Transport coefficients enhanced by suprathermal particles in nonequilibrium heliospheric plasmas

The solar wind is a plasma of a very controversial nature due to the multitude of physical processes which act concomitantly or successively and control not only local properties, but also the variation of plasma parameters with the expansion in heliosphere. Indeed, the solar wind is neither a collision-dominated plasma nor a collisionless one (Maksimovic et al., 1997; Pierrard et al., 2011), and this is most likely the reason why ideal magnetohydrodynamic (MHD) or collision-free (exospheric) models cannot accurately reproduce the global expansion and thermodynamics of the solar wind (Maksimovic et al., 1997; Matteini et al., 2007; Bale et al., 2013). Particle-particle collisions are believed to play an important role at low distances from the Sun in the outer corona, although there are also wave fluctuations and turbulence from MHD to proton and electron scales (Bourouaine et al., 2012). At larger heliocentric distances, for example at 1 AU, low-energy particle (core) populations remain organized by a so-called collisional age (Salem et al., 2003; Štverák et al., 2008; Bale et al., 2009), while the nonequilibrium states of plasma particles with anisotropic velocity (or energy) distributions and suprathermal populations (Maksimovic et al., 2005; Marsch, 2006; Wilson et al., 2019) may be explained by the interaction with wave turbulence and small-scale fluctuations (Štverák et al., 2008; Bale et al., 2009; Alexandrova et al., 2013). There is observational evidence for the preferential heating of plasma particles through resonant wave-particle interactions (Bourouaine et al., 2011; Wilson et al., 2013), as well as for the generation of more diffuse suprathermal (halo) populations by the wave turbulence of a quiet solar wind, or by the instabilities self-induced by the beam (or strahl) populations in the fast solar winds, interplanetary shocks, etc. (Mason & Gloeckler, 2012; Gurgiolo et al., 2016).

Suprathermal populations enhance the high energy tails of the observed velocity distributions (up to a few keV) and are characteristic not only of the solar wind electrons and protons (see Lazar et al. (2012) for a review), but they are also reported for the heavier (minor) ions at 1 AU (Collier et al., 1996) and beyond (Ogilvie et al., 1993). Theoretical approaches use the so-called Kappa distributions to describe these populations (Pierrard & Lazar, 2010) and offer explanations for their origin, mainly based on the energization of plasma particles, for example, electrons, by the interaction with small-scale wave fluctuations, including for instance the electromagnetic electron-cyclotron (whistler) waves (Vocks et al., 2008) or the electrostatic Langmuir plasma waves (Viñas et al., 2000; Yoon et al., 2006). Quasi-linear models involve collisions and a weak plasma turbulence, either with predefined spectral profiles (Roberts & Miller, 1998; Vocks et al., 2008), or self-supported by spontaneous and induced plasma wave fluctuations (Yoon et al., 2006). What matters is that suprathermal populations retain the effects of wave turbulence and fluctuations present in the system, suggesting that a realistic macroscopic theory for the energy and particle transport in space plasmas must rely on the assumption that plasma particles are Kappa-distributed.

Transport processes in a plasma can be studied with a transport equation, such as the (collisional) Boltzmann equation for the evolution of the phase space particle distribution. If a plasma is sufficiently homogeneous, macroscopic equations are obtained by calculating the main velocity moments on the basis of a velocity distribution function describing the plasma particles (Braginskii, 1965; Balescu, 1988). The macroscopic equations are based on a linear relationship between the fluxes and the external forces and gradients (in, e.g., density, temperature, pressure) driving the fluxes. The constants of proportionality are called transport coefficients, and they contain valuable physical information. The collisional term in the transport equation describes the rate of change of the distribution due to collisions, and by adopting simplified models for the collision term (see Sec. 3), one can obtain analytical expressions for the transport coefficients. Because driving gradients and forces lead to deviations from quasi-stationary states, one can assume a small or linear perturbation of the distribution function and linearize the transport equation (Goedbloed et al., 2019). For plasmas near thermal equilibrium, the standard Maxwellian distribution is commonly used, while nonthermal particle populations in space plasmas are well described by the family of Kappa distributions (see Sec. 2).

Recent studies of plasma diffusion, that is, the flux of particles due to a density gradient in the plasma, concerned nonequilibrium plasmas and applied Kappa distributions to determine the diffusion coefficient (Wang & Du, 2017; Ebne Abbasi et al., 2017). In addition to the diffusion, the mobility coefficient describes the flux of charged particles in the presence of an electrical field and was examined involving Maxwellian (Hagelaar & Pitchford, 2005) and Kappa distributions (Wang & Du, 2017). The mobility of charged particles is related to the electric conductivity, setting the current density, and the electric field in the relationship (Du, 2013; Ebne Abbasi et al., 2017). The thermoelectric coefficient relates the electric field to the temperature gradient resulting in electric voltages and currents, and it was derived based on Kappa distributions (Du, 2013; Guo & Du, 2019). The heat flux is driven by a temperature gradient, for which the thermal conductivity is the constant of proportionality, see, for example, Rat et al. (2001) for Maxwellian plasma populations, and Du (2013), Guo & Du (2019), and Ebne Abbasi & Esfandyari-Kalejahi (2019) using Kappa distributions. However, all of these studies involving Kappa distributions invoke a modified approach that drastically differs from the original Olbertian or the standard Kappa distribution introduced by Olbert (1968) and Vasyliūnas (1968). Moreover, recent studies have also pointed to the differences between these two Kappa approaches (Lazar et al., 2015, 2016), clearly stating that a modified Kappa distribution does not allow for a proper comparison with the quasi-thermal core of the observed distributions to outline the suprathermals and their implications (see also Section 2 for more details as well as the discussions in Lazar & Fichtner (2021)).

Motivated by these precedents, in this paper we propose a re-evaluation of transport coefficients based on a standard Kappa approach, which allows for a correct assessment of the contribution of the suprathermal population. Indeed, here we show that a modified approach systematically leads to an underestimation of the effects of these populations on transport coefficients. The paper is structured as follows. In Sec. 2 we discuss the applied Kappa power-law distribution function, which describes the suprathermal tails frequently observed in nonthermal space plasmas. Section 3 presents the calculations for the transport coefficients based on the Kappa distribution, that is to say the electric conductivity, the thermoelectric coefficient, the thermal conductivity, the diffusion coefficient, and the mobility coefficient were derived. We conclude the paper in Sec. 4 with a summary, discussion, and outlook.

In situ measurements by spacecrafts indicate a ubiquitous presence of suparthermal populations in the velocity (or energy) distributions of plasma particles in the heliosphere (Maksimovic et al., 2005; Zouganelis et al., 2005; Marsch, 2006). The low-energy core of the measured distributions is nearly Maxwellian (i.e., quasi-thermal), but high-energy tails are suprathermal, decreasing as power laws and markedly departing from the Maxwellian core. Overall the observed distributions are well fitted by the Kappa distribution functions (Pierrard & Lazar, 2010), which are empirically defined by Olbert (1968) and Vasyliūnas (1968) to describe electrons in the near-Earth solar wind and magnetospheric plasmas. This model distribution was introduced in the following (isotropic) form

where $n$ is the number density, $v$ is the particle speed, $\kappa$ is the power-law exponent describing the high-energy distribution tails, $\Gamma(a)$ denotes the (complete) Gamma function of argument $a$, and $\Theta$ was termed the most probable speed, which relates to the energy of the peak in the differential flux and to the thermal speed of the Maxwellian limit, that is, $\Theta=\Theta_{\mathrm{M}}\equiv\sqrt{2\,k_{\mathrm{B}}\,T_{\mathrm{M}}/m}$ (Olbert, 1968; Vasyliūnas, 1968), also see below. The (kinetic) temperature for Kappa-distributed populations ($T_{\kappa}$) is obtained via the second velocity moment

and is defined for $\kappa>3/2$ (provided that $T_{\kappa}>0$). Here, $m$ denotes the particle mass and $k_{\mathrm{B}}$ is Boltzmann’s constant. This original form of Kappa distribution allows for a straightforward analysis of suprathermals and their effects by a direct contrast with the (quasi-)thermal core (subscript c) of the observed distribution (Lazar et al., 2015, 2016; Lazar & Fichtner, 2021). In this case the core is well reproduced by the Maxwellian limit (subscript M)

with the lower temperature

see Eq. (2), but with the dominant number density $n_{\rm c}\simeq n=n_{\rm total}$ (Lazar, 2017). The kinetic temperature $T_{\kappa}$ increases with decreasing $\kappa$, that is, with an enhancing suprathermal population in the tails. Such a contrasting approach has already been applied to outline the effects of suprathermals in various plasma processes, such as spontaneous or induced fluctuations (instabilities), which show a systematic stimulation in the presence of suprathermals (Viñas et al., 2015; Lazar et al., 2015, 2018; Shaaban et al., 2019, 2020).

Another modified Kappa distribution,

is also invoked in the literature, including studies on instabilities (Lazar et al., 2011, 2013; Viñas et al., 2017) and also evaluations of transport coefficients (Du, 2013; Guo & Du, 2019). The modified Kappa distribution is expressed in terms of Kappa temperature $T_{\kappa}$, which is assumed to be a parameter independent of the $\kappa$ exponent ¹¹1 In this case, parameter $\Theta=\Theta_{\mathrm{M}}$ differs from $\Theta_{\mathrm{m},\kappa}\equiv\sqrt{(1-3/(2\,\kappa))}\,\Theta$.. This implies that the modified Kappa distribution and its Maxwellian limit obtained for $\kappa\to\infty$ are described by the same temperature, that is, $T_{M}=T_{\kappa}$. In that case, the Maxwellian does not reproduce the low-energy core of the modified Kappa distribution. Therefore, it cannot be used by contrast with Kappa to outline the presence of suprathermals in the tails, and, implicitly, their contribution to various physical properties; see also Lazar et al. (2015) and Lazar et al. (2016) for extended comparative analyses. In order to demonstrate how sensitive the evaluations of transport coefficients are to the shape of the distribution, here we adopt a standard (original) Kappa approach (Olbert, 1968; Vasyliūnas, 1968) and compare our results with previous estimations for a Maxwellian-like core, and also for a modified Kappa approach, which underestimates the effects of the suprathermal population.

In order to simplify the computation of the transport coefficients, in Sec. 3 we assume that the plasma consists of two particle species, ions, and Kappa-distributed electrons. For the derivation of the transport coefficients mainly contributed to by the electrons (e.g., in hot but still collisional plasmas whose electron temperature is comparable to the ion temperature), we rewrite Eq. (1) as

in terms of the kinetic energy $\varepsilon=m\,v^{2}/2$, and the normalization constant

The distinction between the two Kappa approaches is captured by the parameter $\eta$, as for $\eta=\kappa$ the distribution (6) represents the standard (original) Kappa from (1), and for $\eta=\kappa-3/2$ it becomes a modified Kappa distribution, as in Eq. (2). For the sake of simplicity, in the following we omit the subscript c for the temperature, and adopt $T_{\mathrm{c}}=T$.

Based on these assumptions, we derived the following transport coefficients: the electric conductivity $\sigma$, the thermoelectric coefficient $\alpha$, the thermal conductivity $\lambda$, the diffusion coefficient $D$, and the mobility coefficient $\mu$. These parameters can be obtained from the following macroscopic laws:

where $f$ is the particle distribution, for example the electron Kappa distribution. Equation (8) is a generalization of Ohm’s law, where $\@vec{E}$ denotes the electric field and $\@vec{j}$ is the current density, which is explicitly defined in Eq. (10). Equation (9) is a generalization of Fourier’s law, where $\@vec{q}$ is the heat flux, defined in Eq. (11), and $\phi$ is the electric potential related to the electric field $\@vec{E}=-\nabla\phi$. The particle flux defined in (12) satisfies an extended Fick’s law in Eq. (13), which includes not only the diffusion due to density gradient, but also the mobility of charged particles in the presence of an electric field. For more details, see Spatschek (1990), Boyd & Sanderson (2003), and Goedbloed et al. (2019).

Transport processes in a plasma are in general described starting from the (collisional) Boltzmann equation for the time-space evolution of the distribution function $f$

We note that $\@vec{E}$ and $\@vec{B}$ are the electric and magnetic fields, including the imposed and self-generated ones, $c$ is the speed of light in vacuum, $q$ is the particle’s charge, and collisional term $\mathcal{C}(f)$ is in general a nonlinear integral. In the absence of a magnetic field and assuming a stationary transport, Eq. (14) simplifies to

The change in the distribution function can be assumed small enough to be described by a linearized velocity distribution

where $f_{0}$ is the stationary nonequilibrium distribution and $f_{1}$ is a small perturbation. If electrons are hot enough (Lorentzian plasma), we can consider only collisions between the mobile electrons and the stationary, heavy ions. For the collisional term, we use a Krook-type collisional operator (Bhatnagar et al., 1954) and assume that the disturbed distribution relaxes under the effects of collisions with a rate $\nu_{\mathrm{ei}}(v)$, which allows us to write

The collision frequency $\nu_{\mathrm{ei}}(v)$ between electrons and ions (subscript ”ei”) can be described by (Helander & Sigmar, 2005)

where $z$ denotes the ion charge number, $L_{\mathrm{ei}}$ is the Coulomb logarithm $L_{\mathrm{ei}}\equiv\ln{\Lambda}$, and $\Lambda$ is the electron Debye length (normalized to the impact parameter). Inserting Eq. (17) into Eq. (15) and setting $f_{0}=f_{\kappa}$ yields

where the spatial and velocity gradients of the perturbation of second order can be neglected. By using Eq. (6), we can transform Eq. (19) into

We note that the middle (energy-free) term with the temperature gradient does not appear in Eq. (10) from Du (2013). This term does not affect the computation of the electric conductivity $\sigma$, the diffusion coefficient $D$, or the mobility coefficient $\mu$, but it can have an important influence on the thermoelectric coefficient $\alpha$ and thermal conductivity $\lambda$, see the Appendix A.

In the present paper we calculated transport coefficients in nonequilibrium plasmas and have shown the consequences of the presence of suprathermal particles on these coefficients. By assuming a plasma with Kappa-distributed electrons and using a particular expression for the collision frequency, we derived expressions for the electric conductivity $\sigma$, the thermoelectric coefficient $\alpha$, the thermal conductivity $\lambda$, the diffusion coefficient $D$ and the mobility coefficient $\mu$. The results indicate significant enhancements in the transport coefficients in the presence of suprathermal particles. For the Kappa distribution, two common approaches are used in literature, the standard Kappa distribution originally introduced by Olbert (1968) and Vasyliūnas (1968), which enables a proper description of suprathermal populations in contrast to the quasi-thermal core, and a modified Kappa distribution, which does not have the same relevance (see section 2). We computed the results for both Kappa approaches and have shown that the modified Kappa approach consistently underestimates the transport coefficients by comparison to the original Kappa approach at small values of $\kappa$, while the transport coefficients approach the Maxwellian limit for both Kappa variants. Furthermore, we were able to compare our results to those found earlier by Du (2013), who used the modified Kappa approach and whose results for the thermoelectric coefficient $\alpha$ and the thermal conductivity $\lambda$ differ from our results due to a missing term in the derivation of the expression for the perturbed distribution function term (see Appendix A).

In the present analysis, we have not incorporated data from measurements or specific values for the temperature or associated quantities, but only evaluated normalized transport coefficients. For normalization, we used the Maxwellian limit that approaches the core of the Kappa distribution, and thus quantifies the contribution of the suprathermal electrons to the transport coefficients. Nonetheless, we have to point out that when using observational data, one needs to be cautious whether the measured temperature is after a Kappa fitting as the second-order velocity moment, or whether it is the lower one obtained after a Maxwellian fitting with the core of the observed distribution. In future work we plan to extend these evaluations of the transport coefficients for plasmas described by the regularized Kappa distribution (RKD; Scherer et al. (2017)), which has been introduced to formulate a contradiction-free macroscopic description of nonthermal populations observed in the solar wind. By contrast to a standard Kappa, the RKD is defined for all values of $\kappa>0$ and so are the macroscopic parameters, drift velocity, temperature, heat flux, etc., avoiding limitations corresponding to singularities (Lazar et al., 2020).