Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T14:28:20.905Z Has data issue: false hasContentIssue false

Nonlinear evolution of firehose-unstable Afvén waves

Published online by Cambridge University Press:  13 March 2009

K. Elsässer
Affiliation:
Max-Planck-Institut für Physik und Astrophysik, München
H. Schamel
Affiliation:
Max-Planck-Institut für Physik und Astrophysik, München

Abstract

The time evolution of firehose-unstable Alfvén waves is calculated within the usual weak-turbulence scheme. In § 2 the amplitude equations are established using a third-order solution of Vlasov's equation. From these a spectrum equation is obtained by discarding four-wave correlations; in addition, we derive an equation which characterizes the truncation error ( §3). Both these equations are integrated numerically, together with the velocity-moment equations (§ 4), forward in time. The results as presented in § 5 correspond to a rapid relaxation of the plasma to equilibrium. For large wave-vectors the relaxation time, as well as the equilibrium wave energy, are in good agreement with the quasilinear treatment, and the truncation error is small. But for low wave-numbers the relaxation is much faster, and the wave energy grows higher as predicted by quasilinear theory. This is because the nonlinear particle current leads to a high effective growth rate, especially at small wave-numbers. In this region the truncation error grows appreciably, and may sometimes reach the order of magnitude of the spectrum. But the overall picture as given by quasilinear theory has been confirmed. In § 6 comparison is made with a macroscopic model of Berezin & Sagdeev (1969), where the plasma noise was simulated by ‘computer noise’. For low amplitudes both methods agree qualitatively.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1972

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

Abraham-Shrauner, B. A. 1970 Phys. Fluids, 13, 837.CrossRefGoogle Scholar
Berezin, Y. A. 1971 J. Exp. Theor. Phys. B 61, 1877.Google Scholar
Berezin, Y. A. & Sagdeev, R. Z. 1969 Soviet Phys. Doklady, 14, 62.Google Scholar
Crew, G. F., Goldberger, M. L. & Low, F. E. 1956 Proc. Roy. Soc. A 236, 112.Google Scholar
Coleman, P. J. 1966 J. Geophys. Res. 71, 5509.CrossRefGoogle Scholar
Daughney, C. C., Holmes, L. S. & Paul, J. W. M. 1970 Phys. Rev. Lett. 25, 497.CrossRefGoogle Scholar
Davidson, R. C. & VÖlk, H. J. 1968 Phys. Fluids, 11, 2259.CrossRefGoogle Scholar
Elsässer, K. 1971 J. Plasma Phys. 5, 31.CrossRefGoogle Scholar
Elsässer, K. 1971 J. Plasma Phys. 5, 39.CrossRefGoogle Scholar
Elsässer, K. & Gräff, P. 1971 Ann. Phys. (N.Y.) 68, 305.CrossRefGoogle Scholar
Forslund, D. W. 1970 J. Geophys. Res. 75, 17.CrossRefGoogle Scholar
Galeev, A. A. & Karpman, V. I. 1963 Soviet Phys. JETP, 17, 403.Google Scholar
Galeev, A. A. & Sagdeev, R. Z. 1966 Int. Centre for Theor. Phys. Rep. IC/66/64.Google Scholar
Hundhausen, A. J., Bame, S. J., Asbridge, J. R. & Sydoriak, S. J. 1970 J. Geophys. Res. 75, 4643.CrossRefGoogle Scholar
Kadomtsev, B. B. 1965 Plasma Turbulence. Academic.Google Scholar
Keilhacker, M. & Steuer, K. 1971 Phys. Rev. Lett. 26, 694.CrossRefGoogle Scholar
Kennel, C. F. & Sagdeev, R. Z. 1967 J. Geophys. Res. 72, 3303.CrossRefGoogle Scholar
Kennel, C. F. & Scraf, F. L. 1968 J. Geophys. Res. 73, 6149.CrossRefGoogle Scholar
Parker, E. N. 1963 Interplanetary Dynamical Processes. Interscience.Google Scholar
Patterson, B. 1971 Phys. Fluids, 14, 1127.CrossRefGoogle Scholar
Pilipp, W. & VÖlk, H. J. 1971 J. Plasma Phys. 6, 1.CrossRefGoogle Scholar
Scraf, F. L. 1970 Space Sci. Rev. 11, 234.Google Scholar
Shapiro, V. D. & Shevchenko, V. I. 1964 Soviet Phys. JETP, 18, 1109.Google Scholar
Zaslavkii, G. M. & Moiseev, S. S. 1962 Prikl. Mekh. i Tekhn. Fiz. 6, 119.Google Scholar