ФИЗИКА ПЛАЗМЫ, 2013, том 39, № 7, с. 622-629
ПЫЛЕВАЯ ^^^^^^^^^^^^^^^^ ПЛАЗМА
УДК 533.9
DUST-ACOUSTIC SOLITARY WAVES IN A FOUR-COMPONENT ADIABATIC MAGNETIZED DUSTY PLASMA
© 2013 г. T. Akhter, A. Mannan, and A. A. Mamun
Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh e-mail: tahminaakhter84@yahoo.com Поступила в редакцию 21.05.2012 г. Окончательный вариант получен 27.09.2012 г.
Theoretical investigation has been made on obliquely propagating dust-acoustic (DA) solitary waves (SWs) in a magnetized dusty plasma which consists of non-inertial adiabatic electron and ion fluids, and inertial negatively as well as positively charged adiabatic dust fluids. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation which admits a solitary wave solution for small but finite amplitude limit. It has been shown that the basic features (speed, height, thickness, etc.) of such DA solitary structures are significantly modified by adiabaticity of plasma fluids, opposite polarity dust components, and the obliqueness of external magnetic field. The SWs have been changed from compressive to rarefactive depending on the value of ц (a parameter determining the number of positive dust present in this plasma model). The present investigation can be of relevance to the electrostatic solitary structures observed in various dusty plasma environments (viz. cometary tails, upper mesosphere, Jupiter's magnetosphere, etc.).
DOI: 10.7868/S0367292113070019
1. INTRODUCTION
There has been a great deal of interest in understanding linear and nonlinear features of the novel dust-acoustic (DA) waves [1], not only because they exist in both space dusty plasmas [2, 3], but also because they triggered a number of remarkable laboratory experiments [4—8]. Rao et al. [1] have first theoretically predicted the existence of this novel extremely low phase velocity (in comparison with the electron and ion thermal velocities) DA waves, where the dust mass provides the inertia and the electron and ion thermal pressures give rise to the restoring force. The prediction of Rao et al. [1] has then conclusively been verified by a number of laboratory experiments [4—6]. The linear features of these novel DA waves have also been extensively studied for some other situations [9—11].
Rao et al., in their seminal work [1], have also studied small, but finite amplitude DA solitary waves (SWs). Various authors [12—14] have then generalized the work of Rao et al. [1] to study arbitrary amplitude SWs. The nonlinear DA waves have also been rigorously investigated by many authors for different dusty plasma situations theoretically [15—29] as well as experimentally [7, 8]. However, all of these works on nonlinear DA waves [1, 12—29] are based on the most commonly used dusty plasma model that assumes negatively charged dust. The consideration of negatively charged dust is due to the fact that in low-temperature laboratory plasmas, collection of plasma particles (viz. electrons and ions) is the only important charging process, and the thermal speeds of electrons
far exceeds that of ions. But, there are some other more important charging processes by which dust grains become positively charged [30—33]. The principal mechanisms by which dust grains become positively charged, are photoemission in the presence of a flux of ultraviolet photons [30, 31], thermo-ionic emission induced by the radiative heating [33], secondary emission of electrons from the surface of the dust grains [32], etc.
There is direct evidence of the co-existence of positively and negatively charged dust in different regions of space, viz. Earth's mesosphere [34—37], cometary tails [38, 39], Jupiter's magnetosphere [38, 40], etc. Chow et al. [32] have theoretically shown that due to the size effect on secondary emission, insulating dust grains with different sizes can have the opposite polarity, smaller ones being positive and larger ones being negative. The opposite situation, i.e. larger (massive) ones being positive and smaller (lighter) ones being negative, is also possible by triboelectric charging [41, 42]. This is predicted from the observations of dipolar electric fields perpendicular to the ground, with negative pole at higher altitudes, generated by dust devils [43, 44] and sand storms [45].
The coexistence of positively and negatively charged dust, with larger (massive) dust being positive and smaller (lighter) dust being negative [46—48] or vice versa [49], is also observed in laboratory devices [46—49] where dust of polymer materials are used. It may be noted here that the coexistence of same sized dust of opposite polarity may also occur by photoemission if the photoemission yields of the dust-material
are very different [50]. Recently, motivated by these theoretical predictions and satellite/experimental observations, a number of authors [41, 51—56] have considered a dusty plasma with dust of opposite polarity, and have investigated linear [41, 52] and nonlinear [51, 53-56] DA waves.
Recently, Tasnim et al. [57] assumed a four-component adiabatic dusty plasma containing adiabatic electrons, adiabatic ions, adiabatic positively and as well as negatively charged warm dust. The above mentioned works [51, 53-58] are limited to one-dimension, which may not be the realistic situation in space and laboratory devices, and the effects of the external magnetic field have not been considered in all of the previous works. We know that, the effect of external magnetic field drastically modifies the properties of electrostatic solitary structures [59-62]. However, using reductive perturbation method the effects of magnetic field on solitary structures of dusty plasma, have been successfully studied by various authors [63-67]. Bedi et al. [64] have considered a dusty plasma system consisting of non-thermal electrons, Maxwellian ions, negatively and positively charged warm adiabatic dust particles, and studied the basic features of the electrostatic solitary structures by deriving the Korteweg-de Vries (K-dV) equation.
It is now well known that the effects of dust fluid temperature and adiabaticity of ion or electron fluid significantly modify all the basic nonlinear features of such ultra-low-frequency DA waves. Therefore, in our present work, we consider more general, more realistic, and consistent magnetized dusty plasma system (containing non-inertial adiabatic electron and ion fluids, and inertial negatively as well as positively charged adiabatic dust fluids, and investigate the basic properties of finite amplitude DA SWs by the reductive perturbation method [68].
The paper is organized as follows. The basic equations governing the dusty plasma system under consideration are given in Section 2. The K-dV equation is derived in Section 3. The solution of K-dV equation is examined in Section 4. A brief discussion is presented in Section 5.
2. GOVERNING EQUATIONS
We consider a magnetized four-component adiabat-ic dusty plasma system consisting of negatively charged warm adiabatic dust, positively charged warm adiabatic dust, and adiabatic electrons as well as ions in the presence of an external static magnetic field B 0 = B0Z (where z is unit vector along the z-direction). Thus, at equilibrium we have ni0 + Zpnp0 = ne0 + Znnn0, where ni0, np0, ne0, nn0, are, respectively, ion, positive dust, electron, and negative dust number density in equilibrium, and Zp (Zn) represents the charge state of positive (negative) dust. This model is relevant to dusty plasmas in cometary tails [38, 39], upper mesosphere
[34-37], and Jupiter's magnetosphere [38, 40], where dust is charged by the secondary emission or photoemission or thermionic emission, and the dust size effect on the latter is important. The dynamics of the three dimensional DA waves [3, 13, 22] in such a dusty plasma system is governed by
^ + V • (n,u s) = 0,
dt
Dnun = Vy - acd(u n x z) - ^ Vpn,
nn
Dpup = -|aVy + itoCd(up x Z) - ^\pn,
np
+ (u, • V)ps +yp, (V • u,) = 0,
dt
V V ■
eT
jeTF Vpj = 0,
qj
(1)
(2)
(3)
(4)
(5)
(6)
V y = Ven - Vn - Vpnp + n„,
where ns is the number density of the species s (s = i for ions, s = e for electrons, s = n for negative dust, and s = p for positive dust) normalized by its equilibrium value ns0, D„ p = d/dt + (up ■ V), unp is the fluid velocity of the negatively and positively charged dust normalized by the speed of the negative dust
Cdn = (ZnkBTi/4nn„0e2Z2n)1/2, Dn^p = d/dt + (uB>p • V), y is the electrostatic wave potential normalized by kBT/e, pp (pn) is the thermal pressure of the positive (negative) dust fluid normalized by its equilibrium value nsokBTs, the time and space variables are in units of the dust plasma period ro-l = (mn /4nnn0e1Z1n )1/2 and the Debye lenth X Dm = (ZJ^bT /4nn„0e2Z„2 )1/2 respectively, a = Te /T, CTi = T„ /ZTi, a 2 = Tpm„ /Z^p, acd = (ZneB0 /mn)/<$d„, is the dust cyclotron frequency normalized to &dn, y(= ye = yl = yp = yn) is the adiabatic coefficient, ^ = Zpm„ /Z„mp, ^ e = n0 /Znnd), V = ni0/Z„n„o, and ^p(= Zpnp0/Z„nM) = 1 + ^e - ^. Z„ (Zp) is the number of electrons (protons) residing on a negative (positive) dust, mn (mp) is the mass of the negative (positive) dust particle, T (Te) is the ion (electron) temperature, Tn (Tp) is the temperature of the negative (positive) dust fluid, kB is the Boltzmann constant, and e is the magnitude of the electron-charge. It should be mentioned here that we have neglected the charge fluctuation since the time period (which is a fraction of a second [3]) of the DA waves under consideration is much larger than the dust charging time (which is order of few micro-second [3]). We note that this appropriate approximation has been used by many authors to study the linear and nonlinear propagation of the DA waves [1, 3, 12, 13, 64, 67].
3. DERIVATION OF K-dV EQUATION
We first investigate the basic features of the small amplitude electrostatic SWs by the reductiv
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