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Pressure-anisotropy-induced nonlinearities in the kinetic magnetorotational instability

  • J. Squire (a1) (a2), E. Quataert (a3) and M. W. Kunz (a4) (a5)

In collisionless and weakly collisional plasmas, such as hot accretion flows onto compact objects, the magnetorotational instability (MRI) can differ significantly from the standard (collisional) MRI. In particular, pressure anisotropy with respect to the local magnetic-field direction can both change the linear MRI dispersion relation and cause nonlinear modifications to the mode structure and growth rate, even when the field and flow perturbations are very small. This work studies these pressure-anisotropy-induced nonlinearities in the weakly nonlinear, high-ion-beta regime, before the MRI saturates into strong turbulence. Our goal is to better understand how the saturation of the MRI in a low-collisionality plasma might differ from that in the collisional regime. We focus on two key effects: (i) the direct impact of self-induced pressure-anisotropy nonlinearities on the evolution of an MRI mode, and (ii) the influence of pressure anisotropy on the ‘parasitic instabilities’ that are suspected to cause the mode to break up into turbulence. Our main conclusions are: (i) The mirror instability regulates the pressure anisotropy in such a way that the linear MRI in a collisionless plasma is an approximate nonlinear solution once the mode amplitude becomes larger than the background field (just as in magnetohyrodynamics). This implies that differences between the collisionless and collisional MRI become unimportant at large amplitudes. (ii) The break up of large-amplitude MRI modes into turbulence via parasitic instabilities is similar in collisionless and collisional plasmas. Together, these conclusions suggest that the route to magnetorotational turbulence in a collisionless plasma may well be similar to that in a collisional plasma, as suggested by recent kinetic simulations. As a supplement to these findings, we offer guidance for the design of future kinetic simulations of magnetorotational turbulence.

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Balbus, S. A. 2004 Viscous shear instability in weakly magnetized, dilute plasmas. Astrophys. J. 616 (2), 857864.
Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks. I – linear analysis. II – nonlinear evolution. Astrophys. J. 376, 214233.
Balbus, S. A. & Hawley, J. F. 1992 A powerful local shear instability in weakly magnetized disks. IV. Nonaxisymmetric perturbations. Astrophys. J. 400, 610621.
Balbus, S. A. & Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70 (1), 1.
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 (2), 211101.
Blackman, E. G. 2012 Accretion disks and dynamos: toward a unified mean field theory. Phys. Scr. 86 (5), 058202.
Bodo, G., Mignone, A., Cattaneo, F., Rossi, P. & Ferrari, A. 2008 Aspect ratio dependence in magnetorotational instability shearing box simulations. Astron. Astrophys. 487 (1), 15.
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205.
Chandra, M., Gammie, C. F., Foucart, F. & Quataert, E. 2015 An extended magnetohydrodynamics model for relativistic weakly collisional plasmas. Astrophys. J. 810 (2), 162.
Chandrasekhar, S., Kaufman, A. N. & Watson, K. M. 1958 The stability of the pinch. Proc. R. Soc. Lond. A 245, 435455.
Chew, C. F., Goldberger, M. L. & Low, F. E. 1956 Proc. R. Soc. Lond. A 236, 112.
Davidson, R. C. & Völk, H. J. 1968 Macroscopic quasilinear theory of the garden-hose instability. Phys. Fluids 11, 22592264.
Ferraro, N. M. 2007 Finite Larmor radius effects on the magnetorotational instability. Astrophys. J. 662 (1), 512516.
Foucart, F., Chandra, M., Gammie, C. F. & Quataert, E. 2015 Evolution of accretion discs around a Kerr black hole using extended magnetohydrodynamics. Mon. Not. R. Astron. Soc. 456 (2), 13321345.
Foucart, F., Chandra, M., Gammie, C. F., Quataert, E. & Tchekhovskoy, A. 2017 How important is non-ideal physics in simulations of sub-Eddington accretion on to spinning black holes? Mon. Not. R. Astron. Soc. 470, 22402252.
Fromang, S., Papaloizou, J., Lesur, G. & Heinemann, T. 2007 MHD simulations of the magnetorotational instability in a shearing box with zero net flux. II. The effect of transport coefficients. Astron. Astrophys. 476 (3), 11231132.
Fromang, S. & Stone, J. M. 2009 Turbulent resistivity driven by the magnetorotational instability. Astron. Astrophys. 507 (1), 1928.
Gary, S. P., McKean, M. E. & Winske, D. 1993 Ion cyclotron anisotropy instabilities in the magnetosheath – theory and simulations. J. Geophys. Res. 98, 39633971.
Gary, S. P., Wang, J., Winske, D. & Fuselier, S. A. 1997 Proton temperature anisotropy upper bound. J. Geophys. Res. 102 (A), 2715927170.
Goodman, J. & Xu, G. 1994 Parasitic instabilities in magnetized, differentially rotating disks. Astrophys. J. 432, 213223.
Hammett, G. W., Dorland, W. & Perkins, F. W. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4, 20522061.
Hammett, G. W. & Perkins, F. W. 1990 Fluid moment models for Landau damping with application to the ion-temperature-gradient instability. Phys. Rev. Lett. 64, 30193022.
Hasegawa, A. 1969 Drift mirror instability of the magnetosphere. Phys. Fluids 12, 26422650.
Hawley, J. F. & Balbus, S. A. 2002 The dynamical structure of nonradiative black hole accretion flows. Astrophys. J. 573, 738748.
Hawley, J. F., Balbus, S. A. & Stone, J. M. 2001 A magnetohydrodynamic nonradiative accretion flow in three dimensions. Astrophys. J. 554, L49L52.
Hawley, J. F., Gammie, C. F. & Balbus, S. A. 1995 Local three-dimensional magnetohydrodynamic simulations of accretion disks. Astrophys. J. 440, 742.
Heinemann, T. & Quataert, E. 2014 Linear Vlasov theory in the shearing sheet approximation with application to the magneto-rotational instability. Astrophys. J. 792 (1), 70.
Hellinger, P. 2007 Comment on the linear mirror instability near the threshold. Phys. Plasmas 14 (8), 082105.
Hellinger, P. & Matsumoto, H. 2000 New kinetic instability: Oblique Alfvén firehose. J. Geophys. Res. 105 (A), 1051910526.
Hellinger, P., Matteini, L., Landi, S., Verdini, A., Franci, L. & Trávníček, P. M. 2015 Plasma turbulence and kinetic instabilities at ion scales in the expanding solar wind. Astrophys. J. 811 (2), L32.
Hellinger, P. & Trávníček, P. M. 2008 Oblique proton fire hose instability in the expanding solar wind: hybrid simulations. J. Geophys. Res. 113, A10.
Hoshino, M. 2013 Particle acceleration during magnetorotational instability in a collisionless accretion disk. Astrophys. J. 773, 118.
Hoshino, M. 2015 Angular momentum transport and particle acceleration during magnetorotational instability in a kinetic accretion disk. Phys. Rev. Lett. 114 (6), 061101.
Johnson, B. M. 2007 Magnetohydrodynamic shearing waves. Astrophys. J. 660 (2), 1375.
Kasper, J. C., Lazarus, A. J. & Gary, S. P. 2002 Wind/SWE observations of firehose constraint on solar wind proton temperature anisotropy. Geophys. Res. Lett. 29 (1), 1839-1.
Kim, H. P., Hwang, J., Seough, J. J. & Yoon, P. H. 2017 Electron temperature anisotropy regulation by whistler instability. J. Geophys. Res. 122, 44104419.
Klein, K. G. & Howes, G. G. 2015 Predicted impacts of proton temperature anisotropy on solar wind turbulence. Phys. Plasmas 22 (3), 032903.
Komarov, S. V., Churazov, E. M., Kunz, M. W. & Schekochihin, A. A. 2016 Thermal conduction in a mirror-unstable plasma. Mon. Not. R. Astron. Soc. 460, 467477.
Kowal, G., Falceta-Gonçalves, D. A. & Lazarian, A. 2011 Turbulence in collisionless plasmas: statistical analysis from numerical simulations with pressure anisotropy. New J. Phys. 13 (5), 053001.
Kulsrud, R. M. 1983 MHD description of a plasma. In Handbook of Plasma Physics (ed. Rosenbluth, M. N. & Sagdeev, R. Z.), vol. 1, p. 115. North Holland.
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 (5), 325810501.
Kunz, M. W., Schekochihin, A. A. & Stone, J. M. 2014a Firehose and mirror instabilities in a collisionless shearing plasma. Phys. Rev. Lett. 112 (2), 205003.
Kunz, M. W., Stone, J. M. & Bai, X.-N. 2014b Pegasus: a new hybrid-kinetic particle-in-cell code for astrophysical plasma dynamics. J. Comput. Phys. 259, 154174.
Kunz, M. W., Stone, J. M. & Quataert, E. 2016 Magnetorotational turbulence and dynamo in a collisionless plasma. Phys. Rev. Lett. 117 (2), 235101.
Latter, H. N., Fromang, S. & Gressel, O. 2010 MRI channel flows in vertically stratified models of accretion discs. Mon. Not. R. Astron. Soc. 406 (2), 848862.
Latter, H. N., Lesaffre, P. & Balbus, S. A. 2009 MRI channel flows and their parasites. Mon. Not. R. Astron. Soc. 394 (2), 715729.
Lesur, G. & Longaretti, P. Y. 2007 Impact of dimensionless numbers on the efficiency of magnetorotational instability induced turbulent transport. Mon. Not. R. Astron. Soc. 378 (4), 14711480.
Longaretti, P. Y. & Lesur, G. 2010 MRI-driven turbulent transport: the role of dissipation, channel modes and their parasites. Astron. Astrophys. 516, 51.
Meheut, H., Fromang, S., Lesur, G., Joos, M. & Longaretti, P.-Y. 2015 Angular momentum transport and large eddy simulations in magnetorotational turbulence: the small Pm limit. Astron. Astrophys. 579, A117.
Melville, S., Schekochihin, A. A. & Kunz, M. W. 2016 Pressure-anisotropy-driven microturbulence and magnetic-field evolution in shearing, collisionless plasma. Mon. Not. R. Astron. Soc. 459 (3), 27012720.
Mikhailovskii, A. B. & Tsypin, V. S. 1971 Transport equations and gradient instabilities in a high pressure collisional plasma. Plasma Phys. 13, 785798.
Mogavero, F. & Schekochihin, A. A. 2014 Models of magnetic field evolution and effective viscosity in weakly collisional extragalactic plasmas. Mon. Not. R. Astron. Soc. 440 (4), 32263242.
Narayan, R., Mahadevan, R. & Quataert, E. 1998 Advection-dominated accretion around black holes. In Theory of Black Hole Accretion Disks (ed. Abramowicz, M. A., Björnsson, G. & Pringle, J. E.), pp. 148182.
Parker, E. N. 1958 Dynamical instability in an anisotropic ionized gas of low density. Phys. Rev. 109, 18741876.
Passot, T., Sulem, P. L. & Hunana, P. 2012 Extending magnetohydrodynamics to the slow dynamics of collisionless plasmas. Phys. Plasmas 19 (8), 082113.
Pessah, M. E., Chan, C.-k. & Psaltis, D. 2006 Local model for angular-momentum transport in accretion disks driven by the magnetorotational instability. Phys. Rev. Lett. 97 (2), 221103.
Pessah, M. E. & Goodman, J. 2009 On the saturation of the magnetorotational instability via parasitic modes. Astrophys. J. 698 (1), L72.
Pokhotelov, O. A., Balikhin, M. A., Alleyne, H. S.-C. K. & Onishchenko, O. G. 2000 Mirror instability with finite electron temperature effects. J. Geophys. Res. 105, 23932402.
Prajapati, R. P. & Chhajlani, R. K. 2010 Effect of pressure anisotropy and flow velocity on Kelvin–Helmholtz instability of anisotropic magnetized plasma using generalized polytrope laws. Phys. Plasmas 17, 112108.
Quataert, E. 1998 Particle heating by Alfvénic turbulence in hot accretion flows. Astrophys. J. 500 (2), 978991.
Quataert, E. 2003 Radiatively inefficient accretion flow models of Sgr A*. Astron. Nachr. Suppl. 324, 435443.
Quataert, E., Dorland, W. & Hammett, G. W. 2002 The magnetorotational instability in a collisionless plasma. Astrophys. J. 577 (1), 524533.
Quataert, E. & Gruzinov, A. 1999 Turbulence and particle heating in advection-dominated accretion flows. Astrophys. J. 520 (1), 248255.
Quataert, E., Heinemann, T. & Spitkovsky, A. 2015 Linear instabilities driven by differential rotation in very weakly magnetized plasmas. Mon. Not. R. Astron. Soc. 447 (4), 33283341.
Remya, B., Reddy, R. V., Tsurutani, B. T., Lakhina, G. S. & Echer, E. 2013 Ion temperature anisotropy instabilities in planetary magnetosheaths. J. Geophys. Res. 118, 785793.
Ressler, S. M., Tchekhovskoy, A., Quataert, E., Chandra, M. & Gammie, C. F. 2015 Electron thermodynamics in GRMHD simulations of low-luminosity black hole accretion. Mon. Not. R. Astron. Soc. 454, 18481870.
Rincon, F., Schekochihin, A. A. & Cowley, S. C. 2015 Non-linear mirror instability. Mon. Not. R. Astron. Soc. 447, L45L49.
Riquelme, M., Osorio, A. & Quataert, E. 2017 Stochastic electron acceleration by the whistler instability in a growing magnetic field. Astrophys. J. 850 (2), 113.
Riquelme, M., Quataert, E. & Verscharen, D.2017 PIC simulations of velocity-space instabilities in a decreasing magnetic field: viscosity and thermal conduction, arXiv:1708.03926.
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 (1), 50.
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 (1), 27.
Riquelme, M. A., Quataert, E. & Verscharen, D. 2016 PIC simulations of the effect of velocity space instabilities on electron viscosity and thermal conduction. Astrophys. J. 824 (2), 123.
Rosenbluth, M. N.1956 Los Alamos Sci. Lab. Rep. LA-2030.
Rosin, M. S. & Mestel, A. J. 2012 Quasi-global galactic magnetorotational instability with braginskii viscosity. Mon. Not. R. Astron. Soc. 425 (1), 7486.
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 (1), 738.
Ryan, B. R., Gammie, C. F., Fromang, S. & Kestener, P. 2017 Resolution dependence of magnetorotational turbulence in the isothermal stratified shearing box. Astrophys. J. 840, 6.
Santos-Lima, R., de Gouveia Dal Pino, E. M., Kowal, G., Falceta-Gonçalves, D., Lazarian, A. & Nakwacki, M. S. 2014 Magnetic field amplification and evolution in turbulent collisionless magnetohydrodynamics: An application to the intracluster medium. Astrophys. J. 781 (2), 84.
Schekochihin, A. A. & Cowley, S. C. 2006 Fast growth of magnetic fields in galaxy clusters: a self-accelerating dynamo. Astron. Nat. 327 (5–6), 599604.
Schekochihin, A. A., Cowley, S. C., Kulsrud, R. M., Rosin, M. S. & Heinemann, T. 2008 Nonlinear growth of firehose and mirror fluctuations in astrophysical plasmas. Phys. Rev. Lett. 100 (8), 081301.
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 (1), 291300.
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39 (1), 129162.
Sharma, P., Hammett, G. W. & Quataert, E. 2003 Transition from collisionless to collisional magnetorotational instability. Astrophys. J. 596 (2), 11211130.
Sharma, P., Hammett, G. W., Quataert, E. & Stone, J. M. 2006 Shearing box simulations of the MRI in a collisionless plasma. Astrophys. J. 637 (2), 952967.
Sharma, P., Quataert, E., Hammett, G. W. & Stone, J. M. 2007 Electron heating in hot accretion flows. Astrophys. J. 667 (2), 714723.
Simon, J. B., Beckwith, K. & Armitage, P. J. 2012 Emergent mesoscale phenomena in magnetized accretion disc turbulence. Mon. Not. R. Astron. Soc. 422 (3), 26852700.
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.
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 (2), 88.
Snyder, P. B., Hammett, G. W. & Dorland, W. 1997 Landau fluid models of collisionless magnetohydrodynamics. Phys. Plasmas 4 (1), 39743985.
Southwood, D. J. & Kivelson, M. G. 1993 Mirror instability. I – physical mechanism of linear instability. J. Geophys. Res. 98, 91819187.
Squire, J. & Bhattacharjee, A. 2014a Magnetorotational instability: nonmodal growth and the relationship of global modes to the shearing box. Astrophys. J. 797 (1), 67.
Squire, J. & Bhattacharjee, A. 2014b Nonmodal growth of the magnetorotational instability. Phys. Rev. Lett. 113 (2), 025006.
Squire, J. & Bhattacharjee, A. 2016 The magnetic shear-current effect: generation of large-scale magnetic fields by the small-scale dynamo. J. Plasma Phys. 82 (2), 535820201.
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 (2), L25.
Squire, J., Schekochihin, A. A. & Quataert, E. 2017 Amplitude limits and nonlinear damping of shear-Alfvén waves in high-beta low-collisionality plasmas. New J. Phys. 19 (5), 055005-34.
Vasil, G. M. 2015 On the magnetorotational instability and elastic buckling. Proc. R. Soc. Lond. A 471, 20140699.
Verscharen, D., Chandran, B. D. G., Klein, K. G. & Quataert, E. 2016 Collisionless isotropization of the solar-wind protons by compressive fluctuations and plasma instabilities. Astrophys. J. 831 (2), 128.
Yoon, P. H., Wu, C. S. & de Assis, A. S. 1993 Effect of finite ion gyroradius on the firehose instability in a high beta plasma. Phys. Fluids B 5 (7), 19711979.
Yuan, F. & Narayan, R. 2014 Hot accretion flows around black holes. Annu. Rev. Astron. Astrophys. 52, 529588.
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