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Astrophysical gyrokinetics: turbulence in pressure-anisotropic plasmas at ion scales and beyond

Part of: Frontiers in Plasma Physics Conference Focus on Plasma Astrophysics

Published online by Cambridge University Press:  12 April 2018

M. W. Kunz Open the ORCID record for M. W. Kunz [Opens in a new window] ,
I. G. Abel Open the ORCID record for I. G. Abel [Opens in a new window] ,
K. G. Klein Open the ORCID record for K. G. Klein [Opens in a new window]  and
A. A. Schekochihin
M. W. Kunz*
Affiliation:
Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA Princeton Plasma Physics Laboratory, PO Box 451, Princeton, NJ 08543, USA
I. G. Abel
Affiliation:
Princeton Center for Theoretical Science, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA Chalmers University of Technology, 41296 Gothenburg, Sweden
K. G. Klein
Affiliation:
CLASP, University of Michigan, Space Research Building, Ann Arbor, MI 48109, USA Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, 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
*
†Email address for correspondence: [email protected]
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Abstract

We present a theoretical framework for describing electromagnetic kinetic turbulence in a multi-species, magnetized, pressure-anisotropic plasma. The turbulent fluctuations are assumed to be small compared to the mean field, to be spatially anisotropic with respect to it and to have frequencies small compared to the ion cyclotron frequency. At scales above the ion-Larmor radius, the theory reduces to the pressure-anisotropic generalization of kinetic reduced magnetohydrodynamics (KRMHD) formulated by Kunz et al. (J. Plasma Phys., vol. 81, 2015, 325810501). At scales at and below the ion-Larmor radius, three main objectives are achieved. First, we analyse the linear response of the pressure-anisotropic gyrokinetic system, and show it to be a generalization of previously explored limits. The effects of pressure anisotropy on the stability and collisionless damping of Alfvénic and compressive fluctuations are highlighted, with attention paid to the spectral location and width of the frequency jump that occurs as Alfvén waves transition into kinetic Alfvén waves. Secondly, we derive and discuss a very general gyrokinetic free-energy conservation law, which captures both the KRMHD free-energy conservation at long wavelengths and dual cascades of kinetic Alfvén waves and ion entropy at sub-ion-Larmor scales. We show that non-Maxwellian features in the distribution function change the amount of phase mixing and the efficiency of magnetic stresses, and thus influence the partitioning of free energy amongst the cascade channels. Thirdly, a simple model is used to show that pressure anisotropy, even within the bounds imposed on it by firehose and mirror instabilities, can cause order-of-magnitude variations in the ion-to-electron heating ratio due to the dissipation of Alfvénic turbulence. Our theory provides a foundation for determining how pressure anisotropy affects turbulent fluctuation spectra, the differential heating of particle species and the ratio of parallel and perpendicular phase mixing in space and astrophysical plasmas.

Keywords

astrophysical plasmas
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 2 , April 2018 , 715840201
Copyright
© Cambridge University Press 2018 

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References

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
Antonsen, T. M. Jr & Lane, B. 1980 Kinetic equations for low frequency instabilities in inhomogeneous plasmas. Phys. Fluids 23, 1205–1214.CrossRefGoogle Scholar
Armstrong, J. W., Coles, W. A., Rickett, B. J. & Kojima, M. 1990 Observations of field-aligned density fluctuations in the inner solar wind. Astrophys. J. 358, 685–692.CrossRefGoogle Scholar
Arzamasskiy, L., Kunz, M. W., Chandran, B. D. G. & Quataert, E.2018 In preparation.Google Scholar
Bale, S. D., Kellogg, P. J., Mozer, F. S., Horbury, T. S. & Reme, H. 2005 Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 94 (21), 215002.Google Scholar
Barnes, A. 1966 Collisionless damping of hydromagnetic waves. Phys. Fluids 9, 1483–1495.CrossRefGoogle Scholar
Bavassano, B., Pietropaolo, E. & Bruno, R. 2004 Compressive fluctuations in high-latitude solar wind. Ann. Geophys. 22, 689–696.CrossRefGoogle Scholar
Belcher, J. W. & Davis, L. Jr 1971 Large-amplitude Alfvén waves in the interplanetary medium, 2. J. Geophys. Res. 76, 3534–3563.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
Boldyrev, S. 2006 Spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 96 (11), 115002.Google Scholar
Boldyrev, S. & Perez, J. C. 2012 Spectrum of kinetic-Alfvén turbulence. Astrophys. J. Lett. 758, L44.CrossRefGoogle Scholar
Brizard, A. J. 1994 Quadratic free energy for the linearized gyrokinetic Vlasov–Maxwell equations. Phys. Plasmas 1, 2473–2479.CrossRefGoogle Scholar
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421–468.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. 1978 Linearized gyro-kinetics. Plasma Phys. 20, 719–722.Google Scholar
Catto, P. J., Tang, W. M. & Baldwin, D. E. 1981 Generalized gyrokinetics. Plasma Phys. 23, 639–650.Google Scholar
Cerri, S. S., Califano, F., Jenko, F., Told, D. & Rincon, F. 2016 Subproton-scale cascades in solar wind turbulence: driven hybrid-kinetic simulations. Astrophys. J. Lett. 822, L12.Google Scholar
Cerri, S. S., Franci, L., Califano, F., Landi, S. & Hellinger, P. 2017a Plasma turbulence at ion scales: a comparison between particle in cell and Eulerian hybrid-kinetic approaches. J. Plasma Phys. 83 (2), 705830202.Google Scholar
Cerri, S. S., Servidio, S. & Califano, F. 2017b Kinetic cascade in solar-wind turbulence: 3D3V hybrid-kinetic simulations with electron inertia. Astrophys. J. Lett. 846, L18.Google Scholar
Chandra, M., Gammie, C. F., Foucart, F. & Quataert, E. 2015 An extended magnetohydrodynamics model for relativistic weakly collisional plasmas. Astrophys. J. 810, 162.CrossRefGoogle Scholar
Chandran, B. D. G., Li, B., Rogers, B. N., Quataert, E. & Germaschewski, K. 2010 Perpendicular ion heating by low-frequency Alfvén-wave turbulence in the solar wind. Astrophys. J. Lett. 720, 503–515.CrossRefGoogle Scholar
Chandran, B. D. G., Schekochihin, A. A. & Mallet, A. 2015 Intermittency and alignment in strong RMHD turbulence. Astrophys. J. 807, 39.Google Scholar
Chandran, B. D. G., Verscharen, D., Quataert, E., Kasper, J. C., Isenberg, P. A. & Bourouaine, S. 2013 Stochastic heating, differential flow, and the alpha-to-proton temperature ratio in the solar wind. Astrophys. J. 776, 45.CrossRefGoogle Scholar
Chen, C. H. K. 2016 Recent progress in astrophysical plasma turbulence from solar wind observations. J. Plasma Phys. 82 (6), 535820602.Google Scholar
Chen, C. H. K., Leung, L., Boldyrev, S., Maruca, B. A. & Bale, S. D. 2014 Ion-scale spectral break of solar wind turbulence at high and low beta. Geophys. Res. Lett. 41, 8081–8088.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.Google Scholar
Chew, G. F., Goldberger, M. L. & Low, F. E. 1956 The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. R. Soc. Lond. A 236, 112–118.Google Scholar
Cho, J. & Lazarian, A. 2004 The anisotropy of electron magnetohydrodynamic turbulence. Astrophys. J. 615, L41.Google Scholar
Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273–282.Google Scholar
Cranmer, S. R. 2014 Ensemble simulations of proton heating in the solar wind via turbulence and ion cyclotron resonance. Astrophys. J. Suppl. 213, 16.Google Scholar
Davidson, R. C. & Völk, H. J. 1968 Macroscopic quasilinear theory of the garden-hose instability. Phys. Fluids 11, 2259–2264.Google Scholar
Dmitruk, P., Matthaeus, W. H. & Seenu, N. 2004 Test particle energization by current sheets and nonuniform fields in magnetohydrodynamic turbulence. Astrophys. J. 617, 667–679.CrossRefGoogle Scholar
Dubin, D. H. E., Krommes, J. A., Oberman, C. & Lee, W. W. 1983 Nonlinear gyrokinetic equations. Phys. Fluids 26, 3524.Google Scholar
Feldman, W. C., Asbridge, J. R., Bame, S. J. & Montgomery, M. D. 1973 Double ion streams in the solar wind. J. Geophys. Res. 78, 2017.Google Scholar
Foucart, F., Chandra, M., Gammie, C. F. & Quataert, E. 2016 Evolution of accretion discs around a kerr black hole using extended magnetohydrodynamics. Mon. Not. R. Astron. Soc. 456, 1332–1345.Google Scholar
Fowler, T. K. 1968 Thermodynamics of Unstable Plasmas. Adv. Plasma Phys. 1, 201.Google Scholar
Franci, L., Landi, S., Matteini, L., Verdini, A. & Hellinger, P. 2015a High-resolution hybrid simulations of kinetic plasma turbulence at proton scales. Astrophys. J. 812, 21.Google Scholar
Franci, L., Landi, S., Matteini, L., Verdini, A. & Hellinger, P. 2016 Plasma beta dependence of the ion-scale spectral break of solar wind turbulence: high-resolution 2D hybrid simulations. Astrophys. J. 833, 91.Google Scholar
Franci, L., Landi, S., Verdini, A., Mattini, L. & Hellinger, P. 2018 Solar wind turbulent cascade from MHD to sub-ion scales: large-size 3D hybrid particle-in-cell simulations. Astrophys. J. 853, 26.Google Scholar
Franci, L., Verdini, A., Matteini, L., Landi, S. & Hellinger, P. 2015b Solar wind turbulence from MHD to sub-ion scales: high-resolution hybrid simulations. Astrophys. J. 804, L39.CrossRefGoogle Scholar
Fried, B. D. & Conte, S. D. 1961 The Plasma Dispersion Function. Academic Press.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.CrossRefGoogle Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2: strong alfvenic turbulence. Astrophys. J. 438, 763–775.Google Scholar
Groselj, D., Mallet, A., Loureiro, N. F. & Jenko, F. 2018 Fully kinetic simulation of 3D kinetic Alfven turbulence. Phys. Rev. Lett. 120, 105101.Google Scholar
Hallatschek, K. 2004 Thermodynamic potential in local turbulence simulations. Phys. Rev. Lett. 93 (12), 125001.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.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. M. 2014 Solar wind protons at 1 AU: trends and bounds, constraints and correlations. Astrophys. J. 784, L15.CrossRefGoogle Scholar
Hollweg, J. V. & Isenberg, P. A. 2002 Generation of the fast solar wind: a review with emphasis on the resonant cyclotron interaction. J. Geophys. Res. 107, 1147.Google Scholar
Horbury, T. S., Forman, M. & Oughton, S. 2008 Anisotropic scaling of magnetohydrodynamic turbulence. Phys. Rev. Lett. 101 (17), 175005.Google Scholar
Hoshino, M. 2015 Angular momentum transport and particle acceleration during magnetorotational instability in a kinetic accretion disk. Phys. Rev. Lett. 114 (6), 061101.Google Scholar
Howes, G. G. 2010 A prescription for the turbulent heating of astrophysical plasmas. Mon. Not. R. Astron. Soc. 409, L104–L108.Google 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. 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
Howes, G. G., Dorland, W., Cowley, S. C., Hammett, G. W., Quataert, E., Schekochihin, A. A. & Tatsuno, T. 2008 Kinetic simulations of magnetized turbulence in astrophysical plasmas. Phys. Rev. Lett. 100 (6), 065004.Google Scholar
Howes, G. G., TenBarge, J. M. & Dorland, W. 2011 A weakened cascade model for turbulence in astrophysical plasmas. Phys. Plasmas 18 (10), 102305.Google Scholar
Hundhausen, A. J., Bame, S. J. & Ness, N. F. 1967 Solar wind thermal anisotropies: Vela 3 and IMP 3. J. Geophys. Res. 72, 5265.Google Scholar
Isenberg, P. A. 2001 Heating of coronal holes and generation of the solar wind by ion-cyclotron resonance. Space Sci. Rev. 95, 119–131.Google Scholar
Kadomtsev, B. B. & Pogutse, O. P. 1974 Nonlinear helical perturbations of a plasma in the tokamak. Sov. J. Exp. Theoret. Phys. 38, 283–290.Google Scholar
Kanekar, A., Schekochihin, A. A., Dorland, W. & Loureiro, N. F. 2015 Fluctuation-dissipation relations for a plasma-kinetic Langevin equation. J. Plasma Phys. 81 (1), 305810104.CrossRefGoogle Scholar
Kasper, J. C., Maruca, B. A., Stevens, M. L. & Zaslavsky, A. 2013 Sensitive test for ion-cyclotron resonant heating in the solar wind. Phys. Rev. Lett. 110 (9), 091102.Google Scholar
Kennel, C. F. & Sagdeev, R. Z. 1967 Collisionless shock waves in high $\unicode[STIX]{x1D6FD}$ plasmas: 1. J. Geophys. Res. 72, 3303–3326.Google Scholar
Kingsep, A. S., Chukbar, K. V. & Yan’kov, V. V. 1990 Reviews of Plasma Physics, vol. 16, p. 243. Consultants Bureau.Google Scholar
Kiyani, K. H., Osman, K. T. & Chapman, S. C. 2015 Dissipation and heating in solar wind turbulence: from the macro to the micro and back again. Phil. Trans. R. Soc. A 373, 20140155.CrossRefGoogle Scholar
Klein, K. G. & Howes, G. G. 2015 Predicted impacts of proton temperature anisotropy on solar wind turbulence. Phys. Plasmas 22 (3), 032903.Google Scholar
Klein, K. G., Howes, G. G. & TenBarge, J. M. 2017 Diagnosing collisionless energy transfer using field-particle correlations: gyrokinetic turbulence. J. Plasma Phys. 83 (4), 535830401.Google Scholar
Kolmogorov, A. N. 1941 Local structure of turbulence in incompressible viscous fluid at very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299.Google Scholar
Kraichnan, R. H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385–1387.Google Scholar
Krommes, J. A. 2012 The gyrokinetic description of microturbulence in magnetized plasmas. Ann. Rev. Fluid Mech. 44, 175–201.Google 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 Press.Google Scholar
Kulsrud, R. M. 1983 MHD description of plasma. In Basic Plasma Physics: Selected Chapters, Handbook of Plasma Physics (ed. Galeev, A. A. & Sudan, R. N.), vol. 1, p. 1. Elsevier.Google Scholar
Kunz, M. W., Schekochihin, A. A., Chen, C. H. K., Abel, I. G. & Cowley, S. C. 2015 Inertial-range kinetic turbulence in pressure-anisotropic astrophysical plasmas. J. Plasma Phys. 81, 325810501.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
Kunz, M. W., Stone, J. M. & Quataert, E. 2016 Magnetorotational turbulence and dynamo in a collisionless plasma. Phys. Rev. Lett. 117 (23), 235101.Google Scholar
Landau, L. 1946 On the vibrations of the electronic plasma. Zh. Exp. Teor. Fiz. 16, 574 (English translation: 1946, J. Phys. USSR, 10, 25).Google Scholar
Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H. & Wong, H. K. 1998 Observational constraints on the dynamics of the interplanetary magnetic field dissipation range. J. Geophys. Res. 103, 4775–4787.CrossRefGoogle Scholar
Leamon, R. J., Smith, C. W., Ness, N. F. & Wong, H. K. 1999 Dissipation range dynamics: kinetic Alfvén waves and the importance of $\unicode[STIX]{x1D6FD}_{e}$ . J. Geophys. Res. 104, 22331–22344.Google Scholar
Lee, W. W. 1983 Gyrokinetic approach in particle simulation. Phys. Fluids 26, 556.Google Scholar
Li, X. & Habbal, S. R. 2000 Electron kinetic firehose instability. J. Geophys. Res. 105, 27377–27386.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. et al. 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
Mallet, A. & Schekochihin, A. A. 2017 A statistical model of three-dimensional anisotropy and intermittency in strong Alfvénic turbulence. Mon. Not. R. Astron. Soc. 466, 3918–3927.Google Scholar
Markovskii, S. A., Vasquez, B. J. & Smith, C. W. 2008 Statistical analysis of the high-frequency spectral break of the solar wind turbulence at 1 AU. Astrophys. J. 675, 1576–1583.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., Rosenbauer, H., Schwenn, R., Muehlhaeuser, K.-H. & Neubauer, F. M. 1982a 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., Schwenn, R., Rosenbauer, H., Muehlhaeuser, K.-H., Pilipp, W. & Neubauer, F. M. 1982b Solar wind protons – three-dimensional velocity distributions and derived plasma parameters measured between 0.3 and 1 AU. J. Geophys. Res. 87, 52–72.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
Matteini, L., Hellinger, P., Landi, S., Trávníček, P. M. & Velli, M. 2012 Ion kinetics in the solar wind: coupling global expansion to local microphysics. Space Sci. Rev. 172, 373–396.CrossRefGoogle Scholar
McComas, D. J., Barraclough, B. L., Gosling, J. T., Hammond, C. M., Phillips, J. L., Neugebauer, M., Balogh, A. & Forsyth, R. J. 1995 Structures in the polar solar wind: plasma and field observations from Ulysses. J. Geophys. Res. 100, 19893–19902.CrossRefGoogle Scholar
Navarro, A. B., Teaca, B., Told, D., Groselj, D., Crandall, P. & Jenko, F. 2016 Structure of plasma heating in gyrokinetic Alfvénic turbulence. Phys. Rev. Lett. 117 (24), 245101.Google Scholar
Obukhov, A. M. 1941 On the distribution of energy in the spectrum of turbulent flow. Izv. Akad. Nauk SSSR Ser. Geogr. 1. Geofiz. 5, 453.Google 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
Parra, F. I.2013 Extension of gyrokinetics to transport time scales. arXiv:1309:7385.Google Scholar
Plunk, G. G. 2013 Landau damping in a turbulent setting. Phys. Plasmas 20 (3), 032304.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
Porazik, P. & Johnson, J. R. 2013 Linear dispersion relation for the mirror instability in context of the gyrokinetic theory. Phys. Plasmas 20, 104501.Google Scholar
Porazik, P. & Johnson, J. R. 2017 Conductivity tensor for anisotropic plasma in gyrokinetic theory. Phys. Plasmas 24, 052121.Google Scholar
Quataert, E. 1998 Particle heating by Alfvénic turbulence in hot accretion flows. Astrophys. J. 500, 978–991.CrossRefGoogle Scholar
Quataert, E., Dorland, W. & Hammett, G. W. 2002 The magnetorotational instability in a collisionless plasma. Astrophys. J. 577, 524–533.Google Scholar
Quataert, E. & Gruzinov, A. 1999 Turbulence and particle heating in advection-dominated accretion flows. Astrophys. J. 520, 248–255.Google Scholar
Rickett, B. J., Kedziora-Chudczer, L. & Jauncey, D. L. 2002 Interstellar scintillation of the polarized flux density in quasar PKS 0405-385. Astrophys. J. 581, 103–126.Google Scholar
Riquelme, M. A., Quataert, E., Sharma, P. & Spitkovsky, A. 2012 Local two-dimensional particle-in-cell simulations of the collisionless magnetorotational instability. Astrophys. J. 755, 50.Google Scholar
Riquelme, M. A., Quataert, E. & Verscharen, D. 2015 Particle-in-cell simulations of continuously driven mirror and ion cyclotron instabilities in high beta astrophysical and heliospheric plasmas. Astrophys. J. 800, 27.Google Scholar
Rincon, F., Schekochihin, A. A. & Cowley, S. C. 2015 Non-linear mirror instability. Mon. Not. R. Astron. Soc. 447, L45.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
Rosin, M. S., Schekochihin, A. A., Rincon, F. & Cowley, S. C. 2011 A non-linear theory of the parallel firehose and gyrothermal instabilities in a weakly collisional plasma. Mon. Not. R. Astron. Soc. 413, 7–38.Google Scholar
Rutherford, P. H. & Frieman, E. A. 1968 Drift instabilities in general magnetic field configurations. Phys. Fluids 11, 569–585.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
Schekochihin, A. A.2017 MHD turbulence in 2017: a biased review.http://www-thphys.physics.ox.ac.uk/research/plasma/JPP/papers17/schekochihin2a.pdf.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., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. 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., Parker, J. T., Highcock, E. G., Dellar, P. J., Dorland, W. & Hammett, G. W. 2016 Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82 (2), 905820212.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
Sharma, P., Hammett, G. W., Quataert, E. & Stone, J. M. 2006 Shearing box simulations of the MRI in a collisionless plasma. Astrophys. J. 637, 952–967.Google Scholar
Sharma, P., Quataert, E., Hammett, G. W. & Stone, J. M. 2007 Electron heating in hot accretion flows. Astrophys. J. 667, 714–723.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.CrossRefGoogle Scholar
Sironi, L. 2015 Electron heating by the ion cyclotron instability in collisionless accretion flows. II. Electron heating efficiency as a function of flow conditions. Astrophys. J. 800, 89.Google Scholar
Sironi, L. & Narayan, R. 2015 Electron Heating by the ion cyclotron instability in collisionless accretion flows. I. Compression-driven instabilities and the electron heating mechanism. Astrophys. J. 800, 88.CrossRefGoogle Scholar
Southwood, D. J. & Kivelson, M. G. 1993 Mirror instability. I. Physical mechanism of linear instability. J. Geophys. Res. 98, 9181–9187.CrossRefGoogle Scholar
Squire, J., Kunz, M. W., Quataert, E. & Schekochihin, A. A. 2017 Kinetic simulations of the interruption of large-amplitude shear-Alfvén Waves in a high- $\unicode[STIX]{x1D6FD}$ plasma. Phys. Rev. Lett. 119 (15), 155101.Google Scholar
Squire, J., Quataert, E. & Schekochihin, A. A. 2016 A stringent limit on the amplitude of Alfvénic perturbations in high-beta low-collisionality plasmas. Astrophys. J. 830, L25.Google Scholar
Stix, T. H. 1992 Waves in Plasmas. American Institute of Physics.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
Taylor, J. B. 1967 Magnetic moment under short-wave electrostatic perturbations. Phys. Fluids 10, 1357–1359.CrossRefGoogle Scholar
Taylor, J. B. & Hastie, R. J. 1968 Stability of general plasma equilibria – I formal theory. Plasma Phys. 10, 479–494.Google Scholar
Told, D., Jenko, F., TenBarge, J. M., Howes, G. G. & Hammett, G. W. 2015 Multiscale nature of the dissipation range in gyrokinetic simulations of Alfvénic turbulence. Phys. Rev. Lett. 115 (2), 025003.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
Verscharen, D., Chen, C. H. K. & Wicks, R. T. 2017 On kinetic slow modes, fluid slow modes, and pressure-balanced structures in the solar wind. Astrophys. J. 840, 106.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
Wilson, L. B. III, Stevens, M. L., Kasper, J. C., Klein, K. G., Maruca, B. A., Bale, S. D., Bowen, T. A., Pulupa, M. P. & Salem, C. S. 2018 The statical properties of solar wind temperature parameters near 1 AU. ArXiv e-prints.Google Scholar
Yuan, F. & Narayan, R. 2014 Hot accretion flows around black holes. Ann. Rev. Astron. Astrophys. 52, 529–588.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 5, 1971–1979.Google Scholar
Zank, G. P. & Matthaeus, W. H. 1992 The equations of reduced magnetohydrodynamics. J. Plasma Phys. 48, 85.Google Scholar