Electron beam-plasma interaction in a dusty plasma with excess suprathermal electrons

arXiv:1108.4573v1 [astro-ph.HE] 23 Aug 2011 Electron beam - plasma interaction in a dusty plasma with excess suprathermal electrons A. Danehkar,1 N. S. Saini,2 M. A. Hellberg,3 and I. Kourakis4 1Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia 2Department of Physics, Guru Nanak Dev University, Amritsar-143005, India 3School of Physics, University of KwaZulu-Natal, Durban 4000, South Africa 4Department of Physics and Astronomy, Queen’s University Belfast, BT7 1NN, UK The existence of large-amplitude electron-acoustic solitary structures is investi- gated in an unmagnetized and collisionless two-temperature dusty plasma penetrated by an electron beam. A nonlinear pseudopotential technique is used to investigate the occurrence of stationary-profile solitary waves, and their parametric dependence on the electron beam and dust perturbation is discussed. PACS numbers: 52.27.Lw, 52.35.Sb, 52.35.Mw, 52.40.-w Keywords: Dusty (complex) plasmas, solitons, nonlinear phenomena, plasma interactions We have previously studied electron-acoustic solitary waves in the presence of a suprather- mal electron component. [1] Our aim here is to investigate the effect of beam electrons and dust on the electrostatic solitary structures. We consider a plasma consisting of cold inertial drifting electrons (the beam), cold in- ertial background electrons, hot suprathermal electrons modeled by a kappa-distribution, stationary ions, and stationary dust (of either positive or negative charge). The dynamics of the cold inertial background electrons and the beam electrons are governed by the following normalized one-dimensional equations: ∂n ∂t + ∂(nu) ∂x = 0, ∂u ∂t + u∂u ∂x = ∂φ ∂x, (1) ∂nb ∂t + ∂(nbub) ∂x = 0, ∂ub ∂t + ub ∂ub ∂x = ∂φ ∂x, (2) ∂2φ ∂x2 = −(η + sδ) + n + βnb + (η + sδ −1 −β)  1 − φ [κ −3 2] −κ+1/2 . (3) 2 2 3 4 5 6 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 κ M1, M2 δ = 0.3 δ = 0.35 δ = 0.4 2 3 4 5 6 0.35 0.4 0.45 0.5 0.55 κ M1, M2 U0 = 0.3 U0 = 0.35 U0 = 0.4 0.2 0.25 0.3 0.35 0.4 0.45 0.44 0.46 0.48 0.5 0.52 U0 M1, M2 β = 0.0005 β = 0.001 β = 0.0015 0 0.1 0.2 0.3 0.4 0.4 0.45 0.5 0.55 0.6 0.65 δ M1, M2 κ = 2 κ = 3 κ = 4 κ = 10 a b d c FIG. 1: Soliton existence region (M1 < M < M2): (a) versus κ for different δ values; (b) versus δ for different κ values; (c) versus κ for different U0 values; (d) versus U0 for different β values. The remaining values are κ = 4.5, δ = 0.3, s = −1, β = 0.001, U0 = 0.4 and η = 4.5, unless values are given. Here, n and nb denote the fluid density variables of the cool electrons and the beam electrons normalized with respect to nc,0 and nb,0. The velocities u and ub, and the equilibrium beam speed U0 = ub,0/cth are scaled by the hot electron thermal speed cth = (kBTh/me)1/2, and the wave potential φ by kBTh/e. Time and space are scaled by the plasma period ω−1 pc = (nc,0e2/ε0me)−1/2and the characteristic length λ0 = (ε0kBTh/nc,0e2)1/2, respectively. We define the hot-to-cold electron charge density ratio α = nh,0/nc,0, the beam-to-cold electron charge density ratio β = nb,0/nc,0, the ion-to-cold electron charge density ratio η = Zini,0/nc,0, and the dust-to-cold electron charge density ratio δ = Zdnd,0/nc,0. Here, suprathermality is denoted by the spectral index κ, and s = ±1 is the sign of the dust charge for positive or negative dust grains. At equilibrium, the plasma is quasi-neutral, so η + sδ = 1 + α + β. Anticipating constant profile solutions of Eqs. (1)–(3) in a stationary frame traveling at a constant normalized velocity M, implying the transformation ξ = x −Mt, we obtain n = (1 + 2φ/M2)−1/2 and nb = [1 + 2φ/(M −U0)2]−1/2. Substituting in Poisson’s equation and integrating yields a pseudo-energy balance equation 1 2 (dφ/dξ)2 + Ψ(φ) = 0, where the Sagdeev pseudopotential Ψ(φ) reads Ψ(φ) = (η + sδ) φ + M2  1 −[1 + 2φ/M2] 1 2  + β(M −U0 )2 ×  1 −[1 + 2φ/(M −U0 )2] 1 2  + (η + sδ −1 −β)  1 −[1 −φ/(κ −3 2)]−κ+ 3 2  . (4) Reality of the density variable implies two limits on the electrostatic potential φmax = −M2/2 and −(M −U0 )2/2 for U0 < 0 and U0 > 0, respectively. In order for solitary waves to exist, 3 two constraints must be satisfied, i.e., F1(M) = −Ψ′′(φ)|φ=0 > 0 and F2(M) = Ψ(φ)|φ=φmax > 0, which yield the solutions for the lower and upper limit in M. As shown in Figure 1, the existence domain for solitons becomes narrower with increasing suprathermal excess (decreasing κ), increasing equilibrium beam speed, and decreasing beam density. Dust charge density shows little effect on the width of the existence domain, but for quasi-Maxwellian electrons, it weakly increases the typical values of M. It was found that both increasing κ and increasing negative dust charge density significantly reduce soliton amplitude at fixed M (not shown here). I. ACKNOWLEDGMENTS AD, NSS and IK thank the Max-Planck Institute for Extraterrestrial Physics for their support. IK acknowledges support from UK EPSRC via S&I grant EP/D06337X/1. [1] A. Danehkar, N. S. Saini, M.A. Hellberg

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