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ISRN Condensed Matter Physics
Volume 2011 (2011), Article ID 896321, 6 pages
http://dx.doi.org/10.5402/2011/896321
Research Article

Generalized Buneman Dispersion Relation in the Longitudinally Dominated Magnetic Field

1Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic
2Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic

Received 5 September 2011; Accepted 13 October 2011

Academic Editor: S. Chromik

Copyright © 2011 Martin Bohata et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The generalized Buneman dispersion relation for two-component plasma is derived in the case of nonzero pressure of both plasma components and longitudinally dominated magnetic field. The derived relation is also valid for other field configurations mentioned in the paper. It can be useful in a variety of plasma systems, for example, in the analyses of plasma jet penetrating into background plasma, in beam-plasma physics, and in tests of various magnetohydrodynamics (MHD) and hybrid numerical codes designed for the magnetized plasmas.

1. Introduction

The two stream instabilities are the most common instabilities in plasmas. Usually plasma consists of two or more kinds of particles moving with different velocities. Under suitable conditions the two-stream instability can occur and particle energy is transferred into the energy of plasma wave excitation [13]. The plasma subcomponents with nonzero initial velocity are thermalized and the plasma temperature increases.

The two-stream instability is observed in plasma jets injected into the background plasma, in beam-plasma and beam-target configurations (e.g., in fusion experiments), and in some other configurations in which the different species have different drift velocities.

For cold unmagnetized plasma with two counter streaming components the problem has been solved by Buneman in his pioneering works in late fifties [4, 5]. He found the fastest stable drift velocity for the electrons and conditions under which the energy is dissipated into the instabilities. The Buneman instability was studied in a variety of experiments, for example, [6].

The amplitude saturation of Buneman instability was discussed in detail much later by Hirose in [7]. The role of the Buneman instability in reconnection processes is very interesting, especially in the vicinity of the plasma neutral sheet [8]. In nonlinear mode the Buneman instability can be the starting mechanism for the onset of ion-acoustic instability [9]. The Buneman instability has many modifications (e.g., two symmetric opposite beams, beam-target instability, etc.) and several related classes of instabilities. First of them is the Farley-Buneman Instability, which develops if electron and ion velocities differ by more than ion acoustic speed. This instability is wellknown in the ionosphere region E [10]. Above some critical value of the current the Buneman instability changes into the Pierce instability which causes sudden nonoscillatory potential drop leading to the double layer formation and capture of ions in the potential well [11]. In the case of anisotropic velocity distribution the so-called Weibel instability develops, which was described in the same time period as the Buneman instability [12]. All these types of instabilities dissipate the particle energy into the plasma waves or other plasma disturbances. In some sense all these instabilities are opposite to the well-known Landau damping predicted in 1946, in which particles can gain the energy from the wave modes [13]. Details of the variety of Buneman-like instabilities can be found in [1, 14, 15].

Buneman-like instabilities are very frequently used as test examples for various MHD and hybrid numerical electrostatic codes [16, 17]. This is advantageous, as the plasma behavior in corresponding regimes is wellknown both theoretically and experimentally. The generalized dispersion relation derived later in this paper could be helpful for similar tests in magnetized plasmas.

2. Basic Assumptions

Let us assume a two-component unbounded plasma under the following conditions.

(1)Plasma is described by nonrelativistic set of equations: 𝜕𝑛𝛼𝑛𝜕𝑡+𝛼𝐮𝛼𝑚=0,(1)𝛼𝑛𝛼𝜕𝐮𝛼𝜕𝑡+𝑚𝛼𝑛𝛼𝐮𝛼𝐮𝛼=𝑝𝛼+𝑞𝛼𝑛𝛼𝐄+𝐮𝛼×𝐁,(2)×𝐄=𝜕𝐁1𝜕𝑡,(3)𝜀0𝜇01×𝐁=𝜀0𝐣+𝜕𝐄𝜕𝑡;𝐣=2𝛼=1𝑞𝛼𝑛𝛼𝐮𝑎𝑝,(4)𝛼=𝐾𝛼𝑛𝛾𝛼𝛼.(5) Expression (1) represents continuity equations, (2) equations of motion, (3) and (4) Maxwell equations for the electric and magnetic fields, and (5) polytrophic formulas for the pressures. Index α denotes the two plasma components (e.g., electrons and ions).(2)Perturbations of all the variables are considered to be linear, that is, 𝜓𝛼(𝑡,𝐱)=𝜓𝛼(0)+𝜆𝜓𝛼(1)(𝜔,𝐤)e𝑖[𝐤𝐱𝜔𝑡]𝜆+𝑂2.(6)(3)All the zero-order solutions 𝜓𝛼(0) are constant in time and space. To ensure this assumption we will suppose that magnetic field component perpendicular to the direction of particle stream motion is weak (magnetic field is longitudinally dominated for both plasma species); that is, the Larmor radii are much greater than arbitrary watched plasma size 𝐿, and therefore gyration is negligible: 𝑚𝛼𝑢(0)𝛼𝑞𝛼𝐵(0)𝐿.(7) The other possibility is to meet the set of conditions resulting from (1)–(5) and Maxwell equation div𝐃=𝜌 (where 𝜌 is the charge density): 𝐄(0)+𝐮𝛼(0)×𝐁(0)𝜌=𝟎,(0)=2𝛼=1𝑞𝛼(0)𝑛𝛼(0)𝐣=0,(0)=2𝛼=1𝑞𝛼(0)𝑛𝛼(0)𝐮𝛼(0)=𝟎.(8) For 𝐄(0)=0 is the first condition in (8) akin to (7).

3. Linearization of the Equations for Arbitrary Frequencies

Perturbing the whole set of (1)–(5) according to (6), we obtain in the first order of the parameter λ a linearized set of algebraic equations:Ω𝛼𝑛𝛼(1)𝑛𝛼(0)𝐤𝐮𝛼(1)=0,(9)𝑖𝑚𝛼𝑛𝛼(0)Ω𝛼𝐮𝛼(1)=𝑖𝐤𝑝𝛼(1)+𝑞𝛼𝑛𝛼(0)𝐄(1)+𝑞𝛼𝑛𝛼(1)𝐄(0)+𝑞𝛼𝑛𝛼(0)𝐮𝛼(0)×𝐁(1)+𝑞𝛼𝑛𝛼(0)𝐮𝛼(1)×𝐁(0)+𝑞𝛼𝑛𝛼(1)𝐮𝛼(0)×𝐁(0),(10)𝐁(1)=1𝜔𝐤×𝐄(1),(11)𝑖𝜔𝐄(1)=𝑖𝑐2𝐤×𝐁(1)𝐣(1)𝜀0,(12)𝐣(1)=2𝛼=1𝑞𝛼𝑛𝛼(0)𝐮𝛼(1)+𝑞𝛼𝑛𝛼(1)𝐮𝛼(0),(13)𝑝𝛼(1)=𝑚𝛼𝑐2𝑠𝛼𝑛𝛼(1),(14) where we have denoted the Doppler shifted frequency:Ω𝛼=𝜔𝐤𝐮𝛼(0),(15) sound velocities:𝑐𝑠𝛼=𝛾𝛼𝑘𝐵𝑇𝛼𝑚𝛼,(16) and the velocity of light:𝑐=1𝜀0𝜇0.(17) Substituting (11) into (12) we achieve the equation for the electric field perturbation in the following form:𝕄𝐄(1)=𝑖𝜔𝜇0𝐣(1)𝜔;𝕄=2𝑐2𝑘2𝕀+𝐤𝐤,(18) where 𝕀denotes the identity matrix. The matrix 𝕄 can be easily inverted as𝕄1=𝑐𝕀2/𝜔2𝐤𝐤𝜔2/𝑐2𝑘2,(19) and the electric field perturbation becomes𝐄(1)=𝑖𝜔𝜇0𝜔2/𝑐2𝑘2𝐣(1)𝑐2𝜔2𝐤𝐣(1)𝐤.(20) Now the pressure perturbations 𝑝𝛼(1) can be derived from (14), perturbations of concentrations 𝑛𝛼(1) from the continuity equations of (9), magnetic field perturbation 𝐁(1) from (11), electric field perturbation 𝐄(1) from (20), and electric current perturbation 𝐣(1) from (13). After substitution of all these perturbations into the equations of motion (10) we finally getΩ2𝛼𝐮𝛼(1)=𝑖𝑚𝛼𝐤𝐮𝛼(1)𝐅𝛼(0)+𝑐2𝑠𝛼𝐤𝐮𝛼(1)𝐤+𝑖𝜉𝛼Ω𝛼𝐮𝛼(1)×𝐁(0)+𝜉𝛼Ω𝛼𝜇0𝑐2𝜔2𝑐2𝑘2×Ω𝛼𝐣(1)+𝐮𝛼(0)𝐣(1)𝑐𝐤2𝜔𝐤𝐣(1)𝐤,𝛼=1,2,(21) where𝐣(1)=2𝛼=1𝑞𝛼𝑛𝛼(0)𝐮𝛼(1)+𝐤𝐮𝛼(1)𝜔𝐤𝐮𝛼(0)𝐮𝛼(0).(22) In expression (21) we have denoted𝜉𝛼=𝑞𝛼𝑚𝛼,𝐅𝛼(0)=𝑞𝛼𝐄(0)+𝐮𝛼(0)×𝐁(0).(23) The quantities 𝜉𝛼, 𝐅𝛼(0) symbolize specific charge and zero-order Lorentz force, respectively. In case of two-component plasma the expression (21) represents 6 equations for velocity perturbations. Both (21) and (22) can be simply written for more than two plasma components, in which case 𝛼=1,,𝑛.

4. Generalized Dispersion Relation in Low-Frequency Limit

In low-frequency limit 𝜔𝑐𝑘, (21) and (22) give usΩ2𝛼𝐮𝛼(1)=𝑖𝑚𝛼𝐤𝐮𝛼(1)𝐅𝛼(0)+𝑐2𝑠𝛼+𝜔2𝑝𝛼𝑘2𝐤𝐮𝛼(1)𝐤+𝜉𝛼Ω𝛼𝜉𝛽Ω𝛽𝜔2𝑝𝛽𝑘2𝐤𝐮𝛽(1)𝐤+𝑖𝜉𝛼Ω𝛼𝐮𝛼(1)×𝐁(0),(24) where the indices 𝛼 and 𝛽 are different, that is, 𝛽𝛼, and furthermore we have designated square of the plasma frequency as𝜔2𝑝𝛼=𝑛𝛼(0)𝑞2𝛼𝑚𝛼𝜀0.(25) The linearized set of (24) can be rewritten in the following form:𝔸𝐮1(1)+𝔹𝐮2(1)=0,𝐮1(1)+𝔻𝐮2(1)=0,(26) or simply𝐮𝔸𝔹𝔻1(1)𝐮2(1)=0,(27) where𝔸=Ω21𝑖𝕀+𝑚1𝐅1(0)+𝑐𝐤2𝑠1+𝜔2𝑝1𝑘2𝐤𝐤+𝑖𝜉1Ω1𝜺𝐁(0),(28)𝜉𝔹=1Ω1𝜔2𝑝2𝜉2Ω2𝑘2𝐤𝐤,(29)𝜉=2Ω2𝜔2𝑝1𝜉1Ω1𝑘2𝐤𝐤,(30)𝔻=Ω22𝑖𝕀+𝑚2𝐅2(0)+𝑐𝐤2𝑠2+𝜔2𝑝2𝑘2𝐤𝐤+𝑖𝜉2Ω2𝜺𝐁(0).(31) The symbol 𝜺 stands for the Levi-Civita tensor. The set of equations of (27) will have nontrivial solution if the determinant of the matrix will be zero:det𝔸𝔹𝔻=0.(32) From definitions (29) and (30) it is obvious thatdet𝔹=0,det=0.(33) Relation (32) can be simply solved if either det𝔸0 or det𝔻0. In the first case we can calculate from the set of equations of(26)𝐮1(1)=𝔸1𝔹𝐮2(1)(34) and substitute this expression into the remaining equation in (26):𝔻𝔸1𝔹𝐮2(1)=0.(35) Nontrivial solution will exist ifdet𝔻𝔸1𝔹=0.(36) The second possibility det𝔻0 leads us to a similar condition:det𝔸𝔹𝔻1=0.(37) After straightforward but rather toilsome calculation both (36) and (37) provide the same dispersion relation:𝜔det𝔸det𝔻2𝑝1𝜔2𝑝2𝑘4Ω41𝑘2Ω21𝜔2𝑐1𝐞𝐵𝐤2×Ω42𝑘2Ω22𝜔2𝑐2𝐞𝐵𝐤2=0,(38) where𝜔𝑐𝛼=𝑞𝛼𝐵(0)𝑚𝛼(39) are the cyclotron frequencies of both species. The unit vector in the zero-order magnetic field direction is designated as𝐞𝐵=𝐁(0)𝐵(0).(40) Evaluation of det𝔸 gives usdet𝔸=Ω61Ω41𝑖𝐅1(0)𝑚1+𝑐𝐤2𝑠1𝑘2+𝜔2𝑝1+𝜔2𝑐1Ω31𝜔𝑐1𝐅1(0)𝑚1×𝐤𝐞𝐵+Ω21𝜔2𝑐1𝐤𝐞𝐵×𝑖𝐅1(0)𝐞𝐵𝑚1+𝑐2𝑠1𝑘2+𝜔2𝑝1𝐤𝐞𝐵𝑘2.(41) Interchanging the indices 1 and 2 in det𝔸 leads us to the value of the det𝔻. Hence the nontrivial solution of (27) exists if2𝛼=1Ω4𝛼Ω2𝛼𝑖𝐅𝛼(0)𝐤𝑚𝛼+𝑐2𝑠𝛼𝑘2+𝜔2𝑝𝛼+𝜔2𝑐𝛼Ω𝛼𝜔𝑐𝛼𝑚𝛼𝐅𝛼(0)×𝐤𝐞𝐵+𝜔2𝑐𝛼𝐤𝐞𝐵𝑖𝐅𝛼(0)𝐞𝐵𝑚𝛼+𝑐2𝑠𝛼𝑘2+𝜔2𝑝𝛼𝐤𝐞𝐵𝑘22𝛼=1𝜔2𝑝𝛼𝑘2Ω2𝛼𝑘2𝜔2𝑐𝛼𝐞𝐵𝐤2=0.(42) In the case of 𝐄(0)=0, 𝐁(0)=0, and cold plasma limit𝑐𝑠𝛼=0, this relation becomes the well-known Buneman dispersion relation:2𝛼=1𝜔2𝑝𝛼𝜔𝐤𝐮𝛼(0)2=1.(43)

5. Some Remarks to the Policy of Computation of the Dispersion Relation for Arbitrary Frequencies

Without the low-frequency limit request, (21) can be rewritten in the matrix form (27) again:𝐮𝔸𝔹𝔻1(1)𝐮2(1)=0.(44) The matrices 𝔸, 𝔹, , and 𝔻 differ from the low-frequency matrixes (28)–(31) and can be determined from (21). The matrix𝐵 has determinant𝜉det𝔹=1𝜉23𝜔6𝑝2𝜔2𝑐2𝑘22Ω51Ω2,(45) which is now nonzero and finite with exception of the discrete frequencies 𝐤𝐮1(0), 𝐤𝐮2(0), and ck. Excusing these frequencies the inverse matrix of 𝔹 can be calculated as𝔹1Ω=𝐴1Ω2+Ω1𝐮2(0)𝐤+Ω2𝐮1(0)+𝐮𝐤1(0)𝐮2(0)𝑐2𝑘2𝕀Ω1𝐮2(0)𝐤Ω2𝐤𝐮1(0)𝐮1(0)𝐮2(0)𝑐2𝐮𝐤𝐤+1(0)×𝐤𝐤×𝐮2(0),(46) where1𝐴=𝜉det𝔹1𝜉2𝜔2𝑝2𝜔2𝑐2𝑘22Ω31Ω2.(47) From matrix equation (44) it follows𝐮2(1)=𝔹1𝔸𝐮1(1),+𝔻𝔹1𝔸𝐮1(1)=0.(48) Dispersion relation for arbitrary frequencies (with exception of the frequencies mentioned above) is represented by the following condition:det+𝔻𝔹1𝔸=0.(49) Analytical calculation is difficult due to the enormous number of terms in this relation, and numerical approach seems to be more advantageous in this case. Maybe some numerical methods could be also applied directly to the set of equations of (21).

6. Results

The new dispersion relation (42) was carefully analyzed numerically. The complex roots were found by the Hubbard-Schleicher-Sutherland algorithm [18]. The code was written in Compaq Visual FORTRAN 6.6.C3. Here we will discuss some results for two identical opposite beams, because they can be easily compared to the well-known Buneman solution. All the variables were converted into the nondimensional form. Both beams were oriented in the direction of the 𝑥-axis, magnetic field in the (𝑥,𝑧) plane; that is, 𝐵𝑧 describes the perpendicular component of the field and 𝐵𝑥 the longitudinal one. There were 5 main input parameters: non-dimensional magnetic field induction as the ratio 𝜔𝑐/𝜔𝑝, nondimensional sound velocity in the combination 𝑐𝑠/𝑢0 (𝑢0 being the beam velocity), and three directional characteristics—the angle between the magnetic field and the 𝑧-axis 𝜃𝐁, the angle between the wave vector and the 𝑧-axis is denoted as 𝜃𝐤, and the (𝑥,𝑦) azimuth of the wave vector 𝜑𝐤. The dispersion relation provides many new features, especially new imaginary branches of the frequency, which imply new types of instabilities. The principal results can be divided into several points.(1)Minimal changes (compared to the Buneman solution) occur in the presence of the longitudinal magnetic field applied in the direction of the wave vector describing the perturbation. All the dispersion relation branches are depicted in the graph on Figure 1(a). The typical “8”-shaped imaginary branch (well known from the Buneman solution) remains unchanged in the (𝜔,𝑘) plane as well as the hyperbola-like real branches. But there are four new linear dependencies𝜔=±𝜔𝑐±𝑢0𝑘. These lines correspond to the Doppler-shifted cyclotron motion of the charged particles.(2)If the magnetic field vector is opposite to the wave vector describing the perturbation, the picture changes completely; see Figure 1(b). Two of the cyclotron lines disappear and the two residual lines change into hyperbola-like shape. Furthermore the portrait of the imaginary parts of the dispersion relation consists of 7 branches corresponding to the new types of instabilities. The original “8”-like imaginary branch broke into two parts and new “8”-like imaginary curve appears in the vertical direction of the (𝜔,𝑘) plane. Furthermore the dispersion relation gives 4 new imaginary branches, which are only very weakly dependent on the perturbation wavelength.(3)The most complicated picture can be seen in the case of perpendicular magnetic field; see Figure 1(c). The original hyperbola-like real parts bifurcate two times in a sequence. Near the origin of the (𝜔,𝑘) plane there are 4 closed imaginary part branches and the bifurcating real part branches. The only “Doppler” curves 𝜔=±𝑢0𝑘 remained unchanged. The behavior of the plasma in this region of parameters seems to be very interesting and deserves detailed experimental investigation. Let us remark that stronger perpendicular fields are not compatible with the assumptions of our analysis.(4)The pressure term also influences the dispersion relation solutions. The imaginary parts are present only in the supersonic velocity of the beams. If the velocity is subsonic, the imaginary solutions completely disappear; see Figure 1(d).

fig1
Figure 1: Numerical solution of the dispersion relation (42) for various parameters. Two opposite beams of the same particles were treated. The dashed red lines represent imaginary parts of the relation, and the solid blue lines the real parts.

7. Conclusion

The generalized Buneman dispersion relation (42) was derived for two-component plasma in longitudinally dominated magnetic field. The plasma was treated as nonrelativistic under the assumption of weak perpendicular component of the magnetic field (gyration is negligible) but the resulting relation is valid for other field configurations as well. The whole calculation was done as usually in low-frequency limit. It describes the behavior of plasmas with macroscopic particle motions (beam-target systems, two beams, etc.) in both nonzero magnetic and electric fields. The derived dispersion relation yields completely new branches of both real and imaginary frequencies, which were found numerically. It can be helpful for the stability analyses; furthermore it can be used in various tests of numerical plasma magnetohydrodynamics and hybrid codes.

Acknowledgment

This research has been supported by the Grant IAA101210801: “Simulations of the DD fusion reactions” of the Grant Agency of the Academy of Sciences of the Czech Republic.

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