Shaping the solar wind temperature anisotropy by the interplay of electron and proton instabilities
aa r X i v : . [ a s t r o - ph . S R ] D ec Shaping the solar wind temperature anisotropy by theinterplay of electron and proton instabilities
S. M. Shaaban • M. Lazar • S. Poedts • A. Elhanbaly Abstract
A variety of nonthermal characteristics likekinetic, e.g., temperature, anisotropies and suprather-mal populations (enhancing the high energy tails ofthe velocity distributions) are revealed by the in-situobservations in the solar wind indicating quasistation-ary states of plasma particles out of thermal equi-librium. Large deviations from isotropy generate ki-netic instabilities and growing fluctuating fields whichshould be more efficient than collisions in limiting theanisotropy (below the instability threshold) and explainthe anisotropy limits reported by the observations. Thepresent paper aims to decode the principal instabilitiesdriven by the temperature anisotropy of electrons andprotons in the solar wind, and contrast the instabilitythresholds with the bounds observed at 1 AU for thetemperature anisotropy. The instabilities are character-ized using linear kinetic theory to identify the appropri-ate (fastest) instability in the relaxation of temperatureanisotropies A e,p = T e,p, ⊥ /T e,p, k = 1. The analysis fo-cuses on the electromagnetic instabilities driven by theanisotropic protons ( A p ≶
1) and invokes for the firsttime a dynamical model to capture the interplay withthe anisotropic electrons by correlating the effects ofthese two species of plasma particles, dominant in thesolar wind.
Keywords plasmas - instabilities - solar wind
S. M. ShaabanM. LazarS. PoedtsA. Elhanbaly Centre for Mathematical Plasma Astrophysics, Celestijnenlaan200B, B-3001 Leuven, Belgium. Theoretical Physics Research Group, Physics Department, Fac-ulty of Science, Mansoura University, 35516 Mansoura, Egypt. Institut f¨ur Theoretische Physik, Lehrstuhl IV: Weltraum- undAstrophysik, Ruhr-Universit¨at Bochum, D-44780 Bochum, Ger-many.
Due to a continuous presence of observational mis-sions in space, the solar wind is currently exploited asa natural laboratory for studying the plasma mecha-nisms and effects including kinetic instabilities drivenlocally by the temperature anisotropy of plasma parti-cles. The existence and viability of these mechanismsof instability are confirmed by a recent combined anal-ysis of the plasma particle distributions and the en-hanced wave fluctuations observed in the solar wind,see Gary et al. (2016). Instead, the back effects of thegrowing fluctuating fields scattering the plasma par-ticles and limiting their anisotropy remain controver-sial, especially the role played in the relaxation pro-cess by the cyclotron electromagnetic fluctuations usu-ally dominating the direction parallel to the station-ary magnetic field (Kasper et al. 2003; Hellinger et al.2006; Bale et al. 2009).Driven by an excess of perpendicular temperature T ⊥ > T k cyclotron modes grow faster than other in-stabilities like mirror instability which may develop inthe same conditions. Resonant interactions, i.e., cy-clotron resonance with plasma particles, are thereforeexpected to be effective in the relaxation of temperatureanisotropy in this case. The interest to explain these ef-fects and their consequences in the solar wind has beenboosted after the simulations suggested that the mirrorinstability is not an effective pitch angle scatterer ofprotons (McKean et al. 1992, 1994; Gary et al. 1993.a)and the cyclotron anisotropy instability may be as-sumed the primary mechanism to constrain the protonanisotropy (Gary and Lee 1994.a). Further confirma-tion of this constraint can also be obtained in a straight-forward way, namely, by fitting the instability thresh-olds predicted by the linear theory with the anisotropylimits reported by the observations (see explanatory ar-guments and references in Gary et al. (1994.b)). How-ever, the instability thresholds derived from simplified models assuming plasma particles bi-Maxwellian dis-tributed and minimizing the effects of electrons con-sidering them isotropic, do not provide a good agree-ment with the observations. Thus, thresholds of theelectromagnetic ion (proton) cyclotron (EMIC) insta-bility simply do not align to the limits of the pro-ton temperature anisotropy in the solar wind, but aremarkedly lower than these limits, which instead appearto be better described by the thresholds of the mirror(aperiodic) modes instability (Hellinger et al. 2006). Inthe opposite case, an excess of parallel temperature T ⊥ < T k may ignite two branches of firehose insta-bility, one destabilizing the electromagnetic cyclotronmodes propagating mainly in the parallel direction butwith an opposite polarization, i.e., right-handed (RH)if driven by protons and left-handed (LH) if driven byelectrons, and the other one destabilizing highly obliqueand aperiodic modes. Again, when the firehose thresh-olds are derived with simplified models the temperatureanisotropy in the solar wind is better constrained bythe aperiodic instability (Hellinger et al. 2006), which,however, cannot undergo cyclotron resonant interac-tions with plasma particles. In both these two cases,the anisotropy thresholds derived for the instabilitiesof cyclotron modes lie below the anisotropy limits re-ported by the observations in the solar wind, seemingthat the instability thresholds may be overestimated byusing simplified models for the velocity distributions ofplasma particles and neglecting their interplay. Twodistinct classes of mechanisms may be at work in the so-lar wind generating temperature anisotropies of plasmaparticles. These are either the large scale mechanismslike adiabatic expansion (leading to A = T ⊥ /T k < A > e and p , respectively), expecting to providea direct correlation between their anisotropies measuredin the solar wind, i.e., both species with A e,p >
1, orboth with A e,p <
1. Binary collisions are not efficientenough to reduce the anisotropy and affect this cor-relation of the electron and proton anisotropies, butthe small scale mechanisms like the wave-particle inter-actions, usually conditioned by the presence of differ-ent wave fluctuations may accelerate plasma particlespreferentially, e.g., in direction perpendicular to themagnetic field by the cyclotron resonance (leading to
A >
A < A e ≷ A p ≶
1, can thereforeresult from local mechanisms involving either microin-stabilities or damping of small scale fluctuations.A quantitative analysis with systematic evidencesand estimations of these correlations between the elec-tron and proton anisotropies in space plasmas is notreported yet, at least to our knowledge, but some qual-itative elements can however be extracted. Thus, animplication of the large scale mechanisms in generat-ing temperature anisotropy of plasma particles seemsto be confirmed by the observations, which show a ra-dial evolution of the temperature anisotropy from anexclusive A = T ⊥ /T k > ∼ . A < ∼ for the first time a dynamical model to include the ef-fects of anisotropic electrons. In Sec. 4 we contrast theinstability thresholds with the observations, and per-form a comparative analysis with the previous resultsfrom simplified approaches. Conclusions are presentedin Sec. 5. For a collisionless and homogeneous electron–protonplasma, the electromagnetic modes in a direction par-allel to the stationary magnetic field ( k k B ) decou-ple from the electrostatic oscillations, and their insta-bilities may display maximum growth rates, e.g., cy-clotron instabilities (Kennel and Petschek 1966). Pro-vided by a linear Vlasov-Maxwell dispersion formalism(Krall and Trivelpiece 1973), the dispersion relationsfor these electromagnetic modes read1 + X α = e,p ω p,α ω (cid:20) ωk u α, k Z α,η (cid:0) ξ ± α,η (cid:1) + ( A α − × (cid:8) ξ ± α,η Z α,η (cid:0) ξ ± α,η (cid:1)(cid:9)(cid:3) = c k ω , (1)where ω is the wave-frequency, k is the wave-number, c is the speed of light, ω p,α = 4 πn α e /m α are the plasmafrequencies for protons (subscript α = p ) and electrons(subscript α = e ), A α = T α, ⊥ /T α, k are the tempera-ture anisotropies, ± denote the circular polarizations,right-handed (RH) and left-handed (LH), respectively. Z α,η (cid:0) ξ ± α,η (cid:1) may denote either the plasma dispersionfunction for (bi)-Maxwellian (subscript η = M ) dis-tributed plasmas (Fried and Conte 1961), or the mod-ified dispersion function for Kappa (subscript η = κ )distributed plasmas as derived in Lazar et al. (2008),and u α, k are the corresponding thermal thermal veloci-ties (Lazar et al. 2015). See appendix A for the explicitdefinitions of these quantities.Since the presence of anisotropic ( A e = 1) electronscan change the dispersive properties of the electromag-netic modes, including those destabilized by the protonanisotropy A p = 1 (Kennel and Scarf 1968; Lazar et al.2011; Michno et al. 2014; Shaaban et al. 2015, 2016),here we propose a dynamical model for the inter-play of the electrons and protons by correlating theirtemperature anisotropies A e,p = 1, as well as theirplasma parallel betas β e,p, k . Previous studies carriedout by Kennel and Scarf (1968); Lazar et al. (2011);Michno et al. (2014); Shaaban et al. (2015, 2016) haveassumed constant values of the electron anisotropy A e and plasma beta β e, k , independent of proton proper-ties. On the other hand, our present analysis includes the effects of the suprathermal populations of electrons,which are ubiquitous in the solar wind.In order to proceed and make the analysis moretransparent, we rewrite the linear dispersion relation(1) in terms of normalized quantities˜ k = A p − A p (˜ ω ± ∓ k p β p, k Z p,η ˜ ω ± k p β p, k ! + µ ( A e −
1) + µ A e (˜ ω ∓ µ ) ± µ ˜ k p µβ e, k Z e,η ˜ ω ∓ µ ˜ k p µβ e, k ! (2)where ˜ ω = ω/ Ω p , ˜ k = kc/ω p,p , µ = m p /m e isthe proton/electron mass ratio, β α, k = 8 πn e k B T α, k /B are the parallel plasma betas for protons (subscript α = p ) or electrons (subscript α = e ). Now we as-sume that the electron and proton temperatureanisotropies are correlated A e = A δp (3) with a correlation index δ , eventually indicatedby the observations. The correlation index may beeither positive δ > B , i.e., if A p ≷ A e ≷
1, or nega-tive δ <
0, when electrons and protons have oppositeanisotropies, i.e., if A p ≷ A e ≶
1. In the follow-ing calculations our reference is the classical case withisotropic electrons, i.e., A e = 1, that here is obtainedfor a correlation index δ = 0 . In this section we examine the electromagnetic insta-bilities driven by proton anisotropies A p = 1 underthe influence of anisotropic electrons by means of thecorrelation-index δ . For the electrons the velocity dis-tributions measured in space plasmas may be consider-ably enhanced by the suprathermal populations and wetherefore consider them (bi-)Kappa distributed. Theproton data invoked in our analysis is measured bySWE/WIND (Ogilvie et al. 1995) with velocities cor-responding to a kinetic energy in the range of 150 eVto 8 keV excluding suprathermal populations. In gen-eral, the presence of suprathermal protons is indeed lesssignificant and we can assume the protons to be morethermalized and bi-Maxwellian distributed.We divide our analysis into two distinct classes ofinstabilities according to the proton anisotropy. Firstwe consider the instabilities developed by an excess ofparallel temperature, i.e., A p >
1. Among these, the
Fig. 1
Effects of δ − index = − . , . , . κ − index=2 , , , ∞ with δ = − . firehose instability is the fastest growing mode and isRH circularly polarized when is propagating in direc-tion parallel to the magnetic field. In the opposite sit-uation, when protons exhibit an excess of perpendic-ular temperature, i.e., A p <
1, the fastest developinginstability is that of the electromagnetic ion cyclotron(EMIC) instability propagating in direction parallel tothe magnetic field and with a LH circular polarization. T p, k > T p, ⊥ The effects of the new correlation-index δ on thegrowth rates of the PFHI are displayed in Figure 1 (a).The unstable solutions are computed, for A p = 0 . β e, k = β p, k = 3, and for different values of δ = 1 . . − .
37 (implying different electron anisotropies, re-spectively, A e = 0 . , . , . δ = 1.0, -0.37. The first peak at low wavenum-bers corresponds to the PFHI, while the second peakmay represent either the electron FHI (EFHI), see the Fig. 2
Wave-frequencies of the unstable modes in Fig. 1.The plasma parameters are mentioned in each panel. solid (red) line, when the correlation-index δ = 1, orthe whistler instability (WI), see the dotted (blue) line,when the correlation-index δ = − .
37. Driven by anelectron anisotropy A e < ω p < ω r ≪ | Ω e | ,while the WI is a RH circularly polarized mode desta-bilized by the anisotropic electrons with A e > ω p < ω r < | Ω e | . Results in Fig-ure 1 (a) show that the growth rates of the PFHI areinhibited by anisotropic electrons described by a nega-tive δ − index (implying A e > δ − index (implying A e < δ < A p or higher values of the par-allel plasma beta β p, k to achieve the same growth rate.We should therefore expect that the PFHI thresholdswill move towards higher values of plasma beta β p, k ,and shape better the anisotropy limits observed in thesolar wind (see Sec. 4). For the opposite case, when δ >
0, the instability is stimulated and threshold con- ditions should diminish towards lower values of plasmabeta β p, k .Based on these premises, the effect of the suprather-mal electrons is shown in Figure 1 (b) only for a neg-ative correlation-index δ = − .
3. The growth ratesare computed for A p = 0 . β e, k = β p, k = 2, andfor different values of the power-index κ e = 2 , , , ∞ .The PFHI growth rates are suppressed by increasingthe suprathermal population, i.e., lowering κ , see thesubplot (zoomed) in Figure 1 (b), while the WI growthrates are enhanced. In other words, the inhibiting ef-fect of the anisotropic electrons described by a nega-tive δ < κ e = 2, only the WI develops (blue dottedline) at electron scales. Note that the growth rateof the WI obtained for A p = 0 .
47 cannot ditinguishfrom that obtained with isotropic protons A p = 1 (notshown here). These new results apparently contra-dict those obtained by Lazar et al. (2011) in a studyof the PFHI cumulatively driven by the anisotropicprotons A p < A e > κ − independent tempera-ture that recently was proven inappropriate for such ananalysis (Lazar et al. 2015). More results on the kineticinstabilities using approaches with a κ − dependent tem-perature, as used in the present paper, can be found inLeubner and Schupfer (2000, 2001); Lazar et al. (2015),and Shaaban et al. (2016). For a complete picture, Figure 2 displays thereal frequencies corresponding to the unstablesolutions in Figures 1. When the anisotropies ofprotons and electrons are correlated by a pos-itive δ = 1 , i.e., A e < , the wave-frequenciesin panel (a) confirm a conversion of the RH-polarized PFH modes to the LH-polarized EFHmodes (solid red line) by changing the signin between the PFHI and EFHI peaks. Atthese low frequencies, the LH and Rh branchesare relatively close to each other making pos-sible a conversion, determined in this case bythe interplay of the PFH and EFH instabili-ties. Otherwise, the RH branch (which is desta-bilized at low frequencies by the PFHI) ex-tends smoothly (monotonically increasing dis-persion) to electron scales (dashed and dottedlines), where the anisotropic electrons with ananti-correlated anisotropy A e > , as given bya negative δ = − . , may drive the instability of whistler modes (WI with dotted line). Inpanel (b) we show that wave-frequencies corre-sponding to the growth rates in Figures 1 (b)are not markedly influenced by the presence ofsuprathermal electrons.3.2 Solar wind protons with T p, ⊥ > T p, k The electromagnetic ion cyclotron (EMIC) instabilityis triggered by the anisotropic protons with A p > T p, ⊥ > T p, k ). This instability develops first in di-rection parallel to the magnetic field, where the EMICmodes are LH circularly polarized. In Figure 3 (a) wedisplay the growth rates of this instability driven by atemperature anisotropy A p = 1 . β e, k = β p, k = 6,and different values of the δ = 1 . , . , − .
0, with men-tion that first value δ = 1 is carefully chosen to pro-duce the same anisotropy of the electrons A e = 1 . δ = − . A e = 0 . δ >
0, implying A e >
1, the WI is absent, as itbelongs to another branch with opposite (RH) polar-ization and much higher frequency ( ω r ≫ Ω p ). TheEMIC instability is inhibited by increasing δ from neg-ative values, implying protons and electrons with anti-correlated anisotropies ( A p > A e <
1) to positive val-ues when these anisotropies are correlated ( A p,e > δ = 0 .
8. Growth-rates are plotted for a protonanisotropy A p = 2 .
7, the same plasma beta parame-ter for protons and electrons β e, k = β p, k = 0 .
1, anddifferent values κ = 1 . , , , ∞ . In the vicinity of theinstability threshold level γ m / Ω p = 10 − the instabil-ity inhibits by increasing the suprathermal populationof electrons. Figure 4 presents the real frequencies corre-sponding to the unstable solutions in Figure 3.Unlike the growth rates, the wave-frequenciesof the EMIC modes are enhanced by increasingthe correlation index δ , see panels (a). How-ever, the anisotropy of protons is modest (notvery large) and after the EMIC saturation thewave-frequency changes the sign, converting tothe RH branch under the influence of electrons,which are expected to manifest important ki-netic effects in this case, due to their high β e, k = 6 . The same high value of β e, k may stim-ulate a LH EFHI to develop (at larger wave-numbers) when the electron anisotropy is anti-correlated, i.e., δ = − (red solid line), and Fig. 3
Effect of δ = 1 , . , − κ -index=1 . , , , ∞ with δ = 0 . this is confirmed by the wave-frequency whichbecomes again positive, i.e., LH-polarized. Inpanel (b) we show that wave-frequency of theEMIC modes are slightly enhanced by the pres-ence of suprathermal electrons, but in this casethe effects of electrons are minimal due to theirsmall β e, k = 0 . . The results in sections 3.1 and 3.2 already suggestthat instabilities thresholds can be enhanced, namely,by a negative correlation-index δ < δ >
In these section we will try to identify the physi-cal mechanisms behind these effects. Basic explana-tions for the electron anisotropy effects on the PFHIare offered by Kennel and Scarf (1968) and later byMichno et al. (2014), namely, that for isotropic elec-trons the protons are weakly resonant, while for
Fig. 4
Wave-frequencies of the unstable modes in Fig. 3.The plasma parameters are mentioned in each panel. anisotropic electrons with A e > | ξ + p | = | (˜ ω + 1) / (˜ k p β p, k ) | givenby the arguments of plasma dispersion function in Ap-pendix A, eq. (7). We call these quantities resonantfactors, and Figures 5 (a) and (b) display them for pro-tons and electrons, respectively, for the same plasmaparameters as in Figure 1 (a). The zoomed plot in Fig-ure 5 (a) provides details on the proton resonant factor | ξ + p | for different values of δ -index at wavenumbers ˜ k corresponding to the peaks of the PFHI growth ratesas ( δ, ˜ k, | ξ + p | ) = ( − . , . , . . , . , . . , . , . A e >
1, the resonant fac-tor increases ( | ξ + p | = 2 . >
1) and the protons becomeless resonant with the resulting PFHI. With increasingthe wave-number the resonant factors | ξ + p | drops downto a minimum value, and then, for large enough ˜ k > δ = 0 . , − . Fig. 5
The resonant conditions for protons | ξ + p | (top) andelectrons | ξ + e | (bottom) for the same plasma parameters inFigure (1-a). son can be made to a simplified approach with isotropicelectrons, i.e., δ = 0 .
0. For a negative correlation-index δ = − .
37 the protons become less resonant ( | ξ + p | > δ = 1 .
0, the EFHI arisesat (slightly) higher wavenumbers, Figure 3 (a), withan opposite (LH) polarization, i.e., the wave-frequencychanges the sign (Michno et al. 2014). The turningpoint (singularity given by the cold-plasma resonance ℜ ( ξ + p ) = 0), where the resonant factor takes a minimumvalue | ξ + p | min = | ˜ γ/ (˜ k p β p, k ) | ≪
1, is followed by theresonance of protons with the EFHI, i.e., | ξ + p | ∼
1, andwith increasing the wave-number the resonant factorincreases again.Figure 5 (b) shows the electron resonant factor | ξ + e | at wave-numbers corresponding to the peaks of theEFHI and WI as ( δ, ˜ k, | ξ + e | ) =(1.0,6.85,3.6),(-0.37,18.5, Fig. 6
The resonant conditions for protons | ξ − p | (top) andelectrons | ξ − e | (bottom) for the same plasma parameters inFigure (3-b). | ξ + e | >
1, and strongly non-resonant nearthe peak of the PFHI where | ξ + e | ≫
1. However, theelectrons become resonant with the WI when | ξ + e | & A e > | ξ − p | ) andelectrons ( | ξ − e | ) are plotted in Figure 6, panels (a) and(b), respectively. These factors are computed for thesame parameters as in Figure 3 (b) to show the influenceof suprathermal electrons quantified by the power-index κ = 1 . , , , ∞ . For protons, four values of the resonantfactor are explicitly given corresponding to the peaks ofthe EMIC growth rates as ( κ e , ˜ k, | ξ − p | ) = (1 . , . . , . . , . . , . resonant ( | ξ − p | = 1 . κ e = 1 . κ e = 3 is near thethreshold level γ m / Ω p = 10 − , and the correspond-ing value of the proton resonant factor | ξ − p | = 1 . | ξ − p | = | (˜ ω − / (˜ k p β p, k ) | decreases with increasingthe wave-number until is drops abruptly down reach-ing a minimum value | ξ − p | min = | ˜ γ/ (˜ k p β p, k ) | ≪ ℜ ( ξ − p ) = 0). Beyond this point the real frequencies aresaturated and the modes are strongly damped with anincreasing damping rate − ˜ γ , making the proton reso-nant factor (as absolute value) to increase.Figure 6 (b) shows that the electrons are highly non-resonant near the peaks of the EMIC instability, with | ξ − e | ≫
1, and remain non-resonant | ξ − e | ≃
10, even forthe wave-numbers corresponding to the electron scales.It becomes also clear that the EMIC branch cannot con-nect to the high-frequency whistler (electron cyclotron)modes, which have a different (RH) polarization andwhich is resonantly destabilized by the (anisotropic)electrons, e.g., the whistler instability (WI) discussedabove, also known as the electron-cyclotron instability.
The anisotropy thresholds may provide a straightfor-ward confirmation for the constraining role playedby the kinetic instability in collision-poor plasmasfrom space, namely, when these thresholds fit thelimits of the temperature anisotropy reported by theobservations. Derived for different levels of maxi-mum growth-rates γ m / Ω p = 10 − , − , − andfor an extended range of the plasma beta parameter0 . β p, k A p ) derived for γ m = 10 − Ω p (a sufficiently low level also adopted in similar investi-gations), and represented as an inverse correlation lawof the proton plasma beta β p, k (Hellinger et al. 2006) A p = 1 + a (cid:0) β p, k − β (cid:1) b . (4)For the instability thresholds derived in Figures 7-9 fit-ting parameters a , b , and β are tabulated in tables 1and 2 in Appendix B. The standard inverse correlationintroduced by (Gary and Lee 1994.a) may be recovered (a) Fig. 7
The influence of the correlation-index δ = 0 . − . , − . κ = 3 , , ∞ with δ = − . γ m = 10 − Ω p ).Thresholds are compared with the proton (core) anisotropyat 1 AU in the solar wind and the plasma parameters areexplicitly given in each panel for β = 0. Thresholds are compared with the observa-tions in the slow solar wind ( v sw km/s ), i.e., pro-tons measured by SWE (Ogilvie et al. 1995) and MFI(Lepping et al. 1995) on the WIND spacecraft at 1 AU(Kasper et al. 2002; Hellinger et al. 2006; Michno et al.2014).The PFHI thresholds are displayed in Figure 7, inpanel (a) for different correlation-indices δ = 0 . , − . − . A e = A δp ),and in panel (b) for different power-indices κ e = 3 , , ∞ and same δ = − .
5. Simplified approaches usuallyadopt δ = 0 . A e = 1 (blue dotted line in Figure 5 (a)). The in-hibiting effect obtained for a negative correlation index δ < A e > δ = − . , − . β p, k . By in-creasing this anti-correlation between protons and elec-trons, i.e., decreasing negative values of δ , the instabil- Fig. 8
Thresholds conditions ( γ m = 10 − Ω p ) forthe EMIC instability compared with the proton (core)anisotropy at 1 AU in the solar wind. The plasma parame-ters are explicitly given in each panel. ity thresholds are enhanced and can markedly improvetheir fit with the limits of the temperature anisotropyobserved in the solar wind. In the second panel (b) wecan observe that these thresholds are further boostedby the suprathermal electrons: for lower values of κ e the PFHI thresholds are moved to higher plasma betaexceeding the limits observed for the proton anisotropyconfirming the results in Figure 1 (b).Figure 8 shows the influence of the anisotropic elec-trons with A e > A e = 1, dotted blue line). For β e, k = β p, k the instabil-ity thresholds do not change much to improve fitting tothe observations, even for δ = 1 (i.e., A p = A e ), see theshort-dashed (orange) line. However, Shaaban et al.(2015, 2016) have recently shown that inhibiting ef-fect induced by the anisotropic electrons is stimu-lated by increasing β e, k . Thus, for an average value β e, k = 1 indicated by the observations (ˇStver´ak et al.2008), also commonly invoked in similar investigations(Hellinger et al. 2006; Matteini et al. 2013), the EMICthreshold can increase considerably constraining moreobservational data, see dashed (green) line. On theother hand, the fit with the observations is considerablyimproved in the presence of suprathermal electrons, i.e.,the instability thresholds enhance with decreasing thepower-index κ e confirming the results in Figure 3. Inthis case, the EMIC thresholds are plotted for A e = A p (i.e., δ = 1), β e, k = 1, κ e = 2 (long-dashed brown line),and κ e = 1 . C e = 2), andHellinger et al. (2006) for the aperiodic ( ω r = 0) insta-bilities, namely, the aperiodic proton firehose (APFH) (a) (b) Fig. 9
Thresholds of the aperiodic ( ω r = 0) instabilities,i.e., mirror and the aperiodic PFH (APFH), as provided byHellinger et al. 2006 , and the instabilities of our periodicmodes, i.e., EMIC and PFH, are compared with the protoncore data at 1 AU in the solar wind. and mirror instabilities. Most interesting appear tobe the new instability thresholds obtained in panel (a)for the EMIC instability in the presence of suprather-mal electrons and with a direct (positive) correlation δ = 1 between the proton and electron anisotropies( A e = A p ). These isocontours show a good alignmentto the observations, similar to the mirror instability. The presence of suprathermal electrons, whichare ubiquitous in the solar wind, is critical andmay considerably enhance the role played bythe EMIC instability in the low-beta regimes(e.g., for κ e = 1 . ). Moreover, these results areobtained for a common (average) value of theelectron beta parameter, namely, β e, k = 1 in-dicated by the observational data cumulation(ˇStver´ak et al. 2008), and for a direct corre-lation of the electron and proton anisotropiesgiven by a positive δ = 1 > . As alreadyexplained in the Introduction we do not dis-pose of systematic observational analyses to con-firm a direct correlation of electron and pro- ton anisotropies, but this condition may in gen-eral be ensured by the mechanisms always atwork in space plasmas, e.g., solar wind expan-sion, magnetic focusing, and it seems there-fore more plausible than an anti-correlation ofthe anisotropies. Based on these arguments,the fits obtained in panel (a) can be consid-ered robust enough to support the implicationof the EMIC instability in constraining the pro-ton temperature anisotropy and explain the ob-servations. In panel (b) we compare the new thresh-olds obtained for the PFHI with those derived byMichno et al. (2014) or Hellinger et al. (2006) for theAPFH instability, and can conclude that anisotropicelectrons may influence the PFHI thresholds and deter-mine them to align better to the observations, e.g., thelong-dashed brown line obtained for δ = − .
8. How-ever, in this case the best-aligned thresholds are con-ditioned by an anti-correlation between the proton andelectron anisotropies as required by a negative δ < In a collisionless electron-proton plasma the tempera-ture anisotropy, and, implicitly, the distribution func-tion should be regulated by the resulting instabili-ties and electromagnetic fluctuations through the wave-particle interaction. For anisotropic protons, the the-ory and simulations predict a dominance of the cy-clotron modes driven unstable by the PFH and EMICinstabilities, which may develop fast enough leading toan important pitch angle scattering of protons towardisotropy (Kennel and Scarf 1968; Gary 1992). How-ever, predictions made by the simplified approaches forthe anisotropy thresholds of these instabilities appearto be overestimated by comparison to the temperatureanisotropy measured in-situ in the solar wind. Instead,these observations seem to be better constrained by theaperiodic instabilities, e.g., APFHI and mirror instabil-ity (Hellinger et al. 2006).In this paper we have aimed to resolve this paradigmand provide more realistic predictions from an advancedmodeling that accounts for the interplay of protons andelectrons, and the presence of suprathermal electrons.New regimes are thus found for the PFH and EMICinstabilities, which are mainly controlled by the cumu-lative effects of protons and electrons by correlating ei-ther their anisotropies, e.g., A e = A δp , via the parameter δ , or/and their plasma beta parameters, β e = β p . Sim-ilarities and differences among these regimes are high-lighted in Section 3, always comparing with the ideal-ized approaches which consider electrons isotropic, i.e., δ = 0. We have studied the effects of these correlationfactors on the growth-rates and provided physical ex-planations by studying the resonant conditions for boththe protons and electrons.A comparative analysis of these new regimes enabledus to identify conditions that may inhibit the instabil-ities and make their thresholds to adjust better to theobservations. Thus, the PFHI is driven by a protonanisotropy A p <
1, and can be inhibited by anisotropicelectrons with anti-correlated anisotropies A e = A δp > δ <
0. In the opposite situa-tion when protons exhibit a temperature anisotropy A p >
1, the EMIC instability is inhibited only for apositive δ > A e = A δp >
1. Moreover, in boththese two cases the inhibiting effect is boosted by thesuprathermal electrons. The explanation is providedby the resonant factors, which indicate that protonsbecome less resonant inhibiting the instability and lead-ing to higher anisotropy thresholds. These results con-firm the expectations from the previous studies carriedout by Kennel and Scarf (1968); Lazar et al. (2011);Michno et al. (2014); Shaaban et al. (2015, 2016).To provide a complete picture, in Section 4 we havestudied the instability thresholds, and recovered theinhibiting effects on the PFH and EMIC instabilitiesfor an extended range of the plasma beta parameter0 . < β p, k < β p, k with decreasing the δ − index and with increasing the suprathermal electronpopulation. For the EMIC instability, the thresholdconditions in the low-beta regimes are only weakly af-fected by the anisotropic electrons, but can be signifi-cantly lowered by increasing the electron plasma beta β e, k = 1 and the presence of suprathermal electrons.To conclude, we have identified the conditions for theinstability thresholds to align and shape the limits ofthe temperature anisotropy reported by the observa-tions. These agreements with the observations can beeven better than those obtained before for the aperi-odic instabilities, but are highly conditioned by the elec-tron properties, i.e., the anisotropy (correlated or anti-correlated with the proton anisotropy), plasma beta pa-rameter, and their suprathermal populations.Suprathermal electrons are ubiquitous in space plas-mas but the main question arising now concerns the ex-istence of the plasma states with protons and electronshaving anisotropies either direct correlated and roughlydescribed by a positive δ >
0, or anti-correlated by anegative δ <
0. As already discussed in the Introduc-tion, we do not dispose of any systematic evidences and estimations of these correlations from the observations,but there are some qualitative indications which appearto be more favorable to direct correlated anisotropies ofelectrons and protons, i.e., A e,p > A e,p < Acknowledgements
The authors acknowledge theuse of WIND SWE (Ogilvie et al. 1995) ion data, andWIND MFI (Lepping et al. 1995) magnetic field datafrom the SPDF CDAWeb service: http://cdaweb.gsfc.nasa.gov/. The authors acknowledge support from theKatholieke Universiteit Leuven. These results were ob-tained in the framework of the projects GOA/2015-014 (KU Leuven), G.0A23.16N (FWO-Vlaanderen),and C 90347 (ESA Prodex). The research leading tothese results has also received funding from IAP P7/08CHARM (Belspo). S.M. Shaaban would like to thankthe Egyptian Ministry of Higher Education for support-ing his research activities.
Appendix A: Distributions and dispersionfunctions
For a plasma of electrons and protons with bi-Maxwellian velocity distribution functions (VDFs) F α,M (cid:0) v k , v ⊥ (cid:1) = 1 π / u α, ⊥ u α, k exp − v k u α, k − v ⊥ u α, ⊥ ! , (5)where thermal velocities u α, k , ⊥ are defined by the com-ponents of the anisotropic temperature T Mα, k = mk B Z d v v k F α ( v k , v ⊥ ) = mu α, k k B (6) T Mα, ⊥ = m k B Z d v v ⊥ F α ( v k , v ⊥ ) = mu α, ⊥ k B , (7)the plasma dispersion function in Eq.(1) takes the stan-dard form (Fried and Conte 1961) Z α,M (cid:16) ξ ± α,M (cid:17) = 1 π / Z ∞−∞ exp (cid:0) − x (cid:1) x − ξ ± α,M dt, ℑ (cid:16) ξ ± α,M (cid:17) > ξ ± α,M = ( ω ± Ω α ) / (cid:0) ku α, k , (cid:1) To include suprathermal population, the electronscan be described by a bi-Kappa VDF (Summers and Thorne1991) F e,κ = 1 π / u e, ⊥ u e, k Γ ( κ e + 1)Γ ( κ e − / " v k κ e u e, k + v ⊥ κ e u e, ⊥ − κ e − (9)which is normalized to unity R d vF e,κ = 1, and is writ-ten in terms of thermal velocities u e, k , ⊥ defined by thecomponents of the effective temperature (for a power-index κ e > / T Ke, k = 2 κ e κ e − m e u e, k k B , T Ke, ⊥ = 2 κ e κ e − m e u e, ⊥ k B . (10)Suprathermals enhance the electron temperature, andimplicitly the plasma beta parameter (Leubner and Schupfer2000, 2001; Lazar et al. 2015) T Ke, k , ⊥ = 2 κ e κ e − T Me, k , ⊥ > T Me, k , ⊥ ,β Ke, k , ⊥ = 2 κ e κ e − β e, k , ⊥ > β e, k , ⊥ , (11)and for the modified Kappa dispersion function (8) weuse in Eq.(1) the form (Lazar et al. 2008) Z e,κ (cid:0) ξ ± e,κ (cid:1) = 1 π / κ / e Γ ( κ e )Γ ( κ e − / × Z ∞−∞ (cid:0) x /κ e (cid:1) − κ e x − ξ ± e,κ dx, ℑ (cid:0) ξ ± e,κ (cid:1) > , (12)of argument ξ ± e,κ = ( ω ± Ω e ) / (cid:0) ku e, k , (cid:1) . Appendix B: Fitting parameters for Eq.(4) Table 1
Fitting parameters for PFH thresholds in Figure 7and 9 (b). κ δ a b β ∞ − .
453 0.467 0.652 ∞ − . − .
733 0.607 0.293 ∞ − . − .
772 0.543 0.7092 − . − .
849 0.865 0.1126 − . − .
990 0.707 0.0
Table 2
Fitting parameters for EMIC thresholds in Fig-ure 8 and 9 (a). κ δ β e, k a b ∞ β p, k ∞ β p, k ∞ References