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Constraints on ion versus electron heating by plasma turbulence at low beta

Part of: Focus on Plasma Astrophysics

Published online by Cambridge University Press:  31 May 2019

A. A. Schekochihin ,
Y. Kawazura  and
M. A. Barnes Open the ORCID record for M. A. Barnes [Opens in a new window]
A. A. Schekochihin*
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Merton College, Oxford OX1 4JD, UK Niels Bohr International Academy, Blegdamsvej 17, 2100 Copenhagen, Denmark
Y. Kawazura
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK
M. A. Barnes
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK University College, Oxford OX1 4BH, UK United Kingdom Atomic Energy Authority, Culham Science Centre, Abington OX14 3DB, UK
*
†Email address for correspondence: [email protected]
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Abstract

It is shown that in low-beta, weakly collisional plasmas, such as the solar corona, some instances of the solar wind, the aurora, inner regions of accretion discs, their coronae and some laboratory plasmas, Alfvénic fluctuations produce no ion heating within the gyrokinetic approximation, i.e. as long as their amplitudes (at the Larmor scale) are small and their frequencies stay below the ion-Larmor frequency (even though their spatial scales can be above or below the ion Larmor scale). Thus, all low-frequency ion heating in such plasmas is due to compressive fluctuations (‘slow modes’): density perturbations and non-Maxwellian perturbations of the ion distribution function. Because these fluctuations energetically decouple from the Alfvénic ones already in the inertial range, the above conclusion means that the energy partition between ions and electrons in low-beta plasmas is decided at the outer scale, where turbulence is launched, and can be determined from magnetohydrodynamic (MHD) models of the relevant astrophysical systems. Any additional ion heating must come from non-gyrokinetic mechanisms such as cyclotron heating or the stochastic heating owing to distortions of ions’ Larmor orbits. An exception to these conclusions occurs in the Hall limit, i.e. when the ratio of the ion to electron temperatures is as low as the ion beta (equivalently, the electron beta is order unity). In this regime, slow modes couple to Alfvénic ones well above the Larmor scale (viz., at the ion inertial or ion sound scale), so the Alfvénic and compressive cascades join and then separate again into two cascades of fluctuations that linearly resemble kinetic Alfvén and ion-cyclotron waves, with the former heating electrons and the latter ions. The two cascades are shown to decouple, scalings for them are derived and it is argued physically that the two species will be heated by them at approximately equal rates.

Keywords

astrophysical plasmasplasma heatingplasma nonlinear phenomena
Type
Research Article
Information
Journal of Plasma Physics , Volume 85 , Issue 3 , June 2019 , 905850303
Copyright
© Cambridge University Press 2019 

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Footnotes

‡

Present address: Frontier Research Institute for Interdisciplinary Sciences and Department of Geophysics, Tohoku University, Aramaki aza Aoba 6-3, Aoba-ku, Sendai 980-8578, Japan.

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, 122509.Google Scholar
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, 116201.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, 165003.Google Scholar
Arzamasskiy, L., Kunz, M. W., Chandran, B. D. G. & Quataert, E.2019 Hybrid-kinetic simulations of ion heating in Alfvénic turbulence. E-print arXiv:1901.11028.Google Scholar
Aschwanden, M. J., Poland, A. I. & Rabin, D. M. 2001 The new solar corona. Annu. Rev. Astron. Astrophys. 39, 175.Google Scholar
Bañón Navarro, A., Teaca, B., Told, D., Grošelj, D., Crandall, P. & Jenko, F. 2016 Structure of plasma heating in gyrokinetic Alfvénic turbulence. Phys. Rev. Lett. 117, 245101.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.Google Scholar
Banerjee, S. & Galtier, S. 2016 Chiral exact relations for helicities in Hall magnetohydrodynamic turbulence. Phys. Rev. E 93, 033120.Google Scholar
Barnes, A. 1966 Collisionless damping of hydromagnetic waves. Phys. Fluids 9, 1483.Google Scholar
Bell, N. & Nazarenko, S.2019 Rotating magnetohydrodynamic turbulence. E-print arXiv:1902.07524.Google Scholar
Bian, N. H. & Tsiklauri, D. 2009 Compressible Hall magnetohydrodynamics in a strong magnetic field. Phys. Plasmas 16, 064503.Google Scholar
Boldyrev, S., Horaites, K., Xia, Q. & Perez, J. C. 2013 Toward a theory of astrophysical plasma turbulence at subproton scales. Astrophys. J. 777, 41.Google Scholar
Boldyrev, S. & Loureiro, N. F.2019 Role of reconnection in inertial kinetic-Alfvén turbulence. E-print arXiv:1901.10096.Google Scholar
Boldyrev, S. & Perez, J. C. 2012 Spectrum of kinetic-Alfvén turbulence. Astrophys. J. 758, L44.Google Scholar
Boozer, A. H. 2018 Why fast magnetic reconnection is so prevalent. J. Plasma Phys. 84, 715840102.Google Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205.Google Scholar
Breech, B., Matthaeus, W. H., Cranmer, S. R., Kasper, J. C. & Oughton, S. 2009 Electron and proton heating by solar wind turbulence. J. Geophys. Res. 114, A09103.Google Scholar
Carter, T. A., Brugman, B., Pribyl, P. & Lybarger, W. 2006 Laboratory observation of a nonlinear interaction between shear Alfvén waves. Phys. Rev. Lett. 96, 155001.Google Scholar
Cerri, S. S., Kunz, M. W. & Califano, F. 2018 Dual phase-space cascades in 3D hybrid-Vlasov–Maxwell turbulence. Astrophys. J. 856, L13.Google Scholar
Chael, A., Narayan, R. & Johnson, M. D. 2019 Two-temperature, magnetically arrested disc simulations of the jet from the supermassive black hole in M87. Mon. Not. R. Astron. Soc. 486, 2873.Google Scholar
Chael, A., Rowan, M., Narayan, R., Johnson, M. & Sironi, L. 2018 The role of electron heating physics in images and variability of the Galactic Centre black hole Sagittarius $\text{A}^{\ast }$ . Mon. Not. R. Astron. Soc. 478, 5209.Google Scholar
Chandran, B. D. G. 2010 Alfvén-wave turbulence and perpendicular ion temperatures in coronal holes. Astrophys. J. 720, 548.Google Scholar
Chandran, B. D. G., Foucart, F. & Tchekhovskoy, A. 2018 Heating of accretion-disk coronae and jets by general relativistic magnetohydrodynamic turbulence. J. Plasma Phys. 84, 905840310.Google 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. 720, 503.Google Scholar
Chandran, B. D. G., Quataert, E., Howes, G. G., Xia, Q. & Pongkitiwanichakul, P. 2009 Constraining low-frequency Alfvénic turbulence in the solar wind using density-fluctuation measurements. Astrophys. J. 707, 1668.Google Scholar
Chaston, C. C., Salem, C., Bonnell, J. W., Carlson, C. W., Ergun, R. E., Strangeway, R. J. & McFadden, J. P. 2008 The turbulent Alfvénic aurora. Phys. Rev. Lett. 100, 175003.Google Scholar
Chen, C. H. K. 2016 Recent progress in astrophysical plasma turbulence from solar wind observations. J. Plasma Phys. 82, 535820602.Google Scholar
Chen, C. H. K. & Boldyrev, S. 2017 Nature of kinetic scale turbulence in the Earth’s magnetosheath. Astrophys. J. 842, 122.Google Scholar
Chen, C. H. K., Boldyrev, S., Xia, Q. & Perez, J. C. 2013 Nature of subproton scale turbulence in the solar wind. Phys. Rev. Lett. 110, 225002.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.Google Scholar
Chen, Q., Chen, S. & Eyink, G. L. 2003 The joint cascade of energy and helicity in three-dimensional turbulence. Phys. Fluids 15, 361.Google Scholar
Cho, J. & Kim, H. 2016 Spectral evolution of helical electron magnetohydrodynamic turbulence. J. Geophys. Res. A 121, 6157.Google Scholar
Cho, J. & Lazarian, A. 2004 The anisotropy of electron magnetohydrodynamic turbulence. Astrophys. J. 615, L41.Google Scholar
Cho, J. & Lazarian, A. 2009 Simulations of electron magnetohydrodynamic turbulence. Astrophys. J. 701, 236.Google Scholar
Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273.Google Scholar
Cranmer, S. R. 2009 Coronal holes. Living Rev. Solar Phys. 6, 3.Google Scholar
Cranmer, S. R., Matthaeus, W. H., Breech, B. A. & Kasper, J. C. 2009 Empirical constraints on proton and electron heating in the fast solar wind. Astrophys. J. 702, 1604.Google Scholar
Davidson, P. A. 2013 Turbulence in Rotating, Stratified and Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Eyink, G. L. 2015 Turbulent general magnetic reconnection. Astrophys. J. 807, 137.Google Scholar
Eyink, G. L. 2018 Cascades and dissipative anomalies in nearly collisionless plasma turbulence. Phys. Rev. X 8, 041020.Google Scholar
Franci, L., Landi, S., Verdini, A., Matteini, 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
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.Google Scholar
Galtier, S. 2014 Weak turbulence theory for rotating magnetohydrodynamics and planetary flows. J. Fluid Mech. 757, 114.Google Scholar
Galtier, S. & Buchlin, E. 2007 Multiscale Hall-magnetohydrodynamic turbulence in the solar wind. Astrophys. J. 656, 560.Google Scholar
Gary, S. P., Smith, C. W. & Skoug, R. M. 2005 Signatures of Alfvén-cyclotron wave-ion scattering: advanced composition explorer (ACE) solar wind observations. J. Geophys. Res. 110, A07108.Google Scholar
Gekelman, W., Vincena, S., van Compernolle, B., Morales, G. J., Maggs, J. E., Pribyl, P. & Carter, T. A. 2011 The many faces of shear Alfvén waves. Phys. Plasmas 18, 055501.Google Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2. Strong Alfvénic turbulence. Astrophys. J. 438, 763.Google Scholar
Goldreich, P. & Sridhar, S. 1997 Magnetohydrodynamic turbulence revisited. Astrophys. J. 485, 680.Google Scholar
Gómez, D. O., Mahajan, S. M. & Dmitruk, P. 2008 Hall magnetohydrodynamics in a strong magnetic field. Phys. Plasmas 15, 102303.Google Scholar
Grošelj, D., Cerri, S. S., Bañón Navarro, A., Willmott, C., Told, D., Loureiro, N. F., Califano, F. & Jenko, F. 2017 Fully kinetic versus reduced-kinetic modeling of collisionless plasma turbulence. Astrophys. J. 847, 28.Google Scholar
Grošelj, D., Mallet, A., Loureiro, N. F. & Jenko, F. 2018 Fully kinetic simulation of 3D kinetic Alfvén turbulence. Phys. Rev. Lett. 120, 105101.Google Scholar
He, J., Marsch, E., Tu, C., Yao, S. & Tian, H. 2011 Possible evidence of Alfvén-cyclotron waves in the angle distribution of magnetic helicity of solar wind turbulence. Astrophys. J. 731, 85.Google Scholar
He, J., Tu, C., Marsch, E. & Yao, S. 2012 Reproduction of the observed two-component magnetic helicity in solar wind turbulence by a superposition of parallel and oblique Alfvén waves. Astrophys. J. 749, 86.Google Scholar
Helander, P. & Sigmar, D. J. 2005 Collisional Transport in Magnetized Plasmas. Cambridge University Press.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, L09101.Google Scholar
Hoppock, I. W., Chandran, B. D. G., Klein, K. G., Mallet, A. & Verscharen, D. 2018 Stochastic proton heating by kinetic-Alfvén-wave turbulence in moderately high- $\unicode[STIX]{x1D6FD}$ plasmas. J. Plasma Phys. 84, 905840615.Google Scholar
Horbury, T. S., Forman, M. & Oughton, S. 2008 Anisotropic scaling of magnetohydrodynamic turbulence. Phys. Rev. Lett. 101, 175005.Google Scholar
Howes, G. G. 2010 A prescription for the turbulent heating of astrophysical plasmas. Mon. Not. R. Astron. Soc. 409, L104.Google Scholar
Howes, G. G. 2011 Prediction of the proton-to-total turbulent heating in the solar wind. Astrophys. J. 738, 40.Google Scholar
Howes, G. G. 2015 The inherently three-dimensional nature of magnetized plasma turbulence. J. Plasma Phys. 81, 325810203.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.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, 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, 065004.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
Kadomtsev, B. B. & Pogutse, O. P. 1974 Nonlinear helical perturbations of a plasma in the tokamak. Sov. Phys. JETP 38, 283.Google Scholar
Kasper, J. C., Lazarus, A. J. & Gary, S. P. 2008 Hot solar-wind helium: direct evidence for local heating by Alfvén-cyclotron dissipation. Phys. Rev. Lett. 101, 261103.Google 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, 091102.Google Scholar
Kawazura, Y. & Barnes, M. 2018 A hybrid gyrokinetic ion and isothermal electron fluid code for astrophysical plasma. J. Comput. Phys. 360, 57.Google Scholar
Kawazura, Y., Barnes, M. & Schekochihin, A. A. 2019 Thermal disequilibration of ions and electrons by collisionless plasma turbulence. Proc. Natl Acad. Sci. USA 116, 771.Google Scholar
Klein, K. G., Howes, G. G., TenBarge, J. M. & Podesta, J. J. 2014 Physical interpretation of the angle-dependent magnetic helicity spectrum in the solar wind: the nature of turbulent fluctuations near the proton gyroradius scale. Astrophys. J. 785, 138.Google Scholar
Krishan, V. & Mahajan, S. M. 2004 Magnetic fluctuations and Hall magnetohydrodynamic turbulence in the solar wind. J. Geophys. Res. 109, A11105.Google Scholar
Kunz, M. W., Abel, I. G., Klein, K. G. & Schekochihin, A. A. 2018 Astrophysical gyrokinetics: turbulence in pressure-anisotropic plasmas at ion scales and beyond. J. Plasma Phys. 84, 715840201.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
Li, T. C., Howes, G. G., Klein, K. G. & TenBarge, J. M. 2016 Energy dissipation and Landau damping in two- and three-dimensional plasma turbulence. Astrophys. J. 832, L24.Google Scholar
Lion, S., Alexandrova, O. & Zaslavsky, A. 2016 Coherent events and spectral shape at ion kinetic scales in the fast solar wind turbulence. Astrophys. J. 824, 47.Google Scholar
Lithwick, Y. & Goldreich, P. 2001 Compressible magnetohydrodynamic turbulence in interstellar plasmas. Astrophys. J. 562, 279.Google Scholar
Loureiro, N. F. & Boldyrev, S. 2017 Collisionless reconnection in magnetohydrodynamic and kinetic turbulence. Astrophys. J. 850, 182.Google Scholar
Loureiro, N. F., Dorland, W., Fazendeiro, L., Kanekar, A., Mallet, A., Vilelas, M. S. & Zocco, A. 2016 Viriato: a Fourier–Hermite spectral code for strongly magnetized fluid-kinetic plasma dynamics. Comput. Phys. Commun. 206, 45.Google Scholar
Loureiro, N. F., Schekochihin, A. A. & Zocco, A. 2013 Fast collisionless reconnection and electron heating in strongly magnetized plasmas. Phys. Rev. Lett. 111, 025002.Google Scholar
Mahajan, S. M. & Yoshida, Z. 1998 Double curl Beltrami flow: diamagnetic structures. Phys. Rev. Lett. 81, 4863.Google Scholar
Mallet, A., Klein, K. G., Chandran, B. D. G., Grošelj, D., Hoppock, I. W., Bowen, T. A., Salem, C. S. & Bale, S. D.2019 Interplay between intermittency and dissipation in collisionless plasma turbulence. J. Plasma Phys. (submitted). arXiv:1807.09301.Google Scholar
Maron, J. & Goldreich, P. 2001 Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J. 554, 1175.Google Scholar
Marsch, E. & Bourouaine, S. 2011 Velocity-space diffusion of solar wind protons in oblique waves and weak turbulence. Ann. Geophys. 29, 2089.Google Scholar
Meyrand, R. & Galtier, S. 2012 Spontaneous chiral symmetry breaking of Hall magnetohydrodynamic turbulence. Phys. Rev. Lett. 109, 194501.Google Scholar
Meyrand, R. & Galtier, S. 2013 Anomalous $k_{\bot }^{-8/3}$ spectrum in electron magnetohydrodynamic turbulence. Phys. Rev. Lett. 111, 264501.Google Scholar
Meyrand, R., Kanekar, A., Dorland, W. & Schekochihin, A. A. 2019 Fluidization of collisionless plasma turbulence. Proc. Natl Acad. Sci. USA 116, 1185.Google Scholar
Meyrand, R., Kiyani, K. H., Gürcan, Ö. D. & Galtier, S. 2018 Coexistence of weak and strong wave turbulence in incompressible Hall magnetohydrodynamics. Phys. Rev. X 8, 031066.Google Scholar
Nazarenko, S. V. & Schekochihin, A. A. 2011 Critical balance in magnetohydrodynamic, rotating and stratified turbulence: towards a universal scaling conjecture. J. Fluid Mech. 677, 134.Google Scholar
Passot, T., Sulem, P. L. & Tassi, E. 2017 Electron-scale reduced fluid models with gyroviscous effects. J. Plasma Phys. 83, 715830402.Google Scholar
Plunk, G. G., Cowley, S. C., Schekochihin, A. A. & Tatsuno, T. 2010 Two-dimensional gyrokinetic turbulence. J. Fluid Mech. 664, 407.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.Google Scholar
Podesta, J. J. & Gary, S. P. 2011 Magnetic helicity spectrum of solar wind fluctuations as a function of the angle with respect to the local mean magnetic field. Astrophys. J. 734, 15.Google Scholar
Quataert, E. 1998 Particle heating by Alfvenic turbulence in hot accretion flows. Astrophys. J. 500, 978.Google Scholar
Quataert, E. 2003 Radiatively inefficient accretion flow models of Sgr A $^{\ast }$ . Astron. Nachr. 324, 435.Google Scholar
Quataert, E. & Gruzinov, A. 1999 Turbulence and particle heating in advection-dominated accretion flows. Astrophys. J. 520, 248.Google Scholar
Ressler, S. M., Tchekhovskoy, A., Quataert, E. & Gammie, C. F. 2017 The disc-jet symbiosis emerges: modelling the emission of Sagittarius A $^{\ast }$ with electron thermodynamics. Mon. Not. R. Astron. Soc. 467, 3604.Google Scholar
Rowan, M. E., Sironi, L. & Narayan, R. 2017 Electron and proton heating in transrelativistic magnetic reconnection. Astrophys. J. 850, 29.Google Scholar
Rowan, M. E., Sironi, L. & Narayan, R. 2019 Electron and proton heating in transrelativistic guide field reconnection. Astrophys. J. 873, 2.Google Scholar
Sahraoui, F., Galtier, S. & Belmont, G. 2007 On waves in incompressible Hall magnetohydrodynamics. J. Plasma Phys. 73, 723.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, 131101.Google Scholar
Salem, C. S., Howes, G. G., Sundkvist, D., Bale, S. D., Chaston, C. C., Chen, C. H. K. & Mozer, F. S. 2012 Identification of kinetic Alfvén wave turbulence in the solar wind. Astrophys. J. 745, L9.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Plunk, G. G., Quataert, E. & Tatsuno, T. 2008 Gyrokinetic turbulence: a nonlinear route to dissipation through phase space. Plasma Phys. Control. Fusion 50, 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.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, 905820212.Google Scholar
Sharma, P., Quataert, E., Hammett, G. W. & Stone, J. M. 2007 Electron heating in hot accretion flows. Astrophys. J. 667, 714.Google Scholar
Smith, C. W., Mullan, D. J., Ness, N. F., Skoug, R. M. & Steinberg, J. 2001 Day the solar wind almost disappeared: magnetic field fluctuations, wave refraction and dissipation. J. Geophys. Res. 106, 18625.Google Scholar
Strauss, H. R. 1976 Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19, 134.Google Scholar
Tatsuno, T., Dorland, W., Schekochihin, A. A., Plunk, G. G., Barnes, M., Cowley, S. C. & Howes, G. G. 2009 Nonlinear phase mixing and phase-space cascade of entropy in gyrokinetic plasma turbulence. Phys. Rev. Lett. 103, 015003.Google Scholar
TenBarge, J. M. & Howes, G. G. 2012 Evidence of critical balance in kinetic Alfvén wave turbulence simulations. Phys. Plasmas 19, 055901.Google Scholar
TenBarge, J. M., Howes, G. G. & Dorland, W. 2013 Collisionless damping at electron scales in solar wind turbulence. Astrophys. J. 774, 139.Google 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.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, 025003.Google Scholar
Turner, L. 1986 Hall effects on magnetic relaxation. IEEE Trans. Plasma Sci. 14, 849.Google Scholar
Vech, D., Klein, K. G. & Kasper, J. C. 2017 Nature of stochastic ion heating in the solar wind: testing the dependence on plasma beta and turbulence amplitude. Astrophys. J. 850, L11.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.Google Scholar
Xu, R. & Kunz, M. W. 2016 Linear Vlasov theory of a magnetised, thermally stratified atmosphere. J. Plasma Phys. 82, 905820507.Google Scholar
Zeman, O. 1994 A note on the spectra and decay of rotating homogeneous turbulence. Phys. Fluids 6, 3221.Google Scholar
Zhdankin, V., Uzdensky, D. A., Werner, G. R. & Begelman, M. C. 2019 Electron and ion energization in relativistic plasma turbulence. Phys. Rev. Lett. 122, 055101.Google Scholar
Zocco, A. & Schekochihin, A. A. 2011 Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas. Phys. Plasmas 18, 102309.Google Scholar