Shaping the solar wind temperature anisotropy by the interplay of electron and proton instabilities

Abstract

A variety of nonthermal characteristics like kinetic, e.g., temperature, anisotropies and suprathermal populations (enhancing the high energy tails of the velocity distributions) are revealed by the in-situ observations in the solar wind indicating quasistationary states of plasma particles out of thermal equilibrium. Large deviations from isotropy generate kinetic instabilities and growing fluctuating fields which should be more efficient than collisions in limiting the anisotropy (below the instability threshold) and explain the anisotropy limits reported by the observations. The present paper aims to decode the principal instabilities driven by the temperature anisotropy of electrons and protons in the solar wind, and contrast the instability thresholds with the bounds observed at 1 AU for the temperature anisotropy. The instabilities are characterized using linear kinetic theory to identify the appropriate (fastest) instability in the relaxation of temperature anisotropies $A_{e,p} = T_{e,p,\perp}/ T _{e,p,\parallel} \ne1$. The analysis focuses on the electromagnetic instabilities driven by the anisotropic protons ($A_{p} \lessgtr1$) and invokes for the first time a dynamical model to capture the interplay with the anisotropic electrons by correlating the effects of these two species of plasma particles, dominant in the solar wind.

Appendices

Appendix A: Distributions and dispersion functions

For a plasma of electrons and protons with bi-Maxwellian velocity distribution functions (VDFs)

$$ F_{\alpha,M} ( v_{\parallel},v_{\perp} ) =\frac{1}{\pi ^{3/2}u_{\alpha,\perp}^{2} u_{\alpha,\parallel}}\exp \biggl( -\frac{v _{\parallel}^{2}}{u_{\alpha,\parallel}^{2}}-\frac{v_{\perp }^{2}}{u _{\alpha,\perp}^{2}} \biggr) , $$

(5)

where thermal velocities $u_{\alpha,\parallel, \perp}$ are defined by the components of the anisotropic temperature

$$\begin{aligned} &T_{\alpha,\parallel}^{M}=\frac{m}{k_{B}} \int d\textbf{v} v_{\parallel}^{2} F_{\alpha}(v_{\parallel}, v_{\perp})=\frac{m u_{\alpha,\parallel}^{2}}{2 k_{B}} \end{aligned}$$

(6)

$$\begin{aligned} &T_{\alpha,\perp}^{M}=\frac{m}{2 k_{B}} \int d\textbf{v} v_{\perp} ^{2} F_{\alpha}(v_{\parallel}, v_{\perp})=\frac{m u_{\alpha,\perp}^{2}}{2 k_{B}}, \end{aligned}$$

(7)

the plasma dispersion function in Eq. (1) takes the standard form (Fried and Conte 1961)

$$ Z_{\alpha,M} \bigl( \xi_{\alpha,M}^{\pm} \bigr) = \frac{1}{\pi^{1/2}} \int_{-\infty }^{\infty}\frac{\exp ( -x^{2} ) }{x-\xi_{\alpha,M}^{\pm}}dt,\quad \Im \bigl( \xi_{\alpha,M}^{\pm} \bigr) >0 $$

(8)

of argument $\xi_{\alpha,M}^{\pm}= ( \omega\pm\varOmega_{ \alpha} ) / ( ku_{\alpha,\parallel,} )$

To include suprathermal population, the electrons can be described by a bi-Kappa VDF (Summers and Thorne 1991)

$$\begin{aligned} F_{e,\kappa} =& \frac{1}{\pi^{3/2}u_{e, \perp}^{2}u_{e, \parallel}} \frac{\varGamma ( \kappa_{e} +1 ) }{\varGamma ( \kappa_{e} -1/2 )} \\ &{}\times\biggl[ 1+\frac{v_{\parallel}^{2}}{\kappa_{e} u_{e, \parallel}^{2}}+\frac{v_{\perp}^{2}}{\kappa_{e} u_{e, \perp}^{2}} \biggr] ^{-\kappa_{e} -1} \end{aligned}$$

(9)

which is normalized to unity $\int d^{3}vF_{e,\kappa}= 1$, and is written in terms of thermal velocities $u_{e,\parallel, \perp}$ defined by the components of the effective temperature (for a power-index $\kappa_{e} >3/2$)

$$ T_{e,\parallel}^{K}=\frac{2 \kappa_{e}}{2 \kappa_{e}-3}\frac{m_{e} u _{e ,\parallel}^{2} }{2 k_{B}}, \quad \quad T_{e,\perp}^{K}=\frac{2 \kappa_{e}}{2 \kappa_{e}-3}\frac{m_{e} u_{e , \perp}^{2}}{2 k_{B}}. $$

(10)

Suprathermals enhance the electron temperature, and implicitly the plasma beta parameter (Leubner and Schupfer 2000, 2001; Lazar et al. 2015)

$$ \begin{aligned} &T_{e,\parallel, \perp}^{K}=\frac{2 \kappa_{e}}{2 \kappa_{e}-3}T_{e,\parallel,\perp}^{M} > T_{e,\parallel,\perp}^{M}, \\ &\beta_{e,\parallel, \perp}^{K}=\frac{2 \kappa_{e}}{2 \kappa_{e}-3} \beta_{e,\parallel,\perp} > \beta_{e,\parallel,\perp}, \end{aligned} $$

(11)

and for the modified Kappa dispersion function (8) we use in Eq. (1) the form (Lazar et al. 2008)

$$\begin{aligned} Z_{e,\kappa} \bigl( \xi_{e,\kappa}^{\pm}\bigr) =& \frac{1}{\pi^{1/2}\kappa_{e}^{1/2}}\frac{\varGamma ( \kappa_{e} ) }{\varGamma ( \kappa_{e} -1/2 )} \\ &{}\times \int_{-\infty}^{\infty}\frac{ ( 1+x^{2}/\kappa_{e} ) ^{-\kappa_{e}}}{x-\xi_{e,\kappa}^{\pm}}dx,\ \Im \bigl(\xi_{e,\kappa}^{\pm} \bigr) >0, \end{aligned}$$

(12)

of argument $\xi_{e,\kappa}^{\pm}= (\omega\pm\varOmega_{e})/(k u_{e,\parallel,})$.

Appendix B: Fitting parameters for Eq. (4)

Table 1 Fitting parameters for PFH thresholds in Figs. 7 and 9(b)

Table 2 Fitting parameters for EMIC thresholds in Figs. 8 and 9(a)