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Large amplitude ionacoustic waves in a plasma with an electron beamYasunori Nejoh and Heiji SanukiCitation: Phys. Plasmas 2, 4122 (1995); doi: 10.1063/1.871035View online: http://dx.doi.org/10.1063/1.871035View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v2/i11Published by theAmerican Institute of Physics.Related ArticlesExcitation of a slow wave structurePhys. Plasmas 19, 123104 (2012)Efficient laser-overdense plasma coupling via surface plasma waves and steady magnetic field generationPhys. Plasmas 18, 102701 (2011)Discontinuous Galerkin particle-in-cell simulation of longitudinal plasma wave damping and comparison to theLandau approximation and the exact solution of the dispersion relationPhys. Plasmas 18, 062111 (2011)Electron flow stability in magnetically insulated vacuum transmission linesPhys. Plasmas 18, 033108 (2011)Generalized matching criterion for electrostatic ion solitary propagations in quasineutral magnetized plasmasPhys. Plasmas 18, 032103 (2011)Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/Journal Information: http://pop.aip.org/about/about_the_journalTop downloads: http://pop.aip.org/features/most_downloadedInformation for Authors: http://pop.aip.org/authors

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Large amplitude ion-acoustic waves in a plasma with an electron beamYasunori NejohHachinohe Institute of Technology, 88-1, Myo-Obiraki, Hachinohe, 031, JapanHeiji SanukiNational Institute for Fusion Science, Furocyo, Chikusa-ku, Nagoya, 464-01, Japan(Received 20 March 1995; accepted 26 July 1995)The nonlinear wave structures of large amplitude ion-acoustic waves are studied in a plasma withan electron beam, by the pseudopotential method. The region of the existence of large amplitudeion-acoustic waves is examined, showing that the condition of the existence sensitively depends onthe parameters such as the electron beam temperature, the ion temperature, the electrostaticpotential, and the concentration of the electron beam density. It turns out that the region of theexistence spreads as the beam temperature increases but the effect of the electron beam velocity isrelatively smalL New findings of large amplitude ion-acoustic waves in a plasma with an electronbeam are predicted. 1995 American Institute of Physics.

I. INTRODUCTIONNonlinear waves may play important roles in rarefiedspace plasmas in the Earth's auroral zone,I-3 the physics of

solar atmosphere,4 and other astrophysical plasmas.56 Therehas been increasing interest in interpreting the low frequencynonlinear broadband electrostatic noise in spaceobservations. 78 Observations convince us of the fact that stationary nonlinear ion-acoustic waves may be excited whenan electron beam is injected into a plasma.9,l0 In the actualsituations, an electron beam component is frequently observed in the region of space where ion-acoustic waves exist.On the other hand, it is known that high-speed electrons havean influence on the excitation of various kinds of nonlinearwaves in the interplanetary space and the Earth'smagnetosphere. I1-14 The nonlinear ion-acoustic solitarywaves have been reported l5 in the plasma with drifting electrons. However, not many theoretical works on these topicshave been done.In this article, we make an attempt to theoretically investigate the possibility of the existence of large amplitude ionacoustic waves under the influence of an electron beam in aplasma consisting of warm ions and hot isothermal electrons.We also demonstrate the region of the existence of the largeamplitude ion-acoustic waves and study the dependence ofthe region of the existence on the parameters such as theelectron beam temperature, the concentration of the beamdensity, and so on.

The layout of this article is as follows. In Sec. II, wepresent the basic equations for a plasma with an electronbeam and derive the pseudopotential for large amplitude ionacoustic waves. In Sec. III, we define the condition of theexistence of large amplitude ion-acoustic waves and illustrate the dependency of the region of the existence region onthe several parameters. The last section is devoted to theconcluding discussion.II. BASIC EQUATIONS AND FORMULATION

We assume a plasma consisting of warm ions and hotisothermal electrons traversed by a warm electron beam, andconsider one-dimensional propagation. We adopt the fluidequations for the ions and beam electrons.

The continuity equation and the equation of motion forions are described by

an a-+ (nu)=Oat ax ' (1)au au 30' an ae/>-+ u -+ n - + - =Oat ax (I+a)2 ax ax ' (2)

where we express the pressure term in Eq. (2) by the thermodynamic equation of state. Here, (J'=T/Te anda=nb01nO' where Ti(Te) and nO(nbO) are the ion (electron)temperature and the unperturbed background electron (electron beam) density, respectively.

We have the following two equations for the beam elec-trons:(3)

au b au b v I an b I ae/>-+Ub-+------=O.at ax p, nb ax p, ax (4)We assume an isothermal equation of state for the beam electrons, because of the existence of the finite beam electrontemperature. In Eq. (4), p,=m/mi and V=TbITe. where me'mi' Tb denote the electron mass, the ion mass, and the beamelectron temperature, respectively.

The electron density follows the Boltzmann distribution,

The Poisson's equation is given bya2 1>ax2 =ne+nb- n .

(5)

(6)The variables neonb, n. Vb' v. and 1> refer to the electrondensity, the beam electron density, the ion density, the beamelectron velocity, the ion velocity, and the electrostatic potential, respectively. The velocities are normalized by the ionsound speed vs= ( K Telmi 12; the time t and the distance xby the ion plasma frequency w;/ = (Eom;lnoe2)112 and theelectron Debye length "'D=( EoKT/noe2) 1/2; and the densi-

4122 Phys. Plasmas 2 (11), November 1995 1070-664Xf95/2( 11 )/412215/$6.00 1995 American Institute of Physics



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ties by the background electron density no, the potential byKTie, where e is the charge of the electron.

In order to solve Eqs. (1)-(6), we introduce the variableg=x-Mt, which is the moving frame with the velocity M.Then the basic equations (1)-(4) and (6) become

an a-M ag+ ag (nv)=O, (7)

av av 30' an acf>-M ag+ U ag+ (l+a)2 n ag+ ag=O, (8)

(9)

(10)

(11)Integrating Eqs. (7) and (8) and using the boundary conditions, cf>---+0, n-41+a, nb-'ta, v-'tO, Vb-"VO at g-'too, weobtain

l+a (12)

Integrating Eqs. (9) and (10) and using the same boundaryconditions, we geta (13)

after the brief calculation.Using Eqs. (5), (12), and (D), the Poisson's equation

(11) reduces to

(14)

where V(cf denotes the pseudopotential. Integration of Eq.(14) gives the Energy law,

I [d cf>] 22" dg +V()=O. (15)From Eq. (14), the pseudopotential V() becomes

-vc)=eXP()- l - [ ~ l [ ~ 1 - 1 - ( I l J V ) 1 ( v o - M ) 2 2 : -1]+ 2(1 ;a ) [ ~ 1 - M22!30' 1].1-(IlJv)(vo-M)2 v M2-30'

The oscillatory solution of the large amplitude nonlinear ionacoustic waves exists when the following two conditions aresatisfied:

(i) The potential V() has the maximum value ifd2V()ld230'+ l+alv ' (17)

where we assumed l ~ ( p J v ) ( v o - M)2. Although the subsonic and supersonic ion-acoustic waves can exist in theplasma under consideration, we are now interested only inthe supersonic wave. It should be noted that V() is real,when

and

The region of the existence of is characterized by theseconditions.Phys. Plasmas, Vol. 2, No. 11, November 1995

(16)

(ii) Nonlinear ion-acoustic waves exist only whenV ( e p M ) ~ O , where the maximum potential epM is determinedby M=(v12)[I-(,u/v)(vo- M)2]. This implies that theinequality

:s;;;av+exp(v12)-I, (18)holds, where we use the approximation l ~ ( p J v ) ( v o - M)2.We show the maximum Mach number as a function of a inFig. 1, where v=0.2 and 0'=0.4. The maximum Mach number and, correspondingly, the maximum amplitude of the ionacoustic wave significantly depends on the parameters a, P,and 0'.

A complete analytical investigation of the ion-acousticsolitons in an electron beam plasma is possible for smallamplitude wave limit ( ~ l ) . The specific results can be obtained by expanding Veep) in powers of and keeping up tothe third-order terms 3. Accordingly, Eq. (16) takes theform

Y. Nejoh and H. Sanuki 4123

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1.6

1.5

1.4M

1 .3

1.2

1.1

10 0.2 0.4 0.6 0.8

aFIG. 1. The maximum Mach number depending on the concentration of theelectron beam density a, in the case of 0-=0.4 and v"'0.2.

-V(tP )0.001o-0.001 .2

FIG. 2. Bird's eye view of the pseudopotential under the conditions ofvo=1.25, M=1.3, 17=0.4, v=O.2. and ,u=1/1836.4124 Phys. Plasmas, Vol. 2. No. 11. November 1995

0.001

0.0005

-0 .0005

-0.001

-0.0015

-0.002

o 0.02 0.04 0.06 0.08 0.1FIG. 3. A pseudopotential curve of large amplitude ion waves for vo= 1.25,M == 1.3, 17=0.4. v=0.2, and a=0.104.It should be noted that the ion-acoustic soliton exists in thelimiting case with 4" 1.

We next consider the region of the existence of the largeamplitude ion-acoustic waves in the plasma with an electronbeam.III. THE REGION OF THE EXISTENCE OF LARGEAMPLITUDE ION-ACOUSTIC WAVES

We show a bird's eye view of - V() in Fig. 2, in thecase of v=O.2, vo"" 1.25, M = 1.3, O"==OA, and ,u= 111836.Figure 3 illustrates the dependence of the pseudopotential- V( ) on the potential when the ratio of the concentrationof the electron beam density to the background electron

0 .2

FIG. 4. Bird's eye view of the pseudopotential under the conditions ofvo=1.25, M=1.3. 17=0.4, v=O.4, and W=1I1836.

Y. Nejoh and H. Sanuki



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-V(q,)

0.005

O ~ ~ ~ - - - - - - - - - - - - - - - - - - - - ~ -0 .005

-0 .01

-0 .015

-0 .02

o 0.05 0.1 0.15 0.2

FIG. 5. A pseudopotentiaI curve oflarge amplitude ion waves for vo= 1.25,M=1.J, u=O.4, v=O.4, and a=0.55.

4. 0

3. 0

a2. 0

1.0

0. 00.0 0. 1 0.2 0. 3 0.4v

riG. 6. The region of the existence of large amplitude ion-acoustic wavesdepending on the concentration of the electron beam density a and theelectron beam temperature v, in the case of vo=1.25, M=1.3, u=O.4, andJ.L= 111836. Large amplitude ion-acoustic waves can exist in the regionbounded by the two curves.Phys. Plasmas, Vol. 2, No. 11, November 1995

0. 2

0. 1

o . o ~ - - - J - - L - - - - L - - ~ - - - - - - ~ - - L - - - ~ 0.0 1.0 2. 0a

FIG. 7. The -a plane where ion-acoustic waves exist, in the case of (a)vo= 1.25, M = 1.3, u=O.4, and v=0.2; (b) vo= 1.25, M = 1.3, 0-=0.4, andv=0.3; (c) vo= 1.25, M= 1.3, 0-=0.4, and v=O.4. Large amplitude ionacoustic waves can exist in the lower region of the curve.

density is fixed to be a=O.104, in the case of v=O.2. In Fig.4, we show a bird's eye view of the pseudopotential whenv=OA, vo=1.25, M=1.3, If=OA, and ,u=1/1836., Thepseudopotential - V(


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temperature to free electron temperature, in the case ofVo= 1.25, M= 1.3, u=OA, and ,u= 111836. Large amplitudeion-acoustic waves propagate in the region bounded by thetwo curves but do not exist in other regions. In addition, weshow the region of the existence of large amplitude ionwaves in the qra plane in Figs. 7(a), 7(b), and 7(c) forv=O.2, 0.3, and 0.4, respectively. Large amplitude ionacoustic waves exist in the lower region of the curves.It turns out that large amplitude nonlinear ion-acousticwaves can exist under proper conditions mentioned above.IV. CONCLUDING DISCUSSION

The nonlinear wave structures of large amplitude ionacoustic waves are studied in a plasma with an electronbeam. We present the region of the existence of the largeamplitude ion-acoustic waves on the basis of the fluid equations for an electron beam-plasma system.We investigate the conditions of the existence for thestationary supersonic ion-acoustic waves, by analyzing thestructure of pseudopotential. Typical results are illustrated inFigs. 1-7. The results are briefly summarized as follows:

(1) The conditions of the existence of large amplitude ion-acoustic waves sensitively depend on the electron beamdensity, the temperature of electron beams, and also theratio of the bulk ion temperature to the electron temperature.(2) The effect of the concentration of the electron beam density reduces the propagation speed of the ion-acousticwaves.(3) The allowable range of the concentration of the electrondensity becomes wider as the beam temperature decreases.(4) The allowable range of the electrostatic potential becomes wider as the beam temperature increases.(5) The effect of the velocity of the electron beam is small,compared with the effects of the electron beam temperature, the beam density and the ion temperature.

4126 Phys. Plasmas, Vol. 2, No. 11, November 1995

(6) The region in qra plane where large amplitude ionacoustic waves exist, spreads as the beam temperatureincreases.The present investigation predicts new findings on largeamplitude nonlinear ion-acoustic waves in a plasma with anelectron beam. In actual situations, large amplitude ionacoustic wave events associated with electron beams are frequently observed in interplanetary space.I,5,8 Hence, refer

ring to the present studies, we can understand the propertiesof large amplitude ion-acoustic waves in space plasmaswhere the electron beam exists. Although we have not referred to any specific observation, the present theory is applicable to analyzing large amplitude ion-acoustic waves,such as shock and solitary waves, associated with electronbeams which may occur in space and laboratory plasmas.ACKNOWLEDGMENTS

This work was partly supported by the Joint ResearchProgram with the National Institute for Fusion Science. Theauthor (Y. Nejoh) wishes to thank the Aomori Foundation forPromotion of Technological Educations and the Special Research Program of Hachinohe Institute of Technology.1M. Temerin, K. Cerny, W. Lotko. and F. S. Moser, Phys. Rev. Lett. 48,1175 (1982).2R. Bostrom, G. Gustafsson. B. Holback, G. Holmgren, and H. Koskinen,Phys. Rev. Lett. 61, 82, (1988).3L. P. Block and C. G. Falhammar, J. Geopbys. Res. 95. 5877 ([990).4H. Alfven, Phys. Today 39, 22 (1986).5H. Alfven. IEEE Trans. Plasma Sci. PS-14, 779 (1986).6D. A. Gurnen and L. A. Frank. J. Geophys. Res. 78, 145 (973).7R. P. H. Chang and M. Porkolab, Phys. Rev. Lett. 25, 1262 (1970).8D. A. Gurnett. L. A. Frank, and R. P. Lepping, J. Geophys. Res. 81, 6059(1976).9E. Witt and W. Lotko, Phys. Fluids 26,2176 (1983).lOy. L. AI'pert. Space Plasma (Cambridge University Press, Cambridge,1990), p. 118.lIy. Nejoh, Phys. Fluids B 4, 2830 (1992).12y. Nejoh, J. Plasma Phys. 51, 441 (1994).l3y. Nejoh and H. Sanuki, Phys. Plasmas 1,2154 (1994).14y. Nejoh and H. Sanuki, Phys. Plasmas 2, 346 (1995).ISH. Sanuki and J. Todoroki, J. Phys. Soc. lpn. 32.517 (1972).

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