Double layers and ion-acoustic waves-- a fully particle-in-cell simulation【推荐论文】 .doc

精品论文Double layers and ion-acoustic waves: a fully particle-in-cell simulationGUO Jun1, LI Bo25(1. School of Mathematics and Physics, Qingdao University of Science and Technology, ShanDong QingDao 266061;2. School of Space science and Physics, Shandong University at Weihai,ShanDong WeiHai 264209)Abstract: Double layers and ion-acoustic waves are investigated by using a one-dimensional10electrostatic particle-in-cell simulation code. Our results show that double layers can be formed even when the drift velocity between electrons and ions is less than the electron thermal velocity. The electron and ion density depressions are clearly seen. Electrons gradually develop a distributioncomprising both background and beam components. In fact, as the initial electron-ion drift velocity isless than the electron thermal velocity, intense ion-acoustic waves can be found only where the electron15beam is located, suggesting that they are excited by the self-consistently developed electron beam.Besides the Langmuir waves and ion-acoustic waves, the beam mode excited by electron beam produced in our simulation has been found clearly.Keywords: electron-ion streaming instability; Langmuir instability; particle-in-cell200IntroductionDouble layers have long been of interest to both laboratory and space plasma communities1-7. The definitive double layer structures were observed by the Viking satellite in themagnetosphere, which suggests that double layers are small. These structures have an extent of100m, amounting to a few tens of Debye lengths, and the plasma density therein is reduced by up8- 9 -25to 50%. The necessary condition for the formation of double layers obtained from the previousworks1-4is that the electron drift velocity exceeds the electron thermal velocity, i.e., doublelayers are a result of the Buneman instability. If the opposite is true, i.e., the relative streaming velocity is sufficiently less than electron thermal speed, the instability is called the ion-acoustic9-12(IA) instability. Both the Buneman16-17and IA instabilities13-15have been investigated carefully.30However, there are observationsthat support the existence of double layers along auroralfield lines where the electron drift speed is much less than the electron thermal speed. Sato and18Okudapoint out that an IA instability can result in the formation of double layers if a system issufficiently long. The initial drift velocity used in their simulation is less than the electron thermal velocity. They also pointed out that the anomalous resistivity generated by the IA instability35causes the buildup of a DC potential which in turn accelerates electrons further to enhance the original instability, leading to the formation of double layers. But a low ion-to-electrons mass ratio (which is equal to 100) is used in their simulation. What consequences will a realistic mass ratio (1836) bring forth?Computer simulations are the most powerful tool to study this dynamical process. For13-15, 19-2340instance, Vlasov simulationshave been in use to study the electron dynamics, and theanomalous resistivity produced therein via the IA instability. Many authors use one- or24-27two-dimensional particle-in-cell codes to examine electrostatic instabilities. To explore theFoundations: National Natural Science Foundation of China (Nos. 40974097, 41204115), Research Fund for theDoctoral Program of Higher Education of China(No.20110131110058)Brief author introduction:GUO Jun, (1973-), Female, Associate Professor, Space Plasma Physics.Correspondance author: LI Bo, (1976-), Male, Professor, Space Plasma Physics. E-mail: bblsdu.edu.cnformation of double layers in general, and the wave and particle dynamics in particular, we use a1D treatment in this study to help isolate the physics we are interested in, by excluding45complications such as oblique wave modes. In this paper, using one-dimensional electrostatic particle simulation, we will present the results of kinetic simulations designed to study thedevelopment and evolution of double layers in a current-carrying plasma. A relatively lowvelocity u /v0the=0.6 is used. The excitation and evolution of ion-acoustic waves will also bediscussed.Figure 1. Evolutionary history of various parameters. (a) The electron drift speed ue versus normalized time, (b)the evolution of electric energy density E2.501SimulationWe have performed one-dimensional electrostatic particle simulation with the system length L=1024D, where D=vthe/pe is the electron Debye length, vthe= (Te/me) 1/2 is the initial electron thermal velocity, pe is the electron plasma frequency. Besides, Te and me are the electron55temperature and mass, respectively. Initially the density is uniform and the electron velocity distribution is a drifting Maxwellian. The ion drift velocity is zero. We normalize velocities by-1initial electron thermal velocity vthe, and time by pe. Periodic boundary conditions are used. Theelectrostatic fields are defined on grids 25-27. Electric fields and potentials are obtained by solving the Poisson equation, while particles move in the electric field. The grid spacing is x=1.0D, and60the time step is pet=0.02. We use u0/vthe =0.6 in the simulation, which means that the initialrelative streaming velocity between ions and electrons is less than the electron thermal speed. The numbers of particles employed for both species are both 409600.Figure 2. Various parameters at . (a) and (c) the electron and ion phase-space distributions; (b)and (d) the electron and ion density profiles; (e) the electric field distribution.A few remarks are necessary on the following parameters we adopt. First, consider the65ion-to-electron mass ratio. We use an mi/me of 1600, which is close to the realistic value (1836).Our simulations practically yield the same result as what one may find with the realistic ratio. Second, let us discuss our choice of electron-to-ion temperature ratio Te/Ti , on which the growth rate of the ion-acoustic instability sensitively depends. The ion-acoustic instability dominates when the electron temperature exceeds the ion temperature Te>Ti. Note that reference 18, the70most relevant to the present study, adopted a temperature ratio Te/Ti of 20 and a mass ratio of 100.To find out what the differences in adopted mass ratios bring out, we adopt a very similar temperature ratio, 16 to be specific.2Simulation ResultsFigure 1 (a) shows the electron drift speed ue versus normalized time. Figure 1 (b) shows the75evolution of electric energy density. In the early stage of wave evolution (pet=0-2500) the total field energy increases almost linearly with time and the electron drift speed ue drops slowly at a roughly constant rate. A drastic reduction of the electron drift speed, and hence a sudden enhancement of the total field energy takes place at pet2700. By the end of the simulation ueapproaches0.43 vthe. Both the electric energy and ue undergo some significant overshoot around80pet=5200 or so.Figures 2 (a) and (c) show the electron and ion phase-space distributions, with the red solid lines representing v/vthe =0; (b) and (d) show the electron and ion density profiles along x, with thered solid lines representing n/n0=1; (e) shows the electric filed distribution at pet =4900 when the double layer has fully developed. A density depression, n/n00.55, is generated for both85electrons and ions at about x/D230. In fig.2 (c), the ion phase-space holes (or whirls) are obvious. Ion phase-space holes have been found in previous simulations 6, 7, but a relative drift velocity that is larger than the electron thermal velocity is used there. Here, we mainly discuss thebiggest whirl region. Along the drift direction, electrons are decelerated and then accelerated in the whirl region. On the contrary, ions are accelerated and then decelerated. The figure also90suggests that the number of energetic electrons decreases while the number of energetic ionsincreases in the whirl region. The electric field with a bipolar structure corresponds to the ion phase-space holes. It is therefore clear that double layers can form through the ion-acoustic instability. In fact, the biggest whirl consists of two double layers.Figure 3. The velocity distributions of electrons and ions at pet =0 (the black curves) and pet =4900 (the red curves).95100105110Figure 3 plots the velocity distributions of electrons and ions at two different times. The black lines give the distributions at pet =0 and the red lines are for those at pet =4900. The electron velocity distribution has undergone some substantial change from the initial drifting Maxwellian by pet =4900. The value of electron f(v) decreases at lower positive phase velocities and increases at lower negative phase velocities, meaning that electrons with a small positive velocity tend to be trapped by the ion-acoustic instability. An electron beam is formed at pet=4900, which is consistent with the previous results 6. This distribution seems to suggest that theion-acoustic instability only affects electrons with small speeds, which is different from that in the simulations of the Buneman instability. The modulation of ion distribution shows that ions are accelerated and heated. A comparison with figure 2 suggests that ions are accelerated in the whirls.Figure 4 shows the time histories of (a) the electron density ne/n0, (b) the ion density ni/n0, (c) electron drift speed ue/vthe, (d) ion drift speed ui/vthe, (e) electric field, and (f) the electrostatic potential e(x,t)/Te. The electron and ion densities show a depression of up to 55%. Fig.4c shows that for the majority of the grid points, the electron drift speed gradually decreases, in much the same way as the one averaged over the simulation domain as shown in Fig.1. Something different appears nonetheless. At some points, corresponding to the ridges in the x-t plane, the drift speed increases instead. In fact, the velocity distributions can be seen as comprising both background and beam electrons. The maximum value is much larger than the initial drift velocity ue/vthe=0.6.115120Fig.4d also shows clearly a ridge, around which the ion drift speed increases substantially, albeit directed in the opposite direction. Figs.4e and 4f show an obvious double layer, which is actuallyconsistent with comprising several double layers. The most intense one can be seen near pet 4900. The maximum and minimum values of the dimensionless electric field are 0.33 and -0.22, respectively. This localized structure propagates along the positive x direction, and its width is about thirty Debye lengths, consistent with observational results 1,8. Compared with the phase-space holes shown in fig.2, the electric filed with a bipolar structure is strong and obvious inthe region where the whirls are most evident.Figure 4. The evolutionary history of (a) the electron density ne/n0, (b) the ion density ni/n0, (c) electron drift speed ue/vthe, (d) ion drift speed ui/vthe, (e) the electric field and (f) electrostatic potential e(x,t)/Te.125To examine the time evolution and excitation of the ion-acoustic waves, we apply a wavelet analysis to the time series of the electric field. Figures 5 (a1), (b1), (c1) and (d1) present the wavelet power spectrum at x/D=100,170, 210 and 230, respectively. Here the vertical and horizontal axes are for frequency and time, respectively. In addition, the thick contour encloses regions where the confidence level exceeds 95%. The horizontal dashed lines correspond to the electron and ion plasma frequencies, pe and pi, respectively. The area below the thick dotted130contour indicates the cone of influence where edge effects become important. Figures 5 (a2), (b2),(c2) and (d2) present the drift velocity evolutions at the corresponding locations.Figure 5. Wavelet power spectra of physical parameters at four locations. (a1), (b1), (c1) and (d1) present electric field spectra at locations x/D being 100,170, 210 and 230, respectively; (a2), (b2), (c2) and (d2) the corresponding drift velocity spectra.135140The ion-acoustic waves shown in fig.5 (a1) are very weak, while the electron drift velocity shown in (a2) shows a rather modest decrease at a roughly constant rate during the entiresimulation. In figure 5 (b1), the ion-acoustic waves are noticeable from pet1200 to 2500.The dominant frequency of the wave is /pe0.019, which is below the ion plasma frequency. Fig. 5 (b2) suggests that the electron drift velocity increases somehow at pet1200 and then decreases. At x/D=210, shown in fig.5 (c1), the waves are significantly more intense. Moreover,the wave power seems to be concentrated at two frequencies, /pe0.011 and /pe0.019,during the interval between pet3500 and 4600. The corresponding relative drift velocity shown in fig.5 (c2) appears to possess two peaks, one being 0.76vthe at pet3700, the other being0.9vthe at pet3800. This suggests that ion-acoustic waves are excited by electron beam. At x/D=230, shown in fig.5 (d1), the spectra of the excited waves become even broader. Correspondingly, the relative drift velocity now possesses one obvious peak. In addition to the ion-acoustic waves, some Langmuir waves (LWs) can be also seen, albeit much weaker.145Figure 6. The -k diagram obtained by Fourier transforming the electric field data. (a) pet= 0 to 409.6, (b)pet= 1800 to 2209.6, (c) pet=3600 to 4009.6 and (d) pet=4800 to 5209.6.150155160165The wave behavior is further examined in Fig.6, where the -k diagram is found by Fourier transforming the electric field. Here frequency is plotted versus wave numbers using a color scale representing wave intensity with the reddish regions indicating the most