Whistler anisotropy instability with a cold electron component: Linear theory

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2012ja017631, 2012 Whistler anisotropy instability with a cold electron component: Linear theory S. Peter Gary, 1 Kaijun Liu, 1 Richard E. Denton, 2 and Shuo Wu 2 Received 21 February 2012; revised 4 June 2012; accepted 5 June 2012; published 6 July [1] The whistler anisotropy instability is driven by an electron temperature anisotropy T? /T k > 1 where? and k denote directions perpendicular and parallel, respectively, to the background magnetic field B o. Here kinetic linear theory in a magnetized, homogeneous, collisionless plasma model is used to study this instability when the electron velocity distribution may be represented as the sum of a hot, anisotropic bi-maxwellian and a cold, isotropic component. The critical b ke, the value at which the maximum growth rate of the instability changes from propagation parallel to B o to oblique propagation, decreases with increasing n c /n e, where n c is the cold electron density and n e is the total electron density. At parallel propagation the maximum growth rate increases with n c /n e up to n c /n e 0.8, but then diminishes with further increases of the relative cold electron density. Introduction of a cold electron component can reduce the hot electron anisotropy necessary to excite this instability by up to a factor of 2. Citation: Gary, S. P., K. Liu, R. E. Denton, and S. Wu (2012), Whistler anisotropy instability with a cold electron component: Linear theory, J. Geophys. Res., 117,, doi: /2012ja Introduction [2] Whistler modes arise in magnetized plasmas and propagate in a frequency range p W lh w r < W e where the lower hybrid frequency W lh ffiffiffiffiffiffiffiffiffiffiffiffiffi W p jw e j, W j represents the cyclotron frequency of the jth species, and p and e denote protons and electrons, respectively. Kinetic linear dispersion theory predicts that, for a single bi-maxwellian electron velocity distribution, at w e / W e > 1 (Here w e denotes the electron plasma frequency), a sufficiently large electron temperature anisotropy T?e /T ke > 1 drives the whistler anisotropy instability [Kennel and Petschek, 1966]. Here? and k denote directions perpendicular and parallel, respectively, to the background magnetic field B o. The instability propagates at W lh w r < W e, at kc/w e < 1, and, at sufficiently large b ke, with maximum growth rate at k B o = 0 [e.g., Gary, 1993, chapter 7]. This is the electromagnetic regime of this instability, because both db and de are transverse to B o. Instability growth leads to enhanced field fluctuations; for example, the whistler anisotropy instability is a likely source of the very large amplitude whistler waves recently observed in the outer radiation belt [Cattell et al., 2008; Cully et al., 2008; Kellogg et al., 2011]. [3] In this electromagnetic regime, the primary consequence of wave-particle interactions is pitch angle scattering. Such scattering reduces the anisotropy on an initially 1 Los Alamos National Laboratory, Los Alamos, New Mexico, USA. 2 Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire, USA. Corresponding author: S. P. Gary, Los Alamos National Laboratory, Mail Stop D466, Los Alamos, NM 87545, USA. (pgary@spacescience.org) American Geophysical Union. All Rights Reserved /12/2012JA bi-maxwellian electron velocity distribution, but generally retains the character of such a distribution; thus the temporal evolution of the electrons can be described in terms of expressions derived from linear dispersion theory based upon bi-maxwellian distributions. As b ke decreases, a successively larger electron anisotropy is needed to excite the instability; this b ke -dependent threshold condition can be written as T?e T ke 1 ¼ S e b ae ke where S e and a e are fitting parameters which vary with the assumed maximum growth rate g m / W e [Gary and Wang, 1996]. The enhanced fluctuating fields arising from the whistler anisotropy instability scatter the electrons, imposing an upper bound on their anisotropy as has been demonstrated both in particle-in-cell (PIC) simulations [Gary and Wang, 1996; Gary et al., 2000] as well as by satellite observations in the terrestrial magnetosheath [Gary et al., 2005] and the magnetosphere [MacDonald et al., 2008]. Other PIC simulations have addressed the whistler anisotropy instability in the electromagnetic regime [Ossakow et al., 1972; Cuperman et al., 1981, and references therein; Devine et al., 1995; Lu et al., 2004, 2010; Liu et al., 2011]. [4] At sufficiently small electron b, the electron anisotropy instability driven by a single bi-maxwellian electron velocity distribution undergoes a fundamental change; kinetic linear dispersion theory shows that, near b ke 0.025, the maximum growth rate shifts from parallel to oblique propagation relative to B o [Hashimoto and Kimura, 1981; Ohmi and Hayakawa, 1986; Gary and Cairns, 1999; Santolík et al., 2010; Gary et al., 2011]. In this regime, the fluctuating electric fields become predominantly electrostatic with k de kde, de k becomes appreciable and PIC ð1þ 1of5

2 Figure 1. Linear theory results for the whistler anisotropy instability at b kh = 1.0 and two values of n c /n e as labeled. The solid and dashed lines represent w r / W e, and the dots represent 5 g/ W e as functions of the parallel wave number. Here T?h /T kh = , corresponding to g m / W e = 0.01 at n c =0. simulations show that magnetic field-aligned heating of electrons becomes important [Gary et al., 2000, 2011; Schriver et al., 2010]. This parallel heating implies that a bi- Maxwellian distribution no longer provides a basis to describe the electrons, and a more general formulation must be developed to explain the late-time electron velocity distributions and the saturation levels of the fluctuating fields. [5] Gary et al. [2011] used kinetic linear dispersion theory for a single bi-maxwellian velocity distribution to examine parametric variations of the critical value of b ke, that is, the value at which the maximum growth rate undergoes a transition from parallel to oblique propagation. They found that this critical b ke is essentially independent of w e / W e as long as w e / W e > 1, and that it is a rather weak function of the instability growth rate, satisfying b ke over g m / W e [6] Spacecraft observations show that electron velocity distributions f e (v) in the magnetosphere are typically more complex than the often-assumed representation of a single bi-maxwellian [e.g., Kellogg et al., 2011]. In particular, plasmaspheric electron velocity distributions are characterized by a hot (few kev), relatively tenuous component, as well as a cold (few ev), denser component [e.g., Santolík et al., 2010; Schriver et al., 2010]. Xiao et al. [2006] carried out linear theory calculations describing the effect of a cold electron component on the threshold of the whistler anisotropy instability. Li et al. [2011a, 2011b] used THEMIS spacecraft observations to study the relationship between total plasma density and linear growth rates of this same instability. They observed that chorus wave amplitudes can be enhanced by both increases and decreases in the total plasma density. Recently, S. Wu and R. E. Denton (Effects of cold electron density on the whistler anisotropy instability, submitted to Journal of Geophysical Research, 2012) have used linear theory to compute how the maximum growth rate of this instability changes in response to variations in the relative cold electron density. But to our knowledge there has been no comprehensive study of how variations in the cold electron density affect the critical b ke of the whistler anisotropy instability. This manuscript addresses this issue, as well as presenting further new results on changes in the instability threshold in response to variations in the relative cold electron density. [7] We pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi denote the jth species plasma frequency as w j 4pn e e 2 =m j, the jth species cyclotron frequency as W j p e j B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o /m j c, the jth component thermal speed as v j k B T kj =m j, b kj 8pn j k B T kj /B 2 o, and ~b kj 8pn e k B T kj =B 2 o.thealfvénspeedisv p A B o = ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pn e m i. Here n e is the total electron density, B o denotes the uniform background magnetic field, and we consider a two-species plasma of electrons (subscript e) and protons (subscript p). [8] In our solutions of the kinetic linear dispersion equation, we take the wave vector k to be real and the complex frequency to be w = w r + ig where g > 0 represents temporal growth of a normal mode of the plasma. For a given set of plasma parameters, g m denotes the maximum growth rate taken over all magnitudes and directions of k. 2. Linear Theory [9] This section describes results from numerical solutions of the full electromagnetic kinetic linear dispersion equation for bi-maxwellian velocity distributions [Gary, 1993] without approximation. The proton velocity distribution is an isotropic Maxwellian, and the electron velocity distribution is the sum of two components, a cold Maxwellian denoted by subscript c and a hot bi-maxwellian (subscript h) with T? /T k 1 corresponding to electrons from several hundred ev to a few tens of kev. Thus n c + n h = n e. We assume m p / m e = 1836 and T kh = 100T c. Furthermore, for n c = 0 we take w e / W e = 4.0, and then increase this parameter as the cold electron density, and therefore n e, is increased. [10] Figure 1 compares the dispersion properties of the whistler anisotropy instability at k B o = 0, corresponding to the direction of propagation at maximum growth rate, at n c = 0 and at n c /n e = The introduction of a substantial cold electron component here does not significantly change the w r / W e versus k k c/w e dispersion, although there is a downward shift in the normalized frequency if the dispersion is plotted as a function of kv e / W e. The presence of cold electrons shifts the maximum growth rate to smaller values of kc/w e but to larger values of kv e / W e and, for these parameters, enhances g m / W e by a factor of 2. The addition of cold electrons, as argued by Brice and Lucas [1971], leads to a decrease in magnetic energy density per particle, reducing the threshold for the whistler anisotropy instability, which corresponds to an increase in the instability growth rate for a fixed hot electron anisotropy. [11] Figure 2 illustrates the critical values of b kh and ~ b kh (that is, the values at which the maximum growth rate changes from parallel to oblique propagation as b kh decreases) for the whistler anisotropy instability at g m / W e = 0.01 as functions of n c /n e. We do not display results as n c /n e approaches unity because the hot electron anisotropy needed to drive the instability in this limit becomes unphysically large. If n c n h, the critical value of ~ b kh remains close to the value of corresponding to a single bi-maxwellian electron model. But this approximation fails as the density of the cold component becomes greater than that of the hot 2of5

3 Figure 2. Linear theory results for the transition of the whistler anisotropy instability from maximum growth at parallel to oblique propagation at g m / W e = The critical values of b kh (indicated by the solid line) and b ~ kh (indicated by the dashed line) are plotted as functions of n c /n e. Above these lines, the maximum instability growth rate lies at k B o = 0, whereas below these lines the maximum growth rate is at propagation oblique to B o. electrons. In contrast, a simple analytic expression provides a good fit to the critical b kh as long as n c /n e < 0.9: b kh ¼ 0:0245½1 1:05ðn c =n e ÞŠ ð2þ We assume b kh > for the remaining calculations described here, so that g m corresponds to k B o =0. [12] We next compute the dependence of g m on the relative cold electron density n c /n e. To do this, we vary the dimensionless parameters so as to maintain the constancy of both the hot electron density, n h and the hot electron temperature. This means that, as we increase n c /n e, we must correspondingly increase w e / W e butholdb kh fixed. Following these prescriptions, Figure 3 illustrates the dependence of g m / W e onn c /n e for two different values of b kh.themaximumgrowthrateat first increases exponentially with n c /n h, then attains a maximum value, and finally decreases as the inertia of the cold component dominates the dynamics and the instability is quenched. Cuperman and Landau [1974] and Cuperman and Salu [1974] provided early demonstrations that the whistler anisotropy instability attains a maximum growth rate as the relative density of the cold electrons is increased. [13] Another way of viewing the effect of cold electrons is, instead of holding the hot electron parameters constant, to compute the instability threshold condition for a fixed maximum growth rate g m / W e. Figure 4 illustrates the linear theory calculation of this threshold at two values of g m / W e for both n c = 0 and n c /n e = 0.80, where the latter value corresponds approximately to the maximum value of g m / W asa function of n c /n e, as illustrated in Figure 3. Over the range b kh 2.5 (which corresponds to a hot electron temperature range of 400 ev to 40 kev for the parameters used here), the threshold condition is well fit by an equation with the same form as equation (1); that is, T?h T kh 1 ¼ S h b ah kh with fitting parameters as given in the caption. ð3þ Figure 3. Linear theory results for the maximum growth rate of the whistler anisotropy instability at k B o =0as a function of the relative cold electron density for two values of b kh as labeled. At b kh = 0.10, the upper dashed line corresponds to T?h /T kh = 2.244, and the lower dashed line corresponds to T?h /T kh = At b kh = 1.0, the upper solid line corresponds to T?h /T kh = 1.363, and the lower solid line corresponds to T?h /T kh = [14] Blum et al. [2009] used linear theory to construct scaling relations as functions of the relative hot proton density for the threshold condition of the Alfvén-cyclotron instability driven by anisotropic protons. Similarly, we here develop scaling relations for S h and a h as functions of the relative cold electron density for the threshold condition of the whistler anisotropy instability. Assuming g m / W e =10 3, we compute a number of instability threshold curves, as in Figure 4, over 0 n c /n e 0.95, and obtain S h and a h as Figure 4. Linear theory results for hot electron anisotropy thresholds of the whistler anisotropy instability at k B o =0. The solid dots represent results for n c = 0 and g m / W e = 0.001; the corresponding solid line plots the fitted curve T? h /T kh 1 = 0.21/b 0.58 kh. The open circles represent results for n c = 0.8 and g m / W e = 0.001; the corresponding dashed line plots the fitted curve T? h /T kh 1 = 0.099/b 0.61 kh. The solid squares represent results for n c = 0 and g m / W e = 0.01; the corresponding solid line plots the fitted curve T?h /T kh 1 = 0.36/b 0.53 kh. The open squares represent results for n c = 0.8 and g m / W e = 0.01; the corresponding dashed line plots the fitted curve T? h /T kh 1 = 0.25/b 0.51 kh. 3of5

4 Figure 5. The fitting parameters S h (black dots) and a h (red squares) of equation (3) for the g m / W e = threshold condition of the whistler anisotropy instability as functions of the relative cold electron density. The dashed lines represent the best fit curves to equation (4) over 0.0 n c /n e 0.95 with the fitting parameters as given in the text. fitting parameters to each of these curves; the results are shown in Figure 5. Then choosing the forms S h ¼ s 0 þ ðn c =n e Þs 1 þ ðn c =n e Þ 2 s 2 a h ¼ a 0 þ ðn c =n e Þa 1 þ ðn c =n e Þ 2 a 2 ð4þ we fit these relations to the results shown in Figure 5, obtaining s 0 = 0.206, s 1 = 0.107, s 2 = , a 0 = 0.574, a 1 = 0.178, and a 2 = These numbers can then be used in equation (3) to experimentally test the validity of the linear theory threshold condition, as has been done by Blum et al. [2009] for the Alfvén-cyclotron instability. [15] Finally, the property that the whistler anisotropy instability has a maximum value of g m / W e as a function of n c /n e suggests that S h should similarly display a minimum as a function of the relative cold electron density. We have added the values of S h and a h corresponding to n c /n e = to Figure 5, thereby confirming this suggestion. Of course, the fitting parameters stated in the above paragraph are not appropriate for n c /n e > 0.95, and for such conditions equation (4) should be modified to represent expansions in the small parameter n h /n e. 3. Conclusions [16] We have used kinetic linear theory to derive scaling relations for the whistler anisotropy instability in a magnetized, homogeneous, collisionless plasma in the presence of a cold electron component of variable relative density. We used linear theory to demonstrate that the critical b kh, the value at which the maximum growth rate of the instability changes from propagation parallel to B o to oblique propagation, decreases monotonically with increasing n c /n e.we confirmed that the maximum instability growth rate attains a maximum as n c /n e varies from zero to unity. We further showed that the hot electron anisotropy necessary to excite the whistler anisotropy instability can be reduced substantially through the introduction of a cold electron component, and then derived a scaling relation for the instability threshold as a function of the relative density of the cold electron component. [17] The conclusions presented in this manuscript were derived from a kinetic linear theory which assumes that the unperturbed part of the hot electron velocity distribution is a bi-maxwellian. Space plasma observations [Lu et al., 2011, and citations therein] suggest that superthermal electrons can exhibit power law, rather than exponential, velocity distributions; such power law distributions act to lower the growth rates of the whistler anisotropy instability and thereby raise the anisotropy thresholds of the instability [Lu et al., 2010]. Thus we expect quantitative changes in our results if power law distributions were used to represent hot electrons. But, as the linear theory and simulations of Lu et al. [2010] have demonstrated, the consequences of non-maxwellian hot electrons are qualitatively the same as for bi-maxwellian distributions; e.g., instability thresholds increase with decreasing hot electron beta. Thus if and when our work is generalized to non-maxwellian distributions, we expect no qualitative changes in our conclusions. [18] A next step in the study of the whistler anisotropy instability would be to use particle-in-cell simulations to determine scalings of the nonlinear properties of the resulting enhanced fluctuations as functions of n c /n e [e.g., Cuperman, 1981]. Questions to be addressed would include: Does the instability saturation level vary in the same way as does the maximum growth rate as a function of n c /n e? What are the conditions which lead to significant heating of the nonresonant cold electrons? Another question worthy of study would be whether the enhanced field fluctuations driven by the whistler anisotropy instability lead to a turbulent cascade, either forward [e.g., Saito et al., 2008; Chang et al., 2011] or inverse. [19] Acknowledgments. The Los Alamos portion of this work was performed under the auspices of the U.S. Department of Energy (DOE). It was supported in part by the Defense Threat Reduction Agency under projects IACRO I and IAA Basic, and in part by the Dynamic Radiation Environment Assimiliation Model (DREAM) Project at Los Alamos National Laboratory. Work at Dartmouth was funded by the Science and Technology Centers Program of the National Science Foundation under Agreement ATM [20] Philippa Browning thanks the reviewers for their assistance in evaluating this paper. References Blum, L. W., E. A. MacDonald, S. P. Gary, M. F. Thomsen, and H. E. Spence (2009), Ion observations from geosynchronous orbit as a proxy for ion cyclotron wave growth during storm times, J. Geophys. Res., 114, A10214, doi: /2009ja Brice, N., and C. Lucas (1971), Influence of magnetospheric convection and polar wind on loss of electrons from the outer radiation belt, J. Geophys. Res., 76, 900. Cattell, C., et al. (2008), Discovery of very large amplitude whistler-mode waves in Earth s radiation belts, Geophys. Res. 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