Double Layers, Spiky Solitary Waves, And Explosive Modes of Relativistic Ion

8/12/2019 Double Layers, Spiky Solitary Waves, And Explosive Modes of Relativistic Ion

1/12

Double layers, spiky solitary waves, and explosive modes of relativistic ion

acoustic waves propagating in a plasma

Yasunori Nejoh

Citation: Physics of Fluids B: Plasma Physics (1989-1993) 4, 2830 (1992); doi: 10.1063/1.860157

View online: http://dx.doi.org/10.1063/1.860157

View Table of Contents: http://scitation.aip.org/content/aip/journal/pofb/4/9?ver=pdfcov

Published by the AIP Publishing

Articles you may be interested inLarge amplitude ion-acoustic solitary waves and double layers in multicomponent plasma with positronsPhys. Plasmas 16, 072307 (2009); 10.1063/1.3175693

Ion-acoustic solitary waves in a relativistic plasmaPhys. Plasmas 14, 022307 (2007); 10.1063/1.2536581

Evolution of nonlinear ion-acoustic solitary wave propagation in rotating plasmaPhys. Plasmas 13, 082303 (2006); 10.1063/1.2245578

Ionacoustic solitary waves in relativistic plasmasPhys. Fluids 30, 2582 (1987); 10.1063/1.866098

Ionacoustic solitary waves in relativistic plasmasPhys. Fluids 28, 823 (1985); 10.1063/1.865050

s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded

IP: 103.27.8.43 On: Thu, 29 May 2014 12:29:08
http://scitation.aip.org/search?value1=Yasunori+Nejoh&option1=authorhttp://scitation.aip.org/content/aip/journal/pofb?ver=pdfcovhttp://dx.doi.org/10.1063/1.860157http://scitation.aip.org/content/aip/journal/pofb/4/9?ver=pdfcovhttp://scitation.aip.org/content/aip?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pop/16/7/10.1063/1.3175693?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pop/14/2/10.1063/1.2536581?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pop/13/8/10.1063/1.2245578?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pof1/30/8/10.1063/1.866098?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pof1/28/3/10.1063/1.865050?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pof1/28/3/10.1063/1.865050?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pof1/30/8/10.1063/1.866098?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pop/13/8/10.1063/1.2245578?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pop/14/2/10.1063/1.2536581?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pop/16/7/10.1063/1.3175693?ver=pdfcovhttp://scitation.aip.org/content/aip?ver=pdfcovhttp://scitation.aip.org/content/aip/journal/pofb/4/9?ver=pdfcovhttp://dx.doi.org/10.1063/1.860157http://scitation.aip.org/content/aip/journal/pofb?ver=pdfcovhttp://scitation.aip.org/search?value1=Yasunori+Nejoh&option1=authorhttp://scitation.aip.org/content/aip/journal/pofb?ver=pdfcov


2/12

Double layers, spiky solitary waves, and explosive modes of relativisticion-acoustic waves propagating in a plasmaYasunori NejohDepartment of Physics, Hachinohe institute of Technology, 88-I, Obiraki, Myo, Hachinohe, 031, Japan(Received 29 April 1991;accepted6 May 1992)The fully relativistic ion fluid equations are presented.These equations are reduced o amixed modified Korteweg-de Vries (MKdV) equation by using the reductive perturbationmethod. The high-speedstreaming ions and the negative cubic nonlinearity of themixed MKdV equation give rise to the ne w nonlinear wave modes, hat is, the compressivedouble ayer, the spiky solitary wave, and the explosive solutions. The double layerand the spiky solitary wave are confined within the specifiedpositive potential region, whilethe explosive solutions are confined within the region where the potential exceeds hemaximum potential or the negative potential region. It is shown for the first time that thedouble-layer hickness narrows as the ion temperature and the relativistic effectsincrease, hat the potential drop of the double layer grows as the ion temperature ncreases,and that the amplitudes of the spiky solitary wave and the explosive solution grow asthe i on temperature effect increases.This investigation relates to the evolution processof thenonlinear wave structure in which these hree modes orm the fine structure in space.

1. INTRODUCTIONIn a historical current of nonlinear wave studies onplasmas,double ayers and solitary waveshave beendraw-ing the attention of many investigators. Electric doublelayers in plasmas have been extensively investigated intheories,-2 the aurora1 zone,34 and laboratoryexperiments concerning he mechanismof particle accel-eration. In nonlinear wave studies the propagation of sol-itary waves s also important as it describes he character-istics of the interaction between he wavesand the plasmas.In the casewhere the velocity of particles is much smallerthan that of light, ion-acoustic wavesexhibit nonrelativisticbehavior, but in the case where the velocity of particlesapproaches hat of light, relativistic effects become domi-nant. Various kinds of nonlinear wavesoccur in relativisticplasmas and thus relativistic Langmuir and electromag-netic waves have been studied as subjects of laser-plasmainteraction7 and spaceplasma phenomena. nterplanetaryspace and the Earths magnetosphereencompassa richvariety of plasma physical processes nd nonlinear wavephenomena.High-speed and energetic streaming ions with the en-

ergies rom 0.1 to 100 MeV are frequently observed n thesolar atmosphereand interplanetary space.Yet, relativisticion-acoustic waves have not been horoughly investigated.When we assume hat the ion energy dependsonly on thekinetic energy, such plasma ons have to attain relativisticspeeds.Thus, by considering relativistic effects where theion velocity is about O.lc, we can describe he relativisticmotion of such ions in the study of nonlinear plasmawaves. Since the ion temperature is very high in solarflare? solar wind, and interplanetary space,i the ratio ofthe ion temperature and the electron temperature s some-times more than 1. In this situation, relativistic and ion

temperatureeffectsare mportant for energetic on-acousticwaves propagating in interplanetary space.It has been suggested hat relativistic double layersmay occur in cosmic and space plasmas, and that theyform the large amplitude electric field and accelerateplasma particles. 2 However, little attention has beengivento relativistic ion-acousticdouble ayers associatedwith theplasma dynamics under the fluid description. We havefound weakly relativistic ion-acoustic double layersI forcold ions in the vicinity of the critical point where thenonlinear coefficient of the Korteweg-de Vries (KdV)equation is negligibly small. The effect of a finite ion t em-perature on relativistic double layers has not yet beenwellinvestigated. On the other hand, relativistic ion-acousticsolitary waves have been found by the KdV equationt3Interesting featuressuch as the formation of the precursorof the ion-acoustic solitary waves, he long wavelength onoscillation modes,l and the associationbetween elativisticion modulati on modest4 re discussed s astrophysical phe-nomena. We show a new spiky solitary waves and an ex-plosive mode of ion-acoustic waves obliquely propagatingin a magnetizedplasma,15 nd also find a new modulation-ally unstable ion-acoustic wave in a relativistic plasma.6Although the relativistic ion-acoustic solitary waves in aspaceplasma are basically describedby the nonlinear evo-lution equation, the effect of the ion temperature is anindispensable actor for ion-acoustic waves in interplane-tary space.It is recogni zed in the recent astrophysical observa-tions that the fine structure of the physical quantities suchas the potential, the electric field, the magnetic lux density,etc., is important in order to understand he properties ofspace plasmas. The fine structure is composed of doublelayers and spiky solitary waves.We present a new point of

2830 Phys. Fluids B 4 (9), September 1992 0899-8213/92/092830-l 1$04.00 @ 1992 American Institute of Physics 2830s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded

IP: 103.27.8.43 On: Thu, 29 May 2014 12:29:08


3/12

view with respect to double layers and spiky solitary waveslater in this paper.The object of this paper is to show the formation ofrelativistic double layers, new spiky solitary waves, andexplosive solutions in an unmagnetized space plasma withhot ions by considering the fully relativistic ion fluid equa-tions. In this paper, it is expected that, by considering theion temperature and the relativistic effects, compressivetype double layers, spiky solitary waves, and explosive so-lutions will be obtained from a mixed modified KdV(MKdV) equation which includes the quadratic and thenegative cubic nonlinearity. The double layers inform us ofthe reason why energetic ions are accelerated during thepropagation in interplanetary space. The spiky solitarywave is closely related to energetic precursor events inspace,3p4 nd the occurrence of these solitary waves areassociated with the disruption of the double layer. Theexplosive solution explains the explosive events such as theejection of solar energetic particles.

the longitudinal ion-acoustic wave propagates parallel tothe electric field, we observe the electrostatic ion-acousticwave in a relativistic plasma. We assume hat the ion flowvelocity is relativistic, and thereby there exist only high-speed streaming ions in an equilibrium state when longitu-dinal ion-acoustic waves propagate in one dimension.The basic equations are described by the fully relativ-istic ion fluid equations for the conservation law of themass, the momentum equation, and the energy equationfor the adiabatic ions. For longitudinal waves, the dynam-ics of the relativistic ion fluid in nondimensional form arewritten as

-g(,,+&

The layout of this paper is as follows. In Sec. II, wederive the MKdV equation at the critical point where thenonlinear term of the KdV equation does not contribute toforming the solitary wave. In addition, we derive a mixedMKdV equation from the basic equations associated withthe fluid descript ion of a relativistic plasma with a finite iontemperature. Section III is divided into two parts. We de-rive relativistic double layers and explosive solutions inSec. III A and show relativistic spiky solitary waves andexplosive solutions in Sec. III B. Dependency of doublelayers and solitary waves on both the ion temperature andthe relativistic effects is illustrated. We derive the relativ-istic oscillatory wave solution in the small wave-numberregion in Sec. IV. The last section (Sec. V) is devoted tothe concluding discussion.

(&+u~)P+3lJP(~+gJ~) u=o,

where the Lorentz factor is y=[l- (u/c)~]-~. The sys-tem of equations is closed with the help of the Poissonsequation,

II. DERIVATION OF A MIXED MKdV EQUATION withWe consider small but finite-amplitude ion-acousticwaves propagating in a collisionless relativistic unmagne-tized plasma. The dimensions of the system are muchlarger than the electron Debye length, and the plasma isquasineutral. Since the electron inertia and the electrontrapping phenomena are neglected for low-frequency oscil-lations of ion-acoustic waves, the electron velocity is can-celed with the help of the Poissons equation; the motionsof electrons can be ignored. The fluid description used in

this study can be ustified by the fact that we are interestedin the macroscopic, average nonlinear behavior of the rel-ativistic plasma rather than the microscopic properties;that is, the motion of individual particles. This justificationcan be further supported by spaceand time scales nvolvedin this nonlinear problem. It is assumed that the velocitydistribution of each species s Maxwellian everywhere andthe relativistic plasma is composed of a mixed fluid withhot and isothermal electrons and hot ion species.To sim-plify the discussion we do not take into account kineticeffects such as the deviation from the Maxwellian distribu-tion, Landau damping, etc. Since we consider the low-frequency motion of the ion-acoustic wave and assume hat

The right-hand side of y is approximated by its expansionup to the second term; n, v, p, 4, n,, and c denote theproper ion density, the ion flow velocity, the pressure, theelectrostatic potential, the electron density, and the veloc-ity of light, respectively. The densities are normalized bythe unperturbed background electron density no, velocitiesby the sound velocity (KTJM) 2, where M is the ionmass; p and 4 by n&T, and KTJe; the time t andthe distance x by the ion plasma frequencyup; = ( e&/noe2) l* and the electron Debye length&, = ( qcTJq,e2> *, respectively. The parameter CYs theratio of the positive ion temperature Ti to the electrontemperature To that is, T/T,.In order to solve Eqs. (la)4 le), we use the reductiveperturbation method. We expand the dependent physicalquantities n, u, p, 4, and n, around the unperturbed uni-form state as power series n terms of the small parametere:

2831 Phys. Fluids B, Vol. 4, No. 9, September 1992

ynv) =o, (la)

(lb)

(ICI

(IdI

(le)

Yasunori Nejoh 2831s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded

IP: 103.27.8.43 On: Thu, 29 May 2014 12:29:08


4/12

m mn=l+ 2 &, u=uo+ C dvl,I=1 I= 1

m m

s=s-VQ,yQ I+ l&/2&yI = 1+ 3@2&y2= 3~~22,7/3 = vQc4.

p=l+ 4%4=0+E&, (2)I=1 I=1m

ne= 1+ C En,.I=1If we expa nd he frequencyo in a power seriesaroundthe wavenumberk=O, the phase actor of the plane wavedepe nds n the small displacement f the wavenumberAk.Whe n we take Ak- O( e*), we introduce the stretchingcoordinates&=P(x--St),P== e3/*t, (3)

whereE measureshe weakness f dispersionand s refers othe phase elocity, respectively.We substitute Eqs. (2) a nd (3) into Eqs. (la)-( le).To the lowest order of E, we obtain#1=n1= 01YQwY* -tY*s)

PI=3Yo(l -b/*s-Y3SMY&y*d (4)

and-ioyI~Z+- 3oyo

Y1- SY2S Y1- 52s ( I-;y*s-y3s fl=O,)where

The solution of the quadratic equation or S iss-,y2* YI + 3QYo ( 3ayoy3s + 3y2d

Yl YOYI -+ cr22

YOYl ( 1i *(5)

When we take the positive sign of Eq. (5), the phasevelocity is obtainedass=uo+ Jl+ 3cyoYQYI

-;(l+vOJzJ 3cycy$yL o ;.(6)

We used the approximations &-O(vQc4) zo,Y*Y3-WW)~oo, y$-0(&2)~0, y;-o(vg/c6)~0,Yz?$-O(~~C~)~O, and (yzy3)2-O(~~~12)~o.To the secondorder of E, we have he KdV equationa#1 d~F~O~??~~2?~4)x+ Nwith

(7)

dw0,yby, ,y4)=~+ ~ (--$$-)2+k (I-----&) (l--f$$+& (*-~s)2-Y2S (ss)

+,ff( 2-$2 Y$) ( +SS+(Yz-sYb) (y,:;zS)I? (8)

and1 1N=-

Y1- iY2s YoYIs+~ M+qoY* . (9)Here we note that the coefficient a( cr,yo,y1,yz,y4) spositive-whatever the values he ion temperatureand therelativistic effects may take.In order to take into accoun t he higher-ordernonlin-earity, we assume hat the displacement f the wave num-ber Ak of the plane wave is of the order of E, that is,2832 Phys. Fluids 6, Vol. 4, No. 9, September 1992

I

Ak - 0( e . We henceuse he following mod ified stretchingcoordinates:f;=e(x-st), (10)7=e3t.

Substituting Eqs. (2) and (10) i nto Eqs. (la)-( le), wehave reducedEqs. (4) to the lowest order in E.To the next order in E, we have(Ila)

Yasunori Nejoh 2832article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to

103.27.8.43 On: Thu, 29 May 2014 12:29:08


5/12

v2=-y,:;2s+$& (;+(-) (yl:~2s~2)~~9 (lib)3Yo( l-iY2S-YP)

I%o( 1 --Y3d

P2= $2+ -(

1+2(y2-y4s)Yl -iY*s Y1-3YzS (y,:;2s~2)+3( l-jyls-y+)(&)

(~YO( 1 --~YzS-Y~S) *

+ ) +2,( 2-~-2y2s) (&)I ;ot (llc)Y1 -?Y*sand

adIWwww24f4M z=O. (12)To the next higher order in E, we can eliminate n3, v3, and &$,/a( by Eq. (5) and haveYI($I) +~2(~~,~2,n2,~2,p2)=0, (13)

wherevs

~I(#,) = Yl -ti*s I Y~YI~+;Y~+%Yo(~+$)]~+,(-;+~+~~Y~-~Y~~(~)~3YS+

(YrsY*s)* (Y2-sY4+ 53+(,,~;2s~2 [Y*(+Y*s)( -I+$3

YOS II2 h a%3oy4(1--2y*s) - hyg+/osjpYl -+Y*si

a 3~0(1-3Y+Y3y3s) ah1 I

2~2W742,n29u21P2 2(?z-s?&&1 =yoS g Wln2) - Y1 +Y*s n*jg + yl-ps+ y*z)/s(Ys)* w4Y2S i )I 36% 1 ?Y*S-Y34& (Au*) + i ah+(y1-2y*S) ___ - 2-$-2y*sYr~*~+Y*-~W v2 -Fig-Yl -;r*s

.,( I-&-Yjr)d+=%j +$--)# i$+,dl--~;;y)sPz~a-YS g (h#2)7

where y4= (Zc*)-.When we substitute Eqs. (4), (6), and (lla)-( llc) into Eq. (13), Eq. (13) is reduced to the MKdV equation@I fl(6YO,Y,,YZ,Y4) 2 a+l 1 a%ar+ 4, pzlyqT=Q (14)2N

wheref%~,yO,y,,~2,~4~ = -2+5 _~-(*)2-qi-2~)-~+[~-4&+(&)i

+~(5-~)+zbi~s-~(5-~+~)(~s)2+~(~)32833 Phys. Fluids B, Vol. 4, No. 9, September 1992 Yasunori Nejoh 2833

article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to

103.27.8.43 On: Thu, 29 May 2014 12:29:08


6/12

+2[ Y20Y0(2-$2Y2S)-(Y2-sY+)( -l+$.)] [ _I+-&+& (-l+S$

-6[ WYo( +%) +?Mj (;+;-r) ,,,( -s+s-zs

Yii$+2a( 1-2Yz.s) -Yr512S *)I

It is noticed that the nonlinear coefficient/3(a,Y0,Y,,y2,y4)of the MKdV equation is always negativewhatever the values the ion temperature and the relativisticeffects may take. Nonlinear waves in the region of/3(o,Yo,Y,,y2,Y4) 0 have not yet been considered. Thus theauthor will discuss relativistic ion-acoustic waves propa-gating in this region.In order to take higher-order nonlinear effects nto ac-count, we derive a mixed MKdV equation with quadraticand negative cubic nonlinearity. Combining Eqs. ( 1 a)-( 1 c) and ( 12), we convert the momentum equation ( lb)to

(15)Assuming that the order of cr(a,Yo,yl,y2,y4)-O(E), theorder of the right-hand side of Eq. ( 15) becomes 0(e4)and is zero in 0(e3). In this situation, the termcr(~,Yo,Yl,Y2,y4)461(d~,/d~)has to be included in the nexthigher-order equation of momentum. In these circum-stances, Eq. (lb) is reduced to-Yts~+oap3+~+(Y,-2Y2s) d (v,q)a{ 3 ai-

(ap2 ai- au1-0 n1-tnz-x. a6 )

au1+y,a7f2y,v:- xa a au2-y$s-g (v*)3+m2~-yIs-+~~+~* at ag a6i 4s

Yl+ T-Y2S a6( ) 2 (v*)2- apIml7g=O. (16)Eliminating third-order terms u3,p3, QIs, nd second-orderterms v2,p2, $z, by using Eqs. (4) and (lla)-( 1 c) andnormalizing r by N, we obtain a mixed MKdV equationfrom Eq. (16),

I(17)

where #=&. We abbreviate a(a,Yo,yl,yz,Y4) andP(a,y~,yt,Yz,y~) to a and P, respectively.III. STATIONARY SOLUTIONS OF EQ. (I?)

In this section, we present a relativistic double layer, anew spiky solitary wave, and an explosive solution associ-ated with the mixed MKdV equation derived in the pre-ceding section. We introduce a variable q=c----UT in a sta-tionary frame, where u is a constant velocity. Inserting 7into Eq, (17) and integrating it twice, we obtain

and+Y(a,P,w$) =O (18)

(19)under the boundary conditions (p, &$/a$ -+0 at 177--* CO,when n=l and 2. Equation (18) with Eq. (19) can beregarded as an equation of motion of a particle with unitmass under the potential function \I/(a,p,u,$) ( (O), or asan equation of an anharmonic oscillator provided that weinterpret q and # as time and space coordinates, respec-tively.A. Relativistic double layers and explosive solutionsWe consider, in this section, that Y (cr,~,u,~) has twodouble roots, In order to obtain solutions of Eq. ( 18) withEq. (19), we transform Eq. (19) to

~(#,cb,) =W6)~2W-4,)2, (20)where

A&,8) = -2a/rJ (214and

u= - (/3/12)&(cr,p) = -a2/3/3. @lb)fasunori Nejoh 2834834 Phys. Fluids B, Vol. 4, No. 9, September 1992


103.27.8.43 On: Thu, 29 May 2014 12:29:08


7/12

FIG. 1. The profile of the relativistic compressive double layer repre-sented by Eq. (23) with u,,/c=O.2 as a function of 7, for the ion temper-ature ratio ~7=0.02 and 0.04, marked on each curve.

6oc

OO I I I I0.01 0.02 0.03 0.04 0.cffFIG. 2. Dependency of the potential drop Ac$of the double layer on theion temperature ratio a, for the relativistic effect ve/c=0.2.2835 Phys. Fluids 6, Vol. 4, No. 9, September1992

Here, 0, denotes he maxi mum potential.For the formation of double layers, the potential mustbehave n the following manner:Y(a,P,u,4) -0 at ~-4 d~Aa,Lk GWd~(dXw$)4 4 at +$m(atP>,d2Y(a,P,4)

d#2-0 at $,(cr,/3)-0.

(22b)(22c)

Integrating Eq. ( 18) with Eq. (20), we give1El (q--rlo)J Qc~~4,j=~lnl~l-We have a solution

C(a,R,?;4,)=1Bl(a.B)[ l-tanh(ll eiX4,(a,P)(17--rlo))I (23)

for0 < $(a,P,q;hJ < dAa,P).

Since (l/2) I( --p/6)12 1 m(a,/3) has to be real in Eq.(23)) it is required that p < 0. The condition p < 0 satisfiesEq. (21a) because l > 0. These results are not inconsistentwith the assumptions. We therefore regard Eq. (23) as acompressive ype of relativistic double layer. The doublelayer is illustrated in Fig. 1. The dependencyof the poten-

25F

I 1 I0.01 0.02 0.03 0.04 0.05 cu

16

FIG. 3. The double-layer thickness as a function of the ion temperatureratio a, for the relativistic effect u,,/c=O.2.YasunoriNejoh 2835


103.27.8.43 On: Thu, 29 May 2014 12:29:08


8/12

65it I------

f

FIG. 4. The profile of the explosive solution (24) with u,,/c=O.2 as afunction of 71, or the ion temperature ratio 0=0.02 and 0.04, marked oneach curve.

tial d rop A+ on the ion temperature ratio ~7s shown n Fig.2. We also show the double-layer thickness L in Fig. 3.For the condition ~(MbphJ > AA@) orCp(cr,P,rl;&J~0, we obtain, from Eq. (18) with Eq. (19),

Equation (24) implies that the potential infinitely grows at7j-rrno, and thereby we regard JSq. (24) as an explosivesolution. We show the profile of Eq. (24) in Fig. 4.B. Relativistic spiky solitary waves and explosivesolutions

We consider the case where Y (c@,u,~) has one dou-ble root as shown in Fig. 5. Then Y (a&,4) of Eq. ( 18)requires~(Q,w$) = WW2W-d,H~-bd,

with(25)

~o,b(a,~,J,u)=(2/P)(-a* &5m, (26)where (b, and & correspond to the positive and negativesign of the right-hand side of Eq. (26)) respectively. Here,when we put u = - a2/4@ > 0, +a and I$~ ake the form of

Q1,= a@>O, Wd

FIG. 5. The profile of the potential function P (a,&&) of Eq. (25) inthe case where V(a,&u,+) has one double root.

&=-3a/&O, (2717)provided that 4, < &.Thus one expects hat Eq. ( 18) with Eq (25) has thefollowing solution:

0 < Q, &: solitary wave solutions,(pCO: explosive type of solutions,becauseY ( CZ,&Q#B)O.These casesare interesting.We integrate Eq. ( 18) with Eq. (25) as

Equation (18) with Eq. (25) has a solution

I$08f . ..^ . . .._.._..-. _-_ ___I

070605 :

\s 0403i0.21 ~

i b?-- ?aFIG. 6. The profile of the relativistic spiky solitary wave representedbyEq. (301, in the case where vdc=O.Z, u=O.O3 and 4,=0.75.

2836 Phys. Fluids B, Vol. 4, No. 9, September 1992 Yasunori Nejoh 2836s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded

IP: 103.27.8.43 On: Thu, 29 May 2014 12:29:08


9/12

I J-l$ (q-q,)=.d(a,P;q$,J -arctanhfor

and

for

whered(a,B;&Jarctanh[(4,-Js)],and

~(a,B;B,)=a~~~o~h[g& (&-~~~)],respectively. Here 4, denotes the peak amplitude of the potential.From Eqs. (28) and (29), we obtain

3o sech2td(a,P;&J - la J-17281 (rl-rldl~(a,8,~m)=-p1+\/3tanh[~(ap~ )-]~~3-1/2/?](~-77~)]J m

for

and3a cosech2t~(a,P;~,)+Ia~~I(rl--o)]

4(a,B,h)=~ l+ticoth[B(a,@+ )+]a\l-1/2fi](~-~o)] mfor

Im-Jzpzgq>y.

(28)

(29)

(30)

(31)

Equations (30) and (3 1) are a spiky solitary wave solutionand an explosive type of solution, respectively. Whenue/c=0.2 and 0=0.03, the maximum potential 4, is 0.75.In this case, we illustrate the spiky solitary wave in Fig. 6,and the dependency of the amplitude of the solitary waveon the ion temperature effect for vc/c=0.2 in Fig. 7. Weshow the explosive solution in Fig. 8, by using the sameparameter.IV. RELATIVISTIC ION-ACOUSTIC OSCILLATIONMODES IN A SMALL WAVE-NUMBER REGION

rl=p(C--p7), (334g=p27, (33b)

where p is the small expanding parameter; S and p aredetermined later.We seek the relativistic ion-acoustic oscillation modes To the first order in p, we obtain the dispersion rela-of the mixed MKdV equation (17) in the small wave- tion2837 Phys. Fluids l3, Vol. 4, No. 9, September 1992

I

number region. We apply the expansion$= f, pn ,=fj ~ *(I1,S)exp[i l(k~--S7)1,co (32)

and the variable transformation

Yasunori Nejoh 2837s article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded

IP: 103.27.8.43 On: Thu, 29 May 2014 12:29:08


10/12

FIG. 7. Dependencyof the amplitude (b,,, f the spiky solitary wave on theion temperature ratio B for the relativistic effect u,,/c=O.2.

ij= +,for componentswith I= f 1 and have

~j=o,for Ilffl.To the second order in cr. he terms I= I1 require

p= -Sk2Jand the term I= 2 requires

?O-60-50-

-s 40-30-20-

FIG. 8. The profile of the explosive solution representedby Eq. ( 3 1) inthe casewhere v,Jc=O.2, u=O.O3, and (p,=O.lS.

#= (a/3k2)($:)2.To the third order in p the term l=O requires(g2 = -(2a/3k21(I#j)/-Cl,

where C is an integration constant independent of q. Fi-nally, to the third order in p the term I= 1 implies thenonlinear Schriidinger equation,dfp 3 a2#)

iT--zkF+-$ a2c#)=o, (34)

where the last term of the left-hand side of Eq. (34) van-ishes by the appropriate variable transformation. If /3< 0,then the modulationally stable relativistic ion-acousticwave can propagate n the small wave-number region,The oscillatory wave solution is obtained asq&(x,0#I p[X-(S-~)+u2~~

Xexp[i[kx-k(s--&k)t]}, (35)where the ordering parameter E associatedwith the wavenumber k is taken to be 1. The phase velocity s is definedby Eq. (6). The frequency part in the phase actor of Eq.(35) agreesprecisely with the dispersion relationti=k[ uo+ Jy:,:e-f (l+~a~~c)

3WoY3 -+-Y2X -$2YOYI Yl

1 1-- 2 E'YoYISS-~~ioYon(1+v~c2)l/(y,--fy2s) k2 ,Iof this system n the small wave-number egion. Hence, Eq.(35) properly expresses he relativistic ion oscillationmode in this system.V. DISCUSSION

We have derived the mixed MKdV equation associatedwith the fluid model and have shown the new nonlinearwave modes, hat is, the relativistic double layer, the spikysolitary wave, and the explosive solutions. It should beemphasized hat, only in the case of fi ~0, the MKdVequation gives rise to the double layer, the spiky solitarywave, and the explosive solutions. We note that the qua-dratic nonlinear coefficienta of the mixed MKdV equationis positive,We have derived, in Sec. III A, the relativistic com-pressivedouble layer and the explosive solution. We haveshown the characteristic feature that the amplitude of thedouble layer grows as the relativistic effect and the ion

2838 Phys. Fluids B, Vol. 4, No. 9, September 1992 Yasunori Nejoh 2838article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to

103.27.8.43 On: Thu, 29 May 2014 12:29:08


11/12

temperature effect increase, and its thickness narrows asboth the effects increase. Moreover the potential drop ofthe double layer grows as the ion temperature increases.These features are shown for the first time for the relativ-istic double layer. The double layer is confined within thespecified region of the positive potential, that is, 0 < 4 < 4,and does not exist in the region of 4 > 4, in Eq. (20). Onthe other hand, the explosive solution associated with thedouble layer is confined within the region of 4 > 4, and theregion of 0 > 4. As a result of this, we can understand thephysical process that both the double layer and the explo-sive mode complementarily generate and disappear.The relativistic spiky solitary wave is confined withinthe region of 0 < C# &,, in Fig. 6 of the Sec. II B, while theexplosive mode associated with this solitary wave existswithin the region of 0 > 4. It is clarified that the amplitudesof the spiky solitary wave and the explosive mode grow asthe ion temperature effect increases. Both modes also existcomplementarily: if the spiky solitary wave exists, the ex-plosive mode disappears. On the contrary, when the explo-sive mode exists, the spiky solitary wave disappears. This isproved by the boundary condition concerning the existenceof both the modes.Many investigators have considered1.7*18hat doublelayers damp and finally disappear when the bell-shapedsolitary wave of the MKdV equation forms. As is seen nspace observations and computer simulations, however,most of the solitary waves which contribute to the gener-ation and the disappearance of double layers are not bellshaped but spiky shaped. We think that the formation ofthe double layer is essentially associated with strong non-linearity and is a nonlinear wave mode in the system gov-erned by the higher-order nonlinearity and dispersiveness.Hence, the solitary waves that contribute to the generationand the disappearance between the solitary wave and thedouble layer need to be spiky shaped and need to possessthe same order of nonlinearity as the order of nonlineari tyof the double layer. We should identify the order of thenonlinearity of the present spiky solitary wave with that ofthe double layer. In addition, since the spiky solitary wavetakes away the energy of the double layer, the solitary wavetakes the place of the double layer. For this reason, thedouble layer which occurs in space disappears when thespiky solitary wave forms. Thus these two modes occurand disappear.Explosive (bursting) phenomena have been discussedas the releasing phenomenon of the energy of the magneticfield by the circuit model for the local system in space,20and investigated as a phenomenon which occurs by thecompression associated with ion beams for space particlesystems.2 In such models, however, they can hardly dis-cuss the energetic nonlinear wave modes and cannot applythem to the actual situations. We present the new explosivemodes associated with the relativistic effect by the fluidmodel. In Ref. 22, Ostriker ef al. treated nonrelativisticexplosive waves and did not at all consider energetic phe-nomena concerning higher-order nonlinear waves. Za-kharov derived the solitary wave and the explosive modewith regard to the collapse of Langmuir waves.23 ts explo-

sive mode is associated with the negative potential. Al-though the results of the present solitary wave and theexplosive mode cannot be compared directly with those ofRef. 23 because we consider the relativistic ion-acousticwave, it is very interesting in that the present explosivemode is analogous to the explosive mode of Ref. 23.Three kinds of nonlinear wave modes presented herehave been independently studied and several examples foreach mode have been reported in space observations.4920Among these studies, the fine structure composed of spikysolitary waves and double layers is observed by satellitestraveling in interplanetary space and it attracts the atten-tion of many researchers.228 Since the particle energyrange in these papers lies in a comparatively low energyrange, we suppose that energetic explosive events such asbursts have not been detected. Recently, the existence ofion beams, such as the streaming ion flux, is worthy ofnotice as a condition of the formation of spiky solitarywaves and double layers.25*26 his is an idea associatedwith the thought that the high-speed streaming ions formsolitary waves and double layers. The present investigationis associated with the point of v iew that not only spikysolitary waves and double layers but also explosive eventsare generated by the high-speed streaming ions.Very little is known about the papers referring to therelation between the nonlinear wave modes and to theirgeneration and disappearance processes. Furthermore,there are hardly any papers that have investigated the gen-eration and the evolution of the nonlinear wave modesassociated with the nonlinear wave structure. Under thesecircumstances, this investigation presents the peculiar fea-ture concerning the evolution process of the nonlinearwave structure in which the relativistic double layer, thespiky solitary wave, and the explosive modes form the finestructure by complementarily generating and disappearingin space.ACKNOWLEDGMENTS

The author would like to thank Professor H. Tagashiraof Hokkaido University and Professor H. Sanuki of theNational Institute for Fusion Science for their valuablesuggestions.The author also thanks the referees or helpfulcomments.S. Torven, Phys. Rev. Lett. 47, 1053 (1981).H. Schamel and S. Bujarbarua, Phys. Fluids 26, 190 ( 1983).3J. S. Wagner, T. Tajima, J. R. Kan, J.-N. Leboeuf, S.-I. Akasofu, and J.Dawson, Phys. Rev. Lett. 45, 803 (1980).M. Temerin, K. Cemy, W. Lotko, and S. F. Moser, Phys. Rev. Lett. 48,1176 (1982).S. Torven, J. Phys. D: Appl. Phys. 15, 1943 (1982).6N. Sato, M. Nakamura, and R. Hatakeyama, Phys. Rev. Lett. 57, 1227(1986).P. K. Shukla, M. Y. Yu, and N. L. Tsintsade, Phys. Fluids 27, 327(1984).5. Arons, Space Sci. Rev. 24, 417 (1979).9D. F. Smith and S. H. Brecht, J. Geophys. Res. 90, 205 (1985).OR. P. Lin, W. K. Levedahl, D. A. Gurnett, and F. L. Scarf, Astrophys.J. 308, 954 (1986).L R Lyons and D. J. Williams, in Quantitative Aspects of Magneto-.spheric Physics (Reidel, Dordrecht, 1984), Chaps. 2 and 5.Y. Nejoh, Phys. Lett. A 123, 245 (1987).

2839 Phys. Fluids B, Vol. 4, No. 9, September 1992 Vasunori Nejoh 2839article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to

103.27.8.43 On: Thu, 29 May 2014 12:29:08


12/12

13Y,Nejoh, J. Plasma Phys. 37, 487 (1987).t4Y. Neioh, Phys. Fluids 31, 2914 (1988). J. Ostriker and C. F. McKee, Rev. Mod. Phys. 60, 1 (1988).z3V. E. Zakharov. Sov. Phvs. USD. 31. 672 (19881.Y. Nejoh, J. Phys. A: Math. Gen. 23, 1973 (1990). . . . I6Y. Nejoh, IEEE Trans. Plasma Sci. PS-20, 80 ( 1992). 24F. S. Moser and M. Temerin, in High Latitude Space Plasma Physics,H. H. Kuehl and K. Imen, I EEE Trans. Plas ma Sci. PS-13, 37 ( 1985). edited by El. Hultqvist and T. Hagfors (Plenum, New York, 1983),*G. Chanteur and M. Raadu, Phys. Fluids 30, 2708 ( 1987). p. 453.T. Sato and H. Okuda, J. Geophys. Res. 86, 3357 (1981). R Bostrlim, B. Holback, G. Holmgren, and H. Koskinen, Phys. Ser. 39,782 (1989).*OH. Alfvbn, Phys. Ser. TZ/l, 10 (1982).2tP. Carlqvist, IEEE Trans. Plasma Sci. PS-14, 794 ( 1986). *R. Bostrom, G. Gustafson, B. Holback, G. Holmgren, and H. Koski-nen, Phys. Rev. Lett. 61, 82 ( 1988).

2840 Phys. Fluids B, Vol. 4, No. 9, September 1992 Yasunori Nejoh 2840s article is copyrighted as indicated in the article Reuse of AIP content is subject to the terms at: http://scitationnew aip org/termsconditions Downloaded

Documents

Double Layers, Spiky Solitary Waves, And Explosive Modes of Relativistic Ion