EXCITATION OF KINETIC ALFVÉN WAVES BY FAST ELECTRON BEAMS

Kinetic Alfvén waves (KAWs) are dispersive Alfvén waves (AWs) with a short perpendicular wavelength λ_⊥ comparable to the ion (or the ion-acoustic) gyroradius ρ_i(ρ_s) and the electron inertial length λ_e, but a parallel wavelength λ_∥ longer than the ion inertial length λ_i. In the 1970s, KAWs were first reported by Chen & Hasegawa (1974a, 1974b) and Hasegawa & Chen (1975, 1976) in the kinetic limit (i.e., λ_e < ρ_i(s) < λ_i or Q < β < 1), in which a positive dispersion is caused by the finite ion (ion-acoustic) gyroradius effect. Observations from the Polar and Cluster satellites have indicated that KAWs are observed throughout the plasma sheet, plasma sheet boundary layer, magnetopause, and magnetopause boundary (Wygant et al. 2002; Keiling et al. 2002; Stasiewicz et al. 2001a, 2001b). While KAWs in the inertial limit (i.e., ρ_i(s) < λ_e < λ_i or β < Q 1) were noted by Goertz & Boswell (1979), in which a negative dispersion is caused by the finite electron inertial effect. Such inertial AWs have also been identified by the Freja and FAST satellites (Stasiewicz et al. 2000; Chaston et al. 2007).

KAWs have a nonzero electric field parallel to the background magnetic filed E_∥ and have strong anisotropy in their polarization state and spatial structure, which means that they can contribute significantly to particle energization as well as to the fine-structure formation in various magneto-plasma environments from laboratory to space and astrophysical plasmas. Since discovered in the 1970s, KAWs have been extensively studied and applied to tokamak plasma heating (Huang et al. 1991), auroral electron acceleration (Wu 2003a, 2003b; Wu & Chao 2003, 2004), solar coronal plasma heating (Wu & Fang 1999, 2003, 2007), and the formation of magneto-plasma fine structures (Brodin et al. 2006; Singh & Sharma 2007). Therefore, the study of their generation and excitation mechanisms becomes an increasingly interesting subject. In general, plasmas in dynamically active states are far from the kinetic as well as dynamic equilibrium. These active plasmas are the most important sources that drive plasma instabilities. Although many excitation mechanisms have been proposed for KAWs, such as temperature anisotropy (Yoon et al. 1993; Yoon 1995; Chen & Wu 2010), field-aligned currents (Chen et al. 2011, 2013; Chen & Wu 2012), flow velocity shearing (Pu & Zhou 1986; Wang et al. 1998; Siversky et al. 2005), ion beams (Voitenko 1996,1998) and inhomogeneities in the density and magnetic field (Duan & Li 2005; Duan et al. 2005), the exact mechanism is still unknown in various plasma environments.

Beams of energetic charged particles are one of the common products in a variety of active phenomena in space and astrophysical plasmas such as auroral energetic electrons from the polar magnetosphere at an altitude of 1–2 R_E (Earth radius), energetic protons or electrons accelerated during the eruption of solar flares, and energetic cosmic rays in supernova blast waves. Non-thermal energetic particle beams with energies exceeding the typical thermal energy of the medium can supply rich sources of free energy to drive various plasma instabilities. The sudden presence of a current carried by a particle beam will lead to an inductive response of the plasma in a way that sets up a charged return-current because the magnetic diffusion time is much longer than the excited timescales of the self-magnetic field formed by the beam. The formation and evolution of the return-current and the electrodynamics of the beam-return current system have been investigated by many authors as a fundamental problem (Knight & Sturrock 1977; Spicer & Sudan 1984; Larosa & Emslie 1989; van den Oord 1990).

In this paper, including the return-current effect of fast electron beams, KAW instability driven by a fast electron beam is investigated in finite-β plasmas. The organization of this paper is as follows. First, a dispersion relation of KAWs in the presence of a fast electron beam is derived in Section 2. Then, based on this dispersion relation, we discuss the growth rate of the dependence of KAWs on the perpendicular wave number k_⊥, the electron beam velocity v_b, and the electron beam number density n_b in Section 3. Finally, our discussion and conclusions are presented in Section 4.

According to the scenario of the beam-return current system, plasma consists of an energetic electron beam with a number density n_b and bulk velocity v_b along the unperturbed ambient magnetic field $\mathbf {B}_0\equiv B_0\mathbf {\hat{z}}$, ambient static protons, and ambient bulk drifting electrons with velocity v_d opposite from the unperturbed ambient magnetic field. For these three components the unperturbed distribution functions are assumed to be written as follows:

$$\begin{eqnarray} f_{b0}&=&{n_b\over 2\sqrt{2}\pi ^{3\over 2}v_{Tb}^3}\mathrm{e}^{-{v_\perp ^2\over 2v_{Tb}^2}}\,\mathrm{e}^{-{(v_\parallel -v_b)^2\over 2v_{Tb}^2}}, \nonumber \\ \nonumber f_{i0}&=&{n_0\over 2\sqrt{2}\pi ^{3\over 2}v_{Ti}^3}\mathrm{e}^{-{v_\perp ^2\over 2v_{Ti}^2}}\,\mathrm{e}^{-{v_\parallel ^2\over 2v_{Ti}^2}}, \\ f_{e0}&=&{n_0-n_b\over 2\sqrt{2}\pi ^{3\over 2}v_{Te}^3}\mathrm{e}^{-{v_\perp ^2\over 2v_{Te}^2}}\,\mathrm{e}^{-{(v_\parallel +v_d)^2\over 2v_{Te}^2}}, \end{eqnarray} \tag{ 1 }$$

where n₀ is the number density of the ambient plasma, v_Ts is the thermal speed for the sth species (s = b, i, or e, where b, i, e represent the electron beams, the ambient protons, and the ambient electrons, respectively), $v_\perp ^2=v_x^2+v_y^2$.

We focus on the low frequency (ω < Ω_i, where Ω_i is the ion gyrofrequency) and magnetically incompressible fluctuations, implying that the parallel component of the fluctuation magnetic field B_1z = 0. Therefore, it is more convenient to introduce two fluctuating electric and magnetic potentials, scalar and ψ, to describe the fluctuation in electric and magnetic fields, E₁ and B₁, in the following form (Hasegawa & Chen 1976):

$$\begin{eqnarray} \mathbf {E}_1&=&-\nabla \varphi -{\partial \psi \over \partial t}\mathbf {\hat{z}}, \nonumber\\ \mathbf {B}_1&=&\nabla \times (\psi \mathbf {\hat{z}}), \end{eqnarray} \tag{ 2 }$$

where the vector potential $\mathbf {A}\equiv \psi \mathbf {\hat{z}}$ satisfies the Coulomb gauge below

$$\begin{equation} \nabla \cdot \mathbf {A}=\nabla \cdot (\psi \mathbf {\hat{z}})=0. \end{equation} \tag{ 3 }$$

The linear density fluctuations can be calculated by n_s1 = ∫f_s1kd³v and the detailed derivation of the fluctuation distribution function f_s1k can be found in Voitenko (1998) and Wu (2012). This results in

$$\begin{eqnarray} n_{b1}&=&{en_{b0}\over m_b}\left({1\over v_{Tb}^2}-i N_b\right)\left(\varphi _k-v_k\psi _k\right), \nonumber\\ n_{i1}&=&{en_{i0}\over m_i}\left[\left({I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}\over v_k^2}+i N_i\right)(\varphi _k-v_k\psi _k)\right. \nonumber\\ &&\left. -\,{1-I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}\over v_{Ti}^2}\varphi _k\right], \nonumber\\ n_{e1}&=&{en_{e0}\over m_e}\left({1\over v_{Te}^2}-i N_e\right)\left(\varphi _k-v_k\psi _k\right), \end{eqnarray} \tag{ 4 }$$

where the plasma β lies in the range Q < β < 1 and β_s ≡ 2μ₀n_sT_s/B², v_k ≡ ω/k_∥ is the parallel phase speed, I₀ is the zero-order modified Bessel function, and

$$\begin{eqnarray} f_{s0}^z&\equiv &2\pi \int J_0^2\left(k_\perp {v_\perp \over \omega _{cs}}\right)f_{s0}(v_\perp,v_\parallel)v_\perp dv_\perp, \nonumber\\ N_{s}&\equiv &{\pi \over n_{s0}}{\partial f_{s0}^z\over \partial v_\parallel }\left| _{v_\parallel =v_k}\right., \end{eqnarray} \tag{ 5 }$$

where J₀ is the zero-order Bessel function.

Similarly, the calculations of the z direction current density fluctuations as velocity space integrals are straightforward. We find

$$\begin{eqnarray} j_{zb1}&=&-e\int v_\parallel f_{b1}d^3\mathbf {v}\nonumber\\ &=&-{e^2n_{b0}\over m_b}\left({1\over v_{Tb}^2}-i N_b\right)\left(\varphi _k-v_k\psi _k\right)v_k, \nonumber\\ j_{zi1}&=&e\int v_\parallel f_{i1}d^3\mathbf {v}\nonumber\\ &=&{e^2n_{i0}\over m_i}\left({I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}\over v_k^2}+i N_i\right)\left(\varphi _k-v_k\psi _k\right)v_k, \nonumber\\ j_{ze1}&=&-e\int v_\parallel f_{e1}d^3\mathbf {v}\nonumber\\ &=&-{e^2n_{e0}\over m_e}\left({1\over v_{Te}^2}-i N_e\right)\left(\varphi _k-v_k\psi _k\right)v_k. \end{eqnarray} \tag{ 6 }$$

From the quasi-neutrality condition and the Ampere law, we have

$$\begin{equation} n_{i1}=n_{e1}+n_{b1}, \end{equation} \tag{ 7 }$$

and

$$\begin{equation} \left(k_x^2+k_z^2\right)\psi _k=\mu _0\left(j_{zi1}+j_{ze1}+j_{zb1}\right), \end{equation} \tag{ 8 }$$

respectively. Substituting Equations (4) and (6) into Equations (7) and (8), the nontrivial solution with _k, ψ_k ≠ 0 leads to the dispersion relation for the real part

$$\begin{equation} \omega =v_Ak_\parallel \sqrt{{1\over 1-I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}}+{T_e\over T_i}}k_\perp \rho _i, \end{equation} \tag{ 9 }$$

and the growth rate

$$\begin{equation} \gamma ={\omega \over 2}{1-I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}\over 1-I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}+{T_i/ T_e}}\sum _s{T_e\over m_s n_0}\pi {\partial f_{s0}^z\over \partial v_{\parallel }}\mid _{v_\parallel =v_k}, \end{equation} \tag{ 10 }$$

where, for low-density beams we can neglect their influence on the real part of the KAW frequency, keeping only the imaginary terms in the first-order number density and current density, which mainly excite KAWs and contribute to the growth rate. The Coulomb collision effect on the wave growth has been neglected for collisionless plasmas, and the less important effect of individual electron acceleration between collisions also has been neglected compared with the effect of quasi-steady bulk electron motion (Voitenko 1998).

Substituting the distribution functions of the beam-return current system of Equation (1) into Equation (10), the growth rate of KAWs is obtained as follows

$$\begin{eqnarray} {\gamma \over \omega } & = & {\sqrt{\pi \over 8}}{1-I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2} \over 1-I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}+{T_i/T_e}} \nonumber\\ & &\times\, \left[{n_b\over n_0}{T_e\over T_b}{v_b-v_k\over v_{Tb}}\mathrm{e}^{-{(v_b-v_k)^2\over 2v_{Tb}^2}}\right.\nonumber\\ &&\left. -I_0\left(k_\perp ^2\rho _i^2\right)\mathrm{e}^{-k_\perp ^2\rho _i^2}{T_e\over T_i}{v_k\over v_{Ti}}\,\mathrm{e}^{-{v_k^2\over 2v_{Ti}^2}}-{v_d\over v_{Te}}-{v_k\over v_{Te}}\right],\nonumber\\ && \end{eqnarray} \tag{ 11 }$$

where $I_0\left(k_\perp ^2\rho _s^2\right)\mathrm{e}^{-k_\perp ^2\rho _s^2}\simeq 1$ for s = b and e, the conditions n_b/T_b < n_e/T_e has been used, implying the beam correction to the KAW dispersion can be neglected. Equation (11) indicates that the growth rate, γ, is directly proportional to the wave frequency as long as ω < Ω_i.

In this expression (11), the first term in the square brackets is contributed by the kinetic resonant interaction of beam electrons. The second term is the Landau damping effect caused by the ambient protons. The third and forth terms are the damping effects (i.e., the return-current effect and the Landau damping effect) caused by the ambient electrons. The drift velocity of ambient electrons v_d has the same order as v_A. By comparing the KAW instability excited by a fast ion beam (Voitenko 1996, 1998; Bell 2004; Malovichko et al. 2014), we can conclude that electron-beam-driven KAWs and proton-beam-driven KAWs have different driving sources. The kinetic resonant interaction of beam electrons (the first term in expression (11)) is the driven source for the KAW instability driven by fast electron beams. However, for resonant instability of KAWs driven by fast ion beams, both the kinetic resonant interaction of beam ions and the return-current are the driving sources once proton velocity overcomes the Alfvén velocity. For non-resonant instability of KAWs driven by fast ion beams, the return-current effect dominates the growth of KAWs only if the beams are very fast, with velocities exceeding several Alfvén velocities.

From expression (11), Figure 1 shows the numerical results of the growth rate γ/ω versus the perpendicular wave number k_⊥ρ_i, where the growth rate and wavelength are normalized by the frequency ω and the ion gyroradius ρ_i, respectively. The solid lines in Figure 1 correspond to the electron beam velocity v_b/v_A = 6, 8, 12. In addition, the parameters n_b/n₀ = 0.3, β_e = β_i = 0.1 and β_b = 0.02 have been used. From Figure 1, it can be found that the growth rate of KAWs has a positive non-zero value in the region of the perpendicular wave number, $0<k_\perp <k_\perp ^u$, implying KAW excitation. Outside this growing region of the perpendicular wave number (i.e., $k_\perp >k_\perp ^u$) the growth rate of KAWs has a negative value, implying the KAWs are damped. From Figure 1 it can be seen that the growth rate reaches its maximum γ^m in the right half of this growing region of the perpendicular wave number, that is, at $k_\perp ^m\ge k_\perp ^u/2$ (specifically, in the range $0.6k_\perp ^u$ to $0.7k_\perp ^u$ for the parameters used in Figure 1). On the other hand, both the maximal growing perpendicular wave number $k_\perp ^m$ and the maximal growth rate γ^m first increase then decrease with the velocity of electron beam v_b/v_A. The growing region of the perpendicular wave number $0<k_\perp <k_\perp ^u$ also first widens then narrows as the velocity of electron beam v_b/v_A increases.

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Clearer variation of the normalized growth rate γ/ω with the normalized velocity of electron beam v_b/v_A is shown in Figure 2, where the solid lines correspond to the electron beam number density n_b/n₀ = 0.28, 0.3, 0.32, respectively. For the sake of comparison, the left panel plots the corresponding growth rate of KAWs, in which the terms of damping effects (the last three terms in expression (11)) have been neglected. The parameters k_⊥ρ_i = 0.5, β_e = β_i = 0.1 and β_b = 0.02 have been used. From the left panel of Figure 2, we can see that KAWs are damped when the velocity of the electron beam is smaller than the parallel phase speed of KAWs, v_b < v_k. The growth rate of KAWs approaches zero when the velocity of the electron beam is equal to the parallel phase speed of KAWs, v_b = v_k. The KAW has a positive non-zero value when v_b > v_k, implying KAW excitation. However, from the right panel of Figure 2, in which all three damping effects have been included, it is clear that both the growth rate and the growing region of KAWs decrease compare with the left panel of Figure 2. The growth rate γ/ω first increases then decreases with the velocity of electron beam v_b/v_A. In addition, the right panel of Figure 2 shows that the growing region of KAWs widens and the growth rate increases as the number density of electron beam n_b/n₀ increases.

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Figure 3 presents the variation of the normalized growth rate γ/ω with the number density of electron beam n_b/n₀, where the solid lines correspond to the perpendicular wave number k_⊥ρ_i = 0.3, 0.5, 0.8, respectively, and the parameters v_b/v_A = 8, β_e = β_i = 0.1, and β_b = 0.02 have been used. One can find that when the number density of electron beam n_b/n₀ satisfies the condition $n_b/n_0>n_b^c/n_0$, the normalized growth rate γ/ω > 0, implying that KAWs become unstable. As shown by Figure 3, the growing ($n_b/n_0>n_b^c/n_0$) and damping ($n_b/n_0<n_b^c/n_0$) regimes can be separated clearly by the critical number density of electron beam $n_b^c/n_0$, which increases with the perpendicular wave number k_⊥ρ_i. In the instable regime of $n_b/n_0>n_b^c/n_0$, the growth rate γ/ω increases with the number density of electron beam n_b/n₀ for the fixed perpendicular wave number k_⊥ρ_i.

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Figure 4 shows the dependence of the normalized maximal growing perpendicular wave number $k_\perp ^m\rho _i$ (left) and the maximal growth rate γ^m/ω at $k_\perp ^m$ (right) on the normalized electron beam velocity v_b/v_A. The solid lines correspond to the electron beam number density n_b/n₀ = 0.28, 0.3, 0.32, respectively. The parameters β_e = β_i = 0.1 and β_b = 0.02 have been used. From Figure 4, it can be seen that both the maximal growing perpendicular wave number $k_\perp ^m\rho _i$ and the maximal growth rate γ^m/ω depend sensitively on the velocity of electron beam v_b. The most favorable beam velocity occurs in the range 8v_A < v_b < 10v_A. On the other hand, the excited KAWs are weakly dispersive with k_⊥ρ_i < 1 and have the maximum growth rate at relatively low perpendicular wave numbers in the range $0.3<k_\perp ^m\rho _i<0.6$ for a beam velocity v_b < 10v_A. It can also be seen that the growing region of KAWs increases as the number density of electron beam n_b/n₀ increases.

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In this paper, including the return-current effect of fast electron beams, KAW instability driven by a fast electron beam in finite-β (Q < β < 1) plasmas is investigated. The results indicate that electron-beam-driven KAWs and proton-beam-driven KAWs have different driven sources. In particular, for electron-beam-driven KAWs, the kinetic resonant interaction of beam electrons is the driving source. However, for resonant instability of KAWs driven by fast ion beams, both the kinetic resonant interaction of beam ions and the return-current are the driven sources once proton velocity overcomes the Alfvén velocity. For non-resonant instability of KAWs driven by fast ion beams, the return-current effect dominates the growth of KAWs only if the beams are very fast with velocities exceeding several Alfvén velocities. The results also show that the KAW instability excited by a fast electron beam has a non-zero growth rate in the range of the perpendicular wave number $0<k_\perp <k_\perp ^u$, and reaches the maximum growth rate γ^m at $k_\perp ^m\ge k_\perp ^u/2$ (specifically, in the range $0.6 k_\perp ^u$ to $0.7 k_\perp ^u$ for the parameters used in Figure 1). As the velocity of electron beam v_b increases, the maximal growing perpendicular wave number $k_\perp ^m$, the maximal growth rate γ^m, and the growing region of the perpendicular wave number $k_\perp ^u$ first increase then decrease. The most favorable beam velocity varies in the range 8v_A < v_b < 10v_A. On the other hand, the excited KAWs are weakly dispersive with k_⊥ρ_i < 1 and have the maximum growth rate at relatively low perpendicular wave numbers in the range $0.3<k_\perp ^m\rho _i<0.6$ for a beam velocity v_b < 10v_A. In the unstable regime of $n_b>n_b^c$, the growth rate increases with the number density of electron beam n_b.

The electron beams that are associated with KAWs have been frequently found by many space satellites, such as Freja and FAST at relatively low altitudes (Boehm et al. 1995; Carlson et al. 1998; Elphic et al. 2000) and Polar and Cluster at relatively high altitudes (Wygant et al. 2002; Morooka et al. 2004). Two different forms of the dispersive AW exist depending on altitude. At low altitudes (i.e., up to 3–4 R_E), the KAW in the inertial limit is the appropriate wave mode. However, at higher altitudes (i.e., above 3–4 R_E), the KAW in the kinetic limit is the appropriate wave mode (Lysak & Carlson 1981). As an example, we briefly discuss the possible application to the electron beams in the terrestrial magnetosphere. Based on the Cluster satellite observations at an altitude of 6 R_E, the measured densities of the intense upward electron beams range from 0.01 to 0.1 cm⁻³ with an energy of about several tens of eV to several keV (Morooka et al. 2004). Estimated by the spacecraft potential and total magnetic field strength ≈10⁵ nT (hence, Ω_i ≈ 95.8 × 10² Hz), the parameters of the plasma density n₀ ≈ 0.1 cm⁻³ and the Alfvén speed v_A ≈ 10⁶ m/s (Morooka et al. 2004). Therefore, this gives a relative number density n_b/n₀ ≈ 0.1 − 1. The ratio of the electron beam velocity to the local Alfvén velocity can be estimated as v_b/v_A ≈ 1–10. From Figure 4, the relative growth rate of the KAW instability can be estimated as γ/ω ~ 0.01 for n_b/n₀ ~ 0.3. This leads to a growth rate γ < 0.01Ω_i and a growth timescale τ = 1/γ > 0.01 s, which is well shorter than the timescale of the wave propagation t = L/v_A ~ 1 s propagating over a distance L ~ several thousand kilometers in the magnetosphere.

It is worth of noting that another important instability driven by a fast electron beam is Langmuir wave (LW) instability. LW instability, however, requires a large threshold of the beam velocity, v_b > v_Te, and we have v_Te 10v_A for the parameters used in this paper. For the KAW instability considered in this paper, the most favorable beam velocity occurs between 8v_A < v_b < 10v_A, in which the LW instability is not important.

This work was supported by NSFC under Grant Nos. 41304136, 11373070, and 11303082; by the NSF of Jiangsu Province under Grant No. BK20131039; by the Key Laboratory of Solar Activity at the National Astronomical Observatories, CAS, under Grant No. KLSA201305; and by MSTC under Grant No. 2011CB811402.