Nonlinear force‐free modeling of the solar coronal magnetic field

[1] The coronal magnetic field is an important quantity because the magnetic field dominates the structure of the solar corona. Unfortunately, direct measurements of coronal magnetic fields are usually not available. The photospheric magnetic field is measured routinely with vector magnetographs. These photospheric measurements are extrapolated into the solar corona. The extrapolated coronal magnetic field depends on assumptions regarding the coronal plasma, for example, force-freeness. Force-free means that all nonmagnetic forces like pressure gradients and gravity are neglected. This approach is well justified in the solar corona owing to the low plasma beta. One has to take care, however, about ambiguities, noise and nonmagnetic forces in the photosphere, where the magnetic field vector is measured. Here we review different numerical methods for a nonlinear force-free coronal magnetic field extrapolation: Grad-Rubin codes, upward integration method, MHD relaxation, optimization, and the boundary element approach. We briefly discuss the main features of the different methods and concentrate mainly on recently developed new codes.

[1]  Testing non-linear force-free coronal magnetic field extrapolations with the Titov-Démoulin equilibrium , 2006, astro-ph/0612650.

[2]  Markus J. Aschwanden.  Physics of the Solar Corona , 2004 .

[3]  Markus J. Aschwanden,et al.  Physics of the Solar Corona: An Introduction with Problems and Solutions , 2005 .

[4]  M. Aschwanden.  Physics of the Solar Corona. An Introduction , 2004 .

[5]  M. Wheatland,et al.  An optimization approach to reconstructing force-free fields from boundary data: II. Numerical results , 1997 .

[6]  K. Yee.  Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media , 1966 .

[7]  J. Aly,et al.  Well posed reconstruction of the solar coronal magnetic field , 2006 .

[8]  C. Munz,et al.  Hyperbolic divergence cleaning for the MHD equations , 2002 .

[9]  C. Schrijver,et al.  The Nonpotentiality of Active-Region Coronae and the Dynamics of the Photospheric Magnetic Field , 2005 .

[10]  T. Neukirch,et al.  Including stereoscopic information in the reconstruction of coronal magnetic fields , 2002, 0801.3234.

[11]  G. A. Gary,et al.  On the numerical computation of nonlinear force-free magnetic fields , 1985 .

[12]  Huaning Wang,et al.  The validity of the boundary integral equation for magnetic field extrapolation in open space above a spherical surface , 2006 .

[13]  Y. Yan,et al.  Properties of the boundary integral equation for solar non-constant-α force-free magnetic fields , 2004 .

[14]  C. Alissandrakis,et al.  On the computation of constant alpha force-free magnetic field , 1981 .

[15]  R. Canfield,et al.  Evolution of magnetic fields and energetics of flares in active region 8210 , 2006 .

[16]  N. Seehafer.  Determination of constant α force-free solar magnetic fields from magnetograph data , 1978 .

[17]  Michael S. Wheatland,et al.  An Optimization Approach to Reconstructing Force-free Fields , 1997 .

[18]  E. Priest,et al.  Aspects of Three-Dimensional Magnetic Reconnection – (Invited Review) , 1999 .

[19]  A. V. Ballegooijen,et al.  Observations and Modeling of a Filament on the Sun , 2004 .

[20]  T. Sakurai.  Calculation of force-free magnetic field with non-constant α , 1981 .

[21]  Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction , 2006, astro-ph/0612641.

[22]  M. Wheatland,et al.  Non-linear Force-free Modeling Of Coronal Magnetic Fields , 2007 .

[23]  Thomas R. Metcalf,et al.  Resolving the 180-degree ambiguity in vector magnetic field measurements: The ‘minimum’ energy solution , 1994 .

[24]  E. Priest.  Heating the Solar Corona by Magnetic Reconnection , 1998 .

[25]  K. Leka,et al.  The Imaging Vector Magnetograph at Haleakalā – II.  Reconstruction of Stokes Spectra , 1999 .

[26]  S. Solanki,et al.  Three-dimensional magnetic field topology in a region of solar coronal heating , 2003, Nature.

[27]  Philip G. Judge,et al.  Spectral Lines for Polarization Measurements of the Coronal Magnetic Field. I. Theoretical Intensities , 1998 .

[28]  M. Aschwanden.  Revisiting the Determination of the Coronal Heating Function from Yohkoh Data , 2001 .

[29]  P. Sturrock,et al.  Force-free magnetic fields - The magneto-frictional method , 1986 .

[30]  R. Canfield,et al.  The imaging vector magnetograph at Haleakala , 1996 .

[31]  A Fast Current-Field Iteration Method for Calculating Nonlinear Force-Free Fields , 2006 .

[32]  Markus J. Aschwanden,et al.  An Evaluation of Coronal Heating Models for Active Regions Based on Yohkoh, SOHO, and TRACE Observations , 2001 .

[34]  J. Aly.  On some properties of force-free magnetic fields in infinite regions of space , 1984 .

[35]  A. Kageyama,et al.  ``Yin-Yang grid'': An overset grid in spherical geometry , 2004, physics/0403123.

[36]  M. Bineau.  On the existence of force-free magnetic fields , 1972 .

[37]  J. P. Goedbloed,et al.  Adaptive Mesh Refinement for conservative systems: multi-dimensional efficiency evaluation , 2003, astro-ph/0403124.

[38]  Jean-Pierre Delaboudiniere,et al.  Three-dimensional Stereoscopic Analysis of Solar Active Region Loops. I. SOHO/EIT Observations at Temperatures of (1.0-1.5) × 106 K , 1999 .

[39]  S. Solanki,et al.  Retrieval of the full magnetic vector with the He I multiplet at 1083 nm. Maps of an emerging flux region , 2004 .

[40]  G. Allen Gary,et al.  Transformation of vector magnetograms and the problems associated with the effects of perspective and the azimuthal ambiguity , 1990 .

[41]  Z. Mikić,et al.  PROBLEMS AND PROGRESS IN COMPUTING THREE-DIMENSIONAL CORONAL ACTIVE REGION MAGNETIC FIELDS FROM BOUNDARY DATA , 1997 .

[42]  Carolus J. Schrijver,et al.  Is the Quiet-Sun Corona a Quasi-steady, Force-free Environment? , 2005 .

[43]  Y. T. Chiu,et al.  Exact Green's function method of solar force-free magnetic-field computations with constant alpha. I - Theory and basic test cases , 1977 .

[44]  S. Solanki,et al.  Comparing magnetic field extrapolations with measurements of magnetic loops , 2005, 0801.4519.

[45]  G. A. Gary,et al.  Plasma Beta above a Solar Active Region: Rethinking the Paradigm , 2001 .

[46]  A New and Fast Way to Reconstruct a Nonlinear Force-free Field in the Solar Corona , 2006 .

[47]  M. Karlický,et al.  The Magnetic Rope Structure and Associated Energetic Processes in the 2000 July 14 Solar Flare , 2001 .

[48]  R. Casini,et al.  Spectral Lines for Polarization Measurements of the Coronal Magnetic Field. IV. Stokes Signals in Current-carrying Fields , 2006 .

[49]  Zoran Mikic,et al.  RECONSTRUCTING THE SOLAR CORONAL MAGNETIC FIELD AS A FORCE-FREE MAGNETIC FIELD , 1997 .

[50]  M. Semel.  Extrapolation functions for constant-alpha force-free fields - Green's method for the oblique boundary value , 1988 .

[51]  Parallel Construction of Nonlinear Force-Free Fields , 2004 .

[52]  G. Simon,et al.  Solar active region evolution : comparing models with observations : proceedings of Fourteenth International Summer Workshop National Solar Observatory/Sacramento Peak, Sunspot, New Mexico 88349, USA, 30 August - 3 September 1993 , 1994 .

[53]  W. Press,et al.  Numerical Recipes in C++: The Art of Scientific Computing (2nd edn)1 Numerical Recipes Example Book (C++) (2nd edn)2 Numerical Recipes Multi-Language Code CD ROM with LINUX or UNIX Single-Screen License Revised Version3 , 2003 .

[54]  B. Low,et al.  Modeling solar force-free magnetic fields , 1990 .

[55]  Yihua Yan,et al.  Direct Boundary Integral Formulation for Solar Non-constant-α Force-free Magnetic Fields , 2006 .

[56]  T. Neukirch,et al.  A Quantitative Method to Optimise Magnetic Field Line Fitting of Observed Coronal Loops , 2003 .

[57]  S. Régnier,et al.  3D magnetic configuration of the Hα filament and X-ray sigmoid in NOAA AR 8151 , 2004 .

[58]  E. DeLuca,et al.  Modeling magnetic flux ropes in the solar atmosphere , 2007 .

[59]  R. Canfield,et al.  Is the solar chromospheric magnetic field force-free? , 1995 .

[60]  T. Wiegelmann.  Optimization code with weighting function for the reconstruction of coronal magnetic fields , 2008, 0802.0124.

[61]  L. Ofman,et al.  Extrapolation of photospheric potential magnetic fields using oblique boundary values: a simplified approach , 1990 .

[62]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[63]  T. Amari,et al.  3D Coronal magnetic field from vector magnetograms: non-constant-alpha force-free configuration of the active region NOAA 8151 , 2002 .

[64]  N. Seehafer.  A comparison of different solar magnetic field extrapolation procedures , 1982 .

[65]  S. Solanki,et al.  How To Use Magnetic Field Information For Coronal Loop Identification , 2005, 0801.4573.

[66]  G. A. Gary,et al.  An Overview of Existing Algorithms for Resolving the 180° Ambiguity in Vector Magnetic Fields: Quantitative Tests with Synthetic Data , 2006 .

[67]  B. Inhester,et al.  Magnetic modeling and tomography: First steps towards a consistent reconstruction of the solar corona , 2003 .

[68]  B. Labonte.  The Imaging Vector Magnetograph at Haleakala –: III. Effects of Instrumental Scattered Light on Stokes Spectra , 2004 .

[69]  M. Aschwanden,et al.  The Coronal Heating Mechanism as Identified by Full-Sun Visualizations , 2004 .

[70]  M. Raadu,et al.  On practical representation of magnetic field , 1972 .

[71]  H. Schmidt.  On the observable effects of magnetic energy storage and release connected with solar flares , 1964 .

[72]  M. Aschwanden,et al.  Evolution of Magnetic Flux Rope in the Active Region NOAA 9077 on 14 July 2000 , 2001 .

[73]  Yihua Yan,et al.  New Boundary Integral Equation Representation for Finite Energy Force-Free Magnetic Fields in Open Space above the Sun , 2000 .

[74]  T. Török,et al.  The evolution of twisting coronal magnetic flux tubes , 2003 .

[75]  Dalton D. Schnack,et al.  Dynamical evolution of a solar coronal magnetic field arcade , 1988 .

[76]  S. Solanki,et al.  Vector tomography for the coronal magnetic field I. Longitudinal Zeeman effect measurements , 2006 .

[77]  Tetsuya Sakurai,et al.  The Solar-B Mission and the Forefront of Solar Physics , 2004 .

[78]  M. Wheatland.  A Better Linear Force-free Field , 1999 .

[79]  A. Pevtsov,et al.  Patterns of Helicity in Solar Active Regions , 1994 .

[80]  Thomas Wiegelmann.  Computing Nonlinear Force-Free Coronal Magnetic Fields in Spherical Geometry , 2006 .

[81]  T. Neukirch,et al.  Computing nonlinear force free coronal magnetic fields , 2003, 0801.3215.

[82]  R. Chodura,et al.  A 3D Code for MHD Equilibrium and Stability , 1981 .

[83]  T. Sakurai.  Green's function methods for potential magnetic fields , 1982 .

[84]  Motokazu Noguchi,et al.  Solar flare telescope at Mitaka , 1995 .

[85]  E. Priest,et al.  Coronal Heating at Separators and Separatrices , 2005 .

[86]  P. Sturrock,et al.  Three-dimensional Force-free Magnetic Fields and Flare Energy Buildup , 1992 .

[87]  C. U. Keller,et al.  SOLIS-VSM Solar Vector Magnetograms , 2006, astro-ph/0612584.

[88]  T. Amari,et al.  An iterative method for the reconstructionbreak of the solar coronal magnetic field. I. Method for regular solutions , 1999 .

[89]  Z. Mikić,et al.  RECONSTRUCTION OF THE THREE-DIMENSIONAL CORONAL MAGNETIC FIELD , 1997 .

[90]  J. Aly,et al.  On the reconstruction of the nonlinear force-free coronal magnetic field from boundary data , 1989 .

[91]  Harold Grad,et al.  HYDROMAGNETIC EQUILIBRIA AND FORCE-FREE FIELDS , 1958 .

[92]  J. Wilcox,et al.  A model of interplanetary and coronal magnetic fields , 1969 .

[93]  Self and mutual magnetic helicities in coronal magnetic configurations , 2005 .

[94]  Yihua Yan,et al.  Topology of Magnetic Field and Coronal Heating in Solar Active Regions , 2001 .

[95]  Yihua Yan,et al.  Topology of Magnetic Field and Coronal Heating in Solar Active Regions – II. The Role of Quasi-Separatrix Layers , 2000 .

[96]  M. Wheatland,et al.  Nonlinear Force-Free Modeling of Coronal Magnetic Fields Part I: A Quantitative Comparison of Methods , 2006 .

[97]  T. Amari,et al.  On the existence of non-linear force-free fields in three-dimensional domains , 2000 .

[98]  Thomas R. Metcalf,et al.  Nonlinear Force-Free Modeling of Coronal Magnetic Fields. II. Modeling a Filament Arcade and Simulated Chromospheric and Photospheric Vector Fields , 2008 .

[99]  J. Kuhn,et al.  Coronal Magnetic Field Measurements , 2004 .

[100]  G. Roumeliotis.  The “Stress-and-Relax” Method for Reconstructing the Coronal Magnetic Field from Vector Magnetograph Data , 1996 .

[101]  L. Driel-Gesztelyi,et al.  Global budget for an eruptive active region - I. Equilibrium reconstruction approach , 2002 .

[102]  T. Sakurai.  Computational modeling of magnetic fields in solar active regions , 1989 .

[103]  M. Molodensky.  Equilibrium and stability of force-free magnetic field , 1974 .

[104]  R. Keppens,et al.  Extrapolation of a nonlinear force-free field containing a highly twisted magnetic loop , 2005 .

[105]  P. Démoulin,et al.  Basic topology of twisted magnetic configurations in solar flares , 1999 .

[106]  P. Démoulin,et al.  Removal of singularities in the Cauchy problem for the extrapolation of solar force-free magnetic fields , 1991 .