Self-Acceleration and Energy Channeling in the Saturation of The

SELF-ACCELERATION AND ENERGY CHANNELING IN THE SATURATION OF THE ION-SOUND INSTABILITY IN A BOUNDED PLASMA Liang Xu Department of Electrical Engineering and Information Science, Ruhr University Bochum, D-44780 Bochum, Germany Andrei Smolyakov and Salomon Janhunen Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E2, Canada Igor Kaganovich Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA April 17, 2020 ABSTRACT A novel regime of the saturation of the Pierce-type ion-sound instability in bounded ion-beam-plasma system is revealed in 1D PIC simulations. It is found that the saturation of the instability is mediated by the oscillating virtual anode potential structure. The periodically oscillating potential barrier separates the incoming beam ions into two groups. One component forms a supersonic beam which is accelerated to an energy exceeding the energy of the initial cold ion beam. The other component is organized as a self-consistent phase space structure of trapped ions with a wide energy spread – the ion hole. The effective temperature (energy spread) of the ions trapped in the hole is lower than the initial beam energy. In the final stage the ion hole expands over the whole system length. Plasma physics has yielded many important results in the typically bounded by walls that introduce important con- theory of nonlinear waves. Earlier theories of coherent straints. In this letter we present a novel mechanism of nonlinear waves, shocks and solitons [1] have been ex- the nonlinear saturation of the ion-sound instability in a panded to include nonlinear structures in phase space, such bounded plasma penetrated by a cold ion beam. as Bernstein-Green-Kruskal (BGK) states [2], electron and In a warm infinite plasma the ion-sound instability of ki- ion holes [3, 4, 5, 6, 7, 8, 9] and clumps[10, 11]. Such netic nature[27, 28] occurs when the current drift veloc- structures have been identified as the nonlinear saturated ity exceeds the phase velocity of the ion sound waves states of the beam-like kinetic instabilities in collision- 2 2 1=2 less and weakly collisional plasmas when the dynamics v0 > cs=(1 + k λDe) , where cs is the ion sound ve- is dominated by a single coherent wave. The saturation locity, λDe is the Debye length and k is the wave number. mechanism via the formation of long lived phase space In a bounded plasma of finite length there exists much structures is a scenario alternative to the quasilinear satura- stronger fluid instability of the Pierce type for v0 < cs tion in the case of a wide spectrum of overlapping modes which has been studied theoretically [29, 30, 31], and has arXiv:2004.07513v1 [physics.plasm-ph] 16 Apr 2020 resulting due to the flattening of the distribution function. been observed in experiments[29]. The linear stage of Phase space holes/clumps have been detected in space the Pierce-type ion beam instability has been well studied plasmas [11, 12, 13, 14, 15] and shown to be important for theoretically. Experiments in double plasma devices have the saturation of fast particle driven instabilities in fusion revealed complex dynamics of Pierce instabilities induced plasmas[16, 10, 17, 18, 19]. Many numerical experiments by the ion and electron beams. Nakamura et. al. [29] [3, 20, 11, 9, 21, 22, 23, 24, 25, 26] have demonstrated have observed ion sound waves and nonlinear harmonics. phase space structures as a result of the saturation of bump- The oscillations were experimentally identified by Mat- on-tail and Buneman instabilities. Most analytical and sukuma et. al. [32]. In the electron-beam-plasma system, numerical studies have dealt with periodic systems. In Iizuka et. al. [33, 34, 35] have demonstrated large scale practical laboratory applications, however, plasmas are A PREPRINT -APRIL 17, 2020 current oscillations, the virtual cathode trapping ions and 0.06 particle simulations particle simulations, ni the double-layer formation. 0.6 b) v 0.05 i theory φ 0.4 theory, ni In this letter, we show that a strong ion sound instability 0.04 vi pi φ ω 0.03 0.2 induced by the ion flow in a bounded plasma saturates / γ 0.02 0 through the formation of the virtual anode potential struc- eigen mode 0.01 -0.2 ture that exhibits coherent large amplitude oscillations. a) 0 -0.4 The large amplitude periodic oscillations of this localized 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 v / c potential on one hand accelerate a fraction of the ion beam 0 s x / λDe (initially subsonic) to supersonic velocity. On the other hand, the oscillating potential barrier stops a fraction of the 30 60 d) ion beam, resulting in the formation of a long-lived large c) 0.5 0.5 28 scale ion hole with a negative potential and a large popula- 50 0 0 40 26 tion of trapped ions. Thus, a fraction of the initial energy s s -0.5 2 -0.5 t/ 30 t/ of the ion beam mini0v0=2 with v0 < cs is channeled to 24 -1 a higher energy creating a lower density supersonic beam 20 -1 22 10 -1.5 with velocity vb > cs , while rest of the beam energy is -1.5 0 20 spread to lower energies forming the phase space vortex of 10 20 30 40 50 1 2 3 4 5 x/ x/ trapped ions with a wide energy spread with an effective De De 2 temperature Ttr < mivb , where mi is the ion mass and ni0 2 0 is the density of injected ions. In our case, Ttr ' mivb =6. e) -5 A simplified 1D collisionless plasma model with a cold -10 ion beam injected to the system is studied using the 1D3V )/eV electrostatic direct implicit particle-in-cell code (EDIPIC) pot -15 [36], which has been comprehensively validated and bench- ln(W marked [37, 38]. We focus only on the ion sound -type -20 low frequency fluctuations !=k vth;e where vth;e is the -25 electron thermal velocity, so that Boltzmann-distributed 0 10 20 30 40 50 60 electrons are used while the ions are computed fully kinet- t/ s ically. The ion and electron dynamics are coupled through Figure 1: a) Linear growth rates and b) linear eigen modes the Poisson equation. of the ion sound instability from PIC simulations and fluid In the simulations, the grounded walls are set at locations theory [31]. c) Potential oscillations in linear and nonlin- x = 0 and x = d, with the electric potential φ(x = 0) = ear regimes. d) Coherent oscillations of the virtual anode φ(x = d) = 0. Ions with constant flux and velocity v0 are potential structure in the nonlinear regime. e) The evolu- injected from the left wall. The ions approaching the walls tion of the potential energy density as a function of time, are absorbed. Initially, the electron density is set to be showing the growth and saturation of the instability. 14 −3 ne0 = 10 m and the (injected) ion density is slightly different to produce a charge perturbation with the value δn = (ni0 − ne0)=ne0 ≈ ±0:1%. The sign of δn was the linear theory [31]. In the simulations of Figs. 1a and found to be important for the instability evolution. In our 1b, the system length is d = 10λDe, the ion beam velocity cases, positive initialization (δn > 0) was used leading is in the range of 0:4 − 1:0c in Fig. 1a; and the ion beam to the ion hole formation. For simplicity, the ion mass s velocity is 0:86cs in Fig. 1b. The linear growth rates and mi is set to be 1 in atomic units. The Debye length in eigen-modes (ion density, velocity and potential) obtained the simulations is calculated to be λDe = 1:3mm and the from the simulations and theory, Figs 1a and 1b, are in rea- simulation box length is set as d = 50λDe with the spatial sonable agreement taking into account that the analytical resolution of ∆x = λDe=10. Therefore, the simulation theory is formulated for the asymptotic limit of finite but domain constitutes of 500 cells and d = 6:5cm. Time step small wave numbers, kλDe 1, while in our simulations is chosen as 15∆t = ∆x=vth;e to fulfill the CFL condition kλ ' 1, see Fig. 1b. We note that according to the during the simulation. Two thousand super-particles per De theory, the ion sound instability occurs only for v0 < cs, cell are used. The electron temperature Te = 3eV is used while the case v > c is stable for this type of instability. throughout this work. 0 s As was shown analytically [31], the instability occurs in the In the linear regime, when the kinetic effects of ion trap- alternating oscillatory (real part of frequency Re(!) 6= 0) ping are not important, our simulations recover the results and aperiodic (Re(!) = 0) zones as the ion beam veloc- of the analytical theory and the fluid simulations of the ion ity changes. In our nonlinear simulations, Figs. 1c, 1d sound instability in a bounded plasma by Koshkarov et. and 1e, the ion beam velocity is v = 0:8c . With this al. [31]. This is shown in Figs. 1a and 1b which compare 0 s value, and d = 50λDe, the instability occurs in the ape- the growth rates and eigenmodes of the the Pierce type ion riodic zone, so the fluctuations grow as a standing wave sound instability obtained from our PIC simulations and as it is clearly seen in Fig.

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