Hostname: page-component-76d6cb85b7-f97m6 Total loading time: 0 Render date: 2026-07-14T17:46:01.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Superdiffusive transport in laboratory and astrophysical plasmas

Part of: Complex Plasma Phenomena

Published online by Cambridge University Press:  01 October 2015

G. Zimbardo ,
E. Amato ,
A. Bovet Open the ORCID record for A. Bovet [Opens in a new window] ,
F. Effenberger ,
A. Fasoli ,
H. Fichtner ,
I. Furno ,
K. Gustafson ,
P. Ricci  and
S. Perri
G. Zimbardo*
Affiliation:
Department of Physics, University of Calabria, Ponte P. Bucci, Cubo 31C, I-87036 Rende, Italy
E. Amato
Affiliation:
INAF, Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy
A. Bovet
Affiliation:
Ecole Polytechnique Federale Lausanne, EPFL, Lausanne, CH-1015, Switzerland
F. Effenberger
Affiliation:
Department of Physics and KIPAC, Stanford University, Stanford, CA 94305, USA Department of Mathematics, University of Waikato, P.B. 3105, Hamilton, New Zealand
A. Fasoli
Affiliation:
Ecole Polytechnique Federale Lausanne, EPFL, Lausanne, CH-1015, Switzerland
H. Fichtner
Affiliation:
Institut für Theoretische Physik IV, Ruhr-Universität Bochum, Universitätsstrasse 150, 44780 Bochum, Germany
I. Furno
Affiliation:
Ecole Polytechnique Federale Lausanne, EPFL, Lausanne, CH-1015, Switzerland
K. Gustafson
Affiliation:
Ecole Polytechnique Federale Lausanne, EPFL, Lausanne, CH-1015, Switzerland
P. Ricci
Affiliation:
Ecole Polytechnique Federale Lausanne, EPFL, Lausanne, CH-1015, Switzerland
S. Perri
Affiliation:
Department of Physics, University of Calabria, Ponte P. Bucci, Cubo 31C, I-87036 Rende, Italy
*
†Email address for correspondence: gaetano.zimbardo@fis.unical.it
Get access Rights & Permissions [Opens in a new window]

Abstract

In the last few years it has been demonstrated, both by data analysis and by numerical simulations, that the transport of energetic particles in the presence of magnetic turbulence can be superdiffusive rather than normal diffusive (Gaussian). The term ‘superdiffusive’ refers to the mean square displacement of particle positions growing superlinearly with time, as compared to the normal linear growth. The so-called anomalous transport, which in general comprises both subdiffusion and superdiffusion, has gained growing attention during the last two decades in many fields including laboratory plasma physics, and recently in astrophysics and space physics. Here we show a number of examples, both from laboratory and from astrophysical plasmas, where superdiffusive transport has been identified, with a focus on what could be the main influence of superdiffusion on fundamental processes like diffusive shock acceleration and heliospheric energetic particle propagation. For laboratory plasmas, superdiffusion appears to be due to the presence of electrostatic turbulence which creates long-range correlations and convoluted structures in perpendicular transport: this corresponds to a similar phenomenon in the propagation of solar energetic particles (SEPs) which leads to SEP dropouts. For the propagation of energetic particles accelerated at interplanetary shocks in the solar wind, parallel superdiffusion seems to be prevailing; this is based on a pitch-angle scattering process different from that envisaged by quasi-linear theory, and this emphasizes the importance of nonlinear interactions and trapping effects. In the case of supernova remnant shocks, parallel superdiffusion is possible at quasi-parallel shocks, as occurring in the interplanetary space, and perpendicular superdiffusion is possible at quasi-perpendicular shocks, as corresponding to Richardson diffusion: therefore, cosmic ray acceleration at supernova remnant shocks should be formulated in terms of superdiffusion. The possible relations among anomalous transport in laboratory, heliospheric, and astrophysical plasmas will be indicated.

Information

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 6 , December 2015 , 495810601
Copyright
© Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abdo, A. A., Ackermann, M., Ajello, M., Baldini, L., Ballet, J., Barbiellini, G., Baring, M. G., Bastieri, D., Baughman, B. M., Bechtol, K. et al. , Fermi Collaboration 2010a Gamma-ray emission from the shell of supernova remnant W44 revealed by the Fermi LAT. Science 327, 1103.Google Scholar
Abdo, A. A., Ackermann, M., Ajello, M., Baldini, L., Ballet, J., Barbiellini, G., Bastieri, D., Baughman, B. M., Bechtol, K., Bellazzini, R. et al. , Fermi Collaboration 2010b Observation of supernova remnant IC 443 with the Fermi large area telescope. Astrophys. J. 712, 459.Google Scholar
Amato, E. 2014 The origin of galactic cosmic rays. Intl J. Mod. Phys. D 23, 1430013.Google Scholar
Amato, E. & Blasi, P. 2005 A general solution to non-linear particle acceleration at non-relativistic shock waves. Mon. Not. R. Astron. Soc. Lett. 364, L76–L80.Google Scholar
Arthur, A. D. & le Roux, J. A. 2013 Particle acceleration at the heliospheric termination shock with a stochastic shock obliquity approach. Astrophys. J. Lett. 772, L26.Google Scholar
Balogh, A., Bykov, A., Lin, R. P., Raymond, J. & Scholer, M.(Eds) 2013 Particle Acceleration in Cosmic Plasmas, Space Science Series of ISSI. Springer.Google Scholar
Bell, A. R. 1978 The acceleration of cosmic rays in shock fronts. I. Mon. Not. R. Astron. Soc. 182, 147.Google Scholar
Blasi, P. 2013 The origin of galactic cosmic rays. Astron. Astrophys. Rev. 21, 70.Google Scholar
Blumen, A., Klafter, J. & Zumofen, G. 1990 A stochastic approach to enhanced diffusion: Lévy walks. Europhys. Lett. 13, 223–229.Google Scholar
Bouchaud, J. P. & Georges, A. 1990 Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293.Google Scholar
Bovet, A., Fasoli, A. & Furno, I. 2014a Time-resolved measurements of suprathermal ion transport induced by intermittent plasma blob filaments. Phys. Rev. Lett. 113, 225001.Google Scholar
Bovet, A., Fasoli, A., Ricci, P., Furno, I. & Gustafson, K. 2015 Nondiffusive transport regimes for suprathermal ions in turbulent plasmas. Phys. Rev. E 91, 041101(R).Google Scholar
Bovet, A., Furno, I., Fasoli, A., Gustafson, K. & Ricci, P. 2012 Investigation of fast ion transport in TORPEX. Nucl. Fusion 52, 094017.Google Scholar
Bovet, A., Furno, I., Fasoli, A., Gustafson, K. & Ricci, P. 2013 Three-dimensional measurements of non-diffusive fast ion transport in TORPEX. Plasma Phys. Control. Fusion 55, 124021.Google Scholar
Bovet, A., Gamarino, M., Furno, I., Ricci, P., Fasoli, A., Gustafson, K., Newman, D. E. & Sánchez, R. 2014b Transport equation describing fractional Lévy motion of suprathermal ions in TORPEX. Nucl. Fusion 54, 104009.Google Scholar
Burresi, M. et al. 2012 Weak localization of light in superdiffusive random systems. Phys. Rev. Lett. 108, 110604.Google Scholar
Bykov, A. M., Ellison, D. C. & Renaud, M. 2012 Magnetic fields in cosmic particle acceleration sources. Space Sci. Rev. 166, 71.Google Scholar
Carreras, B. A., Lynch, V. E. & Zaslavsky, G. M. 2001 Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model. Phys. Plasmas 8, 5096.Google Scholar
del-Castillo-Negrete, D., Carreras, B. A. & Lynch, V. E. 2004 Fractional diffusion in plasma turbulence. Phys. Plasmas 11, 3854–3864.Google Scholar
Chukbar, K. V. 1995 Stochastic transport and fractional derivatives. Zh. Eksp. Teor. Fiz. 108, 1875.Google Scholar
Decker, R.-B., Krimigis, S. M., Roelof, E. C., Hill, M. E., Armstrong, T. P., Gloeckler, G., Hamilton, D. C. & Lanzerotti, L. J. 2008 Mediation of the solar wind termination shock by non-thermal ions. Nature 454, 67–70.Google Scholar
Dendy, R. O., Chapman, S. C. & Paczuski, M. 2007 Fusion, space, and solar plasmas as complex systems. Plasma Phys. Control. Fusion 49, A95.Google Scholar
Dewhurst, J. M., Hnat, B. & Dendy, R. O. 2010 Finite Larmor radius effects on test particle transport in drift wave–zonal flow turbulence. Plasma Phys. Control. Fusion 52, 025004.Google Scholar
Drury, L. O’C. 1983 An introduction to the theory of diffusive shock acceleration of energetic particles in tenuous plasmas. Rep. Prog. Phys. 46, 973–1027.Google Scholar
Duffy, P., Kirk, J.-G., Gallant, Y.-A. & Dendy, R. O. 1995 Anomalous transport and particle acceleration at shocks. Astron. Astrophys. 302, L21.Google Scholar
Effenberger, F. 2014 Energetic particle transport with stochastic differential equations: general methods and the extension to anomalous diffusion regimes. In 8th International Conference of Numerical Modeling of Space Plasma Flows (ASTRONUM 2013) (ed. Pogorelov, N.i V., Audit, E. & Zank, G. P.), ASP Conference Series, vol. 488, p. 201. Astronomical Society of the Pacific.Google Scholar
Fasoli, A., Avino, F., Bovet, A., Furno, I., Gustafson, K., Jolliet, S., Loizu, J., Malinverni, D., Ricci, P., Riva, F. et al. 2013 Basic investigations of electrostatic turbulence and its interaction with plasma and suprathermal ions in a simple magnetized toroidal plasma. Nucl. Fusion 53, 063013.Google Scholar
Florinski, V., Decker, R. B., le Roux, J. A. & Zank, G. P. 2009 An energetic-particle-mediated termination shock observed by Voyager 2. Geophys. Res. Lett. 36, L12101.Google Scholar
Furno, I., Spolaore, M., Theiler, C., Vianello, N., Cavazzana, R. & Fasoli, A. 2011 Direct two-dimensional measurements of the field-aligned current associated with plasma blobs. Phys. Rev. Lett. 106, 245001.Google Scholar
Gardiner, C. W. 2009 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer.Google Scholar
Geisel, T., Nierwetberg, J. & Zacherl, A. 1985 Accelerated diffusion in Josephson junctions and related chaotic systems. Phys. Rev. Lett. 54, 616–619.Google Scholar
Giacalone, J. 2013 Cosmic-ray transport and interaction with shocks. Space Sci. Rev. 176, 73.Google Scholar
Giacalone, J., Jokipii, R. & Mazur, J. E. 2000 Small-scale gradients and large-scale diffusion of charged particles in the heliospheric magnetic field. Astrophys. J. Lett. 532, L75–L78.Google Scholar
Giuliani, A., Cardillo, M., Tavani, M., Fukui, Y., Yoshiike, S., Torii, K., Dubner, G., Castelletti, G., Barbiellini, G., Bulgarelli, A. et al. , AGILE Collaboration 2011 Neutral pion emission from accelerated protons in the supernova remnant W44. Astrophys. J. Lett. 742, 30.Google Scholar
Green, D. A. 2009 A revised Galactic supernova remnant catalogue. Bull. Astron. Soc. India 37, 45.Google Scholar
Gustafson, K. & Ricci, P. 2012 Lévy walk description of suprathermal ion transport. Phys. Plasmas 19, 032304.Google Scholar
Gustafson, K., Ricci, P., Bovet, A., Furno, I. & Fasoli, A. 2012a Suprathermal ion transport in simple magnetized torus configurations. Phys. Plasmas 19, 062306.Google Scholar
Gustafson, K., Ricci, P., Furno, I. & Fasoli, A. 2012b Nondiffusive suprathermal ion transport in simple magnetized toroidal plasmas. Phys. Rev. Lett. 108, 035006.Google Scholar
Helder, E. A., Vink, J., Bykov, A. M., Ohira, Y., Raymond, J. C. & Terrier, R. 2012 Observational signatures of particle acceleration in supernova remnants. Space Sci. Rev. 173, 369–431.Google Scholar
Isichenko, M. B. 1992 Percolation, statistical topography and transport in random media. Rev. Mod. Phys. 64, 961.Google Scholar
Kirk, J. G., Duffy, P. & Gallant, Y. A. 1996 Stochastic particle acceleration at shocks in the presence of braided magnetic fields. Astron. Astrophys. 314, 1010.Google Scholar
Klafter, J., Blumen, A. & Shlesinger, M. F. 1987 Stochastic pathway to anomalous diffusion. Phys. Rev. A 35, 3081–3085.Google Scholar
Klafter, J. & Sokolov, I. M. 2005 Anomalous diffusion spreads its wings. Phys. World (August 2005), 29.Google Scholar
Krommes, J. A., Oberman, C. & Kleva, R. B. 1983 Plasma transport in stochastic magnetic fields. Part 3. Kinetics of test particle diffusion. J. Plasma Phys. 30, 11.Google Scholar
Kusumi, A., Nakada, C., Ritchie, K., Murase, K., Suzuki, K., Murakoshi, H., Kasai, R. S., Kondo, J. & Fujiwara, T. 2005 Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: high-speed single-molecule tracking of membrane molecules. Annu. Rev. Biophys. Biomol. Struct. 34, 351.Google Scholar
Lazarian, A. & Yan, H. 2014 Superdiffusion of cosmic rays: implications for cosmic ray acceleration. Astrophys. J. 784, 38.Google Scholar
Lee, M. A. & Fisk, L. A. 1982 Shock acceleration of energetic particles in the heliosphere. Space Sci. Rev. 32, 205.Google Scholar
Lee, M. A., Mewaldt, R. A. & Giacalone, J. 2012 Shock acceleration of ions in the heliosphere. Space Sci. Rev. 173, 247.Google Scholar
Litvinenko, Y. E. & Effenberger, F. 2014 Analytical solutions of a fractional diffusion–advection equation for solar cosmic-ray transport. Astrophys. J. 796, 125.Google Scholar
Longair, M. S. 1994 High Energy Astrophysics, 2nd edn., vol. 2, pp. 229–251. Cambridge University Press.Google Scholar
Magdziarz, M. & Weron, A. 2007 Competition between subdiffusion and Lévy flights: a Monte Carlo approach. Phys. Rev. E 75 (5), 056702.Google Scholar
Mazur, J. E., Mason, G. M., Dwyer, J. R., Giacalone, J., Jokipii, J. R. & Stone, E. C. 2000 Interplanetary magnetic field line mixing deduced from impulsive solar flare particles. Astrophys. J. Lett. 532, L79–L82.Google Scholar
Metzler, R. & Klafter, J. 2000 The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77.Google Scholar
Metzler, R. & Klafter, J. 2004 Topical review: the restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37, R161–R208.Google Scholar
Mier, J. A., Sánchez, R., Garcia, L., Carreras, B. A. & Newman, D. E. 2008 Characterization of nondiffusive transport in plasma turbulence via a novel Lagrangian method. Phys Rev. Lett. 101, 165001,1–4.Google Scholar
Morlino, G., Amato, E., Blasi, P. & Caprioli, D. 2010 Spatial structure of X-ray filaments in SN 1006. Mon. Not. R. Astron. Soc. Lett. 405, L21–L25.Google Scholar
Onić, D. 2013 On the supernova remnants with flat radio spectra. Astrophys. Space Sci. 346, 3–13.Google Scholar
Orlando, S., Bocchino, F., Reale, F., Peres, G. & Petruk, O. 2007 On the origin of asymmetries in bilateral supernova remnants. Astron. Astrophys. 470, 927.Google Scholar
Perri, S. & Balogh, A. 2010 Stationarity in solar wind flows. Astrophys. J. 714, 937–943.Google Scholar
Perri, S., Yordanova, E., Carbone, V., Veltri, P., Sorriso-Valvo, L., Bruno, R. & Andre, M. 2009 Magnetic turbulence in space plasmas: scale-dependent effects of anisotropy. J. Geophys. Res. 114, A02102.Google Scholar
Perri, S. & Zimbardo, G. 2007 Evidence of superdiffusive transport of electrons accelerated at interplanetary shocks. Astrophys. J. Lett. 671, 177–180.Google Scholar
Perri, S. & Zimbardo, G. 2008 Superdiffusive transport of electrons accelerated at corotating interaction regions. J. Geophys. Res. 113, A03107.Google Scholar
Perri, S. & Zimbardo, G. 2009a Ion and electron superdiffusive transport in the interplanetary space. Adv. Space Res. 44, 465–470.Google Scholar
Perri, S. & Zimbardo, G. 2009b Ion superdiffusion at the solar wind termination shock. Astrophys. J. Lett. 693, L118–L121.Google Scholar
Perri, S. & Zimbardo, G. 2012a Superdiffusive shock acceleration. Astrophys. J. 750, 87.Google Scholar
Perri, S. & Zimbardo, G. 2012b Magnetic variances and pitch-angle scattering times upstream of interplanetary shocks. Astrophys. J. 754, 8.Google Scholar
Perri, S., Zimbardo, G., Effenberger, F. & Fichtner, H. 2015 Parameter estimation of superdiffusive motion of energetic particles upstream of heliospheric shocks. Astron. Astrophys. 578, A2.Google Scholar
Perri, S., Zimbardo, G. & Greco, A. 2011 On the energization of protons interacting with 3-D time-dependent electromagnetic fields in the Earth’s magnetotail. J. Geophys. Res. 116, A05221.Google Scholar
Perrone, D., Dendy, R. O., Furno, I., Sanchez, R., Zimbardo, G., Bovet, A., Fasoli, A., Gustafson, K., Perri, S., Ricci, P. & Valentini, F. 2013 Nonclassical transport and particle–field coupling: from laboratory plasma to the solar wind. Space Sci. Rev. 178, 233–270.Google Scholar
Pommois, P., Zimbardo, G. & Veltri, P. 2007 Anomalous, non-Gaussian transport of charged particles in anisotropic magnetic turbulence. Phys. Plasmas 14, 012311,1–11.Google Scholar
Ragot, B. R. & Kirk, J. G. 1997 Anomalous transport of cosmic ray electrons. Astron. Astrophys. 327, 432.Google Scholar
Rechester, A. B. & Rosenbluth, M. N. 1978 Electron heat transport in a Tokamak with destroyed magnetic surfaces. Phys. Rev. Lett. 40, 38–41.Google Scholar
Reynolds, S. P. 2008 Supernova remnants at high energy. Annu. Rev. Astron. Astrophys. 46, 89.Google Scholar
Reynolds, S. P., Gaensler, B. M. & Bocchino, F. 2012 Magnetic fields in supernova remnants and pulsar-wind nebulae. Space Sci. Rev. 166, 231–261.Google Scholar
Ricci, P., Theiler, C., Fasoli, A., Furno, I., Gustafson, K., Iraji, D. & Loizu, J. 2011 Methodology for turbulence code validation: quantification of simulation–experiment agreement and application to the TORPEX experiment. Phys. Plasmas 18, 032109.Google Scholar
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance–neighbour graph. Proc. R. Soc. Lond. A 110, 709.Google Scholar
Richardson, J., Kasper, J. C., Wang, C., Belcher, J. W. & Lazarus, A. J. 2008 Cool heliosheath plasma and deceleration of the upstream solar wind at the termination shock. Nature 454, 63.Google Scholar
Ritchie, K., Shan, X. Y., Kondo, J., Iwasawa, K., Fujiwara, T. & Kusumi, A. 2005 Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. Biophys. J. 88, 2266–2277.Google Scholar
Saichev, A. I. & Zaslavsky, G. M. 1997 Fractional kinetic equations: solutions and applications. Chaos 7, 753–764.Google Scholar
Samko, S. S. G., Kilbas, A. A. A. & Marichev, O. O. I. 1993 Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach.Google Scholar
Sanchez, R., Newman, D. E., Leboeuf, J.-N., Decyk, V. K. & Carreras, B. A. 2008 Nature of transport across sheared zonal flows in electrostatic ion-temperature-gradient gyrokinetic plasma turbulence. Phys. Rev. Lett. 101, 205002.Google Scholar
Shalchi, A. A. 2010 Unified particle diffusion theory for cross-field scattering: subdiffusion, recovery of diffusion, and diffusion in three-dimensional turbulence. Astrophys. J. 720, L127–L130.Google Scholar
Shalchi, A. & Kourakis, I. 2007 A new theory for perpendicular transport of cosmic rays. Astron. Astrophys. 470, 405–409.Google Scholar
Shklyar, D. R. & Zimbardo, G. 2014 Particle dynamics in the field of two waves in a magnetoplasma. Plasma Phys. Control. Fusion 56, 095002.Google Scholar
Shlesinger, M. F., West, B. J. & Klafter, J. 1987 Levy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58, 1100.Google Scholar
Stern, R., Effenberger, F., Fichtner, H. & Schäfer, T. 2014 The space-fractional diffusion–advection equation: analytical solutions and critical assessment of numerical solutions. Fract. Calc. Appl. Anal. 17 (1), 171–190.Google Scholar
Sugiyama, T. & Shiota, D. 2011 Sign for super-diffusive transport of energetic ions associated with a coronal-mass-ejection-driven interplanetary shock. Astrophys. J. 731, L34–L37.Google Scholar
Tautz, R. C. 2010 Simulation results on the influence of magneto-hydrodynamic waves on cosmic ray particles. Plasma Phys. Control. Fusion 52, 045016.Google Scholar
Tavani, M., Giuliani, A., Chen, A. W., Argan, A., Barbiellini, G., Bulgarelli, A., Caraveo, P., Cattaneo, P. W., Cocco, V., Contessi, T. et al. , AGILE Collaboration 2010 Direct evidence for hadronic cosmic-ray acceleration in the supernova remnant IC 443. Astrophys. J. Lett. 710, 151.Google Scholar
Theiler, C., Furno, I., Ricci, P., Fasoli, A., Labit, B., Muller, S. & Plyushchev, G. 2009 Cross-field motion of plasma blobs in an open magnetic field line configuration. Phys. Rev. Lett. 103, 65001.Google Scholar
Trenchi, L., Bruno, R., Telloni, D., D’Amicis, R., Marcucci, M. F., Zurbuchen, T. H. & Weberg, M. 2013 Solar energetic particle modulations associated with coherent magnetic structures. Astrophys. J. 770, 11.Google Scholar
Trotta, E. M. & Zimbardo, G. 2011 Quasi-ballistic and superdiffusive transport for impulsive solar particle events. Astron. Astrophys. 530, A130.Google Scholar
Vainio, R. & Schlickeiser, R. 1999 Self-consistent Alfvén-wave transmission and test-particle acceleration at parallel shocks. Astron. Astrophys. 343, 303–311.Google Scholar
Vink, J. 2012 Supernova remnants: the X-ray perspective. Astron. Astrophys. Rev. 20, 49.Google Scholar
Webb, G. M., Zank, G. P., Kaghashvili, E. Kh. & le Roux, J. A. 2006 Compound and perpendicular diffusion of cosmic rays and random walk of the field lines. I. Parallel particle transport models. Astrophys. J. 651, 211.Google Scholar
Zaslavsky, G. M. 2002 Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580.Google Scholar
Zimbardo, G. 2005 Anomalous particle diffusion and Lévy random walk of magnetic field lines in three-dimensional solar wind turbulence. Plasma Phys. Control. Fusion 47, B755–B767.Google Scholar
Zimbardo, G., Greco, A., Sorriso-Valvo, L., Perri, S., Vörös, Z., Aburjania, G., Chargazia, K. & Alexandrova, O. 2010 Magnetic turbulence in the geospace environment. Space Sci. Rev. 156, 89.Google Scholar
Zimbardo, G., Greco, A. & Veltri, P. 2000a Superballistic transport in tearing driven magnetic turbulence. Phys. Plasmas 7, 1071–1074.Google Scholar
Zimbardo, G. & Perri, S. 2013 From Lévy walks to superdiffusive shock acceleration. Astrophys. J. 778, 35.Google Scholar
Zimbardo, G., Perri, S., Pommois, P. & Veltri, P. 2012 Anomalous particle transport in the heliosphere. Adv. Space Res. 49, 1633.Google Scholar
Zimbardo, G., Pommois, P. & Veltri, P. 2004 Magnetic flux tube evolution in solar wind anisotropic magnetic turbulence. J. Geophys. Res. 109, A02113.Google Scholar
Zimbardo, G., Pommois, P. & Veltri, P. 2006 Superdiffusive and subdiffusive transport of energetic particles in solar wind anisotropic magnetic turbulence. Astrophys. J. Lett. 639, L91–L94.Google Scholar
Zimbardo, G., Veltri, P. & Pommois, P. 2000b Anomalous, quasilinear, and percolative regimes for magnetic-field-line transport in axially symmetric turbulence. Phys. Rev. E 61, 1940–1948.Google Scholar
Zumofen, G. & Klafter, J. 1993 Scale-invariant motion in intermittent chaotic systems. Phys. Rev. E 47, 851–863.Google Scholar