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On the instability of electrostatic waves in a nonuniform electron–positron magnetoplasma
Physics Letters A 372 (2008) 4067–4075 www.elsevier.com/locate/pla
On the instability of electrostatic waves in a nonuniform electron–positron magnetoplasma W.F. El-Taibany a,∗,1 , W.M. Moslem b,2 , Miki Wadati a , P.K. Shukla b,3 a Department of Physics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan b Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 27 February 2008; accepted 19 March 2008 Available online 22 March 2008 Communicated by F. Porcelli
Abstract The dispersion properties of three-dimensional electrostatic waves in a nonuniform electron–positron (EP) magnetoplasma are analyzed. A new dispersion relation is derived by use of the electron and positron density responses arising from the electron and positron continuity and Poisson equations. In the local approximation, the dispersion relation admits two wave modes with different velocities. The growth rates of various modes are illustrated both analytically and numerically. Considering the temperature gradients produces a linearly stable transverse mode. The growth rate of the slow mode instability due to the density inhomogeneity only is the highest one, though it appears at higher thermal energy. The angle of the wave propagation affects drastically on the instability features in each case. The applications of the present analysis are briefly discussed. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Pair plasmas are composed of charged particles with the same mass and opposite charges, which admit the time and space scales that significantly differ from those of an electron–ion plasma [1–3]. Electron–positron (EP) plasmas are found in the Van Allen belts, in active galactic nuclei (AGN), near the polar cusp regions of the pulsars magnetospheres, in the early universe, and in the inner region of the accretion disks surrounding the central black holes [2,4–7]. Also, EP plasmas may be formed in the magnetospheres of pulsars by the pair-production cascade breeding process [8]. In addition, laboratory experiments have opened up the possibility of creating a nonrelativistic EP plasma. In one scheme, a relativistic electron beam impinges on a high-Z target, where positrons are produced copiously. The pair plasma is then trapped in a magnetic mirror and is expected to cool rapidly by emission of radiations [9]. In another scheme, positrons are accumulated from a radioactive source [10]. EP pair production is also possible during intense ultrashort laser pulse propagation in plasma [11]. Extensive works have been done to investigate the linear and nonlinear waves in pair plasmas (see e.g. Ref. [2,12,13]). Because most laboratory experiments and astrophysical objects contain nonuniform plasmas, it is of practical interest to investigate the instability of nonuniform EP plasmas [1,14]. The plasma collective mode remains stable near the thermodynamic equilibrium state, while it becomes unstable for a non-equilibrium state due to the free energy source availability. There could be different types of free energy sources, such as the streaming of plasma particles [15–17], spatial variations of the equilibrium * Corresponding author.
E-mail addresses: [email protected], [email protected] (W.F. El-Taibany), [email protected], [email protected] (W.M. Moslem), [email protected] (M. Wadati), [email protected] (P.K. Shukla). 1 Permanent address: Department of Physics, Faculty of Science-Damietta, Mansoura University, Damietta El-Gedida, P.O. 34517, Egypt. 2 Permanent address: Department of Physics, Faculty of Education-Port Said, Suez Canal University, 42111, Egypt. 3 Also at School of Physics, University of KwaZulu-Natal, Durban 4000, South Africa; and Department of Physics, Umeå University, SE-90187 Umeå, Sweden. 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.03.024
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number density [15–18], the magnetic field [16,17] and the plasma temperature inhomogeneities [16,18,19]. Such gradients of the physical quantities can drive collective modes unstable [14]. For this purpose several authors have paid attention to explain the enhanced fluctuations which play a significant role in the anomalous transport phenomena such as heat transport across the external magnetic field lines [14–20]. For example, Shukla and Weiland [21] investigated the toroidal electron–temperature–gradient (ETG) driven electromagnetic drift waves. They suggested their model for the soft beta limitation as well as the anomalous energy transport in a tokamak reactor. Employing kinetic ion and fluid electron responses and including the ETG, two broad unstable fluctuations have been derived [22]. Shukla et al. [16] highlighted the combined effects of the ETG and parallel electron velocity gradient on electrostatic instabilities in a nonuniform magnetoplasma. Recently, Shukla and Shukla [17] presented a new purely growing instability in a nonuniform EP plasma with equilibrium density and magnetic field inhomogeneities. In this Letter, we present a study of low-frequency (in comparison with the electron gyrofrequency) electrostatic oscillations in a nonuniform EP magnetoplasma. By employing the two-fluid model for EP plasmas, we derive the dispersion relation taking into account the sheared plasma flow as well as the gradients of the density, the external magnetic field and the plasma temperature. 2. The model-dispersion relation Let us consider a three-dimensional nonuniform magnetoplasma composed of the electrons and positrons in the presence of density, velocity, temperature and magnetic field gradients along the x-axis. The external magnetic field is B0 = B0 (x)ˆz, where zˆ is the unit vector along the z-direction in a Cartesian coordinate system. The electron/positron fluid velocity is U α = uα⊥ + [Uα0 (x) + uαz ]ˆz, where Uα0 , uα⊥ and uαz are the streaming, perpendicular and parallel components to zˆ of the electron and positron fluid velocities, where the subscript α denotes either the positron (α = p) or the electron (α = e). Also, α will be used as a multiplicative factor, denoting the charge sign α = +1(−1), referring to the positron (electron) component, respectively. The charge-neutrality condition for the EP plasma at equilibrium is Ne0 (x) = Np0 (x) = N0 (x), where Ne0 (Np0 ) is the background number density of the electrons (positrons). Also, Te0 (x) = Tp0 (x) = T0 (x), where Te0 (Tp0 ) is the unperturbed temperature of the electrons (positrons). In order to investigate the low-frequency electrostatic waves in comparison with the electron gyrofrequency ωce = eB0 /mc, where e, m and c are the magnitude of the electron charge, the electron/positron mass and the speed of light in vacuum, respectively, we use the basic set of linearized continuity equations [16,17], ∂uαz Dα nα1 + ∇ ⊥ · (N0 uα⊥ ) + N0 = 0, Dt ∂z the momentum equations, −αe 1 1 Dα ∇ ⊥ pα1 , uα⊥ = ∇ ⊥ φ − uα⊥ × B0 zˆ − Dt m c mN0 Dα −1 ∂φ 1 ∂pα1 uαz + uα⊥ · ∇ ⊥ Uα0 = αe + , Dt m ∂z N0 ∂z
(1)
(2) (3)
the energy equation [18,21,23], 2Pα0 ∂uαz Dα Tα1 + uα⊥ .∇ ⊥ T0 + = 0, ∇ ⊥ · uα⊥ + Dt 3N0 ∂z and the Poisson equation, αnα1 = 0. ∇ 2 φ + 4πe
(4)
(5)
α=−1,+1
ˆ ˆ Here Dα /Dt = ∂/∂t + Uα0 ∂/∂z, ∇ ⊥ = x∂/∂x + y∂/∂y with xˆ and yˆ being the unit vectors along the x- and y-axes, nα1 N0 is the small electron and positron density perturbation, Tα1 T0 is the electron and positron temperature perturbation, and φ is the electrostatic potential. In the steady state the following relation should then hold [17,24]: B2 d N0 T0 + 0 = 0. dx 8π For the low-frequency waves, we assume that |Uα0 ∂/∂z|, |∂/∂t| ωce and |∂/∂z| |∇ ⊥ |, and therefore we can neglect the left-hand side of Eq. (2). Now, we shall calculate the perpendicular component of the electron and positron fluid velocities by taking the cross product from the left of Eq. (2) by zˆ and using the identity A × B × C = (A · C)B − (A · B)C to obtain c α uα⊥ = (6) zˆ × ∇ ⊥ (T0 nα1 + Tα1 N0 ) . zˆ × ∇ ⊥ φ + B0 eN0
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Substituting the expressions of uα⊥ into Eqs. (1) and (4), then combining the produced equations, we obtain Dα ∂ ∂φ + aα2 nα1 + aα3 = 0, D˜ α Tα1 − aα1 Dt ∂y ∂y where
∂ Dα D˜ α = − aα4 , Dt ∂y
aα1 =
2T0 , 3N0
aα2 =
T0 aα4 , N0
aα3 = −αeaα4 ,
aα4 =
(7)
αcT0 2 LT − Ln , eB0 3
with the notation Ln = d ln N0 /dx
and LT = d ln T0 /dx.
Also, substitution of (6) into (1) yields ∂uαz ∂Tα1 ∂φ Dˆ α nα1 + N0 + aα5 − aα6 = 0, ∂z ∂y ∂y
(8)
where
T0 ∂ Dα Dˆ α = + aα5 , Dt N0 ∂y
aα5 =
αcN0 LB , eB0
aα6 =
cN0 (Ln − LB ), B0
and LB = d ln B0 /dx. On the other hand, the parallel velocity for the α-component can be evaluated from (3) Dα −Gα T0 nα + Tα1 , uαz = αeφ + Dt m N0 1
(9)
where Gα =
∂ ∂ α ∂Uα0 ∂ ∂ − = + Sα , ∂z ωce ∂x ∂y ∂z ∂y
and Sα = −
α dUα0 . ωce dx
We use Eq. (9) to eliminate uαz from Eq. (8). Then solving the resultant equation with Eq. (7), we obtain the differential equation for α-species number density in terms of the electrostatic potential, φ, ∂ Dα ∂ ∂ Dα ∂ N0 Dα ˆ D˜ α Dα − VT2 Gα + aα5 − Gα aα1 + aα2 nα1 Dt ∂z Dt ∂y m ∂z Dt ∂y Dα ∂ ∂ Dα ∂ 2 ∂2 N0 αeN0 = D˜ α aα6 − (10) + Gα + aα3 aα5 G φ, α Dt ∂y m ∂z Dt ∂y 2 m ∂y∂z √ where VT = T0 /m. Assuming that nα1 and φ are proportional to exp(−iωt + iky y + ikz z), we Fourier transform (10) to get in the local approximation, nα1 = −(Zα /Qα )φ,
(11)
where Zα = Ωα Ωα ky aα6 − kz bα1 (kz + Sα ky ) − bα2 ky2 ,
5 Qα = Ωα Ωα2 − bα3 ky Ωα − VT2 kz (kz + Sα ky ) , 3
with Ωα = ω − kz Uα0 ,
bα1 =
αeN0 , m
bα2 =
−cN0 Ln aα4 , B0
bα3 = −
αcT0 (3LT − 2Ln − 5LB ). 3eB0
Inserting the expressions (11) for ne1 and np1 into (5), we obtain the desired dispersion relation, Ze 4πe Zp − = 0, 1+ 2 Qp Qe k where k 2 = ky2 + kz2 . The following section is devoted to present various limiting cases of the dispersion relation, Eq. (12).
(12)
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3. Limiting cases 3.1. Wave propagation parallel to the magnetic field First, we consider the case of wave propagation along B0 (x)ˆz. Let kz = k cos θ and ky = k sin θ , so in this case, θ = 0 (i.e., ky = 0) and therefore ω = kU0 + 2χωpe , (13) where
1/2 ωpe = 4πe2 N0 /m
and χ = 1 +
5 kVT 2 . 6 ωpe
For simplicity, we assume that Ue0 (x) ≈ Up0 (x) = U0 (x). The produced wave frequency, (13), shows that all the electrostatic waves propagating parallel to the magnetic field are stable and independent of either the magnetic field strength or the plasma inhomogeneity √ frequency is not only modified by the thermal energy but also it grows √ included. Eq. (13) illustrates that the wave by a factor 2. The reason for this modified frequency, 2ωpe , is the electron and positron density-oscillations with the same time scale, but 180◦ out of phase, though the charge densities are in phase [2]. 3.2. Wave propagation transverse to the magnetic field For the transverse mode, θ = 90 (i.e., kz = 0), Eq. (12) leads to the following dispersion relation 8πe be2 ωtr = kbe3 1 + 2 ae6 − , be3 k be3
(14)
and therefore the instability occurs for 8πe (15) (ae6 be3 − be2 ) < 0. k2 Now and thereafter, we let ωj = iγj , where γj is the growth rate and j is an index for the case study. So, the growth rate of this mode is given as
2 + 8πe (b − a b ). γtr = k −be3 (16) e2 e6 e3 k2 Eq. (15) can be rewritten leading to the following inequality, 2 be3 +
ηn− < ηn < ηn+ ,
(17)
where
√ 12(χ − 1) + 5ηB (7 − 4χ) ± 5 ηB [ηB (49 − 40χ) + 24(χ − 1)] ηn± = 8(χ − 1) 5ηB 3 5 + ≈ 1± √ (18) (7 − 4χ ± 49 − 40χ ), 2 8(χ − 1) 49 − 40χ with ηn = Ln /LT and ηB = LB /LT . It is noted here that in order to have a real value for either ηn+ and ηn− , we should verify that ηB > 0 or ηB > 24(1 − χ)/(49 − 40χ). For large and positive value of ηB , the two critical instability boundaries can be approximated as given in the second line of (18). However, for zero density gradient, the transverse mode obeys the rule, −1 kV 2 LT ωtr,Ln =0 = T (19) (3 − 5ηB ) 1 − 5 (χ − 1)(3 − 5ηB ) , 3ωce which exhibits the instability only when 3/5 > ηB > 3(χ − 1)/5χ . On the other hand, the other limit for zero magnetic field inhomogeneity, Eq. (14) leads to a stable mode whose angular frequency is kVT2 LT (20) (3 − 2ηn ). 3ωce This mode describes a linear propagating wave affected by the density gradient and ETG. Fig. 1 shows the variation of γtr against the variation of Ln , LB , and LT . It declares that the transverse mode starts as a growing wave, though after a certain value of LT , its behaviour is changed to be a damping wave. The transition point is affected by the changes of Ln and LT . Increasing LB reduces the growth (damping) rate of the growing (damping) wave. On the other hand, the increase of Ln causes an increase of both the growth and the damping rates of the transverse wave. Now let us classify the other limiting cases categorized according to which inhomogeneity effect(s) is (are) included. ωtr,LB =0 =
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Fig. 1. γtr (in terms of ωpe ) is plotted against LT . Solid curve is for Ln = 0.5 and LB = 0.25, dashed curve is for Ln = 0.5 and LB = 0.35 and dotted curve is for Ln = 1 and LB = 0.25, respectively, where VT /ωce = 0.5 and χ = 1.25.
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Fig. 2. γs (in terms of ωpe ) is plotted against θ (in terms of π ). Solid curve is for Se = 0.4, dashed curve is for Se = 0.7, dotted curve is for Se = 1 and the upper curve is for Se = 1.25, respectively, with χ = 1.25.
3.3. Only shear flow Including only the velocity gradient, i.e. Sα = 0 and Ln = LT = LB = 0, the EP plasma allows a propagating wave whose angular frequency is
ωs = cos θ kU0 + ωpe 2χ − 1 ± 1 + χ(χ − 1)(2Se tan θ )2 . (21) Equation (21) permits two propagating waves; the positive (negative) sign corresponds to the fast (slow) mode. The fast mode frequency can be approximated as ωsf = cos θ kU0 + 2χ ωpe 1 + (χ − 1)(Se tan θ )2 + · · · , (22) while the slower mode is governed by √ ωss = cos θ kU0 + 2ωpe χ 1 + (1 − χ)(Se tan θ )2 + · · · − 1 .
(23)
Equation (21) shows that the argument of the inner square root is always yielding to a positive real value, though the instability may occur due to that of the outer square root. Therefore, the fast mode is stable whatever the value of the shearing flow and/or the angle of propagation, θ . Since, the slow mode suffers from instability when the shear flow satisfies √ √ Se > cotan θ/ χ or Se < −cotan θ/ χ , (24) this condition declares that the instability depends on the angle of propagation as well as the thermal energy parameterized in terms of χ . Eq. (24) is valid for 1 χ(χ − 1)(2Se tan θ )2 . The growth rate of this slow mode is given by
γs = ωpe cos θ 1 − 2χ + 1 + χ(χ − 1)(2Se tan θ )2 . (25) It is obvious from the present case as well as the preceding cases that the shear flow affects only on the oblique propagating EP wave, though the parallel and transverse waves are independent of the velocity gradient effect. Fig. 2 illustrates the dependence of γs on both of Se and θ . As shown, the growth rate first increases with θ , afterward, γs decreases when we go closer to the transverse direction. The increase of Se allows the propagation of this mode at lower angle of propagation. 3.4. Only magnetic field inhomogeneity, LB = 0 Considering only the inhomogeneity due to magnetic field while switch off the other inhomogeneities, we can evaluate the dispersion relation from (12) as
ωB = cos θ kU0 + ωpe 2χ − 1 + A ± 1 + A 2(2χ − 1) + A , (26) where ψ = (VT tan θ/ωce )2 and A = 5χψL2B /3. Similar to the previous case, the fast mode is stable, since the slow mode suffers from instability when LB > LB+
or
LB < LB− ,
with ±ωce cotan θ LB± = VT
6 χ 1 1− −1 . 5 χ χ −1
(27)
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Fig. 3. γB (in terms of ωpe ) is plotted against θ (in terms of π ). Solid curve is for LB = 6, dashed curve is for LB = 10, and dotted curve is for LB = 15, respectively, where VT /ωce = 0.5 and χ = 1.25.
Eq. (27) is verified for small χ values. The growth rate of this instability can be calculated as
γB = ωpe cos θ 1 − 2χ − A + 1 + A 2(2χ − 1) + A . The slow mode here follows the approximated dispersion law A + ··· , ωBs = cos θ kU0 + ωpe 2(χ − 1) + A 2(1 − χ) − 2 while the fast mode obeys
ωBf
(28)
(29)
A = cos θ kU0 + ωpe 2χ + A 2χ + + ··· . 2
(30)
Fig. 3 shows the dependency of the growth rate, γB , on both LB and θ . It illustrates that, for small (higher) values of θ , γB increases (decreases) by the increase of θ . However, as LB increases, γB increases. 3.5. Only temperature gradient, LT = 0 In case of ETG, the EP plasma obeys the following dispersion relation
ωT = cos θ kU0 + ωpe 2χ − 1 + B ± 1 + B 2(2χ − 1) + B ,
(31)
where B = 3ψL2T (χ − 1)/5. Comparison of (26) and (31) reveals that they have the same formula though the difference in the definitions of A and B. Therefore, following the same calculation as what we have done for the case of “only magnetic field inhomogeneity”, we finally get the same dispersion relations for the slow [fast] mode similar to (29) [(30)], respectively; remember to replace A by B. Although, we have evaluated the growth rate, (28), corresponding to the instability occurred for the slower mode in the previous case, the present case does not present any instability for either the fast/slow modes. Therefore, consideration of only ETG results in two different stable modes. 3.6. Only density gradient, Ln = 0 Considering only density gradient, we see that the EP plasma is governed by the following dispersion relation
ωn = cos θ kU0 + ωpe 2χ − 1 + C ± 1 + C 2(8χ − 27) + C , where C = (2ηn /3)2 B. Here, we can separate the dispersion relation for the fast mode to yielding
C 4(4χ − 13) + C + · · · , ωnf = cos θ kU0 + ωpe 2χ + 2 while the slow mode obeys
C ωns = cos θ kU0 + ωpe 2(χ − 1) + 8(7 − 2χ) − C + · · · . 2
(32)
(33)
(34)
As for the previous studied cases, the fast mode is stable while the slow mode suffers from instability when Ln > Ln+
or
Ln < Ln− ,
(35)
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Fig. 4. γn (in terms of ωpe ) is plotted against θ (in terms of π ). The variation of Ln and the curves indication are the same as for LB in Fig. 3 with VT /ωce = 0.5 and χ = 50.
with Ln± =
√ ± 15/2ωce cotan θ , √ VT 2(2χ − 7) + 195 + χ(16χ − 111)
and the growth rate in this case can be calculated as
γn = ωpe cos θ 1 − 2χ − C + 1 + C 2(8χ − 27) + C .
(36)
Fig. 4 shows that by the increase of either θ or Ln , the growth rate increases, though this instability appears for a higher χ values. Also, the growth rate of this mode is remarkably higher than other previous ones. 3.7. General case As Eq. (12) is the exact dispersion relation for the present EP plasma, it is desirable to solve the following algebraic equation to find out the value of ω, d 1 + d2 k2 = 0, + 2 8πe d3 − d42
(37)
where
d1 = d3 (k cos θ )2 be1 + be2 tan2 θ , d2 = d4 k sin θ (−ae6 Ω + Se be1 k cos θ ), 5 5 2 2 2 d4 = k sin θ be3 Ω + VT kSe cos θ . d3 = Ω − (VT k cos θ ) , 3 3 For the ETG, magnetic field, density inhomogeneities are more stronger than Se , then we can assume Se ≈ 0. The dispersion relation, (12), yields to
ωG = cos θ kU0 + ωpe 2(χ − 1) − f1 ± f12 + f2 , (38) with be2 f2 , tan2 θ + f1 = 1 + be1 4 cos2 θ
tan θ 2 8πeae6 f2 = 2(χ − 1) be3 be3 + . VT k2
If f2 f12 , Eq. (38) can be rewritten to give the dispersion relation for the two modes (slow and fast) respectively as f2 ωGs = cos θ kU0 + ωpe 2(χ − 1) − f1 2 + + · · · , 2f12 f2 ωGf = cos θ kU0 + ωpe 2(χ − 1) + + ··· . 2f1
(39)
(40)
For our present EP model, it is proposed that any inhomogeneity effect may vary with either positive or negative gradient. So, Eqs. (39) and (40) state that both of the two modes could suffer from instability depending on the values of f1 and f2 . The numerical analysis does not present any instability for the fast mode, though the slow mode has an instability affected by the variation of f1 and f2 . Therefore, we can say that the slow mode growth rate is
γGs = ωpe cos θ 2(1 − χ) + f1 + f12 + f2 . (41)
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Fig. 5. γG (in terms of ωpe ) is plotted against θ (in terms of π ). Solid curve is for LT = 0.8, dashed curve is for LT = 1.5, and dotted curve is for LT = 3, respectively, where VT /ωce = 0.5, Ln = 0.5, LB = 0.25 and χ = 1.25.
Fig. 5 focuses on the variation of the growth rate against ETG, which is the main point in the present letter, as well as the propagation obliqueness on the magnetic field direction. It illustrates that, for smaller value of θ , the growth rate slightly decreases as θ increases, though as we go closer to the transverse direction (after θ ≈ π/3), it turns to grow to a higher value. Here, the ETG supports the increase of the growth rate, while consideration of ETG only yields two stable wave modes. 4. Conclusion To summarize, we have considered the propagation of the low-frequency electrostatic wave in a nonuniform EP pair magnetoplasma containing density, velocity, temperature and magnetic field inhomogeneities. In the parallel direction to the magnetic field, all the wave modes are stable, though their frequencies are modified because of the thermal energy. The shear flow effect extends the regime of the propagation (cf. Fig. 2). Comparing to the recent work which has been done by Shukla and Shukla [17], the present model reveals two wave modes with different velocities. The inclusion of the electron temperature gradient (ETG) produces two stable wave modes, (31). However, in the other limiting cases, the fast (slow) mode is stable (suffers from instability under certain conditions). The growth rate of each case has been calculated. Considering the density inhomogeneity leads to a remarkable higher growth rate with higher thermal energies, (36). Including all the four-inhomogeneity effects, ETG comes into play to bear an instability for the slower mode. The angle of the wave propagation changes drastically the feature of the growth rate in each case (cf. Figs. 1–5). Mathematically, the fast mode may have an instability, though the numerical analysis with various parameter regimes does not show any instability feature. In general, the instability in the current model comes from the slower modes. The present results may be useful for understanding the properties of low-frequency electrostatic waves in intense laser-plasma interaction experiments [25], as well as in astrophysical objects, such as the neutron stars [26]. In addition, for further extensions, the existence regime of solitary electrostatic waves as well as the effect of dust particle presence in the present EP plasma would be interesting topics to analyze [27]. Acknowledgements W.F.E. and M.W. gratefully acknowledge support from the Inoue Foundation of Science, Tokyo, Japan. W.M.M. thanks the Alexander von Humboldt-Stiftung (Bonn, Germany) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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On the instability of electrostatic waves in a nonuniform electron–positron magnetoplasma W.F. El-Taibany a,∗,1 , W.M. Moslem b,2 , Miki Wadati a , P.K. Shukla b,3 a Department of Physics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan b Institut für Theoretische Physik IV, Fakultät für Physik und Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 27 February 2008; accepted 19 March 2008 Available online 22 March 2008 Communicated by F. Porcelli
Abstract The dispersion properties of three-dimensional electrostatic waves in a nonuniform electron–positron (EP) magnetoplasma are analyzed. A new dispersion relation is derived by use of the electron and positron density responses arising from the electron and positron continuity and Poisson equations. In the local approximation, the dispersion relation admits two wave modes with different velocities. The growth rates of various modes are illustrated both analytically and numerically. Considering the temperature gradients produces a linearly stable transverse mode. The growth rate of the slow mode instability due to the density inhomogeneity only is the highest one, though it appears at higher thermal energy. The angle of the wave propagation affects drastically on the instability features in each case. The applications of the present analysis are briefly discussed. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Pair plasmas are composed of charged particles with the same mass and opposite charges, which admit the time and space scales that significantly differ from those of an electron–ion plasma [1–3]. Electron–positron (EP) plasmas are found in the Van Allen belts, in active galactic nuclei (AGN), near the polar cusp regions of the pulsars magnetospheres, in the early universe, and in the inner region of the accretion disks surrounding the central black holes [2,4–7]. Also, EP plasmas may be formed in the magnetospheres of pulsars by the pair-production cascade breeding process [8]. In addition, laboratory experiments have opened up the possibility of creating a nonrelativistic EP plasma. In one scheme, a relativistic electron beam impinges on a high-Z target, where positrons are produced copiously. The pair plasma is then trapped in a magnetic mirror and is expected to cool rapidly by emission of radiations [9]. In another scheme, positrons are accumulated from a radioactive source [10]. EP pair production is also possible during intense ultrashort laser pulse propagation in plasma [11]. Extensive works have been done to investigate the linear and nonlinear waves in pair plasmas (see e.g. Ref. [2,12,13]). Because most laboratory experiments and astrophysical objects contain nonuniform plasmas, it is of practical interest to investigate the instability of nonuniform EP plasmas [1,14]. The plasma collective mode remains stable near the thermodynamic equilibrium state, while it becomes unstable for a non-equilibrium state due to the free energy source availability. There could be different types of free energy sources, such as the streaming of plasma particles [15–17], spatial variations of the equilibrium * Corresponding author.
E-mail addresses: [email protected], [email protected] (W.F. El-Taibany), [email protected], [email protected] (W.M. Moslem), [email protected] (M. Wadati), [email protected] (P.K. Shukla). 1 Permanent address: Department of Physics, Faculty of Science-Damietta, Mansoura University, Damietta El-Gedida, P.O. 34517, Egypt. 2 Permanent address: Department of Physics, Faculty of Education-Port Said, Suez Canal University, 42111, Egypt. 3 Also at School of Physics, University of KwaZulu-Natal, Durban 4000, South Africa; and Department of Physics, Umeå University, SE-90187 Umeå, Sweden. 0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.03.024
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number density [15–18], the magnetic field [16,17] and the plasma temperature inhomogeneities [16,18,19]. Such gradients of the physical quantities can drive collective modes unstable [14]. For this purpose several authors have paid attention to explain the enhanced fluctuations which play a significant role in the anomalous transport phenomena such as heat transport across the external magnetic field lines [14–20]. For example, Shukla and Weiland [21] investigated the toroidal electron–temperature–gradient (ETG) driven electromagnetic drift waves. They suggested their model for the soft beta limitation as well as the anomalous energy transport in a tokamak reactor. Employing kinetic ion and fluid electron responses and including the ETG, two broad unstable fluctuations have been derived [22]. Shukla et al. [16] highlighted the combined effects of the ETG and parallel electron velocity gradient on electrostatic instabilities in a nonuniform magnetoplasma. Recently, Shukla and Shukla [17] presented a new purely growing instability in a nonuniform EP plasma with equilibrium density and magnetic field inhomogeneities. In this Letter, we present a study of low-frequency (in comparison with the electron gyrofrequency) electrostatic oscillations in a nonuniform EP magnetoplasma. By employing the two-fluid model for EP plasmas, we derive the dispersion relation taking into account the sheared plasma flow as well as the gradients of the density, the external magnetic field and the plasma temperature. 2. The model-dispersion relation Let us consider a three-dimensional nonuniform magnetoplasma composed of the electrons and positrons in the presence of density, velocity, temperature and magnetic field gradients along the x-axis. The external magnetic field is B0 = B0 (x)ˆz, where zˆ is the unit vector along the z-direction in a Cartesian coordinate system. The electron/positron fluid velocity is U α = uα⊥ + [Uα0 (x) + uαz ]ˆz, where Uα0 , uα⊥ and uαz are the streaming, perpendicular and parallel components to zˆ of the electron and positron fluid velocities, where the subscript α denotes either the positron (α = p) or the electron (α = e). Also, α will be used as a multiplicative factor, denoting the charge sign α = +1(−1), referring to the positron (electron) component, respectively. The charge-neutrality condition for the EP plasma at equilibrium is Ne0 (x) = Np0 (x) = N0 (x), where Ne0 (Np0 ) is the background number density of the electrons (positrons). Also, Te0 (x) = Tp0 (x) = T0 (x), where Te0 (Tp0 ) is the unperturbed temperature of the electrons (positrons). In order to investigate the low-frequency electrostatic waves in comparison with the electron gyrofrequency ωce = eB0 /mc, where e, m and c are the magnitude of the electron charge, the electron/positron mass and the speed of light in vacuum, respectively, we use the basic set of linearized continuity equations [16,17], ∂uαz Dα nα1 + ∇ ⊥ · (N0 uα⊥ ) + N0 = 0, Dt ∂z the momentum equations, −αe 1 1 Dα ∇ ⊥ pα1 , uα⊥ = ∇ ⊥ φ − uα⊥ × B0 zˆ − Dt m c mN0 Dα −1 ∂φ 1 ∂pα1 uαz + uα⊥ · ∇ ⊥ Uα0 = αe + , Dt m ∂z N0 ∂z
(1)
(2) (3)
the energy equation [18,21,23], 2Pα0 ∂uαz Dα Tα1 + uα⊥ .∇ ⊥ T0 + = 0, ∇ ⊥ · uα⊥ + Dt 3N0 ∂z and the Poisson equation, αnα1 = 0. ∇ 2 φ + 4πe
(4)
(5)
α=−1,+1
ˆ ˆ Here Dα /Dt = ∂/∂t + Uα0 ∂/∂z, ∇ ⊥ = x∂/∂x + y∂/∂y with xˆ and yˆ being the unit vectors along the x- and y-axes, nα1 N0 is the small electron and positron density perturbation, Tα1 T0 is the electron and positron temperature perturbation, and φ is the electrostatic potential. In the steady state the following relation should then hold [17,24]: B2 d N0 T0 + 0 = 0. dx 8π For the low-frequency waves, we assume that |Uα0 ∂/∂z|, |∂/∂t| ωce and |∂/∂z| |∇ ⊥ |, and therefore we can neglect the left-hand side of Eq. (2). Now, we shall calculate the perpendicular component of the electron and positron fluid velocities by taking the cross product from the left of Eq. (2) by zˆ and using the identity A × B × C = (A · C)B − (A · B)C to obtain c α uα⊥ = (6) zˆ × ∇ ⊥ (T0 nα1 + Tα1 N0 ) . zˆ × ∇ ⊥ φ + B0 eN0
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Substituting the expressions of uα⊥ into Eqs. (1) and (4), then combining the produced equations, we obtain Dα ∂ ∂φ + aα2 nα1 + aα3 = 0, D˜ α Tα1 − aα1 Dt ∂y ∂y where
∂ Dα D˜ α = − aα4 , Dt ∂y
aα1 =
2T0 , 3N0
aα2 =
T0 aα4 , N0
aα3 = −αeaα4 ,
aα4 =
(7)
αcT0 2 LT − Ln , eB0 3
with the notation Ln = d ln N0 /dx
and LT = d ln T0 /dx.
Also, substitution of (6) into (1) yields ∂uαz ∂Tα1 ∂φ Dˆ α nα1 + N0 + aα5 − aα6 = 0, ∂z ∂y ∂y
(8)
where
T0 ∂ Dα Dˆ α = + aα5 , Dt N0 ∂y
aα5 =
αcN0 LB , eB0
aα6 =
cN0 (Ln − LB ), B0
and LB = d ln B0 /dx. On the other hand, the parallel velocity for the α-component can be evaluated from (3) Dα −Gα T0 nα + Tα1 , uαz = αeφ + Dt m N0 1
(9)
where Gα =
∂ ∂ α ∂Uα0 ∂ ∂ − = + Sα , ∂z ωce ∂x ∂y ∂z ∂y
and Sα = −
α dUα0 . ωce dx
We use Eq. (9) to eliminate uαz from Eq. (8). Then solving the resultant equation with Eq. (7), we obtain the differential equation for α-species number density in terms of the electrostatic potential, φ, ∂ Dα ∂ ∂ Dα ∂ N0 Dα ˆ D˜ α Dα − VT2 Gα + aα5 − Gα aα1 + aα2 nα1 Dt ∂z Dt ∂y m ∂z Dt ∂y Dα ∂ ∂ Dα ∂ 2 ∂2 N0 αeN0 = D˜ α aα6 − (10) + Gα + aα3 aα5 G φ, α Dt ∂y m ∂z Dt ∂y 2 m ∂y∂z √ where VT = T0 /m. Assuming that nα1 and φ are proportional to exp(−iωt + iky y + ikz z), we Fourier transform (10) to get in the local approximation, nα1 = −(Zα /Qα )φ,
(11)
where Zα = Ωα Ωα ky aα6 − kz bα1 (kz + Sα ky ) − bα2 ky2 ,
5 Qα = Ωα Ωα2 − bα3 ky Ωα − VT2 kz (kz + Sα ky ) , 3
with Ωα = ω − kz Uα0 ,
bα1 =
αeN0 , m
bα2 =
−cN0 Ln aα4 , B0
bα3 = −
αcT0 (3LT − 2Ln − 5LB ). 3eB0
Inserting the expressions (11) for ne1 and np1 into (5), we obtain the desired dispersion relation, Ze 4πe Zp − = 0, 1+ 2 Qp Qe k where k 2 = ky2 + kz2 . The following section is devoted to present various limiting cases of the dispersion relation, Eq. (12).
(12)
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3. Limiting cases 3.1. Wave propagation parallel to the magnetic field First, we consider the case of wave propagation along B0 (x)ˆz. Let kz = k cos θ and ky = k sin θ , so in this case, θ = 0 (i.e., ky = 0) and therefore ω = kU0 + 2χωpe , (13) where
1/2 ωpe = 4πe2 N0 /m
and χ = 1 +
5 kVT 2 . 6 ωpe
For simplicity, we assume that Ue0 (x) ≈ Up0 (x) = U0 (x). The produced wave frequency, (13), shows that all the electrostatic waves propagating parallel to the magnetic field are stable and independent of either the magnetic field strength or the plasma inhomogeneity √ frequency is not only modified by the thermal energy but also it grows √ included. Eq. (13) illustrates that the wave by a factor 2. The reason for this modified frequency, 2ωpe , is the electron and positron density-oscillations with the same time scale, but 180◦ out of phase, though the charge densities are in phase [2]. 3.2. Wave propagation transverse to the magnetic field For the transverse mode, θ = 90 (i.e., kz = 0), Eq. (12) leads to the following dispersion relation 8πe be2 ωtr = kbe3 1 + 2 ae6 − , be3 k be3
(14)
and therefore the instability occurs for 8πe (15) (ae6 be3 − be2 ) < 0. k2 Now and thereafter, we let ωj = iγj , where γj is the growth rate and j is an index for the case study. So, the growth rate of this mode is given as
2 + 8πe (b − a b ). γtr = k −be3 (16) e2 e6 e3 k2 Eq. (15) can be rewritten leading to the following inequality, 2 be3 +
ηn− < ηn < ηn+ ,
(17)
where
√ 12(χ − 1) + 5ηB (7 − 4χ) ± 5 ηB [ηB (49 − 40χ) + 24(χ − 1)] ηn± = 8(χ − 1) 5ηB 3 5 + ≈ 1± √ (18) (7 − 4χ ± 49 − 40χ ), 2 8(χ − 1) 49 − 40χ with ηn = Ln /LT and ηB = LB /LT . It is noted here that in order to have a real value for either ηn+ and ηn− , we should verify that ηB > 0 or ηB > 24(1 − χ)/(49 − 40χ). For large and positive value of ηB , the two critical instability boundaries can be approximated as given in the second line of (18). However, for zero density gradient, the transverse mode obeys the rule, −1 kV 2 LT ωtr,Ln =0 = T (19) (3 − 5ηB ) 1 − 5 (χ − 1)(3 − 5ηB ) , 3ωce which exhibits the instability only when 3/5 > ηB > 3(χ − 1)/5χ . On the other hand, the other limit for zero magnetic field inhomogeneity, Eq. (14) leads to a stable mode whose angular frequency is kVT2 LT (20) (3 − 2ηn ). 3ωce This mode describes a linear propagating wave affected by the density gradient and ETG. Fig. 1 shows the variation of γtr against the variation of Ln , LB , and LT . It declares that the transverse mode starts as a growing wave, though after a certain value of LT , its behaviour is changed to be a damping wave. The transition point is affected by the changes of Ln and LT . Increasing LB reduces the growth (damping) rate of the growing (damping) wave. On the other hand, the increase of Ln causes an increase of both the growth and the damping rates of the transverse wave. Now let us classify the other limiting cases categorized according to which inhomogeneity effect(s) is (are) included. ωtr,LB =0 =
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Fig. 1. γtr (in terms of ωpe ) is plotted against LT . Solid curve is for Ln = 0.5 and LB = 0.25, dashed curve is for Ln = 0.5 and LB = 0.35 and dotted curve is for Ln = 1 and LB = 0.25, respectively, where VT /ωce = 0.5 and χ = 1.25.
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Fig. 2. γs (in terms of ωpe ) is plotted against θ (in terms of π ). Solid curve is for Se = 0.4, dashed curve is for Se = 0.7, dotted curve is for Se = 1 and the upper curve is for Se = 1.25, respectively, with χ = 1.25.
3.3. Only shear flow Including only the velocity gradient, i.e. Sα = 0 and Ln = LT = LB = 0, the EP plasma allows a propagating wave whose angular frequency is
ωs = cos θ kU0 + ωpe 2χ − 1 ± 1 + χ(χ − 1)(2Se tan θ )2 . (21) Equation (21) permits two propagating waves; the positive (negative) sign corresponds to the fast (slow) mode. The fast mode frequency can be approximated as ωsf = cos θ kU0 + 2χ ωpe 1 + (χ − 1)(Se tan θ )2 + · · · , (22) while the slower mode is governed by √ ωss = cos θ kU0 + 2ωpe χ 1 + (1 − χ)(Se tan θ )2 + · · · − 1 .
(23)
Equation (21) shows that the argument of the inner square root is always yielding to a positive real value, though the instability may occur due to that of the outer square root. Therefore, the fast mode is stable whatever the value of the shearing flow and/or the angle of propagation, θ . Since, the slow mode suffers from instability when the shear flow satisfies √ √ Se > cotan θ/ χ or Se < −cotan θ/ χ , (24) this condition declares that the instability depends on the angle of propagation as well as the thermal energy parameterized in terms of χ . Eq. (24) is valid for 1 χ(χ − 1)(2Se tan θ )2 . The growth rate of this slow mode is given by
γs = ωpe cos θ 1 − 2χ + 1 + χ(χ − 1)(2Se tan θ )2 . (25) It is obvious from the present case as well as the preceding cases that the shear flow affects only on the oblique propagating EP wave, though the parallel and transverse waves are independent of the velocity gradient effect. Fig. 2 illustrates the dependence of γs on both of Se and θ . As shown, the growth rate first increases with θ , afterward, γs decreases when we go closer to the transverse direction. The increase of Se allows the propagation of this mode at lower angle of propagation. 3.4. Only magnetic field inhomogeneity, LB = 0 Considering only the inhomogeneity due to magnetic field while switch off the other inhomogeneities, we can evaluate the dispersion relation from (12) as
ωB = cos θ kU0 + ωpe 2χ − 1 + A ± 1 + A 2(2χ − 1) + A , (26) where ψ = (VT tan θ/ωce )2 and A = 5χψL2B /3. Similar to the previous case, the fast mode is stable, since the slow mode suffers from instability when LB > LB+
or
LB < LB− ,
with ±ωce cotan θ LB± = VT
6 χ 1 1− −1 . 5 χ χ −1
(27)
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Fig. 3. γB (in terms of ωpe ) is plotted against θ (in terms of π ). Solid curve is for LB = 6, dashed curve is for LB = 10, and dotted curve is for LB = 15, respectively, where VT /ωce = 0.5 and χ = 1.25.
Eq. (27) is verified for small χ values. The growth rate of this instability can be calculated as
γB = ωpe cos θ 1 − 2χ − A + 1 + A 2(2χ − 1) + A . The slow mode here follows the approximated dispersion law A + ··· , ωBs = cos θ kU0 + ωpe 2(χ − 1) + A 2(1 − χ) − 2 while the fast mode obeys
ωBf
(28)
(29)
A = cos θ kU0 + ωpe 2χ + A 2χ + + ··· . 2
(30)
Fig. 3 shows the dependency of the growth rate, γB , on both LB and θ . It illustrates that, for small (higher) values of θ , γB increases (decreases) by the increase of θ . However, as LB increases, γB increases. 3.5. Only temperature gradient, LT = 0 In case of ETG, the EP plasma obeys the following dispersion relation
ωT = cos θ kU0 + ωpe 2χ − 1 + B ± 1 + B 2(2χ − 1) + B ,
(31)
where B = 3ψL2T (χ − 1)/5. Comparison of (26) and (31) reveals that they have the same formula though the difference in the definitions of A and B. Therefore, following the same calculation as what we have done for the case of “only magnetic field inhomogeneity”, we finally get the same dispersion relations for the slow [fast] mode similar to (29) [(30)], respectively; remember to replace A by B. Although, we have evaluated the growth rate, (28), corresponding to the instability occurred for the slower mode in the previous case, the present case does not present any instability for either the fast/slow modes. Therefore, consideration of only ETG results in two different stable modes. 3.6. Only density gradient, Ln = 0 Considering only density gradient, we see that the EP plasma is governed by the following dispersion relation
ωn = cos θ kU0 + ωpe 2χ − 1 + C ± 1 + C 2(8χ − 27) + C , where C = (2ηn /3)2 B. Here, we can separate the dispersion relation for the fast mode to yielding
C 4(4χ − 13) + C + · · · , ωnf = cos θ kU0 + ωpe 2χ + 2 while the slow mode obeys
C ωns = cos θ kU0 + ωpe 2(χ − 1) + 8(7 − 2χ) − C + · · · . 2
(32)
(33)
(34)
As for the previous studied cases, the fast mode is stable while the slow mode suffers from instability when Ln > Ln+
or
Ln < Ln− ,
(35)
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Fig. 4. γn (in terms of ωpe ) is plotted against θ (in terms of π ). The variation of Ln and the curves indication are the same as for LB in Fig. 3 with VT /ωce = 0.5 and χ = 50.
with Ln± =
√ ± 15/2ωce cotan θ , √ VT 2(2χ − 7) + 195 + χ(16χ − 111)
and the growth rate in this case can be calculated as
γn = ωpe cos θ 1 − 2χ − C + 1 + C 2(8χ − 27) + C .
(36)
Fig. 4 shows that by the increase of either θ or Ln , the growth rate increases, though this instability appears for a higher χ values. Also, the growth rate of this mode is remarkably higher than other previous ones. 3.7. General case As Eq. (12) is the exact dispersion relation for the present EP plasma, it is desirable to solve the following algebraic equation to find out the value of ω, d 1 + d2 k2 = 0, + 2 8πe d3 − d42
(37)
where
d1 = d3 (k cos θ )2 be1 + be2 tan2 θ , d2 = d4 k sin θ (−ae6 Ω + Se be1 k cos θ ), 5 5 2 2 2 d4 = k sin θ be3 Ω + VT kSe cos θ . d3 = Ω − (VT k cos θ ) , 3 3 For the ETG, magnetic field, density inhomogeneities are more stronger than Se , then we can assume Se ≈ 0. The dispersion relation, (12), yields to
ωG = cos θ kU0 + ωpe 2(χ − 1) − f1 ± f12 + f2 , (38) with be2 f2 , tan2 θ + f1 = 1 + be1 4 cos2 θ
tan θ 2 8πeae6 f2 = 2(χ − 1) be3 be3 + . VT k2
If f2 f12 , Eq. (38) can be rewritten to give the dispersion relation for the two modes (slow and fast) respectively as f2 ωGs = cos θ kU0 + ωpe 2(χ − 1) − f1 2 + + · · · , 2f12 f2 ωGf = cos θ kU0 + ωpe 2(χ − 1) + + ··· . 2f1
(39)
(40)
For our present EP model, it is proposed that any inhomogeneity effect may vary with either positive or negative gradient. So, Eqs. (39) and (40) state that both of the two modes could suffer from instability depending on the values of f1 and f2 . The numerical analysis does not present any instability for the fast mode, though the slow mode has an instability affected by the variation of f1 and f2 . Therefore, we can say that the slow mode growth rate is
γGs = ωpe cos θ 2(1 − χ) + f1 + f12 + f2 . (41)
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Fig. 5. γG (in terms of ωpe ) is plotted against θ (in terms of π ). Solid curve is for LT = 0.8, dashed curve is for LT = 1.5, and dotted curve is for LT = 3, respectively, where VT /ωce = 0.5, Ln = 0.5, LB = 0.25 and χ = 1.25.
Fig. 5 focuses on the variation of the growth rate against ETG, which is the main point in the present letter, as well as the propagation obliqueness on the magnetic field direction. It illustrates that, for smaller value of θ , the growth rate slightly decreases as θ increases, though as we go closer to the transverse direction (after θ ≈ π/3), it turns to grow to a higher value. Here, the ETG supports the increase of the growth rate, while consideration of ETG only yields two stable wave modes. 4. Conclusion To summarize, we have considered the propagation of the low-frequency electrostatic wave in a nonuniform EP pair magnetoplasma containing density, velocity, temperature and magnetic field inhomogeneities. In the parallel direction to the magnetic field, all the wave modes are stable, though their frequencies are modified because of the thermal energy. The shear flow effect extends the regime of the propagation (cf. Fig. 2). Comparing to the recent work which has been done by Shukla and Shukla [17], the present model reveals two wave modes with different velocities. The inclusion of the electron temperature gradient (ETG) produces two stable wave modes, (31). However, in the other limiting cases, the fast (slow) mode is stable (suffers from instability under certain conditions). The growth rate of each case has been calculated. Considering the density inhomogeneity leads to a remarkable higher growth rate with higher thermal energies, (36). Including all the four-inhomogeneity effects, ETG comes into play to bear an instability for the slower mode. The angle of the wave propagation changes drastically the feature of the growth rate in each case (cf. Figs. 1–5). Mathematically, the fast mode may have an instability, though the numerical analysis with various parameter regimes does not show any instability feature. In general, the instability in the current model comes from the slower modes. The present results may be useful for understanding the properties of low-frequency electrostatic waves in intense laser-plasma interaction experiments [25], as well as in astrophysical objects, such as the neutron stars [26]. In addition, for further extensions, the existence regime of solitary electrostatic waves as well as the effect of dust particle presence in the present EP plasma would be interesting topics to analyze [27]. Acknowledgements W.F.E. and M.W. gratefully acknowledge support from the Inoue Foundation of Science, Tokyo, Japan. W.M.M. thanks the Alexander von Humboldt-Stiftung (Bonn, Germany) for financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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