Solitary waves and double layers in complex plasma media
OORIGINAL ARTICLE
Solitary waves and double layers in complex plasma media
A A Mamun a,c and Abdul Mannan a,b a Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh b Institut f¨ur Mathematik, Martin Luther Universit¨at Halle-Wittenberg, D-06099 Halle, Germany
ARTICLE HISTORY
Compiled January 27, 2021
ABSTRACT
A complex plasma medium (containing Cairns nonthermal electron species, adiabaticallywarm inertial ion species, and stationary positively charged dust (PCD) species (making aplasma system very complex) is considered. The effects of PCD species, nonthermal electronspecies, and adiabatic ion-temperature on ion-acoustic (IA) solitary waves (SWs) and doublelayers (DLs) are investigated by the pseudo-potential approach, which is valid for the arbi-trary amplitude time-independent SWs and DLs. It is observed that the presence of the PCDspecies reduces the phase speed of the IA waves, and consequently supports the IA subsoniccompressive SWs in such electron-ion-PCD plasmas. On the other hand, the increase in bothadiabatic ion-temperature and the number of nonthermal or fast electrons causes to reducethe possibility for the formation of the subsonic SWs, and thus convert subsonic SWs intosupersonic ones. It is also observed that after at a certain value of the nonthermal parameter,the IA supersonic SWs with both positive and negative potentials as well as the DLs withonly negative potential exist. The applications of the work in space environments (viz. Earth’smesosphere, cometary tails, Jupiter’s magnetosphere, etc.) and laboratory devices, where thewarm ion and nonthermal electron species along with PCD species have been observed, arebriefly discussed.
KEYWORDS
Positive dust; Non-thermal electrons; Subsonic and supersonic SWs; Double layers
1. Introduction
Nowadays, the existence of positively charged dust (PCD) species in electron-ion plasmasreceived a renewed interest because of their vital role in modifying existing features as well asintroducing new features of linear and nonlinear ion-acoustic (IA) waves propagating in manyspace plasma environments [viz. Earth’s mesosphere [1–3], cometary tails [4,5], Jupiter’ssurroundings [6], Jupiter’s magnetosphere [7], etc.] and laboratory devices [8–10], wherein addition to electron-ion plasmas, the PCD species have been observed. Three principalmechanisms by which the dust species becomes positively charged [11–14] are as follows: • The photoemission of electrons from the dust grain surface induced by the flux of highenergy photons [13]. • The thermionic emission of electrons from the dust grain surface by the intense radia-tive or thermal heating [12]. c Also at Wazed Miah Science Research Centre, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh. Email:mamun − [email protected] author: Abdul Mannan. Email: [email protected]; Telephone: +4915210286280; Fax: +49-345-55-27005 a r X i v : . [ phy s i c s . p l a s m - ph ] J a n The secondary emission of electrons from the dust grain surface by the impact of highenergetic plasma particles like electrons or ions [11].The dispersion relation for the IA waves in an electron-ion-PCD plasma system (containinginertialess isothermal electron species, inertial cold ion species, and stationary PCD species)is given by [15] ω kC i = (cid:113) + µ + k λ D , (1)where ω = π f and k = π / λ in which f ( λ ) is the IA wave frequency (wavelength); C i = ( z i k B T e / m i ) / is the IA speed in which k B is the Boltzmann constant, T e is the elec-tron temperature, and m i is the ion mass; λ D = ( k B T e / π z i n i e ) / is the IA wave-lengthscale in which n i ( z i ) is the number density (charge state) of the ion species at equilibrium,and e is the magnitude of the charge of an electron; µ = z d n d / z i n i with n d ( z d ) being thenumber density (charge state) of the PCD species at equilibrium. This means that µ = µ → ∞ corresponds to electron-dust plasma[5,8–10]. Thus, 0 < µ < ∞ is valid for the electron-ion-PCD plasmas. The dispersion relationdefined by (1) for the long-wavelength limit (viz. λ (cid:29) λ D ) becomes ω kC i (cid:39) (cid:115) + µ . (2)The dispersion relation (2) indicates that the phase speed decreases with the rise of the valueof µ . This new feature of the IA waves (continuous as well as periodic compression andrarefaction or vise-versa of the positive ion fluid) is introduced due to the reduction of thespace charge electric field by the presence of PCD.Recently, based on this new linear feature, Mamun and Sharmin [15] and Mamun [16] haveshown the existence of subsonic shock and SWs, respectively, by considering the assumptionof Maxwellian electron species and cold ion species. The IA waves in different plasma sys-tems composed of ions and electrons have also been studied by a number of authors [17–19].However, the reduction of the IA wave phase speed by the presence of PCD species can alsomake the IA phase speed comparable with the ion thermal speed V Ti = ( k B T i / m i ) / (where T i is the ion-fluid temperature) so that the effect of the ion-thermal pressure cannot be neglected.On the other hand, the electron species in space environments mentioned does not alwaysfollow the Maxwellian electron velocity distribution function. This means that the linear dis-persion relation (2), and the works of Mamun and Sharmin [15] and Mamun [16] are validfor a cold ion fluid ( T i =
0) limit and for the Maxwell electron velocity distribution function,which can be expressed in one dimensional (1D) normalized [normalized by n e / V Te , where V Te = ( k B T e / m e ) / is the thermal speed of the electron species, and v is normalized by V Te ]form as f ( v ) = √ π exp (cid:20) − ( v − φ ) (cid:21) , (3)where φ is the IA wave potential normalized by k B T e / e .To overcome these two limitations, we consider (i) adiabatically warm ion fluid and (ii)nonthermal electron species following Cairns velocity distribution function, which can be2imilarly expressed in 1D normalized form as [20] f ( v ) = + α ( v − φ ) ( + α ) √ π exp (cid:20) − ( v − φ ) (cid:21) , (4)where α is a parameter determining the population of fast (energetic) particles present inthe plasma system under consideration. We note that equation (4) is identical to equation(3) for α =
0. Thus, how the nonthermal parameter α modifies the Maxwell distributionof the electron species is shown mathematically by equation (4) and graphically by the leftpanel of figure 1. On the other hand, including the effects of the Cairns nonthermal electrondistribution ( α ) and the adiabatic ion-temperature ( σ ), the dispersion relation for the longwavelength IA waves can be expressed as ω kC i = (cid:115) + α ( + µ )( − α ) + σ , (5)where σ = T i / z i T e with T i is the ion-temperature at equilibrium. The dispersion relation (5)indicates that as α and σ increase, the phase speed of the IA waves increases. The is due tothe enhancement of the space charge electric field by nonthermal electron species and of theflexibility of the ion fluid by its temperature. The variation of the phase speed of the IA waves[defined by equation (5)] with α and σ is shown in the right panel of figure 1. α = α = α = - - v f ( v ) α σ Figure 1.: The left panel shows the curves representing the Cairns nonthermal velocity dis-tribution function [defined by equation (4)] for φ = . α , whereasthe right panel shows how the normalized phase speed ( ω / kC i ) of the IA waves [defined byequation (5)] varies with σ and α for µ = . . Governing equations To investigate the nonlinear propagation of the IA waves defined by the equation (5), we con-sider an electron-ion-PCD plasma medium. The nonlinear dynamics of the IA waves propa-gating in such an electron-ion-PCD plasma medium is described by ∂ n i ∂ t + ∂∂ x ( n i u i ) = , (6) ∂ u i ∂ t + u i ∂ u i ∂ x = − ∂ φ∂ x − σ n i ∂ P i ∂ x , (7) ∂ P i ∂ t + u i ∂ P i ∂ ξ + γ P i ∂ u i ∂ x = , (8) ∂ φ∂ x = ( + µ ) n e − n i − µ , (9)where n i is the ion number density normalized by n i ; u i is the ion fluid speed normalizedby C i ; P i is the adiabatic ion-thermal pressure normalized by n i k B T i ; γ [= ( + N ) / N ] isthe ion fluid adiabatic index with N being the number of degrees of freedom, which has thevalue 1 (3) for the 1D (3D) case so that in our present work N = γ = t ( x ) is thetime (space) variable normalized by ω − pi ( λ D ); n e is the nonthermal electron number densitynormalized by n e , and is determined by integrating equation (4) with respect to v from − ∞ to + ∞ , i.e. n e can be expressed as [21] n e = ( − β φ + β φ ) exp ( φ ) , (10)with β = α / ( + α ) . We note that for isothermal electron species γ = T i = T i and P i = n i k B T i , equations (6) and (8) are identical.
3. SWs and DLs
To study arbitrary amplitude IA SWs and DLs, we employ the pseudo-potential approach[20–22] by assuming that all dependent variables in equations (6)–(9) depend only on a singlevariable ξ = x − M t , where M is the Mach number (defined by ω / kC i ). This transformation( ξ = x − M t ) along with the substitution of equation (10) into equation (9) and γ = M dn i d ξ − dd ξ ( n i u i ) = , (11) M du i dl ξ − u i du i d ξ = d φ d ξ + σ n i dP i d ξ , (12) M dP i d ξ − u i dP i d ξ − P i du i d ξ = , (13) d φ d ξ = ( + µ ) (cid:0) − β φ + β φ (cid:1) exp ( φ ) − n i − µ . (14)4he appropriate conditions (viz. n i → u i → ξ → ± ∞ ) reduce (11) to u i = M (cid:18) − n i (cid:19) , (15) n i = MM − u i . (16)The substitution of (15) into (13) gives rise to1 n i dP i d ξ + P i dd ξ (cid:18) n i (cid:19) = , (17)which finally reduces to P i = n i , (18)where the integration constant is found to be 1 under the conditions that P i → n i → ξ → ± ∞ . Similarly, the substitution of (15) into equation (12) yields M du i d ξ − u i du i d ξ − σ dP i d ξ + σ M u i dP i d ξ = d φ d ξ . (19)Again, multiplying (13) by σ / M one can write σ dP i d ξ − σ M u i dP i d ξ − P i σ M du i d ξ = . (20)Now, performing (20) − × (19) we obtain3 σ ( P i − ) − σ M ( P i u i ) − M u i + u i + φ = , (21)where the integration constant is found to be 3 σ under the conditions that P i → n i → u i →
0, and φ → ξ → ± ∞ . The substitution of equations (15) and (18) into equation (21)yields 3 σ n i − ( M + σ − φ ) n i + M = . (22)This is the quadratic equation for n i . Thus, the expression for n i can be expressed as n i = √ σ (cid:20)(cid:113) Ψ − (cid:112) Ψ − σ M (cid:21) , (23)where Ψ = M + σ − φ . Now, substituting equation (23) into equation (14), we obtain d φ d ξ = ( + µ ) (cid:0) − β φ + β φ (cid:1) exp ( φ ) − √ σ (cid:20)(cid:113) Ψ − (cid:112) Ψ − σ M (cid:21) − µ , (24)5e finally multiply both side of equation (24) by ( d φ / d ξ ) and integrating the resulting equa-tion with respect to φ , we obtain12 (cid:18) d φ d ξ (cid:19) + V ( φ , M ) = , (25)which represents an energy integral of a pseudo-particle of unit mass, pseudo time ξ , pseudo-position φ and pseudo-potential V ( φ , M ) is defined by V ( φ , M ) = C + µφ − ( + µ ) (cid:20) + α + α (cid:0) − φ + φ (cid:1)(cid:21) exp [ φ ] − √ √ σ (cid:18)(cid:113) Ψ − (cid:112) Ψ − σ M (cid:19) (cid:18) Ψ + (cid:112) Ψ − σ M (cid:19) , (26)where C = ( + µ ) (cid:20) + α + α (cid:21) + σ + M (27)is the integration constant, and it is chosen in such a way that V ( φ , M ) = φ = V ( , M ) = V (cid:48) ( , M ) = V ( φ , M ) with respect to φ . So, the conditions for theexistence of SWs and DLs are: (i) V (cid:48)(cid:48) ( , M ) < V (cid:48) ( φ m , M ) > φ >
0; (iii) V (cid:48) ( φ m , M ) < φ <
0; (iv) V (cid:48) ( φ m , M ) = φ m is theamplitude of the SWs or DLs. Thus, SWs or DLs exist if and only if V (cid:48)(cid:48) ( , M ) <
0, i.e. M > M c , where M c = (cid:115) + α ( + µ )( − α ) + σ . (28)We note that the expression for M c [given by equation (28)] is identical to equation (5). Thephase speed of the IA waves decreases and the possibility for the formation of subsonic IASWs increases as the number of PCD species increases. This is depicted in figure 1(a). Onthe other hand, the possibility for the formation of subsonic (supersonic) IA SWs decreases(increases) with the increase of the values of α and σ . This is shown in figure 1(b). The rangesof the value of M , viz. M c < M < M > M c > M c with µ and α for the fixed valueof σ is graphically shown in figure 2(a), where the shaded (non-shaded) area represents thedomain for the existence of subsonic (supersonic) SWs.It is well known [20,21] that the sign of V (cid:48)(cid:48)(cid:48) ( , M c ) = ( − α ) ( + µ ) [ + α + ( − α )( + µ ) σ ]( + α ) − ( + µ ) (29)determines either the existence of the IA SWs with φ > φ > φ <
0. Thus, the IA SWs with φ > φ < φ >
0] will exist (coexist)if V (cid:48)(cid:48)(cid:48) ( , M c ) > V (cid:48)(cid:48)(cid:48) ( , M c ) < .1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.80.91.01.1 μ ℳ c (a) μ = μ = μ = α σ V ′′′ ( ℳ c )> ′′′ ( ℳ c )< (b) Figure 2.: (a) The variation of M c with µ for σ = .
01 and α = α = . α = . V (cid:48)(cid:48)(cid:48) ( , M c ) = α and σ for differentvalues of µ , viz µ = . µ = . µ = . - ϕ - - - - V ( ϕ ) (a) - ϕ - - - - - V ( ϕ ) (b) Figure 3.: The formation of the potential wells representing the subsonic SWs (a) for α = . µ = . µ = .
75 (dashed curve) µ = . µ = . α = α = .
05 (dashed curve) α = . M = .
985 and σ = . φ > φ > φ < φ > φ < φ > V (cid:48)(cid:48)(cid:48) ( , M c ) > V (cid:48)(cid:48)(cid:48) ( , M c ) < φ >
0. The possibility for the formation of SWs with φ < φ > σ ) increases (decreases) the possibility for existence of SWs with φ > φ < φ > φ m ), which is the intercept onthe positive or negative φ -axis, and width ( φ m / (cid:112) | V m | ), where | V m | is the maximum value of V ( φ ) in the pseudo-potential wells formed in positive or negative φ -axis. Figures 3 and 4indicate the formation of the pseudo-potential wells in the positive φ -axis, which correspondsto the formation of the subsonic IA SWs only with φ >
0, i.e. the subsonic IA SWs with φ < ϕ - - - - - - V ( ϕ ) (a) - ϕ - - - - - - V ( ϕ ) (b) Figure 4.: The formation of the potential wells representing the subsonic SWs (a) for σ = . M = .
95 (solid curve), M = .
97 (dashed curve), M = .
99 (dot-dashed curve); (b)for M = . σ = .
01 (solid curve), σ = .
03 (dashed curve) σ = .
06 (dot-dashed curve).The other parameters, which are kept fixed, are µ = . α = . - - - - ϕ - - - V ( ϕ ) (a) - - - - ϕ - - - - - - V ( ϕ ) (b) Figure 5.: The formation of the potential wells representing the coexistence of supersonicSWs with φ > φ < α = . µ = . µ = .
35 (dashed curve)and µ = . µ = . α = .
26 (solid curve), α = .
27 (dashedcurve) and α = .
28 (dot-dashed curve). The other parameters, which are kept fixed, are M = . σ = . µ ( α and σ ). It is seen that the amplitude (width) of the subsonic IA SWs decreases (increases) as wedecrease the value of µ . On the other hand, the amplitude (width) of subsonic SWs decreases(increases) with increasing the values of α and σ . It is worth to mention that the lower valueof µ and higher values of α and σ convert the subsonic SWs into supersonic ones. It is seenin figures 5 and 6(a) that for M > M c the supersonic SWs with φ > φ < φ > φ < µ . On the other hand, the depth of potential wells (representing thecoexistence of supersonic SWs with φ > φ <
0) decreases with increasing the valuesof σ and α .The IA DLs only with negative potential is formed for M > M c as illustrated in figures6(b) and 7. The rise of the values of µ and σ causes to decrease (increase) the amplitude8 - - - ϕ - - - - - - V ( ϕ ) (a) - - - - - ϕ - - - - - - V ( ϕ ) (b) Figure 6.: The formation of the potential wells representing (a) the coexistence of supersonicSWs with φ > φ < α = . µ = . M = . σ = . σ = .
22 (dashed curve) and σ = .
24 (dot-dashed curve); (b) the existence of DLs with φ < M = . α = .
25 (solid curve), M = . α = .
26 (dashed curve), M = . α = .
27 (dot-dashed curve), µ = .
5, and σ = . - - - - ϕ - - - - - V ( ϕ ) (a) - - - - - ϕ - - - - - - V ( ϕ ) (b) Figure 7.: The formation of the potential wells representing the DLs with φ < M = . σ = .
15 (solid curve), M = . σ = . M = . σ = .
25 (dot-dashed curve), and µ = .
5; (b) for M = . µ = . M = . µ = . M = . µ = . σ = . α = .
26 is kept fixed for both cases.(width) of the DLs (as shown in figure 7). On the other hand, in figure 6, the potential wellsin the negative φ -axis becomes wider as the nonthermal parameter α increases. It means thatthe amplitude of DLs are increased by the effect of nonthermal parameter, but the width ofDLs decreases. It is noted here that for the formation of DLs, the increase in the values of α and σ ( µ ) is required a larger (smaller) value of the Mach number.
4. Discussion
We have considered a complex plasma medium containing Cairns nonthermally distributedelectron species, adiabatically warm ion species, and PCD species, and have investigated theIA SWs and DLs in such a plasma medium. We have employed the pseudo-potential approach9hich is valid for arbitrary or large-amplitude SWs and DLs. The results obtained from thistheoretical work and their applications can be briefly discussed as follows: • The effect of the PCD causes to reduce the IA wave phase speed, and to form subsonicSWs only with positive potential. On the other hand, the effects of Cairns nonthermalelectron distribution and adiabatic ion-temperature cause to enhance the IA wave phasespeed, and to reduce possibility for the formation of the subsonic SWs, and finallyconvert the subsonic SWs into supersonic ones. • The amplitude (width) of the subsonic IA SWs increases (decreases) with the rise of thevalue µ and M , but the amplitude (width) of the subsonic IA SWs decreases (increases)with the rise of the values α and σ . This is due to the fact that the phase speed of theIA waves decreases with rise of the value µ , but increases with the rise of the value of α and σ . • The supersonic IA SWs with φ > φ < α ) in the plasma sys-tem under consideration. However, the increase in the value of σ and µ decreases thepossibility for the formation of the IA SWs with φ < • The amplitude (width) of the supersonic IA SWs (which coexist with φ > φ < µ and M increase, but it decreases (increases) asthe values of α and σ increase. • The height (thickness) of the IA DLs (which exist only with φ <
0) increases (de-creases) as the values of both parameters of its set { M , α } increase. On the otherhand, it decreases (increases) with the rise of the value of both parameters of their sets { M , µ } and { M , σ } .The advantage of the pseudo-potential method [20–22] is that it is valid for arbitrary am-plitude SWs and DLs, but it does not allow us to observe the time evolution of the SWs orDLs. To overcome these limitations, one has to develop a numerical code to solve the basicequations (6) − (10) numerically. This type of simulation will be able to show the time evolu-tion of arbitrary amplitude SWs and DLs. This is, of course, a challenging research problemof recent interest, but beyond the scope of our present work.To conclude, we hope that the results of our present investigation should also be usefulin understanding the basic features of the IA waves and associated nonlinear structures likeSWs and DLs in space environments (viz. Earth’s mesosphere or ionosphere [1–3], cometarytails [4], Jupiter’s surroundings [6,7] and magnetosphere [7], etc.) and laboratory devices [8–10,23]. Data availability
Data sharing is not applicable to this article as no new data were created or analyzed in thisstudy.
Disclosure statement
The authors declare that there is no conflict of interest.
Acknowledgement
A. Mannan gratefully acknowledges the financial support of the Alexander von HumboldtStiftung (Bonn, Germany) through its post-doctoral research fellowship.10 eferences [1] Havnes O, Trøim J, Blix T, et al. First detection of charged dust particles in the earth’s meso-sphere. J Geophys Res. 1996;101(A5):10839.[2] Gelinas LJ, Lynch KA, Kelley MC, et al. First observation of meteoritic charged dust in thetropical mesosphere. Geophys Res Lett. 1998;25(21):4047–4050.[3] Mendis DA, Wong WH, Rosenberg M. On the observation of charged dust in the tropical meso-sphere. Phys Scr. 2004;T113:141.[4] Hor´anyi M. Charged dust dynamics in the solar system. Annu Rev Astron Astrophys. 1996;34:383.[5] Mamun AA, Shukla PK. Dust-acoustic mach cones in magnetized electron-dust plasmas of sat-urn. Geophys Res Lett. 2004;31(L06808):1–4.[6] Tsintikidis D, Gurnett DA, Kurth WS, et al. Micron-sized particles detected in the vicinity ofjupiter by the voyager plasma wave instruments. 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