equations (5),(7) and (8) form a 3rd order system. It is guaranteed invertible to yield the boundary conditions for each

Bi, given measurements of B at bottom boundary, provided the

parameters, 1 ,2 and 3 are distinct.

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11

2

3

( )i

z z

B B

B B

B B

M

Above equations provide the boundary conditions (normal components only at z=0) for each LFFF Bi, given B at certain heights, which then can be solved by an LFFF solver based on FFT (e.g, Alissandrakis, 1981).

• One parameter, 2 has to be set to 0. The parameters, 1 and 3, are determined by optimizing the agreement between calculated (b) and measured transverse magnetic field at z=0, by minimizing

Measurement error Measurement error +Computational error

<0.5?

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• Reduced approach: choose B2=cB’, proportional to a reference field, B’=A’, and B’n=Bn, such that the relative helicity is ABdV-A’B’dV, with A=B1/1+B3/3+cA’.

B’=0, Bz’=Bz, at z=0

Only one layer of vector magnetogram is needed. And the relative helicity of a solar active region can be calculated.(As a special case, B2=0, in Hu and Dasgupta, 2006)

1

1 2

3 1 30 0

1 1

z z

B B B

B B

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• Iterative reduced approach: transverse magnetic field vectors at z=0 (En=0.32): (0) (1) (2) ( )

2 2 2 2 2... ...k B B B B B

k=0

(0)2 min 'cB B

Reduced approach:( )

2 20

k i

iB B

Obtain E(k) and

t t t b B b

If E(k) <

EndY

Nk=k+1

( )2k

t b B

( )knE

n

n

k>kmaxY

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Test Case of Numerical Simulation Test Case of Numerical Simulation Data (Data (Hu et al. 2008, ApJHu et al. 2008, ApJ))

(a) “exact” solution(Courtesy of Prof. J. Buechner)

(b) our extrapolation (12812863)

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• Transverse magnetic field vectors at z=0 (En=0.30):

• Figures of merit (Hu et al., 2008, ApJ): Energy ratios

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• Integrated current densities along field lines:

(a) exact

(b) extrapolated

J_para J_perp 0

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E/Ep_pre=1.26 E/Ep_post=1.30

(Data courtesy of M. DeRosa, Schrijver et al. 2008; Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway).

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Main Features Main Features

• More general non-force free (non-vanishing currents); • Better energy estimate • Fast and easy (FFT-based); Make it much less demanding for computing resources • Applicable to one single-layer measurement (Hu et al. 2008, 2009)• Applicable to flow

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• Homotopy formula for vector magnetic potential (based on ):

1

0d

r' r

A(r) B(r') r'

1

0( )d B r r

12Const B A B r

0 B

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(Berger, M.)

“In topology, two continuous functions …

if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.”

-- Wikipedia

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• Relative magnetic helicity via homotopy formula:

Bz(x,y)

rKr/AU: 3.5x1023 Wb2

Kr/AU (Hu and Dasgupta, 2005):

3.4x1023 Wb2

ˆ2 ' , ' ', 'r t zVK dV B A B A B B z

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Relative magnetic helicity via homotopy formula: B in a 3D volume

r

1

0( ) ( )d A r B r r

( ') ( ') , 'r VK dV A A B B B n B n

(e.g., see Longcope & Malanushenko, 2008)

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• Multi-pole expansion of a potential field:

For each 2k-th pole, B(k), (via a modified homotopy formula)

Dipole:

• Related to spherical harmonic expansion, for example.

• For MDR-based extrapolation:

( )( ) , 1

kk k

k

B rA

A B r

1 32

1 3

B BA A

A simplified vector potential for a potential field?A simplified vector potential for a potential field?

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Outlook• Validate and apply the algorithm for one-layer vector

magnetograms• Validate the theory – proof of MDR by numerical

simulations• Global non-force free extrapolation

Stay tuned!

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32

• MHD states:

p j B 2 ( )p j B

p u u j B

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• Dec. 12-13 2006 Flare and CME (Schrijver et al. 2008)

(KOSOVICHEV & SEKII, 2007)

(SOHO LASCO CME CATALOG http://cdaw.gsfc.nasa.gov/CME_list/)

34

• Reduced approach: transverse magnetic field vectors at z=0 (En=0.7-0.9):

MeasuredComputed

(Data courtesy of M. DeRosa, Schrijver et al. 2008; Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway).

35

( )knE

En=0.28

Post-flare case:

36

En(1,3):

37

38

• Principle of minimum dissipation rate (MDR)

In an irreversible process a system spontaneously evolves to states in which the energy dissipation rate is minimum.

A different variational principleA different variational principle

Suggests a minimizer for our problem

1.1. R. Bhattacharyya and M. S. Janaki,Phys. Plasmas R. Bhattacharyya and M. S. Janaki,Phys. Plasmas 1111, 615 (2004)., 615 (2004).2.2. D. Montgomery and L. Phillips, Phys. Rev. A D. Montgomery and L. Phillips, Phys. Rev. A 3838, 2953, (1988)., 2953, (1988).3.3. B. Dasgupta, P. Dasgupta, M. S. Janaki, T. Watanabe and T. Sato, Phys. Rev. Letts,B. Dasgupta, P. Dasgupta, M. S. Janaki, T. Watanabe and T. Sato, Phys. Rev. Letts, 81 81, ,

3144, (1998)3144, (1998)

39

L Onsager, Phys. Rev, 37, 405 (1931)I. Prigogine, Thermodynamics of Irreversible Processes, Wiley

(1955)-------------------------------------------------------------------------------------------------

-----

A theorem from irreversible A theorem from irreversible thermodynamics:thermodynamics:

Principle of Minimum Entropy Production “The steady state of an irreversible process, i.e., the

state in which thermodynamics variables are independent of time, is characterized by a minimum value of the rate of Entropy Production”

Rate of Minimum Entropy Production is equivalent to Rate of Minimum Dissipation of Energy in most cases.

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Numerical simulation (Shaikh et al., 2007; NG21A-0206 ) showed the evolution of the decay rates associated with the turbulent relaxation, viz, Magnetic Helicity KM, Magnetic Energy WM and the Dissipation Rate R.

A BdV

KM =

2

2B dVWM =

2J dVR=

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• The generalized helicity dissipation rate is time-invariant.

Formulation of the variational problemFormulation of the variational problem(for an open system with external drive, or helicity injection)(for an open system with external drive, or helicity injection)

constraint dL2 ΩuB ( = i, e)

• From the MDR principle, the minimizer is the total energy dissipation rate

2 24 i

i

n q LR J d

c

variational problem 0I

2 2i e e

4 B L ω Ω d B Ω di

i i i i

nqLI J d

c

ii uω

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Euler-Lagrange equationsEuler-Lagrange equations

0m

c

qL

2c

qii

iiei2

ω B

B

0mc

qL

nm2

1

cλm

πnq2 2i

i

iii

iii

2 B ω ω

Eliminating vorticity in favor of the magnetic field

B

B B i

i

ei2

i

22

ii

ii

i

ii

e2

cm

nq2m

c

qL

Lm4cm

nq2

c

q

ii

ei2

i

2

i

2

ii

ii

i

ii

eii

2 cm

nq 2 m

c

qL

Lm4cm

nq2

c

qωωω

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Summary of Summary of ProceduresProcedures

magnetohydrodynamic

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Analytic Test Case: Analytic Test Case: non-force free active non-force free active region model given by Low (1992)region model given by Low (1992)

TopView:

45

A real case: Active Region (AR)8210 (preliminary)A real case: Active Region (AR)8210 (preliminary)

(Choudhary et al. 2001)

Imaging Vector Magnetograph (IVM) at Mees Solar Observatory (courtesy of M. Georgoulis)

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En distribution:

47

• One-fluid Magnetohydrostatic Theory – 2 ½ D: Bz 0– Co-moving frame: DeHoffmann-Teller (HT) frame– No inertia forcep jB, 0 jB, B 0.

Grad-Shafranov (GS) Equation (A=Az):

2 A

x2 2 A

y2 0d

dAp

Bz2

20

0

dPt

dA

z 0

Pt(A)=p(A)+Bz(A)/20 Pt(A)=p(A)+Bz(A)/20

A • Bt = 0

2

( / , / , ( ))zA y A x B A B

GS Reconstruction

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GS Reconstruction Technique1. Find z by the requirement

that Pt(A) be single-valued

2. Transform time to spatial dimensions via VHT, and calculate A(x,0),

3. Calculate Pt(x,0) directly from measurements.

4. Fit Pt(x,0)/Bz(x,0) vs. A(x,0)

by a function, Pt(A)/Bz(A).

A boundary, A=Ab, is chosen. 5. Computing A(x,y) by

utilizing A(x,0), Bx(x,0), and GS equation.

A(x,0) By0

x (,0)d

d VHT xdt

A(x,0) By0

x (,0)d

d VHT xdt^

x: projected s/c path

x: projected s/c path

Am

o:inbound

*:outbound

49

Title: /afs/northstar.dartmouth.edu/ugrad/qianghu/private/delta/matlab/cloud/test/ptalund.eps Creator: MATLAB, The Mathworks, Inc. Preview: This EPS picture was not saved with a preview (TIFF or PICT) included in it Comment: This EPS picture will print to a postscript printer but not to other types of printers • Finding z axis by minimizing residue of Pt(A):

Residue=[∑i(Pt,i – Pt,i )2]

/|max(Pt)-min(Pt)|

A(x,0)

Pt(x,0)

Pt(A)

o: 1st half

*: 2nd half i=1…m

121st 2nd

Enumerating all possible directions in space to find the optimal z axis for which the associated Residue is a minimum. A residue map is constructed to show the uniqueness of the solution with uncertainty estimate.

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• GS Solver:

known.initially are 0at

and )/(),,( where

),()(),(),(

,)()(2

1)()(),(),(

02

2

2

2

,

,2

2

2,2

2

,

ydA

dP

x

A

y

A

ByAyxA

yy

AyxByyxB

yy

Ay

y

AyxAyyxA

t

xyx

yxxx

yxyx

51

• Multispacecraft Test of GS Method

Cluster FTEs (from Sonnerup et al., 2004; see also Hasegawa et al. 2004, 2005, 2006)

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Introduction• Grad-Shafranov (GS) equation: p=j B in 2D

• GS technique: solve GS equation using in-situ data, 1D2D (e.g., Sonnerup and Guo, 1996; Hau and Sonnerup, 1999; Hu and Sonnerup, 2000, 2001, 2002,

2003; Sonnerup et al. 2006)

20 ( )t zA j A

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• Small-scale flux ropes in the solar wind (Hu and Sonnerup, 2001)

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• Features of the GS Reconstruction Technique:

-Fully 2 ½ D solution (less fitting)

-Self-consistent theoretical modeling; boundary definition (less subjective)

-Utilization of simultaneous magnetic and plasma measurements; Non-force free

-Adapted to a fully multispacecraft technique (Hasegawa et al. [2004])

• Limitations (diagnostic measures):

-2D, uncertainty in z (the quality of Pt(A) fitting, Rf)

2D Pt(A), Pt(A) 2D

-Time stationary (quality of the frame of reference)

-Static (evaluating the residual plasma flow)

-Numerical errors limit the extent in y direction (rule of thumb: |y||x|, y«x)

?

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: Wind data

o : Predicted

• ACE-Wind comparison

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• July 11 1998 event

-Apparent magnetic signatures of multiple structures denoted by 1, and 2.

- GS reconstruction is applied to the larger interval (solid vertical lines) and subintervals 1, and 2.

1 2

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Case A

’=(-2º,-90º) Case B

’=(2º,-90º)

r r

Z Z

o: ’ x: Z’

The exact orientation =(0º,-90º); in both cases the results are much better than cylindrical models (Riley et al., 2004).

58

• ACE-Ulysses comparison (Du et al., 2007)

Fluxes at ACE (length D=*1AU):

t Wb

P =44 TWbHelicity:-1.4**1023 Wb^2

At Ulysses (D=5.4AU):

t Wb

P =14 TWbHelicity:-3.7**1021 Wb^2

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2D 3D?

May 22-23, 2007 Event (courtesy of C. Farrugia)

60

(Yurchyshyn et al. 2005, 2007)

at Earth

Prediction?

connection

Multi-wavelength, multi-instrument dataanalysis & modeling

61

CME

x

Connection Between MC flux, and Flux due to Magnetic Reconnection in Low Corona (Qiu et al. 2007)

flare ribbons

Flare loops

reconnection site

(adapted from Forbes & Acton, 1996)

• MC Flux ： MC: mostly poloidal component P

(measured in-situ at 1 AU)• At sun:

MC = pre-existing flux if any + reconnection flux r

P r

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• GS Reconstruction of Locally Toroidal Structure

(Riley et al., 2004)

GS result

63

~ 64% agreement rate (Yurchyshyn et al. 2007)

64

(Gosling et al. 1995)

3D view3D view3D view3D view

65

CME: the flux rope set free

Different scenarios (Gosling et al. 1995):

66

reconnection rate (general)

dAC

BC

BR

dAR

reconnection

Vin

“measure” reconnection rate

v

v

B t

BC dAC =

t

BR dAR

flare at an earlier time flare at a later time MDI magnetogram

dAR

observation: flares and magnetic fields

model: Forbes & Priest 1984

67

The flare-CME connection: some models

loss of equilibrium(Forbes-Priest-Lin)

CME front

filament

flare

magnetic breakout (Antiochos et al. 1999)

overlying flux

post-flare loop

reconnectionemerging flux

Does the CME know the flare?

68

Forbes (2000)

pre-existing flux rope

(Chen 1989, Low 1996, Forbes-Priest-Lin, Fan-Gibson)

What do we see as a magnetic flux rope?

SXR sigmoid

prominence (HAO)

SXT/Yohkoh

Gibson, 2005 Harvey Prize Lecture