Advertisement

The Astronomy and Astrophysics Review

, Volume 17, Issue 4, pp 409–535 | Cite as

Fundamentals of collisionless shocks for astrophysical application, 1. Non-relativistic shocks

  • R. A. TreumannEmail author
Review Article

Abstract

A comprehensive review is given of the theory and properties of nonrelativistic shocks in hot collisionless plasmas—in view of their possible application in astrophysics. Understanding non-relativistic collisionless shocks is an indispensable step towards a general account of collisionless astrophysical shocks of high Mach number and of their effects in dissipating flow-energy, in heating matter, in accelerating particles to high—presumably cosmic-ray—energies, and in generating detectable radiation from radio to X-rays. Non-relativistic shocks have Alfvénic Mach numbers \({{\fancyscript{M}}_A\ll \sqrt{m_i/m_e}(\omega_{pe}/\omega_{ce})}\), where m i /m e is the ion-to-electron mass ratio, and ω pe , ω ce are the electron plasma and cyclotron frequencies, respectively. Though high, the temperatures of such shocks are limited (in energy units) to T < m e c 2. This means that particle creation is inhibited, classical theory is applicable, and reaction of radiation on the dynamics of the shock can be neglected. The majority of such shocks are supercritical, meaning that non-relativistic shocks are unable to self-consistently produce sufficient dissipation and, thus, to sustain a stationary shock transition. As a consequence, supercritical shocks act as efficient particle reflectors. All these shocks are microscopically thin, with shock-transition width of the order of the ion inertial length λ i = c/ω pi (with ω pi the ion plasma frequency). The full theory of such shocks is developed, and the different possible types of shocks are defined. Since all collisionless shocks are magnetised, the most important distinction is between quasi-perpendicular and quasi-parallel shocks. The former propagate about perpendicularly, the latter roughly parallel to the upstream magnetic field. Their manifestly different behaviours are described in detail. In particular, although both types of shocks are non-stationary, they have completely different reformation cycles. From numerical full-particle simulations it becomes evident that, on ion-inertial scales close to the shock transition, all quasi-parallel collisionless supercritical shocks are locally quasi-perpendicular. This property is of vital importance for the particle dynamics near the quasi-parallel shock front. Considerable interest focusses on particle acceleration and the generation of radiation. Radiation from non-relativistic shocks results mainly in wave–wave interactions among various plasma waves. Non-thermal charged particles can be further accelerated to high energies by a Fermi-like mechanism. The important question is whether the shock can pre-accelerate shock-reflected particles to sufficiently high energies in order to create the seed-population of the non-thermal particles required by the Fermi mechanism. Based on preliminary full-particle numerical simulations, this question is answered affirmatively. Such simulations provide ample evidence that collisionless shocks with high-Mach numbers—even when non-relativistic—could probably by themselves produce the energetic seed-particle population for the Fermi-process.

Keywords

Collisionless shocks Supercritical shocks Shock kinetics Shock reformation Shock acceleration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amano T, Hoshino M (2007) Electron injection at high Mach number quasi-perpendicular shocks: surfing and drift acceleration. Astrophys J 661: 190–202ADSCrossRefGoogle Scholar
  2. Bale SD, Mozer FS (2007) Measurement of large parallel and perpendicular electric fields on electron spatial scales in the terrestrial bow shock. Phys Rev Lett 98: 205001. doi: 10.1103/PhysRevLett.98.205001 ADSCrossRefGoogle Scholar
  3. Bale SD et al (2002) Electrostatic turbulence and Debye-scale structures associated with electron thermalization at collisionless shocks. Astrophys J 575: L25–L28ADSCrossRefGoogle Scholar
  4. Bale SD, Mozer FS, Horbury TS (2003) Density-transition scale at quasiperpendicular collisionless shocks. Phys Rev Lett 91: 265004ADSCrossRefGoogle Scholar
  5. Balikhin MA, Krasnoselskikh V, Gedalin MA (1995) The scales in quasiperpendicular shocks. Adv Space Res 15: 247ADSCrossRefGoogle Scholar
  6. Balikhin MA et al (2005) Ion sound wave packets at the quasiperpendicular shock front. Geophys Res Lett 32: L24106. doi: 10.1029/2005GL024660 ADSCrossRefGoogle Scholar
  7. Balogh A, Riley P (2005) Overview of heliospheric shocks. In: Jokipii JR, Sonett CP, Giampapa MS (eds) Cosmic winds and the heliosphere. The University of Arizona Press, Tucson, pp 359–388Google Scholar
  8. Balogh A, Klein KL, Treumann RA (2009) The physics of shock waves in collisionless space plasmas. ISSI SR-10, ESA-ESTEC, Noordwijk, The Netherlands, pp 1–500Google Scholar
  9. Baumjohann W, Treumann RA (1996) Basic space plasma physics. Imperial College Press, LondonGoogle Scholar
  10. Behlke R et al (2003) Multi-point electric field measurements of short large-amplitude magnetic structures (SLAMS) at the Earth’s quasi-parallel bow shock. Geophys Res Lett 30: 1177. doi: 10.1029/2002GL015871 ADSCrossRefGoogle Scholar
  11. Behlke R et al (2004) Solitary structures associated with short large-amplitude magnetic structures (SLAMS) upstream of the Earth’s quasiparallel bow shock. Geophys Res Lett 31: L16805. doi: 10.1029/2004GL019524 ADSCrossRefGoogle Scholar
  12. Biskamp D (1973) Collisionless shock waves in plasmas. Nucl Fusion 13: 719–740Google Scholar
  13. Biskamp D (2003) Magnetohydrodynamic turbulence. Cambridge University Press, CambridgezbMATHGoogle Scholar
  14. Bisnovatyi-Kogan GS, Silich SA (1995) Shock-wave propagation in the nonuniform interstellar medium. Rev Mod Phys 67: 661–712ADSCrossRefGoogle Scholar
  15. Blandford R, Eichler D (1987) Particle acceleration at astrophysical shocks: a theory of cosmic ray origin. Phys Rep 154: 1–75ADSCrossRefGoogle Scholar
  16. Burgess D (1987) Shock drift acceleration at low energies. J Geophys Res 92: 1119–1130ADSCrossRefGoogle Scholar
  17. Burgess D (1995) Collisinless shocks. In: Kivelson MG, Russell CT (eds) Introduction to space physics. Cambridge University Press, Cambridge, pp 129–163Google Scholar
  18. Burlaga LF et al (2008) Magnetic fields at the solar wind termination shock. Nature 454: 75–77. doi: 10.1038/nature07029 ADSCrossRefGoogle Scholar
  19. Coroniti FV (1970) Dissipation discontinuities in hydromagnetic shock waves. J Plasma Phys 4: 265ADSCrossRefGoogle Scholar
  20. Décréau PME et al (2001) Early results from the Whisper instrument on cluster: an overview. Ann Geophysicae 19: 1241–1258ADSGoogle Scholar
  21. De Hoffman F, Teller E (1950) Magneto-hydrodynamic shocks. Phys Rev 80: 692–703ADSCrossRefGoogle Scholar
  22. Dickel JR, Wang S (2004) Non-thermal X-ray and radio emission from the SNR N 157B. Adv Space Res 33: 446–449. doi: 10.1016/j.asr.2003.08.023 ADSCrossRefGoogle Scholar
  23. Diehl R (ed) et al (2002) The astrophysics of galactic cosmic rays. Space Science Series of ISSI, vol 13. Springer, New YorkGoogle Scholar
  24. Dubouloz N, Scholer M (1995) Two-dimensional simulations of magnetic pulsations upstream of the Earth’s bow shock. J Geophys Res 100: 9461–9474ADSCrossRefGoogle Scholar
  25. Edmiston JP, Kennel CF (1984) A parametric survey of the first critical Mach number for a fast MHD shock. J Plasma Phys 32: 429–441ADSCrossRefGoogle Scholar
  26. Eselevich VG (1982) Shock-wave structure in collisionless plasmas from results of laboratory experiments. Space Sci Rev 32: 65–81ADSCrossRefGoogle Scholar
  27. Feldman WC et al (1983) Electron velocity distributions near the earth’s bow shock. J Geophys Res 87: 96–110ADSCrossRefGoogle Scholar
  28. Formisano V, Torbert R (1982) Ion acoustic wave forms generated by ion-ion streams at the earth’s bow shock. Geophys Res Lett 9: 207–210ADSCrossRefGoogle Scholar
  29. Galeev AA, Krasnosel’skikh VV, Lobzin VV (1988) Fine structure of the front of a quasi-perpendicular supercritical collisionless shock wave. Sov J Plasma Phys 14: 697–702Google Scholar
  30. Gedalin MA (1997) Ion heating in oblique low-Mach number shocks. Geophys Res Lett 24: 2511–2514ADSCrossRefGoogle Scholar
  31. Giacalone J (2004) Large-scale hybrid simulations of particle acceleration at a parallel shock. Astrophys J 609: 452–458ADSCrossRefGoogle Scholar
  32. Giacalone J (2005) The efficient acceleration of thermal protons by perpendicular shocks. Astrophys J 628: L37–L40ADSCrossRefGoogle Scholar
  33. Giacalone J, Neugebauer M (2008) The energy spectrum of energetic particles downstream of turbulent collisionless shocks. Astrophys J 673: 629–636ADSCrossRefGoogle Scholar
  34. Ginzburg VL, Zheleznyakov VV (1958) On the possible mechanisms of sporadic solar radio emission ( radiation in an isotropic plasma). Sov Astron 2: 653ADSGoogle Scholar
  35. Goldstein ML, Roberts DA, Matthaeus WM (1995) Magnetohydrodynamic turbulence in the solar wind. Ann Rev Astron Astrophys 33: 283–325ADSCrossRefGoogle Scholar
  36. Goodrich CC, Scudder JD (1984) The adiabatic energy change of plasma electrons and the frame dependence of the cross-shock potential at collisionless magnetosonic shock waves. J Geophys Res 89: 6654–6662ADSCrossRefGoogle Scholar
  37. Gurnett DA (1985) Plasma waves and instabilities. In: Tsurutani BT, Stone RG (eds) Collisionless shocks in the heliosphere: reviews of current research. Amer Geophys Union, Washington DC, pp 207–224Google Scholar
  38. Gurnett DA, Kurth WS (2008) Intense plasma waves at and near the solar wind termination shock. Nature 454: 78–80. doi: 10.1038/nature07023 ADSCrossRefGoogle Scholar
  39. Gurnett DA, Neubauer FM, Schwenn R (1979) Plasma wave turbulence associated with an interplanetary shock. J Geophys Res 84: 541–552ADSCrossRefGoogle Scholar
  40. Hada T, Oonishi M, Lembège B, Savoini P (2003) Shock front nonstationarity of supercritical perpendicular shocks. J Geophys Res 108: 1233CrossRefGoogle Scholar
  41. Hillier DJ et al (1993) The 0.1–2.5-KEV X-ray spectrum of the O4F-STAR Zeta-Puppis. Astron Astrophys 276: 117ADSGoogle Scholar
  42. Hillier DJ et al (1998) An optical and near-IR spectroscopic study of the extreme P Cygni-type supergiant HDE 316285. Astron Astrophys 340: 438ADSGoogle Scholar
  43. Hobara Y et al (2008) Cluster observations of electrostatic solitary waves near the Earth’s bow shock. J Geophys Res 113: A05211. doi: 10.1029/2007JA012789 CrossRefGoogle Scholar
  44. Hoshino M (2001) Nonthermal particle acceleration in shock front region: “Shock Surfing Accelerations”. Progr Theor Phys Suppl 143: 149–181ADSCrossRefGoogle Scholar
  45. Hoshino M, Shimada N (2002) Nonthermal electrons at high Mach number shocks: electron shock surfing acceleration. Astrophys J 572: 880–887ADSCrossRefGoogle Scholar
  46. Hull AJ et al (2006) Large-amplitude electrostatic waves associated with magnetic ramp substructure at Earth’s bow shock. Geophys Res Lett 33: L15104. doi: 10.1029/2005GL025564 ADSCrossRefGoogle Scholar
  47. Jaroschek CH, Lesch H, Treumann RA (2004) Self-consistent diffusive lifetimes of Weibel magnetic fields in gamma-ray bursts. Astrophys J 616: 1065–1071ADSCrossRefGoogle Scholar
  48. Jaroschek CH, Lesch H, Treumann RA (2005) Ultrarelativistic plasma shell collisions in γ-ray burst sources: dimensional effects on the final steady state magnetic field. Astrophys J 618: 822–831ADSCrossRefGoogle Scholar
  49. Jeffrey A, Taniuti T (1964) Nonlinear wave propagation. Academic Press, New YorkGoogle Scholar
  50. Kantrowitz A, Petschek HE (1966) MHD characteristics and shock waves. In: Kunkel WB (eds) Plasma physics in theory and application. McGraw-Hill, New York, pp 148–207Google Scholar
  51. Karpman VI (1964) Structure of shock front propagating at an angle to a magnetic field in a low-density plasma. Sov Phys-Tech Phys 8: 715–719MathSciNetGoogle Scholar
  52. Karpman VI, Sagdeev RZ (1964) The stability of a shock front moving across a magnetic field in a rarefied plasma. Sov Phys Tech Phys 8: 606–611MathSciNetGoogle Scholar
  53. Kennel CF, Sagdeev RZ (1967) Collisionless shock waves in high beta plasma 1 & 2. J Geophys Res 72:3303–3326; 3327–3341Google Scholar
  54. Kennel CF, Edmiston JP, Hada T (1985) A quarter century of collisionless shock research. In: Stone RG, Tsurutani BT (eds) Collisionless shocks in the heliosphere: a tutorial review. Amer Geophys Union, Washington DC, pp 1–36Google Scholar
  55. Kis A et al (2004) Multi-spacecraft observations of diffuse ions upstream of Earth’s bow shock. Geophys Res Lett 31: L20801. doi: 10.1029/2004GL020759 ADSCrossRefGoogle Scholar
  56. Kis A et al (2007) Scattering of field-aligned beam ions upstream of Earth’s bow shock. Ann Geophysicae 25: 785–799ADSGoogle Scholar
  57. Krimigis SM, Zwickl RD, Baker DN (1985) Energetic ions upstream of Jupiter’s bow shock. J Geophys Res 90(A5): 3947–3960ADSCrossRefGoogle Scholar
  58. Kucharek H, Scholer M (1991) Origin of diffuse superthermal ions at quasi-parallel supercritical collisionless shocks. J Geophys Res 96: 21195–21205ADSCrossRefGoogle Scholar
  59. Kudritzki RP, Puls J (2000) Winds from hot stars. Ann Rev Astron Astrophys 38: 613–666ADSCrossRefGoogle Scholar
  60. Landau LD, Lifshitz EM (1959) Fluid mechanics. Addison-Wesley, ReadingGoogle Scholar
  61. Lax PD (1957) Hyperbolic systems of conservation laws II. Commun Pure Appl Math 10: 537zbMATHMathSciNetCrossRefGoogle Scholar
  62. Lee MA (1982) Coupled hydromagnetic wave excitation and ion acceleration upstream of the earth’s bow shock. J Geophys Res 87: 5063–5080ADSCrossRefGoogle Scholar
  63. Lembège B, Savoini P (2002) Formation of reflected electron bursts by the nonstationarity and nonuniformity of a collisionless shock front. J Geophys Res 107(A3): 1037CrossRefGoogle Scholar
  64. Lembège B et al (1999) The spatial sizes of electric and magnetic field gradients in a simulated shock. Adv Space Res 24: 109–112ADSCrossRefGoogle Scholar
  65. Liberman MA, Velikhovich L (1986) Physics of shock waves in gases and plasmas. Springer Verlag, BerlinGoogle Scholar
  66. Lucek EA et al (2002) Cluster magnetic field observations at a quasi-parallel bow shock. Ann Geophysicae 20: 1699–1710ADSCrossRefGoogle Scholar
  67. Marshall W (1955) The structure of magneto-hydrodynamic shock waves. Proc R Soc Lond A 233: 367zbMATHADSCrossRefGoogle Scholar
  68. Mathews WG, Brighenti F (2003) Hot gas in and around elliptical galaxies. Ann Rev Astron Astrophys 41: 191–239. doi: 10.1146/annurev.astro.41.090401.0945542 ADSCrossRefGoogle Scholar
  69. Matsukiyo S, Scholer M (2003) Modified two-stream instability in the foot of high Mach number quasi-perpendicular shocks. J Geophys Res 108: 1459CrossRefGoogle Scholar
  70. Matsukiyo S, Scholer M (2006a) On microinstabilities in the foot of high Mach number perpendicular shocks. J Geophys Res 111: A06104CrossRefGoogle Scholar
  71. Matsukiyo S, Scholer M (2006b) On reformation of quasi-perpendicular collisionless shocks. Adv Space Res 38: 57–63ADSCrossRefGoogle Scholar
  72. Narita Y, Glassmeier K-H, Treumann RA (2006) Wave-number spectra and intermittency in the terrestrial foreshock region. Phys Rev Lett 97: 191101ADSCrossRefGoogle Scholar
  73. Narita Y, Glassmeier K-H, Gary SP, Goldstein ML, Treumann RA (2009) Analysis of wave number spectra through the terrestrial bow shock. J Geophys Res 114 (in press)Google Scholar
  74. Ness NF et al (1979) Magnetic field studies at Jupiter by Voyager 1—preliminary results. Science 204:982–986ADSCrossRefGoogle Scholar
  75. Ohsawa Y (1985a) Strong ion acceleration by a collisionless magnetosonic shock wave propagating perpendicularly to a magnetic field. Phys Fluids 28: 2130–2136ADSCrossRefGoogle Scholar
  76. Ohsawa Y (1985b) Nonlinear magnetosonic fast and slow waves in finite beta plasmas and associated resonant ion acceleration. J Phys Soc Jpn 54: 4073–4076ADSCrossRefGoogle Scholar
  77. Oka M et al (2006) Whistler critical Mach number and electron acceleration at the bow shock: geotail observation. Geophys Res Lett 33: L24104ADSCrossRefGoogle Scholar
  78. Owocki SP, Puls J (1999) Line-driven Stellar winds: the dynamical role of diffuse radiation gradients and limitations to the Sobolev approach. Astrophys J 510: 355ADSCrossRefGoogle Scholar
  79. Papadopoulos K (1988) Electron heating in superhigh Mach number shocks. Astrophys Space Sci 144: 535–547ADSGoogle Scholar
  80. Paschmann G et al (1981) Characteristics of reflected and diffuse ions upstream from the earth’s bow shock. J Geophys Res 86: 4355–4364ADSCrossRefGoogle Scholar
  81. Pickett JS et al (2004) On the generation of solitary waves observed by Cluster in the near-Earth magnetosheath. Nonlinear Process Geophys 12: 181–193ADSGoogle Scholar
  82. Sagdeev RZ (1966) Cooperative phenomena and shock waves in collisionless plasmas. Rev Plasma Phys 4:23–91. Leontovich MA (ed) Consultants Bureau, New YorkGoogle Scholar
  83. Sagdeev RZ, Shapiro VD (1973) Influence of transverse magnetic field on Landau damping. JETP Lett (Engl Transl) 17: 279–283ADSGoogle Scholar
  84. Schmitz H, Chapman SC, Dendy RO (2002) Electron preacceleration mechanisms in the foot region of high Alfvénic Mach number shocks. Astrophys J 570: 637–646ADSCrossRefGoogle Scholar
  85. Scholer M (1993) Upstream waves, shocklets, short large-amplitude magnetic structures and the cyclic behavior of oblique quasi-parallel collisionless shocks. J Geophys Res 98: 47–57ADSCrossRefGoogle Scholar
  86. Scholer M, Burgess D (2006) Transition scale at quasiperpendicular collisionless shocks: full particle electromagnetic simulations. Phys Plasmas 13: 062101ADSCrossRefGoogle Scholar
  87. Scholer M, Burgess D (2007) Whistler waves, core ion heating, and nonstationarity in oblique collisionless shocks. Phys Plasmas 14: 072103ADSCrossRefGoogle Scholar
  88. Scholer M, Kucharek H (1999a) Interaction of pickup ions with quasi-parallel shocks. Geophys Res Lett 26: 29–32ADSCrossRefGoogle Scholar
  89. Scholer M, Kucharek H (1999b) Dissipation, ion injection, and acceleration in collisionless quasi-parallel shocks. Astrophys Space Sci 264: 527–543zbMATHADSCrossRefGoogle Scholar
  90. Scholer M, Matsukiyo S (2004) Nonstationarity of quasi-perpendicular shocks: a comparison of full particle simulations with different ion to electron mass ratio. Ann Geophysicae 22: 2345–2353ADSCrossRefGoogle Scholar
  91. Scholer M, Kucharek H, Giacalone J (2000) Cross-field diffusion of charged particles and the problem of ion injection and acceleration at quasi-perpendicular shocks. J Geophys Res 105: 18285–18293ADSCrossRefGoogle Scholar
  92. Scholer M, Shinohara I, Matsukiyo S (2003) Quasi-perpendicular shocks: length scale of the cross-shock potential, shock reformation, and implication for shock surfing. J Geophys Res 108: 1014. doi: 10.1029/2002JA009515 CrossRefGoogle Scholar
  93. Schwartz SJ, Thomsen MF, Gosling JT (1983) Ions upstream of the earth’s bow shock—a theoretical comparison of alternative source populations. J Geophys Res 88: 2039–2047ADSCrossRefGoogle Scholar
  94. Sckopke N et al (1983) Evolution of ion distributions across the nearly perpendicular bow shock—specularly and non-specularly reflected-gyrating ions. J Geophys Res 88: 6121–6136ADSCrossRefGoogle Scholar
  95. Scudder JD (1995) A review of the physics of electron heating at collisionless shocks. Adv Space Res 15: 181–223ADSCrossRefGoogle Scholar
  96. Scudder JD et al (1986) The resolved layer of a collisionless, high beta, supercritical, quasi-perpendicular shock wave. J Geophys Res 91: 11019–11097ADSCrossRefGoogle Scholar
  97. Shimada N, Hoshino M (2000) Strong electron acceleration at high mach number shock waves: simulation study of electron dynamics. Astrophys J 543: L67–L71ADSCrossRefGoogle Scholar
  98. Shimada N, Hoshino M (2005) Effect of strong thermalization on shock dynamical behavior. J Geophys Res 110: A02105. doi: 10.1029/2004JA010596 CrossRefGoogle Scholar
  99. Shlesinger MF, Zaslavsky GM, Klafter J (1993) Strange kinetics. Nature 363: 31–37ADSCrossRefGoogle Scholar
  100. Sonnerup BUÖ (1969) Acceleration of particles reflected at a shock front. J Geophys Res 74: 1301–1304ADSCrossRefGoogle Scholar
  101. Stone EC et al (2008) An asymmetric solar wind termination shock. Nature 454: 71–74. doi: 10.1038/nature07022 ADSCrossRefGoogle Scholar
  102. Sugiyama T, Terasawa T (1999) A scatter-free ion acceleration process in the parallel shock. Adv Space Res 24: 73–77ADSCrossRefGoogle Scholar
  103. Sugiyama T, Fujimoto M, Mukai T (2001) Quick ion injection and acceleration at quasi-parallel shocks. J Geophys Res 106: 21657–21673ADSCrossRefGoogle Scholar
  104. Terasawa T, Scholer M, Ipavich FM (1985) Anisotropy observation of diffuse ions (greater than 30 keV/e) upstream of the earth’s bow shock. J Geophys Res 90: 249–260ADSCrossRefGoogle Scholar
  105. Tidman DA, Krall NA (1971) Shock waves in collisionless plasmas. Wiley-Interscience, New YorkGoogle Scholar
  106. Treumann RA (2006) The electron cyclotron maser for astrophysical application. Astron Astrophys Rev 13: 229–315ADSCrossRefGoogle Scholar
  107. Treumann RA (2008) A note on the theory of transverse diffusion in shock particle acceleration. Theory of super-diffusion for the magnetopause. Geophys Res Lett 24:1727–1730; arXiv: 0811.3938v1 [astro-ph]Google Scholar
  108. Treumann RA, Baumjohann W (1997) Advanced space plasma physics. Imperial College Press, LondonzbMATHGoogle Scholar
  109. Treumann RA, LaBelle J (1992) Band splitting in solar type II radio bursts. Astrophys J 399: L167–L170ADSCrossRefGoogle Scholar
  110. Trotignon JG et al (2001) How to determine the thermal electron density and the magnetic field strength from the Cluster/Whisper observations around the Earth. Ann Geophysicae 19: 1711–1720ADSCrossRefGoogle Scholar
  111. Tsytovich VN (1970) Nonlinear effects in plasmas. Plenum Press, New YorkGoogle Scholar
  112. Tu C-Y, Marsch E (1995) MHD structures, waves and turbulence in the solar wind: observations and theories. Space Sci Rev 73: 1–210ADSCrossRefGoogle Scholar
  113. Veilleux S, Cecil G, Bland-Hawthorn J (2005) Galactic winds. Ann Rev Astron Astrophys 43: 769–826. doi: 10.11146/annurev.astro.43.072193.150610 ADSCrossRefGoogle Scholar
  114. Weibel ES (1959) Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution. Phys Rev Lett 2: 83–84ADSCrossRefGoogle Scholar
  115. Wu CS (1984) A fast Fermi process—energetic electrons accelerated by a nearly perpendicular bow shock. J Geophys Res 89: 8857–8862ADSCrossRefGoogle Scholar
  116. Wu CC, Kennel CF (1992) Structural relations for time-dependent intermediate shocks. Geophys Res Lett 19: 2087–2090ADSCrossRefGoogle Scholar
  117. Zahibo N et al (2007) Strongly nonlinear steepening of long interfacial waves. Nonlinear Process Geophys 14: 247–256ADSGoogle Scholar
  118. Zeldovich YB, Raizer YP (1966/1967) Physics of shock waves and high-temperature hydrodynamic phenomena, vols 1 & 2. Academic Press, New YorkGoogle Scholar
  119. Ziegler HJ, Schindler K (1988) Structure of subcritical perpendicular shock waves. Phys Fluids 31: 570–576zbMATHADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Geophysics and Environmental Sciences, Geophysics SectionLudwig-Maximilians-University MunichMunichGermany
  2. 2.Department of Physics and AstronomyDartmouth CollegeHanoverUSA
  3. 3.International Space Science InstituteBernSwitzerland

Personalised recommendations