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. Qiu Montana State University, USA

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. Choudhary B. Dasgupta Charlie Farrugia G.A. Gary Yang Liu Dana Longcope Jiong Qiu D. Shaikh R. Skoug C. W. Smith/N.F. Ness W.-L. Teh Bengt U. Ö. Sonnerup Vasyl Yurchyshyn Gary 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

4 Coronal Mass Ejection (CME) (Moore et al. 2007) Simultaneous multi-point in-situ measurements of an Interplanetary CME (ICME) structure ( Adapted from STEREO website, )

5 in-situ spacecraft data Cylindrical flux-rope model fit (Burlaga, 1995; Lepping et al., 1990, etc.) In-situ Detection and Modeling

6 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:

7 Output: 1.Field configuration 2.Spatial config. 3.Electric Current. 4.Plasma pressure p(A). 5.Magnetic Flux  ： - axial (toroidal) flux  t =  B z  x  y - poloidal flux  p =|A b - A m |*L 6.Relative Helicity: K rel =2L  A’· B t dxdy  A’=B z z ^ Reconstruction of ICME Flux Ropes ( 1D  2D) Ab Ab AmAm ACE Halloween event (Hu et al. 2005)

8 z =(0.057, 0.98, -0.18) ± (0.08, 0.01, 0.03) RTN July 11, 1998 [Hu et al, 2004]

9 Field line twist,  Flux rope 1 Flux rope 2 Pink 4.2  2  /AU 4.7  2  /AU Blue 1.9  2  /AU 1.9  2  /AU View towards Sun:  2 ~ Wb 2 z

10 CMEICME Sun at Earth propagation Sun-Earth Connection 1.Orientation of flux rope CME/ICME ( Yurchyshyn et al ) 2.Quantitative comparison of magnetic flux ( Qiu et al )

11 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 view (Gosling et al. 1995) (Moore et al. 2007) Credit: ESA reconnection

12 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 AU). GS method Leamon et al. 04 Lynch et al. 05  P ~  r

13 One existing simple model, variational principle of minimum energy (e.g., Taylor, 1974; Freidberg, 1987): 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) Coronal Magnetic Field Extrapolation ( 2D  3D)

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

15 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 Approach 

16 (5) (7) (8) Equations (5),(7) and (8) form a 3 rd order system. It is guaranteed invertible to yield the boundary conditions for each B i, given measurements of B at bottom boundary, provided the parameters,  1,  2 and  3 are distinct.

17 Above equations provide the boundary conditions (normal components only at z=0) for each LFFF B i, 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?

18 Reduced approach: choose B 2 =cB’, proportional to a reference field, B’=  A’, and B’ n =B n, such that the relative helicity is  A  BdV-  A’  B’dV, with A=B 1 /  1 +B 3 /  3 +cA’.  B’=0, B z ’=B z, 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, B 2 =0, in Hu and Dasgupta, 2006)

19 Iterative reduced approach: transverse magnetic field vectors at z=0 (E n =0.32): k=0 Reduced approach: Obtain E (k) and If E (k) <  End Y N k=k+1 n n k>k max Y

20 Test Case of Numerical Simulation Data ( Hu et al. 2008, ApJ ) (a)“exact” solution (Courtesy of Prof. J. Buechner) (b) our extrapolation (128  128  63)

21 Transverse magnetic field vectors at z=0 ( E n =0.30 ): Figures of merit (Hu et al., 2008, ApJ): Energy ratios

22 Integrated current densities along field lines: (a) exact (b) extrapolated J_para J_perp  0

23 E/E p_pre=1.26 E/E p_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).

24 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

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

26 (Berger, M.) “In topology, two continuous functions …topologycontinuousfunctions if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.” -- Wikipedia

27 Relative magnetic helicity via homotopy formula: B z (x,y) r Kr/AU: 3.5 x Wb 2 Kr/AU ( Hu and Dasgupta, 2005 ): 3.4 x Wb 2

28 Relative magnetic helicity via homotopy formula: B in a 3D volume r (e.g., see Longcope & Malanushenko, 2008)

29 Multi-pole expansion of a potential field: For each 2 k -th pole, B (k), (via a modified homotopy formula) Dipole: Related to spherical harmonic expansion, for example. For MDR-based extrapolation: A simplified vector potential for a potential field?

30 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!

31

32 MHD states:

33 Dec Flare and CME (Schrijver et al. 2008) (KOSOVICHEV & SEKII, 2007) (SOHO LASCO CME CATALOG

34 Reduced approach: transverse magnetic field vectors at z=0 (E n = ): Measured Computed (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 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 principle Suggests a minimizer for our problem 1.R. Bhattacharyya and M. S. Janaki,Phys. Plasmas 11, 615 (2004). 2.D. Montgomery and L. Phillips, Phys. Rev. A 38, 2953, (1988). 3.B. Dasgupta, P. Dasgupta, M. S. Janaki, T. Watanabe and T. Sato, Phys. Rev. Letts, 81, 3144, (1998)

39 L Onsager, Phys. Rev, 37, 405 (1931) I. Prigogine, Thermodynamics of Irreversible Processes, Wiley (1955 ) A theorem from irreversible 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.

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

41 The generalized helicity dissipation rate is time-invariant. Formulation of the variational problem (for an open system with external drive, or helicity injection) constraint (  = i, e) From the MDR principle, the minimizer is the total energy dissipation rate variational problem

42 Euler-Lagrange equations Eliminating vorticity in favor of the magnetic field

43 Summary of Procedures magnetohydrodynamic

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

45 A real case: Active Region (AR)8210 (preliminary) ( Choudhary et al ) Imaging Vector Magnetograph (IVM) at Mees Solar Observatory (courtesy of M. Georgoulis)

46 E n distribution:

47 One-fluid Magnetohydrostatic Theory –2 ½ D: B z  0 –Co-moving frame: DeHoffmann-Teller (HT) frame –No inertia force Grad-Shafranov (GS) Equation (A=A z ): P t (A)=p(A)+B z (A)/2  0  A B t = 0 2 GS Reconstruction

48  GS Reconstruction Technique 1.Find z by the requirement that P t (A) be single-valued 2.Transform time to spatial dimensions via V HT, and calculate A(x,0), 3.Calculate P t (x,0) directly from measurements. 4.Fit P t (x,0)/B z (x,0) vs. A(x,0) by a function, P t (A)/B z (A). A boundary, A=A b, is chosen. 5.Computing A(x,y) by utilizing A(x,0), B x (x,0), and GS equation. ^ x: projected s/c path AmAm o:inbound *:outbound

49 Finding z axis by minimizing residue of P t (A): Residue=[∑ i (P t,i – P t,i ) 2 ] / |max(P t )-min(P t )| A(x,0) P t (x,0) Pt(A)Pt(A) o: 1st half *: 2nd half i=1…m st2nd 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.   

50 GS Solver:

51 Multispacecraft Test of GS Method Cluster FTEs (from Sonnerup et al., 2004; see also Hasegawa et al. 2004, 2005, 2006)

52 Introduction Grad-Shafranov (GS) equation:  p=j  B in 2D GS technique: solve GS equation using in-situ data, 1D  2D (e.g., Sonnerup and Guo, 1996; Hau and Sonnerup, 1999; Hu and Sonnerup, 2000, 2001, 2002, 2003; Sonnerup et al )

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

54 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 P t (A) fitting, R f ) 2D  P t (A), P t (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) ?

55 : Wind data oo : Predicted ACE-Wind comparison

56 July 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. 12

57 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 TWb  Helicity :  -1.4*  *10 23 Wb^2 At Ulysses (D=5.4AU):  t  Wb  P =14 TWb  Helicity :  -3.7*  *10 21 Wb^2

59 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 data analysis & 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

62 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 view

65 CME: the flux rope set free Different scenarios (Gosling et al. 1995):

66 reconnection rate (general) dA C BCBC BRBR dA R reconnection V in “measure” reconnection rate v v flare at an earlier time flare at a later time MDI magnetogram dA R 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 reconnection emerging 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