Generalized Buneman Dispersion Relation in the Longitudinally Dominated Magnetic Field

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 [1–3]. 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: 𝜕𝑛_𝛼𝜕𝑡+∇⋅𝑛^𝛼𝑚𝐮_𝛼=0,(1)_𝛼𝑛_𝛼𝜕𝐮_𝛼𝜕𝑡+𝑚^𝛼𝑛_𝛼𝐮_𝛼⋅∇∇×𝐮𝐄_𝛼==−−∇𝜕𝐁𝑝_𝛼+𝑞_𝛼𝑛_𝛼𝐄+𝐮_𝛼×𝐁,(2)1𝜕𝑡,(3)𝜀₀𝜇₀∇×𝐁=1𝜀₀𝐣+𝜕𝐄𝜕𝑡;𝐣=²_𝛼=1𝑝𝑞_𝛼𝑛_𝛼𝐮_𝑎,(4)_𝛼=𝐾_𝛼𝑛^𝛾𝛼_𝛼.(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, 𝜓_𝛼(𝑡,𝐱)=𝜓^𝛼(0)+𝜆𝜓^𝛼(1)(𝜔,𝐤)e^{𝑖[𝐤⋅𝐱−𝜔𝑡]}+𝑂𝜆².(6)(3)All the zero-order solutions 𝜓^𝛼(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: 𝑚_𝛼𝑢^(0)𝛼⟂𝑞_𝛼𝐵^⟂(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): 𝐄𝜌⁽⁰⁾+𝐮^𝛼(0)×𝐁⁽⁰⁾=𝟎,₍₀₎=²_𝛼=1𝐣𝑞^𝛼(0)𝑛^𝛼(0)=0,₍₀₎=²_𝛼=1𝑞^𝛼(0)𝑛^𝛼(0)𝐮^𝛼(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:Ω_𝛼𝑛^𝛼(1)−𝑛^𝛼(0)𝐤⋅𝐮^𝛼(1)=0,(9)−𝑖𝑚_𝛼𝑛^𝛼(0)Ω_𝛼𝐮^𝛼(1)=−𝑖𝐤𝑝^𝛼(1)+𝑞_𝛼𝑛^𝛼(0)𝐄⁽¹⁾+𝑞_𝛼𝑛^𝛼(1)𝐄⁽⁰⁾+𝑞_𝛼𝑛^𝛼(0)𝐮^𝛼(0)×𝐁⁽¹⁾+𝑞_𝛼𝑛^𝛼(0)𝐮^𝛼(1)×𝐁⁽⁰⁾+𝑞_𝛼𝑛^𝛼(1)𝐮^𝛼(0)×𝐁⁽⁰⁾,(10)𝐁⁽¹⁾=1𝜔𝐤×𝐄⁽¹⁾,(11)−𝑖𝜔𝐄⁽¹⁾=𝑖𝑐²𝐤×𝐁⁽¹⁾−𝐣⁽¹⁾𝜀₀,(12)𝐣⁽¹⁾=²_𝛼=1𝑞^𝛼𝑛^𝛼(0)𝐮^𝛼(1)+𝑞_𝛼𝑛^𝛼(1)𝐮^𝛼(0),(13)𝑝^𝛼(1)=𝑚_𝛼𝑐^2𝑠𝛼𝑛^𝛼(1),(14) where we have denoted the Doppler shifted frequency:Ω_𝛼=𝜔−𝐤⋅𝐮^𝛼(0),(15) sound velocities:𝑐_𝑠𝛼=𝛾_𝛼𝑘_𝐵𝑇_𝛼𝑚_𝛼,(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⁾;𝕄=𝜔²𝑐²−𝑘²𝕀+𝐤⊗𝐤,(18) where 𝕀denotes the identity matrix. The matrix 𝕄 can be easily inverted as𝕄⁻¹=𝕀−𝑐²/𝜔²𝐤⊗𝐤𝜔²/𝑐²−𝑘²,(19) and the electric field perturbation becomes𝐄⁽¹⁾=−𝑖𝜔𝜇⁰𝜔²/𝑐²−𝑘²𝐣⁽¹⁾−𝑐²𝜔²𝐤⋅𝐣⁽¹⁾𝐤.(20) Now the pressure perturbations 𝑝^𝛼(1) can be derived from (14), perturbations of concentrations 𝑛^𝛼(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 getΩ^2𝛼𝐮^𝛼(1)=𝑖𝑚_𝛼𝐤⋅𝐮^𝛼(1)𝐅^𝛼(0)+𝑖𝜉_𝛼+𝑐^2𝑠𝛼𝐤⋅𝐮^𝛼(1)𝐤Ω_𝛼𝐮^𝛼(1)×𝐁⁽⁰⁾+𝜉_𝛼Ω_𝛼𝜇₀𝑐²𝜔²−𝑐²𝑘²×Ω_𝛼𝐣⁽¹⁾+𝐮^𝛼(0)⋅𝐣⁽¹⁾𝐤−𝑐²𝛼=1,2,𝜔𝐤⋅𝐣⁽¹⁾𝐤,(21) where𝐣⁽¹⁾=²_𝛼=1𝑞^𝛼𝑛^𝛼(0)𝐮^𝛼(1)+𝐤⋅𝐮^𝛼(1)𝜔−𝐤⋅𝐮^𝛼(0)𝐮^𝛼(0).(22) In expression (21) we have denoted𝜉_𝛼=𝑞^𝛼𝑚_𝛼,𝐅^𝛼(0)=𝑞_𝛼𝐄⁽⁰⁾+𝐮^𝛼(0)×𝐁⁽⁰⁾.(23) The quantities 𝜉_𝛼, 𝐅^𝛼(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 𝜔≪𝑐𝑘, (21) and (22) give usΩ^2𝛼𝐮^𝛼(1)=𝑖𝑚_𝛼𝐤⋅𝐮^𝛼(1)𝐅^𝛼(0)+𝑐^2𝑠𝛼+𝜔^2𝑝𝛼+𝜉_𝛼𝑘²𝐤⋅𝐮^𝛼(1)𝐤Ω_𝛼𝜉_𝛽Ω_𝛽𝜔^2𝑝𝛽+𝑖𝜉_𝛼𝑘²𝐤⋅𝐮^𝛽(1)𝐤Ω_𝛼𝐮^𝛼(1)×𝐁⁽⁰⁾,(24) where the indices 𝛼 and 𝛽 are different, that is, 𝛽≠𝛼, and furthermore we have designated square of the plasma frequency as𝜔^2𝑝𝛼=𝑛^𝛼(0)𝑞^2𝛼𝑚_𝛼𝜀₀.(25) The linearized set of (24) can be rewritten in the following form:ℂ𝔸𝐮𝐮¹⁽¹⁾₁₍₁₎+𝔹𝐮²⁽¹⁾=0,+𝔻𝐮²⁽¹⁾=0,(26) or simply⎛⎜⎜⎝𝔸ℂ𝔻𝔹⎞⎟⎟⎠⎛⎜⎜⎝𝐮¹⁽¹⁾𝐮²⁽¹⁾⎞⎟⎟⎠=0,(27) where𝔸=−Ω²¹𝕀+𝑖𝑚₁+𝑐𝐅¹⁽⁰⁾⊗𝐤_2𝑠1+𝜔^2𝑝1𝑘²𝐤⊗𝐤+𝑖𝜉¹Ω₁𝜺⋅𝐁⁽⁰⁾,(28)𝔹=𝜉¹Ω₁𝜔^2𝑝2𝜉₂Ω₂𝑘²𝐤⊗𝐤,(29)ℂ=𝜉²Ω₂𝜔^2𝑝1𝜉₁Ω₁𝑘²𝐤⊗𝐤,(30)𝔻=−Ω²²𝕀+𝑖𝑚₂+𝑐𝐅²⁽⁰⁾⊗𝐤_2𝑠2+𝜔^2𝑝2𝑘²𝐤⊗𝐤+𝑖𝜉²Ω₂𝜺⋅𝐁⁽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:det⎛⎜⎜⎝𝔸ℂ𝔻𝔹⎞⎟⎟⎠=0.(32) From definitions (29) and (30) it is obvious thatddeettℂ𝔹==00,.(33) Relation (32) can be simply solved if either det𝔸≠0 or det𝔻≠0. In the first case we can calculate from the set of equations of(26)𝐮¹⁽¹⁾=−𝔸⁻¹𝔹𝐮²⁽¹⁾(34) and substitute this expression into the remaining equation in (26):𝔻−ℂ𝔸⁻¹𝔹𝐮²⁽¹⁾=0.(35) Nontrivial solution will exist ifdet𝔻−ℂ𝔸⁻¹𝔹=0.(36) The second possibility det𝔻≠0 leads us to a similar condition:det𝔸−𝔹𝔻⁻¹ℂ=0.(37) After straightforward but rather toilsome calculation both (36) and (37) provide the same dispersion relation:det𝔸det𝔻−𝜔^2𝑝1𝜔^2𝑝2𝑘⁴Ω⁴¹×Ω₄₂𝑘²−Ω²¹𝜔^2𝑐1𝐞_𝐵⋅𝐤²𝑘²−Ω²²𝜔^2𝑐2𝐞_𝐵⋅𝐤²=0,(38) where𝜔_𝑐𝛼=𝑞^𝛼𝐵⁽⁰⁾𝑚_𝛼(39) are the cyclotron frequencies of both species. The unit vector in the zero-order magnetic field direction is designated as𝐞_𝐵=𝐁⁽⁰⁾𝐵⁽⁰⁾.(40) Evaluation of det𝔸 gives usdet𝔸=Ω⁶¹−Ω⁴¹𝑖𝐅𝑚¹⁽⁰⁾₁⋅𝐤+𝑐^2𝑠1𝑘²+𝜔^2𝑝1+𝜔^2𝑐1−Ω₃₁𝜔_𝑐1𝐅𝑚¹⁽⁰⁾₁×𝐤⋅𝐞_𝐵×+Ω𝑖²¹𝐅𝜔^2𝑐1₁₍₀₎𝐤⋅𝐞_𝐵⋅𝐞_𝐵𝑚₁+𝑐^2𝑠1𝑘²+𝜔^2𝑝1𝐤⋅𝐞^𝐵𝑘².(41) Interchanging the indices 1 and 2 in det𝔸 leads us to the value of the det𝔻. Hence the nontrivial solution of (27) exists if²_𝛼=1Ω^4𝛼−Ω^2𝛼𝑖𝐅^𝛼(0)⋅𝐤𝑚_𝛼+𝑐^2𝑠𝛼𝑘²+𝜔^2𝑝𝛼+𝜔^2𝑐𝛼−Ω_𝛼𝜔_𝑐𝛼𝑚_𝛼𝐅^𝛼(0)×𝐤⋅𝐞_𝐵+𝜔^2𝑐𝛼𝐤⋅𝐞_𝐵𝑖𝐅^𝛼(0)⋅𝐞_𝐵𝑚_𝛼+𝑐^2𝑠𝛼𝑘²+𝜔^2𝑝𝛼𝐤⋅𝐞^𝐵−𝑘²₂_𝛼=1𝜔^2𝑝𝛼𝑘²Ω^2𝛼𝑘²−𝜔^2𝑐𝛼𝐞_𝐵⋅𝐤²=0.(42) In the case of 𝐄⁽⁰⁾=0, 𝐁⁽⁰⁾=0, and cold plasma limit𝑐_𝑠𝛼=0, this relation becomes the well-known Buneman dispersion relation:²_𝛼=1𝜔^2𝑝𝛼𝜔−𝐤⋅𝐮^𝛼(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:⎛⎜⎜⎝𝔸ℂ𝔻𝔹⎞⎟⎟⎠⎛⎜⎜⎝𝐮¹⁽¹⁾𝐮²⁽¹⁾⎞⎟⎟⎠=0.(44) The matrices 𝔸, 𝔹, ℂ, and 𝔻 differ from the low-frequency matrixes (28)–(31) and can be determined from (21). The matrix𝐵 has determinantdet𝔹=𝜉¹𝜉₂³𝜔^6𝑝2𝜔²−𝑐²𝑘²²Ω⁵¹Ω₂,(45) which is now nonzero and finite with exception of the discrete frequencies 𝐤⋅𝐮¹⁽⁰⁾, 𝐤⋅𝐮²⁽⁰⁾, and ck. Excusing these frequencies the inverse matrix of 𝔹 can be calculated as𝔹⁻¹=𝐴+Ω𝐮₁Ω₂+Ω₁𝐮²⁽⁰⁾⋅𝐤+Ω₂𝐮¹⁽⁰⁾⋅𝐤₁₍₀₎⋅𝐮²⁽⁰⁾−𝑐²𝑘²𝕀−Ω¹𝐮²⁽⁰⁾⊗𝐤−Ω₂𝐤⊗𝐮¹⁽⁰⁾−𝐮¹⁽⁰⁾⋅𝐮²⁽⁰⁾−𝑐²𝐤⊗𝐤+𝐮¹⁽⁰⁾×𝐤⊗𝐤×𝐮²⁽⁰⁾,(46) where𝐴=1det𝔹𝜉𝜉¹₂𝜔^2𝑝2𝜔²−𝑐²𝑘²²Ω³¹Ω₂.(47) From matrix equation (44) it followsℂ𝐮+𝔻²⁽¹⁾𝔹=𝔹⁻¹₋₁𝔸𝐮¹⁽¹⁾,𝔸𝐮¹⁽¹⁾=0.(48) Dispersion relation for arbitrary frequencies (with exception of the frequencies mentioned above) is represented by the following condition:detℂ+𝔻𝔹⁻¹𝔸=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, 𝐵_𝑧 describes the perpendicular component of the field and 𝐵_𝑥 the longitudinal one. There were 5 main input parameters: non-dimensional magnetic field induction as the ratio 𝜔_𝑐/𝜔_𝑝, nondimensional sound velocity in the combination 𝑐_𝑠/𝑢₀ (𝑢₀ being the beam velocity), and three directional characteristics—the angle between the magnetic field and the 𝑧-axis 𝜃_𝐁, the angle between the wave vector and the 𝑧-axis is denoted as 𝜃_𝐤, and the (𝑥,𝑦) azimuth of the wave vector 𝜑_𝐤. 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=±𝜔_𝑐±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).

(a)

(b)

(c)

(d)

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.