Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-29T13:29:38.700Z Has data issue: false hasContentIssue false
  • English
  • Français

A Lagrangian theory for nonlinear wave packets in a collisionless plasma

Published online by Cambridge University Press:  13 March 2009

R. L. Dewar
R. L. Dewar
Affiliation:
Conter for Theoretical Physics, University of Maryland
Get access Rights & Permissions [Opens in a new window]

Abstract

A Lagrangian, for the slowly varying complex amplitude of an almost monochromatic electrostatic plasma wave in an unmagnetized plasma, is derived. The method is a variant of the averaged Lagrangian technique of Whitham, adapted for use in plasma physics by employing Low's Lagrangian, together with certain elimination procedures to simplify the Lagrangian to usable form. The expansion in powers of the wave amplitude is carried out to quartic terms, and non-adiabatic terms are also retained. Variations with respect to the amplitude lead to a nonlinear Schrödinger equation for the amplitude, while variations of the particle trajectories lead to a modified Vlasov equation, which includes the nonlinear reaction of the wave on the average trajectories. It is assumed that there are no resonant particles. These equations are coupled through the nonlinear frequency shift. It is found that the particle aspect of the plasma has a profound effect on the stability of the system, due to resonance of the wave envelope with particles moving at the group velocity. This effect, nonlinear Landau damping, is always destabilizing, leading to growth of modulations. In terms of the nonlinear Schrodinger equation, the effect changes the equation from a Hartree-Fock equation with a delta-function interaction to one with a non-momentum-conserving, non-local interaction. In the limit of fairly small wavelength, of the modulations, it is shown that the growth rate approaches that expected from plasma kinetic theory.

Type
Research Article
Information
Journal of Plasma Physics , Volume 7 , Issue 2 , April 1972 , pp. 267 - 284
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

Aamodt, R. E. & Drummond, W. E. 1964 Phys. Fluids 7, 1816.CrossRefGoogle Scholar
Asano, N., Taniuti, T. & Yajima, N. 1969 J. Math. Phys. 10, 2020.CrossRefGoogle Scholar
Benney, D. J. & Newell, A. C. 1967 J. Math. Phys. 46, 133.CrossRefGoogle Scholar
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics. Tnterscience.Google Scholar
Dewar, R. L. 1970 a Phys. Fluids, 13, 2710.CrossRefGoogle Scholar
Dewar, R. L. 1970 b Ph.D. thesis, Princeton University.Google Scholar
Dewar, R. L. 1972 Phys. Fluids. (To be published.)Google Scholar
Dewar, R. L. & Lindl, J. 1972 Phys. Fluids. (To be published.)Google Scholar
Dougherty, J. P. 1970 J. Plasma Phys. 4, 761.CrossRefGoogle Scholar
Frieman, E. & Rutherford, P. 1964 Ann. Phys. (N.Y.) 28, 134.CrossRefGoogle Scholar
Galloway, J. J. & Kim, M. 1971 J. Plasma Phys. 6, 53.CrossRefGoogle Scholar
Goldstein, H. 1950 Classical Mechanics (1st edn.). Addison-Wesley.Google Scholar
Gorbunov, L. M. & Silin, V. P. 1965 Soviet Phys. JETP, 20, 135.Google Scholar
Hasegawa, A. 1970 Phys. Rev. A 1, 1746.CrossRefGoogle Scholar
Henley, E. M. & Thirring, W. 1962 Elementary Quantum Field Theory. McGraw Hill.CrossRefGoogle Scholar
Karpman, V. I. & Krushkal', E. M. 1969 Soviet Phys. JETP, 28, 277.Google Scholar
Low, F. E. 1958 Proc. Roy. Soc. A 248, 282.Google Scholar
Ott, E. & Dum, C. T. 1971 Phys. Fluids, 14, 959.CrossRefGoogle Scholar
Ott, E. & Sudan, R. N. 1969 Phys. Fluids, 12, 2388.CrossRefGoogle Scholar
Tam, C. K. W. 1969 Phys. Fluids, 12, 1028.CrossRefGoogle Scholar
Tam, C. K. W. 1970 J. Plasma Phys. 4, 109.CrossRefGoogle Scholar
Tang, T.-W. & Sivasubramanian, A. 1971 Phys. Fluids, 14, 444.CrossRefGoogle Scholar
Taniuti, T. & Washimi, H. 1968 Phys. Rev. Lett. 21, 209.CrossRefGoogle Scholar
Taniuti, T. & Yajima, N. 1969 J. Math. Phys. 10, 1369.CrossRefGoogle Scholar
Vedenov, A. A. & Rudakov, L. I. 1965 Soviet Phys. Dokl. 9, 1073.Google Scholar
Vedenov, A. A., Gordeev, A. V. & Rudakov, L. I. 1967 Plasma Phys. 9, 719.CrossRefGoogle Scholar
Wentzel, G. 1949 Quantum Theory of Fields (trans. Houtermans, C. and Jauch, J. M.). Interscience.Google Scholar
Whitham, G. B. 1965 J. Fluid Mech. 22, 273.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Soviet Phys. JETP Letters, 7, 226.Google Scholar