Dynamics of the Beam Driven Langmuir Wave Packet in a Plasma with Random Inhomogeneities
K. Musatenko1,2, V. Krasnoselskikh1, V. Lobzin1 and I. Anisimov2
1LPCE/CNRS, 3A, Avenue de la Recherche Scientifique,
45071, Orleans, CEDEX 2, France
2National Taras Shevchenko University of Kyiv, 64, Volodymyrs’ka St.,
01033, Kyiv, Ukraine
Abstract. We consider a model describing the evolution of Langmuir wave packet generated by the electron beam in
randomly inhomogeneous plasma. According to this model the packet changes its form and amplitude due to the linear
instability and angular diffusion of the k-vector. Linear growth takes place when the wave is in the resonance with a
beam, while diffusion on inhomogeneities changes the direction of the k-vector thereby shifting the wave from a
resonance. We present a numerical code simulating a packet dynamics under such conditions and investigate the
dependence of the results on the parameters.
Keywords: beam plasma instability, random density plasma inhomogeneities, Langmuir waves, angular diffusion.
PACS: 94.20.wf, 94.20.Bb, 94.20.wj
The goal of our work is to study the evolution of the wave packets in a randomly inhomogeneous plasma in the
presence of the beam-plasma instability. Such a situation occurs in the source regions of the solar type II and III
radiobursts and in the electron foreshock region.
Density fluctuations are observed in solar wind plasmas during many years [1-4]. Their power spectra and
anisotropy were studied by Celnikier . Using these spectra Kellog demonstrated that their amplitude can reach up
to 5% of the mean density . Such level of fluctuations can influence characteristics of the propagating waves and
affect their interaction with the beams. One of the models describing the wave beam interaction in a plasma with
random inhomogeneities was suggested by Nishikawa and Ryutov . They took into account the small amplitude
fluctuations which effect can be considered as angular diffusion of the wave vector. More extensive study of the
influence of random inhomogeneities on the beam plasma interaction was performed by Goldman and DuBois .
Muschietti with co-authors  carried out the numerical simulation of wave propagation and instability development
in a randomly inhomogeneous plasma using the equation written by Nishikawa and Ryutov . They have shown
the suppression of the beam-plasma instability due to the angular diffusion. Robinson  and Robinson and Cairns
 suggested that the effect of density inhomogeneities can be described as random fluctuations of the increment.
They assumed that probability distribution for the growth rate is Gaussian and concluded that the presence of
random inhomogeneities will result in lognormal distribution for the universal statistics of the Langmuir wave
In our model we consider beam plasma instability development in the presence of the random density
fluctuations taking into account two effects: angular diffusion of the wavevector described in  and the change of
the wave phase velocity due to large amplitude inhomogeneities that shift the wave in or out of the resonance with a
beam. In order to obtain a probability distribution function of the wave amplitudes (PDF) it is necessary to
investigate the evolution of one wave packet crossing one large amplitude inhomogeneity. In this paper we present
results of the numerical modeling of the Langmuir wave packet dynamics in a beam-plasma system with random
We consider a propagation of the Langmuir wave packet resonantly excited by an electron beam. In our study the
packet represents a group of waves having the same frequency and certain angular distribution of the spectral energy
density around the chosen direction that is supposed to be the direction of the propagation of a beam. Our goal is to
describe two physical effects that determine the dynamics of the wave packet due to the presence of the random
density fluctuations. We separate density fluctuations into two groups, small amplitude inhomogeneities that give
rise to the angular diffusion of the wave vector, but do not change its absolute value, and large amplitude density
fluctuations that result in the change of its magnitude and consequently of the phase velocity of the waves under
consideration. In both cases we consider spatial boundary problem, thus the frequency of wave is conserved. We use
the same description of the angular diffusion as the one proposed by Nishikawa and Ryutov in  and that was later
used by Muschietti et al. , and by Goldman and DuBois . The wave crossing such an inhomogeneity can enter
the regions where it is in the resonance.
FIGURE 3. Dependencies of the effective increment on the parameter (a) 1/δ1 for δ2 = 90 and γπ/2 = 40 (pink line), γπ/2 = 50 (blue
line), γπ/2 = 60 (yellow line); (b) 1/δ2 for γπ/2 = 60 and δ1 = 90 (blue line), δ1 = 110 (pink line), δ1 = 130 (yellow line). On both
figures xmax/a = 6000, θ = 0.
They are very close to linear. They indicate that effective growth rate is directly proportional to δ1 (growth to
diffusion ratio) and depends on the size and the amplitude of the resonant region (Fig. 3a). One can see that even if
the beam growth rate is positive the angular diffusion can totally suppress the growth of the wave. For the growth to
occur the characteristic increment threshold value for given diffusion coefficient should be exceeded. One can notice
that even relatively weak angular diffusion (δ1 = γ0/D = 50) can convert the growth into the damping.
The aim of our work was to study the dynamics of the spectral energy density of the Langmuir wave packet
moving in a beam-plasma system with random inhomogeneities. We suggested a model that takes into account two
physical effects: angular diffusion in the phase space and growth/damping caused by variations of the waves phase
velocity. The corresponding equation was solved numerically using Thomas algorithm for the crossing of one
resonant region. Spatial distributions of the spectral energy density for different values of the growth rate and
diffusion coefficient were obtained. We found that growth/damping of the wave amplitude can be characterized by
effective linear increment/decrement coefficient. The dependencies of the effective increment on the model
parameters show that angular diffusion can considerably suppress growth by shifting the wave out of resonance with
a beam. We showed that for the wave growth to occur the characteristic increment threshold value should be
exceeded. This work is a first step of the wave statistical properties studies. Having knowledge of the spectral energy
density dynamics during one inhomogeneity crossing and the information about the distribution of these
inhomogeneities, one can find a probability distribution function of waves propagating in a beam-plasma system
with randomly distributed inhomogeneities to explain the observational data.
The work of M.K. was supported by French Government fellowship. Authors wish to thank I. Cairns and
P.L. Blelly for useful discussions.
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