1D planar, cylindrical and spherical subsonic solitary waves in space electron-ion-positive dust plasma systems
11D planar, cylindrical and spherical subsonic solitary waves in spaceelectron-ion-positive dust plasma systems
A A Mamun ∗ Department of Physics & Wazed Mia Science Research Centre,Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
The space electron-ion-positive dust plasma system containing isothermal inertialess electronspecies, cold inertial ion species, and stationary positive (positivively charged) dust species is con-sidered. The basic features of one dimensional (1D) planar and nonplanar subsonic solitary wavesare investigated by the pseudo-potential and reductive perturbation methods, respectively. It isobserved that the presence of the positive dust species reduces the phase speed of the ion-acousticwaves, and consequently supports the subsonic solitary waves with the positive wave potential insuch a space dusty plasma system. It is observed that the cylindrical and spherical subsonic solitarywaves significantly evolve with time, and that the time evolution of the spherical solitary waves isfaster than that of the cylindrical ones. The applications of the work in many space dusty plasmasystems, particularly in Earth’s mesosphere, cometary tails, Jupiter’s magnetosphere, etc. are ad-dressed.
PACS numbers: 52.27.Lw; 52.35.Sb;94.05.Fg
The existence of the ion-acoustic (IA) waves in aplasma medium was first predicted first by Tonks andLangmur [1] on the basis of the fluid dynamics in 1929.The prediction of Tonks and Langmuir [1] was then ver-ified by Revans [2] in 1933. The well known linear dis-persion relation for the IA waves propagating in a pureelectron-ion plasma containing cold inertial ion fluid andisothermal inertia-less electron fluid is given by ω = kC i (cid:112) k λ D , (1)where ω = 2 πf and k = 2 π/λ in which f ( λ ) is theIA 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 electron temperature, and m i is the ion mass; λ D = ( k B T e / πz i n i e ) / is the IA wave-length scale inwhich n i ( z i ) is the number density (charge state) of theion species at equilibrium, and e is the magnitude of thecharge of an electron. We note that for a pure electron-ion plasma n e = z i n i at equilibrium, where n e is theelectron number density at equilibrium. The dispersionrelation (1) becomes ω (cid:39) kC i for a long-wavelength limit, λ (cid:29) λ D , and ω (cid:39) ω pi for a short wavelength limit, λ (cid:28) λ D , where ω pi = (4 πz i n i e /m i ) / is the angularfrequency of ion plasma oscillations. Thus, the angularfrequency range of the IA waves is 0 > ω > ω pi .The IA waves [1, 2] are found to be modified in anelectron-ion-negative dust plasma system theoretically[3–5] as well as experimentally [6–8]. It has been foundthat the increase in number density and charge of thenegative dust species enhances the phase speed of the IAwaves, and consequently support the supersonic [9–14]solitary waves (SWs). ∗ Corresponding author: mamun [email protected]
There are many space plasma environments, viz.Earth’s mesosphere [15–17], cometary tails [18], Jupiter’ssurroundings [19], Jupiter’s magnetosphere [20], etc.where in addition to electron-ion plasmas, positive dustspecies have been observed [15–20]. There are three prin-cipal mechanisms by which the dust species becomes pos-itively charged [21–24]. These are photo-emission of elec-trons from the dust surface induced by the flux of photons[23], thermionic emission of electrons from the dust grainsurface by the radiative heating [22], and secondary emis-sion of electrons from the dust surface by the impact ofhigh energetic plasma particles [21].The dispersion relation for the IA waves in an electron-ion-positive dust plasma system (containing inertialessisothermal electron species, inertial cold ion species, andstationary positive dust species) is given by ω = 1 √ µ kC i (cid:113) µ k λ D , (2)where µ = z d n d /z i n i with n d ( z d ) being the numberdensity (charge state) of the positive dust species. Thedispersion relation (2) becomes ω (cid:39) kC i / (1 + µ ) for thelong-wavelength limit (viz. λ (cid:29) λ D ). The dispersionrelation ω (cid:39) kC i / √ µ indicates that the phase speed( ω/k ) decreases with the rise of the value of µ . We notethat µ = 0 corresponds to the electron-ion plasma [1, 2],and µ → ∞ corresponds to electron-dust plasma [25, 26].Thus, 0 < µ < ∞ is valid for the electron-ion-positivedust plasma system.To investigate the nonlinear propagation of the modi-fied IA (MIA) waves defined by (2), we consider such anelectron-ion-positive dust plasma system. The nonlineardynamics of the MIA waves (2) is described by ∂n i ∂t + 1 r ν ∂∂r ( r ν n i u i ) = 0 , (3) ∂u i ∂t + u i ∂u i ∂x = − ∂φ∂x , (4) a r X i v : . [ phy s i c s . p l a s m - ph ] S e p r ν ∂∂r (cid:18) r ν ∂φ∂r (cid:19) = (1 + µ ) exp( φ ) − n i − µ, (5)where ν = 0 for one dimensional (1D) planar geometryand ν = 1 (2) for cylindrical (spherical) geometry; theelectron species has is assumed to obey the Boltzmannlaw so that n e = exp( φ ); n e ( n i ) is the electron (ion)number density normalized by n e ( n i ); u i is the ionfluid speed normalized by C i ; φ is the electrostatic wavepotential normalized by k B T e /e ; r and t are normalizedby λ D and ω − pi , respectively. The assumption of station-ary positive dust species is valid because of the mass ofthe positive dust species being extremely high in com-parison with that of the inertial ion species.To study arbitrary amplitude MIA SWs in planar ge-ometry ( ν = 0 and r = x ), we employ the pseudo-potential approach [27, 28] by assuming that all depen-dent variables in (3) − (5) depend only on a single variable ξ = x − M t , where M is the Mach number. This trans-formation and steady state condition allow us to write(3) − (5) as M dn i dξ − ddξ ( n i u i ) = 0 , (6) M du i dlξ − u i du i dξ = dφdξ , (7) d φdξ = (1 + µ ) exp( φ ) − n i − µ (8)The integration of (6) and (7) with respect to ξ , andthe use of appropriate boundary conditions for localizedperturbations (viz. n i → u i →
0, and φ → ξ → ±∞ ) give rise to n i = 1 (cid:113) − φ M . (9)Now substituting (9) into (8), and integrating the re-sulting equation with respect to φ , we obtain an energyintegral [27, 28] in the form12 (cid:18) dφdx (cid:19) + V ( φ ) = 0 , (10) V ( φ ) = C − (1 + µ ) exp( φ ) − M (cid:114) − φ M + µφ. (11)The integration constant C [= 1 + µ + M ] is chosen un-der the condition V (0) = 0. The pseudo-potential V ( φ )allow us to express as[ V ( φ )] φ =0 = 0 , (12) (cid:20) dV ( φ ) dφ (cid:21) φ =0 = 0 , (13) (cid:20) d V ( φ ) dφ (cid:21) φ =0 = 12! (cid:20) M − (1 + µ ) (cid:21) , (14) (cid:20) d V ( φ ) dφ (cid:21) φ =0 = 13! (cid:20) M − (1 + µ ) (cid:21) . (15) It is obvious from (12) and (13) that the MIA SWsexist if and only if [ d V /dφ ] φ =0 <
0, which makesthe fixed point at the origin is unstable [28], and that[ d V /dφ ] φ =0 ( < ) 0 for the existence [28] of the MIA SWswith φ > φ < d V /dφ ] φ =0 = 0yields the critical Mach number M c (minimum value of calM above which the MIA SWs exist). Thus, from (14)we can define M c as M c = 1 √ µ . (16)The variation of M c with µ is graphically shown to findthe range of the values of µ and corresponding M forwhich the subsonic MIA SWs exist. The results aredisplayed in figure 1. It is clear from figure 1 that in FIG. 1: The range of the values of µ and corresponding M for which the subsonic MIA SWs exist. the region above the curve (indicated by the horizon-tal lines) the subsonic MIA SWs are formed, and thatin the region below the curve (indicated by the verti-cal lines) no solitary wave exists. On the other hand,[ d V /dφ ] φ =0 ( M = M c ) = 0 yields S c : The subsonicMIA SWs with φ > φ <
0) exist if S c > S c < S c = µ + 2 /
3. The latter implies that S c > < µ < ∞ , and that the MIA SWsexist only with φ > φ > V ( φ ) vs. φ curves to study the formationof arbitrary amplitude subsonic SWs for which the poten-tial wells are formed in + φ -axis. The numerical resultsare shown in figures 2 and 3. The potential wells in + φ axis in figure 2 or 3 represent the amplitude φ m (value of φ at the point where the V ( φ ) vis. φ curve crosses + φ -axis) and the width W (defined as W = | φ m | / (cid:112) | V m | ,where | V m | is the maximum value of V ( φ ) in the poten-tial wells) of arbitrary amplitude SWs. Thus figures 2and 3 indicate that the amplitude (width) of the sub-sonic SWs increases (decrease) with the rise of the valueof µ and M .We now study the basic features of small amplitude FIG. 2: The potential wells corresponding to arbitrary am-plitude subsonic SWs for M = 0 . µ = 0 . µ = 0 . µ = 0 . µ = 0 . M = 0 .
90 (solid curve), M = 0 .
94 (dotted curve), and M = 0 .
98 (dashed curve). subsonic SWs for which the pseudo-potential V ( φ ) [de-fined by (11)] can be expanded as V ( φ ) = C φ + C φ + · · · , (17) C = 12! (cid:20) M − (1 + µ ) (cid:21) , (18) C = 13! (cid:20) M − (1 + µ ) (cid:21) . (19)It is clear from (17) that the constant and the coeffi-cient of φ in the expansion of V ( φ ) vanish because of thechoice of the integration constant, and the equilibriumcharge neutrality condition, respectively. The approxi-mation V ( φ ) = C φ + C φ , which is valid as long as C n φ n (where n = 4 , , , · · · ) are negligible compared to C φ ), and the condition V ( φ m ) = 0 reduce the solitary wave solution of (10) to φ = (cid:18) − C C (cid:19) sech (cid:32)(cid:114) − C ξ (cid:33) . (20)We have graphically represented (20) for 1 > M > M c and µ > µ and M . Theresults are displayed in figures 4 and 5. It is obvi- FIG. 4: The small amplitude subsonic MIA SWs for M = 0 . µ = 0 . µ = 0 . µ = 0 . µ = 0 . M =0 .
90 (solid curve), M = 0 .
94 (dotted curve), and M = 0 . ous from figures 4 and 5 that stationary positive dustspecies supports the small amplitude subsonic SWs in anelectron-ion-positive dust plasma system. The variationof their amplitudes and width with µ and M are obviousfrom figures 4 and 5. The latter indicate that the ampli-tude (width) of these subsonic SWs increase (decrease) FIG. 6: The time evolution of the subsonic SWs in cylindrical( ν = 1) geometry for U = 0 . µ = 0 . τ = −
20 (solid curve), τ = −
15 (dotted curve), τ = −
10 (dashed curve), and τ = − ν = 2) geometry for U = 0 . µ = 0 . τ = −
20 (solidcurve), τ = −
15 (dotted curve), τ = −
10 (dashed curve), and τ = − with the rise of the value of µ and M as we concludedfrom the direct analysis of the pseudo-potential V ( φ ) infigures 2 and 3.We finally examine the effect of nonplanar geometry ontime dependent supersonic SWs by using the reductiveperturbation method [29] which requires the stretchingthe independent variables [30, 31]: ζ = − (cid:15) ( r + V p t ) , (21) τ = (cid:15) t, (22) expanding the dependent variables [29–31]: n i = 1 + (cid:15)n (1) i + (cid:15) n (2) i + · · · , (23) u i = 0 + (cid:15)u (1) i + (cid:15) u (2) i + · · · , (24) φ = 0 + (cid:15)φ (1) + (cid:15) φ (2) + · · · , (25)and developing equations in various powers of (cid:15) . To thelowest order in (cid:15) , (3) − (5) yield n (1) i = − u (1) i V p , (26) u (1) i = − φ (1) V p , (27) V p = 1 √ µ . (28)We note that V p = M c , and thus the variation of V p is shown in figure 1. To the next higher order in (cid:15) , weobtain a set of equations: ∂n (1) i ∂τ − ∂∂ζ (cid:104) V p n (2) i + u (2) i + n (1) i u (1) i (cid:105) − νu (1) i V p τ = 0 , (29) ∂u (1) i ∂τ − V p ∂u (2) i ∂ζ − u (1) i ∂u (1) i ∂ζ = ∂φ (2) ∂ζ , (30) ∂ φ (1) ∂ζ = (1 + µ ) φ (2) − n (2) i + 12 (1 + µ )[ φ (1) ] . (31)The use of (26) − (31) gives rise to a modified Korteweg-deVries (K-dV) equation in the form ∂φ (1) ∂τ + ν τ φ (1) + A φ (1) ∂φ (1) ∂ζ + B ∂ φ (1) ∂ζ = 0 , (32)where A and B are nonlinear and dispersion coefficients,and are, respectively, given by A = 32 √ µ (cid:18) µ + 23 (cid:19) , (33) B = 12(1 + µ ) . (34)We note that ( ν/ τ ) φ (1) in (32) is due to the effect ofthe nonplanar geometry, and that ν = 0 or | τ | → ∞ corresponds to a 1D planar geometry. Thus, for a largevalue of τ , and for a frame moving with a speed U , thestationary solitary wave solution of (32) is [12] φ (1) = (cid:18) U A (cid:19) sech (cid:20)(cid:114) u B ( ζ − U τ ) (cid:21) . (35)We note that τ < τ → ν/ τ ) φ (1) in (32).We now numerically solve (32) to observe how the SWsevolve with time from past to present by using (35) as theinitial solitary pulse. The results for cylindrical ( ν = 1)and spherical ( ν = 2) geometries are shown in figures 6and 7, respectively. We note that for a large value of τ (viz. τ = 20) we got solid curves in figures 6 and 7,which represent subsonic SWs in 1D planar, cylindrical,and spherical geometries.To summarize, we have considered a space dustyplasma system containing electron, ion and positive dustspecies, and have identified the basic features of the sub-sonic SWs in such space dusty plasma systems. The re-sults obtained from this theoretical investigation are:1. The condition 0 < µ < ∞ causes to form the sub-sonic SWs with φ > µ .2. The amplitude (width) of the subsonic SWs in-creases (decreases) with the rise of the value of µ because of the decrease in the Mach number withthe increase in the value of µ .3. The amplitude of the subsonic SWs is small becauseof their existence for low Mach number ( M < ν/ τ ) φ (1) in the modified K-dV equation.The disadvantage of the pseudo-potential method isthat it does not allow us to observe the time evolutionof the arbitrary amplitude SWs. On the other hand, thereductive perturbation method allows as to observe thetime evolution of the small amplitude SWs, but it is notvalid for the arbitrary amplitude SWs.To overcome this limitation, one has to develop a cor-rect numerical code, and to solve the basic equations(3) − (5) numerically by this numerical code. This typeof numerical analysis will be able to show the time evo-lution of arbitrary amplitude SWs. This is, of course,a challenging research problem, but beyond the scope ofour present work.We finally hope that the results of this work should beuseful for understanding the physics of nonlinear phe-nomena like subsonic solitary like structures in manyspace plasma environments, viz. Earth’s mesosphere[15–17], cometary tails [18], Jupiter’s surroundings [19],Jupiter’s magnetosphere [20], etc. [1] L. Tonks and I. Langmuir, Phy. Rev. , 195 (1929).at[2] R. W. Revans, Phy. Rev.
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