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The development of magnetic field line wander by plasma turbulence

Published online by Cambridge University Press:  22 August 2017

Gregory G. Howes Open the ORCID record for Gregory G. Howes [Opens in a new window]  and
Sofiane Bourouaine
Gregory G. Howes*
Affiliation:
Department of Physics and Astronomy, University of Iowa, Iowa City, IA 54224, USA
Sofiane Bourouaine
Affiliation:
Department of Physics and Astronomy, University of Iowa, Iowa City, IA 54224, USA
*
†Email address for correspondence: gregory-howes@uiowa.edu
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Abstract

Plasma turbulence occurs ubiquitously in space and astrophysical plasmas, mediating the nonlinear transfer of energy from large-scale electromagnetic fields and plasma flows to small scales at which the energy may be ultimately converted to plasma heat. But plasma turbulence also generically leads to a tangling of the magnetic field that threads through the plasma. The resulting wander of the magnetic field lines may significantly impact a number of important physical processes, including the propagation of cosmic rays and energetic particles, confinement in magnetic fusion devices and the fundamental processes of turbulence, magnetic reconnection and particle acceleration. The various potential impacts of magnetic field line wander are reviewed in detail, and a number of important theoretical considerations are identified that may influence the development and saturation of magnetic field line wander in astrophysical plasma turbulence. The results of nonlinear gyrokinetic simulations of kinetic Alfvén wave turbulence of sub-ion length scales are evaluated to understand the development and saturation of the turbulent magnetic energy spectrum and of the magnetic field line wander. It is found that turbulent space and astrophysical plasmas are generally expected to contain a stochastic magnetic field due to the tangling of the field by strong plasma turbulence. Future work will explore how the saturated magnetic field line wander varies as a function of the amplitude of the plasma turbulence and the ratio of the thermal to magnetic pressure, known as the plasma beta.

Keywords

astrophysical plasmasplasma nonlinear phenomenaspace plasma physics
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 4 , August 2017 , 905830408
Copyright
© Cambridge University Press 2017 

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References

Abel, I. G., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2008 Linearized model Fokker–Planck collision operators for gyrokinetic simulations. I. Theory. Phys. Plasmas 15 (12), 122509.Google Scholar
Alexandrova, O., Lacombe, C., Mangeney, A., Grappin, R. & Maksimovic, M. 2012 Solar wind turbulent spectrum at plasma kinetic scales. Astrophys. J. 760, 121.Google Scholar
Alexandrova, O., Saur, J., Lacombe, C., Mangeney, A., Mitchell, J., Schwartz, S. J. & Robert, P. 2009 Universality of solar-wind turbulent spectrum from MHD to electron scales. Phys. Rev. Lett. 103 (16), 165003.Google Scholar
Barnes, M., Abel, I. G., Dorland, W., Ernst, D. R., Hammett, G. W., Ricci, P., Rogers, B. N., Schekochihin, A. A. & Tatsuno, T. 2009 Linearized model Fokker–Planck collision operators for gyrokinetic simulations. II. Numerical implementation and tests. Phys. Plasmas 16 (7), 072107.Google Scholar
Belcher, J. W. & Davis, L. 1971 Large-amplitude Alfvén waves in the interplanetary medium, 2. J. Geophys. Res. 76, 3534–3563.Google Scholar
Bian, N. H., Kontar, E. P. & MacKinnon, A. L. 2011 Turbulent cross-field transport of non-thermal electrons in coronal loops: theory and obser vations. Astron. Astrophys. 535, A18.Google Scholar
Bieber, J. W., Wanner, W. & Matthaeus, W. H. 1996 Dominant two-dimensional solar wind turbulence with implications for cosmic ray transport. J. Geophys. Res. 101, 2511–2522.Google Scholar
Boozer, A. H. 2014 Formation of current sheets in magnetic reconnection. Phys. Plasmas 21 (7), 072907.CrossRefGoogle Scholar
Bourouaine, S., Alexandrova, O., Marsch, E. & Maksimovic, M. 2012 On spectral breaks in the power spectra of magnetic fluctuations in fast solar wind between 0.3 and 0.9 AU. Astrophys. J. 749, 102.Google Scholar
Bourouaine, S. & Howes, G. G. 2017 The development of magnetic field line wander in gyrokinetic plasma turbulence: dependence on amplitude of turbulence. J. Plasma Phys. 83 (3), 905830301.Google Scholar
Bruno, R. & Carbone, V. 2005 The solar wind as a turbulence laboratory. Living Reviews Solar Phys. 2, 4.CrossRefGoogle Scholar
del-Castillo-Negrete, D. & Blazevski, D. 2014 Heat pulse propagation in chaotic three-dimensional magnetic fields. Nucl. Fusion 54 (6), 064009.Google Scholar
del-Castillo-Negrete, D. & Blazevski, D. 2016 Modulated heat pulse propagation and partial transport barriers in chaotic magnetic fields. Phys. Plasmas 23 (4), 042505.Google Scholar
Chandran, B. D. G. 2000 Scattering of energetic particles by anisotropic magnetohydrodynamic turbulence with a Goldreich–Sridhar power spectrum. Phys. Rev. Lett. 85, 4656–4659.Google Scholar
Chandran, B. D. G. & Cowley, S. C. 1998 Thermal conduction in a tangled magnetic field. Phys. Rev. Lett. 80, 3077–3080.Google Scholar
Chandrasekhar, S. 1951 The invariant theory of isotropic turbulence in magneto-hydrodynamics. Proc. R. Soc. Lond. A 204, 435–449.Google Scholar
Chen, C. H. K., Horbury, T. S., Schekochihin, A. A., Wicks, R. T., Alexandrova, O. & Mitchell, J. 2010 Anisotropy of solar wind turbulence between ion and electron scales. Phys. Rev. Lett. 104 (25), 255002.Google Scholar
Cho, J. & Lazarian, A. 2004 The anisotropy of electron magnetohydrodynamic turbulence. Astrophys. J. Lett. 615, L41–L44.Google Scholar
Cho, J. & Lazarian, A. 2009 Simulations of electron magnetohydrodynamic turbulence. Astrophys. J. 701, 236–252.Google Scholar
Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273–282.Google Scholar
Dombre, T., Frisch, U., Henon, M., Greene, J. M. & Soward, A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353–391.Google Scholar
Drake, D. J., Schroeder, J. W. R., Howes, G. G., Kletzing, C. A., Skiff, F., Carter, T. A. & Auerbach, D. W. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. IV. Laboratory experiment. Phys. Plasmas 20 (7), 072901.Google Scholar
Eyink, G., Vishniac, E., Lalescu, C., Aluie, H., Kanov, K., Bürger, K., Burns, R., Meneveau, C. & Szalay, A. 2013 Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence. Nature 497, 466–469.Google Scholar
Filonenko, N. N., Sagdeev, R. Z. & Zaslavsky, G. M. 1967 Destruction of magnetic surfaces by magnetic field irregularities. 2. Nucl. Fusion 7 (4), 253.Google Scholar
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502–508.Google Scholar
Galeev, A. A. & Zeleny, L. M. 1981 Anomalous electron thermal conductivity across the destroyed magnetic surfaces. Physica D 2, 90–101.Google Scholar
Galtier, S., Nazarenko, S. V., Newell, A. C. & Pouquet, A. 2000 A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys. 63, 447–488.CrossRefGoogle Scholar
Giacalone, J. & Jokipii, J. R. 1999 The transport of cosmic rays across a turbulent magnetic field. Astrophys. J. 520, 204–214.Google Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence II. Strong Alfvénic turbulence. Astrophys. J. 438, 763–775.Google Scholar
Gruzinov, A. V., Isichenko, M. B. & Kalda, I. L. 1990 Two-dimensional turbulent diffusion. Sov. Phys. JETP 97, 476–488.Google Scholar
Guest, B. & Shalchi, A. 2012 Random walk of magnetic field lines in dynamical turbulence: A field line tracing method. II. Two-dimensional turbulence. Phys. Plasmas 19 (3), 032902.Google Scholar
Guo, F. & Giacalone, J. 2010 The effect of large-scale magnetic turbulence on the acceleration of electrons by perpendicular collisionless shocks. Astrophys. J. 715, 406–411.CrossRefGoogle Scholar
Guo, F. & Giacalone, J. 2013 The acceleration of thermal protons at parallel collisionless shocks: three-dimensional hybrid simulations. Astrophys. J. 773, 158.Google Scholar
Guo, F. & Giacalone, J. 2015 The acceleration of electrons at collisionless shocks moving through a turbulent magnetic field. Astrophys. J. 802, 97.Google Scholar
Haas, F. A. & Thyagaraja, A. 1986 Conceptual and experimental bases of theories of anomalous transport in Tokamaks. Phys. Rep. 143, 241–276.Google Scholar
Hatch, D. R., Pueschel, M. J., Jenko, F., Nevins, W. M., Terry, P. W. & Doerk, H. 2012 Origin of magnetic stochasticity and transport in plasma microturbulence. Phys. Rev. Lett. 108 (23), 235002.Google Scholar
Hatch, D. R., Pueschel, M. J., Jenko, F., Nevins, W. M., Terry, P. W. & Doerk, H. 2013 Magnetic stochasticity and transport due to nonlinearly excited subdominant microtearing modes. Phys. Plasmas 20 (1), 012307.Google Scholar
Hauff, T., Jenko, F., Shalchi, A. & Schlickeiser, R. 2010 Scaling theory for cross-field transport of cosmic rays in turbulent fields. Astrophys. J. 711, 997–1007.Google Scholar
Howes, G. G. 2015 A dynamical model of plasma turbulence in the solar wind. Phil. Trans. R. Soc. Lond. A 373 (2041), 20140145.Google Scholar
Howes, G. G. 2016 The dynamical generation of current sheets in astrophysical plasma turbulence. Astrophys. J. Lett. 82, L28.Google Scholar
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2006 Astrophysical gyrokinetics: basic equations and linear theory. Astrophys. J. 651, 590–614.Google Scholar
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2008a A model of turbulence in magnetized plasmas: Implications for the dissipation range in the solar wind. J. Geophys. Res. 113 (A12), A05103.Google Scholar
Howes, G. G., Dorland, W., Cowley, S. C., Hammett, G. W., Quataert, E., Schekochihin, A. A. & Tatsuno, T. 2008b Kinetic simulations of magnetized turbulence in astrophysical plasmas. Phys. Rev. Lett. 100 (6), 065004.Google Scholar
Howes, G. G., Drake, D. J., Nielson, K. D., Carter, T. A., Kletzing, C. A. & Skiff, F. 2012 Toward astrophysical turbulence in the laboratory. Phys. Rev. Lett. 109 (25), 255001.Google Scholar
Howes, G. G., Klein, K. G. & TenBarge, J. M. 2014 Validity of the Taylor hypothesis for linear kinetic waves in the weakly collisional solar wind. Astrophys. J. 789, 106.Google Scholar
Howes, G. G. & Nielson, K. D. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. I. Asymptotic solution. Phys. Plasmas 20 (7), 072302.Google Scholar
Howes, G. G., Nielson, K. D., Drake, D. J., Schroeder, J. W. R., Skiff, F., Kletzing, C. A. & Carter, T. A. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. III. Theory for experimental design. Phys. Plasmas 20 (7), 072304.Google Scholar
Howes, G. G., TenBarge, J. M. & Dorland, W. 2011 A weakened cascade model for turbulence in astrophysical plasmas. Phys. Plasmas 18, 102305.Google Scholar
Howes, G. G., TenBarge, J. M., Dorland, W., Quataert, E., Schekochihin, A. A., Numata, R. & Tatsuno, T. 2011 Gyrokinetic simulations of solar wind turbulence from ion to electron scales. Phys. Rev. Lett. 107, 035004.Google Scholar
Huang, Y.-M., Bhattacharjee, A. & Boozer, A. H. 2014 Rapid change of field line connectivity and reconnection in stochastic magnetic fields. Astrophys. J. 793, 106.Google Scholar
Isichenko, M. B. 1992 Percolation, statistical topography, and transport in random media. Rev. Mod. Phys. 64, 961–1043.Google Scholar
Jokipii, J. R. 1966 Cosmic-ray propagation. I. Charged particles in a random magnetic field. Astrophys. J. 146, 480.Google Scholar
Jokipii, J. R. & Parker, E. N. 1968 Random walk of magnetic lines of force in astrophysics. Phys. Rev. Lett. 21, 44–47.Google Scholar
Kiyani, K. H., Chapman, S. C., Khotyaintsev, Y. V., Dunlop, M. W. & Sahraoui, F. 2009 Global scale-invariant dissipation in collisionless plasma turbulence. Phys. Rev. Lett. 103, 075006.Google Scholar
Kiyani, K. H., Osman, K. T. & Chapman, S. C. 2015 Introduction: Dissipation and heating in solar wind turbulence: from the macro to the micro and back again. Phil. Trans. R. Soc. Lond. A 373, 40155.Google Scholar
Klein, K. G., Howes, G. G. & TenBarge, J. M. 2014 The violation of the taylor hypothesis in measurements of solar wind turbulence. Astrophys. J. Lett. 790, L20.Google Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16.Google Scholar
Kowal, G., Lazarian, A., Vishniac, E. T. & Otmianowska-Mazur, K. 2009 Numerical tests of fast reconnection in weakly stochastic magnetic fields. Astrophys. J. 700, 63–85.CrossRefGoogle Scholar
Kowal, G., Lazarian, A., Vishniac, E. T. & Otmianowska-Mazur, K. 2012 Reconnection studies under different types of turbulence driving. Nonlin. Process. Geophys. 19, 297–314.CrossRefGoogle Scholar
Kraichnan, R. H. 1965 Inertial range spectrum of hyromagnetic turbulence. Phys. Fluids 8, 1385–1387.Google Scholar
Krommes, J. A., Oberman, C. & Kleva, R. G. 1983 Plasma transport in stochastic magnetic fields. Part 3. Kinetics of test particle diffusion. J. Plasma Phys. 30, 11–56.Google Scholar
Laval, G. 1993 Particle diffusion in stochastic magnetic fields. Phys. Fluids B 5, 711–721.CrossRefGoogle Scholar
Lazarian, A. & Vishniac, E. T. 1999 Reconnection in a weakly stochastic field. Astrophys. J. 517, 700–718.CrossRefGoogle Scholar
Maron, J. & Goldreich, P. 2001 Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J. 554, 1175–1196.Google Scholar
Matthaeus, W. H., Gray, P. C., Pontius, D. H. Jr. & Bieber, J. W. 1995 Spatial structure and field-line diffusion in transverse magnetic turbulence. Phys. Rev. Lett. 75, 2136–2139.Google Scholar
Matthaeus, W. H., Oughton, S., Pontius, D. H. Jr. & Zhou, Y. 1994 Evolution of energy-containing turbulent eddies in the solar wind. J. Geophys. Res. 99, 19267.CrossRefGoogle Scholar
Montgomery, D. & Matthaeus, W. H. 1995 Anisotropic modal energy transfer in interstellar turbulence. Astrophys. J. 447, 706.CrossRefGoogle Scholar
Montgomery, D. & Turner, L. 1981 Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids 24, 825–831.Google Scholar
Narita, Y., Gary, S. P., Saito, S., Glassmeier, K.-H. & Motschmann, U. 2011 Dispersion relation analysis of solar wind turbulence. Geophys. Res. Lett. 38, L05101.Google Scholar
Nevins, W. M., Wang, E. & Candy, J. 2011 Magnetic stochasticity in gyrokinetic simulations of plasma microturbulence. Phys. Rev. Lett. 106 (6), 065003.Google Scholar
Ng, C. S. & Bhattacharjee, A. 1996 Interaction of shear-Alfven wave packets: implication for weak magnetohydrodynamic turbulence in astrophysical plasmas. Astrophys. J. 465, 845.Google Scholar
Nielson, K. D., Howes, G. G. & Dorland, W. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. II. Numerical solution. Phys. Plasmas 20 (7), 072303.Google Scholar
Numata, R., Howes, G. G., Tatsuno, T., Barnes, M. & Dorland, W. 2010 AstroGK: astrophysical gyrokinetics code. J. Comput. Phys. 229, 9347.Google Scholar
Politano, H. & Pouquet, A. 1998 von Kármán–Howarth equation for magnetohydrodynamics and its consequences on third-order longitudinal structure and correlation functions. Phys. Rev. E 57, 21.Google Scholar
Pueschel, M. J., Kammerer, M. & Jenko, F. 2008 Gyrokinetic turbulence simulations at high plasma beta. Phys. Plasmas 15 (10), 102310.Google Scholar
Qin, G. & Shalchi, A. 2013 The role of the Kubo number in two-component turbulence. Phys. Plasmas 20 (9), 092302.Google Scholar
Ragot, B. R. 2001 Magnetic field line wandering and shock-front acceleration: application to the radio emission of SN 1987A. Astrophys. J. 547, 1010–1023.Google Scholar
Ragot, B. R. 2011 Statistics of field-line dispersal: random-walk characterization and supradiffusive regime. Astrophys. J. 728, 50.Google Scholar
Rappazzo, A. F. & Parker, E. N. 2013 Current sheets formation in tangled coronal magnetic fields. Astrophys. J. Lett. 773, L2.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
Richardson, I. G., von Rosenvinge, T. T., Cane, H. V., Christian, E. R., Cohen, C. M. S., Labrador, A. W., Leske, R. A., Mewaldt, R. A., Wiedenbeck, M. E. & Stone, E. C. 2014 ${>}$ 25 MeV proton events observed by the high energy telescopes on the STEREO A and B spacecraft and/or at earth during the first seven years of the STEREO mission. Solar Phys. 289, 3059–3107.Google Scholar
Roberts, O. W., Li, X. & Jeska, L. 2015 A statistical study of the solar wind turbulence at ion kinetic scales using the $k$ -filtering technique and cluster data. Astrophys. J. 802, 2.Google Scholar
Roberts, O. W., Li, X. & Li, B. 2013 Kinetic plasma turbulence in the fast solar wind measured by cluster. Astrophys. J. 769, 58.Google Scholar
Robinson, D. C. & Rusbridge, M. G. 1971 Structure of turbulence in the zeta plasma. Phys. Fluids 14, 2499–2511.Google Scholar
Rosenbluth, M. N., Sagdeev, R. Z., Taylor, J. B. & Zaslavsky, G. M. 1966 Destruction of magnetic surfaces by magnetic field irregularities. Nucl. Fusion 6 (4), 297.CrossRefGoogle Scholar
Ruffolo, D. & Matthaeus, W. H. 2013 Theory of magnetic field line random walk in noisy reduced magnetohydrodynamic turbulence. Phys. Plasmas 20 (1), 012308.Google Scholar
Sahraoui, F., Goldstein, M. L., Belmont, G., Canu, P. & Rezeau, L. 2010 Three dimensional anisotropic $k$ spectra of turbulence at subproton scales in the solar wind. Phys. Rev. Lett. 105 (13), 131101.Google Scholar
Sahraoui, F., Goldstein, M. L., Robert, P. & Khotyaintsev, Y. V. 2009 Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale. Phys. Rev. Lett. 102 (23), 231102.Google Scholar
Sahraoui, F., Huang, S. Y., Belmont, G., Goldstein, M. L., Rétino, A., Robert, P. & De Patoul, J. 2013 Scaling of the electron dissipation range of solar wind turbulence. Astrophys. J. 777, 15.Google Scholar
Schlickeiser, R. 1989 Cosmic-ray transport and acceleration. I – Derivation of the kinetic equation and application to cosmic rays in static cold media. II – Cosmic rays in moving cold media with application to diffusive shock wave acceleration. Astrophys. J. 336, 243–293.Google Scholar
Schlickeiser, R. & Miller, J. A. 1998 Quasi-linear theory of cosmic-ray transport and acceleration: the role of oblique magnetohydrodynamic waves and transit-time damping. Astrophys. J. 492, 352–378.Google Scholar
Servidio, S., Matthaeus, W. H., Wan, M., Ruffolo, D., Rappazzo, A. F. & Oughton, S. 2014 Complexity and diffusion of magnetic flux surfaces in anisotropic turbulence. Astrophys. J. 785, 56.Google Scholar
Shalchi, A. 2010a A unified particle diffusion theory for cross-field scattering: subdiffusion, recovery of diffusion, and diffusion in three-dimensional turbulence. Astrophys. J. Lett. 720, L127–L130.Google Scholar
Shalchi, A. 2010b Random walk of magnetic field lines in dynamical turbulence: A field line tracing method. I. Slab turbulence. Phys. Plasmas 17 (8), 082902.Google Scholar
Shalchi, A. & Kolly, A. 2013 Analytical description of field-line random walk in Goldreich–Sridhar turbulence. Mon. Not. R. Astron. Soc. 431, 1923–1928.Google Scholar
Shalchi, A. & Kourakis, I. 2007a Analytical description of stochastic field-line wandering in magnetic turbulence. Phys. Plasmas 14 (9), 092903.CrossRefGoogle Scholar
Shalchi, A. & Kourakis, I. 2007b Random walk of magnetic field-lines for different values of the energy range spectral index. Phys. Plasmas 14 (11), 112901.Google Scholar
Shebalin, J. V., Matthaeus, W. H. & Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic field. J. Plasma Phys. 29, 525–547.Google Scholar
Similon, P. L. & Sudan, R. N. 1989 Energy dissipation of Alfven wave packets deformed by irregular magnetic fields in solar-coronal arches. Astrophys. J. 336, 442–453.Google Scholar
Spatschek, K. H. 2008 Aspects of stochastic transport in laboratory and astrophysical plasmas. Plasma Phys. Control Fusion 50 (12), 124027.Google Scholar
Sridhar, S. & Goldreich, P. 1994 Toward a theory of interstellar turbulence. 1: Weak Alfvenic turbulence. Astrophys. J. 432, 612–621.Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476–490.Google Scholar
TenBarge, J. M., Daughton, W., Karimabadi, H., Howes, G. G. & Dorland, W. 2014 Collisionless reconnection in the large guide field regime: Gyrokinetic versus particle-in-cell simulations. Phys. Plasmas 21 (2), 020708.Google Scholar
TenBarge, J. M. & Howes, G. G. 2012 Evidence of critical balance in kinetic Alfvén wave turbulence simulations. Phys. Plasmas 19 (5), 055901.CrossRefGoogle Scholar
TenBarge, J. M. & Howes, G. G. 2013 Current sheets and collisionless damping in kinetic plasma turbulence. Astrophys. J. Lett. 771, L27.Google Scholar
TenBarge, J. M., Howes, G. G. & Dorland, W. 2013 Collisionless damping at electron scales in solar wind turbulence. Astrophys. J. 774, 139.CrossRefGoogle Scholar
TenBarge, J. M., Howes, G. G., Dorland, W. & Hammett, G. W. 2014 An oscillating Langevin antenna for driving plasma turbulence simulations. Comput. Phys. Commun. 185, 578–589.Google Scholar
TenBarge, J. M., Podesta, J. J., Klein, K. G. & Howes, G. G. 2012 Interpreting magnetic variance anisotropy measurements in the solar wind. Astrophys. J. 753, 107.Google Scholar
Tu, C.-Y. & Marsch, E. 1995 MHD structures, waves and turbulence in the solar wind: Observations and theories. Space Sci. Rev. 73, 1–2.Google Scholar
Vishniac, E. T., Pillsworth, S., Eyink, G., Kowal, G., Lazarian, A. & Murray, S. 2012 Reconnection current sheet structure in a turbulent medium. Nonlin. Process. Geophys. 19, 605–610.Google Scholar
Wang, Y., Boldyrev, S. & Perez, J. C. 2011 Residual energy in magnetohydrodynamic turbulence. Astrophys. J. Lett. 740, L36.Google Scholar
Yousef, T. A., Rincon, F. & Schekochihin, A. A. 2007 Exact scaling laws and the local structure of isotropic magnetohydrodynamic turbulence. J. Fluid Mech. 575, 111.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., Basile, G. & Principato, S. 1995 Anomalous diffusion and Lévy random walk of magnetic field lines in three dimensional turbulence. Phys. Plasmas 2, 2653–2663.Google Scholar
Zimbardo, G., Veltri, P. & Pommois, P. 2000 Anomalous, quasilinear, and percolative regimes for magnetic-field-line transport in axially symmetric turbulence. Phys. Rev. E 61, 1940.Google Scholar
Zweben, S. J., Menyuk, C. R. & Taylor, R. J. 1979 Small-scale magnetic fluctuations inside the macrotor tokamak. Phys. Rev. Lett. 42, 1270–1274.Google Scholar