Space Science Reviews

, Volume 198, Issue 1–4, pp 217–266 | Cite as

25 Years of Self-organized Criticality: Numerical Detection Methods

  • R. T. James McAteer
  • Markus J. Aschwanden
  • Michaila Dimitropoulou
  • Manolis K. Georgoulis
  • Gunnar Pruessner
  • Laura Morales
  • Jack Ireland
  • Valentyna Abramenko
Article

Abstract

The detection and characterization of self-organized criticality (SOC), in both real and simulated data, has undergone many significant revisions over the past 25 years. The explosive advances in the many numerical methods available for detecting, discriminating, and ultimately testing, SOC have played a critical role in developing our understanding of how systems experience and exhibit SOC. In this article, methods of detecting SOC are reviewed; from correlations to complexity to critical quantities. A description of the basic autocorrelation method leads into a detailed analysis of application-oriented methods developed in the last 25 years. In the second half of this manuscript space-based, time-based and spatial-temporal methods are reviewed and the prevalence of power laws in nature is described, with an emphasis on event detection and characterization. The search for numerical methods to clearly and unambiguously detect SOC in data often leads us outside the comfort zone of our own disciplines—the answers to these questions are often obtained by studying the advances made in other fields of study. In addition, numerical detection methods often provide the optimum link between simulations and experiments in scientific research. We seek to explore this boundary where the rubber meets the road, to review this expanding field of research of numerical detection of SOC systems over the past 25 years, and to iterate forwards so as to provide some foresight and guidance into developing breakthroughs in this subject over the next quarter of a century.

Keywords

Self organized criticality Numerical methods 

Notes

Acknowledgements

The author team acknowledges the hospitality and partial support for two workshops on Self-Organized Criticality and Turbulence at the International Space Science Institute (ISSI) at Bern, Switzerland, during October 15–19, 2012, and September 16–20, 2013. One of us (JMA) was partially supported by a National Science Foundation Career award, NSF AGS-1255024, and NASA contracts NNH12CG10C and NNX13AE03G. One of us (MJA) was partially supported by NASA contract NNX11A099G and NASA contract NG04EA00C of the SDO/AIA instrument to LMSAL. One of us (MKG) was partially supported by EU FP7 grant PIRG07-GA-2010-268245. One of us (VIA) was partially supported by NASA LWS NNX11AO73G grant and by the Program of the Presidium of Russian Academy of Sciences No. 21. The authors acknowledge the comprehensive and dedicated work of an anonymous referee.

References

  1. V.I. Abramenko, Sol. Phys. 228, 29 (2005a) ADSCrossRefGoogle Scholar
  2. V.I. Abramenko, Astrophys. J. 629, 1141 (2005b) ADSCrossRefGoogle Scholar
  3. V.I. Abramenko, V.B. Yurchyshyn, Astrophys. J. 722, 122 (2010) ADSCrossRefGoogle Scholar
  4. V.I. Abramenko, V.B. Yurchyshyn, H. Wang, T.J. Spirock, P.R. Goode, Astrophys. J. 577, 487 (2002) ADSCrossRefGoogle Scholar
  5. V.I. Abramenko, V.B. Yurchyshyn, H. Wang, T.J. Spirock, P.R. Goode, Astrophys. J. 597, 1135 (2003) ADSCrossRefGoogle Scholar
  6. T.W. Anderson, The Statistical Analysis of Time Series (Wiley, New York, 1971) MATHGoogle Scholar
  7. M.J. Aschwanden, Self-organized Criticality in Astrophysics. The Statistics of Nonlinear Processes in the Universe (Springer, New York, 2011) MATHCrossRefGoogle Scholar
  8. M.J. Aschwanden, Astrophys. J. 757, 94 (2012) ADSCrossRefGoogle Scholar
  9. M.J. Aschwanden, Astrophys. J. 782, 54 (2014) ADSCrossRefGoogle Scholar
  10. M.J. Aschwanden, S.R. Freeland, Astrophys. J. 754, 112 (2012) ADSCrossRefGoogle Scholar
  11. M.J. Aschwanden, J.M. McTiernan, Astrophys. J. 717, 683 (2010) ADSCrossRefGoogle Scholar
  12. M.J. Aschwanden, C.E. Parnell, Astrophys. J. 572, 1048 (2002) ADSCrossRefGoogle Scholar
  13. M.J. Aschwanden, T. Shimizu, Astrophys. J. 776, 132 (2013) ADSCrossRefGoogle Scholar
  14. M.J. Aschwanden, R.W. Nightingale, T.T. Tarbell, C.J. Wolfson, Astrophys. J. 535, 1027 (2000a) ADSCrossRefGoogle Scholar
  15. M.J. Aschwanden, T.T. Tarbell, R.W. Nightingale, C.J. Schrijver, A. Title, C.C. Kankelborg, P. Martens, H.P. Warren, Astrophys. J. 535, 1047 (2000b) ADSCrossRefGoogle Scholar
  16. M.J. Aschwanden, J. Zhang, K. Liu, Astrophys. J. 775, 23 (2013) ADSCrossRefGoogle Scholar
  17. M.J. Aschwanden, N.B. Crosby, M. Dimitropoulou, M.K. Georgoulis, S. Hergarten, R.T.J. McAteer, A.V. Milovanov, S. Mineshige, L. Morales, N. Nishizuka, G. Pruessner, R. Sanchez et al., Space Sci. Rev. (2014, this issue). doi: 10.1007/s11214-014-0054-6, arXiv:1403.6528
  18. F. Auchère, K. Bocchialini, J. Solomon, E. Tison, Astron. Astrophys. 563, A8 (2014) ADSCrossRefGoogle Scholar
  19. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987) ADSMathSciNetCrossRefGoogle Scholar
  20. A.-L. Barabási, H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995) MATHCrossRefGoogle Scholar
  21. M.N. Barber, in Phase Transitions and Critical Phenomena, vol. 8, ed. by C. Domb, J.L. Lebowitz (Academic Press, New York, 1983), pp. 145–266 Google Scholar
  22. G.I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics (Cambridge University Press, Cambridge, 1996) MATHCrossRefGoogle Scholar
  23. J.-L. Barrat, J.-P. Hansen, Basic Concepts for Simple and Complex Liquids (Cambridge University Press, Cambridge, 2003) CrossRefGoogle Scholar
  24. M. Batty, P. Longley, S. Fotheringham, Environ. Plan. A 21, 1447–1472 (1989) CrossRefGoogle Scholar
  25. A. Bemporad, W.H. Matthaeus, G. Poletto, Astrophys. J. Lett. 677, L137 (2008) ADSCrossRefGoogle Scholar
  26. A.O. Benz, S. Krucker, Astrophys. J. 568, 413 (2002) ADSCrossRefGoogle Scholar
  27. J. Berges, N. Tetradis, C. Wetterich, Phys. Rep. 363(4–6), 223 (2002) ADSMathSciNetMATHCrossRefGoogle Scholar
  28. D.A. Biesecker Ph.D. thesis, University of New Hampsire, 1994 Google Scholar
  29. K. Binder, Z. Phys. B 43, 119 (1981) ADSCrossRefGoogle Scholar
  30. G. Boffetta, V. Carbone, P. Giuliani, P. Veltri, A. Vulpiani, Phys. Rev. Lett. 83, 4662 (1999) ADSCrossRefGoogle Scholar
  31. J.A. Bonachela, M. Muñoz, Physica A 384, 89 (2007) ADSCrossRefGoogle Scholar
  32. E. Buchlin, S. Galtier, M. Velli, Astron. Astrophys. 436, 355 (2005) ADSCrossRefGoogle Scholar
  33. E. Buchlin, J.C. Vial, P. Lemaire, Astron. Astrophys. 451, 1091 (2006) ADSCrossRefGoogle Scholar
  34. P. Charbonneau, S.W. McIntosh, H.L. Liu, T.J. Bogdan, Sol. Phys. 203, 321 (2001) ADSCrossRefGoogle Scholar
  35. C. Chatfield, The Analysis of Time Series an Introduction (Chapman & Hall, London, 1996) MATHGoogle Scholar
  36. K. Christensen, Z. Olami, J. Geophys. Res. 97, 8729 (1992a) ADSCrossRefGoogle Scholar
  37. K. Christensen, Z. Olami, J. Geophys. Res. 97(B6), 8729 (1992b) ADSCrossRefGoogle Scholar
  38. K. Christensen, N. Farid, G. Pruessner, M. Stapleton, Eur. Phys. J. B 62(3), 331 (2008) ADSCrossRefGoogle Scholar
  39. A. Clauset, C.R. Shalizi, M.E.J. Newman, SIAM Rev. 51(4), 661 (2009) ADSMathSciNetMATHCrossRefGoogle Scholar
  40. P.A. Conlon, P.T. Gallagher, R.T.J. McAteer, J. Ireland, C.A. Young, P. Kestener, R.J. Hewett, K. Maguire, Sol. Phys. 248, 297 (2008) ADSCrossRefGoogle Scholar
  41. P.A. Conlon, R.T.J. McAteer, P.T. Gallagher, L. Fennell, Astrophys. J. 722, 577 (2010) ADSCrossRefGoogle Scholar
  42. G. Consolini, E. Berrilli, E. Pietropaolo, R. Bruno, V. Carbone, B. Bavassano, G. Ceppatelli, in Magnetic Fields and Solar Processes. ESA, vol. SP-448 (ESA, Paris, 1999), p. 209 Google Scholar
  43. J.C. Crocker, D.G. Grier, J. Colloid Interface Sci. 179, 298 (1996) CrossRefGoogle Scholar
  44. N.B. Crosby Ph.D. thesis, Universite Paris VII, Paris, 1996 Google Scholar
  45. C.E. DeForest, H.J. Hagenaar, D.A. Lamb, C.E. Parnell, B.T. Welsch, Astrophys. J. 666, 576 (2007) ADSCrossRefGoogle Scholar
  46. R.O. Dendy, P. Helander, M. Tagger, Astron. Astrophys. 337, 962 (1998) ADSGoogle Scholar
  47. D. Dhar (1999), arXiv:cond-mat/9909009
  48. M. Dimitropoulou, H. Isliker, L. Vlahos, M.K. Georgoulis, Astron. Astrophys. 529, 101 (2011) ADSCrossRefGoogle Scholar
  49. M. Dimitropoulou, H. Isliker, L. Vlahos, M.K. Georgoulis, Astron. Astrophys. 553, 65 (2013) ADSCrossRefGoogle Scholar
  50. B. Drossel, F. Schwabl, Phys. Rev. Lett. 69, 1629 (1992) ADSCrossRefGoogle Scholar
  51. S.F. Edwards, D.R. Wilkinson, Proc. R. Soc. A 381(1780), 17 (1982) ADSMathSciNetCrossRefGoogle Scholar
  52. B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans (SIAM, Philadelphia, 1982) CrossRefMATHGoogle Scholar
  53. K.J. Falconer, Fractal Geometry (Wiley, London, 2003) MATHCrossRefGoogle Scholar
  54. R.C. Falconer, R.E. Woods, Digital Image Processing (Pearson Education, Upper Saddle River, 2008) Google Scholar
  55. F. Family, T. Vicsek, J. Phys. A, Math. Gen. 18(2), L75 (1985) ADSCrossRefGoogle Scholar
  56. J. Feder, Fractals (Plenum, New York, 1988) MATHCrossRefGoogle Scholar
  57. U. Frisch, Turbulence, the Legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, 1995) MATHGoogle Scholar
  58. M.K. Georgoulis, Sol. Phys. 276, 161 (2012) ADSCrossRefGoogle Scholar
  59. M.K. Georgoulis Ph.D. thesis, Aristotelian University, Thessaloniki, Greece, 2000 Google Scholar
  60. M.K. Georgoulis, L. Vlahos, Astrophys. J. 469, L135 (1996) ADSCrossRefGoogle Scholar
  61. M.K. Georgoulis, L. Vlahos, Astrophys. J. 336, 721 (1998) ADSGoogle Scholar
  62. M.K. Georgoulis, D.M. Rust, P.N. Bernasconi, B. Schmieder, Astrophys. J. 575, 506 (2002) ADSCrossRefGoogle Scholar
  63. G. Grinstein, in Scale Invariance, Interfaces, and Non-equilibrium Dynamics, ed. by A. McKane, M. Droz, J. Vannimenus, D. Wolf (Plenum, New York, 1995), pp. 261–293. See also in NATO Advanced Study Institute on Scale Invariance, Interfaces, and Non-Equilibrium Dynamics, Cambridge, UK, Jun 20–30 (1994) CrossRefGoogle Scholar
  64. G.R. Gupta, Astron. Astrophys. 568, A96 (2014) ADSCrossRefGoogle Scholar
  65. D. Hamon, M. Nicodemi, H.J. Jensen, Astron. Astrophys. 387, 326 (2002) ADSCrossRefGoogle Scholar
  66. S. Havlin, A. Bunde, Fractals and Disordered Systems (Springer, Berlin, 1996) MATHGoogle Scholar
  67. R.J. Hewett, P.T. Gallagher, R.T.J. McAteer, Sol. Phys. 248, 311C (2008) ADSCrossRefGoogle Scholar
  68. C. Holm, W. Janke, Phys. Rev. B 48(2), 936 (1993) ADSCrossRefGoogle Scholar
  69. T. Hwa, M. Kardar, Phys. Rev. Lett. 62(16), 1813 (1989) ADSCrossRefGoogle Scholar
  70. J. Ireland, R.T.J. McAteer, A.R. Inglis, Astrophys. J. 798, 1 (2015) ADSCrossRefGoogle Scholar
  71. Ž. Ivezić, S. Tabachnik, R. Rafikov, R.H. Lupton, T. Quinn, M. Hammergren, L. Eyer, J. Chu, J.C. Armstrong, X. Fan, K. Finlator, T.R. Geballe et al., Astron. J. 122, 2749 (2001) ADSCrossRefGoogle Scholar
  72. L.P. Kadanoff, Phys. Today 44, 9 (1991) CrossRefGoogle Scholar
  73. L.P. Kadanoff, S.R. Nagel, L. Wu, S.-M. Zhou, Phys. Rev. A 39, 6524 (1989) ADSCrossRefGoogle Scholar
  74. P. Kestener, P.A. Conlon, A. Khalil, L. Fennell, R.T.J. McAteer, P.T. Gallagher, A. Arneodo, Astrophys. J. 717, 995 (2010) ADSCrossRefGoogle Scholar
  75. M.S. Kirk, K.S. Balasubramaniam, J. Jackiewicz, R.T.J. McAteer, R.O. Milligan, Astrophys. J. 750, 145 (2012) ADSCrossRefGoogle Scholar
  76. M.S. Kirk, K.S. Balasubramaniam, J. Jackiewicz, B.J. McNamara, R.T.J. McAteer, Sol. Phys. 283, 97 (2013) ADSCrossRefGoogle Scholar
  77. A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941) ADSGoogle Scholar
  78. A.N. Kolmogorov, in Mecanique de la turbulence. Coll. Intern. du CNRS a Marseille, vol. 447 (1962a) Google Scholar
  79. A.N. Kolmogorov, J. Fluid Mech. 13, 82 (1962b) ADSMathSciNetMATHCrossRefGoogle Scholar
  80. S. Krucker, A.O. Benz, Astrophys. J. 501, L213 (1998) ADSCrossRefGoogle Scholar
  81. J. Krug, Adv. Phys. 46(2), 139 (1997) ADSMathSciNetCrossRefGoogle Scholar
  82. D.A. Lamb, C.E. DeForest, H.J. Hagenaar, C.E. Parnell, B.T. Welsch, Astrophys. J. 674, 520 (2008) ADSCrossRefGoogle Scholar
  83. D.A. Lamb, C.E. DeForest, H.J. Hagenaar, C.E. Parnell, B.T. Welsch, Astrophys. J. 720, 1405 (2010) ADSCrossRefGoogle Scholar
  84. J.K. Lawrence, A.C. Cadavid, A.A. Ruzmaikin, Astrophys. J. 417, 805 (1993) ADSCrossRefGoogle Scholar
  85. S. Lise, J. Phys. A, Math. Gen. 35(22), 4641 (2002) ADSMathSciNetMATHCrossRefGoogle Scholar
  86. W. Liu, L. Morales, P. Charbonneau, V. Uritsky, J. Geophys. Res. 116, A03213 (2010) ADSGoogle Scholar
  87. J.M. López, Phys. Rev. Lett. 83(22), 4594 (1999) ADSCrossRefGoogle Scholar
  88. E.T. Lu, R.J. Hamilton, Astrophys. J. 380, L89 (1991) ADSCrossRefGoogle Scholar
  89. E.T. Lu, R.J. Hamilton, J.M. McTiernan, K.R. Bromund, Astrophys. J. 412, 841 (1993) ADSCrossRefGoogle Scholar
  90. S. Lübeck, Int. J. Mod. Phys. B 18(31/32), 3977 (2004) ADSCrossRefGoogle Scholar
  91. S. Lübeck, P.C. Heger, Phys. Rev. E 68(5), 056102 (2003) (11 pp.) ADSCrossRefGoogle Scholar
  92. A.T.Y. Lui, K. Liou, P.T. Newell, C.I. Meng, S. Ohtani, R.W. McEntire, N.J. Fox, R.P. Lepping, W.R. Paterson, J.B. Sigwarth, L.A. Frank, S. Kokubun et al., Geophys. Res. Lett. 27, 1831 (2000) ADSCrossRefGoogle Scholar
  93. B.B. Mandelbrot, Les objets fractals, forme, hasard et dimension (Flammarion, Paris, 1975) MATHGoogle Scholar
  94. S.S. Manna, J. Phys. A, Math. Gen. 24(7), L363 (1991) ADSMathSciNetCrossRefGoogle Scholar
  95. McAteer, R.T.J.: In: Aschawnden, M. (ed.) SOC and Fractal Geometry in Self Organized Criticality Systems, vol. 3, p. 73 (2013). http://ojs.antek666.website.pl/SOC3.pdf Google Scholar
  96. R.T.J. McAteer, Sol. Phys. (2015, in press) Google Scholar
  97. R.T.J. McAteer, Low frequency oscillations of the solar atmosphere. Ph.D. thesis, Queen’s University Belfast, 2003 Google Scholar
  98. R.T.J. McAteer, D.S. Bloomfield, Astrophys. J. 776, 66 (2013) ADSCrossRefGoogle Scholar
  99. R.T.J. McAteer, P.T. Gallagher, D.R. Williams, M. Mathioudakis, K.J.H. Phillips, F.P. Keenan, Astrophys. J. 567, L165 (2002) ADSCrossRefGoogle Scholar
  100. R.T.J. McAteer, P.T. Gallagher, D.R. Williams, M. Mathioudakis, D.S. Bloomfield, K.J.H. Phillips, F.P. Keenan, Astrophys. J. 587, 806 (2003) ADSCrossRefGoogle Scholar
  101. R.T.J. McAteer, P.T. Gallagher, J. Ireland, Astrophys. J. 631, 628 (2005) ADSCrossRefGoogle Scholar
  102. R.T.J. McAteer, C.A. Young, J. Ireland, P.T. Gallagher, Astrophys. J. 662, 691 (2007) ADSCrossRefGoogle Scholar
  103. R.T.J. McAteer, P.T. Gallagher, P.A. Conlon, Adv. Space Res. 45, 1067 (2010) ADSCrossRefGoogle Scholar
  104. S.W. McIntosh, P. Charbonneau, Astrophys. J. Lett. 563, L165 (2001) ADSCrossRefGoogle Scholar
  105. A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics, vol. 2 (MIT Press, Cambridge, 1975) Google Scholar
  106. L. Morales, P. Charbonneau, Geophys. Res. Lett. 35, 4108 (2008a) ADSCrossRefGoogle Scholar
  107. L. Morales, P. Charbonneau, Astrophys. J. 682, 654 (2008b) ADSCrossRefGoogle Scholar
  108. M. Muñoz, R. Dickman, A. Vespignani, S. Zapperi, Phys. Rev. E 59(5), 6175 (1999) ADSCrossRefGoogle Scholar
  109. J.V. Neumann, The Theory of Self-reproducing Automata (University of Illinois Press, Champaign, 1966) Google Scholar
  110. M. Newman, Contemp. Phys. 46, 323 (2005a) ADSCrossRefGoogle Scholar
  111. M. Newman, Contemp. Phys. 46(5), 323 (2005b). doi: 10.1080/00107510500052444 ADSCrossRefGoogle Scholar
  112. M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, New York, 1999) MATHGoogle Scholar
  113. D. Newman, B.A. Carreras, P.H. Diamond, T.S. Hahm, Phys. Plasmas 3, 1858 (1996) ADSCrossRefGoogle Scholar
  114. Z. Olami, H.J.S. Feder, K. Christensen, Phys. Rev. Lett. 68, 1244 (1992) ADSCrossRefGoogle Scholar
  115. M. Paczuski, S. Boettcher, Phys. Rev. Lett. 77(1), 111 (1996) ADSCrossRefGoogle Scholar
  116. C.E. Parnell, Mon. Not. R. Astron. Soc. 335, 389 (2002) ADSCrossRefGoogle Scholar
  117. C.E. Parnell, P.E. Jupp, Astrophys. J. 529, 554 (2000) ADSCrossRefGoogle Scholar
  118. C.E. Parnell, C.E. DeForest, H.J. Hagenaar, B.A. Johnston, D.A. Lamb, B.T. Welsch, Astrophys. J. 698, 75 (2009) ADSCrossRefGoogle Scholar
  119. G. Pearce, A.K. Rowe, J. Yeung, Astrophys. Space Sci. 208, 99 (1993) ADSCrossRefGoogle Scholar
  120. J. Perrin, Atoms (Constable, London, 1920). Translated by D. Ll. Hammick Google Scholar
  121. L. Pickering, G. Pruessner, K. Christensen, Avalanche size moments in soc models are linear in the driving (2015, in preparation) Google Scholar
  122. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, 2nd edn. (Cambridge University Press, New York, 1992) MATHGoogle Scholar
  123. G. Pruessner, Phys. Rev. E 67(3), 030301(R) (2003) (4 pp.) ADSCrossRefGoogle Scholar
  124. G. Pruessner, Phys. Rev. Lett. 92(24), 246101 (2004a) (4 pp.) ADSCrossRefGoogle Scholar
  125. G. Pruessner, J. Phys. A, Math. Gen. 37(30), 7455 (2004b) ADSMathSciNetMATHCrossRefGoogle Scholar
  126. G. Pruessner, New J. Phys. 10(11), 113003 (2008) (13 pp.) ADSCrossRefGoogle Scholar
  127. G. Pruessner, Self-organised Criticality (Cambridge University Press, Cambridge, 2012) CrossRefGoogle Scholar
  128. G. Pruessner, Int. J. Mod. Phys. B 27(5), 1350009 (2013) ADSMathSciNetCrossRefGoogle Scholar
  129. G. Pruessner, The field theory of self-organised criticality (2015, in preparation) Google Scholar
  130. K.P. Reardon, F. Lepreti, V. Carbone, A. Vecchio, Astrophys. J. Lett. 683, L207 (2008) ADSCrossRefGoogle Scholar
  131. R. Sanchez, D.E. Newman, B.A. Carreras, Phys. Rev. Lett. 88, 6 (2002) Google Scholar
  132. W. Schottky, Ann. Phys. 362(23), 541 (1918). ISSN: 1521-3889 CrossRefGoogle Scholar
  133. R.R. Schroeder, Fractals, Chaos, Power Laws (Freeman, New York, 2000) MATHGoogle Scholar
  134. A. Sokolov, A. Melatos, T. Kieu, R. Webster, Physica A 428, 295 (2015). doi: 10.1016/j.physa.2015.02.001 ADSCrossRefGoogle Scholar
  135. E. Spada, V. Carbone, R. Cavazzana, L. Fattorini, G. Regnoli, N. Vianello, V. Antoni, E. Martines, G. Serianni, M. Spolaore, L. Tramontin, Phys. Rev. Lett. 86, 3032 (2001) ADSCrossRefGoogle Scholar
  136. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971) Google Scholar
  137. J. Starck, F. Murtagh, Astronomical Image and Data Analysis (Springer, Berlin, 2006) CrossRefMATHGoogle Scholar
  138. G. Stolovitzky, K. Sreenivasan, Phys. Rev. E 48, R33 (1992) ADSCrossRefGoogle Scholar
  139. W.A. Strauss, Partial Differential Equations (Wiley, Chichester, 2007) Google Scholar
  140. D.L. Turcotte, R.F. Smalley Jn, S.A. Solla, Nature 313, 604 (1985) CrossRefGoogle Scholar
  141. V.M. Uritsky, A.J. Klimas, D. Vassiliadis, D. Chua, G. Parks, J. Geophys. Res. Space Phys. 107, 1426 (2002) ADSCrossRefGoogle Scholar
  142. V.M. Uritsky, M. Paczuski, J. Davilla, S.I. Jones, Phys. Rev. Lett. 99, 025001 (2007) ADSCrossRefGoogle Scholar
  143. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1992). Third impression 2001, enlarged and revised MATHGoogle Scholar
  144. S. Vaughan, Mon. Not. R. Astron. Soc. 402, 307 (2010) ADSCrossRefGoogle Scholar
  145. M.S. Wheatland, Astrophys. J. Lett. 536, L109 (2000) ADSCrossRefGoogle Scholar
  146. M.S. Wheatland, Y.E. Litvinenko, Sol. Phys. 211, 255 (2002) ADSCrossRefGoogle Scholar
  147. M.S. Wheatland, P.A. Sturrock, J.M. McTiernan, Astrophys. J. 509, 448 (1998) ADSCrossRefGoogle Scholar
  148. E.P. White, B.J. Enquist, J.L. Green, Ecology 89(4), 905 (2008) CrossRefGoogle Scholar
  149. T. Wiegelmann, J. Geophys. Res. 113, A03S02 (2008) ADSGoogle Scholar
  150. H.A. Zebker, E.A. Marouf, G.L. Tyler, Icarus 64, 531 (1985) ADSCrossRefGoogle Scholar
  151. A.V.D. Ziel, Physica 16(4), 359 (1950). ISSN: 0031-8914 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • R. T. James McAteer
    • 1
  • Markus J. Aschwanden
    • 2
  • Michaila Dimitropoulou
    • 3
  • Manolis K. Georgoulis
    • 4
  • Gunnar Pruessner
    • 5
  • Laura Morales
    • 6
  • Jack Ireland
    • 7
  • Valentyna Abramenko
    • 8
    • 9
  1. 1.Solar Physics and Space Weather, Department of AstronomyNew Mexico State UniversityLas CrucesUSA
  2. 2.Lockheed Martin, Solar and Astrophysics Laboratory (LMSAL)STAR LabsPalo AltoUSA
  3. 3.Dept. PhysicsKapodistrian University of AthensAthensGreece
  4. 4.Research Center Astronomy and Applied MathematicsAcademy of AthensAthensGreece
  5. 5.Dept. MathematicsImperial College LondonLondonUnited Kingdom
  6. 6.Departamento de Física, Facultad de Ciencias Exactas y Naturales, Instituto de Física Plasmas (CONICET)Universidad de Buenos AiresBuenos AiresArgentina
  7. 7.ADNET Systems, Inc.NASA Goddard Space Flight CenterGreenbeltUSA
  8. 8.Space Weather Prediction Laboratory, Department of Solar PhysicsCentral Astronomical Observatory of Russian Academy of Science at PulkovoSt. PetersburgRussia
  9. 9.Big Bear Solar Observatory of NJITBig Bear CityUSA

Personalised recommendations