Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T13:50:44.283Z Has data issue: false hasContentIssue false

Collisional behaviors of astrophysical collisionless plasmas

Published online by Cambridge University Press:  27 February 2015

Antoine Bret*
Affiliation:
ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus Universitario de Ciudad Real, 13071 Ciudad Real, Spain
*
Email address for correspondence: [email protected]

Abstract

In collisional fluids, a number of key processes rely on the frequency of binary collisions. Collisions seem necessary to generate a shock wave when two fluids collide fast enough, to fulfill the Rankine–Hugoniot (RH) relations, to establish an equation of state or a Maxwellian distribution. Yet, these seemingly collisional features are routinely either observed or assumed, in relation with collisionless astrophysical plasmas. This article will review our current answers to the following questions: How do colliding collisionless plasmas end-up generating a shock as if they were fluids? To which extent are the RH relations fulfilled in this case? Do collisionless shocks propagate like fluid ones? Can we use an equation of state to describe collisionless plasmas, like MHD codes for astrophysics do? Why are Maxwellian distributions ubiquitous in particle-in-cell simulations of collisionless shocks? Time and length scales defining the border between the collisional and the collisionless behavior will be given when relevant. In general, when the time and length scales involved in the collisionless processes responsible for the fluid-like behavior may be neglected, the system may be treated like a fluid.

Type
Research Article
Copyright
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

REFERENCES

Alves, E. P., Grismayer, T., Fonseca, R. A. and Silva, L. O. 2014 Electron-scale shear instabilities: magnetic field generation and particle acceleration in astrophysical jets. New J. Phys. 16 (3), 035007.Google Scholar
Anile, A. M. and Russo, G. 1987 Linear stability for plane relativistic shock waves. Phys. 30, 10451051.Google Scholar
Arad, I. and Lynden-Bell, D. 2005 Inconsistency in theories of violent relaxation. Mon. Not. R. Astron. Soc. 361, 385395.Google Scholar
Baalrud, S. D. 2010 Kinetic theory of instability-enhanced collective interactions in plasma. PhD thesis, University of Wisconsin-Madison.Google Scholar
Baalrud, S. D., Callen, J. D. and Hegna, C. C. 2009 Instability-enhanced collisional effects and Langmuir's paradox. Phys. Rev. Lett. 102, 245005.Google Scholar
Baalrud, S. D., Callen, J. D. and Hegna, C. C. 2010 Kinetic theory of instability-enhanced collisional effects. Phys. Plasmas 17 (5), 055704.Google Scholar
Balbus, S. A. and Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70, 153.Google Scholar
Bale, S. D., Mozer, F. S. and Horbury, T. S. 2003 Density-transition scale at quasiperpendicular collisionless shocks. Phys. Rev. Lett. 91, 265004.Google Scholar
Balogh, A. and Treumann, R. A. 2013 Physics of Collisionless Shocks: Space Plasma Shock Waves. New York: Springer.Google Scholar
Bamba, A., Yamazaki, R., Ueno, M. and Koyama, K. 2003 Small-scale structure of the SN 1006 shock with chandra observations. Astrophys. J. 589, 827837.Google Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press.Google Scholar
Bates, J. W. and Montgomery, D. C. 2000 The d'yakov-kontorovich instability of shock waves in real gases. Phys. Rev. Lett. 84, 1180.Google Scholar
Bell, A. R. 2004 Turbulent amplification of magnetic field and diffusive shock acceleration of cosmic rays. Mon. Not. R. Astron. Soc. 353, 550.Google Scholar
Bell, A. R. 1978 The acceleration of cosmic rays in shock fronts. I. Mon. Not. R. Astron. Soc. 182, 147.Google Scholar
Binney, J. and Tremaine, S. 2011 Galactic Dynamics, 2nd edn.Princeton: Princeton University Press.Google Scholar
Blandford, R. and Eichler, D. 1987 Particle acceleration at astrophysical shocks: a theory of cosmic ray origin. Phys. Rep. 154 (1), 175.CrossRefGoogle Scholar
Blandford, R. D. and Ostriker, J. P. 1978 Particle acceleration by astrophysical shocks. Astrophys. J. 221, L29.Google Scholar
Bouquet, S., Romain, T. and Chieze, J. P. 2000 Analytical study and structure of a stationary radiative shock. Astrophys. J. 127, 245252.Google Scholar
Bret, A. and Deutsch, C. 2006 A fluid approach to linear beam plasma electromagnetic instabilities. Phys. Plasmas 13 (4), 042106.Google Scholar
Bret, A., Dieckmann, M. and Deutsch, C. 2006 Oblique electromagnetic instabilities for a hot relativistic beam interacting with a hot and magnetized plasma. Phys. Plasmas 13, 082109.Google Scholar
Bret, A., Gremillet, L., Bénisti, D. and Lefebvre, E. 2008 Exact relativistic kinetic theory of an electron-beam plasma system: hierarchy of the competing modes in the system-parameter space. Phys. Rev. Lett. 100, 205008.Google Scholar
Bret, A., Gremillet, L. and Dieckmann, M. E. 2010 Multidimensional electron beam-plasma instabilities in the relativistic regime. Phys. Plasmas 17, 120501.Google Scholar
Bret, A., Stockem, A., Fiúza, F., Ruyer, C., Gremillet, L., Narayan, R. and Silva, L. O. 2013 Collisionless shock formation, spontaneous electromagnetic fluctuations, and streaming instabilities. Phys. Plasmas 20, 042102.Google Scholar
Bret, A., Stockem, A., Narayan, R. and Silva, L. O. 2014 Collisionless Weibel shocks: full formation mechanism and timing. Phys. Plasmas 21, 072301.Google Scholar
Broderick, A. E., Chang, P. and Pfrommer, C. 2012 The Cosmological impact of luminous TeV blazars. I. Implications of plasma instabilities for the Intergalactic magnetic field and extragalactic gamma-ray background. Astrophys. J. 752, 22.Google Scholar
Buneman, O. 1964 Models of collisionless shock fronts. Phys. Fluids (1958-1988) 7, S3S8.Google Scholar
Bykov, A. M. and Treumann, R. A. 2011 Fundamentals of collisionless shocks for astrophysical application, 2. Relativistic shocks. Astron. Astrophys. Rev. 19, 167.Google Scholar
Caprioli, D. 2014 Hybrid simulations of particle acceleration at shocks. Nuclear Phys. B Proc. Suppl. 256, 4855.Google Scholar
Caprioli, D., Pop, A. R. and Spitkovsky, A. 2015 Simulations and Theory of Ion Injection at Non-Relativistic Collisionless Shocks. Astrophys. J. Lett. 798, L28.Google Scholar
Caprioli, D. and Spitkovsky, A. 2014 Simulations of ion acceleration at non-relativistic shocks. I. Acceleration efficiency. Astrophys. J. 783, 91.Google Scholar
Cercignani, C. 1976 Theory and Application of the Boltzmann Equation. New York: Elsevier.Google Scholar
Chandrasekhar, S. 1943 Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 189.Google Scholar
Chang, P., Spitkovsky, A. and Arons, J. 2008 Long-term evolution of magnetic turbulence in relativistic collisionless shocks: electron-positron plasmas. Astrophys. J. 674, 378387.CrossRefGoogle Scholar
Chen, H.et al. 2010 Relativistic quasimonoenergetic positron jets from intense laser-solid interactions. Phys. Rev. Lett. 105, 015003.Google Scholar
Chew, G. F., Goldberger, M. L. and Low, F. E. 1956 The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. R. Soc. Lond. Proc. Ser. A 236, 112118.Google Scholar
Davis, S. P., Capdessus, R., d'Humières, E., Jequier, S., Andriyash, I. and Tikhonchuk, V. 2013 Numerical simulations of energy transfer in counter-streaming plasmas. High Energy Density Phys. 9 (1), 231238.Google Scholar
Dieckmann, M. E. and Bret, A. 2009 Particle-in-cell simulation of a strong double layer in a nonrelativistic plasma flow: electron acceleration to ultrarelativistic speeds. Astrophys. J. 694, 154.CrossRefGoogle Scholar
Dieckmann, M. E., Ahmed, H., Sarri, G., Doria, D., Kourakis, I., Romagnani, L., Pohl, M. and Borghesi, M. 2013a Parametric study of non-relativistic electrostatic shocks and the structure of their transition layer. Phys. Plasmas 20 (4), 042111.Google Scholar
Dieckmann, M. E., Drury, L. O. C. and Shukla, P. K. 2006 On the ultrarelativistic two-stream instability, electrostatic turbulence and Brownian motion. New J. Phys. 8 (3), 40.Google Scholar
Dieckmann, M. E., Sarri, G., Doria, D., Ahmed, H. and Borghesi, M. 2014 Evolution of slow electrostatic shock into a plasma shock mediated by electrostatic turbulence. New J. Phys. 16 (7), 073001.Google Scholar
Dieckmann, M. E., Sarri, G., Doria, D., Pohl, M. and Borghesi, M. 2013b Modification of the formation of high-Mach number electrostatic shock-like structures by the ion acoustic instability. Phys. Plasmas 20 (10), 102112.Google Scholar
Dubinov, A. E., Dubinova, I. D. and Gordienko, V. A. 2006 Solitary electrostatic waves are possible in unmagnetized symmetric pair plasmas. Phys. Plasmas 13, 082111.Google Scholar
Dunkel, J. and Hänggi, P. 2006 Relativistic Brownian motion: from a microscopic binary collision model to the Langevin equation. Phys. Rev. E 74, 051106.Google Scholar
D'yakov, S. P. 1954 Stability of shock waves in inert gases. Zh. Eksp. Teor. Fiz. 27, 288.Google Scholar
Farris, M. H., Russell, C. T., Thomsen, M. F. and Gosling, J. T. 1992 ISEE 1 and 2 observations of the high beta shock. J. Geophys. Res.: Space Phys. 97, 19121.Google Scholar
Fermi, E. 1949 On the origin of the cosmic radiation. Phys. Rev. 75, 11691174.Google Scholar
Feynman, R. F. and Leighton, R. B. 1977 The Feynman Lectures on Physics, Vol. 1. Reading, Massachusetts: Addison-Wesley.Google Scholar
Forslund, D. W. and Shonk, C. R. 1970 Formation and structure of electrostatic collisionless shocks. Phys. Rev. Lett. 25, 1699.CrossRefGoogle Scholar
Fried, B. D. 1959 Mechanism for instability of transverse plasma waves. Phys. Fluids (1958-1988) 2, 337.Google Scholar
Gabor, D., Ash, E. A. and Dracott, D. 1955 Langmuir's paradox. Nature 176, 916919.Google Scholar
Gedalin, M., Smolik, E., Spitkovsky, A. and Balikhin, M. 2012 Electron heating by filamentary instability. EPL 97, 35002.Google Scholar
Ghavamian, P., Schwartz, S. J., Mitchell, J., Masters, A. and Laming, J. M. 2013 Electron-ion temperature equilibration in collisionless shocks: the supernova Remnant-Solar Wind connection. Space Science Rev. 178, 633.Google Scholar
Godfrey, B. B., Shanahan, W. R. and Thode, L. E. 1975 Linear theory of a cold relativistic beam propagating along an external magnetic field. Phys. Fluids 18, 346.Google Scholar
Gould, R. W., O'Neil, T. M. and Malmberg, J. H. 1967 Plasma wave echo. Phys. Rev. Lett. 19, 219222.Google Scholar
Grismayer, T., Alves, E. P., Fonseca, R. A. and Silva, L. O. 2013 dc-magnetic-field generation in unmagnetized shear flows. Phys. Rev. Lett. 111, 015005.Google Scholar
Gruzinov, A. 2008 GRB: magnetic fields, cosmic rays, and emission from first principles? ArXiv:0803.1182.Google Scholar
Gurnett, D. A. and Bhattacharjee, A. 2005 Introduction to Plasma Physics: With Space and Laboratory Applications. Cambridge: Cambridge University Press.Google Scholar
Hawley, J. F., Smarr, L. L. and Wilson, J. R. 1984 A numerical study of nonspherical black hole accretion. II - Finite differencing and code calibration. Astrophys. J. 55, 211246.Google Scholar
Hoyaux, M. F. 1968 Arc Physics. New York: Springer-Verlag.Google Scholar
Ichimaru, S. 1973 Basic Principles of Plasma Physics: A Statistical Approach. New York: W.A. Benjamin.Google Scholar
Kadomtsev, B. B. 1965 Plasma Turbulence. London and New York: Academic Press.Google Scholar
Kahn, F. D. 1958 Collision of two highly ionized clouds of gas. Rev. Mod. Phys. 30, 10691072.Google Scholar
Karimabadi, H., Omidi, N. and Quest, K. B. 1991 Two-dimensional simulations of the ion/ion acoustic instability and electrostatic shocks. Geophys. Res. Lett. 18 (10), 18131816.Google Scholar
Kasper, J. C., Lazarus, A. J. and Gary, S. P. 2008 Hot solar-wind helium: direct evidence for local heating by Alfvén-cyclotron dissipation. Phys. Rev. Lett. 101, 261103.Google Scholar
Kato, T. N. and Takabe, H. 2010 Electrostatic and electromagnetic instabilities associated with electrostatic shocks: two-dimensional particle-in-cell simulation. Phys. Plasmas 17 (3), 032114.CrossRefGoogle Scholar
Klimontovich, Yu. L. 1967 The Statistical Theory of Non-equilibrium Processes in Plasma. Cambridge, MA: M.I.T. Press.Google Scholar
Kontorovich, V. M. 1958 Concerning the stability of shock waves. Sov. J. Exp. Theor. Phys. 6, 1179.Google Scholar
Krall, N. A. 1997 What do we really know about collisionless shocks? Adv. Space Res. 20, 715724.Google Scholar
Krymskii, G. F. 1977 A regular mechanism for the acceleration of charged particles on the front of a shock wave. Sov. Phys. Dokl. 22, 327.Google Scholar
Kunz, M. W., Schekochihin, A. A. and Stone, J. M. 2014 Firehose and mirror instabilities in a collisionless shearing plasma. Phys. Rev. Lett. 112, 205003.Google Scholar
Landau, L. D. and Lifshitz, E. M. 1980 The Classical Theory of Fields. Oxford: Elsevier Science.Google Scholar
Landau, L. D. and Lifshitz, E. M. 2013 Fluid Mechanics. Oxford: Elsevier Science.Google Scholar
Langmuir, I. 1925 Scattering of electrons in ionized gases. Phys. Rev. 26, 585613.Google Scholar
Lapenta, G., Markidis, S. and Kaniadakis, G. 2009 Computer experiments on the relaxation of collisionless plasmas. J. Stat. Mech.: Theory Exp. 2009 (02), P02024.Google Scholar
Lapenta, G., Markidis, S., Marocchino, A. and Kaniadakis, G. 2007 Relaxation of relativistic plasmas under the effect of wave-particle interactions. Astrophys. J. 666 (2), 949.Google Scholar
Lembège, B. and Dawson, J. M. 1987 Self-consistent study of a perpendicular collisionless and nonresistive shock. Phys. Fluids (1958-1988) 30, 17671788.Google Scholar
Lembège, B., Giacalone, J., Scholer, M., Hada, T., Hoshino, M., Krasnoselskikh, V., Kucharek, H., Savoini, P. and Terasawa, T. 2004 Selected problems in collisionless-shock physics. Space Sci. Rev. 110, 161226.Google Scholar
Lembège, B. and Savoini, P. 1992 Nonstationarity of a two-dimensional quasiperpendicular supercritical collisionless shock by self-reformation. Phys. Fluids B: Plasma Phys. (1989-1993) 4, 35333548.Google Scholar
Lemoine, M. 2015 Non-linear collisionless damping of Weibel turbulence in relativistic blast waves. J. Plasma Phys. 81, 455810101.Google Scholar
Lemoine, M., Pelletier, G., Gremillet, L. and Plotnikov, I. 2014 A fast current-driven instability in relativistic collisionless shocks. Europhysics Letters 106, 55001.Google Scholar
Lemoine, M., Pelletier, G. and Revenu, B. 2006 On the efficiency of fermi acceleration at relativistic shocks. Astrophys. J. Lett. 645, L129.Google Scholar
Lynden-Bell, D. 1967 Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 136, 101.Google Scholar
Matsukiyo, S. 2010 Mach number dependence of electron heating in high Mach number quasiperpendicular shocks. Phys. Plasmas 17, 042901.Google Scholar
Mazelle, C., Lembège, B., Morgenthaler, A., Meziane, K., Horbury, T. S., Génot, V., Lucek, E. A. and Dandouras, I. 2010 Self-reformation of the quasi-perpendicular shock: CLUSTER observations. In: 12th Int. Solar Wind Conf., AIP Conference Proceedings, Vol. 1216, 471–474.Google Scholar
Mikhailovskii, A. B. 1974 Theory of Plasma Instabilities, Vol. 1. New York: Consultant Bureau.CrossRefGoogle Scholar
Murdin, P. and Murdin, L. 1985 Supernovae. Cambridge: Cambridge University Press.Google Scholar
Nakar, E., Bret, A. and Milosavljević, M. 2011 Two-stream-like instability in dilute hot relativistic beams and astrophysical relativistic shocks. Astrophys. J. 738, 93.Google Scholar
Niemiec, J., Pohl, M., Bret, A. and Wieland, V. 2012 Nonrelativistic parallel shocks in unmagnetized and weakly magnetized plasmas. Astrophys. J. 759, 73.Google Scholar
Oohara, W., Date, D. and Hatakeyama, R. 2005 Electrostatic waves in a paired fullerene-ion plasma. Phys. Rev. Lett. 95, 175003.CrossRefGoogle Scholar
Oohara, W. and Hatakeyama, R. 2003 Pair-ion plasma generation using fullerenes. Phys. Rev. Lett. 91, 205005.CrossRefGoogle ScholarPubMed
Peeters, A. G. and Strintzi, D. 2008 The Fokker-Planck equation, and its application in plasma physics. Ann. Phys. 520, 142157.Google Scholar
Pitaevskii, L. P. and Lifshitz, E. M. 1981 Physical Kinetics. Oxford: Elsevier Science.Google Scholar
Plotnikov, I., Pelletier, G. and Lemoine, M. 2013 Particle transport and heating in the microturbulent precursor of relativistic shocks. Mon. Not. R. Astron. Soc. 430, 1280.CrossRefGoogle Scholar
Quataert, E., Dorland, W. and Hammett, G. W. 2002 The magnetorotational instability in a collisionless plasma. Astrophys. J. 577 (1), 524.Google Scholar
Raymond, J. C., Winkler, P., Frank, B., William, P., Lee, J.-J. and Park, S. 2010 Non-Maxwellian hα profiles in Tycho's supernova remnant. Astrophys. J. 712 (2), 901.Google Scholar
Riquelme, M., Quataert, E. and Verscharen, D. 2015 PIC Simulations of continuously driven mirror and ion cyclotron instabilities in high beta astrophysical and heliospheric plasmas. Astrophys. J. 800, 27.Google Scholar
Riquelme, M. A., Quataert, E., Sharma, P. and Spitkovsky, A. 2012 Local two-dimensional particle-in-cell simulations of the collisionless magnetorotational instability. Astrophys. J. 755, 50.Google Scholar
Sadowski, A., Narayan, R., McKinney, J. C. and Tchekhovskoy, A. 2014 Numerical simulations of super-critical black hole accretion flows in general relativity. Mon. Not. R. Astron. Soc. 439, 503520.Google Scholar
Sagdeev, R. Z. 1966 Cooperative phenomena and shock waves in collisionless plasmas. Rev. Plasma Phys. 4, 23.Google Scholar
Sagdeev, R. Z. and Kennel, C. F. 1991 Collisionless shock waves. Sci. Am. 264, 106113.Google Scholar
Sarri, G.et al. 2013 Table-top laser-based source of femtosecond, collimated, ultrarelativistic positron beams. Phys. Rev. Lett. 110, 255002.Google Scholar
Schwartz, S. J., Henley, E., Mitchell, J. and Krasnoselskikh, V. 2011 Electron temperature gradient scale at collisionless shocks. Phys. Rev. Lett. 107, 215002.Google Scholar
Shaisultanov, R., Lyubarsky, Y. and Eichler, D. 2012 Stream instabilities in relativistically hot plasma. Astrophys. J. 744, 182.CrossRefGoogle Scholar
Sharma, P., Hammett, G. W., Quataert, E. and Stone, J. M. 2006 Shearing box simulations of the MRI in a collisionless plasma. Astrophys. J. 637, 952.Google Scholar
Silva, L. O., Fonseca, R. A., Tonge, J. W., Mori, W. B. and Dawson, J. M. 2002 On the role of the purely transverse Weibel instability in fast ignitor scenarios. Phys. Plasmas 9, 2458.Google Scholar
Sironi, L. 2014 Private Communication.Google 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. and Giannios, D. 2014 Relativistic pair beams from TeV blazars: a source of reprocessed GeV emission rather than intergalactic heating. Astrophys. J. 787, 49.Google Scholar
Sironi, L. and 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.Google Scholar
Sironi, L. and Spitkovsky, A. 2009 Particle acceleration in relativistic magnetized collisionless pair shocks: dependence of shock acceleration on magnetic obliquity. Astrophys. J. 698, 1523.Google Scholar
Sironi, L. and Spitkovsky, A. 2011 Particle acceleration in relativistic magnetized collisionless electron-ion shocks. Astrophys. J. 726, 75.Google Scholar
Sironi, L., Spitkovsky, A. and Arons, J. 2013 The maximum energy of accelerated particles in relativistic collisionless shocks. Astrophys. J. 771, 54.Google Scholar
Spitkovsky, A. 2008 Particle acceleration in relativistic collisionless shocks: Fermi process at last? Astrophys. J. Lett. 682, L5L8.Google Scholar
Spitzer, L. 2013 Physics of Fully Ionized Gases: Second Revised Edition. Dover Publications.Google Scholar
Stil, J. M. and Irwin, J. A. 2001 GSH 1380194: an old supernova remnant in the far outer galaxy. Astrophys. J. 563 (2), 816.Google Scholar
Stockem, A., Fiúza, F., Bret, A., Fonseca, R. A. and Silva, L. O. 2014a Exploring the nature of collisionless shocks under laboratory conditions. Sci. Rep. 4, 3934.CrossRefGoogle ScholarPubMed
Stockem, A., Fiúza, F., Fonseca, R. A. and Silva, L. O. 2012 The impact of kinetic effects on the properties of relativistic electron-positron shocks. Plasma Phys. Control. Fusion 54, 125004.CrossRefGoogle Scholar
Stockem, A., Grismayer, T., Fonseca, R. A. and Silva, L. O. 2014b Electromagnetic field generation in the downstream of electrostatic shocks due to electron trapping. Phys. Rev. Lett. 113, 105002.Google Scholar
Stone, J. M. and Norman, M. L. 1992 ZEUS-2D: a radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. I - The hydrodynamic algorithms and tests. Astrophys. J. 80, 753790.Google Scholar
Takabe, H.et al. 2008 High-mach number collisionless shock and photo-ionized non-LTE plasma for laboratory astrophysics with intense lasers. Plasma Phys. Control. Fusion 50, 124057.CrossRefGoogle Scholar
Tidman, D. A. and Krall, N. A. 1971 Shock Waves in Collisionless Plasmas. New York: Wiley-Interscience.Google Scholar
Treumann, R. A. 2009 Fundamentals of collisionless shocks for astrophysical application, 1. Non-relativistic shocks. Astron. Astrophys. Rev. 17, 409535.Google Scholar
Usov, V. V. 1992 Millisecond pulsars with extremely strong magnetic fields as a cosmological source of gamma-ray bursts. Nature 357, 472474.Google Scholar
Vedenov, A. A. 1963 Quasi-linear plasma theory (theory of a weakly turbulent plasma). J. Nuclear Energy. Part C, Plasma Phys. Accelerators, Thermonuclear Res. 5 (3), 169.Google Scholar
Verheest, F. 2005 On the nonexistence of large amplitude stationary solitary waves in symmetric unmagnetized pair plasmas. Nonlinear Process. Geophys. 12, 569574.Google Scholar
Villani, C. 2014 Particle systems and nonlinear landau damping. Phys. Plasmas 21 (3), 030901.Google Scholar
Vink, J., Broersen, S., Bykov, A. and Gabici, S. 2014 On the electron-ion temperature ratio established by collisionless shocks. ArXiv:1407.4499.Google Scholar
Weibel, E. S. 1959 Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution. Phys. Rev. Lett. 2, 83.Google Scholar
Wouchuk, J. G. and López Cavada, J. 2004 Spontaneous acoustic emission of a corrugated shock wave in the presence of a reflecting surface. Phys. Rev. E 70, 046303.Google Scholar
Yang, Z. W., Lu, Q. M., Lembège, B. and Wang, S. 2009 Shock front nonstationarity and ion acceleration in supercritical perpendicular shocks. J. Geophys. Res.: Space Phys. 114 (A3), A03111.Google Scholar
Yuan, F. and Narayan, R. 2014 Hot accretion flows around black holes. Annu. Rev. Astron. Astrophys. 52, 529.Google Scholar
Zel'dovich, Ya B. and Raizer, Yu P. 2002 Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Mineola, NY: Dover Publications.Google Scholar