Alternative high plasma beta regimes of electron heat-flux instabilities in the solar wind
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Alternative high plasma beta regimes of electron heat-flux instabilities in the solar wind
R. A. L´opez, M. Lazar,
2, 3
S. M. Shaaban,
2, 4
S. Poedts,
2, 5
P. S. Moya,
2, 6 Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Santiago, Chile Centre for mathematical Plasma Astrophysics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium Institut f¨ur Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universitt Bochum, D-44780 Bochum, Germany Theoretical Physics Research Group, Physics Department, Faculty of Science, Mansoura University, 35516, Mansoura, Egypt Institute of Physics, University of Maria Curie-Skodowska, PL-20-031 Lublin, Poland Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile (Received; Revised; Accepted)
Submitted to ApJABSTRACTThe heat transport in the solar wind is dominated by the suprathermal electron populations, i.e.,a tenuous halo and a field-aligned beam/strahl, with high energies and antisunward drifts along themagnetic field. Their evolution may offer plausible explanations for the rapid decrease of the heatflux with the solar wind expansion, typically invoked being the self-generated instabilities, or theso-called heat flux instabilities (HFIs). The present paper provides a unified description of the fullspectrum of HFIs, as prescribed by the linear kinetic theory for high beta conditions ( β e ≫ . U ) of the suprathermals. HFIs of different nature are distinguished, i.e.,electromagnetic, electrostatic or hybrid, propagating parallel or obliquely to the magnetic field, etc.,as well as their regimes of interplay (co-existence) or dominance. These alternative regimes of HFIscomplement each other and may be characteristic to different relative drifts of suprathermal electronsand various conditions in the solar wind, e.g., in the slow or fast winds, streaming interaction regionsand interplanetary shocks. Moreover, these results strongly suggest that heat flux regulation may notinvolve only one but several HFIs, concomitantly or successively in time. Conditions for a single, welldefined instability with major effects on the suprathermal electrons and, implicitly, the heat flux, seemto be very limited. Whistler HFIs are more likely to occur but only for minor drifts (as also reportedby recent observations), which may explain a modest implication in their regulation, shown already inquasilinear studies and numerical simulations. Keywords: solar wind – electron strahl – heat-flux – wave instabilities – methods: kinetic – numerical AN INTRODUCTORY MOTIVATIONThe solar wind heat flux is mainly attributed to theenergetic suprathermal electrons, a diffuse halo presentat all pitch-angles, and an electron beam, or strahl, di-rected along the interplanetary magnetic field away fromthe sun. Suprathermals may not exceed 10% of the totaldensity, but have high energies (much higher than ther-mal or core electrons) and significant antisunward drifts(Pilipp et al. 1987; Wilson III et al. 2019). The strahl is
Corresponding author: R. A. L´[email protected] in general responsible for a major velocity shift betweenthe core and suprathermal electrons (Rosenbauer et al.1977; Pilipp et al. 1987; Wilson III et al. 2019), but re-cent studies also reveal a relative drift of the halo(Wilson III et al. 2019) to be taken into account in cer-tain circumstances; for instance, in the low-speed windsthe strahl can be almost absent (Gurgiolo & Goldstein2017) and the heat is transported by the halo electrons(Pilipp et al. 1987; Pagel et al. 2005; Bale et al. 2013).However, if the strahl is observed most of the solar windheat flux is carried by the strahl electrons (Pilipp et al.1987; Pagel et al. 2005; Graham et al. 2017; Lazar et al.2020).
L´opez et al.
The modifications of suprathermal electrons with thesolar wind expansion can be directly linked to the vari-ations of heat flux, and are expected to explain theobserved dropouts and an accelerated decrease of theheat flux, more rapid than predicted by an adiabaticdecrease of the main plasma parameters. Indeed, theobservations reveal an important erosion of the strahl,which decline in relative density and drift, and broadentheir pitch-angle distribution with increasing heliocen-tric distance (Maksimovic et al. 2005; Pagel et al. 2007;Anderson et al. 2012; Graham et al. 2017; Berˇciˇc et al.2019). The effect of binary collisions on suprathermals isinsignificant, but these evolutions may be explained bythe so-called heat flux instabilities (HFIs), self-generatedby the relative drifts and beaming velocity of suprather-mal electrons (Gary & Feldman 1977; Gary et al. 1999a;Pavan et al. 2013; Shaaban et al. 2018a; Shaaban et al.2019a; Vasko et al. 2019; Verscharen et al. 2019a). Theresulting wave fluctuations can induce a diffusionof suprathermals in velocity space, contributing totheir relaxation, as already shown in numerical sim-ulations (Dum & Nishikawa 1994; Gary & Saito 2007;Kuzichev et al. 2019; L´opez et al. 2019a).In this paper we provide a comparative analysis of thefull spectrum of HFIs prescribed by the linear kinetictheory for high plasma beta conditions ( β ≫ .
1) anddifferent relative drifts ( U ) of the suprathermal popula-tions. Such a unified analysis offers new and multipleperspectives for the implication of HFIs in the evolutionof suprathermals and, implicitly, of the solar wind heatflux. Current way of thinking that a single instabilitycan be identified as the principal mechanism of regula-tion of the heat flux in the solar wind may need majorupgrades, to include the interplay and/or succession oftwo or more instabilities.In section 2 we introduce the kinetic formalism oftenadopted in studies of plasma wave dispersion and sta-bility, in our case a typical plasma with two asymmetriccounter-drifting populations of electrons. A short de-scription is also provided for the numerical solver allow-ing us to determine the full spectrum of the unstable so-lutions, covering all ranges of frequencies, wave-numbersand angles of propagation. Models assumed for the zero-th order velocity distribution are drifting-Maxwellian,which enable a standard and simple parameterization ofthe solar wind electron-proton plasma populations. Weare aware of the existence of other more realistic repre-sentations, like Kappa models (Shaaban et al. 2018a) forthe halo, or more asymmetric combinations of driftingMaxwellians for a more skewed strahl (Horaites et al.2018), which would only complicate our analysis but ul-timately would lead to similar results and conclusions. Adopting drifting Maxwellian keeps at this stage theanalysis simple and enables straightforward interpreta-tions of the HFIs, their nature, interplay and dominance.Moreover, such a dual model can reproduce the slowwind core-halo distribution, in the absence of strahl, butmay also be relevant for the fast wind core-strahl config-uration if the less drifting halo is assimilated to the corepopulation . The results are presented and discussed indetail in section 3, considering each alternative unstableregime in part. These regimes have a wide relevance,covering lower drifts and higher thermal spreads repro-ducing better the halo electrons, or higher drifts andlower thermal spreads specific to the strahl population,and, nevertheless, a series of intermediary states whichmay be associated with the relaxation of strahl and theformation or/and enhancement of halo (Hammond et al.1996; Anderson et al. 2012; Graham et al. 2017). Thelast section summarizes our results and formulates a se-ries of conclusions, which should help in understandingthe observations and make realistic interpretations ofHFIs and their implications. DISPERSION AND STABILITYWe consider a collisionless quasi-neutral plasma ofprotons and two electron populations, namely, a densecentral or core component (subscript “ c ”) and a ten-uous suprathermal population (subscript “ s ”) counter-drifting along the ambient magnetic field, assumed con-stant over at least a few maximum wave-lengths ofthe instabilities considered here (e.g., Shaaban & Lazar2020 and references therein) f e (cid:0) v ⊥ , v k , (cid:1) = n c n e f c (cid:0) v ⊥ , v k (cid:1) + n s n e f s (cid:0) v ⊥ , v k (cid:1) , (1)where n e ≈ n p is the total electron number density,and n c and n s are the number densities of the core andstrahl populations, respectively, satifying n c + n s = n e .In the next this suprathermal population will be calledgenerically ’strahl’, but the analysis may also apply to acore-halo configuration, as explained already above. Forboth the core ( j = c ) and strahl ( j = s ) populations weadopt a simple standard description (widely used in sim-ilar studies) as drifting bi-Maxwellians (Saito & Gary2007b; Verscharen et al. 2019b) f j ( v ⊥ , v k ) = π − / α ⊥ j α k j exp ( − v ⊥ α ⊥ j − ( v k − U j ) α k j ) , (2)where α ⊥ , k ,j = (2 k B T ⊥ , k ,j /m e ) / are components ofthermal velocities perpendicular ( ⊥ ) and parallel ( k ) to High beta ( β e > .
5) instabilities may not be significantly alteredby the inclusion of halo in this case (Horaites et al. 2018)
Figure 1.
Growth rates γ/ Ω e (top), and wave-frequency ω/ Ω e (bottom), for β c = 2 . U s /v A = 15,135, 150, and 180. the background magnetic field, and U j are drift veloci-ties, which preserve a zero net current n s U s + n c U c = 0.For simplicity, protons are assumed isotropic ( T p ⊥ = T p k ), nondrifting ( U p = 0), and Maxwellian distributed.We preset a general dispersion and stability analy-sis covering the full wave-vector spectrum of (unsta-ble) plasma modes propagating at arbitrary angles θ with respect to the background magnetic field ( B = B ˆ z ). Without loss of generality the wave-vector k = k ⊥ ˆ x + k k ˆ z is chosen in the x – z plane ( k k = k cos θ and k ⊥ = k sin θ ). Our analysis is based on the kineticVlasov-Maxwell dispersion formalism, as provided byStix (1992), and the unstable solutions are found numer-ically, providing accurate description for the full spec-trum of instabilities (e.g., electrostatic, electromagneticor hybrid), and various regimes of their co-existenceand dominance. We use a complex root finder basedon the M¨uller’s method to locate the solutions of theplasma dispersion tensor. Solutions provided by this code have been validated in previous studies for variouskinetic instabilities (Shaaban et al. 2019b; L´opez et al.2019b; Lazar et al. 2019), and using PIC simulations inthe low and high-frequency regimes, and also for multi-component plasmas (L´opez et al. 2017; L´opez & Yoon2017; L´opez et al. 2020; Micera et al. 2020).Present study focuses on the solar wind high plasmabeta conditions, i.e., for β c ≫ . β c & n s /n e = 1 − n c /n e = 0 .
05, tempera-ture contrast T s /T c = 4, plasma beta β c = 2, frequencyratio ω pe / | Ω e | = 100 and a realistic proton-electron massratio m p /m e = 1836. L´opez et al.
Table 1.
Plasma parameters used in the present study.Strahl ( s ) Core electrons ( c ) Protons ( i ) n j /n i T j, k /T i, k m j /m i T j, ⊥ /T j, k Note —Other parameters are: ω pe / Ω e = 100, β c =8 πn e T c /B = 2 We characterize the HFIs as primarily defined bythe main plasma eigen-modes destabilized by the rel-ative drift of suprathermal electron population, e.g., (1)fast-magnetosonic/whistler (FM/W) waves, RH-circularpolarized when propagating in parallel direction, (2)Alfv´enic modes, LH-circular polarized in parallel direc-tion, and (3) electrostatic beaming instabilities. Highbeta electrons ( β eff = 8 πn e k B T eff /B = 8 πk B ( n c T c + n s T s ) /B = β c + β s > .
1) present in the solar wind areexpected to excite moderate and high frequency modesof these branches. The unstable FM/W modes withhigh frequencies in the range Ω p < ω r < | Ω e | will sim-ply be named whistler heat-flux instabilities (WHFIs),but making however distinction between the (quasi-)parallel and oblique branches of WHFIs (Gary et al.1994; Wilson III et al. 2009; Russell et al. 2009). Theinstability mechanisms imply resonant or nonresonantinteractions with plasma particles, especially electrons,and may determine linear interplay and conversions be-tween different branches of plasma modes. Even in theabsence of instabilities, the wave dispersion of electro-magnetic (EM) modes decouples from electrostatic (ES)oscillations only for parallel propagation ( θ = 0). Theseaspects will be discussed in the next, in an attempt toaccurately identify the regimes of HFIs, and characterizethe transition between these regimes. RESULTSWe perform a spectral analysis of the unstable modesin ( ck/ω pe , θ ) − space, where ck/ω pe is the wave-numbernormalized to the electron inertial length, and θ is thepropagation angle. Upper panels in Figure 1 display thefull range of the growth rates γ/ | Ω e | > U s /v A = 15 (left panel), U s /v A = 135 (middle left), U s /v A = 150 (middle right), U s /v A = 171 (right). For anominal value v A = 20 km/s for the Alfv´en speed (usu-ally between 10 and 50 km/s at 1 AU) the highest driftsassumed in Figure 1 correspond to the limit values mea-sured for the relative drift of the electron beam/strahl, Figure 2.
Polarization, Re { i ( E x /E y )Sign( ω r ) } , for the lasttwo cases in Fig. 1, U s /α s = 1 .
24 (lef) and U s /α s = 1 . see Wilson III et al. (2019). The corresponding wavefrequency ω/ | Ω e | > U s with, α s , the thermal speedof strahl electrons, which is particularly important in thestudy of kinetic instabilities, directly conditioning theirthresholds and dominance regimes, e.g., for the WHFI(Gary 1985; Shaaban et al. 2018a) and the electrostaticinstabilities (Gary 1993).3.1. Whistler heat flux instabilities
The left panels in Figure 1 describe the (quasi-)parallelWHFI (Gary 1985; Shaaban et al. 2018a,b; Tong et al.2019b), which is solely predicted for the parameters cho-sen in this case, i.e., less energetic strahls with a low drift U s = 15 v A = 0 . α s < α s , lower than thermal speedof the suprathermal drifting electrons. Although theWHFI also extends to small oblique angles, the fastestgrowing mode propagates in direction parallel to thebackground magnetic field, i.e., θ = 0 ◦ . These modesare RH circularly polarized, as showin by the positivepolarization (green) in Figure 2. Here the polarizationis defined as Pol = Re { i ( E x /E y )Sign( ω r ) } , see Gary(1993).With increasing the drift velocity the growth rate ofthe parallel WHFI decreases and this mode becomeseventually damped, see Figure 1, the middle and rightpanels, for respectively, U s /α S = 1.11, 1.24, and 1.41.Middle-left panels in Figure 1 present the unstable solu-tions for a higher beaming speed, U s = 135 v A = 1 . α s ,exceeding the thermal speed. The WHFI restrains,but for oblique angles of propagation we find anotherwhistler-like instability, known already as the oblique Figure 3.
Transition from the WHFI regime to thedominance of O-WHFI. Growth rate (top) and polariza-tion Re { i ( E x /E y )Sign( ω r ) } (bottom) as a function of wave-number for β c = 8 and various drift velocities. Dotted blackline indicates the contour of minimum polarization ( ≃ . WHFI (O-WHFI) (Sentman et al. 1983; Tokar et al.1984; Wong & Smith 1994; Verscharen et al. 2019b).This oblique mode has a wave frequency dispersion (bot-tom panels) quite similar to that of parallel whistlers,specific wave-frequencies (Ω p < ω < | Ω e | ) and wave-numbers, and a RH elliptic (positive) polarization forall directions. Polarization is computed (only for theunstable modes, γ >
0) as Re { i ( E x /E y )Sign( ω r ) } andis mapped in Figure 2 and bottom panels of Fig-ure 3. By contrast to the WHFI, the O-WHFI ispurely oblique and may reach much higher growth rates.In this case maximum growth rates of the O-WHFI( γ max / Ω e = 1 . × − ) are obtained for θ = 54 . ◦ and ck/ω pe = 0 .
26. The growth rates of this instabilityare markedly enhanced by only slightly increasing thedrift, see the next two cases in Figure 1. The peak-ing maximum of the growth rates moves toward higherwave-numbers and larger angles of propagation as thedrift velocity increases, i.e., γ max / Ω e = 6 . × − at θ = 60 . ◦ and ck/ω pe = 0 . U s /α s = 1 .
24, and γ max / Ω e = 1 . × − at θ = 66 . ◦ and ck/ω pe = 0 . U s /α s = 1 . Figure 4.
Electric and magnetic field powers for the fastestgrowing O-WHFI, θ = 60 . ◦ , in Fig. 1, third case for U s /α s = 1 .
24 ( U s /v A = 150). Here the directions longi-tudinal (L) and transverse (T) are defined with respect tothe wave-vector, δ E L = ( δ E · k ) k . Dashed lines show themagnetic/electric powers of WHFI at θ = 0 ◦ . oblique angles forms and detaches from the standardWHFI which remains at lower angles. These obliquewhistlers can be destabilized by the asymmetric counter-drifting populations of electrons specific to the upstreamconditions of the interplanetary shocks (Sentman et al.1983; Tokar et al. 1984; Wong & Smith 1994) and to thefast winds (Verscharen et al. 2019b). In simulations ofa predefined low-scale whistler turbulence the obliquewhistlers were found able to strongly interact with strahlelectrons, contributing to their pitch-angle and energyscattering (Saito et al. 2008). Typical fluctuations ofoblique whistlers were also reported by the observationsin the magnetosphere during magnetically active peri-ods (Wilson III et al. 2011), in association with electronbeams in interplanetary high- β shocks (Breneman et al.2010; Wilson III et al. 2012; Ram´ırez V´elez et al. 2012)and recently, collocated with magnetic field holes in theouter-corona (Agapitov et al. 2020).Figure 4 displays the wave-number dispersion of theelectric and magnetic powers for the fastest growing O-WHFI ( θ = 60 . ◦ ) in Figure 1, the third case ( U s /α s =1 . L´opez et al. ( x, y, z ) representation (bottom), and with respect tothe wave-vector k , the longitudinal (subscript L ) ortransverse (subscrit T ) components (top). Dashed linescorrespond to the WHFI at θ = 0 ◦ , as expected for thepurely transverse (electric and magnetic) fields propa-gating in parallel direction. Based on this understand-ing, we can claim that O-WHFI can be driven cumula-tively by the resonant interactions with beaming elec-trons, via their Landau and transit time resonanceswith longitudinal (electrostatic) component E L , and ananomalous cyclotron resonance with transverse (elec-tromagnetic) component E T . The wave-particle reso-nant mechanisms governing this instability (Tokar et al.1984) can be identified following the same wave-numberdispersion of the arguments of plasma dispersion func-tion (absolute values) | ξ ( m ) s | , known as “resonant fac-tors” (Gary et al. 1975b). These arguments are com-puted in Figure 5 for the fastest growing O-WHFI,the same third case in Figure 1 ( U s = 1 . α s ). Thegrowth rate is overplotted with a solid red line. Forwave-numbers corresponding to the maximum growthrate both resonance conditions are well satisfied, i.e., | ξ (0) s | → | ξ ( ± s | → E x field component in Figure 4). Instead,highly oblique whistlers are mainly destabilized by theinteraction of beaming electrons with the electrostaticand compressive components, through, respectively, aLandau resonance with E z (which is minor but increaseswith increasing the wave-number in Figure 4, bottompanels), and a transit time resonance with B z (which isnot minor and shows the similar enhancement with in-creasing the wave-number in Figure 4, bottom panels).For more explanations see Gary et al. (1975b), or thetextbook of Gary (1993) and more references therein.3.2. Firehose-like instabilities of Alfv´enic waves
Another unstable solution obtained for higher drifts,e.g., the last two cases in Figure 1, for U s /α s = 1 . Figure 5.
Arguments of plasma dispersion functions | ξ ( m ) s | (absolute value) quantifying Landau and transit time reso-nances | ξ (0) s | →
1, and cyclotron resonances | ξ ( ± s | →
1, forthe fastest growing O-WHFI in Figure 1, third panel. Thegrowth rate is overplotted with a solid red line. wave-numbers and low frequencies. Growth rates are ingeneral lower than those of the O-WHFI, and maximumspeak at θ = 0 o . New detailed descriptions of the parallelFHFI, including comparisons with the WHFI and the ef-fects of suprathermal electrons present in the solar wind,can be found in Shaaban et al. (2018a,b). Last case inFigure 1 ( U s /α s = 1 .
41) shows the growth rates of FHFIextending to more oblique angles and overlaping withthe O-WHFI. However, distinction can easily be madebetween the LH-polarization of FHFI, i.e., negative val-ues, and the RH-polarization of the O-WHFI, positivevalues, in Figure 2 and 3. Moreover, the O-WHFI isby far dominant, exhibiting much higher growth ratesthan FHFI. Middle panels in Figure 1 identify with theregime of dominance of the O-WHFI, when this instabil-ity exhibit growth rates much higher than all the othermodes, e.g., WHFI or FHFI. However, for higher drifts,e.g., the last case in Figure 1 (for U s = 1 . α s ), theO-WHFI is already competed by the electrostatic in-stabilities, showing maximum growth rates for parallelpropagation. 3.3. Electrostatic instabilities
The electrostatic (ES) plasma modes are destabilizedwhen the relative drift of electron strahl is large enough,e.g., U s > α s > α c , to ensure Landau resonance withelectrons satisfying γ ∝ ∂f s /∂v k >
0. Thus, the theorypredicts a bump-on-tail instability of Langmuir wavesfor U s /α s < ( n e /n s ) / , or a more reactive electronbeam instability (EBI) for U s /α s > ( n e /n s ) / (Gary1993). For highly contrasting electron populations with T s > T c the electron acoustic waves become a nor-mal mode, and can be destabilized by a relative core-strahl drift several times higher than thermal speed ofthe core electrons (Gary 1987, 1993). These instabil- Figure 6.
Linear growth rates, γ/ Ω e , ω/ Ω e , for β c = 2 . U s /α s = 1 .
49 (left) and U s /α s = 2 . Figure 7.
Wave-frequency and growth rate dispersion ofthe ES instabilities: EAI (solid lines with γ > γ > U s /α s = 2 .
06 (top), and 3.30(bottom). ities are widely invoked in space plasma applications,to explain electron acoustic emissions detected in theEarth’s bow shock (Lin et al. 1985), radio bursts asso-ciated with bump-on-tail instability of coronal or inter-planetary shock-reflected electrons (Nindos et al. 2008),and broadening of solar wind strahls by self-generatedLangmuir waves (Pavan et al. 2013) or fast-growing elec-tron beam modes (An et al. 2017; Lee et al. 2019).The last case in Figure 1 shows the electron acous-tic instability (EAI) within built-in panels, with growthrates peaking at θ = 0 o ( γ max / Ω e = 1 . × − ) and competing with those of the O-WHFI. In this case driftvelocity is U s /α = 1 . < ( n e /n s ) / ≃ .
71 and satis-fies also conditions for a Langmuir wave instability (LWI- not shown in Figure 1) with growth rates much lowerthan EAI, see Figure 6. The first panel in Figure 6shows the unstable solutions for a slightly higher drift U s /α s =1.49, with the EAI in a narrow wavenumber in-terval but with growth rates much higher than both theO-WHFI and LWI. Note also that FHFI extends to evenlarger angles but maximum growth rates remain muchless than those of the O-WHFI. The LWI and EAI ex-cite waves with frequencies close to the electron plasmafrequency ( ω ∼ ω pe ≃ ω pc ), but wave-numbers specificto EAI are one order of magnitude higher, see Figures 6and 7.Figure 7 describes the unstable ES modes for U s /α s =2.06 (top, the same with the right panel of Fig-ure 6), and for U s /α s =3.03 (bottom). Specific to moreenergetic flows and coronal ejections these high drifts areassumed to be at the origin of coronal and interplane-tary bursts. In Figure 7 we show the wave frequency(left) and imaginary frequency (right) for various an-gles of propagation, this time normalized by the elec-tron plasma frequency. It becomes thus clear that thefastest growing mode is obtained for parallel propaga-tion, and characteristic frequencies are around the elec-tron plasma frequency. These details enable us to clarifythe differences shown by the peaking growth rates in Fig-ure 6. With increasing the drift, maximum growth ratesremain in parallel direction, but extend to lower wave-numbers and lower frequencies characteristic to the EBI( ω r ≃ kU s ). The most unstable modes result from theinterplay of EAI and EBI at low angles, and EAI remainssolely responsible for the lower growth rates obtained atoblique angles only. In the second case ( U s /α s = 3 . γ , which correspond to the EBI and EAIwhen γ > L´opez et al.
Figure 8.
Maximum growth rates (color codded) as a function of β c and U s /α s (top panels), or − U c /v A and U s /v A (lowerpanels) for the WHFI (left), FHFI (middle) and O-WHFI (right panels). ble regimes for the highest drifts considered in Figure 7(bottom panels) satisfying U s /α s > ( n e /n s ) / ≃ . Drift and beta instability thresholds
We have already identified and characterized a seriesof alternative regimes of HFIs, as predicted by the the-ory for different relative drifts of the electron strahl (sat-isfying the zero net-current condition). The parametricanalysis is completed here with a description of the in-stability thresholds, which highly depend on the electronplasma beta (limiting to high beta conditions, β > . γ max / Ω e ,which are derived in terms of drift velocities for thestrahl ( U s ) or core ( U c ) and the core plasma beta ( β c ).Note that these contours have no information about θ or k , as they represent the maximum growth rates from thefull spectrum of unstable modes (including all frequen-cies, wave-numbers and angles of propagation) obtainedfor each combination of drift and electron plasma beta.Figure 8 presents contours of maximum growth ratesfor the WHFI (left), FHFI (middle) and O-WHFI (rightpanels). These are derived in terms of the core elec-tron beta ( β c ) and the drift velocity, expressed as U s /α s (top), or − U c /v A and U s /v A (lower panels). There are unstable regimes which appear in both cases, but com-plementary regimes are also shown, for instance, thosehidden by a direct dependence of β c on v A (via thedensity and magnetic field) are shown in top panels,while those hidden by a more subtle dependence of β c on α s (due to a fixed core-strahl temperature contrast T c /T s = 1 /
4, see Table 1, leading to α c = ( T c /T s ) / α s )appear in bottom panels. On the other hand, the varia-tions of relative drifts with respect to thermal speed α s (top panels) may have an extended physical relevance,helping us not only to delimit complementary regimescorresponding to different instabilities, e.g., WHFI fromFHFI, or even from ES instabilities, but to understandthe difference between physical mechanisms responsiblefor these instabilities (as discussed already above).Left panels in Figure 8 show a non-monotonous varia-tion of the growth rate of WHFI with the drift velocity,as the growth rate increases and then decreases with in-creasing the drift. Consequently, the most unstable solu-tions of WHFI are located in-between the lower and up-per thresholds, as also found by Shaaban et al. (2018a,b)for lower β c . Figure 9.
Maximum growth rate as a function of coreplasma beta and beam velocity, β c vs. U s /α s (top panel),and − U c /v A and also U s /v A (lower panel), for all the in-stabilities discussed, WHFI, FHFI, O-WHFI and EAI (plusEBI). Dashed white line indicates U s /α s = √ with the drift velocity, and the core plasma beta ( β c ).The maximum growth rate γ max / Ω e of FHFI increaseswith the drift velocity, but decreases as β c increases(bottom panels). The most unstable FHFI is locatedat large drifts ( U c ) and low β c . Secondly, right panelsin Figure 8 show the O-WHFI, mostly overlapping withthe parametric regime of FHFI, but the O-WHFI ex-hibits much higher maximum growth rates than FHFIand WHFI. Similar to FHFI, the maximum growth rateof the O-WHFI is, in general, a monotonous function ofthe drift velocity and core plasma beta. The O-WHFI isstimulated by increasing the drift velocity and decreas-ing the core plasma beta. For low beta the most unstableO-WHFI is located at large drifts, but with increasingthe plasma beta this instability becomes operative for Figure 10.
Maximum growth rate as a function of coreplasma beta and beam velocity, − U c /v A and U s /v A , for dif-ferent temperature ratios between core and strahl, T s /T c = 3and 5. lower drift velocities. The lowest drifts remain suscepti-ble only to WHFI.The alternative regimes of EM instabilities describedin Figure 8 are contrasted in Figure 9 with the very highgrowth rates of ES instabilities. The range of plasmabeta is extended to the interval 0 . ≤ β c ≤
10, to in-clude lower beta conditions. For moderately high valuesof beta (e.g., β c = 2), WHFI and O-WHFI are com-plementary, their regimes, respectively, at the lowestor higher drifts velocities, being well delimited by thelowest contour levels of γ max . For higher values of β c these two regimes overlap, in-between defining a transi-tion where WHFI and O-WHFI interplay and may com-pete to each other. The lower beta part of the figureis dominated by the ES instabilities, which involve theEAI and for higher drifts the EBI. These instabilities ex-hibit very high growth rates, which explains the abrupt0 L´opez et al.et al.
10, to in-clude lower beta conditions. For moderately high valuesof beta (e.g., β c = 2), WHFI and O-WHFI are com-plementary, their regimes, respectively, at the lowestor higher drifts velocities, being well delimited by thelowest contour levels of γ max . For higher values of β c these two regimes overlap, in-between defining a transi-tion where WHFI and O-WHFI interplay and may com-pete to each other. The lower beta part of the figureis dominated by the ES instabilities, which involve theEAI and for higher drifts the EBI. These instabilities ex-hibit very high growth rates, which explains the abrupt0 L´opez et al.et al. transition to the O-WHFI. Marked with white-dashedlines at about U s /α s ≃ √
2, these narrow threshold con-ditions are characteristic to the interplay of O-WHFIand EAI described in the last case of Figure 1. Forour parameterization characteristic to the solar wind,the growth rate of FHFI is always smaller than the O-WHFI or EAI, and we could not find any regime whereFHFI can develop.Finally, in Figure 10 we show the effect of the strahl-core temperature ratio, contrasting maximum growthrates obtained for lower and higher values, respectively, T s /T c = 3 in the top panel and T s /T c = 5 in the bottompanel. These values are in the range of solar wind mea-surements (Wilson III et al. 2019). Major differencesare observed for the WHFI thresholds. For higher ratios T s /T c , the region of dominance of the WHFI extends tolower betas and higher drifts (as also shown in Figure 8for the case T s /T c = 4) covering a larger portion of theparameter space. The region of co-existence of WHFIand O-WHFI also extends to lower values of β c , whilethe O-WHFI region of dominance remains almost un-changed, although the maximum growth rates of thisinstability decrease as the temperature ratio increases. CONCLUSIONSWe have provided a unified description of the full spec-trum of heat-flux instabilities (HFIs) driven by the rela-tive drift of suprathermal electron populations under thehigh-beta solar wind conditions. Their nature, wave dis-persion, stability and polarization highly depend on therelative drift (or beaming) velocity of suprathermal elec-trons and the plasma beta parameter. The zeroth ordercounter-drifting distributions are modeled with standarddrifting-Maxwellians, which enable simple parameteri-zations and straightforward analysis and interpretationof HFIs in various conditions typically encountered inthe solar wind, e.g., a drifting halo in the slow wind,or the electron strahl carrying the heat flux of the highspeed flows.The unstable solutions have been derived and ex-amined in terms of their main features, i.e., wave fre-quencies, growth-rates, wave-numbers and propagationangles, and in terms of plasma (electron) parametersdefining the instability conditions, thresholds, etc. Pre-dicted are three electromagnetic instabilities, namely,the quasi-parallel whistler heat-flux instability (WHFI),the firehose heat-flux instability (FHFI) and the obliqueWHFI (O-WHFI), and a series of electrostatic insta-bilities destabilizing Langmuir waves (LW), the electronacoustic (EA) modes, or the more reactive electron beaminstability (EBI). We can identify three alternative regimes, each ofthem characterized by a well defined instability, solelypredicted by the theory or with (maximum) growthrates much higher than other unstable modes. Thus,for relatively low drifts of suprathermal electrons, i.e., U s < α s , the WHFI is the only operative, with maxi-mum growth rates associated with parallel propagation.This regime is characteristic to the low drifts and thelarge quasithermal spread of halo electrons, and seemsto be controlled exclusively by the WHFI (Gary 1985;Shaaban et al. 2018a; Scime et al. 1994). Typical WHFfluctuations associated with drifting suprathermal pop-ulations are confirmed by the solar wind observations,see Wilson III et al. (2013); Tong et al. (2019b,a).For higher drifts the dominance shifts to the O-WHFI,which are hybrid modes triggered unstable by cyclotronresonance, mainly at small angles of propagation, com-bined with Landau and transit time resonances, domi-nant at larger angles (Sentman et al. 1983; Tokar et al.1984; Wong & Smith 1994; Verscharen et al. 2019b).With increasing the drift the instabilities become morespecific to a core-strahl configuration, switching froma kinetic nature near the threshold to a more reactivetype for higher drifts. The growth rate of O-WHFIincreases with the drift, and is in general higher (oreven much higher) than that of the WHFI. The wavefluctuations resembling oblique whistlers are indeed as-sociated with electron beams in the solar wind obser-vations (Breneman et al. 2010; Wilson III et al. 2012;Ram´ırez V´elez et al. 2012). The increase of tempera-ture contrast ( T s /T c ) slightly inhibits the growth ratesof O-WHFI, but stimulates the WHFI extending the in-stability conditions to lower plasma betas. The effect issimilar to that caused by a decrease of the relative drift,leading to a regime more specific to the halo electrons.Theoretically, the electrostatic modes can be desta-bilized already for U s > α s , for instance, conditionsfor a bump-on-tail instability can be satisfied for α s √ α s (when β e = 2). Near thethreshold this instability strongly compete with the O-WHFI, while for slightly higher drifts the growth ratesof EAI become already very large, with peaking valuesat least one order of magnitude higher than those ofthe O-WHFI. For even more energetic beams satisfying U s > ( n e /n s ) / α s ( ≃ . α s ), which are relevant for1the fast outflows in the outer corona (also coronal massejections), the theory predicts an additional EBI. Nearthe threshold of this instability we found four unstablemodes, O-WHFI, LWI, EAI ad EBI, but only the lasttwo have chances to develop, with comparable growthrates much higher than those of the other two modes.These kind of electrostatic instabilities are widely in-voked in space plasma applications, but the resultinghigh amplitude fluctuations may undergo rapid nonlin-ear decays and are ultimately witnessed by the elec-tromagnetic or radio emissions, see Nindos et al. (2008)and refs therein.Summarizing, our results identify three complemen-tary regimes of HFIs, associated to three distinct insta-bilities, the parallel WHFI, the O-WHFI or the EAI,and interlinked by a series of transitory regimes. Foreach transition the theory predicts the interplay or co-existence of at least two distinct instabilities, for in-stance, the interplay of parallel and oblique whistlers forlower drifts, and with increasing the drift a mixing theO-WHFI and EAI, or the limit case where the EAI andEBI can develop concomitantly. These findings stronglysuggest that heat flux regulation may not involve onlyone but several HFIs, concomitantly or successively intime. Conditions for a single, well defined instabilitywith major effects on the suprathermal electrons and,implicitly, the heat flux, may be very limited. WhistlerHFIs are more likely to occur but only for minor drifts (as also reported by recent observations), which mayexplain a modest implication in their regulation, shownalready in quasilinear studies (Shaaban et al. 2019a,b)and numerical simulations (L´opez et al. 2019a).We can finally conclude stating that a realistic plasmaparameterization combined with a selective spectralanalysis can be crucial for understanding the nature andorigin of HFIs and their implication in the regulation ofthe solar wind heat flux. Our theoretical predictions areexpected to stimulate further investigations using fullkinetic simulations, and confirm the existence of thesealternative regimes, not only in the initial linear phaseof HFIs but also during their quasi- or non-linear growthin time, which involves a relaxation of the relative driftand, implicitly, changes to different successive regimesof HFIs corresponding to lower drifts.ACKNOWLEDGMENTSThese results were obtained in the framework of theprojects SCHL 201/35-1 (DFG-German Research Foun-dation), C14/19/089 (C1 project Internal Funds KULeuven), G.0A23.16N (FWO-Vlaanderen), and C 90347(ESA Prodex). R.A.L thanks the support of AFOSRgrant FA9550-19-1-0384. S.M. Shaaban acknowledgessupport by a FWO Postdoctoral Fellowship, grantNo. 12Z6218N. P.S. Moya is grateful for the support ofKU Leuven BOF Network Fellowship NF/19/001, andANID Chile through FONDECyT Grant No. 1191351.REFERENCES Agapitov, O. V., de Wit, T. D., Mozer, F. 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