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Burgers' equation and cosmic ray shocks

Published online by Cambridge University Press:  13 March 2009

G. M. Webb  and
J. F. McKenzie
G. M. Webb
Affiliation:
Max-Planck-Institut für Kernphysik, Postfach 10 39 80, D-6900 Heidelberg, Federal Republic of Germany
J. F. McKenzie
Affiliation:
Department of Mathematics and Applied Mathematics, University of Natal, King George V Avenue, Durban, Natal, South Africa
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Abstract

The time evolution of weak shocks in the classical theory of shock waves may be described by Burgers' equation, which is a small-amplitude, long-wavelength, but nonlinear wave equation. The present paper derives Burgers' equation for three different hydrodynamical models of cosmic ray shocks. The main development of the paper concerns the hydrodynamical model in which Alfvénic effects are neglected. For this model it is shown that the steady-state solution of Burgers' equation is equivalent to the weak, but smooth, transition solutions of the shock structure equation obtained previously. It is shown that Burgers' equation may be regarded as a wave energy equation, in which the interaction of the wave with the background medium is taken into account. Burgers' equation is also derived for Alfvénic models in which the Alfvén waves that scatter the cosmic rays are generated by the cosmic ray streaming instability, and propagate down the cosmic ray pressure gradient, at the Alfvén velocity relative to the background fluid.

Type
Research Article
Information
Journal of Plasma Physics , Volume 31 , Issue 3 , June 1984 , pp. 337 - 368
Copyright
Copyright © Cambridge University Press 1984

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References

REFERENCES

Achterberg, A. 1981 Astron. Astrophys. 98, 161.Google Scholar
Achterberg, A. 1983 Astron. Astrophys. 119, 274.Google Scholar
Axford, W. I. 1980 I.A.U. Symp. No. 94, 334.Google Scholar
Axford, W. I., Leer, E. & Skadron, G. 1977 Proceedings of 15th International Conference on Cosmic Rays, Plovdiv, vol. 11, p. 132.Google Scholar
Axford, W. I., Leer, E. & McKenzie, J. F. 1982 Astron. Astrophys. 111, 317.Google Scholar
Bell, A. R. 1978 a M.N.R.A.S. 182, 147.CrossRefGoogle Scholar
Bell, A. R. 1978 b M.N.R.A.S. 182, 443.CrossRefGoogle Scholar
Ben, Israel I., Piran, T., Eviator, A. & Weinstock, J. 1975 Astrophys. Space Sci. 38, 125.Google Scholar
Blandford, R. D. 1980 Astrophys. J. 238, 410.CrossRefGoogle Scholar
Boyd, T. J. & Sanderson, J. J. 1969 Plasma Dynamics. Barnes and Noble.Google Scholar
Bretherton, F. P. & Garrett, C. J. R. 1969 Proc. Roy. Soc. A, 529.Google Scholar
Burgers, J. 1948 Advances in Applied Mechanics, vol. 1, p. 171. Academic.Google Scholar
Cole, J. D. 1951 Quart. Appl. Maths. 9, 225.CrossRefGoogle Scholar
Dewar, R. L. 1970 Phys. Fluids, 13, 2710.CrossRefGoogle Scholar
Dolginov, A. Z. & Toptygin, I. N. 1966 Soviet Phys. JETP, 51, 1771.Google Scholar
Dolginov, A. Z. & Toptygin, I. N. 1967 Geomag. Aeron 7, 785.Google Scholar
Dolginov, A. Z. & Toptygin, I. N. 1968 Icarus, 8, 54.CrossRefGoogle Scholar
Drury, L. O'C. 1983 Rep. Prog. Phys. 46, 973.CrossRefGoogle Scholar
Drury, L. O'C. & Völk, H. J. 1981 Astrophys. J. 248, 344.CrossRefGoogle Scholar
Drury, L. O'C., Axford, W. I. & Summers, D. 1982 M.N.R.A.S. 198, 833.CrossRefGoogle Scholar
Eichler, D. 1979 Astrophys. J. 228, 419.CrossRefGoogle Scholar
Eichler, D. 1981 Phys. Rev. Lett. 47, 1560.CrossRefGoogle Scholar
Forman, M. A. & Morfill, G. 1979 Proceedings of 16th International Conference on Cosmic-Rays, Kyoto, vol. 5, p. 328.Google Scholar
Gleeson, L. J. & Axford, W. I. 1967 Astrophys. J 149, L115.CrossRefGoogle Scholar
Goldstein, M. L. 1976 Astrophys. J. 204, 900.CrossRefGoogle Scholar
Heavens, A. 1983 M.N.R.A.S. 204, 699.CrossRefGoogle Scholar
Holman, G. D., Ionson, J. A. & Scott, J. S. 1979 Astrophys. J. 228, 576.CrossRefGoogle Scholar
Hopf, E. 1950 Comm. Pure Appl. Maths. 3, 201.CrossRefGoogle Scholar
Jokipii, J. R. & Parker, E. N. 1970 Astrophys. J. 160, 735.CrossRefGoogle Scholar
Jones, F. C., Birmingham, T. J. & Kaiser, T. B. 1978 Phys. Fluids, 21, 347.CrossRefGoogle Scholar
Krymsky, G. F. 1981 Izv. An. Sssr. Ser. Phys., 45, 461.Google Scholar
Kulsrud, R. M. & Pearce, W. 1969 Astrophys. J. 156, 445.CrossRefGoogle Scholar
Lee, M. A. 1982 J. Geophys. Res. 87, 5063.CrossRefGoogle Scholar
Lee, M. A. & Völk, H. J. 1973 Astrophys. Space Sci. 24, 31.CrossRefGoogle Scholar
Leer, E., Skadron, G. & Axford, W. I. 1976 EOS, 57, 980.Google Scholar
Lerche, I. 1967 Astrophys. J. 147, 689.CrossRefGoogle Scholar
Lundgren, T. S. 1963 Phys. Fluids, 6, 898.CrossRefGoogle Scholar
Mckenzie, J. F. 1982 J. Plasma Phys. 28, 317.CrossRefGoogle Scholar
Mckenzie, J. F. & Völk, H. J. 1981 Proceedings of 17th International Conference on Cosmic-Rays, Paris, vol. 9, p. 242.Google Scholar
Mckenzie, J. F. & Völk, H. J. 1982 Astron. Astrophys. 116, 191.Google Scholar
Mckenzie, J. F. & Webb, G. M. 1984 J. Plasma Phys. 31, 275.CrossRefGoogle Scholar
Morrison, P. J., Scott, J. S., Holman, G. D. & Ionson, J. A. 1980 Preprint No. 316 Steward Observatory Tucson Arizona (Astrophys. J., submitted).Google Scholar
Newcomb, W. A., 1962 Nucl. Fusion Suppl. Pt. 2, 451.Google Scholar
Parker, E. N. 1965 Planet. Space Sci. 13, 9.CrossRefGoogle Scholar
Ptuskin, V. S. 1981 Astrophys. Space Sci. 76, 265.CrossRefGoogle Scholar
Schwarz, S. J. & Skilling, J. 1978 Astron. Astrophys. 70, 607.Google Scholar
Skilling, J. 1975 a M.N.R.A.S. 172, 557.CrossRefGoogle Scholar
Skilling, J. 1975 b M.N.R.A.S. 173, 255.CrossRefGoogle Scholar
Su, C. H. & Gardner, C. S. 1969 J. Math. Phys. 10, 536.CrossRefGoogle Scholar
Taniuti, T. & Yajima, N. 1969 J. Math. Phys. 10, 1369.CrossRefGoogle Scholar
Völk, H. J. 1973 Astrophys. Space Sci. 25, 471.CrossRefGoogle Scholar
Völk, H. J. & Mckenzie, J. K. 1981 Proceedings of 17th International Conference on Cosmic-Rays, Paris, vol. 9, p. 246.Google Scholar
Völk, H. J. & Cesarsky, C. J. 1982 Z. Naturforsch. 37 a, 809CrossRefGoogle Scholar
Völk, H. J., Drury, L. O'C. & Mckenzie, J. K. 1983 Astron. Astrophys. 130, 19.Google Scholar
Webb, G. M. & Gleeson, L. J. 1979 Astrophys. Space Sci. 60, 335.CrossRefGoogle Scholar
Webb, G. M. 1983 Astron. Astrophys. 127, 97.Google Scholar
Wentzel, D. G. 1968 Astrophys. J. 152, 987.CrossRefGoogle Scholar
Wentzel, D. G. 1974 Ann. Rev. Astron. Astrophys. 12, 71.CrossRefGoogle Scholar
Whitham, G. B. 1965 J. Fluid Mech. 22, 273.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Non-Linear Waves. Wiley Interscience.Google Scholar
Zweibel, E. G. 1979 Proceedings of A.I.P. Conference, La Jolla, (ed. J. Arons and C. McKee), p. 319.Google Scholar