Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T05:45:45.748Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial-range kinetic turbulence in pressure-anisotropic astrophysical plasmas

Part of: Present Achievements in SpacePlasmas

Published online by Cambridge University Press:  16 July 2015

M. W. Kunz ,
A. A. Schekochihin ,
C. H. K. Chen ,
I. G. Abel  and
S. C. Cowley
M. W. Kunz*
Affiliation:
Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA
A. A. Schekochihin
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK Merton College, Merton Street, Oxford OX1 4JD, UK
C. H. K. Chen
Affiliation:
Department of Physics, Imperial College London, London SW7 2AZ, UK
I. G. Abel
Affiliation:
Princeton Center for Theoretical Science, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA
S. C. Cowley
Affiliation:
Department of Physics, Imperial College London, London SW7 2AZ, UK EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon OX14 3DB, UK
*
†Email address for correspondence: mkunz@princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical framework for low-frequency electromagnetic (drift-)kinetic turbulence in a collisionless, multi-species plasma is presented. The result generalises reduced magnetohydrodynamics (RMHD) and kinetic RMHD (Schekochihin et al., Astrophys. J. Suppl. Ser., vol. 182, 2009, pp. 310–377) to the case where the mean distribution function of the plasma is pressure-anisotropic and different ion species are allowed to drift with respect to each other – a situation routinely encountered in the solar wind and presumably ubiquitous in hot dilute astrophysical plasmas such as the intracluster medium. Two main objectives are achieved. First, in a non-Maxwellian plasma, the relationships between fluctuating fields (e.g. the Alfvén ratio) are order-unity modified compared to the more commonly considered Maxwellian case, and so a quantitative theory is developed to support quantitative measurements now possible in the solar wind. Beyond these order-unity corrections, the main physical feature of low-frequency plasma turbulence survives the generalisation to non-Maxwellian distributions: Alfvénic and compressive fluctuations are energetically decoupled, with the latter passively advected by the former; the Alfvénic cascade is fluid, satisfying RMHD equations (with the Alfvén speed modified by pressure anisotropy and species drifts), whereas the compressive cascade is kinetic and subject to collisionless damping (and for a bi-Maxwellian plasma splits into three independent collisionless cascades). Secondly, the organising principle of this turbulence is elucidated in the form of a conservation law for the appropriately generalised kinetic free energy. It is shown that non-Maxwellian features in the distribution function reduce the rate of phase mixing and the efficacy of magnetic stresses, and that these changes influence the partitioning of free energy amongst the various cascade channels. As the firehose or mirror instability thresholds are approached, the dynamics of the plasma are modified so as to reduce the energetic cost of bending magnetic-field lines or of compressing/rarefying them. Finally, it is shown that this theory can be derived as a long-wavelength limit of non-Maxwellian slab gyrokinetics.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 5 , October 2015 , 325810501
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.)

References

Abel, I. G., Plunk, G. G., Wang, E., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2013 Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows. Rep. Prog. Phys. 76 (11), 116201.Google Scholar
Alexandrova, O., Lacombe, C. & Mangeney, A. 2008 Spectra and anisotropy of magnetic fluctuations in the Earth’s magnetosheath: cluster observations. Ann. Geophys. 26, 3585–3596.Google Scholar
Antonsen, T. M. Jr & Lane, B. 1980 Kinetic equations for low frequency instabilities in inhomogeneous plasmas. Phys. Fluids 23, 1205–1214.CrossRefGoogle Scholar
Asbridge, J. R., Bame, S. J., Feldman, W. C. & Montgomery, M. D. 1976 Helium and hydrogen velocity differences in the solar wind. J. Geophys. Res. 81, 2719–2727.Google Scholar
Bale, S. D., Kasper, J. C., Howes, G. G., Quataert, E., Salem, C. & Sundkvist, D. 2009 Magnetic fluctuation power near proton temperature anisotropy instability thresholds in the solar wind. Phys. Rev. Lett. 103, 211101.CrossRefGoogle ScholarPubMed
Barnes, A. 1966 Collisionless damping of hydromagnetic waves. Phys. Fluids 9, 1483–1495.Google Scholar
Barnes, A. 1979 Hydromagnetic waves and turbulence in the solar wind. In Solar System Plasma Physics (ed. Parker, E. N., Kennel, C. F. & Lanzerotti, L. J.), vol. 1, p. 249. North-Holland.Google Scholar
Bavassano, B., Pietropaolo, E. & Bruno, R. 1998 Cross-helicity and residual energy in solar wind turbulence: radial evolution and latitudinal dependence in the region from 1 to 5 AU. J. Geophys. Res. 103, 6521.CrossRefGoogle Scholar
Bavassano, B., Pietropaolo, E. & Bruno, R. 2004 Compressive fluctuations in high-latitude solar wind. Ann. Geophys. 22, 689–696.Google Scholar
Belcher, J. W. & Davis, L. Jr 1971 Large-amplitude Alfvén waves in the interplanetary medium. Part 2. J. Geophys. Res. 76, 3534–3563.Google Scholar
Bhattacharjee, A., Ng, C. S. & Spangler, S. R. 1998 Weakly compressible magnetohydrodynamic turbulence in the solar wind and the interstellar medium. Astrophys. J. 494, 409–418.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.CrossRefGoogle Scholar
Boldyrev, S., Perez, J. C., Borovsky, J. E. & Podesta, J. J. 2011 Spectral scaling laws in magnetohydrodynamic turbulence simulations and in the solar wind. Astrophys. J. Lett. 741, L19.Google Scholar
Boldyrev, S., Perez, J. C. & Wang, Y. 2012 Residual energy in weak and strong MHD turbulence. In Numerical Modeling of Space Plasma Flows (ASTRONUM 2011) (ed. Pogorelov, N. V., Font, J. A., Audit, E. & Zank, G. P.), Astronomical Society of the Pacific Conference Series, vol. 459, p. 3. Astronomical Society of the Pacific.Google Scholar
Borovsky, J. E. 2012 The velocity and magnetic field fluctuations of the solar wind at 1 AU: statistical analysis of Fourier spectra and correlations with plasma properties. J. Geophys. Res. 117, A05104.Google Scholar
Brizard, A. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421.Google Scholar
Bruno, R., Bavassano, B. & Villante, U. 1985 Evidence for long period Alfvén waves in the inner solar system. J. Geophys. Res. 90, 4373–4377.Google Scholar
Bruno, R. & Carbone, V. 2005 The solar wind as a turbulence laboratory. Living Rev. Solar Phys. 2, 4.Google Scholar
Burlaga, L. F., Scudder, J. D., Klein, L. W. & Isenberg, P. A. 1990 Pressure-balanced structures between 1 AU and 24 AU and their implications for solar wind electrons and interstellar pickup ions. J. Geophys. Res. 95, 2229–2239.Google Scholar
Catto, P. J., Tang, W. M. & Baldwin, D. E. 1981 Generalized gyrokinetics. Plasma Phys. 23, 639–650.Google Scholar
Chandrasekhar, S., Kaufman, A. N. & Watson, K. M. 1958 The stability of the pinch. Proc. R. Soc. Lond. A 245, 435–455.Google Scholar
Chen, C. H. K., Bale, S. D., Salem, C. S. & Maruca, B. A. 2013 Residual energy spectrum of solar wind turbulence. Astrophys. J. 770, 125.Google Scholar
Chen, C. H. K., Mallet, A., Yousef, T. A., Schekochihin, A. A. & Horbury, T. S. 2011 Anisotropy of Alfvénic turbulence in the solar wind and numerical simulations. Mon. Not. R. Astron. Soc. 415, 3219–3226.CrossRefGoogle Scholar
Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273–282.Google Scholar
Daughton, W. & Gary, S. P. 1998 Electromagnetic proton/proton instabilities in the solar wind. J. Geophys. Res. 103, 20613–20620.Google Scholar
Davidson, R. C. 1983 Kinetic waves and instabilities in a uniform plasma. In Basic Plasma Physics: Selected Chapters, Handbook of Plasma Physics, vol. 1 (ed. Galeev, A. A. & Sudan, R. N.), p. 229.Google Scholar
Davidson, R. C. & Völk, H. J. 1968 Macroscopic quasilinear theory of the garden-hose instability. Phys. Fluids 11, 2259–2264.CrossRefGoogle Scholar
Dubin, D. H. E., Krommes, J. A., Oberman, C. & Lee, W. W. 1983 Nonlinear gyrokinetic equations. Phys. Fluids 26, 3524.Google Scholar
Elsasser, W. M. 1950 The hydromagnetic equations. Phys. Rev. 79, 183.Google Scholar
Fowler, T. K. 1968 Thermodynamics of unstable plasmas. Adv. Plasma Phys. 1, 201.Google Scholar
Fried, B. D. & Conte, S. D. 1961 The Plasma Dispersion Function. Academic.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
Gary, S. P. 1991 Electromagnetic ion/ion instabilities and their consequences in space plasmas: a review. Space Sci. Rev. 56, 373–415.Google Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. Part II. Strong Alfvénic turbulence. Astrophys. J. 438, 763–775.Google Scholar
Goldstein, B. E., Neugebauer, M. & Smith, E. J. 1995 Alfvén waves, alpha particles, and pickup ions in the solar wind. Geophys. Res. Lett. 22, 3389–3392.Google Scholar
Goldstein, B. E., Neugebauer, M., Zhang, L. D. & Gary, S. P. 2000 Observed constraint on proton–proton relative velocities in the solar wind. Geophys. Res. Lett. 27, 53–56.CrossRefGoogle Scholar
Grappin, R., Velli, M. & Mangeney, A. 1991 ‘Alfvénic’ versus ‘standard’ turbulence in the solar wind. Ann. Geophys. 9, 416–426.Google Scholar
Hallatschek, K. 2004 Thermodynamic potential in local turbulence simulations. Phys. Rev. Lett. 93 (12), 125001.Google Scholar
Hammett, G. W., Dorland, W. & Perkins, F. W. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4, 2052–2061.Google Scholar
Hastie, R. J., Taylor, J. B. & Haas, F. A. 1967 Adiabatic invariants and the equilibrium of magnetically trapped particles. Ann. Phys. 41, 302–338.CrossRefGoogle Scholar
Hazeltine, R. D. 1973 Recursive derivation of drift-kinetic equation. Plasma Phys. 15, 77–80.Google Scholar
Hellinger, P. 2007 Comment on the linear mirror instability near the threshold. Phys. Plasmas 14 (8), 082105.Google Scholar
Hellinger, P. & Matsumoto, H. 2000 New kinetic instability: oblique Alfvén fire hose. J. Geophys. Res. 105, 10519–10526.Google Scholar
Hellinger, P., Trávníček, P., Kasper, J. C. & Lazarus, A. J. 2006 Solar wind proton temperature anisotropy: linear theory and WIND/SWE observations. Geophys. Res. Lett. 33, 9101.CrossRefGoogle Scholar
Horbury, T. S., Forman, M. & Oughton, S. 2008 Anisotropic scaling of magnetohydrodynamic turbulence. Phys. Rev. Lett. 101 (17), 175005.Google Scholar
Horbury, T. S., Wicks, R. T. & Chen, C. H. K. 2012 Anisotropy in space plasma turbulence: solar wind observations. Space Sci. Rev. 172, 325–342.CrossRefGoogle Scholar
Howes, G. G., Bale, S. D., Klein, K. G., Chen, C. H. K., Salem, C. S. & TenBarge, J. M. 2012 The slow-mode nature of compressible wave power in solar wind turbulence. Astrophys. J. Lett. 753, L19.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
Kadomtsev, B. B. & Pogutse, O. P. 1974 Nonlinear helical perturbations of a plasma in the tokamak. Sov. J. Expl Theor. Phys. 38, 283–290.Google Scholar
Kennel, C. F. & Sagdeev, R. Z. 1967 Collisionless shock waves in high ${\it\beta}$ plasmas. Part 1. J. Geophys. Res. 72, 3303–3326.Google Scholar
Kraichnan, R. H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385–1387.Google Scholar
Krommes, J. A. 1993 Dielectric response and thermal fluctuations in gyrokinetic plasma. Phys. Fluids B 5, 1066.Google Scholar
Krommes, J. A. 1999 Thermostatted ${\it\delta}$ f. Phys. Plasmas 6, 1477–1494.Google Scholar
Krommes, J. A. 2012 The gyrokinetic description of microturbulence in magnetized plasmas. Annu. Rev. Fluid Mech. 44, 175–201.Google Scholar
Krommes, J. A. & Hu, G. 1994 The role of dissipation in the theory and simulations of homogeneous plasma turbulence, and resolution of the entropy paradox. Phys. Plasmas 1, 3211–3238.CrossRefGoogle Scholar
Kruskal, M. D. 1958 The Gyration of a Charged Particle. Project Matterhorn Publications and Reports.Google Scholar
Kulsrud, R. M. 1964 Teoria dei Plasmi (ed. Rosenbluth, M. N.), p. 54. Academic.Google Scholar
Kulsrud, R. M. 1983 MHD description of plasma. In Basic Plasma Physics: Selected Chapters, Handbook of Plasma Physics, vol. 1 (ed. Galeev, A. A. & Sudan, R. N.), p. 1. North-Holland.Google Scholar
Kunz, M. W., Schekochihin, A. A., Cowley, S. C., Binney, J. J. & Sanders, J. S. 2011 A thermally stable heating mechanism for the intracluster medium: turbulence, magnetic fields and plasma instabilities. Mon. Not. R. Astron. Soc. 410, 2446–2457.Google Scholar
Kunz, M. W., Schekochihin, A. A. & Stone, J. M. 2014 Firehose and mirror instabilities in a collisionless shearing plasma. Phys. Rev. Lett. 112 (20), 205003.Google Scholar
Landau, L. 1946 On the vibrations of the electronic plasma. Zh. Eksp. Teor. Fiz. 16, 574 (English translation: J. Phys. USSR (1946), 10, 25).Google Scholar
Lee, W. W. 1983 Gyrokinetic approach in particle simulation. Phys. Fluids 26, 556.Google Scholar
Lithwick, Y. & Goldreich, P. 2001 Compressible magnetohydrodynamic turbulence in interstellar plasmas. Astrophys. J. 562, 279–296.Google Scholar
Maksimovic, M., Pierrard, V. & Lemaire, J. F. 1997a A kinetic model of the solar wind with Kappa distribution functions in the corona. Astron. Astrophys. 324, 725–734.Google Scholar
Maksimovic, M., Pierrard, V. & Riley, P. 1997b Ulysses electron distributions fitted with Kappa functions. Geophys. Res. Lett. 24, 1151–1154.Google Scholar
Maksimovic, M., Zouganelis, I., Chaufray, J.-Y., Issautier, K., Scime, E. E., Littleton, J. E., Marsch, E., McComas, D. J., Salem, C., Lin, R. P. & Elliott, H. 2005 Radial evolution of the electron distribution functions in the fast solar wind between 0.3 and 1.5 AU. J. Geophys. Res. 110, 9104.Google Scholar
Maron, J. & Goldreich, P. 2001 Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J. 554, 1175–1196.Google Scholar
Marsch, E. 2006 Kinetic physics of the solar corona and solar wind. Living Rev. Solar Phys. 3, 1.Google Scholar
Marsch, E. & Livi, S. 1987 Observational evidence for marginal stability of solar wind ion beams. J. Geophys. Res. 92, 7263–7268.Google Scholar
Marsch, E., Rosenbauer, H., Schwenn, R., Muehlhaeuser, K.-H. & Denskat, K. U. 1981 Pronounced proton core temperature anisotropy, ion differential speed, and simultaneous Alfvén wave activity in slow solar wind at 0.3 AU. J. Geophys. Res. 86, 9199–9203.Google Scholar
Marsch, E., Rosenbauer, H., Schwenn, R., Muehlhaeuser, K.-H. & Neubauer, F. M. 1982 Solar wind helium ions: observations of the HELIOS solar probes between 0.3 and 1 AU. J. Geophys. Res. 87, 35–51.Google Scholar
Marsch, E. & Tu, C.-Y. 1990 On the radial evolution of MHD turbulence in the inner heliosphere. J. Geophys. Res. 95, 8211–8229.Google Scholar
Marsch, E. & Tu, C. Y. 1993 Correlations between the fluctuations of pressure, density, temperature and magnetic field in the solar wind. Ann. Geophys. 11, 659–677.Google Scholar
Maruca, B. A., Kasper, J. C. & Gary, S. P. 2012 Instability-driven limits on helium temperature anisotropy in the solar wind: observations and linear Vlasov analysis. Astrophys. J. 748, 137.Google Scholar
Matthaeus, W. H. & Goldstein, M. L. 1982 Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind. J. Geophys. Res. 87, 6011–6028.CrossRefGoogle Scholar
Müller, W.-C. & Grappin, R. 2005 Spectral energy dynamics in magnetohydrodynamic turbulence. Phys. Rev. Lett. 95 (11), 114502.Google Scholar
Neugebauer, M., Goldstein, B. E., Bame, S. J. & Feldman, W. C. 1994 ULYSSES near-ecliptic observations of differential flow between protons and alphas in the solar wind. J. Geophys. Res. 99, 2505–2511.Google Scholar
Numata, R., Howes, G. G., Tatsuno, T., Barnes, M. & Dorland, W. 2010 AstroGK: Astrophysical gyrokinetics code. J. Comput. Phys. 229, 9347–9372.Google Scholar
Oughton, S., Dmitruk, P. & Matthaeus, W. H. 2003 Coronal heating and reduced MHD. In Turbulence and Magnetic Fields in Astrophysics (ed. Falgarone, E. & Passot, T.), Lecture Notes in Physics, vol. 614, pp. 28–55. Springer.CrossRefGoogle Scholar
Oughton, S., Priest, E. R. & Matthaeus, W. H. 1994 The influence of a mean magnetic field on three-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 280, 95–117.Google Scholar
Parker, E. N. 1958 Dynamical instability in an anisotropic ionized gas of low density. Phys. Rev. 109, 1874–1876.CrossRefGoogle Scholar
Parra, F. I.2013 Extension of gyrokinetics to transport time scales. arXiv:1309.7385.Google Scholar
Perez, J. C. & Chandran, B. D. G. 2013 Direct numerical simulations of reflection-driven, reduced magnetohydrodynamic turbulence from the sun to the Alfvén critical point. Astrophys. J. 776, 124.Google Scholar
Perri, S. & Balogh, A. 2010 Differences in solar wind cross-helicity and residual energy during the last two solar minima. Geophys. Res. Lett. 37, 17102.Google Scholar
Podesta, J. J. 2009 Dependence of solar-wind power spectra on the direction of the local mean magnetic field. Astrophys. J. 698, 986–999.CrossRefGoogle Scholar
Podesta, J. J., Roberts, D. A. & Goldstein, M. L. 2007 Spectral exponents of kinetic and magnetic energy spectra in solar wind turbulence. Astrophys. J. 664, 543–548.Google Scholar
Pokhotelov, O. A., Treumann, R. A., Sagdeev, R. Z., Balikhin, M. A., Onishchenko, O. G., Pavlenko, V. P. & Sandberg, I. 2002 Linear theory of the mirror instability in non-Maxwellian space plasmas. J. Geophys. Res. 107, 1312.Google Scholar
Pouquet, A., Frisch, U. & Leorat, J. 1976 Strong MHD helical turbulence and the nonlinear dynamo effect. J. Fluid Mech. 77, 321–354.CrossRefGoogle Scholar
Rincon, F., Schekochihin, A. A. & Cowley, S. C. 2015 Nonlinear mirror instability. Mon. Not. R. Astron. Soc. 447, L45–L49.Google Scholar
Riquelme, M., Quataert, E. & Verscharen, D.2014 PIC simulations of continuously driven mirror and ion cyclotron instabilities in high beta astrophysical and heliospheric plasmas. arXiv:1402.0014.Google Scholar
Roberts, D. A. 1990 Heliocentric distance and temporal dependence of the interplanetary density-magnetic field magnitude correlation. J. Geophys. Res. 95, 1087–1090.Google Scholar
Roberts, D. A., Klein, L. W., Goldstein, M. L. & Matthaeus, W. H. 1987 The nature and evolution of magnetohydrodynamic fluctuations in the solar wind: Voyager observations. J. Geophys. Res. 92, 11021–11040.Google Scholar
Rosenbluth, M. N.1956 LANL Report LA-2030.Google Scholar
Rosin, M. S., Schekochihin, A. A., Rincon, F. & Cowley, S. C. 2011 A nonlinear theory of the parallel firehose and gyrothermal instabilities in a weakly collisional plasma. Mon. Not. R. Astron. Soc. 413, 7–38.Google Scholar
Salem, C., Mangeney, A., Bale, S. D. & Veltri, P. 2009 Solar wind magnetohydrodynamics turbulence: anomalous scaling and role of intermittency. Astrophys. J. 702, 537–553.Google Scholar
Sanders, J. S. & Fabian, A. C. 2013 Velocity width measurements of the coolest X-ray emitting material in the cores of clusters, groups and elliptical galaxies. Mon. Not. R. Astron. Soc. 429, 2727–2738.Google Scholar
Schekochihin, A. A. & Cowley, S. C. 2006 Turbulence, magnetic fields, and plasma physics in clusters of galaxies. Phys. Plasmas 13 (5), 056501.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Plunk, G. G., Quataert, E. & Tatsuno, T. 2008a Gyrokinetic turbulence: a nonlinear route to dissipation through phase space. Plasma Phys. Control. Fusion 50 (12), 124024.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182, 310–377.Google Scholar
Schekochihin, A. A., Cowley, S. C., Kulsrud, R. M., Hammett, G. W. & Sharma, P. 2005 Plasma instabilities and magnetic field growth in clusters of galaxies. Astrophys. J. 629, 139–142.Google Scholar
Schekochihin, A. A., Cowley, S. C., Kulsrud, R. M., Rosin, M. S. & Heinemann, T. 2008b Nonlinear growth of firehose and mirror fluctuations in astrophysical plasmas. Phys. Rev. Lett. 100 (8), 081301.Google Scholar
Schekochihin, A. A., Cowley, S. C., Rincon, F. & Rosin, M. S. 2010 Magnetofluid dynamics of magnetized cosmic plasma: firehose and gyrothermal instabilities. Mon. Not. R. Astron. Soc. 405, 291–300.Google Scholar
Scott, B. 2010 Derivation via free energy conservation constraints of gyrofluid equations with finite-gyroradius electromagnetic nonlinearities. Phys. Plasmas 17 (10), 102306.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
Snyder, P. B., Hammett, G. W. & Dorland, W. 1997 Landau fluid models of collisionless magnetohydrodynamics. Phys. Plasmas 4, 3974–3985.Google Scholar
Southwood, D. J. & Kivelson, M. G. 1993 Mirror instability. Part I. Physical mechanism of linear instability. J. Geophys. Res. 98, 9181–9187.Google Scholar
von Steiger, R., Geiss, J. & Gloeckler, G. 1997 Composition of the solar wind. In Cosmic Winds and the Heliosphere (ed. Jokipii, J. R., Sonett, C. P. & Giampapa, M. S.), p. 581.Google Scholar
Stix, T. H. 1962 The Theory of Plasma Waves. McGraw-Hill.Google Scholar
Strauss, H. R. 1976 Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19, 134–140.Google Scholar
Strauss, H. R. 1977 Dynamics of high beta Tokamaks. Phys. Fluids 20, 1354–1360.Google Scholar
Štverák, Š., Trávníček, P., Maksimovic, M., Marsch, E., Fazakerley, A. N. & Scime, E. E. 2008 Electron temperature anisotropy constraints in the solar wind. J. Geophys. Res. 113, 3103.Google Scholar
Sugama, H., Okamoto, M., Horton, W. & Wakatani, M. 1996 Transport processes and entropy production in toroidal plasmas with gyrokinetic electromagnetic turbulence. Phys. Plasmas 3, 2379–2394.Google Scholar
Summers, D. & Thorne, R. M. 1991 The modified plasma dispersion function. Phys. Fluids B 3, 1835–1847.Google Scholar
Summers, D. & Thorne, R. M. 1992 A new tool for analyzing microinstabilities in space plasmas modeled by a generalized Lorentzian (kappa) distribution. J. Geophys. Res. 97, 16827.Google Scholar
Taylor, J. B. 1967 Magnetic moment under short-wave electrostatic perturbations. Phys. Fluids 10, 1357–1359.Google Scholar
Tu, C.-Y., Marsch, E. & Thieme, K. M. 1989 Basic properties of solar wind MHD turbulence near 0.3 AU analyzed by means of Elsasser variables. J. Geophys. Res. 94, 11739–11759.Google Scholar
Vasyliunas, V. M. 1968 A survey of low-energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3. J. Geophys. Res. 73, 2839–2884.Google Scholar
Vedenov, A. A. & Sagdeev, R. Z. 1958 On some properties of a plasma with an anisotropic ion-velocity distribution in a magnetic field. Sov. Phys. Dokl. 3, 278.Google Scholar
Verscharen, D., Bourouaine, S. & Chandran, B. D. G. 2013 Instabilities driven by the drift and temperature anisotropy of alpha particles in the solar wind. Astrophys. J. 773, 163.Google Scholar
Watanabe, T.-H. & Sugama, H. 2004 Kinetic simulation of steady states of ion temperature gradient driven turbulence with weak collisionality. Phys. Plasmas 11, 1476–1483.Google Scholar
Wicks, R. T., Horbury, T. S., Chen, C. H. K. & Schekochihin, A. A. 2010 Power and spectral index anisotropy of the entire inertial range of turbulence in the fast solar wind. Mon. Not. R. Astron. Soc. 407, L31–L35.Google Scholar
Wicks, R. T., Roberts, D. A., Mallet, A., Schekochihin, A. A., Horbury, T. S. & Chen, C. H. K. 2013 Correlations at large scales and the onset of turbulence in the fast solar wind. Astrophys. J. 778, 177.Google Scholar
Yoon, P. H., Wu, C. S. & de Assis, A. S. 1993 Effect of finite ion gyroradius on the fire-hose instability in a high beta plasma. Phys. Fluids B 5, 1971–1979.Google Scholar
Zank, G. P. & Matthaeus, W. H. 1992a The equations of reduced magnetohydrodynamics. J. Plasma Phys. 48, 85.Google Scholar
Zank, G. P. & Matthaeus, W. H. 1992b Waves and turbulence in the solar wind. J. Geophys. Res. 97, 17189.Google Scholar
Zhuravleva, I., Churazov, E., Schekochihin, A. A., Allen, S. W., Arévalo, P., Fabian, A. C., Forman, W. R., Sanders, J. S., Simionescu, A., Sunyaev, R., Vikhlinin, A. & Werner, N. 2014 Turbulent heating in galaxy clusters brightest in X-rays. Nature 515, 85–87.Google Scholar