Modulational Interactions in Plasmas
Modulational Interactions in Plasmas is the first book to present all the basic considerations relevant to the topic. It adopts a simple and universal approach, based on new methods developed for the description of modulation interactions in arbitrary med
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ASTROPHYSICS AND SPACE SCIENCE LIBRARY VOLUME201
Executive Committee W. B. BURTON, Sterrewacht, Leiden, The Netherlands C. DE JAGER, Foundation Space Research, Utrecht, The Netherlands E. P. J. VAN DEN HEUVEL, Astronomical Institute, University ofAmsterdam, The Netherlands H. VANDERLAAN, Astronomical Institute, University of Utrecht, The Netherlands
Editorial Board I. APPENZELLER, Landessternwarte Heidelberg-Konigstuhl, Germany J. N. BAHCALL, The Institute for Advanced Study, Princeton, U.SA. F. BERTOLA, Universita di Padova, Italy W. B. BURTON, Sterrewacht, Leiden, The Netherlands J.P. CASSINELLI, University ofWisconsin, Madison, U.S A. C. J. CESARSKY, Centred' Etudes de Saclay, Gif-sur-Yvette Cedex, France C. DE JAGER, Foundation Space Research, Utrecht, The Netherlands R. McCRAY, University of Colorado, JIIA, Boulder, U.S A. P. G. MURDIN, Royal Greenwich Observatory, Cambridge, U.K. F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India F. H. SHU, University of California, Berkeley, U.SA. B. V. SOMOV, Astronomical Institute, Moscow State University, Russia S. TREMAINE, CITA, University of Toronto, Canada Y. TANAKA, Institute of Space & Astronautical Science, Kanagawa, Japan E. P. J. VAN DEN HEUVEL, Astronomical Institute, University ofAmsterdam, The Netherlands H. VANDERLAAN, Astronomical Institute, University of Utrecht, The Netherlands N. 0. WEISS, University of Cambridge, U.K.
MODULATIONAL INTERACTIONS IN PLASMAS by
SERGEY V. VLADIMIROV General Physics Institute, Moscow, Russia
VADIM N. TSYTOVICH General Physics Institute, Moscow, Russia
SERGEY I. POPEL Institute for Dynamics ofGeospheres, Moscow, Russia
and
FOTEKH KH. KHAKIMOV Tajik State University, Dushanbe, Tajikistan
WKAP ARCHIEF
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data Modulational lnteraetions In plasmas / by Sergey V. Vladlmirov ... [et al.
l.
p. em. -- ISBN 978-90-481-4556-0 ISBN 978-94-017-2306-0 (eBook) DOI 10.1007/978-94-017-2306-0
1. Plasma + f(l) + !(2 ) + . . . ' (2.35) where is the initial unperturbed distribution function, and j(l), j< 2l, etc. are the small perturbations (linear, quadratic, etc. in the electric field) of the initial distribution. It is commonly supposed that
2.2. NONLINEAR RESPONSES
73
Using the Fourier components, we find !(1)
k,w
=
e
i(w- k·v)
Ek
,w
(!_. ~~) lkl Bp '
(2.36)
(2.37)
(2.38)
According to the definition of the nonlinear responses S and I: given by Eq. (1.5) we have 47re
ilkl
I(
dp (2 ) _ 21r)3fk,w-
I
d12S1,2E1E2,
(2.39)
and (2.40) Next we symmetrize the nonlinear responses obtained from Eqs. (2.39) and (2.40) and write
S
X {
I (21r)3w1 +w2- (k1 +k2)·v 1'2 _
dp
27rie 3
1 (k1·~) Op W2- k2·V (k2·~) Op + (k2·~) Op
(2.41)
74
CHAPTER 2. POTENTIAL ELECTRIC FIELDS
(2.42) Note that for the third-order response the used symmetrization is not the full symmetrization on all its three indices.
2.2.2
Effective third-order response
Approximate expressions for the eff
