Skip to main content Accessibility help

Linear Vlasov theory of a magnetised, thermally stratified atmosphere

  • Rui Xu (a1) and Matthew W. Kunz (a1) (a2)


The stability of a collisionless, magnetised plasma to local convective disturbances is examined, with a focus on kinetic and finite-Larmor-radius effects. Specific application is made to the outskirts of galaxy clusters, which contain hot and tenuous plasma whose temperature increases in the direction of gravity. At long wavelengths (the ‘drift-kinetic’ limit), we obtain the kinetic version of the magnetothermal instability (MTI) and its Alfvénic counterpart (Alfvénic MTI), which were previously discovered and analysed using a magnetofluid (i.e. Braginskii) description. At sub-ion-Larmor scales, we discover an overstability driven by the electron-temperature gradient of kinetic-Alfvén drift waves – the electron MTI (eMTI) – whose growth rate is even larger than the standard MTI. At intermediate scales, we find that ion finite-Larmor-radius effects tend to stabilise the plasma. We discuss the physical interpretation of these instabilities in detail, and compare them both with previous work on magnetised convection in a collisional plasma and with temperature-gradient-driven drift-wave instabilities well known to the magnetic-confinement-fusion community. The implications of having both fluid and kinetic scales simultaneously driven unstable by the same temperature gradient are briefly discussed.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Linear Vlasov theory of a magnetised, thermally stratified atmosphere
      Available formats

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Linear Vlasov theory of a magnetised, thermally stratified atmosphere
      Available formats

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Linear Vlasov theory of a magnetised, thermally stratified atmosphere
      Available formats


Corresponding author

Email address for correspondence:


Hide All
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.
Antonsen, T. M. Jr. & Lane, B. 1980 Kinetic equations for low frequency instabilities in inhomogeneous plasmas. Phys. Fluids 23, 12051214.
Avara, M. J., Reynolds, C. S. & Bogdanović, T. 2013 Role of magnetic field strength and numerical resolution in simulations of the heat-flux-driven buoyancy instability. Astrophys. J. 773, 171180.
Balbus, S. A. 2000 Stability, instability, and ‘backward’ transport in stratified fluids. Astrophys. J. 534, 420427.
Balbus, S. A. 2001 Convective and rotational stability of a dilute plasma. Astrophys. J. 562, 909917.
Barnes, A. 1966 Collisionless damping of hydromagnetic waves. Phys. Fluids 9, 14831495.
Bogdanović, T., Reynolds, C. S., Balbus, S. A. & Parrish, I. J. 2009 Simulations of magnetohydrodynamics instabilities in intracluster medium including anisotropic thermal conduction. Astrophys. J. 704, 211225.
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205.
Brunt, D. 1927 The period of simple vertical oscillations in the atmosphere. Q. J. R. Meteorol. Soc. 53, 30.
Carilli, C. L. & Taylor, G. B. 2002 Cluster magnetic fields. Annu. Rev. Astron. Astrophys. 40, 319348.
Catto, P. J. & Simakov, A. N. 2004 A drift ordered short mean free path description for magnetized plasma allowing strong spatial anisotropy. Phys. Plasmas 11, 90102.
Catto, P. J., Tang, W. M. & Baldwin, D. E. 1981 Generalized gyrokinetics. Plasma Phys. 23, 639650.
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, 112118.
Coppi, B., Rosenbluth, M. N. & Sagdeev, R. Z. 1967 Instabilities due to temperature gradients in complex magnetic field configurations. Phys. Fluids 10, 582587.
Cowley, S. C., Kulsrud, R. M. & Sudan, R. 1991 Considerations of ion-temperature-gradient-driven turbulence. Phys. Fluids B 3, 27672782.
Dimits, A. M., Bateman, G., Beer, M. A., Cohen, B. I., Dorland, W., Hammett, G. W., Kim, C., Kinsey, J. E., Kotschenreuther, M., Kritz, A. H. et al. 2000 Comparisons and physics basis of tokamak transport models and turbulence simulations. Phys. Plasmas 7, 969983.
Dorland, W., Jenko, F., Kotschenreuther, M. & Rogers, B. N. 2000 Electron temperature gradient turbulence. Phys. Rev. Lett. 85, 55795582.
Fried, B. D. & Conte, S. D. 1961 The Plasma Dispersion Function. Academic.
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502508.
Goodrich, L. C., Durney, C. H. & Grow, R. W. 1972 General recurrence relation for use in evaluating moments of the integrand of the plasma dispersion function. Phys. Fluids 15, 715716.
Heinemann, T. & Quataert, E. 2014 Linear Vlasov theory in the shearing sheet approximation with application to the magneto-rotational instability. Astrophys. J. 792, 7092.
Hellinger, P. & Trávníček, P. M. 2008 Oblique proton fire hose instability in the expanding solar wind: hybrid simulations. J. Geophys. Res. 113 (A12), A10109.
Hellinger, P. & Trávníček, P. M. 2015 Proton temperature-anisotropy-driven instabilities in weakly collisional plasmas: hybrid simulations. J. Plasma Phys. 81 (1), 305810103.
Horton, W. 1999 Drift waves and transport. Rev. Mod. Phys. 71, 735778.
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, 590614.
Islam, T. 2014 The collisionless magnetoviscous-thermal instability. Astrophys. J. 787, 5363.
Kaufman, A. N. 1960 Plasma viscosity in a magnetic field. Phys. Fluids 3, 610616.
Kim, J. Y., Horton, W. & Dong, J. Q. 1993 Electromagnetic effect on the toroidal ion temperature gradient mode. Phys. Fluids B 5, 40304039.
Kingsep, A. S., Chukbar, K. V. & Yan’kov, V. V. 1990 Rev. Plasma Phys. 16, 243.
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.
Kulsrud, R. M. 1964 Teoria dei plasmi (ed. Rosenbluth, M. N.), p. 54. Academic.
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. Atomizdat (U.S.S.R.) and North-Holland Publication.
Kunz, M. W. 2011 Dynamical stability of a thermally stratified intracluster medium with anisotropic momentum and heat transport. Mon. Not. R. Astron. Soc. 417, 602616.
Kunz, M. W., Bogdanović, T., Reynolds, C. S. & Stone, J. M. 2012 Buoyancy instabilities in a weakly collisional intracluster medium. Astrophys. J. 754, 122141.
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.
Landau, L. 1946 On the vibrations of the electronic plasma. Zh. Exp. Teor. Fiz. 16, 574 (English translation: 1946, J. Phys. U.S.S.R., 10, 25).
Latter, H. N. & Kunz, M. W. 2012 The HBI in a quasi-global model of the intracluster medium. Mon. Not. R. Astron. Soc. 423, 19641972.
McCourt, M., Parrish, I. J., Sharma, P. & Quataert, E. 2011 Can conduction induce convection? On the non-linear saturation of buoyancy instabilities in dilute plasmas. Mon. Not. R. Astron. Soc. 413, 12951310.
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, 27012720.
Mikellides, I. G., Tassis, K. & Yorke, H. W. 2011 2D Magnetohydrodynamics simulations of induced plasma dynamics in the near-core region of a galaxy cluster. Mon. Not. R. Astron. Soc. 410, 26022616.
Mikhailovskii, A. B. 1962 Dielectric properties of an inhomogeneous plasma. Nucl. Fusion 2 (3–4), 162.
Mikhailovskii, A. B. 1967 Oscillations of an inhomogeneous plasma. Rev. Plasma Phys. 3, 159.
Mikhailovskii, A. B. 1974 Theory of Plasma Instabilities, Vol. 2, Instabilities of an Inhomogeneous Plasma. Consultants Bureau.
Mikhailovskii, A. B. 1992 Electromagnetic Instabilities in an Inhomogeneous Plasma. IOP.
Mikhailovskii, A. B. & Tsypin, V. S. 1971 Transport equations and gradient instabilities in a high pressure collisional plasma. Plasma Phys. 13, 785798.
Mikhailovskii, A. B. & Tsypin, V. S. 1984 Transport equations of plasma in curvilinear magnetic field. Beiträge aus der Plasmaphysik 24, 335354.
Neronov, A. & Vovk, I. 2010 Evidence for strong extragalactic magnetic fields from fermi observations of Tev blazars. Science 328, 73.
Parrish, I. J., McCourt, M., Quataert, E. & Sharma, P. 2012 The effects of anisotropic viscosity on turbulence and heat transport in the intracluster medium. Mon. Not. R. Astron. Soc. 422, 704718.
Parrish, I. J. & Quataert, E. 2008 Nonlinear simulations of the heat-flux-driven buoyancy instability and its implications for galaxy clusters. Astrophys. J. Lett. 677, L9L12.
Parrish, I. J., Quataert, E. & Sharma, P. 2009 Anisotropic thermal conduction and the cooling flow problem in galaxy clusters. Astrophys. J. 703, 96108.
Parrish, I. J. & Stone, J. M. 2007 Saturation of the magnetothermal instability in three dimensions. Astrophys. J. 664, 135148.
Parrish, I. J., Stone, J. M. & Lemaster, N. 2008 The magnetothermal instability in the intracluster medium. Astrophys. J. 688, 905917.
Planck Collaboration, Ade, P. A. R., Aghanim, N., Arnaud, M., Arroja, F., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J. et al. 2016 Planck 2015 results. XIX. Constraints on primordial magnetic fields. Astron. Astrophys. 594, 19.
Quataert, E. 2008 Buoyancy instabilities in weakly magnetized low-collisionality plasmas. Astrophys. J. 673, 758762.
Quataert, E., Dorland, W. & Hammett, G. W. 2002 The magnetorotational instability in a collisionless plasma. Astrophys. J. 577, 524533.
Quataert, E., Heinemann, T. & Spitkovsky, A. 2015 Linear instabilities driven by differential rotation in very weakly magnetized plasmas. Mon. Not. R. Astron. Soc. 447, 33283341.
Ramos, J. J. 2005 General expression of the gyroviscous force. Phys. Plasmas 12 (11), 112301.
Reynders, J. V. W. 1994 Finite- $\unicode[STIX]{x1D6FD}$ modification of the ion-temperature-gradient-driven instability in a sheared slab geometry. Phys. Plasmas 1, 19531961.
Riquelme, M., Quataert, E. & Verscharen, D. 2016 PIC simulations of the effect of velocity space instabilities on electron viscosity and thermal conduction. Astrophys. J. 824, 123133.
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, 2743.
Rudakov, L. I. & Sagdeev, R. Z. 1961 On the instability of a nonuniform rarefied plasma in a strong magnetic field. Dokl. Akad. Nauk SSSR 138, 581.
Rutherford, P. H. & Frieman, E. A. 1968 Drift instabilities in general magnetic field configurations. Phys. Fluids 11, 569585.
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, 291300.
Schwarzschild, K. 1906 On the equilibrium of the sun’s atmosphere. Nach. Gesell. Wiss. Göttingen 195, 41.
Sharma, P., Hammett, G. W. & Quataert, E. 2003 Transition from collisionless to collisional magnetorotational instability. Astrophys. J. 596, 11211130.
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, 88111.
Snyder, P. B. & Hammett, G. W. 2001 Electromagnetic effects on plasma microturbulence and transport. Phys. Plasmas 8, 744749.
Stix, T. H. 1962 The Theory of Plasma Waves. McGraw-Hill.
Taylor, J. B. & Hastie, R. J. 1968 Stability of general plasma equilibria - I formal theory. Plasma Phys. 10, 479494.
Väisälä, V. 1925 Über die Wirkung der Windschwankungen auf die pilot Beobachtungen. Soc. Sci. Fennica, Commentaliones Phys. Math II 2, 46.
Watson, G. N. 1966 A Treatise on the Theory of Bessel Functions. Cambridge University Press.
MathJax is a JavaScript display engine for mathematics. For more information see


Linear Vlasov theory of a magnetised, thermally stratified atmosphere

  • Rui Xu (a1) and Matthew W. Kunz (a1) (a2)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed