Electrostatic harmonic Langmuir waves are virtual modes excited in weakly turbulent plasmas, first observed in early laboratory beam-plasma experiments as well as in rocket-borne active experiments in space. However, their unequivocal presence was confirmed through computer simulated experiments and subsequently theoretically explained. The peculiarity of harmonic Langmuir waves is that while their existence requires nonlinear response, their excitation mechanism and subsequent early time evolution are governed by essentially linear process. One of the unresolved theoretical issues regards the role of nonlinear wave-particle interaction process over longer evolution time period. Another outstanding issue is that existing theories for these modes are limited to one-dimensional space. The present paper carries out two dimensional theoretical analysis of fundamental and (first) harmonic Langmuir waves for the first time. The result shows that harmonic Langmuir wave is essentially governed by (quasi)linear process and that nonlinear wave-particle interaction plays no significant role in the time evolution of the wave spectrum. The numerical solutions of the two-dimensional wave spectra for fundamental and harmonic Langmuir waves are also found to be consistent with those obtained by direct particle-in-cell simulation method reported in the literature. Published by AIP Publishing.

Electrostatic waves with frequencies that are integer multiples of the electron plasma frequency have been observed since the early days of laboratory experiments on beam-plasma interactions, and also in experiments made in the space environment. These waves have also appeared in numerical experiments, and can be explained in the context of weak turbulence theory. This paper presents results obtained by numerical solution of the equations of weak turbulence theory, which show the coupled time evolution of the amplitudes of harmonic waves and of the amplitudes of Langmuir and ion acoustic waves, and the time evolution of the electron distribution function. The results are obtained considering a two-dimensional geometry, considering harmonics up to n ¼ 5, and are consistent with earlier results obtained by one-dimensional analyses.