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.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6DFC211BBBC516771A7DF89C92424EE9/S002237781400035Xa.pdf/div-class-title-modulational-stability-of-electron-plasma-wave-spectra-div.pdf

Modulational stability of electron plasma wave spectra

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.

/pdf/the-impact-of-turbulence-on-the-ionosphere-and-magnetosphere-422qjv87cy.pdf

The Impact of Turbulence on the Ionosphere and Magnetosphere

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.

Stability of electron wave spectra in weakly magnetized plasmas

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.

Parametric decay of wide band Langmuir wave spectra

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).

https://iopscience.iop.org/article/10.1088/0031-8949/33/3/012/pdf

Langmuir Turbulence: Proposal for a Closure

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

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022377800001124

D. Majumder

Papers

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