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Physics of Collisionless Shocks pp 45–123Cite as

Basic Equations and Models

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Part of the book series: ISSI Scientific Report Series ((ISSI,volume 12))

Abstract

This chapter presents the basic theoretical background of collisionless shock physics. It states the basic process of shock formation as the growth of a small disturbance in the plasma by the action of the intrinsic nonlinearity of flow, independent of the cause of the initial disturbance. The latter can be an external driver like a piston or a blast, it can also be an internal instability. Shocks form when nonlinearity causes steeping (or steepening) of the disturbance in space and some process exists which prevents breaking of the steep wave. Such processes are of dissipative or dispersive nature and are discussed in ascending importance. An intermediate step is the evolution of solitary waves based on the Sagdeev pseudo-potential. After this fundamental discussion, the plasma kinetic equations are given and the Rankine-Hugoniot jump conditions at shocks are derived with the shock solutions explicitly given. Critical Mach numbers are defined beyond which dissipation is unable to prevent wave breaking. The relevant wave instabilities causing initial disturbances, dispersion and dissipation are discussed at length. Transport ratios are given, and anomalous transport is reviewed. Finally, shock particle reflection is identified as the basic process of shock stabilisation preventing breaking. The last section provides a cursory and incomplete briefing on numerical simulation techniques.

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Notes

  1. 1.

    It might be of interest to note that to find a method that solves this general time-dependent equation analytically took more than a century, and for this purpose it was necessary to develop the whole apparatus of non-relativistic quantum mechanics. This was done in a seminal paper by Gardner et al [1967].

  2. 2.

    In fact there is one exception to this statement. There exist free space electromagnetic mode branches (radiation) above and even below the electron cyclotron frequency on which electron excited electromagnetic waves could in principle propagate. The mechanism to excite them is the electron cyclotron maser instability [for a recent review see, e.g, Treumann, 2006], which is a very particular instability that becomes awakened under conditions which to our knowledge are barely satisfied in the non-relativistic shock environment.

References

  • Bale SD, Hull A, Larson DE, Lin R P, Muschietti L, Kellogg PJ, Goetz K, Monson SJ (2002) Electrostatic turbulence and Debye-scale structures associated with electron thermalization at collisionless shocks. Astrophys J 6:L25–L28. doi:10.1086/342609

    Article  ADS  Google Scholar 

  • Baumjohann W, Treumann RA (1996) Basic space plasma physics. Imperial College Press, London. Revised edn. 2012

    Google Scholar 

  • Berthomier M, Pottelette R, Treumann RA (1999) Parametric study of kinetic Alfvén waves in a two electron temperature plasma. Phys Plasmas 6:467–475. doi:10.1063/1.873213

    Article  ADS  Google Scholar 

  • Birdsall CK, Langdon AB (1985) Plasma physics via computer simulation. McGraw-Hill, New York

    Google Scholar 

  • Buneman O (1958) Instability, turbulence, and conductivity in current-carrying plasma. Phys Rev Lett 1:8–9. doi:10.1103/PhysRevLett.1.8

    Article  ADS  Google Scholar 

  • Davidson RC (1972) Methods in nonlinear plasma theory. Academic Press, New York

    Google Scholar 

  • Davidson RC (1978) Quasi-linear stabilization of lower-hybrid-drift instability. Phys Fluids 21:1375–1380. doi:10.1063/1.862379

    Article  ADS  MATH  Google Scholar 

  • Davidson RC, Gladd NT, Wu CS, Huba JD (1977) Effects of finite plasma beta on the lower-hybrid-drift instability. Phys Fluids 20:301–310. doi:10.1063/1.861867

    Article  ADS  Google Scholar 

  • Davidson RC, Krall NA (1977) Anomalous transport in high-temperature plasmas with applications to solenoidal fusion systems. Nucl Fusion 17:1313–1372

    Article  ADS  Google Scholar 

  • Davis L, Lüst R, Schlüter A (1958) The structure of hydromagnetic shock waves. I. Non linear hydromagnetic waves in a cold plasma. Z Naturforsch 13a:916–936

    ADS  Google Scholar 

  • Dawson JM (1983) Particle simulation of plasmas. Rev Mod Phys 55:403–447. doi:10.1103/RevModPhys.55.403

    Article  ADS  Google Scholar 

  • Dawson JM (1995) Computer modeling of plasma: past, present, and future. Phys Plasmas 2:2189–2199. doi:10.1063/1.871304

    Article  ADS  Google Scholar 

  • Decker G, Robson AE (1972) Instability of the whistler structure of oblique hydromagnetic shocks. Phys Rev Lett 29:1071–1073. doi:10.1103/PhysRevLett.29.1071

    Article  ADS  Google Scholar 

  • Drummond WE, Rosenbluth MN (1962) Anomalous diffusion arising from microinstabilities in plasma. Phys Fluids 5:1507–1513. doi:10.1063/1.1706559

    Article  ADS  MATH  Google Scholar 

  • Dubouloz N, Pottelette R, Malingre M, Treumann RA (1991) Generation of broadband electrostatic noise by electron acoustic solitons. Geophys Res Lett 18:155–158. doi:10.1029/90GL02677

    Article  ADS  Google Scholar 

  • Dum CT, Chodura R (1979) Anomalous transition from Buneman to ion sound instability. In: Palmadesso PJ, Papadopoulos K (eds) Wave instabilities in space plasmas. Astroph space sci library, vol 74. Reidel, Dordrecht, p 135

    Chapter  Google Scholar 

  • Dupree TH, Wagner CE, Manheimer WM (1975) Observation of clumps in simulation plasma. Phys Fluids 18:1167–1172. doi:10.1063/1.861278

    Article  ADS  Google Scholar 

  • Friedman A, Langdon AB, Cohen BI (1981) Implicit time integration for plasma simulation. Comments Plasma Phys Control Fusion 6:225

    Google Scholar 

  • Galeev AA (1976) Collisionless shocks. In: Williams DJ (ed) Physics of solar planetary environment. AGU, Washington, pp 464–493

    Google Scholar 

  • Galeev AA, Karpman VI (1963) Turbulence theory of a weakly nonequilibrium low-density plasma and structure of shock waves. Sov Phys JETP, Engl Transl 17:403–409

    MATH  Google Scholar 

  • Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg-de Vries equation. Phys Rev Lett 19:1095–1098. doi:10.1103/PhysRevLett.19.1095

    Article  ADS  MATH  Google Scholar 

  • Gary SP (1993) Theory of space plasma microinstabilities. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Gary SP, Omidi N (1987) The ion-ion acoustic instability. J Plasma Phys 37:45–61. doi:10.1017/S0022377800011983

    Article  ADS  Google Scholar 

  • Gary SP, Sanderson JJ (1979) Electrostatic temperature gradient drift instabilities. Phys Fluids 22:1500–1509. doi:10.1063/1.862769

    Article  ADS  Google Scholar 

  • Goldstein ML, Wong HK, Vinas AF, Smith CW (1985) Large-amplitude MHD waves upstream of the Jovian bow shock reinterpretation. J Geophys Res 90:302–310. doi:10.1029/JA090iA01p00302

    Article  ADS  Google Scholar 

  • Hasegawa A (1979) Excitation of Alfven waves by bouncing electron beams – an origin of nightside magnetic pulsations. Geophys Res Lett 6:664–666. doi:10.1029/GL006i008p00664

    Article  ADS  Google Scholar 

  • Hasegawa A, Mima K (1976) Exact solitary Alfvén waves. Phys Rev Lett 37:690–693. doi:10.1103/PhysRevLett.37.690

    Article  ADS  Google Scholar 

  • Kakutani T (1966) Non-linear hydromagnetic waves propagating along a magnetic field in a cold collision-free plasma. J Phys Soc Jpn 23:385–391. doi:10.1143/JPSJ.21.385

    Article  ADS  Google Scholar 

  • Karimabadi H, Omelchenko Y, Driscoll J, Fujimoto R, Perumala K, Krauss-Varban D (2005) A new simulation technique for study of collisionless shocks: self-adaptive simulations. In: Li G, Zank GP, Russell CT (eds) The physics of collisionless shocks. AIP Conf Proc. AIP, New York, pp 56–63. doi:10.1063/1.2032675

    Google Scholar 

  • Karpman VI (1964) Structure of the shock front propagating at an angle to the magnetic field in a low-density plasma. Sov Phys Tech Phys, Engl Transl 8:715–719

    Google Scholar 

  • Karpman VI, Sagdeev RZ (1964) The stability of the structure of a shock-wave front moving across a magnetic field in a rarified plasma. Sov Phys Tech Phys, Engl Transl 8:606–611

    MathSciNet  Google Scholar 

  • Kennel CF, Petschek HE (1966) Limit on stably trapped particle fluxes. J Geophys Res 71:1–28

    Article  ADS  Google Scholar 

  • Kindel JM, Kennel CF (1971) Topside current instabilities. J Geophys Res 76:3055–3065. doi:10.1029/JA076i013p03055

    Article  ADS  Google Scholar 

  • Krall NA, Trivelpiece AW (1973) Principles of plasma physics. McGraw-Hill, New York

    Google Scholar 

  • Landau LD, Lifshitz EM (1980) Statistical physics3, vol 1. Pergamon-Elsevier, Oxford

    Google Scholar 

  • Leroy MM, Mangeney A (1984) A theory of energization of solar wind electrons by the earth’s bow shock. Ann Geophys 2:449–456

    ADS  Google Scholar 

  • Lichtenberg AJ, Lieberman MA (1992) Regular and chaotic dynamics. Appl math sci, vol 38. Springer, New York

    MATH  Google Scholar 

  • Macmahon A (1965) Finite gyro-radius corrections to the hydromagnetic equations for a Vlasov plasma. Phys Fluids 8:1840–1845. doi:10.1063/1.1761116

    Article  MathSciNet  ADS  Google Scholar 

  • Matsukiyo S, Scholer M (2003) Modified two-stream instability in the foot of high Mach number quasi-perpendicular shocks. J Geophys Res 108:1459. doi:10.1029/2003JA010080

    Article  Google Scholar 

  • Matsukiyo S, Treumann RA, Scholer M (2004) Coherent waveforms in the auroral upward current region. J Geophys Res 109:A06212. doi:10.1029/2004JA00477

    Article  Google Scholar 

  • McBride JB, Ott E, Boris JP, Orens JH (1972) Theory and simulation of turbulent heating by the modified two-stream instability. Phys Fluids 15:2367–2383. doi:10.1063/1.1693881

    Article  ADS  Google Scholar 

  • Moiseev SS, Sagdeev RZ (1963) Collisionless shock waves in a plasma in a weak magnetic field. J Nucl Energy Pt C 43–47. doi:10.1088/0368-3281/5/1/309

  • Montgomery DC (1959) Development of hydromagnetic shocks from large-amplitude Alfven waves. Phys Rev Lett 2:36–37. doi:10.1103/PhysRevLett.2.36

    Article  ADS  Google Scholar 

  • Montgomery DC, Tidman DA (1964) Plasma kinetic theory. McGraw-Hill, New York

    Google Scholar 

  • Newman DL, Goldman MV, Ergun RE (2002) Evidence for correlated double layers, bipolar structures, and very-low-frequency saucer generation in the auroral ionosphere. Phys Plasmas 9:2337–2343. doi:10.1063/1.1455004

    Article  ADS  Google Scholar 

  • Newman DL, Goldman MV, Ergun RE, Mangeney A (2001) Formation of double layers and electron holes in a current-driven space plasma. Phys Rev Lett 87:255001. doi:10.1103/PhysRevLett.87.255001

    Article  ADS  Google Scholar 

  • Ott E, Sudan R (1969) Nonlinear theory of ion acoustic waves with Landau damping. Phys Fluids 12:2388–2394. doi:10.1063/1.1692358

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Papadopoulos K (1985) Microinstabilities and anomalous transport. In: Stone RG, Tsurutani BT (eds) Collisionless shocks in the heliosphere: a tutorial review. AGU, Washington, pp 59–90

    Chapter  Google Scholar 

  • Rodriguez P, Gurnett DA (1975) Electrostatic and electromagnetic turbulence associated with the earth’s bow shock. J Geophys Res 80:19–31. doi:10.1029/JA080i001p00019

    Article  ADS  Google Scholar 

  • Sagdeev RZ (1966) Cooperative phenomena and shock waves in collisionless plasmas. In: Leontovich MA (ed) Rev plasma phys, vol 4. Consultants Bureau, New York, pp 23–91

    Google Scholar 

  • Sagdeev RZ (1979) The 1976 Oppenheimer lectures: critical problems in plasma astrophysics. I. Turbulence and nonlinear waves, II. Singular layers and reconnection. Rev Mod Phys 51:1–20. doi:10.1103/RevModPhys.51.1

    Article  ADS  Google Scholar 

  • Schamel H (1972) Non-linear electrostatic plasma waves. J Plasma Phys 7:1–12. doi:10.1017/S0022377800006371

    Article  ADS  Google Scholar 

  • Tidman DA, Krall NA (1971) Shock waves in collisionless plasmas. Wiley, New York

    Google Scholar 

  • Treumann RA (2006) The electron-cyclotron maser for astrophysical application. Rev Astron Astrophys 13:229–315. doi:10.1007/s00159-006-0001-y

    Article  ADS  Google Scholar 

  • Treumann RA, Güdel M, Benz AO (1990) Alfven wave solitons and solar intermediate drift bursts. Astron Astrophys 236:242–249

    ADS  Google Scholar 

  • Treumann RA, LaBelle J, Pottelette R (1991) Plasma diffusion at the magnetopause – the case of lower hybrid drift waves. J Geophys Res 96:16009–16013. doi:10.1029/91JA01671

    Article  ADS  Google Scholar 

  • Vekshtein GE, Ryutov DD, Sagdeev RZ (1970) Asymptotic solution in the problem of anomalous plasma resistance. Sov Phys JETP Lett 12:291–294

    ADS  Google Scholar 

  • Winske D (1985a) Hybrid simulation codes with application to shocks and upstream waves. Space Sci Rev 42:53–66. doi:10.1007/BF00218223

    Article  ADS  Google Scholar 

  • Winske D (1985b) Microtheory of collisionless shock current layers. In: Tsurutani BT, Stone RG (eds) Collisionless shocks in the heliosphere: reviews of current research. AGU, Washington, pp 225–236

    Chapter  Google Scholar 

  • Winske D, Omidi N (1995) Diffusion at the magnetopause: hybrid simulations. J Geophys Res 100:11923–11933. doi:10.1029/94JA02730

    Article  ADS  Google Scholar 

  • Witham GB (1966) Linear and nonlinear waves. Wiley, New York

    Google Scholar 

  • Woods LC (1971) On double-structured, perpendicular, magneto-plasma shock waves. Plasma Phys 13:289–302. doi:10.1088/0032-1028/13/4/302

    Article  ADS  Google Scholar 

  • Wu CS (1984) A fast Fermi process – energetic electrons accelerated by a nearly perpendicular bow shock. J Geophys Res 89:8857–8862. doi:10.1029/JA089iA10p08857

    Article  ADS  Google Scholar 

  • Wu CS, Winske D, Tanaka M, Papadopoulos K, Akimoto K, Goodrich CC, Zhou YM, Tsai ST, Rodriguez P, Lin CS (1984) Microinstabilities associated with a high Mach number, perpendicular bow shock. Space Sci Rev 37:63–109. doi:10.1007/BF00213958

    Article  ADS  Google Scholar 

  • Yoon PH, Lui ATY (2007) Anomalous resistivity by fluctuation in the lower-hybrid frequency range. J Geophys Res 112:A06207. doi:10.1029/2006JA012209

    Article  Google Scholar 

  • Yoon PH, Wang CB, Wu CS (2007) Ring-beam driven maser instability for quasiperpendicular shocks. Phys Plasmas 14:022901. doi:10.1063/1.2437118

    Article  ADS  MATH  Google Scholar 

  • Zahibo N, Slunyaev A, Talipova T, Pelinovsky E, Kurkin A, Polukhina O (2007) Strongly nonlinear steepening of long interfacial waves. Nonlinear Process Geophys 14:247–256. nonlin-processes-geophys.net/14/247/2007/

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. The Blackett Laboratory, Imperial College London, London, UK

    Prof. Dr. André Balogh

  2. International Space Science Institute ISSI, Bern, Switzerland

    Prof. Dr. André Balogh & Prof. Dr. Rudolf A. Treumann

  3. Department of Geophysicsand Environmental Sciences, Munich University, Munich, Germany

    Prof. Dr. Rudolf A. Treumann

  4. Dartmouth College, Hanover, NH, USA

    Prof. Dr. Rudolf A. Treumann

Authors
  1. Prof. Dr. André Balogh
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  2. Prof. Dr. Rudolf A. Treumann
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Balogh, A., Treumann, R.A. (2013). Basic Equations and Models. In: Physics of Collisionless Shocks. ISSI Scientific Report Series, vol 12. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6099-2_3

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