Nonplanar dust-ion-acoustic double layers in a dusty nonthermal plasma

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, do: /2011ja017016, 2011 Nonplanar dust-on-acoustc double layers n a dusty nonthermal plasma A. A. Mamun 1 and S. Islam 1 Receved 21 July 2011; revsed 25 September 2011; accepted 6 October 2011; publshed 22 December [1] The possblty for the formaton of the dust-on-acoustc (DIA) double layers (DLs) n a four-component dusty nonthermal plasma (contanng nonthermal electrons, nertal ons, and statonary postvely as well as negatvely charged dust) has been theoretcally nvestgated by dervng a modfed Gardner (MG) equaton. The latter has been numercally analyzed n order to dentfy the basc features of the nonplanar (cylndrcal and sphercal) DIA DLs. The DIA DLs (that are shown to be assocated only wth negatve potental) have been found to exst for a sutable value of a or m, where a (m) s the parameter determnng the number of nonthermal electrons (net dust charge densty) n the plasma system under consderaton. The mplcatons of our results n some space dusty plasma envronments have been dscussed. Ctaton: Mamun, A. A., and S. Islam (2011), Nonplanar dust-on-acoustc double layers n a dusty nonthermal plasma, J. Geophys. Res., 116,, do: /2011ja Introducton [2] Nowadays, the populaton of fast or energetc or nonthermal partcles and ther dstrbuton have receved a great deal of nterest n understandng the nonlnear electrostatc dsturbances n space plasma envronments, partcularly n the aurora acceleraton regon [Temern et al., 1982; Boström et al., 1988], n the upper Martan onosphere [Lundn et al., 1989], n the lower part of magnetosphere [Boström, 1992; Dovner et al., 1994], n/around the Earth s bow shock [Matsumoto et al., 1994], etc. About ffteen years ago, motvated by the dfferent spacecraft/satellte observatons [Temern et al., 1982; Boström et al., 1988; Lundn et al., 1989; Boström, 1992; Dovner et al., 1994; Matsumoto et al., 1994], Carns et al. [1995] ntroduced a dstrbuton functon (now known as Carns dstrbuton ) representng the populaton of the fast or energetc or nonthermal partcles, whch s able to explan some specal features (partcularly the exstence of rarefactve on-acoustc soltons or densty cavtons that are observed by the Freja Satellte [Dovner et al., 1994] and Vkng spacecraft [Boström, 1992]) of the on-acoustc (IA) soltary structures. After ths semnal work of Carns et al., a sgnfcant number of the theoretcal nvestgatons have been made on the role of ths nonthermal dstrbuton of electrons n modfyng the basc features of the nonlnear IA waves, e.g., those by Mamun [1997, 2000], Sahu and Roychoudhury [2005], Cho et al. [2010], etc. Mamun [1997, 2000] has studed the effects of nonthermal electron dstrbuton on tme-ndependent [Mamun, 1997] and tme-dependent [Mamun, 2000] arbtrary ampltude IA soltary waves (SWs) ncludng [Mamun, 1997] or excludng 1 Department of Physcs, Jahangrnagar Unversty, Dhaka, Bangladesh. Copyrght 2011 by the Amercan Geophyscal Unon /11/2011JA [Ghosh and Bharuthram, 2008; Mamun, 2000] the effect of the on-temperature; Sahu and Roychoudhury [2005] have examned the effects of the nonplanar (cylndrcal and sphercal) geometres on small ampltude IA SWs ncludng the effects of nonthermal electron dstrbuton and ontemperature; Cho et al. [2010] have nvestgated the possblty for the formaton of tme-ndependent double layers (DLs) whch are due to the combned effects of the nonthermal electron dstrbuton and heavy ons. [3] The effects of the nonthermal electron dstrbuton on the nonlnear propagaton of dust-on-acoustc (DIA) waves of Shukla and Sln [1992] (whch are, n fact, IA waves n a plasma wth charged dust) have also been nvestgated by several authors [Trbeche and Berbr, 2008; Xue, 2004; Alnejad, 2010]. Trbeche and Berbr [2008] have studed the effects of nonthermal electron dstrbuton on onedmensonal (1D) planar DIA soltary and shock waves; Xue [2004] has consdered nonplanar cylndrcal and sphercal geometres [Mamun and Shukla, 2002] and examned the nteracton between the compressve and rarefactve DIA SWs n dusty plasma wth nonthermal electrons; Alnejad [2010] has ncluded the effect of the nonthermal electron dstrbuton n dust-chargng current, and studed the DIA soltary and shock waves. All of these works are lmted to negatvely charged dust [Trbeche and Berbr, 2008; Xue, 2004; Alnejad, 2010] and to soltary [Trbeche and Berbr, 2008; Xue, 2004] or shock [Alnejad, 2010] structures. The consderaton of negatvely charged dust s vald when dust chargng process by collecton of plasma partcles (vz. electrons and ons) s much more mportant than other chargng processes [Rosenberg and Mends, 1995; Fortov et al., 1998; Chow et al., 1993; Rosenberg et al., 1999]. But, there are some other more mportant chargng processes by whch the dust grans become postvely charged [Rosenberg and Mends, 1995; Fortov et al., 1998; Chow 1of6

2 et al., 1993; Rosenberg et al., 1999]. The prncpal mechansms by whch dust grans become postvely charged are photoemsson n the presence of a flux of ultravolet photons [Rosenberg and Mends, 1995; Fortov et al., 1998], thermonc emsson nduced by radatve heatng [Rosenberg et al., 1999], secondary emsson of electrons from the surface of the dust grans [Chow et al., 1993], etc. [4] There are drect evdence of the coexstence of postvely and negatvely charged dust n dfferent regons of space, vz. Earth s mesosphere [Havnes et al., 1996], cometary tals [Horány, 1996; Ells and Neff, 1991], Jupter s magnetosphere [Horány et al., 1993; Horány, 1996], etc. Chow et al. [1993] have theoretcally shown that due to the sze effect on secondary emsson, nsulatng dust grans wth dfferent szes can have the opposte polarty, smaller ones beng postve and larger ones beng negatve. The opposte stuaton,.e. larger (massve) ones beng postve and smaller (lghter) ones beng negatve, s also possble by trboelectrc chargng [Shukla and Rosenberg, 2006; Lacks and Levandovsky, 2007]. Ths s predcted from the observatons of dpolar electrc felds perpendcular to the ground, wth negatve pole at hgher alttudes, generated by dust devls [Farrell et al., 2004; Merrson et al., 2004] and sand storms [Stow, 1969]. The formaton of these dpolar electrc felds means that negatvely charged smaller (.e. lghter) dust are blown upward n the convecton, whle postvely charged larger (more massve) dust reman the surface due to gravty. It s shown by experments usng Mars dust analogous n Mars smulaton wnd tunnel that m-szed dust can carry net charges of around 10 5 e, and there could be almost equal quanttes of postvely and negatvely charged dust n the suspenson [Shukla and Rosenberg, 2006; Merrson et al., 2004]. [5] Recently, Mamun and Shukla [2009] have analyzed the effects of nonthermal dstrbuton of electrons and polarty of net dust charge densty on nonplanar DIA SWs by dervng the Korteweg-de Vres (K-dV) equaton whch cannot provde any nformaton about the possblty for the formaton of the DIA DLs. Ths means that to study the fnte ampltude tme-dependent DIA DLs, one must resort the other type of hgher order nonlnear dynamcal equaton whch can provde the nformaton about the possblty for the formaton of the DLs. Therefore, n our present work, we consder a four-component dusty nonthermal plasma (contanng nonthermal electrons, nertal ons, and statonary postvely as well as negatvely charged dust), and derve a hgher order nonlnear equaton (known as Gardner equaton [Lee, 2009]), and examne the basc features of DLs whch are formed due to the presence of nonthermal electrons or charged dust partcles n such a dusty nonthermal plasma (DNP). [6] The manuscrpt s organzed as follows. The modfed Gardner (MG) equaton s derved n secton 2. The numercal solutons of MG equaton are analyzed n secton 3. A bref dscusson s fnally presented n secton Dervaton of MG Equaton [7] We consder the nonlnear propagaton of fnte ampltude nonplanar (cylndrcal and sphercal) DIA waves n a four-component collsonless, unmagnetzed, DNP composed of nonthermal electrons, nertal ons, and statonary postvely as well as negatvely charged dust. Thus, at equlbrum, we have n 0 + Z dp n dp = n e0 + Z dn n dn, where n 0, n e0, n dp, and n dn are, respectvely on, electron, postve dust, and negatve dust number densty at equlbrum, and Z dp (Z dn ) represents the charge state of postve (negatve) dust. We also assume that the electrons follow Carns dstrbuton [Carns et al., 1995]. Thus, n presence of the purely electrostatc waves, one can express the electron number densty n e as [Carns et al., 1995; Mamun, 1997] n e ¼ n e0 1 by þ by 2 expðyþ; ð1þ where y s the electrostatc wave potental normalzed by k B T e /e (where k B s the Boltzmann constant, T e s the electron temperature, and e s the magntude of the electron charge); b =4a/(1 + 3a) wth a beng the parameter determnng the number of nonthermal (fast or energetc) electrons present n our DNP system. We note that a = 0 corresponds to the Boltzmann dstrbuton of electrons. The nonlnear dynamcs of the low-frequency, purely electrostatc nonplanar DIA waves n the DNP under consderaton s descrbed by n t þ 1 r n ð r rn n u Þ ¼ 0; ð2þ u t þ u u r ¼ y r ; 1 r n r r n y r ¼ r; r ¼ ð1 jmþ 1 by þ by 2 expðyþ n þ jm; where n = 0 for 1D planar geometry and n = 1 (2) for a nonplanar cylndrcal (sphercal) geometry, n s the on number densty normalzed by ts equlbrum value n 0, u s the on flud speed normalzed by the on-acoustc speed C = (k B T e /m ) 1/2, r s the net surface charge densty normalzed by n 0 e, m = Z dn n dn Z dp n dp /n 0, and j = 1 for Z dn n dn > Z dp n dp and j = 1 for Z dn n dn < Z dp n dp. We note that j = 1 (j = 1) represents the postve (negatve) net dust charge. The tme and space varables are n unts of the on plasma perod w 1 p =(m /4pn 0 e 2 ) 1/2, and the Debye-radus l Dm = (k B T e /4pn 0 e 2 ) 1/2, respectvely. We note that (5) s vald for statonary or statc (postve and negatve) dust. Ths s a correct approxmaton for the DIA waves whose frequency s much larger than both the postve-dust-plasma frequency (w pdp ) and negatve-dust-plasma frequency (w pdn ),.e. w w pdp, w pdn, where w s the DIA wave frequency. [8] To study fnte ampltude DIA DLs n a DNP by usng (2) (5) by the reductve perturbaton method [Washm and Tanut, 1966], we frst ntroduce the stretched coordnates [Lee, 2009]: x ¼ r V p t ; ð6þ t ¼ 3 t; ð3þ ð4þ ð5þ ð7þ 2of6

3 We note that (15) and (16) are obtaned from (4) and (5) by takng the co-effcent of 2. It s obvous from (15) (17) that A = 0 snce f 0. The soluton of A = 0 yelds a = a c, where pffffffffffffffffffffffffffffffffffffff 31 ð jmþ 1 a c ¼ pffffffffffffffffffffffffffffffffffffff 31 ð jmþþ 3 ; ð18þ Fgure 1. The varaton of a c (obtaned from A(a = a c )=0) wth m for j = 1 and j = 1. The dash (sold) lne represents ths varaton for j =1( 1). where s a small parameter (0 < < 1) measurng the weakness of the dsperson, and V p (normalzed by C )s the phase speed of the perturbaton mode, and expand all the dependent varables (vz. n, u, y, and r) n power seres of : It s obvous that (15) (17) are satsfed for a = a c.we have numercally shown how a c vares wth m. The results are dsplayed n Fgure 1, whch, n fact, represents the parametrc regmes correspond to A =0,A > 0, and A <0. Our present nterest n the parametrc regme where A s not exactly zero, but t s so small that a a c =,.e. A 0, but A = A 0, where A 0 can be expressed as A 0 s A ja a c j¼sa a ; ð19þ a a¼a c where A a = 24(a 1)(1 jm) 2 /(1 + 3a) 3, and s = 1 for a > a c and s = 1 for a < a c. So, for a a c, one can express r (2) as n ¼ 1 þ n ð1þ þ 2 n ð2þ þ 3 n ð3þ þ ; ð8þ r ð2þ 1 2 sa af 2 : ð20þ u ¼ 0 þ u ð1þ þ 2 u ð2þ þ 3 u ð3þ þ ; ð9þ y ¼ 0 þ y ð1þ þ 2 y ð2þ þ 3 y ð3þ þ ; r ¼ 0 þ r ð1þ þ 2 r ð2þ þ 3 r ð3þ þ : ð10þ ð11þ Now, expressng (2) (5) n terms of x and t, and substtutng (8) (11) nto the resultng equatons ((2) (5) expressed n terms of x and t), one can easly develop dfferent sets of equatons n varous powers of. To the lowest order n, one obtans u ð1þ ¼ f ; n ð1þ ¼ f V p Vp 2 ; r ð1þ ¼ 0; ð12þ 1 V p ¼ p ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ; ð13þ 1 jm ð1 jmþb where f = y (1). The expresson (13) represents the lnear dsperson relaton for the DIA waves propagatng n the DNP under consderaton. To the next hgher order n, one obtans another set of equatons, whch, after usng (12) (13), can be smplfed as u ð2þ ¼ f2 2Vp 3 þ y ð2þ ð Þ ; n 2 V p 0 ¼ r ð2þ ; r ð2þ ¼ 1 2 Af2 ; ¼ 3f2 2V 4 p þ y ð2þ Vp 2 ; ð14þ ð15þ ð16þ A ¼ 3 Vp 4 ð1 jmþ: ð17þ Ths means that for a a c, r (2) must be ncluded n the thrd order Posson s equaton. To the next hgher order n, one obtans the thrd set of equatons: n ð1þ t V p n ð3þ u ð1þ t V p x þ u ð 3 Þ x þ F n x þ n u ð3þ x þ F u x þ y ð 3 Þ x V p t u ð1þ ¼ 0; ð21þ ¼ 0; ð22þ 2 f x 2 þ 1 2 sa af 2 ð1 jmþ y ð3þ þ fy ð2þ þ 1 6 f3 þ ð1 jmþb y ð3þ 1 2 f3 þ n ð3þ ¼ 0; ð23þ where F n = n (1) (2) u + n (2) (1) u and F u = u (1) u (2). Now, usng (12) (17) and (21) (23), one fnally obtans a nonlnear dynamcal equaton of the form: f t þ n f f f þ pf þ qf2 2t x x þ q 3 f 0 x 3 ¼ 0; where p = sa a q 0, q = p 0 q 0, and p 0 ¼ 15 2V 6 p ð24þ 1 1 jm 2 ð Þ 3 ð1 jmþb; ð25þ 2 q 0 ¼ V 3 p 2 ; ð26þ Equaton (24) s the MG equaton. The modfcaton s due to the extra term, (n/2t)f, whch arses due to the effects of the nonplanar geometry. We have already mentoned that n = 0 corresponds to a 1D planar 3of6

4 Fgure 2. The parametrc regme for the exstence of DLs (obtaned from the solutons of p 0 = 0 for a) for j =1 and j = 1. The dash (sold) lne represents ths varaton for j =1( 1). Fgure 4. The effects of sphercal geometry on DIA negatve DLs for a = 0.071, m = 0.5, and U 0 = 0.1 for j =1. geometry whch reduces (equaton (24)) to a standard Gardner (SG) equaton. 3. Numercal Analyss of MG Equaton [9] Our am s now to numercally analyze MG equaton. However, for clear understandng, we frst brefly dscuss the statonary DL soluton of ths SG equaton (.e., (24) wth n = 0). The statonary DL soluton of ths SG equaton s obtaned by consderng a frame (movng wth speed U 0 ) z = x U 0 t, and mposng all approprate boundary condtons for the DL solutons, ncludng f 0, df/dz 0, d 2 f/dz 2 0 at z +. Thus, one can express the statonary DL soluton of ths SG equaton as f ¼ f m 1 þ tanh z ; ð27þ 2 D where the ampltude (f m ) and the wdth (D) of the DLs are f m ¼ 6sU 0 A a q 0 ; ð28þ sffffffffffffffffffffffffffffff D ¼ 24 f 2 m p : 0 ð29þ It s clear from (27) and (29) that DLs exst f and only f p 0 < 0,.e. a > a D, where a D s represented by the p 0 =0 plot shown n Fgure 2. On the other hand, snce q 0 > 0 and U 0 > 0, (27) and (28) ndcate that the DLs are assocated wth negatve potental f s = 1,.e. a > a c, and assocated wth postve potental f s = 1,.e. a < a c. It s obvous from Fgures 1 and 2 that a D > a c whch confrm us that DLs are assocated wth negatve potental only. The parametrc regmes for the exstence of negatve DLs are shown n Fgure 2, and DLs exst for parameters correspondng to any ponts above the p 0 = 0 plot. [10] Now, we have numercally solved (24), and have studed the effects of cylndrcal and sphercal geometres on tme-dependent DIA DLs. The results are depcted n Fgures 3 6. Fgure 3 (Fgure 4) shows how the effects of cylndrcal (sphercal) geometry modfy the DIA DLs for j = 1. Fgure 5 (Fgure 6) shows how the effects of cylndrcal (sphercal) geometry modfy the DIA DLs for j = 1. [11] The numercal solutons of (24) (dsplayed n Fgures 3 6) reveal that for a large value of t (e.g. t = 30), the cylndrcal (n = 1) and sphercal (n = 2) DLs are almost smlar to 1D planar (n = 0) structures. Ths s because for a large value of t, the term (n/2t)f, whch s due to the effects of the cylndrcal or sphercal geometry, s no longer domnant. However, as the value of t decreases, the term (n/2t)f becomes domnant, and the cylndrcal and sphercal DL structures dffer from 1D planar ones. It s found that as the Fgure 3. The effects of cylndrcal geometry on DIA negatve DLs for a = 0.071, m = 0.5, and U 0 = 0.1 for j =1. Fgure 5. The effects of cylndrcal geometry on DIA negatve DLs for a = 0.25, m = 0.5, and U 0 = 0.1 for j = 1. 4of6

5 Fgure 6. The effects of sphercal geometry on DIA negatve DLs for a = 0.25, m = 0.5, and U 0 = 0.1 for j = 1. value of t decreases, the ampltude of these localzed pulses ncreases. It s also found that the ampltude of cylndrcal DIA DL structures s larger than those of 1D planar ones, but smaller than that of the sphercal ones. 4. Dscusson [12] We have consdered a four-component collsonless, unmagnetzed, dusty nonthermal plasma system consstng of electrons followng nonthermal dstrbuton [Carns et al., 1995], cold moble nertal ons, and postvely as well negatvely charged dust, and have examned the basc features of the planar and nonplanar tme-dependent DIA DLs. The reductve perturbaton method has been employed n order to derve the MG equaton whch s vald beyond the K-dV lmt (correspondng to the vanshng of the nonlnear coeffcent of the K-dV equaton,.e. a a c n our present stuaton). The results, whch have been found from the numercal solutons of the MG equaton, can be summarzed as follows. [13] 1. The DLs are found to be formed when a > a c, a D, where a c and a D depend on m, and decrease (ncrease) wth the ncrease of m for j =1( 1) as obvous from Fgures 1 and 2. Ths means that, j =1( 1) s n favor (dsfavor) of the formaton of DLs. [14] 2. The DLs are assocated wth negatve potental only for both j =1( 1). We note that for m = 0.5, a c and a D for j = 1 and a c and a D for j = 1. [15] 3. The basc features (vz. ampltude, wdth, speed, etc.) are sgnfcantly modfed by the nonthermal electrons as well as by the polarty and number densty of the charged dust. [16] 4. The DLs can be formed at a a c for whch the K-dV equaton does not vald because of the vanshng of the nonlnear coeffcent. [17] 5. The magntude of the ampltude of the DLs ncreases wth both a and U 0 for both j = 1 and j = 1. [18] 6. The wdth of the DLs decreases wth both a and U 0 for both j = 1 and j = 1. [19] 7. The numercal analyss of the MG equaton dctates that for a very large value of t the planar and nonplanar DIA DLs are dentcal, but the magntude of the ampltude of both cylndrcal and sphercal DIA DLs ncreases wth the decrease of the value of t. [20] 8. The ampltude of the cylndrcal DIA DLs s larger than that of 1D planar DLs, but smaller than that of the sphercal ones. [21] We have used a wde range of the plasma parameters (vz. m = and a =0 0.5) n our numercal analyss. Thus, the dust-plasma parameters are wthn the approprate ranges for space envronments [Havnes et al., 1996; Horány et al., 1993; Horány, 1996; Ells and Neff, 1991; Farrell et al., 2004; Merrson et al., 2004]. [22] We note that our present theory s vald for collsonless, and unmagnetzed plasma wth constant dust charge, and for small but fnte ampltude tme dependent DL structures. Ths means that our present theory s vald as long as n c, w c, n ch w (where n c s the on-neutral collson frequency, w c s the on cyclotron frequency, and n ch s the dust chargng frequency). However, for some space and laboratory dusty plasma stuatons, the effects of on-neutral collson frequency, external magnetc feld, and dust charge fluctuaton on the arbtrary ampltude tme-dependent DIA DL structures may also be problems of great mportance, but beyond the scope of our present work. [23] We fnally hope that our results may eventually be appled more drectly to the nterpretaton of space data from future mssons. We also suggest to perform a laboratory experment to test the theory that we presented n ths work. [24] Acknowledgments. The research grant for research equpment from the Thrd World Academy of Scences (TWAS), ICTP, Treste, Italy s gratefully acknowledged. [25] Phlppa Brownng thanks the revewers for ther assstance n evaluatng ths paper. References Alnejad, H. 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