A&A 494, 311-315 (2009)
DOI: 10.1051/0004-6361:200811109
M. Lazar - S. Poedts
Center for Plasma Astrophysics, Celestijnenlaan 200B, 3001 Leuven, Belgium
Received 8 October 2008 / Accepted 21 November 2008
Abstract
Context. The existence of suprathermal charged particle populations in space plasma is frequently confirmed by interplanetary missions. In general, the velocity distribution functions are anisotropic, field aligned (gyrotropic) with two temperatures, parallel (
)
and perpendicular ()
to the ambient magnetic field .
Aims. Here, the dispersion properties of the firehose instability, which relaxes an anisotropic electron distribution function (
)
of bi-kappa type, are investigated for the first time.
Methods. The Solar wind is generally accepted to be a collisionless plasma and, therefore, the dispersion formalism is constructed on the basis of the kinetic Vlasov-Maxwell equations. The general dispersion relations are derived in terms of the modified plasma dispersion function.
Results. Simple analytical forms are obtained for the dispersion relation of the firehose instability and the instability criterion is derived. The exact numerical evaluation shows a significant departure of the dispersion curves from those obtained for a bi-Maxwellian plasma.
Conclusions. While the maximum growth rate is slightly diminished, the instability extends to large wave-numbers in the presence of suprathermal particles. Thus, this instability is more likely to be found in space plasmas with an anisotropic distribution of bi-kappa type. If all other parameters are known, measuring the instability growth time enables the determination of the spectral index .
Key words: plasmas - instabilities - Sun: coronal mass ejections (CMEs) - Sun: flares - Sun: solar wind
There is consistent observational evidence that nonthermal particle populations are ubiquitous in the near Earth space plasma (for an overview, see in Zouganelis et al. 2005, and the references therein). Suprathermal populations are described by the so-called distribution function (introduced by Vasyliunas 1968), i.e. a power law in particle speed, with high-energy tails, and which degenerates into a Maxwellian distribution function as the spectral index becomes very large ( ). Moreover, such deviations from the Maxwellian distribution are expected to exist in any low-density plasma in our Universe, where binary collisions are sufficiently rare. Instead, the wave-particle interactions can be responsible for the energization of plasma particles and the occurrence of distributions. In an ambient magnetic field, plasma particles gain energy through the cyclotron resonance and transit time damping (Landau resonance) of the linear waves (Fisk 1976). Thus, the small (linear) amplitude MHD modes, Alfvén and fast magnetosonic waves are able to accelerate both the electrons and the protons in the solar flares (Miller 1991, 1997), and in the inner magnetosphere (Summers & Ma 2000), while the whistler mode can enhance the energy of the electrons in the Earth's foreshock (Ma & Summers 1998). In the presence of large amplitude waves and plasma turbulence the nonlinear mechanisms of dissipation (e.g., the nonlinear Landau damping) also can be responsible for the energization of plasma particles (Miller 1991, Leubner 2000, Shizgal 2007). However, the MHD turbulent cascades channel the energy to small scales where the quasi-linear interactions with particles are again possible (Miller 1991, 1997). The non-thermal features of the particle distribution functions can also result from heat flows and temperature anisotropies (Leubner & Viñas 1986). The use of the generalized kappa-type distribution functions enables a more realistic description of the turbulent fields and the acceleration of plasma particles in such environments (Summers & Thorne 1991; Maksimovic et al. 1997; Mace & Hellberg 2003; Lazar et al. 2008b).
In magnetized interplanetary plasmas, the particle distribution functions are in general anisotropic, i.e. depend on the direction in velocity space (Pilipp et al. 1990; Salem et al. 2003), and, therefore, unstable to the excitation of plasma instabilities (Paesold & Benz 1999; Fahr & Siewert 2007, for a review of the fundamental properties of anisotropic plasma instabilities and their applications in space plasma physics, we recommend the textbook of Gary 1993). Although considerable progress has been made, questions still exist about the origin of the particle velocity anisotropies and the relaxation mechanisms in space plasmas (Stverak et al. 2008). Recently it has been shown how the bulk energy of the solar wind ions is converted by developing pronounced pitchangle anisotropies when the ions pass over the solar wind MHD termination shock (Fahr & Siewert 2007). Such anisotropic distribution functions are mirror-mode unstable and up to 60 percent of the thermal ion energy can be converted by this instability into magnetoacoustic turbulence energy. However, at large solar distances mechanisms of adiabatic cooling can be identified (e.g., the conservation of the two magnetic CGL invariants), which prevent the solar wind from developing strong pressure anisotropies (Fahr & Siewert 2008).
The preference for the acceleration of plasma particles along the background magnetic field is a common feature of the different acceleration models in solar flares and in the solar wind (see the Introduction of Paesold & Benz 1999, for a brief description of these mechanisms of acceleration). Thus, the velocity distribution function becomes anisotropic with a substantial amount of free energy residing in the direction of the magnetic field that may give rise to the excitation of the firehose instability.
The electron firehose mode propagates along the ambient magnetic
field and the instability typically arises when the parallel
temperature is much larger than the transverse temperature
(
). In this case, the approximate
instability criterion of Hollweg & Völk (1970) is
fulfilled:
Since the frequency of the firehose mode is in the range of the ion gyrofrequency, the electrons are non-resonant and the instability is non-resonantly driven by the electron temperature anisotropy (Hollweg & Völk 1970). The ions, however, are resonant and they can gain energy making the firehose instability responsible for the transfer of the electron energy to the ions. The first simulations of the electron firehose instability have also shown that the generated magnetic field fluctuations scatter the electrons, reducing their anisotropy (Gary & Nishimura 2003; Paesold & Benz 2003).
Presently, it is believed that the electron firehose instability presents two branches (Li & Habbal 2000): one with a finite real frequency, and quasi-parallel propagation ( ), and another one that is non-resonant, with and quasi-perpendicular propagation ( ). Both of these branches have recently been confirmed by two-dimensional Particle-In-Cell (PIC) simulations with a physical mass-ratio (Camporeale & Burges 2008). Here, we should remark that the second branch describing a purely-growing and quasi-perpendicular instability is a Weibel-like instability (Weibel 1959), as long as it is non-resonantly excited by a temperature anisotropy and has the wave-vector normal to the high temperature axis.
Potential applications of the kinetic electron firehose instability to solar flares or to the solar wind are discussed in Paesold & Benz (1999) and Li & Habbal (2000). The main idea is that this instability can constrain the increase of the electron temperature anisotropy, thus explaining the observations. Indeed, the anisotropic velocity distribution can be subjected to a pitchangle isotropisation by quasilinear interactions of electrons with plasma waves and other dissipative processes. The electron firehose instability itself can be the most efficient mechanism of temperature isotropisation in solar flares limiting electron anisotropy and thus providing the necessary condition for further acceleration (Messmer 2002).
In the present paper, we briefly describe the dispersion properties of the electron firehose instability driven by an anisotropic electron distribution function ( ) of the bi-kappa type, and propagating along the ambient magnetic field ( ). The plasma is assumed to be quiescent, without large amplitude oscillations and turbulence, so that we restrict this discussion to a quasilinear description of this instability.
For the transverse waves propagating along the ambient magnetic field, i.e.,
,
(along the y-axis in the present model), we take the general dispersion relation from Tautz & Schlickeiser (2005)
We consider a bi-kappa velocity distribution function of plasma
particles
The electron firehose instability arises when the plasma is hotter along the direction of the stationary magnetic field, i.e. when . This instability is left-handed (LH) circularly polarized at , and the electrons are non-resonant, . Only the ions are cyclotron resonant, but their contribution is minimized here assuming that, initially, the ion distribution function is isotropic. The ion firehose instability develops for similar conditions of the ion temperature anisotropy, , and saturates at maximum growth rates comparable to the electron firehose instability (viz. of the order of the ion gyrofrequency). However, because the ion response could be either inhibited or enhanced by the resonant dissipation of the electron firehose instability, the interplay of these two instabilities is more complex and it will be investigated in detail separately.
We now insert the distribution function (3) in Eq. (2) and derive the dispersion relation of the LH transverse modes
Consider a plasma with anisotropic electrons and isotropic ions (protons),
.
The dispersion relation (7) then simplifies to
Here we follow the same formalism as Hollweg & Völk
(1970) and generalize the firehose instability criterion for
anisotropic particle velocity distributions of the bi-kappa
type. We use some reasonable simplifications based on what we are
looking for: the firehose instability has a frequency (
)
in the vicinity of the ion cyclotron
frequency, and therefore
,
and
.
Moreover, for simplicity, the protons can be
considered Maxwellian (as sustained by many observations) and
isotropic in the velocity space without any contribution to the
instability. Besides that, the protons are resonant and we can
assume a small argument f_{i} < 1 of the dispersion function
(Maxwellian) that in this case approaches
.
With these
presumptions, the dispersion relation (18) becomes
While Eq. (22) gives the instability criterion in a plasma with a bi-kappa
temperature anisotropy, for an infinitely large
we find the exact criterion of firehose instability for a bi-Maxwellian distribution
Keeping an ion distribution function of the kappa-type (as suggested by the recent works of Fisk & Gloeckler 2006, 2007), only the numerical factor in the last term of the dispersion relation (19) will increase slightly, for example, to 2.02 and 2.82 for and , respectively. But this term has no influence on the instability criterion in Eq. (22). This is confirmed in the next section by comparing the exact numerical solutions.
Figure 1: The real and imaginary solutions of Eq. (25) are plotted with solid lines for three values of , and with dashed lines for an infinitely large . The electron-proton plasma parameters are cm^{-3}, K, , G. | |
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In the previous section, we derived the analytical (approximative)
forms of the frequency and the growth rate. Here, however, to be
rigorous, we present the exact numerical solutions of the firehose
instabilities. If the protons are Maxwellian, Eq. (18)
transforms in terms of the standard dispersion function (Fried &
Conte 1961) into
In this case, the maximum growth rates cannot be obtained by simply
imposing d
in Eq. (21), or
Eq. (25), because the instability is saturated by two
mechanisms at the same time, either by scattering electrons or by
heating ions (Messmer 2002; Paesold & Benz 2003).
However, the growth rates shown in Fig. 1 are only slightly
diminished by the non-Maxwellian distributions of the electrons.
Contrary to this, the range of instability is extended to larger
wave-numbers. Again, the so-called cutoff wave number, ,
which
limits the existence of the firehose instability to wave numbers
,
cannot be found by simply imposing condition
in Eq. (25), because the growth rate decreases drastically
but it does not vanish completely at
.
Therefore, here, we
only suppose that
can be derived from the resonance condition
for the ions,
,
in the dispersion Eq. (25),
and we leave the exact calculation of the maximum growth rates and
the cutoff wave numbers for a detailed analysis elsewhere.
Figure 2: Comparison of numerical solutions given by the Eqs. (18) (dotted lines) and (25) (solid lines) for an index . The plasma parameters are the same as in Fig. 1. | |
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In Fig. 2, with dotted and solid lines, the dispersion curves (a) and the growth rates (b) given by Eqs. (18) and (25), respectively, are compared. We have assumed that the electrons are described by a bi- anisotropic distribution function, and in Eq. (18), the ions are described by an isotropic distribution function. A small deviation is observed only for the growth rates, which are slightly decreased by the non-Maxwellian ions.
The dispersion relations provided above, for example Eqs. (18) and (25), allow for a more extensive analysis of the dispersion properties and the stability of different plasma systems if we combine electron and ion distribution functions with diverse spectral indexes as given by the observations.
Acknowledgements
The authors acknowledge financial support from the Research Foundation Flanders - FWO Belgium. These results were obtained in the framework of the projects GOA/2009-009 (K.U. Leuven), G.0304.07 (FWO-Vlaanderen) and C 90205 (ESA Prodex 9). Financial support by the European Commission through the SOLAIRE Network (MTRN-CT-2006-035484) is gratefully acknowledged. The numerical results were obtained on the HPC cluster VIC of the K.U. Leuven. We are grateful to the referee (Prof. Fahr) for his insightful comments and the constructive suggestions.