Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T12:33:18.036Z Has data issue: false hasContentIssue false
  • English
  • Français

Ultra-relativistic geometrical shock dynamics and vorticity

Published online by Cambridge University Press:  14 May 2008

JEREMY GOODMAN  and
ANDREW MACFADYEN
JEREMY GOODMAN
Affiliation:
Institute for Advanced Study, Princeton, NJ 08544, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA
ANDREW MACFADYEN
Affiliation:
Institute for Advanced Study, Princeton, NJ 08544, USA Department of Physics, New York University, New York, NY 10003, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Geometrical shock dynamics, also called CCW theory, yields approximate equations for shock propagation in which only the conditions at the shock appear explicitly; the post-shock flow is presumed approximately uniform and enters implicitly via a Riemann invariant. The non-relativistic theory, formulated by G. B. Whitham and others, matches many experimental results surprisingly well. Motivated by astrophysical applications, we adapt the theory to ultra-relativistic shocks advancing into an ideal fluid whose pressure is negligible ahead of the shock, but is one third of its proper energy density behind the shock. Exact results are recovered for some self-similar cylindrical and spherical shocks with power-law pre-shock density profiles. Comparison is made with numerical solutions of the full hydrodynamic equations. We review relativistic vorticity and circulation. In an ultra-relativistic ideal fluid, circulation can be defined so that it changes only at shocks, notwithstanding entropy gradients in smooth parts of the flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 604 , 10 June 2008 , pp. 325 - 338
Copyright
Copyright © Cambridge University Press 2008

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

Barenblatt, G. I. 1996 Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
Baskar, S. & Prasad, P. 2005 Propagation of curved shock fronts using shock ray theory and comparison with other theories. J. Fluid Mech. 523, 171–198.CrossRefGoogle Scholar
Blandford, R. D. & McKee, C. F. 1976 Fluid dynamics of relativistic blast waves. Phys. Fluids 19, 1130–1138.CrossRefGoogle Scholar
Bryson, A. E. & Gross, R. W. F. 1961 Diffraction of strong shocks by cones, cylinders, and spheres. J. Fluid Mech. 10, 1–16.CrossRefGoogle Scholar
Chester, W. 1954 The quasi-cylindrical shock tube. Phil. Mag. 45 (7), 1293–1301.CrossRefGoogle Scholar
Chisnell, R. F. 1957 The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves. J. Fluid Mech. 2, 286–298.CrossRefGoogle Scholar
Eshraghi, H. 2003 On the vortex dynamics in fully relativistic plasmas. Phys. Plasmas 10, 3577–3583.CrossRefGoogle Scholar
Johnson, M. H. & McKee, C. F. 1971 Relativistic hydrodynamics in one dimension. Phys. Rev. D 3, 858–863.Google Scholar
Kulsrud, R. M. 2003 Plasma Physics for Astrophysics. Princeton University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1980 Statistical Physics. Pergamon.Google Scholar
Li, H. & Ben-Dor, G. 1998 Modified CCW theory for detonation waves. Combust. Flame 113, 1–12.CrossRefGoogle Scholar
Martí, J. M. & Müller, E. 1994 The analytical solution of the Riemann problem in relativistic hydrodynamics. J. Fluid Mech. 258, 317–333.CrossRefGoogle Scholar
Medvedev, M. V. & Loeb, A. 1999 Generation of magnetic fields in the relativistic shock of gamma-ray burst sources. Astrophys. J. 526, 697–706.CrossRefGoogle Scholar
Mészáros, P. 2006 Gamma-ray bursts. Rep. Prog. Phys. 69, 2259–2321.CrossRefGoogle Scholar
Moeckel, W. E. 1952 Interaction of oblique shock waves with regions of variable pressure, entropy, and energy. Tech. Rep. 2725. Natl. Adv. Comm. Aero, Washington.Google Scholar
van Paradijs, J., Kouveliotou, C. & Wijers, R. A. M. J. 2000 Gamma-ray burst afterglows. Ann. Rev. Astron. Astrophys. 38, 379–425.CrossRefGoogle Scholar
Perna, R. & Vietri, M. 2002 A self-similar solution for the propagation of a relativistic shock in an exponential atmosphere. Astrophys. J. 569, L47–L50.CrossRefGoogle Scholar
Piran, T. 2005 Magnetic fields in gamma-ray bursts: a short overview. In AIP Conf. Proc. 784: Magnetic Fields in the Universe: From Laboratory and Stars to Primordial Structures. (ed. Pino, E. M. de Gouveia dal, Lugones, G. & Lazarian, A.), pp. 164–174.Google Scholar
Sari, R. 2006 First and second-type self-similar solutions of implosions and explosions containing ultrarelativistic shocks. Phys. Fluids 18, 027106.CrossRefGoogle Scholar
Schutz, B. F. 1990 A First Course in General Relativity. Cambridge University Press.Google Scholar
Schwendeman, D. W. 1988 Numerical scheme for shock propagation in three dimensions. Proc. R. Soc. Londo. A 416, 179–198.Google Scholar
Sironi, L. & Goodman, J. 2007 Production of magnetic energy by macroscopic turbulence in GRB afterglows. Astrophys. J. 671, 1858–1867.CrossRefGoogle Scholar
Taylor, E. F. & Wheeler, J. A. 1966 Spacetime Physics. Freeman.Google Scholar
Whitham, G. B. 1957 A new approach to problems of shock dynamics. Part 1. Two-dimensional problems. J. Fluid Mech. 2, 146–171.CrossRefGoogle Scholar
Whitham, G. B. 1958 On the propagation of shock waves through regions of non-uniform area or flow. J. Fluid Mech. 4, 337–360.CrossRefGoogle Scholar
Whitham, G. B. 1959 New approach to problems of shock dynamics. Part 2. Three-dimensional problems. J. Fluid Mech. 5, 369–386.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar