# Effect of finite spectral width on the modulational instability of Langmuir waves

TL;DR: In this article, the effets de the largeur spectrale finie sur l'instabilite de modulation des ondes de Langmuir sont etudies en appliquant une methode developpee par Alber.

Abstract: Les effets de la largeur spectrale finie sur l'instabilite de modulation des ondes de Langmuir sont etudies en appliquant une methode developpee par Alber pour obtenir l'equation de transport pour la densite spectrale. Les resultats numeriques montrent que le spectre est stable vis-a-vis de la perturbation de modulation quand le taux de croissance spectral depasse une valeur critique

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TL;DR: In this article, an analytical model for weakly nonlinear electron plasma waves is considered in order to obtain dynamic equations for the space-time evolution of their local power spectra.

Abstract: Analytical models for weakly nonlinear electron plasma waves are considered in order to obtain dynamic equations for the space-time evolution of their local power spectra. The model contains the wave kinetic equation as a limiting case for slow, long wavelength modulations. It is demonstrated that a finite spectral width in wavenumbers has a stabilizing effect on the modulational instability. The results invite a simple heuristic relation between the spectral width and the root-mean-square amplitude of stable stationary turbulent Langmuir wave spectra. A non-local average dispersion relation is derived as a limiting form by using the formalism developed for the spectral dynamics.

10 citations

### Cites background or methods from "Effect of finite spectral width on ..."

...We rewrite (2.3) for two sets of variables r1, t1 and r2, t2 (Alber 1978; Marcuvitz 1980; Bhakta and Majumder 1983; Dysthe et al. 1986)....

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...3) for two sets of variables r1, t1 and r2, t2 (Alber 1978; Marcuvitz 1980; Bhakta and Majumder 1983; Dysthe et al. 1986)....

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TL;DR: A mini-review of the description of plasma turbulence with particular attention to wave phenomena that contribute to anomalous resistivity and diffusion can be found in this paper, where the authors discuss the role of wave phenomena in anomalous transport in space plasmas.

Abstract: An important property associated with turbulence in plasmas and fluids is anomalous transport. Plasma, being a good conductor, can in addition be affected by turbulence through anomalous resistivity that can significantly exceed its classical counterpart. While turbulent transport may be adequately described in configuration space, some aspects of the anomalous resistivity are best accounted for in phase space. Kinetic phenomena like electron and ion phase space vortices can thus act as obstacles for the free flow of slow charged particles. Plasma instabilities and large amplitude plasma waves are candidates for contributions to the anomalous resistivity by generating such structures. Langmuir waves can be relevant, but also others, such as upper- as well as lower-hybrid waves in magnetized plasmas. Often these anomalous resistivity effects can be small, but due to the large spatial and temporal scales involved in space plasmas, planetary ionosphere and magnetosphere in particular, even such moderate effects can be important. This mini-review is discussing elements of the description of plasma turbulence with particular attention to wave phenomena that contribute to anomalous resistivity and diffusion. Turbulence effects can have relevance for space weather phenomena as well, where ground based and airborne activities relying on for instance Global Positioning and Global Navigation Satellite Systems are influenced by plasma conditions in geospace.

7 citations

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TL;DR: In this paper, a model for nonlinear electron plasma waves in weakly magnetized plasmas was developed for single and multi-mode conditions, with continuous wave spectra being a limiting case.

Abstract: Analytical models for nonlinear electron plasma waves in weakly magnetized plasmas are developed for single as well as multi-mode conditions, with continuous wave spectra being a limiting case. The conditions for wave decay as well as modulational instabilities are analysed. Our results demonstrate that slow or nearly stationary plasma density variations can be found for weakly magnetized plasmas even for weakly nonlinear electron plasma waves without involving cavitation of large amplitude plasma waves. A reduction of the growth rates for decay as well as modulational instabilities are found when the spectral width of the wave spectrum is increased. Some of our results are relevant for the interpretation of the nonlinearly enhanced ion acoustic lines often observed in non-equilibrium ionospheres.

6 citations

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TL;DR: In this article, the authors extended the results obtained for modulational instability of a Langmuir wave spectrum to account also for the Langevinear decay of a single-mode wave spectrum and two-wave models, where several combinations are considered: one wave is modulationally unstable, another decay unstable and one where both waves are unstable with respect to decay.

Abstract: Previous results obtained for modulational instability of a Langmuir wave spectrum are extended to account also for the Langmuir wave decay. The general model is tested by considering first the parametric decay of single-mode Langmuir waves, and also two-wave models, where several combinations are considered: one wave is modulationally unstable, another decay unstable and one where both waves are unstable with respect to decay. For the general case with continuous wave spectra it is found that distribution of the Langmuir wave energy over a wide wavenumber band reduces the decay rate when the correlation length for the spectrum becomes comparable to the wavelength of the most unstable sound wave among the possible decay products.

5 citations

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TL;DR: In this article, the Wigner formalism is applied to the standard equations for weakly nonlinear Langmuire waves and a closed set of equations is obtained to describe the statistical evolution of the average field intensity and the intensity correlation function.

Abstract: The Wigner formalism is applied to the standard equations for weakly nonlinear Langmuire waves. On the basis of this formulation, a closed set of equations is obtained to describe the statistical evolution of the average field intensity and the intensity correlation function. The analysis allows a straightforward generalization to include features of the random coupling model (one version of the direct interaction approximation).

5 citations

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TL;DR: In this paper, a theoretical analysis of the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes is presented, where the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.

Abstract: The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if
\[
0 < \delta \leqslant (\sqrt{2})ka,
\]
where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

2,109 citations

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TL;DR: In this paper, a pair of coupled, nonlinear, partial differential equations which describe the evolution of low-frequency, large-scale-length perturbations propagating parallel, or nearly parallel, to the equilibrium magnetic field in high-β plasma have been obtained.

Abstract: A pair of coupled, nonlinear, partial differential equations which describe the evolution of low‐frequency, large‐scale‐length perturbations propagating parallel, or nearly parallel, to the equilibrium magnetic field in high‐β plasma have been obtained. The equations account for irreversible resonant particle effects. In the regime of small but finite propagation angles, the pair of equations collapses into a single Korteweg‐de Vries equation (neglecting irreversible terms) which agrees with known results.

252 citations

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TL;DR: In this paper, the authors studied the dynamics of weak Langmuir turbulence and showed that the number of polynomial conserved densities (p.c.d.) of solitons in a three-dimensions soliton-like structures is a function of the density of the cavitons.

238 citations

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TL;DR: In this article, the effect of self-phase modulation of a pulse propagating in a long-geometry waveguide was investigated and it was shown that for long waveguides with relatively small group-velocity dispersion but finite nonlinear coefficient, the pulse will develop a sizable asymmetric frequency and temporal spectra.

Abstract: We calculate the effect of self-phase modulation of a pulse propagating in a long-geometry waveguide. Our calculations go beyond the usual theory, which does not take into account the envelope time variation in the nonlinear term of the wave equation. We show that for long waveguides with relatively small group-velocity dispersion but finite nonlinear coefficient ${n}_{2}$, the pulse will develop a sizable asymmetric frequency and temporal spectra.

232 citations

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TL;DR: In this paper, a simplified nonlinear spectral transport equation for narrowband Gaussian random surface wavetrains, slowly varying in space and time, is derived from the weakly nonlinear equations of Davey & Stewartson.

Abstract: A simplified nonlinear spectral transport equation, for narrowband Gaussian random surface wavetrains, slowly varying in space and time, is derived from the weakly nonlinear equations of Davey & Stewartson. The stability of an initially homogeneous wave spectrum, to small oblique wave perturbations is studied for a range of spectral bandwidths, resulting in an integral equation for the amplification rate of the disturbance. It is shown for random deep water waves that instability of the wavetrain can exist, as in the corresponding deterministic Benjamin-Feir (B-F) problem, provided that the normalized spectral bandwidth $\sigma /k_{0}$ is less than twice the root mean square wave slope, multiplied by a function of the perturbation wave angle $\phi $ = arctan (m/l). A further condition for instability is that the angle $\phi $ be less than 35.26 degrees. It is demonstrated that the amplification rate, associated with the B-F type instability, diminishes and then vanishes as the correlation length scale of the random wave field (ca. $1/\sigma)$) is reduced to the order of the characteristic length scale for modulational instability of the wave system.

181 citations