The Interplay of the Solar Wind Core and Suprathermal Electrons: A Quasilinear Approach for Firehose Instability

The in situ measurements collected from different missions, e.g., Ulysses, Helios 1, and Cluster II, unveil electron distributions in the slow winds (V_SW ≤ 500 km s⁻¹) with a dual structure combining a thermal dense core (subscript c), and a dilute suprathermal halo (subscript h) (Maksimovic et al. 2005; Štverák et al. 2008)

$$\begin{eqnarray}&&{f}_{e}\left({v}_{\parallel },{v}_{\perp }\right)={\eta }_{c}\,{f}_{c}\left({v}_{\parallel },{v}_{\perp }\right)+{\eta }_{h}\,{f}_{h}\left({v}_{\parallel },{v}_{\perp }\right).\end{eqnarray} \tag{ 1 }$$

Here, η_h = n_h/n₀ and η_c = 1 − η_h are relative densities of the halo and core, respectively, and n₀ is the total electron number density. The core population is well fitted by a bi-Maxwellian distribution (Štverák et al. 2008)

$$\begin{eqnarray}&&{f}_{c}\left({v}_{\parallel },{v}_{\perp }\right)=\displaystyle \frac{1}{{\pi }^{3/2}{\alpha }_{\perp ,c}^{2}\,{\alpha }_{\parallel ,c}}\exp \left(-\displaystyle \frac{{v}_{\parallel }^{2}}{{\alpha }_{\parallel ,c}^{2}}-\displaystyle \frac{{v}_{\perp }^{2}}{{\alpha }_{\perp ,c}^{2}}\right),\end{eqnarray} \tag{ 2 }$$

with thermal velocities ${\alpha }_{\parallel ,\perp ,c}\equiv {\alpha }_{\parallel ,\perp ,c}(t)$ (varying in time t in our QL approach) defined by the corresponding temperature components (as the second-order moments of the distribution)

$$\begin{eqnarray}&&{T}_{\parallel ,c}=\displaystyle \frac{{m}_{e}}{{k}_{B}}\int d{\boldsymbol{v}}{v}_{\parallel }^{2}{f}_{c}({v}_{\parallel },{v}_{\perp })=\displaystyle \frac{{m}_{e}\,{\alpha }_{\parallel ,c}^{2}}{2{k}_{B}},\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&{T}_{\perp ,c}=\displaystyle \frac{{m}_{e}}{2{k}_{B}}\int d{\boldsymbol{v}}{v}_{\perp }^{2}{f}_{c}({v}_{\parallel },{v}_{\perp })=\displaystyle \frac{{m}_{e}\,{\alpha }_{\perp ,c}^{2}}{2{k}_{B}}.\end{eqnarray} \tag{ 3b }$$

The halo component is described by an anisotropic bi-Kappa distribution function

$$\begin{eqnarray}\begin{array}{rcl}{f}_{h}\left({v}_{\parallel },{v}_{\perp }\right) & = & \displaystyle \frac{1}{{\pi }^{3/2}{\alpha }_{\perp ,h}^{2}\,{\alpha }_{\parallel ,h}}\displaystyle \frac{{\rm{\Gamma }}\left(\kappa +1\right)}{{\rm{\Gamma }}\left(\kappa -1/2\right)}\\ & & \times \,{\left[1+\displaystyle \frac{{v}_{\parallel }^{2}}{\kappa {\alpha }_{\parallel ,h}^{2}}+\displaystyle \frac{{v}_{\perp }^{2}}{\kappa {\alpha }_{\perp ,h}^{2}}\right]}^{-\kappa -1},\end{array}\end{eqnarray} \tag{ 4 }$$

with parameters ${\alpha }_{\parallel ,\perp ,h}\equiv {\alpha }_{\parallel ,\perp ,h}(t)$ (varying in time t in our QL approach) defined by the components of the anisotropic temperature

$$\begin{eqnarray}&&{T}_{\parallel ,h}=\displaystyle \frac{2\kappa }{2\kappa -3}\displaystyle \frac{{m}_{e}}{2{k}_{B}}{\alpha }_{\parallel ,h}^{2},\,{T}_{\perp ,h}=\displaystyle \frac{2\kappa }{2\kappa -3}\displaystyle \frac{{m}_{e}}{2{k}_{B}}{\alpha }_{\perp ,h}^{2}.\end{eqnarray} \tag{ 5 }$$

Such a plasma system we parameterize from the observations reported by Štverák et al. (2008), from roughly 120,000 events detected in the ecliptic at different distances (0.3– 3.95 au) from the Sun. We choose only the slow wind data (V_SW ≤ 500 km s⁻¹), for which the density of the strahl population is faint enough to not affect the instability conditions. These data have been used in a series of recent studies to characterize the electron core and halo populations (Pierrard et al. 2016; Lazar et al. 2017a). For slow wind conditions, the suprathermal halo tails are fitted by a Kappa with $2.5\lt \kappa \lt 9.5$. Comparing to the core population, the halo is hotter (T_h > T_c) but less dense, with a number density η_h ≤ 0.10 (Maksimovic et al. 2005). Both the core and halo populations may exhibit temperature anisotropy, i.e., $A\,={T}_{\perp }/{T}_{\parallel }\ne 1$, which can be at the origin of kinetic instabilities, such as whistler (if A > 1) or firehose instability (if A < 1). Their anisotropies show a prevalent tendency for a direct (linear) correlation, like ${A}_{c}={{CA}}_{h}$, with C ≃ 1 (Pierrard et al. 2016). In the present study we are interested in the conditions most favorable to firehose instability. For the interplay of the core and suprathermal halo these conditions are those associated with similar sub-unitary anisotropies A_c = A_h < 1. We use the observations to evaluate the relative densities η_h and η_c and the ratio of parallel plasma betas ${\beta }_{c}/{\beta }_{h}$ (where ${\beta }_{\parallel }\equiv 8\pi {{nk}}_{B}{T}_{\parallel }/{B}^{2}$), corresponding to two different representative values of κ-index. Figure 1 displays a density histogram for β_c versus β_h with a color bar counting the number of events. For a low κ = 3 (2.5 ≤ κ ≤ 3.5) we find relevant β_c/β_h ≃ 2.236, and for high κ = 8 (7.5 ≤ κ ≤ 8.5) we consider β_c/β_h ≃ 3.809. Contrary to other recent studies which assume β_c < β_h (e.g., Sarfraz 2018 and references therein), here we consider β_c > β_h as suggested by the observations. A similar linear correlation between plasma beta parameters β_c and β_h is found in Lazar et al. (2015) using a different set of data from Ulysses during the slow winds. In Section 3 we show how important this ratio is for contrasting the core and halo parallel betas for the instability conditions.

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For a collisionless and homogeneous plasma, the general, linear dispersion relation of the electromagnetic modes propagating parallel to the background magnetic field $({{\boldsymbol{B}}}_{0})$, i.e., k × B₀ = 0) reads

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{k}^{2}{c}^{2}}{{\omega }^{2}} & = & 1+\displaystyle \sum _{a=p,c,h}\displaystyle \frac{{\omega }_{p,a}^{2}}{{\omega }^{2}}\displaystyle \int d{\boldsymbol{v}}\displaystyle \frac{{v}_{\perp }}{2\left(\omega -{{kv}}_{\parallel }\pm {{\rm{\Omega }}}_{a}\right)}\\ & & \times \,\left((\omega -{{kv}}_{\parallel })\displaystyle \frac{\partial {f}_{a}}{\partial {v}_{\perp }}+{{kv}}_{\perp }\displaystyle \frac{\partial {f}_{a}}{\partial {v}_{\parallel }}\right),\end{array}\end{eqnarray} \tag{ 6 }$$

where k is the wave number, c is the speed of light, ${\omega }_{p,\alpha }=\sqrt{4\pi {n}_{a}{e}^{2}/{m}_{a}}$ and ${{\rm{\Omega }}}_{a}={{eB}}_{0}/{m}_{a}c$ are, respectively, the non-relativistic plasma frequency and the gyro-frequency of species a, and ±denotes the right-handed (RH) and left-handed (LH) circular polarization, respectively.

For the LH electron firehose (EFH) unstable modes in our plasma system the instantaneous dispersion relation (6) reduces to

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{k}}^{2} & = & {A}_{p}-1+\left(\displaystyle \frac{{A}_{p}\,\tilde{\omega }-\left({A}_{p}-1\right)}{\tilde{k}\sqrt{{\beta }_{p}}}\right){Z}_{p}\left(\displaystyle \frac{\tilde{\omega }-1}{\tilde{k}\sqrt{{\beta }_{p}}}\right)\\ & & +\,{\eta }_{c}\,\mu \left[{A}_{c}-1+\left(\displaystyle \frac{{A}_{c}\,\tilde{\omega }+\left({A}_{c}-1\right)\mu }{\tilde{k}\sqrt{\mu \,{\beta }_{c}/{\eta }_{c}}}\right)\right.\\ & & \left.\times \,{Z}_{c}\left(\displaystyle \frac{\tilde{\omega }+\mu }{\tilde{k}\sqrt{\mu \,{\beta }_{c}/{\eta }_{c}}}\right)\right]+{\eta }_{h}\,\mu \left[\Space{0ex}{3.80ex}{0ex}{A}_{h}-1\right.\\ & & \left.+\,\left(\displaystyle \frac{{A}_{h}\,\tilde{\omega }+\left({A}_{h}-1\right)\mu }{\tilde{k}\sqrt{\mu \,{\chi }_{h}\,{\beta }_{h}/{\eta }_{h}}}\right){Z}_{h}\left(\displaystyle \frac{\tilde{\omega }+\mu }{\tilde{k}\sqrt{\mu \,{\chi }_{h}\,{\beta }_{h}/{\eta }_{h}}}\right)\right]\end{array}\end{eqnarray} \tag{ 7 }$$

where $\tilde{k}={kc}/{\omega }_{p,p}$ is the normalized wave-number, $\tilde{\omega }=\omega /{{\rm{\Omega }}}_{p}$ is the normalized wave frequency, μ = m_p/m_e is the proton–electron mass contrast, ${A}_{a}=\,{T}_{\perp ,a}/{T}_{\parallel ,a}\equiv \,{\beta }_{\perp ,a}/{\beta }_{\parallel ,a}$ and ${\beta }_{\parallel ,\perp ,a}=8\pi {n}_{a}{k}_{B}{T}_{\parallel \perp ,a}/{B}_{0}^{2}$ are, respectively, the temperature anisotropy and plasma beta parameters for protons (subscript "a = p"), electron core (subscript "a = c"), and electron halo (subscript "a = h") populations, χ_h = (κ − 1.5)/κ, η_h = n_h/n₀, η_c = 1 − η_h are the halo and the core density contrast, respectively,

$$\begin{eqnarray}&&{Z}_{a}\left({\xi }_{a}^{\pm }\right)=\displaystyle \frac{1}{\sqrt{\pi }}{\int }_{-\infty }^{\infty }\displaystyle \frac{\exp \left(-{x}^{2}\right)}{x-{\xi }_{a}^{\pm }}{dt},\ \ {\mathfrak{I}}\left({\xi }_{a}^{\pm }\right)\gt 0,\end{eqnarray} \tag{ 8 }$$

is the plasma dispersion function (Fried & Conte 1961) and

$$\begin{eqnarray}\begin{array}{rcl}{Z}_{h}\left({\xi }_{h}^{+}\right) & = & \displaystyle \frac{1}{{\pi }^{1/2}{\kappa }^{1/2}}\displaystyle \frac{{\rm{\Gamma }}\left(\kappa \right)}{{\rm{\Gamma }}\left(\kappa -1/2\right)}\\ & & \times \,{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{\left(1+{x}^{2}/\kappa \right)}^{-\kappa }}{x-{\xi }_{h}^{+}}{dx},{\mathfrak{I}}\left({\xi }_{h}^{\pm }\right)\gt 0.\end{array}\end{eqnarray} \tag{ 9 }$$

is the generalized (Kappa) dispersion function (Lazar et al.2008).

The time evolution of the velocity distributions are described by the particle kinetic equation in the diffusion approximation

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {f}_{a}}{\partial t} & = & \displaystyle \frac{{{ie}}^{2}}{4{m}_{a}^{2}{c}^{2}\,{v}_{\perp }}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{dk}}{k}\left[\left({\omega }^{* }-{{kv}}_{\parallel }\right)\displaystyle \frac{\partial }{\partial {v}_{\perp }}+{{kv}}_{\perp }\displaystyle \frac{\partial }{\partial {v}_{\parallel }}\right]\\ & & \times \,\displaystyle \frac{{v}_{\perp }\delta {B}^{2}(k,\omega )}{\omega -{{kv}}_{\parallel }-{{\rm{\Omega }}}_{a}}\left[\left(\omega -{{kv}}_{\parallel }\right)\displaystyle \frac{\partial {f}_{a}}{\partial {v}_{\perp }}+{{kv}}_{\perp }\displaystyle \frac{\partial {f}_{a}}{\partial {v}_{\parallel }}\right],\end{array}\end{eqnarray} \tag{ 10 }$$

where the energy density of the fluctuations δB²(k) is described by the wave equation

$$\begin{eqnarray}&&\displaystyle \frac{\partial \,\delta {B}^{2}(k)}{\partial t}=2{\gamma }_{k}\delta {B}^{2}(k),\end{eqnarray} \tag{ 11 }$$

with growth rate γ_k of the EFH instability. Equation (10) is used to derive perpendicular and parallel velocity moments for protons, and electron core and halo populations, as follows:

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{{dT}}_{\perp p}}{{dt}}=-\displaystyle \frac{{e}^{2}}{2{m}_{p}{c}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{dk}}{{k}^{2}}\langle \,\delta {B}^{2}(k)\,\rangle \\ \quad \times \,\left\{\left(2{A}_{p}-1\right){\gamma }_{k}+\mathrm{Im}\displaystyle \frac{2i\gamma -{{\rm{\Omega }}}_{p}}{k{\alpha }_{\parallel p}}\,{F}^{-}\left({A}_{p},{{\rm{\Omega }}}_{p},{Z}_{p}\right)\right\}\end{array}\end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dT}}_{\parallel p}}{{dt}} & = & \displaystyle \frac{{e}^{2}}{{m}_{p}{c}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{dk}}{{k}^{2}}\langle \,\delta {B}^{2}(k)\,\rangle \\ & & \times \,\left\{{A}_{p}\,{\gamma }_{k}+\mathrm{Im}\displaystyle \frac{\omega -{{\rm{\Omega }}}_{p}}{k{\alpha }_{\parallel p}}\,{F}^{-}\left({A}_{p},{{\rm{\Omega }}}_{p},{Z}_{p}\right)\right\}\end{array}\end{eqnarray} \tag{ 12b }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dT}}_{\perp c}}{{dt}} & = & -\displaystyle \frac{{e}^{2}}{2{m}_{e}{c}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{dk}}{{k}^{2}}\langle \,\delta {B}^{2}(k)\,\rangle \\ & & \times \,\left\{\left(2{A}_{c}-1\right){\gamma }_{k}+\mathrm{Im}\displaystyle \frac{2i\gamma +{{\rm{\Omega }}}_{e}}{k{\alpha }_{\parallel c}}\,{F}^{+}\left({A}_{c},{{\rm{\Omega }}}_{e},{Z}_{c}\right)\right\}\end{array}\end{eqnarray} \tag{ 12c }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dT}}_{\parallel c}}{{dt}} & = & \displaystyle \frac{{e}^{2}}{{m}_{e}{c}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{dk}}{{k}^{2}}\langle \,\delta {B}^{2}(k)\,\rangle \\ & & \times \,\left\{{A}_{c}\,{\gamma }_{k}+\mathrm{Im}\displaystyle \frac{\omega +{{\rm{\Omega }}}_{e}}{k{\alpha }_{\parallel c}}\,{F}^{+}\left({A}_{c},{{\rm{\Omega }}}_{e},{Z}_{c}\right)\right\}\end{array}\end{eqnarray} \tag{ 12d }$$

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{{dT}}_{\perp h}}{{dt}}=-\displaystyle \frac{{e}^{2}}{2{m}_{e}{c}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{dk}}{{k}^{2}}\langle \,\delta {B}^{2}(k)\,\rangle \\ \quad \times \,\left\{\left(2{A}_{h}-1\right){\gamma }_{k}+\mathrm{Im}\displaystyle \frac{2i\gamma +{{\rm{\Omega }}}_{e}}{k{\alpha }_{\parallel h}}\,{F}^{+}\left({A}_{h},{{\rm{\Omega }}}_{e},{Z}_{h}\right)\right\}\end{array}\end{eqnarray} \tag{ 12e }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dT}}_{\parallel h}}{{dt}} & = & \displaystyle \frac{{e}^{2}}{{m}_{e}{c}^{2}}{\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{{dk}}{{k}^{2}}\langle \,\delta {B}^{2}(k)\,\rangle \\ & & \times \,\left\{{A}_{h}\,{\gamma }_{k}+\mathrm{Im}\displaystyle \frac{\omega +{{\rm{\Omega }}}_{h}}{k{\alpha }_{\parallel h}}\,{F}^{+}\left({A}_{h},{{\rm{\Omega }}}_{e},{Z}_{h}\right)\right\}\end{array}\end{eqnarray} \tag{ 12f }$$

with

$$\begin{eqnarray*}&&{F}^{\pm }({A}_{a},{{\rm{\Omega }}}_{a},{Z}_{a})=\left[{A}_{a}\,\omega \pm {{\rm{\Omega }}}_{a}\left({A}_{a}-1\right)\right]{Z}_{a}\left(\displaystyle \frac{\omega \pm {{\rm{\Omega }}}_{a}}{k{\alpha }_{\parallel a}}\right).\end{eqnarray*}$$

In terms of the normalized quantities these equations can be rewritten, respectively, as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{\beta }_{\perp p}}{d\tau } & = & -\displaystyle \int \displaystyle \frac{d\tilde{k}}{{\tilde{k}}^{2}}W(\tilde{k})\left\{\left(2{A}_{p}-1\right)\tilde{\gamma }+\mathrm{Im}\displaystyle \frac{2i\tilde{\gamma }-1}{\tilde{k}\sqrt{{\beta }_{\parallel p}}}\,\right.\\ & & \left.\times \,\left[{A}_{p}\,\tilde{\omega }-\left({A}_{a}-1\right)\right]{Z}_{p}\left(\displaystyle \frac{\tilde{\omega }-1}{\tilde{k}\sqrt{{\beta }_{\parallel p}}}\right)\right\}\end{array}\end{eqnarray} \tag{ 13a }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{\beta }_{\parallel p}}{d\tau } & = & 2\displaystyle \int \displaystyle \frac{d\tilde{k}}{{\tilde{k}}^{2}}W(\tilde{k})\left\{{A}_{p}\,\tilde{\gamma }+\mathrm{Im}\displaystyle \frac{\tilde{\omega }-1}{\tilde{k}\sqrt{{\beta }_{\parallel p}}}\,\right.\\ & & \left.\times \,\left[{A}_{p}\,\tilde{\omega }-\left({A}_{a}-1\right)\right]{Z}_{p}\left(\displaystyle \frac{\tilde{\omega }-1}{\tilde{k}\sqrt{{\beta }_{\parallel p}}}\right)\right\}\end{array}\end{eqnarray} \tag{ 13b }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{\beta }_{\perp c}}{d\tau } & = & -{\eta }_{c}\displaystyle \int \displaystyle \frac{d\tilde{k}}{{\tilde{k}}^{2}}W(\tilde{k})\left\{\mu \left(2{A}_{c}-1\right)\tilde{\gamma }+\mathrm{Im}\displaystyle \frac{2i\tilde{\gamma }+\mu }{\tilde{k}\sqrt{{\beta }_{\parallel c}/{\eta }_{c}}}\,\,\right.\\ & & \left.\times \,\tilde{F}({A}_{c},{Z}_{c},{\beta }_{c},{\eta }_{c})\Space{0ex}{3.85ex}{0ex}\right\}\end{array}\end{eqnarray} \tag{ 13c }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{\beta }_{\parallel c}}{d\tau } & = & 2\,{\eta }_{c}\displaystyle \int \displaystyle \frac{d\tilde{k}}{{\tilde{k}}^{2}}W(\tilde{k})\left\{\mu \,{A}_{c}\,\tilde{\gamma }+\mathrm{Im}\displaystyle \frac{\tilde{\omega }+\mu }{\tilde{k}\sqrt{{\beta }_{\parallel c}/{\eta }_{c}}}\,\,\right.\\ & & \left.\times \,\tilde{F}({A}_{c},{Z}_{c},{\beta }_{c},{\eta }_{c})\Space{0ex}{3.85ex}{0ex}\right\}\end{array}\end{eqnarray} \tag{ 13d }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{\beta }_{\perp h}}{d\tau } & = & -{\eta }_{h}\displaystyle \int \displaystyle \frac{d\tilde{k}}{{\tilde{k}}^{2}}W(\tilde{k})\left\{\mu \left(2{A}_{h}-1\right)\tilde{\gamma }+\mathrm{Im}\displaystyle \frac{2i\tilde{\gamma }+\mu }{\tilde{k}\sqrt{{\beta }_{\parallel h}/{\eta }_{h}}}\,\right.\\ & & \left.\times \,\tilde{F}({A}_{h},{Z}_{h},{\chi }_{h},{\beta }_{h},{\eta }_{h})\Space{0ex}{3.85ex}{0ex}\right\}\end{array}\end{eqnarray} \tag{ 13e }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{\beta }_{\parallel h}}{d\tau } & = & 2\,{\eta }_{h}\displaystyle \int \displaystyle \frac{d\tilde{k}}{{\tilde{k}}^{2}}W(\tilde{k})\left\{\mu \,{A}_{h}\,\tilde{\gamma }+\mathrm{Im}\displaystyle \frac{\tilde{\omega }+\mu }{\tilde{k}\sqrt{{\beta }_{\parallel h}/{\eta }_{h}}}\,\right.\\ & & \left.\times \,\tilde{F}({A}_{h},{Z}_{h},{\chi }_{h},{\beta }_{h},{\eta }_{h})\Space{0ex}{3.85ex}{0ex}\right\}\end{array}\end{eqnarray} \tag{ 13f }$$

with

$$\begin{eqnarray*}\begin{array}{rcl}\tilde{F}({A}_{a},{Z}_{a},{\chi }_{a},{\beta }_{a},{\eta }_{a}) & = & \sqrt{\mu }\left[{A}_{a}\,\tilde{\omega }+\left({A}_{a}-1\right)\mu \right]\\ & & \times \,{Z}_{a}\left(\displaystyle \frac{\tilde{\omega }+\mu }{\tilde{k}\sqrt{\mu \,{\chi }_{a}\,{\beta }_{\parallel a}/{\eta }_{a}}}\right),\end{array}\end{eqnarray*}$$

and

$$\begin{eqnarray}&&\displaystyle \frac{\partial \,W(\tilde{k})}{\partial \tau }=2\,\tilde{\gamma }\,W(\tilde{k}).\end{eqnarray} \tag{ 14 }$$

where $W(\tilde{k})=\delta {B}^{2}(\tilde{k})/{B}_{0}^{2}$ is the wave energy density, τ = Ω_p t and χ_c = 1.

For the normalization, we keep the distinction between the plasma frequencies of the core (subscript c) and halo (subscript h) electrons, i.e., ${\omega }_{p,a}=\sqrt{4\pi {n}_{a}{e}^{2}/{m}_{e}}$ is defined in terms of the number density n_a, which is expected to play an important role mainly triggering the effects of these components on instabilities. Normalization to a total plasma frequency ${\omega }_{p,e}=\sqrt{4\pi {n}_{o}{e}^{2}/{m}_{e}}$, where the total number density n₀ = n_c + n_h, invoked in similar studies (e.g., Sarfraz 2018 and references therein), may introduce an artificial coupling between the core and halo electrons, and therefore alter their QL relaxation under the effect of firehose fluctuations.

For a linear analysis we use the dispersion relation (7). Figures 2 and 3 illustrate unstable EFH solutions for case 1 and case 2, respectively. Case 1 (κ = 8, A_c,h = 0.1), allows us to compare growth rates for different core plasma betas β_∥,c = 5, 7, 10 (implying different halo plasma betas β_∥,h = 1.312, 1.84, 2.63). Increasing the plasma beta may slightly enhance the the maximum growth rates (peaks), but markedly diminishes the range of unstable wavenumbers. These counter-balancing effects are also observed for case 2 (not shown here).

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In Figure 3 we study the effect of the electron anisotropies A_c,h(0) = 0.1, 0.2, 0.3 considering the plasma parameters in case 2 (κ = 3); as for case 1 the observed variations are similar. The maximum (peaking) growth rates increase as the electron anisotropy increases, but the range of unstable wavenumbers decreases. The EFH instability becomes more operative at lower wavenumbers by increasing the plasma beta parameters and/or electron anisotropies. The corresponding wave frequencies are increasing with increasing plasma betas and/or the electron anisotropy, see panels (b) in Figures 2 and 3.

We can already highlight the importance of these results and motivate the present study by a comparative analysis between theoretical predictions from linear theory and observations in the solar wind. We use the same set of slow wind (V_sw ≤ 500 km s⁻¹) data mentioned in Section 2 to compare the observed anisotropies with theoretical thresholds of the instability, see Figure 4. Data are displayed in Figure 4 using a histogram plot counting the number of events with a color logarithmic scale. Thresholds of EFH unstable solutions are derived for a maximum growth rate γ_m = 10⁻³Ω_p and are fitted to an inverse power law (Lazar et al. 2018a)

$$\begin{eqnarray}&&{A}_{a}=1-\displaystyle \frac{s}{{\beta }_{\parallel ,a}^{\alpha }}\end{eqnarray} \tag{ 17 }$$

for the core (a = c, panel a) and the halo (a = h, panel b), with fitting parameters s and α given in Table 1.

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Table 1.  Fitting Parameters for Thresholds γ_m/Ω_p = 10⁻²

κ	Core		Halo		SBM
	s	α	s	α	s	α
3	1.32	0.994	0.6	1.03	⋯	⋯
8	1.45	1.0	0.35	0.88	⋯	⋯
$\infty$	⋯	⋯	⋯	⋯	1.75	1.04

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The new instability thresholds from the interplay of the core and halo population (red for κ = 3 and blue for κ = 8) are contrasted with those obtained for a single bi-Maxwellian component (dotted black). We should observe that these new thresholds markedly decrease, approaching and shaping much better the limits of the halo anisotropy (panel b). The lower threshold is obtained in this case for a higher κ = 8 (blue, case 1), under the influence of a more dense halo (compared to case 2). We can also admit minor (but still visible) changes of the instability thresholds constraining the core anisotropy (panel a). The instability driven by the core populations is stimulated by the anisotropic halo, but lower thresholds are obtained for a more dense core (red, case 2).

Beyond the linear theory, the QL analysis describes the temporal evolution of electromagnetic fluctuations, which are enhanced by the instability and change the main features of the distributions, such as the temperature components (${T}_{\perp ,\parallel }$), the anisotropy (A_a = T_⊥/T_∥), and implicitly the plasma betas (${\beta }_{\perp }/{\beta }_{\parallel }\equiv {T}_{\perp }/{T}_{\parallel }$). However, we assume no exchange of electrons between core and halo, i.e., η_h/η_c is constant, and the shape of the distributions is preserved; for instance, the protons and electron core remain Maxwellian, while the electron halo remains Kappa-distributed with the same value of power index. The recent comparisons of QL theory with the particle-in-cell simulation by Yoon et al. (2017) and Lazar et al. (2018b) show a reasonable agreement for the temporal evolution of the electron VDFs. There are also some indications from Vlasov simulations that the power index does not change much in the QL relaxation, though these results are still limited to studies of different instabilities (Lazar et al. 2017c). We resolve the system of QL Equations (13) and (14) for the same plasma parameters introduced here above as cases 1 and 2.

Figure 5 describes the time evolution of the perpendicular (red) and parallel (blue) plasma betas for the electron components, core (top) and halo (middle-first), thermal protons (middle-second), and the corresponding variation of the wave energy density $\delta {B}^{2}/{B}_{0}^{2}$ (bottom) as functions of the normalized time τ = Ω_p t for case 1 (κ = 8) and different initial plasma betas β_c(0) = 5.0 (left), 7.0 (middle), and 10.0 (right). The excitation of the EFH instability from the interplay of the anisotropic core and halo can regulate the initial temperature anisotropy for both these two components and also for protons, through the cooling and heating mechanisms reflected by the parallel (blue) and perpendicular (red) plasma betas. The level of saturated fluctuations is increasing with increasing the initial plasma beta, i.e., β_c(0) = 10.0, confirming a stimulation of the instability predicted for the peaking growth rates by the linear theory in Figure 2(a). Nevertheless, both the core and halo components show similar intervals of relaxation, but the halo remains slightly less anisotropic than the core after the instability saturation, i.e., τ_m = 12. Linear theory cannot describe quantitatively the energy transfer between plasma particles mediated by the instability. However, from a QL analysis we can estimate these exchanges induced by the free energy of electrons, which is transferred during the instability growing to the protons: the third row of panels shows the time evolution of the parallel and perpendicular plasma betas for protons (${\beta }_{\perp ,\parallel ,p}$), which are initially isotropic, i.e., A_p(0) = 1. Both beta components are increased as τ increases, but the protons are heated more in the perpendicular direction and become anisotropic at later stages A_p(τ_m) > 1.0. Their anisotropy increases with increasing the intial plasma beta. These evolutions confirm (see Paesold & Benz 1999; Messmer 2002; Paesold & Benz 2003) that a finite amount of free energy is transferred by the growing fluctuations to the protons, especially in the direction perpendicular to the background magnetic field.

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Figure 6 presents temporal profiles of the temperature anisotropy and parallel plasma betas in a (A_a, β_∥,a)-space, for both the core (subscript "a = c," in panel (a)) and halo (subscript "a = h," in panel b). Black circles indicate the initial positions, while the gray circles mark the limit states after saturation. The associated wave energy density $\delta {B}^{2}/{B}_{0}^{2}$ is color-coded. As expected, the final states align along the anisotropy thresholds predicted by the linear theory, with unstable regimes located below the thresholds. This time evolutions clearly shows deviations from isotropy decrease toward less unstable regimes closer to marginal stability. These relaxations of the core and halo anisotropies are a direct consequence of the EFH instability and the enhanced electromagnetic fluctuations, which scatter the particles toward quasi-stationary states. This is confirmed by the increase of the wave energy density as the electrons become less anisotropic.

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The results in Figure 7 are obtained for case 2 (κ = 3), with an initial β_c(0) = 5.0 and different anisotropies A_c(0) = A_h(0) = 0.3 (left), 0.2 (middle), 0.1 (right). The main effects, associated with the heating or cooling of different particle populations by the enhanced fluctuations, are reflected by the temporal profiles of the beta components for core, halo, and protons. These effects are similar to those in case 1, but here are mainly triggered by the different anisotropies, e.g., for higher anisotropies, i.e., A_c,h(0) = 0.1 (right column), the resulting magnetic wave energy $\delta {B}^{2}/{B}_{0}^{2}$ increases as an effect of the enhancement of the instability growth rates predicted by the linear theory in Figure 3(a). This may also explain the increase of the energy transferred to the initially isotropic protons (A_p(0) = 1.0), which reach moderate anisotropies A_p(τ_m) > 1.0 after saturation.

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In Figure 8 we display dynamical paths of the temperature anisotropy and the parallel plasma beta in a (A_a, β_∥,a)-space for both the core (panel a) and halo (panel b). The initial anisotropies A_a(0) are marked with black circles, while anisotropies A_a(τ_m) at final stages are indicated by gray circles. The levels of the magnetic wave energy $\delta {B}^{2}/{B}_{0}^{2}$ are coded with colors. As in case 1 (Figure 6), the initial deviations from isotropy are reduced in time, toward less unstable states. It is obvious that final states tend to align to the instability thresholds, and larger initial anisotropies, e.g., A_a(0) = 0.1, will determine longer dynamical paths for the particles. Nevertheless, in Figure 8 the final states of electron anisotropies reach a more stable regime by comparison to Figure 6. One plausible explanation for this physical behavior can be found in the increase of the suprathermal population, which stimulates the energy exchange during the instability development to protons, enhancing their anisotropy (T_⊥ > T_∥), while the core and halo electrons become less anisotropic. This explanation is also supported by the results in Figure 5, left panels, and Figure 7, right panels, which are obtained for β_c(0) = 5.0 and A_e(0) = 0.1, but different values of the power index κ = 8 (Figure 5) and 3 (Figure 7). It is obvious that at later stages both the proton anisotropy and the associated magnetic wave energy obtained for κ = 3, i.e., $\delta {B}^{2}/{B}_{0}^{2}=0.24$ and A_p(τ_m) = 1.08, are higher than those obtained for κ = 8, i.e., $\delta {B}^{2}/{B}_{0}^{2}=0.2$ and A_p(τ_m) = 1.016.

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In both these cases the initial proton plasma beta β_∥p(0) = β_∥c(0)/3 > 1.0 is relatively large, and transfer of energy from electrons to protons remains modest. Dynamical paths of the proton anisotropy are very short in these cases, and we have not displayed them in Figures 6 and 8. Seeking completeness, we extend the analysis for a series of new cases in Figures 9 and 10, which allow us to examine the mutual effects between the electron populations and protons with relatively low initial β_∥p(0) = 0.05.

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Figure 9 presents by comparison temporal profiles of temperature anisotropy for the core (top) and halo (middle), as well as the associated magnetic wave energy (bottom) for the same plasma parameters in case 1, but for two different initial proton plasma betas, either large ${\beta }_{\parallel ,p}(0)={\beta }_{\parallel c}(0)/3\gt 1.0$ (dotted lines), or small β_∥p(0) = 0.05 (solid lines). It is clear that protons with lower beta stimulate the cooling process of both populations of electrons, making it faster, and the core and halo electrons become less anisotropic in this case (solid blue lines). The associated wave energy is enhanced with decreasing proton parallel beta, confirming the enhanced relaxation of the anisotropies. The dynamical paths of the core, halo, and proton anisotropies for the new case with lower β_∥p(0) = 0.05 are displayed in Figure 10. For the core and halo the paths are orientated in the direction of stable regimes and end up at or close to the anisotropy thresholds (blue lines) predicted by the linear theory in Figure 4. This behavior suggests that for low plasma betas β_∥,p = 0.05, the free energy of electrons is transferred via the growing fluctuations to the (resonant) protons, and especially in the perpendicular direction. Thus, at later stages after saturation both core and halo become less anisotropic, while the initially isotropic protons become strongly anisotropic A_p(τ_m) > 1, which is evident in Figure 10. These variations are stimulated by increasing the initial plasma beta, i.e., β_∥c(0) = 10.0. Protons with temperature anisotropy A_p > 1 may trigger LH polarized electromagnetic ion cyclotron (EMIC) instability with a maximum growth rate in the parallel direction to the background magnetic field (Shaaban et al. 2015, 2016). Shaaban et al. (2017) show that anisotropic electrons with A_e < 1.0 and their suprathermal populations may stimulate the EMIC unstable modes. These predictions from linear theory seem to be confirmed and explained by the present QL analysis. For visual guidance Figure 10 displays also the EMIC anisotropy threshold (black line) derived by Shaaban et al. (2017), with fitting parameters s = −1.221 and α = 0.579 in Equation(17). For β_∥c(0) = 0.05, protons (with A_p(0) = 1) gain anisotropy in the perpendicular direction and move toward the EMIC thresholds, and then their anisotropy decreases to settle down well below the instability threshold after saturation.

From a comparison of our results in Figure 10 with those in a recent study by Sarfraz (2018, Figure 4 therein), we find a reasonable agreement for the dynamical paths of the proton anisotropies, but not for the instability thresholds and paths predicted for the core and halo electrons by the linear and QL approaches. Contrary to Sarfraz (2018) the halo anisotropy threshold is lower than that obtained for the core anisotropy threshold, and extends to lower parallel plasma beta. Our results, from both linear and the QL analyses, show an excellent agreement with the observational limits of the core and halo anisotropies, and this is mainly explained by the more realistic plasma parametrization used in our analysis.