TY - JOUR
AU - Qureshi, M. N., S.
AB - Abstract In situ observations reveal the existence of electron velocity distribution function in the solar wind, where the net distribution can be modelled by a combination of core, halo and strahl. These components often possess a relative drift and with respective temperature anisotropies. The relative drift between the core and halo components leads to heat flux (HF) instability, while temperature anisotropies drive electromagnetic electron-cyclotron (EMEC) instability. These instabilities have been separately studied in the literature, but for the first time, the present study combines both unstable modes in the presence of two free energy sources, namely, excessive parallel pressure and excessive perpendicular temperature. HF instability (which is a left-hand circularly polarized mode) is effectively similar to electron firehose instability, except that the free energy is provided by net relative drift among two component electrons in the background of protons. The HF instability is discussed here along with (the right-hand polarized) EMEC instability driven by temperature anisotropy. The unstable HF mode is conventionally termed the ‘whistler’ HF instability, but it is actually polarized in the opposite sense to the whistler wave. EMEC mode, on the other hand, reduces to the proper whistler wave in the absence of free energy source. The present combined analysis clarifies the polarization characteristics of these two modes in an unambiguous manner. instabilities, plasmas, methods: analytical, solar wind 1 INTRODUCTION This paper is a continuation of our companion paper on the heat flux (HF) instability (Sundas et al. 2017), and is intended to explore the properties of the left-hand circularly polarized HF mode and the right-handed circularly polarized electromagnetic electron-cyclotron (EMEC) mode. Conventionally, HF instability is known as the ‘whistler’ HF instability, which implies that the basic mode is associated with the right-hand polarized whistler wave. However, in our companion paper (Sundas et al. 2017), we have shown that HF mode is polarized in the opposite sense to whistler wave, and is caused by the net relative drift of two component electrons drifting in opposite directions in the background protons. We have also shown that the basic mechanism of instability involves unstable coupling of the left-hand circularly polarized ion-cyclotron wave and whistler-beam mode; hence, it is not entirely improper to call it ‘whistler’ HF instability. Nevertheless, the polarization is opposite to that of classic whistler wave. In contrast, EMEC mode is a right-hand mode excited by temperature anisotropy, and it is this mode that reduces to the customary whistler wave in the absence of free energy. This paper further elucidates these points and compares the two unstable modes when the two free energy sources, namely, net relative drifts of core-halo two electron components and perpendicular temperature anisotropy exist concomitantly. The importance of this study is that these instabilities may be intimately related to the solar wind dynamics. Solar wind plasma, which comprises of electrons, protons, admixture of a few per cent alpha particles, and other less abundant heavy ions, is the only media for which we have various in situ observations. These observations have been used for many decades in order to understand the naturally existing phenomena in the solar wind and the underlying physics behind it. Observed solar wind electrons are found in non-thermodynamic equilibrium. Observations from 0.3 to 1 AU show a noticeable break in electron velocity distribution function (EVDF) at kinetic energy of a few tens of electron volts, suggesting that these electron distributions in the solar wind can be modelled as superposition of roughly three or four electron populations: relatively cool denser core, less dense and hotter halo (Montgomery, Bame & Hundhausen 1968; Feldman et al. 1975; Pilipp et al. 1987) and largely field-aligned strahl (Maksimovic et al. 2005) moving in anti-sunward direction, as well as energetic superhalo electrons that exist in all solar conditions (Wang et al. 2012; Che & Goldstein 2014). The strahl component, which is generally associated with the fast solar wind, overlaps the halo in energy range. As a lowest order approximation, the strahl and halo can be treated together by simply absorbing the strahl into halo into a single component, but by allowing a net drift associated with halo. Core electrons carry energies of tens of eV, while halo electrons are characterized by energy range of ∼102–103 eV (Yoon 2015). Origin of suprathermal particles in interstellar medium may be related to intensity, velocity and source properties of solar wind. In situ observations reveal that there are three types of solar wind flow: fast, slow and transient. Fast solar wind originates from open magnetic field lines in polar coronal holes (Hassler et al. 1999; Xia, Marsch & Wilhelm 2004; Wiegelmann, Xia & Marsch 2005), while slow solar wind comes from opening loops and active regions in the corona. Transient flow has roots in solar maximum activity in the form of coronal mass ejections (CMEs) (Marsch 2006). For slow solar wind plasma, core-halo configuration with very low relative drift generally systematizes the EVDF (Feldman et al. 1975). Solar wind observations reveal that the main ∼95 per cent colder core population is surrounded by 4–5 per cent hotter halo electrons in velocity space. Together they make up most of the EVDF. Solar wind data taken from Imp 6, 7 and 8 (Feldman et al. 1975) show EVDFs possessing a relative drift to solar wind protons. Relative drift between these two components leads to the HF along ambient magnetic field. Observations reveal that the HF instability in solar wind up to 1 AU is governed by local parameters of the plasma (Feldman et al. 1975; Gary & Feldman 1977) instead of thermal conductivity equation and to a large extent, this HF is carried by halo electrons. In the frame of solar wind protons, core electrons are observed to drift slightly in sunward direction while halo and strahl electrons are observed to drift in anti-sunward direction along the local magnetic field satisfying current neutrality (Tong et al. 2015). HF is the fundamental property of solar wind, providing an energy flow channel from hot solar source to cooler heliospheric region. HF instabilities possess the potential to regulate and maintain the heat flow in the solar wind (Forslund 1970). At minimum, there are three instabilities associated with HF: whistler, magnetosonic and Alfvén instabilities (Gary et al. 1975; Gary & Feldman 1977; Gary 1985; Gary & Li 2000). Making use of drifting bi-Maxwellian EVDF with zero net current, Gary (1985) studied these electromagnetic electron beam instabilities along with their thresholds. However, in our companion paper (Sundas et al. 2017), we highlighted that this conventionally known whistler HF instability is actually polarized in the opposite sense to the right-hand circularly polarizes whistler wave, and is essentially similar to the parallel electron firehose instability. The excessive parallel pressure in the case of HF instability is provided by the counter-streaming core-halo electrons in the proton rest frame. Temperature anisotropies defined with respect to ambient magnetic field direction is another macroscopic parameter that characterizes core and halo components. Magnitude of core and halo anisotropies could offer different clues about macro or microscopic processes at work in solar plasma. The foremost reliable information of temperature anisotropy of solar wind electrons date back to the late 1960s and 1970s (Serbu 1969). Other systematic observational studies were subsequently carried out by Scudder, Lind & Ogilvie (1973), Feldman et al. (1975) and Pierrard et al. (2016). Double adiabatic or CGL theory predicts that for typically slow solar wind near 1 AU, electron anisotropy could be more than 30 (Phillips & Gosling 1990). However, this contradicts observations. The averaged observed electron anisotropy is nearly isotropic T⊥/T∥ = 1.5 (Feldman et al. 1975), although higher anisotropies are also reported in the literature (Lazar et al. 2015; Pierrard et al. 2016). Various plasma processes occurring in the space plasma effectively transfer internal kinetic energy of solar wind electrons from parallel to perpendicular direction, which counteract the adiabatic expansion. There are three major kinetic processes that limit the unchecked increase of anisotropies. One of them is a class of anisotropy-driven instabilities, while the rest are HF skewness and Coulomb collisions. Whistler, or equivalently, EMEC instability is excited in environment where there is an excess of perpendicular temperature T⊥ > T∥ (Gary et al. 1999; Lazar et al. 2015). Electron firehose and ordinary-mode instabilities arise for excessive parallel temperature T∥ > T⊥. Lazar, Poedts & Schlickeiser (2014) investigated the excitation of EMEC instability by halo anisotropy and showed that the instability is suppressed by thermal spread of isotropic core. Further, they show that increasing relative core-halo density inhibits the instability. Ulysses observations show that even for large heliocentric distances R > 1 AU, core temperature remains hot enough and kinetic energies of both core and halo reach comparable values (Lazar et al. 2015). Consequently, for more general conditions of the solar wind, it is appropriate to consider the situation where both core and halo anisotropies are considered. As it will be shown, electron-cyclotron mode becomes unstable even for relatively low anisotropies of core T⊥/T∥ ∝ 1.2, and when the core anisotropy increases from 1.2 to 2 the instability growth increases by an order of magnitude. The purpose of this paper is to investigate combined HF and EMEC instabilities by considering the effects of temperature anisotropy and net parallel drift between core and halo electrons. Our companion paper (Sundas et al. 2017) focuses only on HF instability. Sundas et al. (2017) first investigate the primary as well secondary and tertiary unstable modes in the framework of cold fluid theory. We also demonstrated that while the secondary and tertiary instabilities are easily stabilized by thermal spreads, the primary HF mode remains quite robust and is not easily stabilized by finite beta effects. The analysis (Sundas et al. 2017) also show that HF instability vanishes if single-electron component is considered or ion dynamics are ignored. These basics properties of HF instability have not been fully discussed in the literature. Extending our preceding work, we now examine the interplay of HF and EMEC instabilities by allowing temperature anisotropies for both halo and core electrons. The organization of the paper is as follows. Section 2 describes model of distribution function. Section 3 covers theoretical formalism and dispersion relations of HF and EMEC modes. Section 4 describes numerical results. Section 5 summarizes the findings, together with conclusions and discussion. 2 MODEL OF DISTRIBUTION FUNCTION Since the availability of spacecraft data, it is well known that EVDFs at higher electron energies up to 102–103 eV deviate from the Maxwellian form and possess energetic population called suprathermal tails. Typically, EVDFs are phenomenologically treated as a superposition of relatively cool dense core and relatively hotter halo electrons, along with field-aligned strahl and minor populations of superhalo electrons (much less than 1 per cent of total number density with energy up to 100 keV) (Wang et al. 2012, 2015; Yoon 2015). According to Che & Goldstein (2014), the origin of superhalo electrons with energy ranging from 2 to 20 keV that exist even in quite time solar periods may lie in coronal nanoflares. These suprathermal tails and superhalo electrons are well described by (r, q) and kappa-like power-law distributions (Pierrard, Maksimovic & Lemaire 2001; Qureshi et al. 2004; Maksimovic et al. 2005; Pierrard & Lazar 2010). Strahl is associated with fast solar wind and becomes less pronounced for large heliocentric distances and slow solar wind (Maksimovic et al. 2005; Lazar et al. 2015). Although drifting components of EVDFs are also associated with high energy streams, currently in our model we ignore strahl as a separate component in the distribution, but include its presence in drifting halo component. Thus, our two component electron model can be considered as zeroth-order approach for describing the core and halo-strahl situations. That is, instead of treating strahl as a separate component, enhanced drifting halo mimics the presence of strahl. In our companion paper on HF instability (Sundas et al. 2017), we showed the minimal difference in bi-kappa and bi-Maxwellian methodology for realistic solar wind parameters, at least for kappa index, which is typical for solar wind halo distribution, namely κ ≈ 8 or so. Thus, here in this paper, we will follow bi-Maxwellian approach and model our distribution functions by superposition of drifting Maxwellian anisotropic core and drifting Maxwellian anisotropic halo in isotropic proton rest frame for both HF and EMEC instabilities. Contrary to this, Lazar et al. (2014) studied EMEC by using bi-Maxwellian core and bi-kappa halo model, and state that EMEC instability is more influenced by kappa index. However, they considered a range of κ index including a somewhat unrealistically low value such as κ = 2. For reasonably typical value such as κ = 8, we believe that the results are virtually indistinguishable from bi-Maxwellian calculation. The dispersion analysis along with its marginal stability condition is already surveyed by Lazar et al. (2014), so our purpose is not to revisit EMEC per se, but to understand the difference between HF and EMEC instabilities and their interplay. We consider the linear theory for collisionless, magnetized and homogenous bi-Maxwellian distributed plasmas possessing field aligned drifts. Isotropic proton velocity distribution function represented in Maxwellian form is given by \begin{equation} f_p({\boldsymbol v})=\frac{1}{\pi ^{3/2}\alpha _p^3}\exp \left( -\frac{v^2}{\alpha _p^2}\right), \end{equation} (1) where fp(⁠|$\boldsymbol v$|⁠) is normalized according to |$\int \text{d}{\bf \boldsymbol v}f_p({\boldsymbol v})=1$|⁠. The quantity αp and mp represent the proton thermal speed and proton mass, respectively, where |$T_p=m_p\alpha _p^2/2$| is the temperature of protons expressed in energy units; therefore, Boltzmann constant does not appear. The core and halo populated electrons are modelled as bi-Maxwellians, \begin{eqnarray} f_e(v_\parallel ,v_\perp ) &=& \frac{n_c}{n_0}\, f_c(v_\parallel ,v_\perp )+\frac{n_h}{n_0}\, f_h(v_\parallel ,v_\perp ), \nonumber \\ f_a(v_\parallel ,v_\perp ) &=& \frac{1}{\pi ^{3/2} \alpha _{\perp a}^2 \alpha _{\parallel a}} \exp \left(-\frac{v_\perp ^2}{\alpha _{\perp a}^2} -\frac{(v_\parallel -U_a)^2}{\alpha _{\parallel a}^2}\right), \end{eqnarray} (2) where (a = c, h) represent core and halo populations, nc and nh designate the core and halo electron number densities, respectively, and n0 is the total number density. Electron thermal speeds perpendicular and parallel to the ambient magnetic field are denoted by α⊥a and α∥a, where temperatures for each electron species are defined by |$T_{\perp a}=m_e\alpha _{\perp a}^2/2$| and |$T_{\parallel a}=m_e\alpha _{\parallel a}^2/2$|⁠. Here, Ua is net parallel drift speed, satisfying the current neutrality condition, ∑a = c, hnaUa = 0 that gives Uc = nhUh/(n0 − nh). 3 THEORETICAL FORMALISM AND DISPERSION RELATIONS In order to numerically investigate HF and EMEC instabilities in a combined manner, we consider here two species, two electron-component counter-streaming bi-Maxwellian plasma with zero net currents for propagation at |${\boldsymbol k}\times {\boldsymbol B}_0$| = 0. As elaborated by Gary et al. (1975) and Gary (1985), HF instability has its maximum growth rate for |${\boldsymbol k} \parallel {\boldsymbol B}_0$| and EMEC mode has also maximum growth rate for propagation parallel to ambient magnetic field |${\boldsymbol k} \parallel {\boldsymbol B}_0$| (Lazar et al. 2015). Owing to these assessments, we hereby just limit ourselves to parallel propagation. The general dispersion equation for electromagnetic modes propagating parallel to the ambient magnetic field is given by \begin{eqnarray} &&{\left[1-\frac{c^2k^2}{\omega ^2} + \sum _{a=p,c,h} \frac{n_a}{n_0}\frac{\omega _{pa}^2}{\omega ^2}\int \text{d}{\boldsymbol v}\,\frac{v_\perp /2}{\omega -kv_\parallel \pm \Omega _a}\right.} \nonumber \\ &&{\quad\left.\times \left((\omega -kv_\parallel ) \frac{\mathrm{\partial} f_a}{\mathrm{\partial} v_\perp } + kv_\perp \frac{\mathrm{\partial} f_a}{\mathrm{\partial} v_\parallel } \right)\right]\frac{E_{k,\omega }^x\mp iE_{k,\omega }^y}{2}=0.} \end{eqnarray} (3) The electric field eigenfunction in equation (3), namely, |$\left(E_{k,\omega }^x\mp iE_{k,\omega }^y\right)/2$| defines the wave electric field vector specifying right- and left-hand polarizations. The upper sign in equation (3) is for right-hand polarization and lower sign is for left-hand circular polarization. The term polarization defines the handedness of rotation with respect to time, at a fixed point in space, when viewed parallel to magnetic field. However, the sense of spatial twist associated with the wave electric field either in terms of right- or left-handedness defines positive or negative helicity, respectively. Here, ω is frequency and k is wavenumber. The quantity Ωa = eaB0/mac is cyclotron frequency, where c is speed of light and |$\omega _{pa}=(4\pi n_0e_a^2/m_a)^\frac{1}{2}$| is plasma frequency. Unit charge ea = −e is for electrons and charge ea = +e is for protons. Also, np = n0. Making an allowance for bi-Maxwellian drifting distribution function (2) for two plasma species, we obtain the dispersion relation in terms of plasma dispersion function Z(ζ) (Fried & Conte 1961) as \begin{eqnarray} \frac{c^2k^2}{\omega ^2} &=& 1+\sum _{a=p,c,h} \frac{\omega _{pa}^2}{\omega ^2} \left\lbrace \frac{\alpha _{\perp a}^2}{\alpha _{\parallel a}^2}-1 +\left[\frac{\alpha _{\perp a}^2}{\alpha _{\parallel a}^2} \,(\omega -kU_a)\right.\right. \nonumber \\ && \left.\left.\pm \left(\frac{\alpha _{\perp a}^2}{\alpha _{\parallel a}^2}-1\right)\Omega _a\right] \frac{1}{k\alpha _{\parallel a}} \,Z(\zeta _a^\pm )\right\rbrace , \end{eqnarray} (4) where \begin{eqnarray} \zeta _a^\pm = \frac{\omega \pm \Omega _a-kU_a}{k\alpha _{\parallel a}},\qquad \xi _a=\frac{\omega -kU_a}{k\alpha _{\parallel a}}, \end{eqnarray} (5) and \begin{eqnarray} Z(\zeta ) = \frac{1}{\sqrt{\pi }}\int ^\infty _{-\infty } \frac{e^{-x^2}\,\text{d}x}{x-\zeta },\qquad {\rm Im}\zeta >0. \end{eqnarray} (6) We may ignore the displacement current, the unity on the right-hand side of equation (4), as we are not interested in superluminal R/L mode waves here. Upper sign of equation (4) gives us the unstable solution for right-hand circularly polarized EMEC mode, which is excited by perpendicular temperature anisotropy along with relative drift effects. Lower sign of equation (4) is for HF mode. Cold limit characteristics of HF mode has already been discussed in our companion paper (Sundas et al. 2017), where we have also shown that HF instability is suppressed if ion dynamics are ignored and only gives real solution. We also showed that HF instability is highly analogous to electron firehose instability, which is also a left-hand mode. Moreover, we also demonstrated that HF instability is only triggered if both core-halo drifting components are considered. Otherwise (unlike Buneman instability, which is operative with only one component electrons drifting in the background of protons), the instability vanishes. Introducing the following dimensionless quantities: \begin{eqnarray} z &=& \frac{\omega }{\Omega _e},\qquad q=\frac{ck}{\omega _{pe}}, \nonumber \\ \delta &=& \frac{n_h}{n_0},\qquad M=\frac{m_i}{m_e}, \nonumber \\ u_h &=& \frac{\omega _{pe}}{\Omega _e}\frac{U_h}{c} =\frac{(4\pi n_0m_e)^{1/2}U_h}{B_0}, \nonumber \\ u_c &=& \frac{\omega _{pe}}{\Omega _e}\frac{U_c}{c} =\frac{\delta u_h}{1-\delta }, \nonumber \\ \beta _i &=& \frac{8\pi n_0T_i}{B_0^2},\qquad \beta _c=\frac{8\pi n_0T_c}{B_0^2},\qquad \beta _h=\frac{8\pi n_0T_h}{B_0^2} \nonumber \\ A_c &=& \frac{\beta _{\perp c}}{\beta _{\parallel c}}-1,\qquad A_h=\frac{\beta _{\perp h}}{\beta _{\parallel h}}-1 \nonumber \\ \xi _i &=& \frac{z}{q(M\beta _i)^{1/2}},\qquad \zeta _i^\pm =\frac{Mz\pm 1}{q(M\beta _i)^{1/2}}, \nonumber \\ \zeta _c^\pm &=& \frac{z\mp 1+qu_c}{q\beta _c^{1/2}},\qquad \zeta _h^\pm =\frac{z\mp 1-qu_h}{q\beta _h^{1/2}}, \nonumber \\ \eta _c^\pm &=& \frac{(A_c+1)(z+qu_c)\mp A_c}{q\beta _c^{1/2}}, \nonumber \\ \eta _h^\pm &=& \frac{(A_h+1)(z-qu_h)\mp A_h}{q\beta _h^{1/2}}. \end{eqnarray} (7) Equation (4) can be represented in dimensionless quantities as \begin{eqnarray} 0 &=& q^2-\xi _i\,Z(\zeta _i^\pm )-(1-\delta ) \left[A_c + \eta _c^\pm \,Z(\zeta _c^\pm )\right] \nonumber \\ &&-\,\delta \left[A_h+\eta _h^\pm \,Z(\zeta _h^\pm )\right]. \end{eqnarray} (8) 3.1 Realistic solar wind parameters Both core and halo components in the solar wind feature anisotropic temperature profiles (Phillips et al. 1989; Lazar et al. 2015; Pierrard et al. 2016). Observations reveal that core temperature anisotropy drops faster with distance from the Sun as compared with total temperatures, T∥ and T⊥ (Pilipp et al. 1990). This core temperature anisotropy also varies steadily as a function of electron densities and get diminished for large densities (Phillips et al. 1989; Phillips & Gosling 1990). The core anisotropy could also manifest itself in either excess of parallel T∥c > T⊥c or perpendicular T∥c < T⊥c temperature (Marsch 2012). Halo component of EVDFs, as with the core, also contains anisotropy, and is actually found to be more anisotropic than the core, which can be understood since halo electrons have longer Coulomb scattering time than core electrons. As with core component, halo component also has both profiles, T∥h < T⊥h or T⊥h < T∥h, characterizing excessive parallel or perpendicular temperature anisotropies (Marsch 2012). Correlations with plasma stream structures show that electron temperatures associated with slow solar wind are generally higher than in fast wind (Phillips & Gosling 1990) with excessive perpendicular temperature anisotropy, T⊥ > T∥, whereas excessive parallel temperature anisotropy, T∥ > T⊥ has been observed for fast solar wind. This excess of parallel temperature increases with the speed of solar wind flow for halo electrons, probably for the reason that strahl energy overlaps in halo component (Pierrard et al. 2016). For numerical investigation of HF and EMEC modes, we choose here observed solar wind parameters. Electron densities for distinct components of core and halo are provided by Ulysses/SWOOPS electron data for the heliocentric distances 1.34–5.41 AU, as well as average bulk solar wind properties. Ulysses 1-h averaged measurements of magnetic field and solar wind bulk speed data for the period 1994–2004 is used here (Lazar et al. 2015), while temperature anisotropies for solar wind data are discussed by Lazar et al. (2014). Lazar et al. (2015) and Pierrard et al. (2016) also investigated the solar wind data for temperature and temperature anisotropies of core and halo components for heliocentric distances 0.3–4 AU. Phillips et al. (1989) reported core anisotropy with an excess of parallel temperature T∥c/T⊥c = 1.5, as observed by International Sun Earth Explorer 3 for solar wind data near 1 AU. Pilipp et al. (1987) also reported EVDF with highly anisotropic core T∥/T⊥ > 3. The plasma beta of core population in the solar wind, β∥c, generally ranges from 0.05 to 1, while plasma beta of halo population, β∥h, lies within a range of 0.3–5 (Lazar et al. 2015). Making use of all these observations, instead of plotting our results for each time period, we choose the most probable parameters for solar wind observed data. For large number of events, core component of the solar wind EVDF is found to possess 95 per cent of the total density. So here, for our numerical analysis, we fix core-halo relative number density ratio, δ = nc/n0 = 0.95. If we choose the most appropriate realistic thermal velocities of core and halo electrons as, αc ∼ 0.01c and αh ∼ 3αc, respectively, and choose ωpe/Ωe ∼ 100 as typical of 1 AU, then making use of the quasi-neutrality condition for Uc ∼ 10−3c, we have Uh = 0.019c. This ultimately gives us \begin{equation} u_h=1.9,\qquad \beta _c=1,\qquad \beta _h=9. \end{equation} (9) For, now we take βc = βi ∼ 0.1. However, for lower thermal velocities of core population, αc = 0.002c, we obtain \begin{eqnarray} u_c &=& \frac{(1-\delta )u_c}{\delta }=0.038, \nonumber \\ u_h &=& 10^2 (0.038)=3.8,\qquad \nonumber \\ \beta _c &=& (10^2 (2)\times 10^{-3})^2=0.04, \nonumber \\ \beta _h &=& (10^2(6)\times 10^{-3})^2=0.36. \end{eqnarray} (10) 4 NUMERICAL ANALYSIS Making use of the defined realistic observed parameters discussed above, we have numerically solved equation (8) for HF and EMEC modes, under variation of magnitudes of two free energy sources. Fig. 1 is a reference case that shows how EMEC mode behaves when there is no net relative drift. One group of solutions is for the case when core temperature anisotropy is T⊥c/T∥c = 1 and halo anisotropy T⊥h/T∥h varies from 2 to 4, and the other group of solutions are for the case when halo is assumed to be isotropic and core anisotropy varies as T⊥c/T∥c = 2, 3, 4. Other input parameters are specified in the figure. Results show that growth rate of EMEC mode is higher when the anisotropy resides with core, although halo electrons anisotropy also contributes to the growth of EMEC instability. Figure 1. Open in new tabDownload slide EMEC instability: real frequencies (inset) and growth rates versus wavenumber, for zero net relative drift uh = 0. Red curves depict the case with core anisotropy (and isotropic halo), while blue curves indicate the situation depicting anisotropic halo (and isotropic core). Figure 1. Open in new tabDownload slide EMEC instability: real frequencies (inset) and growth rates versus wavenumber, for zero net relative drift uh = 0. Red curves depict the case with core anisotropy (and isotropic halo), while blue curves indicate the situation depicting anisotropic halo (and isotropic core). Fig. 2 displays solutions to equation (8) for both plus and minus signs for most probable realistic solar wind parameters. The core temperature anisotropy is taken as T⊥c/T∥c = 2.25 and the halo is considered to be isotropic. The lower sign of equation (8) is for HF instability and upper sign corresponds to EMEC instability. Fig. 2 is plotted for varying values of net relative drift speed for both HF and EMEC in order to see the effect of variation of the two-component relative drift speed on both instabilities and to show that HF instability is not associated with the whistler mode proper, which is a right-hand EMEC mode, but rather the HF instability operates on the left-hand mode branch. Figure 2. Open in new tabDownload slide Real frequencies (upper panel) and growth rates (lower panel) of HF and EMEC instability for varying halo drift speed uh and for fixed values of input parameters: Ac = 1.25, Ah = 0, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. Figure 2. Open in new tabDownload slide Real frequencies (upper panel) and growth rates (lower panel) of HF and EMEC instability for varying halo drift speed uh and for fixed values of input parameters: Ac = 1.25, Ah = 0, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. As it can be seen in the upper panel of Fig. 2, where both the left-hand polarized HF and right-hand polarized EMEC modes are plotted in combined real-frequency plot, EMEC mode (inset) is none other than the whistler electron-cyclotron mode, which is a right-hand mode. The HF mode, on the other hand, is a left-hand mode, and characterized by long wavelength on the order of ion scale, and it is of low frequency. Hence, the two modes operate on distinct spatial and temporal scales. As the magnitudes of net relative drift decreases from 3.8 to 2.6, the growth rate of HF instability decreases and get diminished for magnitude below than 2.6. However, EMEC instability, which is excited by large T⊥ anisotropy, is not affected by these varying magnitude of relative core-halo drift (all curves virtually overlap) for isotropic halo and for fixed value of core's anisotropy. In Fig. 3, the core is taken isotropic and halo component is chosen to be anisotropic. The anisotropy of halo component T⊥h/T∥h is allowed to vary from minimum magnitude 2–4 for fixed value of net relative drift, uh = 3.8. A reference case is drawn for value of zero net relative drift uh = 0, and for halo anisotropy T⊥h/T∥h = 1.5, shown in black line. The purpose of this black line (EMEC instability) is to show that EMEC mode grows for small magnitude of halo anisotropy T⊥h/T∥h =1.5, when no net relative drift is considered. However, when the net relative drift is increased and taken to be uh = 3.8, EMEC mode does not show any growth even for larger values of halo anisotropy T⊥h/T∥h = 2, 2.5, 3, 3.5, 3.8, 4. (In order to avoid the figure becoming too dense, we choose T⊥h/T∥h = 1, 2, 3, 4 for real frequency, and T⊥h/T∥h = 1, 2, 3.5, 3.8, 4 for growth rate plot.) This is because the halo drift speed of uh = 3.8 (and the associated counter-streaming core drift) provides large ‘effective’ parallel kinetic velocity spread, thereby generating large parallel pressure. On the other side, HF instability for relative drift u = 3.8 and for isotropic core does not show any major effect on its growth for the values T⊥h/T∥h = 2, 2.5 and 3. For high anisotropy of halo component T⊥h/T∥h = 3.5, 3.8, 4, HF instability growth rate shows two peaks, as shown in corresponding colour coded curves. Figure 3. Open in new tabDownload slide Growth rates (upper panel) and real frequencies (lower panel) of HF and EMEC instability for isotropic core and for varying magnitude of halo anisotropy, T⊥h > T∥h, where other input parameters are held constant: uh = 3.8, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. Figure 3. Open in new tabDownload slide Growth rates (upper panel) and real frequencies (lower panel) of HF and EMEC instability for isotropic core and for varying magnitude of halo anisotropy, T⊥h > T∥h, where other input parameters are held constant: uh = 3.8, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. In the upper panel, real frequencies are plotted for both HF and EMEC instabilities. For zero drift case, uh = 0, the real frequency is shown with black curve, as a benchmark. Frequency of EMEC wave for zero drift is shown to be generally higher than the cases with finite drift. For the latter, the real frequency shifts towards lower frequency, since for finite drift, uh = 3.8, the doppler shift affects the right-hand whistler mode, as illustrated by Sundas et al. (2017). In contrast, real frequency of HF mode does not show much sensitivity with the change in the halo temperature anisotropy for this comparatively large value of relative drift u = 3.8. It can be seen that the real frequency of HF mode is highly reminiscent of the real frequency associated with the electron firehose instability that is also caused by parallel free energy source. Similar results are found that are not shown here. If plotted for the same parameters of varying halo temperature anisotropies and plasma betas, as in Fig. 3 for relative drift u = 3.8, except with different choices of core temperature anisotropy up to ∼2.3. Fig. 4 represents a case when relative drift u is chosen to be comparatively low, i.e. u = 1.6, while all other parameters are the same as Fig. 3, except that halo anisotropy is chosen with slightly different combinations. This figure is plotted to see whether HF and EMEC modes behave the same if relative drift u is considered to be small. In the case when relative drift u = 0 and Ah = 0 (reference case), we observe no HF or EMEC instability, as expected. We plot the reference case with black curves. When finite relative drift speed is chosen, specifically, u = 1.6, we find that no EMEC instability exists even for the largest magnitude of halo anisotropy considered, namely, Ah = T⊥h/T∥h − 1 = 5. We also find that the left-hand circularly polarized HF mode is stabilized. However, we find that the backward-propagating, right-hand circularly polarized HF instability is excited. We know that the mode is backward propagating because the real frequency is in the negative domain for positive wavenumber. For ωr < 0 and k > 0, we also know that the mode polarization is in the sense of right hand for the lower sign in the dispersion relation (equation 8). This new instability is suppressed when the halo anisotropy is zero, so that the new instability is the result of the combination of finite drift speed and finite positive halo anisotropy. Figure 4. Open in new tabDownload slide Real frequencies (upper panel) and growth rates (lower panel) of backward-propagating right-hand circularly polarized HF and EMEC modes for isotropic core and for varying magnitude of halo anisotropy, T⊥h > T∥h, where other input parameters are held constant: uh = 1.6, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. Figure 4. Open in new tabDownload slide Real frequencies (upper panel) and growth rates (lower panel) of backward-propagating right-hand circularly polarized HF and EMEC modes for isotropic core and for varying magnitude of halo anisotropy, T⊥h > T∥h, where other input parameters are held constant: uh = 1.6, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. The upper and lower panels of Fig. 5 depict the combined real frequency and growth rates for HF and EMEC modes, by first considering core (top panels) and subsequently halo (lower panels) isotropies, respectively. The relative drift speed is held constant at u = 2.7. Note that when the anisotropy is provided by the core electrons, both HF and EMEC modes are unstable. The real frequency for the HF mode is not affected by the halo anisotropy so all the curves overlap, but the real frequency for EMEC mode shows some dependence on Ac (top left). The HF and EMEC mode growth rates are operative in distinct wavenumber ranges, so in order to show the dependence of HF instability growth rate on Ah, we show the close-up in the top-right panel. When the anisotropy is provided by the halo electrons, it is seen that both HF and EMEC modes show little change as far as the real frequency is concerned (bottom left). The EMEC mode is stable for all cases of Ah, and the HF mode is unstable only for Ah = 1. For all higher Ah, HF instability vanishes (bottom right). The case of varying Ah (bottom panels) can be compared with Fig. 4. The two cases are similar except in Fig. 4, we adopted lower drift speed, uh = 1.6, while in Fig. 5, we choose higher halo drift speed, uh = 2.7. In the case of Fig. 4, we observed that the backward-propagating HF instability (thus, right-hand mode) was excited. However, by increasing uh, it appears that this mode is stabilized. This finding implies that the interplay of halo drift speed versus core and halo temperature anisotropies is quite complex and it deserves further parametric study. However, such a work is beyond the scope of this paper. Figure 5. Open in new tabDownload slide Real frequency (left-hand panels) and growth/damping rates (right-hand panels) for isotropic halo and increasing core anisotropy Ac = T⊥c/T∥c − 1 and (upper panels), and for isotropic core and for increasing halo anisotropy Ah = T⊥h/T∥h − 1 (lower panels). Other input parameters are uh = 2.7, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. Figure 5. Open in new tabDownload slide Real frequency (left-hand panels) and growth/damping rates (right-hand panels) for isotropic halo and increasing core anisotropy Ac = T⊥c/T∥c − 1 and (upper panels), and for isotropic core and for increasing halo anisotropy Ah = T⊥h/T∥h − 1 (lower panels). Other input parameters are uh = 2.7, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. In Fig. 6, halo anisotropies are taken to be fixed T⊥h/T∥h = 2 and core anisotropy is varied from 1.5 to 2.5. For anisotropy T⊥c/T∥c = 1.5, EMEC instability does not get excited. The excitation starts from T⊥c/T∥c = 2 and the growth rate rapidly goes on increasing as the magnitude of core temperature ratio T⊥c/T∥c is increased from 2.2 to 2.5. This shows that EMEC mode is highly dependent on core anisotropy magnitude as discussed in the literature (Lazar et al. 2014). Although HF instability is not much affected by varying magnitude of core or halo anisotropy if large magnitude of relative drift is considered, but still there is a small but noticeable change for real frequencies. Same results are obtained for these parameters if halo is assumed to be isotropic and all parameters are taken to be the same as Fig. 4 for relative drift u = 3.8 that are not shown here. However, for smaller values of relative drift, varying magnitude of anisotropic core against isotropic halo also slightly suppresses the unstable magnitude of the HF mode as shown in Fig. 5. Figure 6. Open in new tabDownload slide Real frequencies (upper panel) and growth rates (lower panel) for halo anisotropy T⊥h/T∥h = 2 and for increasing core temperature ratio T⊥c/T∥c with the same input parameters: uh = 3.8, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. Figure 6. Open in new tabDownload slide Real frequencies (upper panel) and growth rates (lower panel) for halo anisotropy T⊥h/T∥h = 2 and for increasing core temperature ratio T⊥c/T∥c with the same input parameters: uh = 3.8, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. As observations illustrate, halo component sometimes show excessive parallel temperature in the fast solar wind. Consequently, taking this observation into account, Fig. 7 is plotted for temperature anisotropies, T⊥h/T∥h = 0.04, 0.05, 0.06 by fixing the core anisotropy T⊥c/T∥c = 0.3, for fixed value of relative drift u = 3.8. These parameters are meant to depict the case for solar wind where both core and halo exist with excessive parallel temperatures. For such parameters, HF instability shows no change in the magnitude of growth rate, and EMEC mode remains stable, the reason being that EMEC is driven by large perpendicular temperature anisotropies and we are considering both halo and core with excessive parallel temperatures. Even if large perpendicular temperature anisotropy exists for the core, that is, even when the temperature ratio of the order of ∼2 is considered, which is not shown here, and parameters are adopted as the same of Fig. 7, we found that HF instability does not show any dependence on the variation of excessive parallel halo anisotropy. In contrast, we found weak growth for EMEC instability because of the perpendicular anisotropy associated with the core, T⊥c/T∥c ∼ 2. Nevertheless, EMEC mode again remains unaffected for varying degrees of effective parallel anisotropy associated with halo. This is in agreement with our early finding (Sundas et al. 2017) in that the essential properties of the HF instability is already captured by cold fluid theory and that finite beta (or thermal) effects do not significantly alter the basic fluid picture. Figure 7. Open in new tabDownload slide Growth rates (upper panel) and real frequencies (lower panel) for slightly anisotropic core T⊥c/T∥c = 0.3 and for parallel excess energy T∥h > T⊥h of halo component by keeping other parameters constant: uh = 3.8, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. Figure 7. Open in new tabDownload slide Growth rates (upper panel) and real frequencies (lower panel) for slightly anisotropic core T⊥c/T∥c = 0.3 and for parallel excess energy T∥h > T⊥h of halo component by keeping other parameters constant: uh = 3.8, δ = 0.05, βp = β∥c = 0.04, β∥h = 0.36. 5 CONCLUSION AND DISCUSSIONS HF instability and other temperature instabilities remain an integral part of study on solar wind for many decades. Observations taken from satellites provide a peek inside interstellar medium, which permit scientists to understand the phenomena near and far beyond the Earth. The motivation for this paper is within this context. The present analysis has been carried out for the solar wind regions where core and halo components associated with the electrons are found to possess relative drift motion. This drift is caused when electrons emanating from the solar source move faster than heavy ion particles, being less massive, leave behind the ions. The drift of the electrons in relation to solar wind background protons cause HF instability. As observed in situ, solar wind plasma does not generally exist in thermal equilibrium. Among nonthermal features are temperature anisotropies. In this paper, we choose to study the combined effects of relative drift and temperature anisotropies on EMEC and HF instabilities. Both of these instabilities has been studied separately in the literature, but the combined EMEC and HF instabilities have not been systematically investigated thus far. Of the two types of instabilities driven unstable by different free energy source, the HF instability is less well understood. In fact, in the literature, the said instability is often termed the ‘whistler’ HF instability, which may be somewhat misleading, the classic whistler wave is a right-hand mode, while the unstable HF mode has opposite polarization. Our companion paper (Sundas et al. 2017) provided an insight on HF instability, but the discussion thereof assumed isotropic temperatures. This paper is a generalization in that both drift and temperature anisotropy are considered. Our results show that HF (left-hand mode) becomes unstable, if a critical value of net relative drift is achieved. Beyond this value, HF instability growth rate monotonically increases with the increase of net relative drift of two component electrons. The instability vanishes if this value falls below the magnitude u = 2.6. Effects of the drift is negligible on EMEC instability, if halo is assumed to be isotropic and if core anisotropy is taken to be fixed. This is demonstrated in Fig. 2 where decreasing relative drift is seen to lead to decreasing growth rate of HF instability, but is seen to be not effecting the growth of EMEC mode, which is driven by core anisotropy. Further, if core is assumed to be isotropic and change is made in halo anisotropy, then our results show that EMEC mode is stable in the presence of relative drift u = 3.8 even for values of halo anisotropy that would have made EMEC mode unstable in the absence of drift. This is interpreted as the relative drift, giving rise to an effective parallel pressure, thereby suppressing the EMEC instability. As for HF instability, for relative drift u = 3.8, it is generally insensitive to the change in halo anisotropy until it reaches T⊥h/T∥h = 4, at which, secondary unstable solution appears – see Fig. 3. However, if relative drift is small for example, if u = 1.6 or u = 2.7 and core is considered to be isotropic then varying values of halo anisotropy effects the growth rate of HF mode in a complicated way, for in the case of u = 2.7, increasing halo temperature anisotropy suppresses the growth rate of HF mode (Fig. 5), but in the other case, namely, u = 1.6 (Fig. 4), increasing halo temperature anisotropy has the opposite effect increasing the growth rate. This shows that the combined effect of drift speed and halo temperature anisotropy deserves further more systematic parametric study, which is beyond the scope of this paper. However, for EMEC the trend remains the same – even with the largest magnitude of halo anisotropy, up to 5 and 6, and for isotropic core, EMEC is not excited in presence of small relative drift. Further, for the choice of core anisotropy T⊥c/T∥c = 2.3, we did not find any profound difference for HF instability when compared against Fig. 3 for relative drift u = 3.8. However, it does slightly affect the HF instability if relative drift is small (see Fig. 5). So, we can generalize the statement as follows. If core or halo is considered to be isotropic for comparatively smaller values of relative drift, and change in the other component's anisotropy is made, then it does affect either unstable modes as shown in Figs. 4 and 5. For fix net relative drift u = 3.8 and for isotropic halo, the change in core anisotropy is seen to lead to substantial change in the magnitude of EMEC instability growth rate. With the increasing magnitude of core anisotropy T⊥c/T∥c, the EMEC growth rate is seen to increase quite rapidly. This consistent with the discussion found in the literature. We also considered other possible solar wind conditions where excess of parallel kinetic energies exists. Thus, we allowed parallel core and halo anisotropy T⊥ < T∥. As the free energy source of EMEC is not in parallel kinetic energy, we do not find any instability in EMEC mode, and also we did not find any profound changes in the growth rate of HF mode as parallel effective temperatures are varied (see Fig. 7). To conclude, the present findings may be useful for solar wind research in two respects. One is that the HF instability is insensitive to the change in temperature anisotropies of core or halo, especially if the magnitude of relative drift is not small, otherwise change in temperature anisotropies slightly decreases the growth of HF instability, but it depends primarily on the magnitude of the relative drift between the two electron species. Such a parametric dependence has not clearly been elucidated in the literature, since most of the discussions on HF instability assume isotropic temperature at the outset. The second significance of this paper is to unambiguously demonstrate that the HF mode is not associated with whistler branch of the dispersion relation, but it is indeed associated with left-hand circularly polarized mode, since we solved dispersion relations for both branches and showed that EMEC mode, which is the same as whistler wave in the case of isotropic electrons, and HF mode operate on distinct spatial and temporal scales, and are independent solutions. In an overall sense, our investigation contributes towards better understanding of these instabilities that are known to be important for solar wind research. Acknowledgments SS and MS acknowledges support from Higher Education Commission (HEC), Pakistan. MNSQ acknowledges the Higher Education Commission (HEC), Pakistan grant 20-1886/R&D10. PHY acknowledges NSF grant no. AGS1550566 to the University of Maryland, and the BK21 plus program from the National Research Foundation (NRF), Korea, to Kyung Hee University. 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TI - Characteristics of heat flux and electromagnetic electron-cyclotron instabilities driven by solar wind electrons
JF - Monthly Notices of the Royal Astronomical Society
DO - 10.1093/mnras/stx049
DA - 2017-05-01
UR - https://www.deepdyve.com/lp/oxford-university-press/characteristics-of-heat-flux-and-electromagnetic-electron-cyclotron-0VVHydGmgO
SP - 4928
VL - 466
IS - 4
DP - DeepDyve
ER -