Reply to J. F. McKenzie et al.'s comment on "Obliquely propagating large amplitude solitary waves in charge neutral plasmas"

Abstract

I note the comments by McKenzie et al. (2007), but it may be wise to remember that the plasma approximation, in which quasi-neutrality holds with n i n e but ∇·E =0 (Chen, 1974), is precisely that: an approximation. Arguments given by McKenzie et al. (2007), and indeed by several other authors, indicating how reasonable the plasma approximation is, are almost all based on linear wave theory and on keeping the relativistic effects small. In addition, it should be borne in mind that the plasma approximation is true only for lowfrequency motions where the electron inertia is not a factor (Chen, 1974;Nicholson, 1983), and, as has been noted before (Verheest, 2007), the concept of frequency might be borrowed from the linear counterparts of the nonlinear waves studied, but it is not a well defined property of stationary modes studied in their own reference frames.Although the plasma approximation has been used extensively in linear and small amplitude work, it is a legitimate question to ask what its effects might be for large nonlinear structures. In this context, the step was taken (Verheest, 2007) of considering the extreme case in which n i =n e and ∇·E=0.An indication of what might happen is given by the reductive perturbation treatment of moderately nonlinear electromagnetic waves propagating parallel to the ambient field. As is well known from many papers since the original derivation by Rogister (1971), these waves are governed by the derivative nonlinear Schrödinger (DNLS) equation. If one starts the description from the full set of Maxwell's equations, including Poisson's equation and the displacement current, one finds that to lowest order the parallel electric field vanishes, by combining the equations of motion to that order and without having assumed charge neutrality or relied upon Poisson's equation (see e.g. Verheest, 2000). Charge neutrality Correspondence to: F. Verheest (frank.verheest@ugent.be) then follows to lowest order from Poisson's equation, without having been assumed a priori.Continuing the iterative procedure, one eliminates the higher-order electromagnetic fields by combining in a judicious way the results obtained from the continuity and momentum equations and plugging these into Ampère's law, to arrive at the DNLS equation governing the perpendicular wave magnetic field. Usually the discussion then focuses on the solutions of the DNLS equation and the higher-order fields are not further investigated. The DNLS equation admits only envelope solitons, where the amplitude of the wave magnetic field shows the profile of a traditional solitary wave, but the phase increases linearly on the slow timescale.However, once the wave magnetic field obeys the DNLS equation, one can evaluate the higher-order electric fields and finds that a parallel electric field has been generated to that order by the nonlinearities (Verheest, 2004). This subsequently tells us that, although at the linear level there is automatically charge neutrality, this cannot be maintained to higher order. Since the cited treat...

Comment on “Plasma ionization by annularly bounded helicon waves” [Phys. Plasmas 13, 063501 (2006)]

“…Re-examination of the plasma equations of motion with consideration of the convective terms for the case of vanishing mass flow, charge density, and pressure gradients leads to the familiar Ohm's law of conductive material, indicating that only the usual electromagnetic radiation propagates through such regions. A similar discussion on the existence of whistler oscillitons [5][6][7][8] in geophysical plasmas is noted.…”