Electromagnetic instabilities of low-beta alpha/proton beams in space plasmas
EElectromagnetic instabilities of low-beta alpha/protonbeams in space plasmas
M.A. Rehman • S.M. Shaaban , (cid:70) • P.H.Yoon , , • M. Lazar , • S. Poedts , Abstract
Relative drifts between different species orparticle populations are characteristic to solar plasmaoutflows, e.g., in the fast streams of the solar winds,coronal mass ejections and interplanetary shocks. Thispaper characterizes the dispersion and stability of thelow-beta alpha/proton drifts in the absence of any in-trinsic thermal anisotropies, which are usually invokedin order to stimulate various instabilities. The disper-sion relations derived here describe the full spectrumof instabilities and their variations with the angle ofpropagation and plasma parameters. The results un-veil a potential competition between instabilities of theelectromagnetic proton cyclotron and alpha cyclotronmodes. For conditions specific to a low-beta solar wind,e.g., at low heliocentric distances in the outer corona,the instability operates on the alpha cyclotron branch.The growth rates of the alpha cyclotron mode are sys-tematically stimulated by the (parallel) plasma betaand/or the alpha-proton temperature ratio. One can
M.A. Rehman • S.M. Shaaban , (cid:70) • P.H. Yoon , , • M. Lazar , • S. Poedts , Department of Physics, GC University, Kachery Road, Lahore54000, Pakistan Centre for Mathematical Plasma Astrophysics, Celestijnenlaan200B, B-3001 Leuven, Belgium. ∗ E-mail: [email protected] Theoretical Physics Research Group, Physics Department, Fac-ulty of Science, Mansoura University, 35516, Egypt. Institute for Physical Science and Technology, University ofMaryland, College Park, MD 20742, USA. Korea Astronomy and Space Science Institute, Daejeon 34055,Korea;. School of Space Research, Kyung Hee University, Yongin,Gyeonggi 17104, Korea. Institut f¨ur Theoretische Physik, Lehrstuhl IV: Weltraum- undAstrophysik, Ruhr-Universit¨at Bochum, D-44780 Bochum, Ger-many. Institute of Physics, University of Maria Curie-Sklodowska, 20-400 Lublin, Poland. therefore expect that this instability develops even inthe absence of temperature anisotropies, with poten-tial to contribute to a self-consistent regulation of theobserved drift of alpha particles.
Plasma outflows released by the Sun in interplanetaryspace consist mainly of electrons, protons and minorions, among which the alpha particles are dominantwith an average abundance of about 5% of the totalnumber density of ions. Owing to their mass density,typically 20% of the total ion mass density, alpha-ionsmay have important implications in the solar wind dy-namics (Robbins et al. 1970; Marsch 2006; Kasper et al.2007; Maruca et al. 2012; Maneva et al. 2014). The in-situ measurements in the fast wind and regions not toodistant from the Sun, i.e., (cid:46) a r X i v : . [ phy s i c s . s p ace - ph ] J un the Alfv´en/ion cyclotron waves and the fast magne-tosonic/whistler modes (Revathy 1978; Gary et al.2000b; Li and Habbal 1999). In turn, these fluctua-tions regulate the ion VDFs through the same wave-particle interactions, which determine a diffusion in ve-locity space and a thermalisation and relaxation of thealpha beam (Marsch and Livi 1987; Gary et al. 2000a).The observations confirm that the alphas are heated inperpendicular direction and cool more slowly than whatwould be expected from adiabatic expansion (Reisen-feld et al. 2001; Stansby et al. 2019).The investigations of alpha-ion beams have initiallyconcluded that high plasma beta conditions, e.g., β (cid:62) (cid:62) T ⊥ /T (cid:107) < (cid:107) , ⊥ being directions withrespect to the magnetic field), while isotropic beams,with, e.g., T α, ⊥ /T α, (cid:107) (cid:39)
1, may drive another instabilityof the parallel-propagating Alfv´en/ion-cyclotron waves(Verscharen et al. 2013). The ion-beam instabilitiesare also highly dependent on the plasma beta, whichdecreases towards the Sun and in the outer corona( ∼ . β p,α = 8 πn p,α T p,α /B < p ) or alpha particles (sub-script α ) (Marsch et al. 1982). Here, n p,α denoteproton/alpha number densities, T p,α their respectivetemperatures (in the unit of energy), and B denotesthe ambient magnetic field intensity. The low-beta al-pha/proton beams appear to be more susceptible toAlfv´enic instabilities, but again, it is not clear if theseinstabilities are driven by the alpha/proton drift or bythe anisotropic temperature T ⊥ /T (cid:107) >
1, as such a con-dition is always assumed for protons or alpha particlesin the literature (Li and Habbal 1999; Gomberoff andValdivia 2003). In the outer corona an excess of temper-ature (kinetic energy) in perpendicular direction maybe associated only to the proton core, while suprather-mal populations including proton halo or beams of pro-tons and alpha particles rather show an opposite, rela-tively small anisotropy T ⊥ /T (cid:107) (cid:46) T ⊥ /T (cid:107) =1. The dispersion and stability properties of theplasma system are derived on the basis of a general ki-netic approach, which cover the full spectrum of wave-frequencies, wave-numbers and angles of propagation with respect to the magnetic field. The organizationof our paper is the following: In section 2, we de-rive the general dispersion relation for the electromag-netic modes propagating at an arbitrary angle. Growthrates of the unstable solutions are obtained for standardrepresentations of the particle velocity distribution asdrifting-Maxwellians in section 3. The unstable solu-tions are discussed by numerical means in Section 4,providing also an analysis of their variation with themain plasma parameters. Section 5 summarizes ourpresent results. In a collision-poor plasma the general linear dispersionrelation for electromagnetic modes propagating at anarbitrary angle θ with respect to the uniform back-ground magnetic field ( B = B ˆ z ) is given by0 = det (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) ij ( k , ω ) − c k ω (cid:18) δ ij − k i k j k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (1)with a dielectric tensor defined by (cid:15) ij = δ ij + (cid:88) a ω pa ω (cid:90) d v (cid:20) v (cid:107) (cid:18) ∂∂v (cid:107) − v (cid:107) v ⊥ ∂∂v (cid:107) (cid:19) b j f a + ∞ (cid:88) n = −∞ V i V ∗ j ω − n Ω a − k (cid:107) v (cid:107) × (cid:18) n Ω a v ⊥ ∂∂v ⊥ + k (cid:107) ∂∂v (cid:107) (cid:19) f a (cid:21) , (2)where ω is the wave frequency, b i = B /B = ˆ z isthe unit vector along the direction of ambient mag-netic field vector, f a ( v ⊥ , v (cid:107) ) is the velocity distri-bution function for particle species labeled a ( a = e, p, α , etc., for electrons, protons, alpha particles, etc.), ω pa = (cid:112) πn a e /m a and Ω a = e a B /m a are theplasma and cyclotron frequencies for species a , and V i = (cid:18) v ⊥ nJ n ( b ) b , − iv ⊥ J (cid:48) n ( b ) , v (cid:107) J n ( b ) (cid:19) i ,b = k ⊥ v ⊥ Ω a . (3)Here, J n is a Bessel function of the first kind, of order n ,with argument b . In the above, e , n a , m a , and c standfor unit electric charge, ambient density for plasma par-ticle species a , their mass, and the speed of light invacuo , respectively.In the present analysis we adopt the cold plasmaapproximation for the real frequency, combined withwarm plasma growth/damping rate expression in the context of weak growth/damping formula, or equiva-lent, the dissipative instability formalism. The weakgrowth or dissipative instability theory is also knownas the weak kinetic instability theory, as opposed tothe reactive or fluid instability theory. We take thegeneral cold plasma dielectric tensor elements given incomponent form by (cid:15) xx = 1 − ω pe ω − Ω e − (cid:88) a ω pa ω − Ω a = (cid:15) yy ,(cid:15) xy = − i Ω e ω ω pe ω − Ω e + i (cid:88) a Ω a ω ω pa ω − Ω a = − (cid:15) yx ,(cid:15) xz = (cid:15) zx = 0 = (cid:15) yz = (cid:15) zy ,(cid:15) zz = 1 − ω pe ω − (cid:88) a ω pa ω , (4)and further approximate the situation by consideringtwo species ions, namely, protons and alpha particles,and ignore the displacement current, that is, the unityon the right-hand side of Eq. (4), which is valid for lowfrequency waves. We also assume ω (cid:28) Ω e , in order tosimplify the dielectric tensor elements, (cid:15) xx ≈ − ω pp ω − Ω p − ω pα ω − Ω α ,(cid:15) xy = i ω pp ω Ω p (cid:32) n n p + Ω p ω − Ω p (cid:33) + i Ω α ω ω pα ω − Ω α ,(cid:15) zz = − m p m e n n p ω pp ω , (5)where Ω p = eB / ( m p c ), Ω α = Ω p / ω pp = 4 πn p e /m p , ω pe = m p n ω pp / ( m e n p ), and ω pα = ( n α /n p ) ω pp . Notethat in (cid:15) zz the other ion terms can be neglected com-pared to m p /m e (cid:29) q = sx (1 + µ ) x ∓ σ (cid:112) (1 + µ ) x + stµ , (6)where q = ckω pp , x = ω Ω p , µ = cos θ, δ = n α n p , (7)are dimensionless wave number, frequency, cosine of thewave propagation angle, and the alpha-to-proton den- Fig. 1
Dispersion relation plotted as q = c k /ω pp versus x = ω/ Ω p curves for three different values of θ correspondingto 1 ◦ (solid line), 60 ◦ (dash-dotted), and θ = 80 ◦ (dotted). sity ration, and other quantities are defined by s = 2( xQ − + δP − Q + )( xQ + + δP + Q − ) D ,σ = | D | D , D = Q + Q − + δP + P − ,t = 2 P + P − Q + Q − D ,P ± = x ± , Q ± = x ± . (8)Figure 1 plots q versus x for three values of θ cor-responding to 1 ◦ , 60 ◦ , and 80 ◦ . The dispersion curvesshow that they are composed of three distinct branches.The first branch (topmost, color coded in red) denotesthe low frequency mode that begins as Alfv´en wave forlow frequency ( ω (cid:28) Ω p ), but gradually turns into theresonant mode (where k → ∞ ) at the alpha cyclotronfrequency, ω = Ω α = Ω p /
2. For quasi parallel prop-agation angle ( θ → ω < Ω p ) starts off in low frequency regime( ω (cid:28) Ω p ) as fast/magnetosonic mode, and this modeis right-hand circularly polarized for quasi parallel angleof propagation. For nonzero θ , this mode switches overto the proton cyclotron mode, which becomes resonant( k → ∞ ) at the proton cyclotron frequency ( ω = Ω p ).The switchover takes place in the vicinity of the alphacyclotron frequency. A third brach that begins as thealpha cyclotron mode turns into the fast/magnetosonicmode above Ω α and beyond. The third branch is plot-ted with red color. In general, modes designated withred color corresponds to the lower sign in the dispersionrelation (6), while the blue curves belong to the uppersign. In the growth rate calculation to be discussed subse-quently, we will denote instabilities operative on thetopmost branch as the “alpha cyclotron” instability,while instabilities operative on the middle, blue-coloredmode will be designated as the “proton cyclotron” in-stability. It turns out that the third, bottommost curve– that is, the fast/magnetosonic mode – remains stable.
Assuming weak growth/damping rate ( | γ | (cid:28) ω ), andfollowing the standard method of derivation (Mel-rose 1986) we obtain an explicit expression for thegrowth/damping rate, γ = (cid:88) a πω pa (cid:104) − | e ( k ) · ˆ k | (cid:105) [ ∂ ( ω N ) /∂ω ] × (cid:90) d v ∞ (cid:88) n = −∞ | e ( k ) · V | δ ( ω − n Ω a − k (cid:107) v (cid:107) ) × (cid:18) n Ω a v ⊥ ∂∂v ⊥ + k (cid:107) ∂∂v (cid:107) (cid:19) f a (cid:0) v ⊥ , v (cid:107) (cid:1) , (9)where N = ck/ω is the index of refraction, V is de-fined in Eq. (3), and e represents the unit electricfield vector, which is discussed in Appendix A. Mak-ing use of the dispersion relation (6) one may alsocompute ∂ ( ω N ) /∂ω explicitly, which is given in Ap-pendix B. In Eq. (9) f a = f a ( v ⊥ , v (cid:107) ) represents thevelocity distribution function for ion species labeled a .The following is the resulting explicit expression forthe growth/damping rate after taking into account thedispersive wave properties associated with the low fre-quency modes: γ Ω p = (cid:88) a = p,α n a n p m p m a π (1 + M x µ ) R (cid:90) ∞ dv ⊥ v ⊥ × ∞ (cid:88) n = −∞ (cid:12)(cid:12)(cid:12)(cid:12) M x n J n b + ( J (cid:48) n ) − M x nJ n J (cid:48) n b (cid:12)(cid:12)(cid:12)(cid:12) × (cid:18) n Ω a kv ⊥ µ ∂f a ( v ⊥ , v r ) ∂v ⊥ + ∂f a ( v ⊥ , v r ) ∂v r (cid:19) ,v r = ω − n Ω a kµ , (10)where the argument of the Bessel function b = k ⊥ v ⊥ / Ω a applies for proton ( a = p ) and alpha particles ( a = α ),and M and R are defined in Eqs. (A5) and (B2), re-spectively.For drifting Maxwellian distributed plasmas we re-place the arbitrary distribution function f a in the growth rate expression (10) by f a ( v ⊥ , v (cid:107) ) = 1 π / v T a exp (cid:18) − v ⊥ v T a − ( v (cid:107) − V a ) v T a (cid:19) , (11)where v T a = (2 T a /m a ) / stands for thermal speed,and V a represents the average drift speed. After somestraightforward mathematical manipulations it can beshown that the growth rate expression reduces to γ Ω p = − (cid:88) a n a n p m p m a π / (1 + M µ ) R ∞ (cid:88) n = −∞ × (cid:18) n (1 + M x )2 [ I n − ( λ a ) − I n +1 ( λ a )] e − λ a +( M nx + λ a ) [2 I n ( λ a ) − I n − ( λ a ) − I n +1 ( λ a )] e − λ a (cid:19) ξ a exp (cid:2) − ( ζ an ) (cid:3) , (12)where I n is the modified Bessel function of the first kindof order n , and ξ a = ω − kV a µkv T a µ ,ζ an = ω − n Ω a − kV a µkv T a µ ,λ a = k v T a (1 − µ )2Ω a . (13) The growth rate of low frequency modes is a function offrequency ω and angle of propagation θ . It also dependsimplicitly on alpha-proton number density ratio n α /n p ,alpha particle drift velocity V α (we assume zero drift forthe protons, V p = 0), plasma beta parameters β p and β α , where β p = 8 πn p T p /B and β α = 8 πn α T α /B , re-spectively. In the fast solar wind alpha particles possessan average density of 5% of the total number densityand are drifting with respect to the protons with a typ-ical speed on the order of local Alfv´en speed, V α = v A (Marsch et al. 1982; Reisenfeld et al. 2001). Conse-quently, in the present analysis, for alpha-proton rela-tive number density we consider n α /n p = 0 .
05, and foralpha drift velocity, we fix the value at V α = v A . Forlow corona the beta values are relatively low. We thusconsider low value of β α = 0 .
01 and a slightly higherproton beta of β p = 2 β α , as an example. Note thatwhile β α and β p are comparable, this actually repre-sent much high alpha particle temperature, since thealpha particle number density is much lower. This isconsistent with observation. Fig. 2
Dispersion surfaces corresponding to alpha cyclotron modes [left], and proton cyclotron mode, which includes themagnetosonic mode branch [right]. The color-coded growth rates for each mode is shown, for n α /n p = 0 . T α /T p = 2,and β α = 0 .
01. Note that the alpha cyclotron instability is an order of magnitude higher than that for the proton cyclotroninstability.
Figure 2 displays on the left, the dispersion sur-face or manifold, corresponding to Alfv´en-alpha cy-clotron mode, while the right-hand panel plots the dis-persion surface, depicting the proton cyclotron branch(which also includes fast/magnetosonic mode brachin the lower frequency regime). Vertical axis repre-sents normalized wave number, q = ck/ω pp , while thetwo horizontal axes denote normalized (real) frequency x = ω/ Ω p , and wave propagation angle θ , respectively.We indicate the region of wave growth on each surfaceas well as the magnitude by color scheme. As indicatedby color bars, however, it is apparent that the alphacyclotron instability growth rate is almost an order ofmagnitude higher than that of proton cyclotron branch.The instability for both branches take place over nar-row bands of frequencies and along extended domainsof propagation angles. However, only the most unstablealpha cyclotron modes take place along quasi paralleldirection, while the peaking growth rates of the protoncyclotron instability peak appear at oblique angles.In Figure 3 we plot the maximum growth rate for theunstable alpha cyclotron modes, which was determinedby surveying the entire frequency and angle space fora given set of input parameters β p and β α . We havethen systematically varied both β p and β α , for fixed δ = 0 .
05 and V α = v A , until we covered the two dimen-sional parameter space ( β p , β α ). Figure 3 shows thatthe alpha cyclotron beam instability becomes more un-stable as β α decreases, for fixed value of β p . Of course,one cannot indefinitely decrease β α , since the presentweak growth/damping rate formalism assumes that thedistribution function has a relatively mild of velocityspace gradient, ∂f α /∂v ⊥ and ∂f α /∂v (cid:107) . For very low β α values, the distribution will have a sufficiently high ve-locity derivative so that the assumption of weak growthrate is violated. Such a caveat notwithstanding, it is in-teresting to note that the beam-driven alpha cyclotroninstability becomes more unstable for lower beta valuesfor alpha particles. Figure 3 shows that the instabil-ity is suppressed as the proton beta (or, equivalently,proton temperature) increases.Figure 4 exhibits the maximum growth rate for pro-ton cyclotron instability in the same format as Fig. 3.Conditions relevant for a low-beta solar wind are in gen-eral confined around the solid blue line that correspondsto the sample case assumed in Figure 2. Note that themaximum growth rate is lower in magnitude than thatof the alpha cyclotron instability by an order of magni-tude in an overall sense. It is interesting to note that themaximum proton cyclotron beam instability has a peakvalue around β α ∼ − , but for both higher and lower β α , the maximum growth rate decreases. This behavioris in contrast to that of the alpha cyclotron instability,where the alpha cyclotron instability monotonically in-creases in magnitude of the maximum growth rate as β α is decreased (until, presumably, the assumption ofweak growth rate is eventually violated). Note thatthe proton cyclotron instability is also suppressed byincreasing proton beta, which is similar to that of theunstable alpha cyclotron mode.To summarize, we find that the relative proton-alphabeam driven cyclotron instabilities of both the alphacyclotron and proton cyclotron mode branches are gen-erally confined to low beta regime, which in generalconforms with measurements in the solar wind at lowaltitudes (Matteini et al. 2013; Maruca et al. 2012). Fig. 3
Maximum growth rate for alpha cyclotron modebranch versus β p and β α . Other parameters are fixed, n α /n p = 0 .
05 and V α = v A . Of the two modes, however, the dominant instability isthat of the alpha cyclotron branch, so that in the non-linear stage, we expect that the alpha cyclotron modewill dominate the dynamics. For higher beta values, thebeam driven cyclotron instabilities are generally sup-pressed, which might explain why in the literature, theproton-alpha drift instabilities in the high beta regimehave been typically studied in combination with tem-perature anisotropies of either the protons or alpha par-ticles.
The alpha/proton beam instabilities can play an im-portant role in constraining the beaming velocity of al-pha particles, and may explain the deceleration of alphaparticles in the solar wind with increasing distance fromthe Sun. Previous studies have explored in much de-tail the high beta plasma regime of these instabilities,clarifying the role of internal temperature anisotropiesof protons or alpha particles, which may switch fromAlfv´enic instabilities for isotropic beams to an instabil-ity of fast-magnetosonic/whistler mode if alpha beamexhibit T ⊥ /T (cid:107) < T ⊥ /T (cid:107) for protons oralpha particles, which stimulates Alfv´enic instabilitiesand prohibits any other instabilities to develop. Con-trary to that, here we assumed alpha and proton beams Fig. 4
Maximum growth rate for proton cyclotron modebranch versus β p and β α . Other parameters are fixed, n α /n p = 0 .
05 and V α = v A . with isotropic temperatures T ⊥ /T (cid:107) = 1, in order toprovide basic insights on the dispersion and stability ofalpha/proton beams.In Section 2, we have derived the general dispersionrelations of electromagnetic waves propagating at arbi-trary angle ( θ ) with respect to the magnetic field. InSection 3, we formulated the weak growth rate theoryfor plasma particles that are assumed to be distributedaccording to standard (Maxwellian) statistics, withprotons and alpha particles considered to be counter-drifting Maxwellians. In section 4 we have examineda sample growth rate calculation associated with theelectromagnetic modes: fast-magnetosonic/whistler,proton-cyclotron and alpha-cyclotron waves, for a givenset of alpha-to-proton density ratio, n α /n p , alpha andproton beta’s, β α and β p , and alpha-proton relativedrift speed, V α , which is fixed at v A . The results un-veiled a potential competition between instabilities ofproton and alpha cyclotron modes, but the sample cal-culation also showed that the alpha cyclotron modemay reach growth rates one order of magnitude higherthan that of the proton cyclotron mode. We have thenproceeded with the calculation of maximum growthrates for each mode as we continuously varied β α and β p . It was shown that for the low beta solar wind condi-tions in the outer corona, the alpha cyclotron instabil-ity is the dominant mode, with growth rates increasingwith decreasing β α . In contrast, the less unstable pro-ton cyclotron mode has a local peak associated withthe maximum growth rate around β α = 10 − , but themode is suppressed for either decreasing or increasing β α around this peaking value. Both modes, however,are stabilized by increasing proton beta, or equivalently,proton temperature. The consequence of the excitation of these unsta-ble modes on the solar wind proton and alpha particledynamics cannot be understood purely on the basis oflinear theory. In order to address such an issue, wewill carry out quasilinear analysis in the near future.The impact of instability excitation on the radially ex-panding solar wind condition can also be studied in thefuture where, the the effects of radial expansion canbe counter balanced by the wave-particle relaxation byinstabilities. Such a task is a subject of our ongoingresearch.
Acknowledgements
The authors acknowledge sup-port from the Katholieke Universiteit Leuven (GrantNo. SF/17/007, 2018), Ruhr-University Bochum,and Alexander von Humboldt Foundation. These re-sults were obtained in the framework of the projectsSCHL 201/35-1 (DFG–German Research Foundation),GOA/2015-014 (KU Leuven), G0A2316N (FWO-Vlaa-nderen), and C 90347 (ESA Prodex 9). M.A.R.acknowledges Punjab Higher Education Commission(PHEC) Pakistan for granted Postdoctoral FellowshipFY 2017-18. S.M.S. gratefully acknowledges supportby a Postdoctoral Fellowship (Grant No. 12Z6218N)of the Research Foundation Flanders (FWO-Belgium).P.H.Y. acknowledges NASA Grant NNH18ZDA001N-HSR and NSF Grant 1842643 to the University ofMaryland, and the BK21 plus program from the Na-tional Research Foundation (NRF), Korea, to KyungHee University.
A Polarization vector
For an ambient magnetic field vector directed along z axis, ˆ b = B / | B | = ˆ z and the wave vector lying in xz plane, k = k ⊥ ˆ x + k (cid:107) ˆ z = ˆ x k sin θ + ˆ z k cos θ , we definethree orthogonal unit vectors, following (Melrose 1986),ˆ κ = ˆ x sin θ + ˆ z cos θ , ˆ a = ˆ y , and ˆ t = ˆ x cos θ − ˆ z sin θ .Then the unit wave electric field vector is given byˆ e ( k ) = δ E | δE | = K ˆ κ + T ˆ t + i ˆ a ( K + T + 1) / . (A1)Making use of linear wave equation, (cid:2) (cid:15) ij − N (cid:0) δ ij − k i k j /k (cid:1)(cid:3) δE j = 0 , (A2)it is possible to obtain δE x = (cid:15) xx − N (cid:15) xy δE y ,δE z = − N sin θ cos θ(cid:15) zz − N sin θ (cid:15) xx − N (cid:15) xy δE y . (A3)Upon direct comparison with Eq. (A1) one may identify K = − i sin θ ( (cid:15) zz − N ) (cid:15) xy (cid:15) xx (cid:15) zz − N A ,T = − i cos θ (cid:15) zz (cid:15) xy (cid:15) xx (cid:15) zz − N A , (A4)where A = (cid:15) xx sin θ + (cid:15) zz cos θ . Upon making use ofEq. (5), we further obtain K = − M sin θ, T = − M cos θ,M = x Q + Q − + δP + P − ( x + ) x D + q P + P − Q + Q − µ , (A5)where various quantities, P ± , Q ± , and D , as well asnormalized wave number and frequency, q = ck/ω pp and x = ω/ Ω p , are defined in Eq. (8). B Parameter R In the growth rate expression (9) appears a quantity ∂ ( ω N ) /∂ω , which in normalized form, is defined by R = Ω p ω pp ∂ ( ω N ) ∂ω = ∂q ∂x . (B1) Making use of Eqs. (6) and (8), the desired quantity R can readily be computed as R = x P + P − Q + Q − q µ + D (1 + µ ) x × (cid:20)(cid:18) Q + Q − P + P − + δ P + P − Q + Q − (cid:19) q (1 + µ )+2 δ P + P − Q + Q − + 2 x ( x − Q + Q − P + P − + δx x (2 x −
5) + 3(9 x − P + P − Q + Q − (cid:21) . (B2) References
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