In the present study, we analyze the data sets produced by a one-dimensional Vlasov-Poisson simulation of the weak electron beam-plasma instability to clarify the nonlinearity of the Langmuir turbulence excited by the weak-beam interaction. The growth of wave number modes is analyzed by using the momentum equation of the whole electrons. The analysis shows that the primary Langmuir wave mode is almost linear, while the nonlinear terms play important roles in the growth of the lower harmonic

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... and the secondary higher harmonic mode. After the linear growth saturates, while the wave power of the primary mode is much larger than the other modes, linear and nonlinear interactions occurring in both lower harmonic and secondary higher harmonic modes are more active than those in the primary mode. Nonlinearity in the system comes from the advection rather than the ponderomotive forces. The amplitude-modulated Langmuir waves are frequently observed in space plasmas. 1-3 The past numerical studies have suggested that some of the Langmuir turbulence in space plasmas and in type III solar radio bursts are consequences of electron beam-plasma instabilities. [4] [5] [6] [7] [8] 12, 10 In the case of weak electron beam-plasma interactions, the mechanism of the amplitude modulation is thought to be due to the nonlinear trapping of beam electrons by excited Langmuir waves. 5, 6 On the other hand, the previous Vlasov simulations of a weak electron beam-plasma interaction have shown that Langmuir waves are not directly modulated by the nonlinear trapping but are modulated by the nonlinear interaction between the most unstable primary Langmuir mode and its sideband modes. 8-10 Silin et al. 10 has suggested that the temporal change of velocity distribution function ͑VDF͒ of beam electrons due to the nonlinear trapping and the plateau formation lead to a broadband wave number spectrum, which corresponds to the envelope modulation in the real space, due to higher wave number shift of the linearly unstable modes. The purpose of this Brief Communication is to clarify the nonlinearity of the Langmuir turbulence excited by the weak-beam interaction. We analyze the data sets produced by a one-dimensional Vlasov-Poisson simulation of the weak electron beam-plasma instability with the same parameters as Silin et al. 10 We adopt the "splitting method" 11 and the positive interpolation for hyperbolic conservation laws scheme 12 suggested for time advancement of the one-dimensional electrostatic Vlasov equation, where f e is the VDF of electrons, m e is the electron mass, and e is the elementary electric charge, v x and E x are the velocity and electric field along the simulation domain ͑x͒, respectively. In our run, two electron components, which are a very weak electron beam and background major electrons, are initially given with the same plasma parameters as the previous study. 10 The beam and background electrons have the equal thermal velocity v th = ͱ ͑k B T e ͒ / m e = 0.125v d , where k B is the Boltzmann factor, T e is the electron temperature, and v d ͑=1͒ is the beam drift velocity, respectively. The density ratio of the beam component is R = n b / ͑n c + n b ͒ϳ0.003, where the subscripts c and b represent background core electrons and beam electrons, respectively. The total electron plasma frequency e , m e , n e ͑=n c + n b ͒, and e are also assumed to be unity. Since ion dynamics is negligible for the evolution of the system with the above parameters, as shown in the previous study, 10 we consider ions as an immobile background in the present study. The simulation domain is taken along an ambient magnetic field. We use periodic boundary conditions in the real space, and open boundary conditions in the velocity space. The number of cells is N x = 2048 in the x direction and is N vx = 400 in the v x direction over a velocity range from v max = 1.5v d to v min = −1.5v d . The grid spacing is ⌬x = 0.25v d / e = 0.03125v th / e , and the time step is ⌬t = 0.02/ e . We initially impose a seed density perturbation ␦n e ϳ Ϯ 10 −10 as a white noise. 13 Silin et al. 10 modeled the nonlinear deformations of the VDFs by using a simple analytic function, which represent the transition from the initial VDFs ͓s = 1 in Eq. ͑2͔͒ to the stationary Maxwellian with a flat shoulder ͓s = 6 in Eq. ͑2͔͒ as follows: 10 th 2 ZЈ͑z e ͒ + 1 s ͚ n=1 s R 2v th 2 ZЈ͑z bn ͒, ͑2͒ where Z is the plasma dispersion function, ZЈ͑z ‫ء‬ ͒ =−2͓1 + z ‫ء‬ Z͑z ‫ء‬ ͔͒, z e = / ͑ ͱ 2kv th ͒, and z bn = ͕ + ͓v d − ͑n −1͒v th ͔k͖ / ͑ ͱ 2kv th ͒.