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

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 insta

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

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Secondary instabilities in the collisionless Rayleigh-Taylor instability: Full kinetic simulation

Phys Plasmas 23, 112117 (2016); 10.1063/1.4967859

Fluid theory and kinetic simulation of two-dimensional electrostatic

streaming instabilities in electron-ion plasmas

C.-S.Jao1and L.-N.Hau1,2

1

Institute of Space Science, National Central University, Taoyuan City, Taiwan

2

Department of Physics, National Central University, Taoyuan City, Taiwan

(Received 3 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

mag-netic field For both instabilities, the fluid theory gives rise to the linear growth rates and the

wave-lengths 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

two-dimensional 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

seed-ing roles in the kinetic evolution of electrostatic streamseed-ing instabilities.V C 2016 Author(s) All article

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.19As shown in many studies based on

particle-in-cell simulations, ESWs may easily be generated by

the electrostatic streaming instability,10–13 including both

bi-streaming and bump-on-tail instabilities associated with the

ESWs observed in the auroral region and magnetotail plasma

sheet boundary layer, respectively.37 In multi-dimensional

systems and via streaming instabilities, solitary structures,

however, may form in a steady manner only for certain

mag-nitudes of background magnetic field and for unmagnetized or

weakly magnetized plasmas the formed solitary structures

may inevitably be destroyed in the nonlinear evolution

pro-cess.14,15The simulation results also show that the formed

sol-itary structures may tend to evolve into one- or

two-dimensional 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

exis-tence of one-dimensional ESWs Both the linear kinetic

the-ory 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

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

Goldmanet al.17and Oppenheimet al.18have 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 Miyakeet al.16later 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 inter-comparisons between the fluid theory and one-dimension par-ticle simulations for electrostatic streaming instabilities to infer the role of linear fluid theory in nonlinear kinetic simula-tions.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.20We have further shown that thepqcvalue is significantly increased at the nonlinear satu-ration accompanying with the particle trapping effect.19,20 In light of the important application of ESWs in geospace plasma

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

electro-static solitary waves in multi-dimensional streaming

instabil-ities The plasma system under consideration consists of

electrons and ions embedded in the background static

mag-netic 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

reso-nance between the two wave modes associated with

station-ary 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

x2H¼1

2 x2cþ x2

pþ ck2v2th

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2þ x2þ ck2v2

th

 4x2ðx2þ ck2v2

thÞ cos2h r

x2

L¼1

2 x2

cþ x2

pþ ck2v2th



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2þ x2þ ck2v2

th

 4x2ðx2þ ck2v2

thÞ cos2h r

(1)

for which xHand xL denote the high and low frequency modes, respectively For parallel propagation (h¼ 0), the above relations reduce to the Langmuir mode of x2¼ x2

pþ ck2v2thand cyclotron mode of x2¼ x2

c, while for perpendicular propa-gation (h¼ 90), there exists only the high frequency mode with x2H¼ x2

pþ ck2v2thþ x2

c Figure 1shows the wave fre-quency versus wave number for the cases of xc=xp¼ 0:5 (panel (a)) and xc=xp¼ 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> 1 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 fre-quency mode and the large wave number portion of the higher frefre-quency 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 disper-sion relation derived from the set of fluid equations with the adiabatic energy law is23

x2 p;i x2 x2

c;icos2h

x2 ck2v2

th;i

x2 x2 c;i

 x2 c;ick2v2 th;isin2h

þ

x2 p;e1 x2 x2

c;ecos2h

x2 ck2v2

th;e1

x2 x2 c;e

c;eck2v2 th;e1sin2h

2 p;e2 ðx kue2cos hÞ2 x2cos2h

x kue2cos h

th;e2

x kue2cos h

c;e

 x2 c;eck2v2 th;e2sin2h

FIG 1 The wave frequency versus wave number for the cases with

xc=xp¼ 0:5 (panel (a)) and x c =xp¼ 2:0 (panel (b)) with h ¼ 0  (solid line), 30(dashed line), 60(dot-dashed line), and 90(dotted line) In both panels, x H and x L denote the high and low frequency roots in equation (1) , respectively.

for which the subscripts “e1,” “e2,” and “i” are, respectively,

for the background stationary electrons, streaming electrons,

and background stationary ions; and v2

th;a kBTa=ma,

x2

p;a na0q2a=e0ma, and xca qaB0=ma In the following

calculations, Equation (2) is solved numerically for the

bi-steaming instability withne1;0¼ ne2;0andvth;e1¼ vth;e2while

for the bump-on-tail instability we set ne1;0=ne2;0¼ 9:0 and

vth;e1¼ 5vth;e2 Other parameters such asvth;i¼ 0:1vth;e2and

ue2¼ 10vth;e2(along the background magnetic field) are

cho-sen The c value is set to be 5/3 in the fluid model

consider-ing that in the simulation all particles possess three degrees

of freedom

Figure2shows the imaginary part of wave frequency xi

as functions of wave numberkx andkyfor both bi-streaming

(panel (a)) and bump-on-tail instabilities (panel (b)) with the

parameter values of xc=xp¼ 0:5 and mi=me¼ 1836 As

expected, the most unstable mode occurs for parallel

propaga-tion (ky¼ 0) with the growth rate of xi¼ 0:343x1

kx¼ 0:095xp=vth and xi¼ 0:171x1

p at kx¼ 0:181xp=vth

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 bi-streaming instability exists within the wave number range of

FIG 2 The imaginary part of the wave frequency x i 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 coupling, cyclotron coupling,

Langmuir-ion coupling, cyclotron-Langmuir coupling, and cyclotron-cyclotron

cou-pling, respectively.

FIG 3 The imaginary part of the wave frequency x i as functions of wave number kx and ky derived from the linear fluid theory for the bi-streaming instability with x c =xp¼ 0:0 (panel (a)), x c =xp¼ 1:0 (panel (b)), and

x c =x p ¼ 2:0 (panel (c)) In the figure, LLM, LIM, CLM, and CCM denote the unstable modes induced by the Langmuir coupling, Langmuir-ion coupling, cyclotron-Langmuir coupling, and cyclotron-cyclotron coupling, respectively.

kx¼ 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 propaga-tion, 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.23A 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 back-ground electrons (CLM) There is also another non-propagating 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 spec-trum 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 propaga-tion 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 Figure2(b)

Figure 3shows xi as functions of kx and ky for the bi-streaming instability with various background magnetic fields,

FIG 4 The imaginary part of the wave frequency x i as functions of wave

number kxand kyderived from the linear fluid theory for the bump-on-tail

instability with x c =xp¼ 0:0 (panel (a)), x c =xp¼ 1:0 (panel (b)), and

x c =x p 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 5 The time evolution of logarithm of maximum electric potential for the bi-streaming instability with xc=xp¼ 0:0 (circle symbol), x c =xp¼ 0:5 (cross symbol), and x c =xp¼ 2:0 (plus symbol) from two-dimensional sim-ulations 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.

xc=xp¼ 0:0 (panel (a)), xc=xp¼ 1:0 (panel (b)), and

xc=xp¼ 2:0 (panel (c)) As expected, the most unstable

modes occur for the parallel direction in all cases For the

unmagnetized case (Figure3(a)), there exists no any resonant

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 Figures2(a),

FIG 6 The spatial distributions of electric potential /ðx; yÞ at t ¼ 25x 1

p for the bi-streaming instability with x c =x p ¼ 0:0 (panel (a)), x c =x p ¼ 0:5 (panel (b)), and x c =x p ¼ 2:0 (panel (c)) from two-dimensional simulations, and the corresponding wavenumber spectrum of electric potential /ðk x ; k y Þ for the cases with xc=xp¼ 0:0 (panel (d)), x c =xp¼ 0:5 (panel (e)), and x c =xp¼ 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.

3(b), and3(c), it is shown that the instability is obviously

sup-pressed in the oblique direction with increasing background

magnetic field For strongly magnetized system (xc=xp¼ 2:0;

Figure3(c)), the major portion of the instability occurs within

the wave number range ofk < 0:175xp=vthand h 30

Figure 4shows the same analyses for the bump-on-tail

instability with various background magnetic fields of

xc=xp¼ 0:0 (panel (a)), xc=xp¼ 1:0 (panel (b)), and

xc=xp¼ 2:0 (panel (c)) Since the instability regime caused

by the Langmuir-Langmuir mode resonance (LLM) is

intrin-sically narrow for the bump-on-tail instability, the

suppres-sion 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=xp¼ 0:0 (Figure 4(a)) and xc=xp¼ 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=xp¼ 2:0), the main instability

portion occurs in the narrow region ofky< 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

instabil-ities.10,15,17,18,21,22 In the kinetic model, the most unstable

modes occur mainly fork < 1:0xp=vthand fork 1:0xp=vth

(i.e., the wavelength is about the Debye length kD¼ vth=xp)

the linear growth rate becomes much smaller than 0:1x1p 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=vthand the linear growth rate fork 1:0xp=vthis

an order of magnitude smaller than 0:1x1p

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.23In the

calcula-tions, dimensionless units are used with the plasma

fre-quency of total electrons being xp¼ 1:0 and the thermal

velocity of beam electrons beingvth¼ 1:0 The grid size is

chosen to be Dx Dy ¼ 1kD 1kDfor the simulation box of

Lx Ly¼ 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 Dt¼ 0:01x1

p is typically adopted with ini-tial thermal velocity distributions of all particles being

described by the isotropic Maxwellian function For

compar-isons, the parameters used for the particle simulations of

bi-streaming 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

bi-streaming instability in one- and two-dimensional systems

Figure5shows the time evolution of logarithm of maximum electric potential based on the one- (triangle symbol) and two-dimensional simulations with xc=xp¼ 0:0 (circle sym-bol), xc=xp¼ 0:5 (cross symbol), and xc=xp¼ 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 (t¼ 10  20x1

p ) The dashed straight line in Figure5 corresponds to the growth rate of the most unstable mode derived from the linear theory (xi¼ 0:343x1p ), which is in good agreement with the simulation results Figure6shows the distribution of electric potential /ðx; yÞ (panels (a)–(c)) and the corresponding wave number spectrum /ðkx; kyÞ (panel (d)–(f)) att¼ 25x1

p for the cases with xc=xp ¼ 0:0,

xc=xp¼ 0:5, and xc=xp¼ 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=xp¼ 0:5 (Figure 6(b)) and xc=xp¼ 2:0 (Figure 6(c)) is more one-dimensional The wave number spectra (Figures6(d)–6(f)) show that the major perturbations are all parallel to the background magnetic field (i.e., ky¼ 0) with the wave number being kx 0:095xp=vth, which is consis-tent 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=xp¼ 0:0 (Figure6(d)) are more evident than the case of

xc=xp¼ 2:0 (Figure6(f)), which is also consistent with the

FIG 7 The time evolution of logarithm of maximum electric potential for the bump-on-tail instability with xc=xp¼ 0:0 (circle symbol), x c =xp¼ 0:5 (cross symbol), and x c =xp¼ 2:0 (plus symbol) from two-dimensional sim-ulations 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 lin-ear theory.

tendency obtained from the linear fluid theory (Figures2(a)

and3)

As shown in Figure 5, for the unmagnetized

two-dimensional system (xc=xp¼ 0:0; circle symbol), the electric

structures begin to damp in the nonlinear evolution process

(t > 60x1p ) while for one-dimensional (triangle symbol) and

two-dimensional magnetized systems (cross symbol for

xc=xp¼ 0:5 and plus symbol for xc=xp¼ 2:0) the saturated electric potentials are shown to be steady in the early nonlin-ear stage (t < 256x1p ) These results are consistent with the previous simulation studies for the multi-dimensional unmag-netized system.14Indeed, the results are similar for the rela-tively weakly magnetized system such as xc=xp ¼ 0:2 and consistent with the early study on the nonlinear evolution

FIG 8 The spatial distributions of electric potential /ðx; yÞ at t ¼ 40x 1

p for the bump-on-tail instability with x c =x p ¼ 0:0 (panel (a)), x c =x p ¼ 0:5 (panel (b), and x c =x p ¼ 2:0 (panel (c)) from two-dimensional simulations, and the corresponding wavenumber spectrum of electric potential /ðk x ; k y Þ for the cases with xc=xp¼ 0:0 (panel (d)), x c =xp¼ 0:5 (panel (e)), and x c =xp¼ 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.

process of bi-streaming instability reported in several

literatures.14,16–18,22

We now discuss the simulation results on the

bump-on-tail instability Figure7shows the time evolution of logarithm

of maximum electric potential based on one-dimensional

(tri-angle symbol) and two-dimensional simulations with

xc=xp¼ 0:0 (circle symbol), xc=xp¼ 0:5 (cross symbol),

and xc=xp¼ 2:0 (plus symbol) As indicated in Figure7, 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

(t¼ 20  40x1

p ) The slope of the corresponding straight

lines in Figure7is the growth rate of the most unstable mode

derived from the linear theory (xi¼ 0:171x1

p ), indicating that the two-dimensional simulation results are in accordance

with the linear theory (Figures2(b)and4) Note that the

one-dimensional simulation result (triangle symbol in Figure 7)

shows smaller growth rate as compared to the

two-dimensional 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 8 shows the electric potential

/ðx; yÞ after the linear growing stage (t ¼ 48x1

p ) for the cases of xc=xp¼ 0:0, xc=xp¼ 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

(Figure8(b)) as compared to the unmagnetized (Figure8(a))

or strongly magnetized cases (Figure8(c)) The corresponding

wave number spectra /ðkx; kyÞ are shown in Figures

8(d)–8(f) As shown by the linear fluid theory (Figures2(b)

and 4), oblique modes with larger growth rate are seen in

strongly magnetized systems (Figure8(f)) While for weakly

magnetized cases (Figure8(e)), the unstable modes are more

along the magnetic field direction as compared to the

unmag-netized case (Figure8(d)) The results shown in Figure8based

on the kinetic simulations are thus more or less consistent

with the linear fluid theory For all the cases presented in

Figure8, the major perturbations are along the drifting

direc-tion, in good agreement with the linear theory, but the most

unstable mode occurs for kx 0:095xp=vth, instead of kx

¼ 0:181xp=vthpredicted by the linear theory (red star symbol

in panels (d)–(f)) It is interesting to note that the wave

num-berkx 0:095xp=vth corresponds closely to another unstable

mode (CCM) with the same order of growth rate

(xi> 0:13x1p ) 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

IV EFFECTS OF ION-TO-ELECTRON MASS RATIO

In multi-dimensional kinetic simulations, unreal

ion-to-electron 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

effects of the ion-to-electron mass ratio, mi=me, on the streaming instabilities shown above for the electron-proton plasma Figure9shows xias functions ofkxandkyfrom the linear theory of bi-streaming instability with mi=me¼ 1

FIG 9 The imaginary part of the wave frequency xias functions of wave number k x and k y derived from the linear fluid theory for the bi-streaming instability with m i =m e ¼ 1 (panel (a)) and m i =m e ¼ 100 (panel (b)) In the figure, LLM, CLM, and CCM denote the unstable modes induced by the Langmuir-Langmuir coupling, Langmuir coupling, and cyclotron-cyclotron coupling, respectively.

FIG 10 The time evolution of logarithm of maximum electric potential for the bi-streaming instability with m i =m e ¼ 1 (circle symbol), m i =m e ¼ 1836 (cross symbol), and m i =m e ¼ 100 (plus symbol) from two-dimensional sim-ulations The slope of the corresponding straight line is the maximum growth rates of the unstable mode derived from the linear theory.

(panel (a)) andmi=me¼ 100 (panel (b)) By comparing the

three cases with mi=me¼ 1 (Figure 9(a)), mi=me¼ 1836

(Figure2(b)), andmi=me¼ 100 (Figure9(b)), it is seen that,

as expected, the unstable modes induced by the beam and

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 decreasingmi=mevalues

FIG 11 The spatial distributions of electric potential /ðx; yÞ at t ¼ 256x 1

p for the bi-streaming instability with x c =x p ¼ 0:5, u e2 ¼ 10:0, and m i =m e ¼ 100 (panel (a)), x c =x p ¼ 2:0, u e2 ¼ 10:0, and m i =m e ¼ 100 (panel (b)), and x c =x p ¼ 0:5, u e2 ¼ 20:0, and m i =m e ¼ 100 (panel (c)) from two-dimensional simula-tions, and the corresponding wavenumber spectrum of electric potential /ðk x ; kyÞ for the cases with x c =xp¼ 0:5, u e2 ¼ 10:0, and m i =me¼ 100 (panel (d)),

x =x ¼ 2:0, u ¼ 10:0, and m =m ¼ 100 (panel (e)), and x =x ¼ 0:5, u ¼ 20:0, and m =m ¼ 100 (panel (f)).