THE SPATIO-TEMPORAL EVOLUTION OF SOLAR FLARES OBSERVED WITH AIA/SDO: FRACTAL DIFFUSION, SUB-DIFFUSION, OR LOGISTIC GROWTH?

We explore the spatio-temporal evolution of solar flares by fitting a radial expansion model r(t) that consists of an exponentially growing acceleration phase, followed by a deceleration phase that is parameterized by the generalized diffusion function r(t)∝κ(t – t 1)β/2, which includes the logistic growth limit (β = 0), sub-diffusion (β = 0-1), classical diffusion (β = 1), super-diffusion (β = 1-2), and the linear expansion limit (β = 2). We analyze all M- and X-class flares observed with Geostationary Operational Environmental Satellite and Atmospheric Imaging Assembly/Solar Dynamics Observatory (SDO) during the first two years of the SDO mission, amounting to 155 events. We find that most flares operate in the sub-diffusive regime (β = 0.53 ± 0.27), which we interpret in terms of anisotropic chain reactions of intermittent magnetic reconnection episodes in a low plasma-β corona. We find a mean propagation speed of v = 15 ± 12 km s–1, with maximum speeds of v max = 80 ± 85 km s–1 per flare, which is substantially slower than the sonic speeds expected for thermal diffusion of flare plasmas. The diffusive characteristics established here (for the first time for solar flares) is consistent with the fractal-diffusive self-organized criticality model, which predicted diffusive transport merely based on cellular automaton simulations.

[1]  Diffusion entropy and waiting time statistics of hard-x-ray solar flares. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  J. Harvey,et al.  Characteristic size and diffusion of quiet sun magnetic patterns , 1995 .

[3]  Tang,et al.  Self-Organized Criticality: An Explanation of 1/f Noise , 2011 .

[4]  Stefano Casertano,et al.  Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution , 2004, astro-ph/0402512.

[5]  R. Hoover,et al.  Observation and Modeling of Soft X-Ray Bright Points. II. Determination of Temperature and Energy Balance , 1997 .

[6]  G. S. Vaiana,et al.  A survey of soft X-ray limb flare images: the relation between their structure in the corona and other physical parameters. , 1977 .

[7]  J. Brown,et al.  Collective plasma effects and the electron number problem in solar hard X-ray bursts , 1977 .

[8]  N. Sheeley,et al.  Magnetic Flux Transport on the Sun , 1989, Science.

[9]  L. Porter,et al.  Soft X-Ray Loops and Coronal Heating , 1995 .

[10]  Harold P. Furth,et al.  Finite‐Resistivity Instabilities of a Sheet Pinch , 1963 .

[11]  D. Nguyen,et al.  DISK BRAKING IN YOUNG STARS: PROBING ROTATION IN CHAMAELEON I AND TAURUS-AURIGA , 2009, 0902.0001.

[12]  T. Yokoyama,et al.  Single and Multiple Solar Flare Loops: Hydrodynamics and Ca XIX Resonance Line Emission , 1998 .

[13]  S. Cranmer,et al.  Hyperdiffusion as a Mechanism for Solar Coronal Heating , 2008, 0802.1751.

[14]  T. Metcalf,et al.  A Test of a New Flare Loop Scaling Law Using YOHKOH SXT and GOES Observations , 1996 .

[15]  John K. Tomfohr,et al.  Lecture Notes on Physics , 1879, Nature.

[16]  Zoran Mikic,et al.  Coronal Mass Ejection: Initiation, Magnetic Helicity, and Flux Ropes. II. Turbulent Diffusion-driven Evolution , 2003 .

[17]  J. Lawrence.  Diffusion of magnetic flux elements on a fractal geometry , 1991 .

[18]  E. Lu,et al.  Avalanches and the Distribution of Solar Flares , 1991 .

[19]  Markus J. Aschwanden,et al.  Flare Plasma Cooling from 30 MK down to 1 MK modeled from Yohkoh, GOES, and TRACE observations during the Bastille Day Event (14 July 2000) , 2001 .

[20]  Lévy flights and Lévy walks revisited , 1999 .

[21]  C. J. Wolfson,et al.  The Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (SDO) , 2011 .

[22]  Markus J. Aschwanden,et al.  Self-Organized Criticality in Astrophysics , 2011 .

[23]  Markus J. Aschwanden,et al.  AUTOMATED SOLAR FLARE STATISTICS IN SOFT X-RAYS OVER 37 YEARS OF GOES OBSERVATIONS: THE INVARIANCE OF SELF-ORGANIZED CRITICALITY DURING THREE SOLAR CYCLES , 2012, 1205.6712.

[24]  B. Kliem.  Fragmentary energy release due to tearing and coalescence in coronal current sheets , 1990 .

[25]  Strong bursts and diffusion in avalanching systems , 2007 .

[26]  T. Tajima,et al.  Loop coalescence in flares and coronal x-ray brightening , 1982 .

[27]  The evolution of reconnection along an arcade of magnetic loops , 2005, astro-ph/0504436.

[28]  Clare E. Parnell,et al.  Nanoflare Statistics from First Principles: Fractal Geometry and Temperature Synthesis , 2002 .

[29]  R. Ellis,et al.  Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae , 1998, astro-ph/9812133.

[30]  M. Phillips,et al.  Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant , 1998, astro-ph/9805201.

[31]  Robert C. Earnshaw,et al.  Brownian motion : theory, modelling and applications , 2012 .

[32]  M. Aschwanden,et al.  Self-Organized Criticality in Astrophysics: The Statistics of Nonlinear Processes in the Universe , 2011 .

[33]  T. Yokoyama,et al.  Statistical Study of the Reconnection Rate in Solar Flares Observed with Yohkoh SXT , 2006, astro-ph/0605712.

[34]  Harry P. Warren,et al.  Time Variability of the “Quiet” Sun Observed with TRACE. II. Physical Parameters, Temperature Evolution, and Energetics of Extreme-Ultraviolet Nanoflares , 2000 .

[35]  TURBULENT DIFFUSION IN THE PHOTOSPHERE AS DERIVED FROM PHOTOSPHERIC BRIGHT POINT MOTION , 2011 .

[36]  P. Sturrock.  Model of the High-Energy Phase of Solar Flares , 1966, Nature.

[37]  H. Garcia.  Reconstructing the Thermal and Spatial Form of a Solar Flare from Scaling Laws and Soft X-Ray Measurements , 1998 .

[38]  R. Leighton.  Transport of Magnetic Fields on the Sun. , 1964 .

[39]  C. Schrijver,et al.  Properties of the large- and small-scale flow patterns in and around AR 19824 , 1990 .

[40]  E. A. Jackson,et al.  Perspectives of nonlinear dynamics , 1990 .

[41]  R. Leighton.  A Magneto-Kinematic Model of the Solar Cycle , 1969 .

[42]  H. Zirin.  The rise and fall of sunspot group 18962 - A case of magnetic submergence , 1985 .

[43]  T. Tajima,et al.  Dynamic magnetic x-points , 1981 .

[44]  M. Aschwanden,et al.  Solar Flare Geometries. II. The Volume Fractal Dimension , 2008 .

[45]  V. Nakariakov,et al.  SLOW MAGNETOACOUSTIC WAVES IN TWO-RIBBON FLARES , 2011 .

[46]  CROSS-FIELD DIFFUSION OF ELECTRONS IN TANGLED MAGNETIC FIELDS AND IMPLICATIONS FOR CORONAL FINE STRUCTURE , 2006 .

[47]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[48]  E. Quataert,et al.  A MAGNETIC RECONNECTION MECHANISM FOR ION ACCELERATION AND ABUNDANCE ENHANCEMENTS IN IMPULSIVE FLARES , 2009 .

[49]  C. Cadavid,et al.  Spectral Properties of Solar Convection and Diffusion , 1996 .

[50]  Säm Krucker,et al.  Energy Distribution of Heating Processes in the Quiet Solar Corona , 1998 .

[51]  Diffusion dominated dynamics of avalanching systems , 2008 .

[52]  B. Low.  Resistive diffusion of force-free magnetic fields in a passive medium. III. Acceleration of flare particles , 1974 .

[53]  Huaning Wang,et al.  A logistic model for magnetic energy storage in solar active regions , 2009 .

[54]  M. Aschwanden,et al.  Solar Flare Geometries. I. The Area Fractal Dimension , 2008 .

[55]  Y. Litvinenko.  REGULAR VERSUS DIFFUSIVE PHOTOSPHERIC FLUX CANCELLATION , 2011 .

[56]  B. Dennis,et al.  THE RELATIONSHIP BETWEEN HARD X-RAY PULSE TIMINGS AND THE LOCATIONS OF FOOTPOINT SOURCES DURING SOLAR FLARES , 2012, 1303.6309.

[57]  Brian R. Dennis,et al.  Logistic Avalanche Processes, Elementary Time Structures, and Frequency Distributions in Solar Flares , 1998 .

[58]  Andrew R. Jones,et al.  Dynamics of the Chromospheric Network: Mobility, Dispersal, and Diffusion Coefficients , 1996 .

[59]  B. Kliem.  Coupled magnetohydrodynamic and kinetic development of current sheets in the solar corona , 1995 .

[60]  M. Aschwanden.  Do EUV Nanoflares Account for Coronal Heating? , 1999 .

[61]  R. Brown.  XXVII. A brief account of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies , 1828 .

[62]  S. Tsuneta,et al.  Scaling Law of Solar Coronal Loops Obtained with YOHKOH , 1995 .

[63]  S. Wu,et al.  Magnetic diffusion and flare energy buildup , 1992 .

[64]  M. Aschwanden.  A statistical fractal-diffusive avalanche model of a slowly-driven self-organized criticality system , 2011, 1112.4859.

[65]  Leon Golub,et al.  Dynamic Responses to Magnetic Reconnection in Solar Arcades , 1998 .

[66]  A. Einstein.  Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [AdP 17, 549 (1905)] , 2005, Annalen der Physik.

[67]  M. Hesse,et al.  The Diffusion Region in Collisionless Magnetic Reconnection , 1999 .

[68]  Edward J. Beltrami,et al.  Mathematics for Dynamic Modeling , 1987 .

[69]  Takeo Kosugi,et al.  Electron Time-of-Flight Distances and Flare Loop Geometries Compared from CGRO and YOHKOH Observations , 1996 .