Pis'ma v ZhETF, vol.95, iss.6, pp.310-316
© 2012 March 25
Cylindrical and spherical electron-acoustic Gardner solitons and double layers in a two-electron-temperature plasma with nonthermal ions
S. T. ShuchyV, A. Mannan, A. A. Mamun
Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh Submitted 19 January 2012
Cylindrical and spherical Gardner solitons (GSs) and double layers (DLs) in a two-electron-temperature plasma system (containing cold electrons, hot electrons obeying a Boltzmann distribution, and hot ions obeying a nonthermal distribution) are studied by employing the reductive perturbation method. The modified Gardner (MG) equation describing the nonlinear propagation of the electron-acoustic (EA) waves is derived, and its nonplanar GS- and DL-solutions are numerically analyzed. The parametric regimes for the existence of GSs, which are associated with both positive and negative potential, and DLs which are associated with positive potential, are obtained. The basic features of nonplanar EA GSs, and DLs, which are found to be different from planar ones, are also identified. The implications of our results in space and laboratory plasmas are briefly discussed.
1. Introduction. The idea of electron-acoustic (EA) mode had been conceived by Fried and Gould [1] during numerical solutions of the linear electrostatic Vlasov dispersion equation in an unmagnetized, homogenous plasma. It is basically an acoustic-type of waves [2] in which the inertia is provided by the cold electron mass, and the restoring force is provided by the hot electron thermal pressure. The ions play the role of a neutralizing background only. The spectrum of the linear EA-waves, unlike that of the well-known Lang-muir waves, extends only up to the cold electron plasma frequency wpc = (inncoe2¡trie)1/2, where nco is the unperturbed cold electron number density, e is magnitude of the electron charge, and me is the mass of an electron. This upper wave frequency limit (w ~ wpc) corresponds to a short-wavelength EA-wave and depends on the unperturbed cold electron number density nco■ On the other hand, the dispersion relation of the linear EA-waves in the long-wavelength limit (in comparison with the hot electron Debye radius A¿h = (feT^/A-nrihoe2)1/2, where T^ is the hot electron temperature, fee is the Boltzmann constant, and n^o is the unperturbed hot electron number density) is w ~ kCe, where k is the wave number and Ce = {ncok-B,Th/nhonig)1^2 is the EA speed [3]. Besides the well-known Langmuir and ion-acoustic waves, they noticed the existence of a heavily damped acoustic-like solution of the dispersion equation. It was later shown that in the presence of two distinct groups (cold and hot) of electrons and immobile ions, one indeed obtains a weakly damped EA-mode [2], the properties of which significantly differ from those of the Langmuir waves. Gary and Tokar
1) e-mail: shuchy_phys0yahoo.com
[3] performed a parameter survey and found conditions for the existence of the EA-waves. The most important condition is Tc -C Th, where Tc (T^) is the temperature of cold (hot) electrons. The propagation characteristics of the EA-waves have also been studied by Yu and Shukla [4], Mace and Hellberg [5-7] and Mace et al. [8].
Two-electron-temperature plasmas are known to occur both in laboratory experiments [9, 10] and in space environments [11-17]. The propagation of the EA-waves has received a great deal of renewed interest not only because the two-electron-temperature plasma is very common in laboratory experiments and in space, but also because of the potential importance of the EA-waves in interpreting electrostatic component of the broadband electrostatic noise (BEN) observed in the cusp of the terrestrial magnetosphere [12, 18], in the geomagnetic tail [19], in auroral region [11, 13, 15], etc.
The EA-mode has been used to explain various wave emissions in different regions of the Earth's magnetosphere [11, 15]. It was first applied to interpret the hiss emissions observed in the polar cusp region in association with low-energy 100 eV) upward moving electron beams [20]. The EA-mode was also utilized to interpret the generation of the BEN-emissions detected in the plasma sheath [19] as well as in the dayside auroral zone [11, 15]. Dubouloz et al. [11] rigorously studied the BEN observed in the dayside auroral zone and showed that because of the very high electric field amplitudes (lOOmV/m) involved, the nonlinear effects must play a significant role in the generation of the BEN in the dayside auroral zone. Dubouloz et al. [11, 15] also explained the short-duration (<ls) burst of the BEN in terms of electron acoustic solitary waves (EA-SWs):
such EA-SWs passing the satellite would generate electric field spectra. To study the properties of EA solitary structures, Dubouloz et al. [11] considered a one-dimensional, unmagnetized collisionless plasma consisting of cold electrons, Maxwellian hot electrons, and stationary ions. El-Shewy [21] has investigated the propagation of linear and nonlinear EA-SWs in a plasma containing cold electrons, nonthermal hot electrons, and stationary ions. The effects of arbitrary amplitude EA-SWs and electron acoustic double layers (EA-DLs) in a plasma consisting of cold electrons, superthermal hot electrons, and stationary ions has been considered by Sahu [22]. The EA-SWs in a two-electron-temperature plasma where ions form stationary charge neutral background has been observed by Dutta [23]. El-Wakil et al. [24] considered cold electrons, nonthermal hot electrons, and stationary ions, and studied the nonlinear properties of EA-SWs by using time-fractional Korteweg-de Vries (K-dV) equation. However, all of these studies [915,17,21-24] are limited to one-dimensional (ID) planar geometry, which may not be the realistic situation in space and laboratory devices, since the waves observed in space (laboratory devices) are certainly not infinite (unbounded) in one-dimension [25]. Thus, there are some space, where the energetic ions are observed. The energetic ions are described by a new distribution called nonthermal distribution [26-28]. The latter is now being common feature of the Earth's atmosphere and in general it is turning out to be a characteristic feature of space plasmas [29]. To the best of our knowledge no attempt has been made in order to study the effect of nonthermal (energetic or fast) ions on the electron acoustic Gardner solitons (EA-GSs) and EA-DLs. Therefore, in our present work, we consider a plasma system (consisting of cold electrons, hot electrons obeying a Boltzmann distribution, and hot ions following nonthermal distribution) and a more general geometry (which is valid for both planar, cylindrical and spherical geometries), and theoretically study the basic features of the EA-GSs and EA-DLs that are found to exist in such a realistic nonthermal plasma system.
2. Derivation of the MG-equation. We consider a nonplanar (cylindrical or spherical) geometry, and nonlinear propagation of the EA-waves in a nonplanar, collisionless unmagnetized plasma system consisting of cold electrons, hot electrons obeying a Boltzmann distribution, and hot ions obeying a nonthermal distribution. Thus, at equilibrium we have n« = n^o + nco, where n« is the nonthermal ion number density at equilibrium. The nonlinear dynamics of the EA-waves propagating in such a nonplanar plasma system is governed
by
dnc 1 d dt rv dr duc duc dt c dr i d
(:rvncuc) = 0,
dr' = -P,
(1) (2)
r"dr\' drj (3)
p = (1 + /3a<f> + - pe* - (1 - p)nc, (4)
where v = 0 for ID planar geometry, and v = 1 (2) for a nonplanar cylindrical (spherical) geometry; nc is the cold electron number density normalized by its equilibrium value nco, uc is the cold electron speed normalized by Ce, 4> is the wave potential normalized by Th/e, p is the surface charge density normalized by enho, p = nh0/ni0, a = Th/Th 7* is the ion temperature, and ¡3 = 4a/(1 + 3a) in which a is the nonthermal parameter [26-28]. The time variable t is normalized by w^1, and the space variable r is normalized by Adh-
To study finite amplitude EA-GSs and EA-DLs by the reductive perturbation method [30, 31], we first introduce the stretched coordinates:
c = e(r - vpt),
t = e3t,
(5)
(6)
where e is a small parameter (0 < c < 1) measuring the weakness of the dispersion, and Vp (normalized by Ce) is the phase speed of the perturbation mode, and expand all the dependent variables (viz. nc, uc, <j>, and p) in power series of e:
e4»
nc = 1 + uc = 0 + <j) = 0 + e^1) ■ (i)
p = 0 + ep,
- e2t42)
e2<^2) -
• eV2) -
• c3^3) -
• eV3) -
(7)
(8) (9)
(10)
Now, expressing (l)-(4) in terms of ( and r by using (5), (6), and substituting (7)-(10) into the resulting equations, one can easily develop different sets of equations in various powers of e. To the lowest order in e one obtains
uj, ^ — , Tlr ^ —
V2'' "p
■M
pw =0 V2 = —
(11) (12)
where = The expression for Vp in (12) represents the linear dispersion relation for the EA-waves propagating in a plasma under consideration. To the next higher order in c, we obtain another set of equations, which, after using (11), (12), can be simplified as
u(2) ~ c ~
4>2 (j>w
n(2) _ vp' c 2v;
V2 ' "p
(13)
,(2) = ±Ai/>2 = 0, A = a2
vp
It is obvious from (14) that A = 0, since i{> ^ 0. The solution of A = 0 for a is given by
a = ar =
6(a2 - p2) + 3p(l + a)2 ± Aarj 18/t2 + 9p(a - l)2 - 6a2 '
(15)
where rj = y/3(p — l)(/t — cr2). We have numerically shown how qc varies with p and cr. The result is displayed in Fig. 1 which, in fact, represents the A = 0 sur-
Fig. 1. (Color online) Showing how ac varies with a and ¡i for A(a = ac) = 0
face plot, and provides us the parametric regimes (which correspond to above or below the A = 0 surface plot) of our present interest. So, for a around its critical value (qc), i.e. for |a — ac\ = e corresponding to A = A0, we can express A0 as
A0 ~ s
8A da
| a
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