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On the Calculation of Magnetic Helicity of a Solar Active Region and a Cylindrical Flux Rope

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 qh0001@uah.edu. Acknowledgement. Collaborators :. D. Shaikh R. Skoug

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On the Calculation of Magnetic Helicity of a Solar Active Region and a Cylindrical Flux Rope

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  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 qh0001@uah.edu

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

  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) 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) (Moore et al. 2007)

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

  6. 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 ½D • self-consistent • non-force free • flux rope boundary definition • multispacecraft actual result: x: projected s/c path

  7. Reconstruction of ICME Flux Ropes (1D2D) • Output: • Field configuration • Spatial config. • Electric Current. • Plasma pressure p(A). • Magnetic Flux ： • - axial (toroidal) flux • t= Bzxy • - poloidal flux • p=|Ab - Am|*L • Relative Helicity: • Krel=2L A’· Btdxdy • A’=Bzz Am • Ab ^ ACE Halloween event (Hu et al. 2005)

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

  9. View towards Sun: 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 2 ~ 1025-1026 Wb2 z

  10. Sun-Earth Connection CME ICME propagation Sun at Earth • Orientation of flux rope CME/ICME (Yurchyshyn et al. 2007) • Quantitative comparison of magnetic flux (Qiu et al. 2007)

  11. 3D view poloidal or azimuthal magnetic flux P: the amount of twist along the field lines poloidal flux P (Moore et al. 2007) The helical structure, in-situ formed flux rope, results from magnetic reconnection. Longcope et al (2007) reconnection reconnection (Gosling et al. 1995) reconnection flux r Credit: ESA ribbons toroidal or axial magnetic flux t

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

  13. Coronal Magnetic Field Extrapolation (2D3D) • One existing simple model, variational principle of minimum energy (e.g., Taylor, 1974; Freidberg, 1987): Linear force-free field (LFFF,const) Or, Nonlinear FFF ( varies) 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.

  14. An alternative... • 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) • 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

  15. New Approach • 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

  16. (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.

  17. Above equations provide the boundary conditions (normal components only at z=0) for each LFFF Bi, given Bat 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 <0.5? Measurement error + Computational error Measurement error

  18. Reduced approach: choose B2=cB’, proportional to a reference field, B’=A’, and B’n=Bn, such that the relative helicity is ABdV-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)

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

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

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

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

  23. 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).

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

  27. Relative magnetic helicity via homotopy formula: Bz(x,y) r

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

  29. 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: 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. MHD states:

  32. Dec. 12-13 2006 Flare and CME (Schrijver et al. 2008) (SOHO LASCO CME CATALOG http://cdaw.gsfc.nasa.gov/CME_list/) (KOSOVICHEV & SEKII, 2007)

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

  34. Post-flare case: En=0.28

  35. En(1,3):

  36. A different variational principle Suggests a minimizer for our problem • R. Bhattacharyya and M. S. Janaki,Phys. Plasmas 11, 615 (2004). • D. Montgomery and L. Phillips, Phys. Rev. A 38, 2953, (1988). • B. Dasgupta, P. Dasgupta, M. S. Janaki, T. Watanabe and T. Sato, Phys. Rev. Letts, 81, 3144, (1998) • Principle of minimum dissipation rate (MDR) In an irreversible process a system spontaneously evolves to states in which the energy dissipation rate is minimum.

  37. 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 ofan 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.

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

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

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

  41. Summary of Procedures magnetohydrodynamic

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

  43. A real case: Active Region (AR)8210 (preliminary) Imaging Vector Magnetograph (IVM) at Mees Solar Observatory (courtesy of M. Georgoulis) (Choudhary et al. 2001)

  44. En distribution:

  45. GS Reconstruction • One-fluid Magnetohydrostatic Theory • 2 ½ D: Bz 0 • Co-moving frame: DeHoffmann-Teller (HT) frame • No inertia force Grad-Shafranov (GS) Equation (A=Az): A • Bt = 0 2 Pt(A)=p(A)+Bz(A)/20

  46.  GS Reconstruction Technique • Find z by the requirement that Pt(A) be single-valued • 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. x: projected s/c path o:inbound Am ^ *:outbound

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

  48. GS Solver:

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