fluid theory and kinetic simulation of two dimensional electrostatic streaming instabilities in electron ion plasmas

Fluid theory and kinetic simulation of two-dimensional electrostatic streaming instabilities in electron-ion plasmas C.-S Jao and L.-N Hau Citation: Phys Plasmas 23, 112110 (2016); doi: 10.1063/1.4967283 View online: http://dx.doi.org/10.1063/1.4967283 View Table of Contents: http://aip.scitation.org/toc/php/23/11 Published by the American Institute of Physics Articles you may be interested in Secondary instabilities in the collisionless Rayleigh-Taylor instability: Full kinetic simulation Phys Plasmas 23, 112117 (2016); 10.1063/1.4967859 PHYSICS OF PLASMAS 23, 112110 (2016) Fluid theory and kinetic simulation of two-dimensional electrostatic streaming instabilities in electron-ion plasmas C.-S Jao1 and L.-N Hau1,2 Institute of Space Science, National Central University, Taoyuan City, Taiwan Department of Physics, National Central University, Taoyuan City, Taiwan (Received August 2016; accepted 21 October 2016; published online 10 November 2016) Electrostatic streaming instabilities have been proposed as the generation mechanism for the electrostatic solitary waves observed in various space plasma environments Past studies on the subject have been mostly based on the kinetic theory and particle simulations In this paper, we extend our recent study based on one-dimensional fluid theory and particle simulations to two-dimensional regimes for both bi-streaming and bump-on-tail streaming instabilities in electron-ion plasmas Both linear fluid theory and kinetic simulations show that for bi-streaming instability, the oblique unstable modes tend to be suppressed by the increasing background magnetic field, while for bump-on-tail instability, the growth rates of unstable oblique modes are increased with increasing background magnetic field For both instabilities, the fluid theory gives rise to the linear growth rates and the wavelengths of unstable modes in good agreement with those obtained from the kinetic simulations For unmagnetized and weakly magnetized systems, the formed electrostatic structures tend to diminish after the long evolution, while for relatively stronger magnetic field cases, the solitary waves may merge and evolve to steady one-dimensional structures Comparisons between one and twodimensional results are made and the effects of the ion-to-electron mass ratio are also examined based on the fluid theory and kinetic simulations The study concludes that the fluid theory plays crucial seedC 2016 Author(s) All article ing roles in the kinetic evolution of electrostatic streaming instabilities V content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4967283] I INTRODUCTION Observational evidences for the existence of electrostatic solitary waves (ESWs) in space plasma environments have been overwhelming.1–9 As shown in many studies based on particle-in-cell simulations, ESWs may easily be generated by the electrostatic streaming instability,10–13 including both bistreaming and bump-on-tail instabilities associated with the ESWs observed in the auroral region and magnetotail plasma sheet boundary layer, respectively.3–7 In multi-dimensional systems and via streaming instabilities, solitary structures, however, may form in a steady manner only for certain magnitudes of background magnetic field and for unmagnetized or weakly magnetized plasmas the formed solitary structures may inevitably be destroyed in the nonlinear evolution process.14,15 The simulation results also show that the formed solitary structures may tend to evolve into one- or twodimensional structures in the nonlinear evolution process of two- and three-dimensional particle simulations.16 Based on the linear kinetic theory and two-dimensional particle simulations, Miyake et al.15 have examined the effects of the background magnetic field on the bump-on-tail instability It is shown that the presence of the background magnetic field is the necessary condition for the steady existence of one-dimensional ESWs Both the linear kinetic theory and particle simulations show the unstable modes with maximum growth rate occurring for parallel drifting along the background magnetic field While for oblique propagation, the growth rate of unstable modes is increased with increasing 1070-664X/2016/23(11)/112110/13 background magnetic field Their simulation results also show that one-dimensional electric potential structures always form after the linear growth stage but may be destroyed by thermal fluctuations in the nonlinear stage for the relatively weak background magnetic field Goldman et al.17 and Oppenheim et al.18 have examined the evolution of bi-steaming instability in two-dimensional simulations for strongly magnetized plasmas It is shown that one-dimensional electrostatic structures initially form in the linear stage and become diminished with the generation of electrostatic whistler waves Miyake et al.16 later investigated the evolution of bi-steaming instability in a relatively weakly magnetized system and showed that the electric structures formed in the linear growing stage are one-dimensional, becoming two-dimensional in transient caused by the lower hybrid waves associated with the ion dynamics, and after the long evolution may evolve to steady one-dimensional ESWs again Recently, we have attempted to carry out intercomparisons between the fluid theory and one-dimension particle simulations for electrostatic streaming instabilities to infer the role of linear fluid theory in nonlinear kinetic simulations.19,20 It is shown that the linear fluid theory is in good agreement with the initial evolution of both bi-streaming and bump-on-tail instabilities in terms of the growth rate and wavelength of unstable modes.20 We have further shown that the pqÀc value is significantly increased at the nonlinear saturation accompanying with the particle trapping effect.19,20 In light of the important application of ESWs in geospace plasma 23, 112110-1 C Author(s) 2016 V 112110-2 C.-S Jao and L.-N Hau Phys Plasmas 23, 112110 (2016) environments, in this study we extend the model calculations to two-dimensionality and oblique propagation to examine the role of fluid theory in the evolution and formation of electrostatic solitary waves in multi-dimensional streaming instabilities The plasma system under consideration consists of electrons and ions embedded in the background static magnetic field, which has important applications to various space plasma environments as already shown in many studies based on the kinetic simulations For the applications to the observed ESWs occurring in the magnetosphere, two types of electron streaming instability with bi-streaming and bump-on-tail velocity distributions are examined in this study.3,10,14–18,20–22 Comparisons between one and two-dimensional results from linear theory and nonlinear simulations are also made The effects of the ion-to-electron mass ratio on the results will also be examined The study may help to clarify the role of fluid theory in the complex evolution of kinetic streaming instabilities II LINEAR FLUID THEORY It is well known that based on the linear fluid theory the electrostatic streaming instability is induced by the resonance between the two wave modes associated with stationary and streaming plasmas For oblique propagation with h being the angle between the background magnetic field and wave vector, the dispersion relations for two electrostatic electron waves are FIG The wave frequency versus wave number for the cases with xc =xp ¼ 0:5 (panel (a)) and xc =xp ¼ 2:0 (panel (b)) with h ¼ 0 (solid line), 30 (dashed line), 60 (dot-dashed line), and 90 (dotted line) In both panels, xH and xL denote the high and low frequency roots in equation (1), respectively " ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi#   r 2 x2H ẳ x2c ỵ x2p ỵ ck2 v2th þ x2c þ x2p þ ck2 v2th À 4x2c ðx2p þ ck2 v2th Þ cos2 h " ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi#  r 2  2 2 2 2 xL ẳ xc ỵ xp ỵ ck vth x2c ỵ x2p ỵ ck vth 4x2c x2p ỵ ck vth ị cos2 h (1) for which xH and xL denote the high and low frequency modes, respectively For parallel propagation (h ¼ 0 ), the above relations reduce to the Langmuir mode of x2 ẳ x2p ỵ ck2 v2th and cyclotron mode of x2 ¼ x2c , while for perpendicular propagation (h ¼ 90 ), there exists only the high frequency mode with x2H ẳ x2p ỵ ck2 v2th ỵ x2c Figure shows the wave frequency versus wave number for the cases of xc =xp ¼ 0:5 (panel (a)) and xc =xp ¼ 2:0 (panel (b)) with various h values As indicated, for xc =xp < 1, both the high (Langmuir) and low (cyclotron) frequency modes can still be identified, while for xc =xp > the Langmuir and cyclotron modes can no longer be clearly distinguished by simply the high or low wave modes The frequency variations are sensitive to the wave number only in the small wave number portion of the lower frequency mode and the large wave number portion of the higher frequency mode, which has similar characteristics to the Langmuir mode For streaming instability and assuming that the beam plasma is drifting along the background magnetic field, the dispersion relation derived from the set of fluid equations with the adiabatic energy law is23 h i h i x2p;i x2 À x2c;i cos2 h x2p;e1 x2 À x2c;e cos2 h i i h ih ỵh x2 ck2 v2th;e1 x2 À x2c;e À x2c;e ck2 v2th;e1 sin2 h x2 À ck2 v2th;i x2 À x2c;i À x2c;i ck2 v2th;i sin2 h h i x2p;e2 ðx À kue2 cos hị x2c cos2 h i ih ẳ1 ỵh ðx À kue2 cos hÞ2 À ck2 v2th;e2 ðx À kue2 cos hÞ2 À x2c;e À x2c;e ck2 v2th;e2 sin2 h (2) 112110-3 C.-S Jao and L.-N Hau Phys Plasmas 23, 112110 (2016) for which the subscripts “e1,” “e2,” and “i” are, respectively, for the background stationary electrons, streaming electrons, and background stationary ions; and v2th;a  kB Ta =ma , x2p;a  na0 q2a =e0 ma , and xca  qa B0 =ma In the following calculations, Equation (2) is solved numerically for the bisteaming instability with ne1;0 ¼ ne2;0 and vth;e1 ¼ vth;e2 while for the bump-on-tail instability we set ne1;0 =ne2;0 ¼ 9:0 and vth;e1 ¼ 5vth;e2 Other parameters such as vth;i ¼ 0:1vth;e2 and ue2 ¼ 10vth;e2 (along the background magnetic field) are chosen The c value is set to be 5/3 in the fluid model considering that in the simulation all particles possess three degrees of freedom Figure shows the imaginary part of wave frequency xi as functions of wave number kx and ky for both bi-streaming (panel (a)) and bump-on-tail instabilities (panel (b)) with the parameter values of xc =xp ¼ 0:5 and mi =me ¼ 1836 As expected, the most unstable mode occurs for parallel propagaat tion (ky ¼ 0) with the growth rate of xi ¼ 0:343xÀ1 p kx ¼ 0:095xp =vth and xi ¼ 0:171xÀ1 at k ¼ 0:181x =v x p th p for the bi-streaming (panel (a)) and bump-on-tail instabilities (panel (b)), respectively, which can also be inferred from the earlier one-dimensional fluid theory.20 In Figure 2(a), it is shown that the major unstable mode (xi > 0:3xp ) of the bistreaming instability exists within the wave number range of FIG The imaginary part of the wave frequency xi as functions of wave number kx and ky derived from the linear fluid theory for bi-streaming (panel (a)) and bump-on-tail instabilities (panel (b)) In the figure, LLM, LCM, LIM, CLM, and CCM denote the unstable modes induced by the Langmuir-Langmuir coupling, Langmuir-cyclotron coupling, Langmuirion coupling, cyclotron-Langmuir coupling, and cyclotron-cyclotron coupling, respectively FIG The imaginary part of the wave frequency xi as functions of wave number kx and ky derived from the linear fluid theory for the bi-streaming instability with xc =xp ¼ 0:0 (panel (a)), xc =xp ¼ 1:0 (panel (b)), and xc =xp ¼ 2:0 (panel (c)) In the figure, LLM, LIM, CLM, and CCM denote the unstable modes induced by the Langmuir-Langmuir coupling, Langmuirion coupling, cyclotron-Langmuir coupling, and cyclotron-cyclotron coupling, respectively 112110-4 C.-S Jao and L.-N Hau Phys Plasmas 23, 112110 (2016) kx ¼ 0:075 $ 0:175xp =vth and ky < 0:35xp =vth (i.e., h 63:5 ), which is due to the Langmuir-Langmuir mode resonance (LLM) between the streaming and background electrons The phase velocity of the Langmuir-Langmuir resonance mode has a dependence of cos h and is thus decreased with increasing h value For the parallel propagation, we have shown that the unstable regime and linear growth rate are increased with increasing drifting velocity since the drifting plasma with larger drifting velocity would resonate with the background plasma in longer wavelengths from the aspect of fluid theory.23 A similar tendency has also been found for oblique propagation cases Two other unstable modes also appear in the oblique direction In particular, the unstable mode with k < 0:075xp =vth occurring for any h is induced by the cyclotron-cyclotron mode resonance between the beam and background electrons (CCM) while the unstable mode with the large propagation angle is induced by the cyclotron-Langmuir resonance between the beam and background electrons (CLM) There is also another nonpropagating unstable mode arising from the resonance between the drifting Langmuir wave and background ions (LIM) in the system, but it has some overlap with the cyclotron-Langmuir mode (CLM) in the wave number spectrum As for the bump-on-tail instability indicated in Figure 2(b), the Langmuir-Langmuir (LLM), cyclotron-cyclotron (CCM), cyclotron-Langmuir (CIM), and Langmuir-ion (LIM) mode resonance are also coexistent in the system, and due to the warm background electrons the major regime for the Langmuir-Langmuir coupling mode occurs for small propagation angle of h 27 For the bump-on-tail instability, a new unstable mode induced by the Langmuir-cyclotron coupling (LCM) between the beam and background electrons is also present in Figure 2(b) Figure shows xi as functions of kx and ky for the bistreaming instability with various background magnetic fields, FIG The imaginary part of the wave frequency xi as functions of wave number kx and ky derived from the linear fluid theory for the bump-on-tail instability with xc =xp ¼ 0:0 (panel (a)), xc =xp ¼ 1:0 (panel (b)), and xc =xp ¼ 2:0 (panel (c)) In the figure, LLM, LCM, LIM, CLM, and CCM denote the unstable modes induced by the Langmuir-Langmuir coupling, Langmuir-cyclotron coupling, Langmuir-ion coupling, cyclotron-Langmuir coupling, and cyclotron-cyclotron coupling, respectively FIG The time evolution of logarithm of maximum electric potential for the bi-streaming instability with xc =xp ¼ 0:0 (circle symbol), xc =xp ¼ 0:5 (cross symbol), and xc =xp ¼ 2:0 (plus symbol) from two-dimensional simulations for two different time ranges The curve with the triangle symbol is the result from one-dimensional simulations The slope of the dashed straight line is the maximum growth rates of the unstable mode derived from the linear theory 112110-5 C.-S Jao and L.-N Hau xc =xp ¼ 0:0 (panel (a)), xc =xp ¼ 1:0 (panel (b)), and xc =xp ¼ 2:0 (panel (c)) As expected, the most unstable modes occur for the parallel direction in all cases For the unmagnetized case (Figure 3(a)), there exists no any resonant Phys Plasmas 23, 112110 (2016) mode about the cyclotron wave and the non-propagating Langmuir-Ion mode is present in the quasi-perpendicular direction (h $ 80 ) with a smaller growth rate of xi < 0:1xp By comparing the magnetized cases presented in Figures 2(a), FIG The spatial distributions of electric potential /x; yị at t ẳ 25x1 p for the bi-streaming instability with xc =xp ¼ 0:0 (panel (a)), xc =xp ¼ 0:5 (panel (b)), and xc =xp ¼ 2:0 (panel (c)) from two-dimensional simulations, and the corresponding wavenumber spectrum of electric potential /ðkx ; ky Þ for the cases with xc =xp ¼ 0:0 (panel (d)), xc =xp ¼ 0:5 (panel (e)), and xc =xp ¼ 2:0 (panel (f)) The black star on panel (d), (e), and (f) denotes the wave number of the most unstable mode derived from the linear theory 112110-6 C.-S Jao and L.-N Hau 3(b), and 3(c), it is shown that the instability is obviously suppressed in the oblique direction with increasing background magnetic field For strongly magnetized system (xc =xp ¼ 2:0; Figure 3(c)), the major portion of the instability occurs within the wave number range of k < 0:175xp =vth and h 30 Figure shows the same analyses for the bump-on-tail instability with various background magnetic fields of xc =xp ¼ 0:0 (panel (a)), xc =xp ¼ 1:0 (panel (b)), and xc =xp ¼ 2:0 (panel (c)) Since the instability regime caused by the Langmuir-Langmuir mode resonance (LLM) is intrinsically narrow for the bump-on-tail instability, the suppression by increasing background magnetic field on the Langmuir-Langmuir resonance mode (LLM) in the oblique propagation direction is not as evident as the bi-streaming instability Nevertheless, by comparing the cases with xc =xp ¼ 0:0 (Figure 4(a)) and xc =xp ¼ 1:0 (Figure 4(b)), the slight suppression of the Langmuir-Langmuir resonance mode (LLM) by increasing background magnetic field can still be observed As for the stronger magnetic field case shown in Figure 6(c) (xc =xp ¼ 2:0), the main instability portion occurs in the narrow region of ky < 0:125xp =vth but for the broader propagation angles of h 60 , in contrast to the bi-streaming instability The correlation tendency between the linear grow rate and background magnetic field for the bump-on-tail instability has also been shown in the linear kinetic Vlasov-Poisson model.15 Note that the Vlasov-Poisson model has also been adopted for the linear analysis of streaming instabilities.10,15,17,18,21,22 In the kinetic model, the most unstable modes occur mainly for k < 1:0xp =vth and for k $ 1:0xp =vth (i.e., the wavelength is about the Debye length kD ¼ vth =xp ) the linear growth rate becomes much smaller than 0:1xÀ1 p for both bi-streaming and bump-on-tail instabilities We have found similar results in the linear fluid model As shown in Figures 2–4, the main unstable modes for both bi-streaming and bump-on-tail instabilities occur in a narrow regime of k < 0:4xp =vth and the linear growth rate for k $ 1:0xp =vth is an order of magnitude smaller than 0:1xÀ1 p Phys Plasmas 23, 112110 (2016) Figure shows the time evolution of logarithm of maximum electric potential based on the one- (triangle symbol) and two-dimensional simulations with xc =xp ¼ 0:0 (circle symbol), xc =xp ¼ 0:5 (cross symbol), and xc =xp ¼ 2:0 (plus symbol) As indicated in Figure 5, in all cases the electric potential grows at the same time with similar growth rate and approaching to the same amplitude in the linear growing stage (t ¼ 10 $ 20xÀ1 p ) The dashed straight line in Figure corresponds to the growth rate of the most unstable mode derived from the linear theory (xi ¼ 0:343xÀ1 p ), which is in good agreement with the simulation results Figure shows the distribution of electric potential /ðx; yÞ (panels (a)–(c)) and the corresponding wave number spectrum /kx ; ky ị (panel (d)(f)) at t ẳ 25x1 p for the cases with xc =xp ¼ 0:0, xc =xp ¼ 0:5, and xc =xp ¼ 2:0 Note that in this study the wave number spectra are all presented in logarithmic scale As indicated, the electric structures are formed in all cases after the linear growing stage with similar wavelength (Figures 6(a)–6(c)), and the solitary structure for the cases with xc =xp ¼ 0:5 (Figure 6(b)) and xc =xp ¼ 2:0 (Figure 6(c)) is more one-dimensional The wave number spectra (Figures 6(d)–6(f)) show that the major perturbations are all parallel to the background magnetic field (i.e., ky ¼ 0) with the wave number being kx $ 0:095xp =vth , which is consistent with the most unstable mode derived from the linear fluid analysis (black star symbol in panels (d), (e), and (f)) The perturbations in the oblique direction for the case of xc =xp ¼ 0:0 (Figure 6(d)) are more evident than the case of xc =xp ¼ 2:0 (Figure 6(f)), which is also consistent with the III PARTICLE SIMULATIONS A two-dimensional electrostatic particle-in-cell code has been developed to carry out the kinetic simulations of streaming instabilities reported in this study.23 In the calculations, dimensionless units are used with the plasma frequency of total electrons being xp ¼ 1:0 and the thermal velocity of beam electrons being vth ¼ 1:0 The grid size is chosen to be Dx  Dy ¼ 1kD  1kD for the simulation box of Lx  Ly ¼ 1024kD  1024kD and the boundary conditions for both particles and fields are set to be periodic The total number density of particles is 128 per cell over the system for all cases and Dt ¼ 0:01xÀ1 p is typically adopted with initial thermal velocity distributions of all particles being described by the isotropic Maxwellian function For comparisons, the parameters used for the particle simulations of bistreaming and bump-on-tail instabilities are the same as those used in the linear fluid calculations We first present the particle simulation results for the bistreaming instability in one- and two-dimensional systems FIG The time evolution of logarithm of maximum electric potential for the bump-on-tail instability with xc =xp ¼ 0:0 (circle symbol), xc =xp ¼ 0:5 (cross symbol), and xc =xp ¼ 2:0 (plus symbol) from two-dimensional simulations for two different time ranges The curve with triangle symbol is the result from one-dimensional simulations The slope of the dashed straight line is the maximum growth rates of the unstable mode derived from the linear theory 112110-7 C.-S Jao and L.-N Hau tendency obtained from the linear fluid theory (Figures 2(a) and 3) As shown in Figure 5, for the unmagnetized twodimensional system (xc =xp ¼ 0:0; circle symbol), the electric structures begin to damp in the nonlinear evolution process (t > 60xÀ1 p ) while for one-dimensional (triangle symbol) and two-dimensional magnetized systems (cross symbol for Phys Plasmas 23, 112110 (2016) xc =xp ¼ 0:5 and plus symbol for xc =xp ¼ 2:0) the saturated electric potentials are shown to be steady in the early nonlinear stage (t < 256xÀ1 p ) These results are consistent with the previous simulation studies for the multi-dimensional unmagnetized system.14 Indeed, the results are similar for the relatively weakly magnetized system such as xc =xp ¼ 0:2 and consistent with the early study on the nonlinear evolution FIG The spatial distributions of electric potential /x; yị at t ẳ 40x1 p for the bump-on-tail instability with xc =xp ¼ 0:0 (panel (a)), xc =xp ¼ 0:5 (panel (b), and xc =xp ¼ 2:0 (panel (c)) from two-dimensional simulations, and the corresponding wavenumber spectrum of electric potential /ðkx ; ky Þ for the cases with xc =xp ¼ 0:0 (panel (d)), xc =xp ¼ 0:5 (panel (e)), and xc =xp ¼ 2:0 (panel (f)) The red and black stars on panel (d), (e), and (f) denote the wave number of the most and the second unstable modes, respectively, derived from the linear theory 112110-8 C.-S Jao and L.-N Hau process of bi-streaming instability reported in several literatures.14,16–18,22 We now discuss the simulation results on the bump-ontail instability Figure shows the time evolution of logarithm of maximum electric potential based on one-dimensional (triangle symbol) and two-dimensional simulations with xc =xp ¼ 0:0 (circle symbol), xc =xp ¼ 0:5 (cross symbol), and xc =xp ¼ 2:0 (plus symbol) As indicated in Figure 7, for two-dimensional systems with various magnetic field strengths the electric potentials all grow to the same amplitude with similar growth rate in the early evolution stage (t ¼ 20 $ 40xÀ1 p ) The slope of the corresponding straight lines in Figure is the growth rate of the most unstable mode derived from the linear theory (xi ¼ 0:171xÀ1 p ), indicating that the two-dimensional simulation results are in accordance with the linear theory (Figures 2(b) and 4) Note that the onedimensional simulation result (triangle symbol in Figure 7) shows smaller growth rate as compared to the twodimensional linear theory and simulation results Our earlier one-dimensional study has shown the consistency between the fluid theory and kinetic simulations, especially for relatively larger drift velocity.20 Figure shows the electric potential /x; yị after the linear growing stage (t ẳ 48x1 p ) for the cases of xc =xp ¼ 0:0, xc =xp ¼ 0:5, and xc =xp ¼ 2:0 It is seen that the electric potential structures in the perpendicular direction are most pronounced in the weakly magnetized case (Figure 8(b)) as compared to the unmagnetized (Figure 8(a)) or strongly magnetized cases (Figure 8(c)) The corresponding wave number spectra /ðkx ; ky Þ are shown in Figures 8(d)–8(f) As shown by the linear fluid theory (Figures 2(b) and 4), oblique modes with larger growth rate are seen in strongly magnetized systems (Figure 8(f)) While for weakly magnetized cases (Figure 8(e)), the unstable modes are more along the magnetic field direction as compared to the unmagnetized case (Figure 8(d)) The results shown in Figure based on the kinetic simulations are thus more or less consistent with the linear fluid theory For all the cases presented in Figure 8, the major perturbations are along the drifting direction, in good agreement with the linear theory, but the most unstable mode occurs for kx $ 0:095xp =vth , instead of kx ¼ 0:181xp =vth predicted by the linear theory (red star symbol in panels (d)–(f)) It is interesting to note that the wave number kx $ 0:095xp =vth corresponds closely to another unstable mode (CCM) with the same order of growth rate (xi > 0:13xÀ1 p ) predicted by the linear fluid theory (black star symbol in panels (d)–(f)) It is conjectured that due to the kinetic damping the competing CCM with relatively longer wavelength may tend to dominate the evolution in particle simulations As indicated in Figure 7, for the unmagnetized system (xc =xp ¼ 0:0; circle symbol), after the long evolution the formed electric structures are eventually destroyed which is consistent with the past studies.15 Phys Plasmas 23, 112110 (2016) FIG The imaginary part of the wave frequency xi as functions of wave number kx and ky derived from the linear fluid theory for the bi-streaming instability with mi =me ¼ (panel (a)) and mi =me ¼ 100 (panel (b)) In the figure, LLM, CLM, and CCM denote the unstable modes induced by the Langmuir-Langmuir coupling, cyclotron-Langmuir coupling, and cyclotroncyclotron coupling, respectively effects of the ion-to-electron mass ratio, mi =me , on the streaming instabilities shown above for the electron-proton plasma Figure shows xi as functions of kx and ky from the linear theory of bi-streaming instability with mi =me ¼ IV EFFECTS OF ION-TO-ELECTRON MASS RATIO In multi-dimensional kinetic simulations, unreal ion-toelectron mass ratios are sometimes adopted to achieve faster calculations for the electron-proton plasma system.16,21,24–26 Here, we shall carry out some calculations to examine the FIG 10 The time evolution of logarithm of maximum electric potential for the bi-streaming instability with mi =me ¼ (circle symbol), mi =me ¼ 1836 (cross symbol), and mi =me ¼ 100 (plus symbol) from two-dimensional simulations The slope of the corresponding straight line is the maximum growth rates of the unstable mode derived from the linear theory 112110-9 C.-S Jao and L.-N Hau (panel (a)) and mi =me ¼ 100 (panel (b)) By comparing the three cases with mi =me ¼ (Figure 9(a)), mi =me ¼ 1836 (Figure 2(b)), and mi =me ¼ 100 (Figure 9(b)), it is seen that, as expected, the unstable modes induced by the beam and Phys Plasmas 23, 112110 (2016) background electrons are not affected by the ion-to-electron mass ratio but the Langmuir-ion mode (LIM) shows clear differences with the unstable regime occurring for the broader wave number spectrum for decreasing mi =me values FIG 11 The spatial distributions of electric potential /x; yị at t ẳ 256x1 p for the bi-streaming instability with xc =xp ¼ 0:5, ue2 ¼ 10:0, and mi =me ¼ 100 (panel (a)), xc =xp ¼ 2:0, ue2 ¼ 10:0, and mi =me ¼ 100 (panel (b)), and xc =xp ¼ 0:5, ue2 ¼ 20:0, and mi =me ¼ 100 (panel (c)) from two-dimensional simulations, and the corresponding wavenumber spectrum of electric potential /ðkx ; ky Þ for the cases with xc =xp ¼ 0:5, ue2 ¼ 10:0, and mi =me ¼ 100 (panel (d)), xc =xp ¼ 2:0, ue2 ¼ 10:0, and mi =me ¼ 100 (panel (e)), and xc =xp ¼ 0:5, ue2 ¼ 20:0, and mi =me ¼ 100 (panel (f)) 112110-10 C.-S Jao and L.-N Hau The growth rate of the most unstable mode is only slightly increased with decreasing mi =me value Figure 10 shows the time evolution of logarithm of maximum electric potential based on two-dimensional nonlinear calculations with mi =me ¼ (circle symbol), mi =me ¼ 1836 (cross symbol), and mi =me ¼ 100 (plus symbol) As indicated, in all cases the electric potential grows at the same time with similar growth rate as predicted by the linear theÀ1 ory (xi ¼ 0:343xÀ1 p ) After long evolution (t > 120xp ), the saturated electric potential becomes steady for the case of mi =me ¼ (circle symbol) and mi =me ¼ 1836 (cross symbol) but begins to damp for the case of mi =me ¼ 100 (plus symbol) Figure 11 shows the distribution of /ðx; yÞ at t ¼ 256xÀ1 p , indicating that for mi =me ¼ 100 (Figure 11(a)) the solitary structures are destroyed As pointed out by Miyake et al., in the nonlinear stage the lower hybrid waves may result in the change of electric structures from one to two-dimensions for mobile ions with mi =me ¼ 100 which does not occur for the case with immobile ions (mi =me ¼ 1) FIG 12 The imaginary part of the wave frequency xi as functions of wave number kx and ky derived from the linear fluid theory for the bump-on-tail instability with mi =me ¼ (panel (a)) and mi =me ¼ 100 (panel (b)) In the figure LLM, LCM, LIM, CLM, and CCM denote the unstable modes induced by the Langmuir-Langmuir coupling, Langmuir-cyclotron coupling, Langmuir-ion coupling, cyclotron-Langmuir coupling, and cyclotroncyclotron coupling, respectively Phys Plasmas 23, 112110 (2016) in their calculations.16 In their study, the formed electric structures, however, may exist steadily after long evolution for both cases of mobile and immobile ions Note that larger drift velocity ue2 ¼ 10vth;e2 is adopted in this study while ue2 ¼ 4vth;e2 is used by Miyake et al.16 for which mi =me ¼ 100 is adopted to represent the mobile ion case Our calculations show similar results for mi =me ¼ 1836 and mi =me ¼ in the nonlinear evolution process Since the strength of the background magnetic field and the drift velocity of beam electrons are crucial parameters for the steady existence of solitary electric structures,14,16–18,22 we have examined different parameter values of xc =xp and u0;e2 As indicated in Figure 11, for mi =me ¼ 100, the formed electric structures are destroyed for the case with xc =xp ¼ 2:0 and ue2 ¼ 10vth;e2 (Figure 11(b)) but may exist in a steady manner for xc =xp ¼ 1:0 and ue2 ¼ 20vth;e2 (Figure 11(c)) The corresponding wave number spectrum of electric potential /ðkx ; ky Þ is also presented in Figures 11(d)–11(f) Due to the relatively smaller mass ratio (mi =me ¼ 100), the lower hybrid waves with larger ky values become apparent in all cases, and especially pronounced in panel (e) for the case with relatively stronger magnetic field (xc =xp ¼ 2:0) The perpendicular lower hybrid mode may tend to suppress the formation of the solitary structure as shown in panels (a), (b), (d), and (e), but for relatively larger drift velocity the solitary waves may again form as shown in panels (c) and (f) for the case of xc =xp ¼ 0:5 and ue2 ¼ 20:0 Figure 12 shows xi as functions of kx and ky from the linear theory of bump-on-tail instability with mi =me ¼ (panel (a)) and mi =me ¼ 100 (panel (b)) As indicated, the effects on the Langmuir-ion mode (LIM) are similar to the bi-streaming instability Figure 13 shows the corresponding time evolution of logarithm of maximum electric potential for two-dimensional simulations of bump-on-tail instability with mi =me ¼ (circle symbol), mi =me ¼ 1836 (cross symbol), and mi =me ¼ 100 (plus symbol), indicating similar results for different mass ratios Figure 14 shows the corresponding distributions of /ðx; yÞ (panel (a)–(c)) and /ðkx ; ky Þ (panels (d)–(f)) at the nonlinear stage (t ¼ 512xÀ1 p ) for FIG 13 The time evolution of logarithm of maximum electric potential for the bump-on-tail instability with mi =me ¼ (circle symbol), mi =me ¼ 1836 (cross symbol), and mi =me ¼ 100 (plus symbol) from twodimensional simulations The slope of the corresponding straight line is the maximum growth rates of the unstable mode derived from the linear theory 112110-11 C.-S Jao and L.-N Hau Phys Plasmas 23, 112110 (2016) FIG 14 The spatial distributions of electric potential /x; yị at t ẳ 512x1 p for the bump-on-tail instability with mi =me ¼ (panel (a)), mi =me ¼ 1836 (panel (b)), and mi =me ¼ 100 (panel (c)) from two-dimensional simulations, and the corresponding wavenumber spectrum of electric potential /ðkx ; ky Þ for the case of mi =me ¼ (panel (d)), mi =me ¼ 1836 (panel (e)), and mi =me ¼ 100 (panel (f)) mi =me ¼ 1, mi =me ¼ 1836, and mi =me ¼ 100 It is seen that for larger mi =me the electric structure is more aligned along the direction perpendicular to the magnetic field (drift velocity) while for the case of mi =me ¼ 100 (panel (f)) the perturbations in the perpendicular and parallel directions are in nearly the same order (panel (f)) due to the lower hybrid mode Based on the simulation studies with open boundary, Umeda et al.21 have also examined the mass ratio effects on 112110-12 C.-S Jao and L.-N Hau the bump-on-tail instability and our results with different boundary conditions are in consistent with their study V CONCLUSION In this study, we have carried out a systematic investigation of electrostatic streaming instabilities in twodimensional magnetized electron-ion plasmas based on the linear fluid theory and particle-in-cell simulations Two types of streaming instability, electron bi-streaming and bump-ontail instabilities, are considered which have been proposed as the generation mechanisms for the electrostatic solitary waves widely observed in space plasma environments Past studies, however, are mostly based on the kinetic theory and particle simulations.3,10,14–22 In this study, the fluid theory is applied to infer the unstable modes and the growth rate for various parameter values and compared to the kinetic simulations From the fluid aspect, the electrostatic streaming instabilities are induced by the wave mode resonance between the stationary and streaming plasmas which in our model include the unstable modes induced by the LangmuirLangmuir coupling (LLM), cyclotron-cyclotron coupling (CCM), cyclotron-Langmuir coupling (CLM), and Langmuir -cyclotron coupling (LCM) as well as the Langmuir-ion coupling (LIM) For both bi-streaming and bump-on-tail instabilities, the most unstable mode is the Langmuir-Langmuir coupling between the streaming and background electrons along the background magnetic field and drift velocity direction For the bi-streaming instability, the effect of background magnetic field based on the fluid theory and kinetic simulations is found to decrease the growth rate of oblique unstable modes The growth rate and the wavelength of unstable modes calculated from the simulation results are also consistent with the linear theory For the bump-on-tail instability, we have shown that for unmagnetized or weakly magnetized systems the major perturbations in the linear growing stage are mostly along the background magnetic field (drifting velocity) direction which is consistent with the findings by Miyake et al.15 based on the linear Vlasov-Poisson model and particle simulation results For the bump-on-tail instability, the linear theory also gives rise to the correct growth rate and wavelength of unstable modes inferred from the simulations for which the second most unstable mode with similar growth rate as the most unstable mode seems to dominate in the system In this study, the effects of the ion-to-electron mass ratio on linear fluid theory and nonlinear simulation results are also examined due to the consideration that unrealistic mi =me values have been adopted in some multi-dimensional simulations of electron-proton plasmas which involve the physics of electrostatic solitary waves.24–26 The fluid theory shows that for both bi-streaming and bump-on-tail instabilities the unstable regime in terms of the wave number for the Langmuir-ion mode (LIM) is increased with the decreasing ion-to-electron mass ratio In the nonlinear calculation, the effects of ion dynamics are not apparent in the linear growÀ1 ing stage (e.g., t < 25xÀ1 p and t < 40xp for bi-streaming Phys Plasmas 23, 112110 (2016) and bump-on-tail instabilities, respectively) but become pronounced after the long evolution time For the bi-streaming instability, Miyake et al.16 have examined the evolution of solitary structures with mobile (mi =me ¼ 100) and immobile ions (mi =me ¼ 1) in the simulations and concluded that transient solitary structures may be observed in the nonlinear evolution processes when the ion dynamics is involved and steady one-dimensional solitary structures are observed in the final stage for both mi =me ¼ 100 and mi =me ¼ cases In our study, both mi =me ¼ 1836 and mi =me ¼ 100 correspond to the mobile ion case 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