Microscopic Lagrangian description of warm plasmas: 2. Nonlinear wave interactions

Abstract

The averaged-Lagrangian method is applied to wave-wave interactions in an infinite, homogeneous, collisionless, warm magnetoplasma. The amplitudes of the waves are assumed to vary slowly in time and space, due to coupling between them. Euler-Lagrange equations are obtained from the contributions to the averaged microscopic Lagrangian second and third order in perturbation, by variation with respect to the wave amplitudes. These are the coupled-mode equations. The phase variation yields the action transfer equation. As applications of the method, coupled-mode equations are derived in explicit forms for all possible interactions among waves propagating nearly parallel, and among those propagating exactly perpendicular, to the static magnetic field. Some of the coupling coefficients are new. Where comparisons with previous iterative analyses are possible, the advantages of the method are discussed. INTRODUCTIONThe linear theory developed in Part 1 of this three-part paper assumes that an arbitrary perturbation can be expressed as a superposition of non-iteracting eigenmodes. A characteristic result of nonlinearity is the occurrence of coupling among these modes. If the coupling is weak, then arbitrary perturbations may still be expressed approximately as a superposition of eigenmodes, but whose amplitudes are now assumed to vary slowly in time and space. By 'slowly' we mean that the scale of the variation due to the wave-wave interaction is much longer in time and space than the period and wavelength. We shall assume that this scale •s comparable to the ratio of the amplitude of a perturbation to that of the zero-order quantity, and denote it by a small parameter E. We shall also assume that interaction occurs coherently, i.e., at this point we shall not treat interaction in the random phase approximation. We shall consider all possible types of three-wave interactions in the infinite, collisionless, warm magnetoplasma, with an immobile ion background, considered in Part 1. The procedure to be followed is similar to that used in the analysis of nonlinear wave interaction in a cold plasma [Dougherty, 1970;Dysthe, 1974]. The averaged-Lagrangian is first obtained in section 2 for waves propagating at an oblique angle to the static magnetic field. It was demonstrated in Part Copyright (•) 1977 by the American Geophysical Union. 1, section 3, that action is conserved in the linear theory. When nonlinear interaction occurs, action is exchanged among waves due to the wave coupling. The action-transfer equations, and the coupled-mode equations, are derived in section 3. In section 4, these equations are specialized to the case of waves propagating nearly parallel to the static magnetic field. In section 5, interactions among longitudinal and ordinary cyclotron harmonic waves propagating perpendicular to the static magnetic field are considered. (Coupled-mode equations for these cases are presented in explicit forms suitable for computation in appendix A.) A comment on general solutions to the coupled mode equations is made ...

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