Opposite Dust on Linear Dust-acoustic Waves

Transcription

1 Intrnatonal Journal of Engnrng Rsarch an Dvlopmnt -ISSN: 78-67X, p-issn: 78-8X, Volum, Issu 4 (Aprl 4), PP.53-6 Oppost Dust on Lnar Dust-acoustc Wavs Sanjt Kumar Paul Abstract:- Th lnar propagaton of th ust-acoustc wavs (L-DAWs) n a usty plasma consstng of Boltzmann-strbut lctrons an ons, mobl charg fluctuatng postv an ngatv ust charg fluctuatng statonary postv ust an charg fluctuatng statonary ngatv ust has bn thortcally nvstgat Kywors: Charg fluctuaton, lnar ust-acoustc wavs, usty plasmas, shock wavs. I. INTRODUCTION Th wav propagaton n usty plasmas has rcv much attnton n th rcnt yars bcaus of ts vtal rol n unrstanng ffrnt typs of collctv procsss n spac nvronmnts, namly, lowr an uppr msosphr, comtary tals, plantary rngs, plantary magntosphr, ntrplantary spacs, ntrstllar ma, tc. []-[6]. Th usty plasmas hav also notcabl applcatons n laboratory vcs [7]-[]. Th consraton of charg ust grans n plasmas os not only mofy th xstng plasma wav spctra []-[3], but also ntroucs a numbr of novl gnmos, such as th ust on-acoustc (DIA) wavs, th ust-acoustc (DA) wavs, th ust lowr-hybr (DLH) wavs, th ust lattc (DL) wavs, tc [4]-[8]. Most of th stus n usty plasmas hav bn confn n consrng th ust as ngatvly charg grans n aton to lctrons an postvly charg ons as th plasma spcs [4]-[6], [9]-[4]. It has bn foun that thr ar som plasma systms, partcularly n spac plasma nvronmnts, namly, comtary tals []-[3], [5], [6], uppr msosphr [7], Juptr's magntosphr [8], tc. whr postvly charg ust grans play sgnfcant rols. Thr ar bascally thr mchansms by whch th ust grans n th plasma systms mnton abov can b postvly charg. Ths mchansms ar th followng: () photo msson n th prsnc of a flux of ultravolt (UV) photons; () thrmonc msson nuc by raatv hatng; an () sconary msson of lctrons from th surfac of th ust grans. In ths papr, w hav consr usty plasma contanng mobl charg fluctuatng postv ust, charg fluctuatng statonary ngatv ust, Boltzmann-strbut lctrons an ons, an hav stu th lnar propagaton of DA wavs. Ths papr s organz as follows. Th basc quatons scrbng our usty plasma mol ar prsnt n Scton II. I hav rv th sprson rlaton n Scton III.I hav analyz numrcally th sprson proprts of th DA wav mo n Scton IV, whr I hav sn that ngatv ust-charg fluctuaton s a sourc of lnar growth nstablty. Fnally, a brf scusson s gvn n Scton V. II. GOVERNING EQUATIONS: W consr an unmagntz collsonlss usty plasma systm consstng of charg fluctuatng ngatvly charg mobl ust, charg fluctuatng statonary ngatv ust an Boltzmann-strbut lctrons an ons. Thus at qulbrum, w hav n o + zon o = n o + zon o whr n o ( n o ) s th qulbrum lctron (on) numbr nsty, n o s th ngatv ust numbr nsty, z o s th qulbrum charg stat of th postv ust componnt, z o s th qulbrum charg stat of th ngatv ust componnt. Th ynamcs of th DA wavs of such a usty plasma systm n on-mnsonal form s gvn by n ( nu ) t x () u u z u t x m x () 4 n (3) n z n z n x 53

2 j Oppost Dust on Lnar Dust-acoustc Wavs whr n s th numbr nsty of th plasma spcs j (j quals for ons, for lctrons), n ( n ) s th numbr nsty of postv (ngatv) ust. u ( u ) s th postv (ngatv) ust flu sp. z ( z ) charg stat of th postv (ngatv) ust componnt. s th lctrostatc wav potntal. Th lctron an th on nsts ar assum to follow th Boltzmann strbuton: n n no xp (4) kt B n o xp (5) kt B whr k B s th Boltzmann constant an that ust s charg by photo-msson currnt s th T s th lctron tmpratur, T s th on tmpratur. W assum P I, th thrmonc msson currnt I t an th lctron absorpton currnt I, th lctron currnt for ngatv charg ust I, th on currnt for ngatv charg ust I only. All othr chargng procsss ar nglct. Th charg stat z componnt s not constant, but vars accorng to th followng quatons: z z I I I t t z I I t P t u (6) (7) whr I z p r JY xp kbr T ph 3 mkbtp 8kT B P z z W r xp m kbr Tp kbr Tp kbtp It (9) (8) I kt 8kT B P B P z r no m kbr T p kt k B BT z 4 o xp m kbr T I r n kt k B BT z 4 o m kbr T I r n whr s th Planck s constant, T ph s th photon tmpratur, W s th work functon, J s th UV photon flux, Y s th yl of photons (typcal valus of W, J an Y ar. V, 5. 4 photonscm s, an., rspctvly), an r s th ust raus. Now,usng q z (whr th charg stat z s th numbr of lctrons rsng on th ust gran spac). Introucng th followng normalz varabls: 54 () () ()

3 Oppost Dust on Lnar Dust-acoustc Wavs N = n n o, p + + U = u C, Φ=φkBT, =z zo = z z, + + T= t ω, λ = k T 4πz n +, D B o o On can ruc quaton () to (7) as = z z o, X= xλ D, + + C = zokbt m an p o o ω = 4πz n m. N T X U U U T X X (4) N X (5) NU (3) T T U P Q R (6) X X X (7) Whr z zo r k T, T T, B ph X 4 r o p zo r k T r z, B p o, kt B no m 3 mkbtp 8kT B P Q m, p X W kt B P z n z n o o o o, zo r k T, B kt B no m n, o zono, P = JY, 8kT B p, R no m zo r k T. B n, o zono III. DERIVATION OF THE LINEAR DISPERSION RELATION: To rv a ynamcal quaton for th lnar propagaton of th DA shock wavs n a usty plasma, I frst xprss our pnnt varabls N, U,,, an n trms of thr qulbrum an prturb parts as () () N N N (8) () () U U U ----(9) () () --() () () --() () () () Now, substtutng (8)-() nto (3)-(7), w vlop quatons n varous powr of w hav 55

4 Oppost Dust on Lnar Dust-acoustc Wavs () () N U T X (3) U T X (4) () () () () () ( ) N X (5) () () () [( Q P) R( ) ] T (6) () T () () [( X X ) ( X X X X ) ] (7) Now assumng that all prturb quantts ar proportonal to xp( T k X ),.. takng T an X k, whr an k ar th wav angular frquncy an th propagaton constant rspctvly n Eqs.(3)-(7), w obtan Now usng (3)-(34), on can lmnat followng quaton: N, U,, () () () () an (), an can fnally obtan th () () () () A C (35) whr th nonlnar coffcnt A an th sspaton coffcnt C ar gvn by B A = A C C = - A (36) (37) A (38) V f P Q Q 3 f B V 3 3 P Q Q P R R f R R P Q Q P R V X X X X X X X X X 4 X X X (39) 56

5 Oppost Dust on Lnar Dust-acoustc Wavs C V f P Q Q P R 3 3 X X X V, (4) Equaton (35) s th wll-known Burgrs quaton scrbng th nonlnar propagaton of th DA shock wavs n th usty plasma unr consraton. It s obvous from (35) an (37) that th sspatv trm,.. th rghthan s of (35) s u to th prsnc of th charg fluctuatng ust. IV. NUMERICAL ANALYSIS: W ar now ntrst n lookng for th statonary shock wav soluton of (35) by ntroucng th varabls U an, whr U s th shock wav sp (n th rfrnc fram) normalz by C, s normalz by, an s normalz by p D. Ths las us to wrt (35), unr th stay stat conton ( U A C () () () () It can b asly shown that [36], [37] that (4) scrbs shock wavs whos sp U (n th rfrnc fram) s rlat to th xtrm valus th conton that () ( ) an () ( ) by (4) ), as () ( ) - () ( ) = U A. Thus, unr () s boun at, th shock wav soluton of (4) can b wrttn as () [ tanh( )] (4) whr A (43) U C U (44) ar rspctvly, th hght an thcknss of th shock wavs movng wth th sp U. It s obvous from (4) to (44) that th shock wavs ar u to th prsnc of th charg fluctuatng ust, an th shock structurs ar assocat wth th ngatv potntal ( A ) as wll as wth postv potntal ( A ).To fn th paramtrc rgms for whch postv an ngatv shock wav (potntal) profls xst, w hav numrcally analyz A an obtan A = ( -D ) curvs for γ =. to.6 an μ = to 3.8. Th A = curv s shown n Fg.. It shows that w can hav postv shock wav (potntal) profls for th paramtrs whos valus l abov A = curv an ngatv shock wav (potntal) profls for th paramtrs whos valu l blow th A = curv. Ths ar shown n Fgs.-3. Fgs an 3 show th postv an ngatv shock potntal profls rspctvly. V. DISCUSSION W hav stu th nonlnar propagaton of DA wavs n an unmagntz usty plasma contanng Boltzmann-strbut lctrons an ons, mobl charg fluctuatng postv ust an charg fluctuatng statonary ngatv ust. W hav shown hr how th basc faturs of th nonlnar DA wavs ar mof by th prsnc of th charg fluctuatng ust n usty plasmas. Th rsults, whch hav bn obtan from ths nvstgaton, can b summarz as follows: Th ust charg fluctuaton s a sourc of sspaton an s rsponsbl for th formaton of DA shock wavs n th usty plasma. Th shock structurs ar assocat wth th ngatv potntal (A<) as wll as postv potntal (A<). It s shown that th hght (normalz by kbt ) of th potntal structurs n th 57

6 Oppost Dust on Lnar Dust-acoustc Wavs form of th shock wavs s rctly proportonal to th shock sp U, an t s also foun that th thcknss (normalz by λ D ) of ths shock structurs s nvrsly proportonal to th shock sp U. Th paramtrc rgms for th xstnc of postv as wll as ngatv shock structurs ar shown n Fg.. Fgs. an 3 show th postv an ngatv shock potntal profls of shock wavs rspctvly. It s to b mnton hr that th paramtrs w hav chosn n our numrcal analyss ar vry much rlvant to th plasma n th msosphr [7]. W strss that th rsults of th prsnt nvstgaton coul b usful n unrstanng th proprts of localz DA wavs of usty plasmas n th msosphr. REFERENCES []. D. A. Mns an M. Rosnbrg, Cosmc usty plasma, Annu. Rv. Astron. Astrophys., vol.3, pp , Sp.994. []. M. Horany an D. A. Mns, Th ynamcs of charg ust n th tal of comt Gacobn-nnr, J. Gophys. Rs., vol.9, pp , Jan [3]. M. Horany, Charg ust ynamcs n th solar systm", Annu.Rv. Astrophys, vol.34, pp , Sp.996. [4]. F. Vrhst, Wavs n Dusty Plasmas, Kluwr Acamc Publshrs, Dorrcht, Th Nthrlans.. [5]. P. K. Shukla, A survy of usty plasma physcs", Phys. Plasmas., vol. 8, no.5, pp , May. [6]. P. K. Shukla an A. A. Mamun, Introucton to Dusty Plasma Physcs, Insttut of Physcs Publshng Lt., Brstol, U.K.. [7]. A. Barkan, R. L. Mrlno, an N. D'Anglo, Laboratory obsrvaton of th ust-acoustc mo", Phys. Plasmas, vol.,no., pp , Oct [8]. A. Barkan, N. D'Anglo, an R. L. Mrlno, Exprmnts on on-acoustc wavs n usty plasmas", Plant. Spac Sc., vol 44, no.3, pp. 39-4, Mar.996. [9]. R. L. Mrlno, A. Barkan, C. Thompson, an N. D'Anglo, Laboratory stus of wavs an nstablts n usty plasmas", Physcs of Plasmas, vol. 5, no.5, pp , May, 998. []. A. Homann, A. Mlzr, S. Ptrs, an A. Pl, Dtrmnaton of th ust scrnng lngths by lasrxct lattc wavs", Phys. Rv. E, vol. 56, no.6, pp , Apr []. P. V. Blokh an V. V. Yaroshnko, Elctrostatc wavs n Saturn rngs", Sov. Astron., vol.9, pp , 985. (Engl.Trasl). []. U. Angls, V. Formsano, an M. Gorano, Ion plasma wavs n usty plasmas", J. Plasma Phys, vol.4, no.3 pp , Dc.988. [3]. P. K. Shukla an L. Stnflo, Stmulat scattrng of lctromagntc wavs n usty plasma", Astrophys. Spac Sc., vol.9, no. pp 3-3, Apr.99. [4]. P. K. Shukla an V. P. Sln, Dust on acoustc wav", Physca Scrpta, vol. 45, pp. 58, 99. [5]. N. N. Rao, P. K. Shukla, an M. Y. Yu, Dust-acoustc wavs n usty plasmas", Plant. Spac Sc., vol. 38, no.4 pp , May,99. [6]. F. Mlanso, Lattc wavs n ust plasma crystals, Phys. Plasmas, Vol.3, no.. pp , Nov [7]. Y. Nakamura, H. Balung an P. K. Shukla, Obsrvaton of on-acoustc shocks n a usty plasma", Phys. Rv. Ltt., vol. 83, no.8. pp.6-65,aug. 999 [8]. R. L. Mrlno an J. Gor, Dust vortx mos n a nonunform usty plasma", Phys. Toay, vol. 57, p.3-39, 4. [9]. M.R.Amn, an G.E.Morfll an P.K.Shukla, Moulatonal nstablty of ust-acoustc an ust-onacoustc wavs", Phys. Rv. E. Stat. Phys. Plasmas Flus Rlat.Intrscp.Top., vol.58, no.5. p.p , Fb 998. []. R. Bharuthram an P. K. Shukla, Larg ampltu on-acoustc soltons n a usty plasma", Plant. Spac Sc., vol.4, no.7, pp , Jul.99. []. S. I. Popl an M. Y. Yu, Ion acoustc soltons n mpurty-contanng plasmas", Contrb. Plasma Phys. vol.35, no, pp.3-8, 995. []. J. X. Ma an J. Lu, Dust-acoustc solton n a usty plasma, Phys. Plasmas. vol.4, no., pp , Fb [3]. A. A. Mamun, Arbtrary ampltu ust-acoustc soltary structurs n a thr-componnt usty plasma", Plant. Spac Sc. vol. 68, no.4, pp , Nov.999. [4]. P. K. Shukla, M. R. Amn, an G.E.Morfll, Instablty of ust-acoustc wavs n partally onz collsonal usty gass", Phys.Scr. vol. 59, no.5. pp ,

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8 Oppost Dust on Lnar Dust-acoustc Wavs P = 5. cm. Q =.93 cm s. R =.48 cm s wth α = 4.77, β = = 4.5 (Sol Curv). = 5.6 (Dott curv), = 7.5 (ash curv). Fg.3 showng ngatv potntal ( vs. ) curvs for th paramtrs P = 5. cm. Q =.93 cm s. R =.48 β = =.5 (Sol Curv). =. (Dott curv), cm s wth =.3 (ash curv). α = 4.77, 6