Electron kinetic firehose instability

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 15, NO. A1, PAGES 7,377-7,385, DECEMBER 1, Electron kinetic firehose instability Xing Li and Shadia Rifai Habbal Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts Abstract. The linear dispersion equation describing electromagnetic waves propagating in a homogeneous electron-proton plasma along arbitrary directions relative to the direction of the background magnetic field is solved numerically for bi-maxwellian particle distributions. It is found that in the presence of an electron temperature anisotropy Tñ < and a sufficiently warm plasma (/ > ), several purely growing modes (zero real frequency) and a quasi-parallel electron firehose instability develop. While the quasi-parallel mode is unstable for both parallel and oblique propagation, the zero frequency modes are unstable only for oblique propagation. Comparison of these modes further shows that the propagation angle for maximum growth rate and the maximum growth rate are larger for the purely growing modes than the quasi-parallel electron firehose while the threshold is lower. Potential application of the kinetic electron firehose instability to the slow solar wind is briefly discussed. 1. Introduction The development of electron temperature anisotropies is common in laboratory and space plasmas. In the slow solar wind, for example, observations of a strong electron temperature anisotropy Teñ < Tell, where _L and [[ denote directions perpendicular and parallel to the background magnetic field Bo, respectively, have been reported [Feldman et al., 1975; Gosling et al., 1987; Phillips et al., 1989; Phillips and Gosling, 199]. Using an analytical approach for solving the kinetic plasma wave dispersion relation in a Vlasov (collisionless) plasma, Hollweg and Vb'lk [197] found that a parallel propagating electron firehose instability with nonzero real frequency can be excited by an electron temperature anisotropy in a sufficiently warm plasma (i.e., an electron beta value greater than ). This ki- netic electron firehose instability has larger growth rate than the conventional magnetohydrodynamic (MHD) firehose instability. Hollweg and VSlk suggested that this instability could transfer the electron thermal energy to protons in the slow solar wind. Parallel propagating firehose modes were also derived from the exact solution of the Vlasov plasma dispersion relation [Pilipp and Vb'lk, 1971]. Gary and Madland [1985] gave the parametric dependence of the maximum growth rates of these modes. If the electron anisotropy is not very large, these modes are only resonant with ions. However, if the electron parallel temperature is much greater than the perpendicular temperature, the electrons also Copyright by the American Geophysical Union. Paper number JA63. 148-7 / / JA 6359. 7,377 become resonant [Pilipp and Bentz, 1977]. While Pilipp and Bentz [1977] only considered parallel propagation, Paesold and Benz [1999] demonstrated that the maximum growth rate of these modes is larger for oblique propagation [see also Gary and Madland, 1985]. According to MHD theory, only purely growing (zero real frequency) firehose instabilities can be excited in a sufficiently warm plasma when the plasma parallel thermal pressure is greater than the perpendicular component [see Hasegawa, 1975; Treumann and Baumjohann, 1997]. The unstable zero frequency firehose modes predicted by MHD theory have often been interpreted as a consequence of the fluid approach [see Treumann and Baumjohann, 1997]. Hence these zero frequency firehose modes were not treated and compared with the nonzero frequency modes in the studies with the kinetic approach mentioned above. Yoon et al. [1993] investigated the effect of finite ion gyroradius on the proton firehose instability in a high-/ plasma under the as- sumption that [;[ << FFp, and (kzva/ai) <</ [li, where Fti and VA are the ion gyrofrequency and Alfv6n speed, respectively, and/ [lis the parallel ion beta. They found that the effect of finite ion gyroradius results in a significant enhancement of the growth rate over a large range of wave numbers. Unstable zero real frequency modes are found at intermediate propagation angles, while the unstable nonzero real frequency modes are at near parallel and larger propagation angles. The goal of this paper is to conduct a detailed study of the electron firehose instability by including the zero frequency firehose modes. As will be shown, the two main assumptions made by Yoon et al. [1993] regarding the proton firehose instability are not applicable to the electron firehose instability. We show that the kinetic plasma wave theory predicts the existence of

7,378 LI AND HABBAL: ELECTRON KINETIC FIREHOSE INSTABILITY purely growing electron firehose modes. This study is limited to the electron firehose instability.. Basic Wave Equations We consider a homogeneous, collisionless, and charge neutral plasma in a uniform background magnetic field Bo with no ambient electric field. We will assume, without loss of generality, that Bo is in the z direction and that the wave vector k is in the x z plane. The plasma wave equation for wave vector k and complex frequency co may be written as [Stix, 199] is given by D'E= 5xx 5xy 5xz + x z 5yz 5yy TI, x TI, z 5yz 5zx + z x 5zy 5zz -- x parallel and perpendicular to the background magnetic field. The following symbols are defined for the sth species: the cyclotron frequency, Fts - qsbo/msc; the plasma frequency, cops _ 4 rnsq /ms; the parallel and perpendicular thermal velocity, Wsll - (kstsll/ms) 1/ and wsñ = (kstsñ/ms)l/; the parallel and perpendicular temperature, Tsll and Tsñ; plasma beta, s - 8 rnskstsll/b' ; the flow velocity, Vs. The mass, electric charge and number density are ms, qs and ns, respectively, and ks is the Boltzmann constant. The plasma dispersion function Z [Fried and Conte, 1961] Z( ) - 71.-1/ dxexp(-x) (8) The cyclotron resonant factor j is -o, - kllv - -, j #, (9) where D is the dispersion tensor, n = kc/co is the refrac- tive index, c is the speed of light, and E is the electric field of the wave. The dielectric tensor s is related to the susceptibility Ks of the sth plasma component through the relation, ' " -I+Zxs. () The dispersion relation of a plasma wave is then determined.by ;. det D =. (3) -. For the familiar case in which the particle velocity distribution is assumed to be bi-maxwellian, the susceptibility Ks is given by where Xs - e.e.cokllw _ V + cop y. e-xyj(,x) (4) co j=-c J-- -Aj -ijcaj - - Bj Yj(A) - ijcaj FAj i ' CBj, kllw Ij = Ij(X) is the modified Bessel function with ar- gument X - kzwz /, F- Ij. + X[j -, C = Ij - 5, and Ij = (d/dx)ij(x). One may also find that kllw[ I (co - kll V - jgt)ta_ + jfttii Z( j), (6) cozjl -jet kllaj ' s (7) where kll and kñ are the components of the wave vector and the Landau resonant factor is - co - kll V We consider a two-species electron-proton plasma, denoting the protons by the subscript p, and the elec- trons by e. The Alfv(m speed is VA = Bo/(47rnpmp) 1/. Since we take the zeroth-order plasma to be charge and current free, ne = r p, and V = Vp =. For a given value of the wave number k and = arctan(/ca_/kll), the angle of propagation with respect to the magnetic field, we wish to find the complex frequency co = cot + it satisfying the dispersion relation (). The ratios VA/C and rap/me are set at 1-3 and 1836, respectively in this study. The major difficulty in finding the roots of the dispersion relation () is to evaluate the plasma dispersion function Z. To calculate Z, we adopt a method similar to that given by Fried and Conte [1961]. 3. Solutions of the Linear Dispersion Equation 3.1. Electron Firehose Instability Figure 1 shows the real frequency (left) and growth rate (right) of the electron firehose instability as a function of normalized wave number at parallel (Figure la) and oblique propagation (Figures lb-le). In this example, protons are isotropic, electrons have an anisotropy T a_/t [ I =.5 and are hotter than protons: = 5, p = 1. The parameters of Figure 1 will be used as a basis for the parameter space to be explored in this paper. The results for parallel propagation are similar to earlier published investigations [Pilipp and Vb'lk, 1971; Gary and Madland, 1985; Paesold and Benz; 1999]. In this case, the frequency of the electron firehose instabil-

... LI AND HABBAL: ELECTRON KINETIC FIREHOSE INSTABILITY 7,379 (o) ø. 1.5 "'1. "'1 8 4 (b) 8-17 ø. 1.5 1..5. 1 a 8 1 a 8 (c) 19= ø (d) e= ø 8 6 8 6 %4 1 8 4 4 6 8 %4 8 6 4 6 /fl Figure 1. The (left) real frequency and (right) growth rate of the electron firehose instability as a function of wave number for propagation angle t? equal to (a) ø, (b) 17 ø, (c) ø, (d) ø, and (e) 4 ø. Here/3p - 1, ] e-- 5, Vpñ/Vp[[- 1, and Veñ/Ve[[ --.5. ity at small wave numbers is very small (but not zero). For oblique propagation the unstable mode at small wave numbers splits into two branches (see : 17 ø in Figure lb), and the real frequency of both branches be- comes zero. Hence the two branches are different from the nonzero real frequency unstable mode in which the wave number kva/ p is between 1.85 and 7 in this example. The maximum growth rate of the nonzero real frequency electron firehose instability at oblique angles is greater than that at parallel propagation [Paesold and Benz, 1999]. MHD theory predicts the existence of zero real frequency firehose modes at both parallel and oblique directions. However, we find that the unstable zero real frequency electron firehose modes occur only for oblique propagation. As increases further, the growth rate of the upper branch of the purely growing mode increases much more rapidly than the nonzero real frequency mode while the wave number range of the lower branch narrows, as seen in Figures 1 c-ld for increasing. The nonzero real frequency electron firehose mode has a maximum growth rate when is around a small oblique angle ø. For this reason, we will call the nonzero real frequen cy electron firehose mode "quasi-parallel electron firehose" for sire-

., 7,38 LI AND HABBAL: ELECTRON KINETIC FIREHOSE INSTABILITY... I... I... I... I... I ß ' Im(P)/ I I. t ii.o.. =17 ø --... I... I... I... I... 1 3 4 5 6 Figure. The polarization (real part) of the electron firehose instability as a function of wave number for = ø and 17 ø, for the parameters of Figure 1. For = 17 ø, R and the imaginary part of P for the upper dispersion branch are shown as dotted and dashed lines, respectively. plicity. When = 4 ø, two zero real frequency branches virtually become one, and the wave number range of the quasi-parallel electron firehose is very narrow. As the propagation becomes more oblique, the quasi-parallel mode eventually stabilizes. Following Stix [199] and Gary [1986], the polarization of waves can be defined as P - (11) When r is, P = i Ex/Ey. Figure shows the polarization of electron firehose instabilities for the examples of Figure 1. The polarization (1) is a complex number, the real part of P is shown as solid lines. When Re(P) is negative (positive), it corresponds to the left- (right-) handed sense. At parallel propagation the electron firehose instability goes through a transition from almost right- to left-hand circular polarization when the wave number increases [Gary and Madland, 1985]. The imaginary part of P at parallel propagation is zero and not shown. When = 17 ø, the real part of the polarization of the two zero real frequency branches is zero, which means that the instability in this case is a static structure and the wave vector does not rotate. However, the imaginary part of P, which is E /Ey, is a finite real number (dashed line) and represents the real t '... I... I... I... I... I... I... I... I... I 't =7 ø 15 6O 1 75 ø 5O 5 ø 4O 4 8 1 4 8 3 36 4 Figure 3. Growth rate of the oblique electron firehose mode for large angles of propagation for the same parameters as Figure 1.

LI AND HABBAL: ELECTRON KINETIC FIREHOSE INSTABILITY 7,381 polarization of these purely g rowing modes. One can the normalized wave number kv 4/glp is between 1.5 and also define parameter R - I k. E] /IEI to show the 3.8, left-hand ion cyclotron waves are unstable. Hownature of a wave. For a purely transverse electromagnetic wave, R -, and for a purely electrostatic wave, R - 1. In Figure, R is shown as a dotted line for ever, oblique electron firehose instabilities occur both at small (kva/glp 1.5) and large (kva/glp 3.8)wave numbers. At large, all the unstable branches eventu- - 17 ø (the upper branch). The instability is basically ally become one purely growing branch as in Figure 3. electromagnetic (R 1) at small k and becomesubstantially electrostatic at larger k (R.5), which is expected. When the two branches touch, the electron firehose instability is left-hand polarized. Figure 3 shows the growth rate of the electron firehose instability for large for the same parameters as in Figure 1. For 3 ø, only the zero frequency Figure 5a shows the maximum growth rate of the oblique electron firehose, the propagation direction and normalized wave number at the maximum growth rate as a function of electron anisotropy Teñ/Tell. Figure 4 demonstrates that at some propagating angles the maximum growth rate of the quasi-parallel electron firehose appears at a point where quasi-parallel and purely electron firehose mode is unstable. The zero real fre- growing firehose branches touch. At this point the frequency electron firehose instability also has a maximum growth rate close to 7 ø (68.3 ø as will be shown later in Figure 5). We will term this purely growing mode as "oblique electron firehose." Table 1 shows a summary of the quasi-parallel and oblique electron firehose instabilities. It is obvious that the maximum growth rate of the oblique electron firehose is almost orders of magnitude greater than the maximum growth rate of the quasi-parallel electron firehose. When electrons are more anisotropic, the plasma can support more unstable branches for moderate values (Figure 4) than for a weaker anisotropy (compare with Figure 1). Figure 4 shows an example for Teñ/Tell =. with the other parameters being the same as in Figure 1. At - 3 ø, there are five unstable branches. When quency (real part) is zero, therefor, it is not very meaningful to plot similar curves like those in Figure 5 for the quasi-parallel electron firehose mode. As electrons become more anisotropic, Figure 5a shows how the purely growing mode at maximum growth rate shifts to shorter wavelength. Figure 5a suggests that when Teñ << Tell, the maximum growth rate of the oblique electron firehose instability occurs at an angle almost perpendicular to the magnetic field. The oblique electron firehose instability is a fast growing instability. The maximum growth rate reaches an appreciable fraction of the electron cyclotron frequency. To demonstrate which species is in resonance with the electron firehose instability, the electron and proton cyclotron resonant factors Re( e ) and Re( p ) are shown 1.6 = - 1..8.4. 1 1 8 4 4 6 8 Figure 4. When electrons are more anisotropic, more branches are unstable at moderate angles. Here Teñ/T ii -., while the other parameters are the same as in Figure 1.

. 7,38 LI AND HABBAL: ELECTRON KINETIC FIREHOSE INSTABILITY 1 oo i, i lo 3 8 (o) 1-11 - 6O 4O 1 O- '),, o,,/["l p \ \ \ % 1-1-1 _ 1- _ 1-3 _ o o.o -4 1..4.6.8...4.6.8 '. /r.,. /., Figure 5. (a) The maximum growth rate (thick solid line) of zero real frequency electron firehose instability, the oblique angle (dashed line) and the normalized wave number (thin solid line) at the maximum growth rate as a function of re_l/tell. (b) The electron and proton cyclotron resonant factor at the maximum growth rate as a function of re_k/r 11. Other parameters are the same as in Figure 1. in Figure 5b. Since the phase velocity of these purely growing modes is zero, there is no Landau resonance between these instabilities with either electrons or protons. The value of lq.e(( ) is quite constant (_>.1) in the entire range of electron anisotropy considered. Hence electrons are only weakly cyclotron resonant at the maximum growth rate of the purely growing mode: only electrons in the tail of the distribution function are resonant. For smaller wave number, there are even less electrons in resonance. Hence this instability is basically nonresonant in nature. Since Re((p ) < 1, protons seem to be highly cyclotron resonant. In principle, the protons should be heated when the electron firehose instability develops. However, the role of protons is quite complicated as will be shown in Figure 6. Figure 6 shows the maximum growth rate of the electron firehose instability as a function of proton beta value. We find that the maximum growth rate of the electron firehose is almost independent of the proton temperature when the electron anisotropy is strong (see Figure 6, T _L/T 11 -.6). However, when electrons are weakly anisotropic, the oblique electron firehose is marginally unstable, the maximum growth rate increases as protons become increasingly hot (see curve corresponding to reñit, ll -.75). This is different from the quasi-parallel electron firehose instability, the growth rate is a slowly increasing function as protons become cold [Garry and Madland, 1985], since an increased p will enhance the proton cyclotron resonance. For the purely growing modes we note that -- k lw I -- k /pp can be as large as 3 when p >.1, where pp -- p/wa is the perpendicular proton thermal Larmor radius. This means that the proton perpendicular thermal Larmor radius is greater than the perpendicular wavelength. (However, the electron perpendicular thermal Larmor radius is still signi[ icantly smaller than the perpendicular wavelength.) In this situation, increasing p suggests a decreasing con- Table 1. Electron Firehose Instability Name Frequencies Wave Numbers at -... -... Condition and Polarization Quasi-parallel electron firehose Oblique electron firehose < wr << qe kv 4/Sqp > 1 < 3 ø, left-hand kv 4/Sqp >> 1 > 3 ø, plane

. --. - _. - _. _ LI AND HABBAL' ELECTRON KINETIC FIREHOSE INSTABILITY 7,383 loo i i i i iiii 7. o./fi,, T,,/T,,,=.6 8 6 4 o 17,.,o,,/fi,, T,,j./T,,=.75 i i i, illi 1-6 1-5 1-4 1-3 1-1-1 1 1 Figure 6. The maximum growth rate of the zero real frequency electron firehose as a function of proton beta for different electron anisotropies and for/ e - 5. tribution of protons to the dielectric tensor as can be cal solutions of the dispersion relation we find that (11) seen from (4). However, the primary reason is that gives only an approximate description of the electron protons are unmagnetized since -/>> 'lp[gary, 1993]. firehose instability threshold and is an overestimate of When protons are unmagnetized for a mode, protons T ñ/t [ I necessary for the onset of the electron firehose. cannot damp the mode. This is why/ p does not change For example, if/ is set equal to $ as in Figure 1, for "/ x when Teñ/Tei I =.5. As long as the growth rate the electron firehose instability to occur, (11) requires satisfies -/>> p and protons are unmagnetized, these T ñ/te[i <.6. In fact, when Te_L/T [i _>.668 and other parameters in Figure 1 are unchanged, all quasiparallel modes are stable. However, the threshold of oblique electron firehose modes is different. Zero real high growth rate instabilities will not heat protons. On the other hand, when the growth rate of the purely growing modes is small at very small propagation angles or small wave numbers (and very large wave num- bers), protons are still magnetized as seen from Figures 1 and 3. These modes of small growth rate will heat protons when the instability develops. From our calculation we also find (not shown here) that varying does not change the oblique angle and wave number at the maximum growth rate. 3.. Instability Threshold In this section, we show that not only does the electron firehose instability have a much greater maximum growth rate than the quasi-parallel instability, as shown in the previous section, but it has a lower instability threshold as well. Hollweg and VSlk [197] gave the following approximate instability threshold for the parallel electron firehose when protons are isotropic 1 +/ (T ñ/t [[- 1)/ <, (1) while Hase#awa [1975] demonstrated that the same instability threshold applied to all propagation directions for MHD firehose modes. However, from our numeri- frequency modes will stabilize when T ñ/t 11 >.733 in this case. Figure 7 shows the growth rates of the oblique electron firehose mode as a function of normalized wave number at different for T ñ/tei I =.668, while the other parameters are the same as in Figure 1. Even though the plasma cannot support an unstable quasiparallel electron firehose mode, the growth rate of the oblique electron firehose is still very significant. Unlike Figure 3, the unstable oblique electron firehose cannot extend to very small wave numbers. Hence the oblique electron firehose has a lower threshold than the quasiparallel electron firehose. 4. Conclusion The oblique and quasi-parallel electron firehose instabilities in a warm plasma are investigated in this paper. It is found that an electron temperature anisotropy of T ñ/t [ I < 1 can excite several branches of purely growing modes in addition to the quasi-parallel electron firehose instability. The essential point of this paper is that

7,384 LI AND HABBAL: ELECTRON KINETIC FIREHOSE INSTABILITY 5O 55 ø 4O 5O 3O =6 ø 1 3O 64.8" 5 1 15 5 Figure 7. Same as Figure 3 but with rea_/rell -.668. the purely growing electron firehose has a much greater / p: maximum growth rate and lower instability threshold than the quasi-parallel electron firehose instability. The electron firehose instabilities also have a lower threshold than the conventional MHD firehose instability. This study may have potential applications for the slow solar wind and other space plasma environments. The approximate instability threshold (11) can be writ- 1 as illustrated in the last section. In this case, the oblique electron firehose mode is unstable when the ratio in the left-hand side of (1) is larger than.88. Hence the potential role of the electron firehose instability, especially the purely growing mode, in regulating the electron temperature anisotropy may need further investigation. In the undisturbed low-latitude slow solar wind near 1 AU the electron velocity distribution is frequently well ten as Pell > 1, (13) represented as two components, a more dense, cooler Pe_L + B/4 r core and a hotter, more tenuous halo [Feldman et al., where Pell = nekbtell and Pe_L : nekbt _L are the elec- 1975]. In addition, protons also often have double tron parallel and perpendicular pressure, respectively. streams. The electron halo component and proton high- In the slow solar wind, TelI/Te_L is typically 1 to 1.5 at energy stream may shift the purely growing electron 1AU [Phillips et al., 1989]. On the basis of the instabil- firehose mode to a propagating one and could potenity threshold (1), Phillips et al. [1989] analyzed ISEE tially further decrease the electron firehose instability 3 data and found that the firehose modes are unlikely to threshold. contribute significantly to the regulation of the electron distribution since (1) is not satisfied in most of the Acknowledgments. This work is supported by NASA observed events (see Figure 9 of Phillips et al. [1989]). grant NAG5-671 to the Smithsonian Astrophysical Obser- However, (1) is an overestimate of the electron firehose vatory. instability threshold. For example, the oblique mode Janet G. Luhmann thanks S. Peter Gary and another is unstable when Te_L/Tell _<.733 for fie = 5, and referee for their assistance in evaluating this paper.

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