Arbitrary amplitude dust ion acoustic solitary waves and double layers in a plasma with non-thermal electrons

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1 Inan Journal of Pure & Apple Physcs Vol. 50, February 01, pp Arbtrary ampltue ust on acoustc soltary waves an ouble layers n a plasma wth non-thermal electrons Runmon Gogo a *, Rajkumar Roychouhury b & anoranjan Khan c a Department of Instrumentaton Scence & Centre for Plasma Stues, Jaavpur Unversty, Kolkata , Ina * Present aress: Assam own town Unversty, Pankhat, Guwahat , Assam, Ina b Physcs an Apple athematcal Unversty, Inan Statstcal Insttute, Kolkata , Ina E-mal: a runmon@gmal.com, b rajaju@reffmal.com c mkhan_ju@yahoo.com Receve September 011; accepte 15 November 011 Sageev s pseuo potental metho s employe to stuy ust on-acoustc soltary waves n an unmagnetze plasma wth non-thermal electrons. An exact analytcal expresson s obtane for the pseuo potental. The range of parameters for the exstence of soltary waves an ouble layers usng the analytcal expresson of the Sageev potental has been foun. It s observe that, epenng on the values of the plasma parameters lke on temperature σ, non-thermal electron parameter β an the value of the ust gran charge µz, both rarefactve an compressve soltary waves may exst. Crtcal values of these parameters, beyon whch soltary waves woul cease to exst, are obtane for some partcular cases. Double layer solutons are also obtane for certan parametrc values. Exact numercal results are obtane for arbtrary ampltue soltons an ouble layers. Keywors: Non-thermal electron, Sageev potental, Soltary wave, Double layers 1 Introucton Dusty plasma attracte the attenton of many researchers ue to ts ubqutous nature. It s present n cometary tals, astero zones, planetary rngs, nterstellar meum, the lower part of the Earth s onosphere an magnetosphere 1-3. Occurrence of the ust n laboratory plasmas lke flames, plasmas n fuson evces, plasmas use n nustral laboratores etc. also makes ths object extremely popular. It has been observe that the presence of massve ust grans n plasmas gves rse to new egenmoes such as the ust acoustc moe, ust on acoustc (DIA) moe, ust lower hybr moe, ust rft moe, ust Bernsten-Green-Kruskal moes etc Out of them, the ust on-acoustc wave moe has attracte broa attenton together wth expermental confrmaton n several low temperature usty plasma evces 7. In the last 0 years, ust acoustc soltary wave an ust on acoustc soltary waves have been stue by several researchers both expermentally 8-10 an theoretcally Verheest et al 14. apple the flu ynamc approach to nvestgate the exstng contons for postve an negatve potental soltons n DIA soltons an ust-acoustc soltons. agnetc effects n usty plasmas have been of partcular nterest. atra an Roychouhury 15 stue the propertes of non-lnear on acoustc waves n a usty plasma subjecte to an external magnetc fel usng the Sageev 16 potental metho. Varous stues clearly ncate the presence of energetc electrons n a varety of astrophyscal plasma envronments an measurements of ther strbuton functons an shown to be hghly nonthermal 17. The non-thermal electron an on 18,19 strbutons are turnng out to be a characterstc feature of space plasmas, whereas coherent non-lnear waves an structures lke soltons, shocks, ouble layers (DL), vortces, etc., are expecte to play an mportant role. Recently, non-thermal strbuton of electrons has been observe n space plasmas by the Vkng satellte 0 an Freja satellte 1. Non-thermal electrons are consere by numerous researchers -4 n varous stuatons to observe fferent phenomena n plasma. Carns et al. 5 suggeste that the presence of non-thermal strbuton of electrons may change the on acoustc soltary structures. Conserng the non-thermal strbuton of electrons, Sngh an Lakhna 6 have stue electron acoustc soltary waves an observe that ncluson of non-thermal electrons prects negatvely charge potental structures. Conserng an external magnetc fel, Saha an Chatterjee have stue the DIA soltary waves wth the help of Sageev s pseuopotental metho wth an expanson up to Φ 4. They nvestgate

2 GOGOI et al.: ARBITRARY APLITUDE DUST ION ACOUSTIC SOLITARY WAVES 111 a range of parameter for whch soltary waves exst. Takng both statc an ynamc ust grans, Boltzmann electrons, ynamcs of col on, Bharuthram an Shukla 7 stue large ampltue on-acoustc soltons n an unmagnetze plasma, usng the Sageev potental metho. They have examne the effect of ust partcles on the ampltue an ach number of the soltons, n partcular. Usng the Sageev potental metho, arbtrary ampltue ust-acoustc ouble layers n a non-thermal plasma was observe by aharaj et al 8. Very recently, Berbr an Trbeche 9 have consere non-thermal electrons to nvestgate weakly non-lnear DIA shock waves an entfy the contons that favour ther exstence. They have apple Reuctve Perturbaton Technque (RPT) to erve the mofe Korteweg-e Vres (mkv) equaton, whch s sutable only for small ampltue soltary waves. It seems that many problems n plasmas are taken care of by perturbaton technque 30. However, hgher orer approxmatons cannot be neglecte n case of the large ampltue waves specally ouble layers an perturbaton technque s not aequate to stuy such waves. Sageev 16 was the frst to use non-perturbatve approach to stuy soltary waves n plasmas n Exact soluton of the fferental equatons escrbng non-lnear waves can be obtane through a stanar metho known as the Sageev s Pseuopotental metho. To stuy large ampltue soltary waves or ouble layers 31,3, the Sageev s Pseuo potental metho s a sutable metho. Usng Sageev s technque, t s foun that n nature both the rarefactve an compressve type soltons are possble. Also, ths formulaton prects the exstence of ouble layers, conserng the nonthermal electrons usng non-perturbatve approach. It may be note that n some cases, the Sageev s potental cannot be obtane n close form. But even then one can obtan, n prncple, the pseuo potental up to any orer n Φ, the electrc potental. However, n the present paper, exact analytcal expresson s obtane for the Sageev s potental. Also exact numercal results were obtane for arbtrary ampltue soltary waves an ouble layers. Our assumpton for constant ust charge s base on the followng logc. In the case of small but fnte on to electron temperature rato, the sspatve tme scale 33 s (λ D /a) (T /T e ) ω p 1, where λ D s the on Debye length, ω p s the on plasma frequency an a s the gran raus. If the characterstc tme scale for ynamcal processes (that for the DIA solton s ω p 1 ) s much larger than τ (the anomalous sspaton tme scale), the ust partcle chargng process can be consere to take place promptly. Uner ths conton, ω p τ <<1, the ust charge, wth a goo accuracy, coul be assume constant. The expermental observatons of the Freja satellte 1 have brought to notce the fact that ouble layers 34 are actually soltary structures wth ensty epleton n plasmas where the electrons are nonthermal. It must be mentone here that to stuy ouble layers small ampltue approxmaton woul be naequate. So here we conser an analyss of the non-lnear structure n an unmagnetze usty plasma wth warm ons an non-thermal electrons, wthout any approxmaton. Snce ust partcles are present n space an foun n the vcnty of artfcal satelltes, space statons etc.,35, the fnngs reporte here on the stuy of non-lnear plasma wave structures s lkely to be useful n explanng the expermental ata. Basc Equatons an Dervaton of the Sageev Potental Usually, the ust partcles acqure a negatve charge n many space an laboratory plasmas. Uner specfc contons, the overall charge on the ust partcles can also become postve. For example, photoelectrc emsson from ust grans n the presence of UV lght can result n ust partcles acqurng a postve charge. Here, we have consere non-thermal electrons, nertal warm ons an mmoble negatve charge ust grans n an unmagnetze usty plasma. We assume that all the grans have the same charge equal to q =-ez, wth z negatvely charge ust an e s the elementary charge. Thus, after normalzaton, the basc equatons for one mensonal DIA waves can be wrtten as follows (assumng ons to be aabatc): For the ons: N t + ( NV) = 0 x V V P V σ + = t x x N x The pressure law s: P N γ, For aabatc ons, γ = 3 whch mples P = N Thus, gves Eq. () as: 3 (1) () (3)

3 11 INDIAN J PURE & APPL PHYS, VOL 50, FEBRUARY 01 V + V V = 3σ N N t x x x (4) For the electrons: e = e / e0 = (1 Φ + Φ )exp( Φ) N n n β β (5) where β=4α/(1+3α) an α s a parameter etermnng the rato of non-thermal electron present n our plasma moel. The parameter β cannot excee a certan range (0 β 1.3), beyon whch α s negatve whch s not acceptable. When β 0, Eq. (5) gves the Boltzmann strbuton of electrons. It may be mentone that the strbuton gven n Eq. (5) s use by Carns et al 5. However, t shoul be kept n mn that t s one of the possble non-thermal strbutons. We close the set of Eqs (1-3) wth the Posson s equaton: Φ = (1 µ Z ) N e N + µ Z x Charge neutralty at equlbrum requres: e0 0 (6) µ Z = 1 n / n (7) We have normalze the on ensty N by n 0 (equlbrum on ensty) an ust enstes n by n 0 (equlbrum ust ensty), the on flu velocty V by C s = (T e /m ) 1/, tme t by nverse of on plasma frequency ω p -1 = (m /4πn 0 e ) 1/, the space varable x by a Debye length λ = (T e /4πn 0 e ) 1/, the electrostatc potentals Φ by T e /e. Here σ=t /T e, µ=n 0 /n 0, µz s the fracton of the negatve charge n the plasma whch reses on the ust grans. For travellng wave soluton, we take the transformaton ξ = x t an get the followng expressons, after ntegraton, from Eq.(1): V =, or V N = N (8) From Eq. (), we get: Φ = (9) σ σ N N Solvng Eq. (8), we can easly get N as explct functon of φ an s gven by: N = ( ) Φ + 3σ Φ + 3σ 1σ Next our job s to fn For that, we substtute: Φ N 0 6σ Φ Φ + 3σ = 1σ coshθ (10) whch, after some algebra, gves us the followng result: ϕ 0 1 N Φ = 3 6σ Φ + 3σ ( σ ) Φ + 3 1σ ( Φ + 3σ ) ( Φ + 3 σ ) σ + + 1σ (11) Durng ntegraton, we have use the bounary contons N 1, V, Φ 0 as ξ ±, Eq. (6) can be obtane n the form: ( β β β ) 1 Φ Φ + Φ exp( Φ) = (1 µ Z) ξ (1 + 3 β ) N Φ + µ Z Φ 0 Φ (1) Usng Eq. (11) n Eq. (1), Eq. (1) can be wrtten n the form: 1 Φ + V ( Φ ) = 0 ξ where the Sageev potental V(Φ) s gven by: (13) (1 + 3 β ) V ( Φ ) = (1 µ Z ) ( 1+ 3β 3βΦ + βφ ) exp( Φ) µ Z Φ 1 Φ + 3σ 3 6σ ( Φ + 3σ ) 1σ { ( 3 σ ) ( 3σ ) 1σ } Φ + Φ + σ + + (14)

4 GOGOI et al.: ARBITRARY APLITUDE DUST ION ACOUSTIC SOLITARY WAVES 113 Eq. (14) s an exact analytcal expresson for the Sageev potental wthout any approxmaton. It can be seen from Eq. (14) that wth β = 0, n the lmt σ 0, Eq. (14) reuces to: { } V ( Φ ) = (1 µ Z ) 1 exp( Φ) µ Z Φ + Φ (15) Ths s n agreement wth the result obtane by Bharuthram an Shukla 7 prove we wrte the symbols as 1 µz =N e, whch mples µz =1 N e =N. The analytcal soluton of Eq. (13) can be obtane n the small but fnte ampltue lmt, where one can neglect terms of O(Φ 5 ) on the rght han se of Eq. (1). Next we shall scuss the contons for exstence of soltary waves an ouble layers. From Eq. (13), t s clear that V(Φ) must be negatve to get real soluton. The contons for soltary waves are: () V(Φ) = 0 at Φ = 0 an Φ = Φ m ; () V ( Φ) V ( Φ) = 0 an < 0 Φ = 0 Φ = 0 () V(Φ) <0 for Φ lyng between 0 an Φ m,.e. for Φ m < Φ <0, rarefactve soltary waves exst an for 0<Φ <Φ m compressve soltary waves exst. Here Φ m s the ampltue of the soltary wave. For exstence of ouble layers, the contons are : () V(Φ) = 0 at Φ = 0 an Φ = Φ m ; () V ( Φ) V ( Φ) = 0 an = 0 V ( Φ) () < 0 Φ = 0 Φ = Φ m Φ = Φ m In the present work, to obtan exact numercal solutons for the formaton of soltary waves an ouble layers, we have chosen parameters n such a way that the nequalty gven n Eq. (16) s satsfe..1 Arbtrary ampltue soltary waves an ouble layers In our present plasma moel wth warm on ynamcs, statc ust an non-thermal electrons, we have observe both rarefactve an compressve soltary waves as observe by Bharuthram an Shukla 7 wth both statc an ynamc ust, Boltzmann electrons an col on ynamcs. In aton, we have also nvestgate the formaton of arbtrary ampltue rarefactve ouble layers. The exstence of large ampltue soltary waves can be etermne by plottng equaton V(Φ) aganst Φ for fferent values of parameters. Durng our search for exstence of soltary waves an ouble layers, we choose the parameter ranges keepng n mn expermental as well as space an astrophyscal stuatons 7,9. From the charge neutralty conton gven n Eq. (7), t s clear that µz <1. Range of β cannot excee a certan value (0 β 1.3). In ths work, the on to electron temperature rato σ s taken as 0.15< σ <0.18 because no soltons or ouble layer are foun to exst except ths range. Whle explorng for soltary wave or ouble layer exstence, we woul keep three parameters fxe an vary one parameter to stuy the effect of each nvual parameter. The non-thermal electron parameter β plays an mportant role n the formaton of soltary waves n a plasma, as can be seen n Fg. 1, where we fxe parameters, σ, µz an vare β. Here compressve soltons are foun an the ampltue of the soltary wave ecreases as β ncreases. Ampltue ecreases n such a way that beyon a crtcal β, soltary wave Usng solton contons () an (), we can get an analytcal conton to fn a range of the ach number for the exstence of soltary waves an ouble layers as well. It can be easly checke that the contons V(Φ) = 0 an V (Φ) = 0 at Φ = 0 are satsfe. The conton V (Φ) < 0 at Φ = 0, gves rse to the conton: > 3σ + (1 µ Z )(1 β ) 1 (16) Fg. 1 Plot of V(Φ) versus Φ showng the formaton of both rarefactve soltons for µz = 0.5, = 1.9, σ = an fferent values of β = 0(----), β = 0.1(...)

5 114 INDIAN J PURE & APPL PHYS, VOL 50, FEBRUARY 01 solutons cease to exst. For example, for the parameter set of Fg. 1, soltary wave exsts up to β = 0.34 (curve not nclue n Fgure) an above that t oes not exst. Wth β lyng between 0.31 to 0.34, an nterestng stuaton s observe whch s explane n Fg. 3. The ashe curve represents a soltary wave formaton for β = 0, whch mples Boltzmannan electrons. Wth non-thermal electron an warm ons, the fference we have notce s that ampltue of the soltary wave s smaller than that of Boltzmann electrons. Fgure shows the co-exstence of both rarefactve an compressve soltary waves for µz = 0.5, = 1.75, σ = , an fferent values of β = 0 (----), β = 0.05 ( ). For ths parameter set, t s very clear from Fg. that ampltues of rarefactve soltary waves are much greater than the ampltues of compressve soltary waves as mentone by Bharuthram an Shukla 7 also. As β ncreases Fg. Plot of V(Φ) versus Φ showng the formaton of both rarefactve an compressve soltons for µz = 0.5, = 1.75, σ = , an fferent values of β = 0 (----),β = 0.05( ) ampltues of both rarefactve an compressve soltary waves ecrease. Snce compressve soltary waves are weaker they lost ther exstence faster. But the case s completely fferent for the parameter set µz = 0.5, = 1.9, σ = an fferent values of β = 0.31 (----), β = 0.33 ( ), β = 0.38 (...) as shown n Fg. 3. Here, for β = 0.31 only rarefactve soltary wave exst an compressve soltary wave tens to exst. As β s slghtly ncrease, compressve soltary wave exsts as well. So the curve wth β = 0.33 shows the exstence of both rarefactve an compressve soltary wave. Contrary to Fg., now t s notce that ampltue of rarefactve soltary wave s smaller than the ampltue of compressve soltary waves. Here also as β ncreases ampltues of both rarefactve an compressve soltary waves ecrease. But as rarefactve soltary waves are weaker they lost ther exstence faster at some value of β, only compressve solton exst an above β = 0.38 nether rarefactve nor compressve solton exst. Thus, as non-thermal electron parameter β an ach number ncrease, compressve soltary waves become stronger than rarefactve soltons. Fgure 4 shows the plot of V(Φ) aganst Φ for =.5, β = 0., σ = 0.17 an µz = 0.64, (----), µz = 0.7 ( ). Dashe curve shows the formaton of rarefactve soltary wave. Ampltue of the rarefactve soltary wave ecreases as µz ncreases. Whle observng for the exstence of non-lnear structures, we have notce that beyon a crtcal µz they cease to exst. The sol curve of Fg. 4 for µz = 0.7 oes not gve a soltary wave soluton as the curve goes slghtly above the Φ axs at Φ = 0. Thus, for the exstence of soltary wave, µz <0.65. In a smlar manner, we have fxe Fg. 3 Plot of V(Φ) versus Φ showng the formaton of nonlnear structures for µz = 0.5, = 1.9, σ = an fferent values of β = 0.31 (----), β = 0.33 ( ), β = 0.38 (...) Fg. 4 Plot of V(Φ) versus Φ showng the formaton of rarefactve soltons for =.5, β = 0., σ = 0.17 an fferent values of µz = 0.64 (----), µz = 0.7 ( )

6 GOGOI et al.: ARBITRARY APLITUDE DUST ION ACOUSTIC SOLITARY WAVES 115 Fg. 5 Plot of V(Φ) versus Φ showng the formaton of nonlnear structures for β = 0., σ = , µz = 0.5 an fferent values of = 1.8 ( ), = 1.83 (----), = 1.87 (...) Fg. 6 Plot of V(Φ) versus Φ showng the formaton of nonlnear structures for =.635, σ = 0.167, µz = 0.4 an fferent values of β = 0.59 (----), β = 0.6 ( ), β = 0.61 (...) parameters β, σ, µz an vare the ach number, as can be seen n Fg. 5 from whch we have foun that ampltue of the rarefactve soltary wave ncreases as ncreases. Also, t s notce that rarefactve an compressve soltary wave coexst for 1.74 < < Above = 1.86, only rarefactve soltary wave exsts. For ths parameter set, soltary wave oes not exst when < Fgure 6 shows fferent non-lnear structures for fxe =.635, σ = 0.167, µz = 0.4 an fferent values of β. Nature of the curves changes as soon as the values of β ncreases. For example, for a value of β = 0.59, the ashe curve nether represents a soltary wave nor a ouble layer. Sol curve shows the exstence of rarefactve ouble layers for β = 0.6 an otte curve shows the exstence of rarefactve soltary wave for β = It can be observe that ampltue of the soltary wave s smaller as compare to ampltue of ouble layers. Therefore, there woul Fg. 7 Plot of V(Φ) versus Φ showng the formaton of ouble layers for β = 0.6, σ = (a) µz = 0.6, =3.3 (----); (b) µz = 0.7, =3.875 ( ) be a crtcal value of β beyon whch these non-lnear structures wll not exst. As observe by numerous researchers before, we also observe that DL structures are very senstve to the ensty of nonthermal electrons. From Fg. 7, we have observe that as we ncrease µz, t also emans the ncrease of ach number for the exstence of a ouble layer, whch can be easly unerstoo from the parameters β = 0.6, σ = that we fxe for both the curves wth (a) µz = 0.6, =3.3 (----) an (b) µz = 0.7, =3.875 ( ). For the sol curve as we ncrease the value of µz, we nee to ncrease the value of to get a DL soluton. Fnally, nvestgaton reporte here may be useful to stuy the exstence of non-lnear structures, where non-thermal electron populaton were observe, such as space an astrophyscal plasmas. 3 Conclusons A pseuo potental approach has been apple to stuy the arbtrary ampltue ust on-acoustc soltary waves an ouble layers n plasma wth nonthermal electrons an warm ons. An analytcal ervaton of pseuo potental s obtane n terms of Φ wthout any approxmaton, whch s val for arbtrary ampltue soltons. The analytcal soluton of Eq. (13) can be obtane n terms of small but fnte ampltue lmt. To obtan the soluton one can expan the pseuo potental up to any esre orer of Φ, so that hgher orer non-lneartes can be taken nto account. It has been observe that the nonthermal electron, on temperature, ust gran charge play sgnfcant role n fnng the exstence of nonlnear structures.

7 116 INDIAN J PURE & APPL PHYS, VOL 50, FEBRUARY 01 Acknowlegement One of the authors RG acknowlege the CSIR, Ina for provng fnancal support FILE No. 9/96(0579)K9-ER-I. Thanks are also ue to Prof R Gupta, Centre for Plasma Stues, Faculty of Scence, Jaavpur Unversty, Jaavpur an S Ghosh, Calcutta Unversty, Kolkata for ther help an suggestons. References 1 Whpple E C, Northrop T G & ens D A, J Geophys Res, 90 (1985) Goertz C K, Rev Geophys, 7 (1989) ens D A, Plasma Sources Sc Technol, 11 (00) A19. 4 Rao N N, Shukla P K & Yu Y, Planet Space Sc, 38(4) (1990) Shukla P K & Sln V P, Physca Scrpta, 45 (199) Trbeche, Ham R & Zergun T H, Phys Plasmas, 7 (000) erlno R L, Barkan A, Thompson C & Angelo N D, Phys Plasmas 5, 1607 (1998); Plasma Phys Controlle Fuson, 39 (1997) A41. 8 Barkan A, Angelo N D & erlno R L, Planet Space Sc, 44 (1996) Nakamura Y & Sharma A, Phys Plasmas, 8(9) (001) Nakamura Y, Balung H & Shukla P K, Phys Rev Lett, 83 (1999) Roychouhury R & atra S, Phys Plasmas, 9 (00) Forlan A, e Angels U & Tsytovch V, Physca Scrpta, 45 (199) Gogo R, Roychouhury R & Khan, Inan J Pure & Appl Phys, 49 (011) Verheest F, Cattaert T & Hellberg A, Phys Plasmas, 1 (005) atra S & Roychouhury R, Phys Plasmas, 13 (006) Sageev R Z, Revews of Plasma Physcs, New York Consultant Bureau, 4 (1966) Golman V, Oppenhem & Newman D L, Nonlnear Processes Geophys, 6 (1999) Ghosh S, Bharuthram R, Khan & Gupta R, Phys Plasmas, 11 (004) Roychouhury R, J Plasmas Phys, 67 (00) Bostrom R, IEEE Trans Plasma Sc, 0 (199) Dovner P O, Erksson A I, Bostrom R & Holback B, Geophys Res Lett, 1 (1994) 187. Saha T & Chatterjee P, Phys Plasmas,16 (009) Chatterjee P, onal G, Roy K, unany S V, Yap S L & Wong C S, Phys Plasmas, 16 (009) Berbr A & Trbeche, Phys Plasmas, 16 (009) Carns R A, amun A A, Bngham R, Deny R, Bostrom R, Nars C R C & Shukla P K, Geophys Res Lett, (1995) Sngh S V & Lakhna G S, Non-lnear Processes Geophys, 11 (004) Bharuthram R & Shukla P K, Planet Space Sc, 40 (7) (199) aharaj et al, J Plasma Phys, 7 (006) Berbr A & Trbeche, Phys Plasmas, 16 (009) Gogo R, Dev N & Das G C, Inan J Pure & Appl Phys, 46 (008) Gogo R & Dev N, Phys Plasmas, 15 (008) Gogo R & Khan, Phys Plasmas, 17 (010) Popel S I, Golub A P, Losseva T V, Ivlev A V, Khrapak S A & orfll G, Phys Rev E, 67 (003) Temern, Cerny K, Lotko W & ozer F S, Phys Rev Lett, 48 (17) (198) Northrop T G, Phys Scr, 45 (199) 475.