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Modified non-linear Burgers' equations and cosmic ray shocksA reductive perturbation scheme is used to derive a generalized non-linear Burgers' equation, which includes the effects of dispersion, in the long wavelength regime for the two-fluid hydrodynamical model used to describe cosmic ray acceleration by the first-order Fermi process in astrophysical shocks. The generalized Burger's equation is derived for both relativistic and non-relativistic cosmic ray shocks, and describes the time evolution of weak shocks in the theory of diffusive shock acceleration. The inclusion of dispersive effects modifies the phase velocity of the shock obtained from the lower order non-linear Burger's equation through the introduction of higher order terms from the long wavelength dispersion equation. The travelling wave solution of the generalized Burgers' equation for a single shock shows that larger cosmic ray pressures result in broader shock transitions. The results for relativistic shocks show a steepening of the shock as the shock speed approaches the relativistic cosmic ray sound speed. The dependence of the shock speed on the cosmic ray pressure is also discussed.
Document ID
19880039876
Acquisition Source
Legacy CDMS
Document Type
Reprint (Version printed in journal)
Authors
Zank, G. P.
(Max-Planck-Institut fuer Kernphysik, Heidelberg, Federal Republic of Germany; Natal, University Durban, Republic of South Africa, United States)
Webb, G. M.
(Arizona, University Tucson, United States)
Mckenzie, J. F.
(Natal, University Durban, Republic of South Africa, United States)
Date Acquired
August 13, 2013
Publication Date
January 1, 1988
Publication Information
Publication: Astronomy and Astrophysics
Volume: 189
Issue: 1-2
ISSN: 0004-6361
Subject Category
Space Radiation
Accession Number
88A27103
Funding Number(s)
CONTRACT_GRANT: NSF ATM-83-17701
CONTRACT_GRANT: NSG-7101
Distribution Limits
Public
Copyright
Other

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