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Parametric decay of plasma waves near the upper-hybrid resonance I. Y. Dodin1 and A. V. Arefiev2 1Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA 2Center for High Energy Density Science, The University of Texas, Austin, Texas 78712, USA An intense X wave propagating perpendicularly to dc magnetic field is unstable with respect to aparametric decay into an electron Bernstein waveand a lower-hybridwave. A modified theory of thiseffect is proposed that extendsto thehigh-intensityregime, wheretheinstability rateγ ceases to be a linear function of the incident-wave amplitude. An explicit formula for γ is derived and 7 expressedintermsofcold-plasmaparameters. Theorypredictionsareinreasonableagreementwith 1 theresults of theparticle-in-cell simulations presented in a separate publication. 0 2 PACSnumbers: 52.35.-g,52.35.Mw,52.35.Hr n a J I. INTRODUCTION II. NOTATION 6 . Thesymbol=willbeusedfordefinitions. Hats(ˆ)are ] The increase in the computer power and advances in h used to denote nonlocal (differential or integral) opera- numerical modeling have recently made it possible to p tors. Also,foranya,thenotation“δa:”willdenotethat simulatethepropagationofradiofrequencyplasmawaves - thecorrespondingequationrepresentsanEuler-Lagrange m using first-principle particle-in-cell (PIC) algorithms. In equation(ELE)obtainedbyextremizingtheactionfunc- particular,therehasbeenagrowinginterestinPICmod- s tional with respect to a. Also, for any E(q), where q a eling of the X-B conversion [1, 2], i.e., a transformation S pl ofanexternally-launchedelectromagneticXwaveintoan denotes the wave type (X, EBW, LHW), we use the following convention for the complex representation: . electron Bernstein wave (EBW), which is useful for de- s c positing energy into the dense core of tokamak plasmas i [3,4]. InRef.[2],aninstabilitywasobservedinsuchsim- E(q) =ReE(q), E(q) = (q)eiθq. (1) s c c E y ulations that results in a parametric decay of an X wave h intoanEBWandalower-hybridwave(LHW).Theexist- Here (q) isthewaveenvelope,andθ istherapidphase. q p ing theory of this instability [5, 6] is limited to relatively The cEorresponding frequency and wave vector are de- [ lowamplitudesandassumesthattheEBWandLHWare fined as ω(q) =. ∂tθq and k(q) =. θq. In particular, 1 inhomogeneous and propagate at nonzero group veloci- when a wave is−stationary and ho∇mogeneous, one has v ties. (For other relevant studies, see Refs. [7–9].) This θ (t,x)= ω(q)t+k(q) x+const. q 0 involves delicate assumptions about how thermal effects − · For each given species s, q and m denote the par- 9 enter the wave dispersion relation. But the simulation s s ticle charge and mass, n is the unperturbed density, 6 results in Ref. [2] indicate that the instability is a robust . s 1 cold-plasma effect and is experienced by homogeneous ωps = (4πns.qs2/ms)1/2 is the corresponding plasma fre- 0 quency, Ωs = qsB0/msc is the cyclotron frequency, B0 wavestoo. Hence, adifferenttheory is neededto explain . is the magnitude of the background dc magnetic field, 1 thoseresultsexplicitlyandalsotoprovideamorerobust and c is the speed of light. For simplicity, we consider 0 description of the effect in general. 7 a single type of ions, denoted with index i, but it is 1 Here we propose such theory. By using a variational straightforward to generalize the theory to multiple ion v: approach, we derive the instability rate γ through the types. Electr.ons are denoted with index e, and we also i cold-plasma linear susceptibility. This approach is ad- introduce e = qe . The symbols ωLH and ωUH denote X vantageous in the sense that nonlinear ponderomotive the lower-hybri|d (|LH) and upper-hybrid (UH) frequen- r forces on the plasma, which are somewhat complicated, cies(AppendixA).Notably,theconstantωLH isnotnec- a donotneedtobecalculated. Ourtheoreticalpredictions essarily the same as ω(lh), which is the actual frequency forγ areinreasonableagreementwithsimulationresults of the LHW that, in general, can depend on the plasma in Ref. [2]. We also extend the theory to higher ampli- temperature and also on k(lh). tudes, when the wave interaction is not quite resonant, and γ is a nonlinear function of the X-wave amplitude. The work is organized as follows. In Sec. II, we de- III. BASIC APPROACH fine the basic notation used in this paper. In Sec. III, weintroduceourgeneralapproach. InSec.IV,wederive A. Resonance conditions theinstabilityrate. InSec.V,wepresentestimatescom- paring our theory with simulations in Ref. [2] and also summarize the main results of our paper. Some auxil- We consideraprocessinwhichanXwave(a“pump”) iary calculations are reported in appendices. scatters into an EBW and a LHW approximately under 2 the conditions of the three-wave resonance: ˆǫ is the linear dielectric tensor in the operator form, 0 χˆ(int) is the LHW-driven perturbation to the linear- ω(x) ω(ebw)+ω(lh), (2) c susceptibility operator, and ≈ k(x) k(ebw)+k(lh). (3) ≈ ωˆ =. i∂ , kˆ =. i . (8) t That being said, we allow the LHW envelope to − ∇ evolve in time at a rate comparable to ω(lh). The Usingthese,onecanconsiderthedielectrictensor(which temporal resonance (2) is assumed satisfied only in isHermitianintheabsenceofdissipation[15])asapseu- tωh(elh)senωse thatωω(x) ≈ω(ωeb(ewb)w.),[Tbheecsauupsee,rpoinsitaionnyofcatshee, dodifferential operator; i.e., ˆǫ0 =ǫ0(t,x,ωˆ,kˆ), where ǫ0 ∼ LH ≪ UH ∼ is a tensor function. (The prefix “pseudo”indicates that X-wave and EBW fields can be considered as a quasi- the expansion of ǫ in ωˆ and kˆ can contain infinite pow- monochromaticfield,henceforthcalledUHfield.] Incon- 0 ers; i.e., although expressed in terms of derivatives, such trast, all the three wave vectors are allowed to be com- operator can be essentially nonlocal.) Similarly, we in- parable to each other. Assuming ω(lh) is close to ω irrespective of k(lh), troduce a tensor function D via Dˆ =D(t,x,ωˆ,kˆ). LH Eq. (2) determines ω(ebw). In a one-dimensional (1D) Note that ELEs derived from the variationalprinciple problem,thissetsk(ebw) throughtheEBWdispersionre- are manifestly Lagrangian. In particular, they conserve lation;then,Eq.(3)setsk(lh). Inthismodel,Eqs.(2)and thetotalnumberofhigh-frequencyquanta,asguaranteed by the fact that Dˆ is Hermitian [18]. It is to be noted (3) can always be satisfied exactly. (In multiple dimen- sions, ensuring the resonance is even easier.) However, that the theory proposed in Ref. [5, 6] does not have thisproperty,foritreliesonthefalseassumptionthatˆǫ when thermal effects are taken into account, the avail- 0 ability of anexactresonancecanbe a subtle issue, so we can be inferred from the homogeneous-plasma dielectric allow for nonzero detuning ∆ω =. ω(x) ω(ebw) ω(lh) tensor ˆǫ0,h simply by replacing k(q) with kˆ. Fixing this treated as a free parameter. − − issue would require a derivation of ˆǫ0 without assuming the geometrical-optics approximation, because ˆǫ does 0,h notcontainenoughinformationinprinciple[19]. Itisnot B. Variational principle our goal to make such revision in the present paper. In- stead, we are interestedin modifying the theory in other respects(e.g.,extendingittohigheramplitudes),sohere In contrast with Refs. [5, 6], where field equations are we limit our consideration to homogeneous plasmas. derived by calculating ponderomotive on plasma particles, we propose an arguably more transparent formu- lation in terms of the plasma linear susceptibility. The fact that ponderomotive forces can be inferred from the C. Reduced problem linear susceptibility is widely known,for example, as the K-χ theorem [10–14]. We adopt a variational approach The simplifiedsystemthat westudy hereis asfollows. to utilize this link efficiently. We assume dissipation to Weassumethatthebackgroundplasmaisstationaryand be negligible for simplicity, but the generalmethod used homogeneous,so L does not contain explicit dependence hereisextendable todissipativewavestoo[15]. Itis also on(t,x). WealsoconsidertheXwaveasprescribed;then to be noted that a related calculation was proposed re- L(x) can be dropped. We also adopt the electrostatic cently in Ref. [16] in application to Raman scattering. approximation for the EBW and LHW (yet not for the (For earlier applications of the variational approach to X wave), so E(ebw) k(ebw) and E(lh) k(lh); besides, the three-wave interactions, see, e.g., Ref. [17].) dispersionoperatorkreducestoDˆ =ǫˆ k . Wealsoassume Assumingthatdissipationisnegligible,thewaveinter- 0,xx 1D propagation along the x axis, which is transverse to action can be described using the least action principle the dc magnetic field B = e B (e is a unit vector δ = 0. The functional can be adopted in the form 0 z 0 j S= Ldtd3x (the MinkSowski metric is assumed), and along the jth axis). Then, Sthe LRagrangiandensity L has the form L=L(ebw)+L(lh)+L(int), (9) L=L(x)+L(ebw)+L(lh)+L(int), (4) 1 L(q) = 1 E(q)∗ Dˆ E(q), (5) L(q) = 16π Ec(q)∗DÊc(q), (10) 16π c · · c 1 L(int) = 161π RenE(cx)∗·χˆ(cint)·E(cebw)o. (6) L(int) = 16π Ren(cid:2)E(cx)∗·χˆ(cint)·ex(cid:3)Ec(ebw)o. (11) (If this is not obvious, see Appendix C.) Here, Dˆ is the Since L(int) is small, it is enough to calculate χˆ(int) c linear dispersion operator given by approximately, so we adopt the cold-plasma approximation for that. In this approximation,the susceptibility is Dˆ =. c2 kˆkˆ 1(kˆ kˆ) +ˆǫ , (7) entirely determined by the particle densities and by the ωˆ2 − · 0 magnetic field. But the LHW is assumed electrostatic, (cid:2) (cid:3) 3 so it does not perturb the magnetic field. Thus, Accordingly, we can use χˆ(int) = ∂χˆ(suh) n(lh) = n(sl,hc) χˆ(uh), (12) e−iθqDÊc(q) =D ω(q)+ωˆ,k(q) E(q) ≈Dω(q)ωÊ(q), c Xs ∂ns s,c Xs ns s where D(q) =. ∂ D(ω(cid:0)(q),k(q)). This(cid:1)gives ω ω wherethesummationistakenoverspecies,andχˆ(uh) are the corresponding unperturbed susceptibility opserators 16πL=E(ebw)∗D(ωebw)i∂tE(ebw)+E(lh)∗D(ωlh)i∂tE(lh) acting on the UH field; hence the index (uh). Assuming +iβ (ebw) (lh)ei∆ωt iβ∗ (ebw)∗ (lh)∗e−i∆ωt. theUHfieldisquasimonochromatic,wecanapproximate E E − E E them with χ (ω ). Also, n are the corresponding un- The corresponding ELEs are as follows: s UH s perturbedsusceptibilityoperatorsandunperturbed den- sciotmiesp,leaxndrenpr(sl,ehcs)enartaettiohne.LFHrodmensthiteycpoerrrteusrpboantdioinngs icnonthtie- δE(ebw) : ∂tE(ebw)∗ = DβE(ωe(blhw)) ei∆ωt, (20) nuity equations, the latter are found to be β∗ (ebw)∗ δ (lh)∗ : ∂ (lh) = E e−i∆ωt (21) n(lh) = kˆ(lh)χˆ(sl,hx)x E(lh). (13) E tE D(ωlh) s,c 4πiqs c (plusthetwoequationsadjointtothese,whichwedonot need to consider). These can be combined into a single We added the index (lh) to emphasize that the cor- equation for (lh): responding operators act on the LH field. Since E χχˆˆ(e(l,uhxh)x)∼χˆχˆ(i,l(xhux)h).(ATphpisenmdeixansAt)h,aot,nienhEaqs.(n1e2),∼itinsie.noBuguht ∂t2E(lh)+i(∆ω)∂tE(lh)−γ02E(lh) =0, (22) toereta≫injuist the electroncontribution. (In other words, where γ0 is a constant given by theponderomotiveforceonionsisneglected.) Thatgives . β γ = | | . (23) 0 χˆ(int) kˆ(lh)χˆ(el,hx)x E(lh)χ (ω ). (14) D(ωebw)D(ωlh) c ≈ 4πiq n c s UH q e e In particular, for (lh) exp( iωt), one gets Hence, E ∝ − 1 ∆ω (∆ω)2 L(int) ≈ 16π Ren2iE(ebw)(cid:2)eiθebw−iθxβÊc(lh)(cid:3)o, (15) ω1,2 = 2 ±r 4 −γ02. (24) βˆ=. E(x)∗ χ (ω ) e kˆ(lh)χˆ(el,hx)x. (16) Clearly,γ0 isreal,becausethesignofDω(q) coincideswith −(cid:2) · s UH · x(cid:3) 8πqene thatofthemodeenergy[20]andthewavesinvolvedhave positive energies. Hence, if ∆ω < 2γ , an instability Using Eq. (A12), one can simplify the expression in the 0 square brackets down to (x)∗. Also, χˆ(lh) ω2 /Ω2, develops with the rate −Ex e,xx ≈ pe e which is just a constant. Below, we also assume for sim- (∆ω)2 plicitythatwavesarespatiallymonochromatic. Thisim- γ = γ2 . (25) r 0 − 4 plies that the condition of the spatial resonance (3) is satisfied exactly and kÊc(q) =k(q)Ec(q). Hence, the oper- In order to explicitly calculate γ0 that enters here, we ator βˆ can be replaced with the following constant: invoke Eqs. (19), (A14), and (A23). This leads to β = k(lh) ωp2e. (17) D(ebw) 2ωUH, D(lh) 2ωp2i, (26) 8πq n Ω2 ω ≈ ω2 ω ≈ ω3 e e e pe LH Then, the formula for L(int) is summarized as follows: where we neglected thermal corrections. We also invoke the condition of plasma neutrality, namely, ω2 /Ω = 1 pe e L(int) ≈ 16π Reh2iβE(ebw)E(lh)e−i(θx−θebw−θlh)i. (18) −ωp2i/Ωi. Then, Eq. (23) gives γ ω ωpeωpi ωLH k(lh)Ex(x) . (27) IV. INSTABILITY RATE 0 ≈ LH |ΩeΩi|rωUH (cid:12)(cid:12)16πene(cid:12)(cid:12) Although the effect calculated here is due to the ex- A. Weak pump istence of EBW, which implies a nonzero temperature T, the instability rate (25) is insensitive to T (except, First, suppose that the pump amplitude (x) is small of course, at large enough T). In this sense, the insta- E enough. Then, the EBWand the LHW oscillateapprox- bility can be understood as a cold-plasma effect. Also imately at the unperturbed frequencies that satisfy note that, since γ0 k(lh) = kx(x) kx(ebw) , the insta- ∝| | | − | bility rate is somewhat larger for backscattering, when D(ω(q),k(q))=0. (19) k(x) and k(ebw) have opposite signs. x x 4 B. Strong pump 2 HaL 0.2 HbL Now let us consider the case when the pump is strong 1 0.1 e(TnohuegEhBsWo tihsestLilHlaWssucmeaesdesqutoasbimeoqnuoacshimroomnaotcihcrboemcaautisce. (cid:144)ΩΩLH 0 (cid:144)ΩΩLH 0.0 it has a much higher frequency.) In this case, we adopt Re Im -0.1 -1 the Lagrangiandensity in the form -0.2 J=0.3ΩLH J=0.3ΩLH 16πL=E(ebw)∗D(ωebw)i∂tE(ebw)+E¯(lh)∗DÊ¯(lh) -20 1 2 3 4 0 1 2 3 4 +iβ (ebw)¯(lh)eiϑt iβ∗ (ebw)∗¯(lh)e−iϑt. Γ0(cid:144)ΩLH Γ0(cid:144)ΩLH E E − E E . Here,ϑ=ω(x) ω(ebw) =ω(lh)+∆ω,andweintroduced 2 0.6 ¯(lh)(t)=. Ec(lh)−(t,x)exp( ik(lh) x)thatisnotnecessar- HcL 0.4 HdL iEly slow in time but has−no dep·endence on x. Rather 1 than expanding the dispersionoperatorfor the LHW, as ΩLH ΩLH 0.2 in Sec. IVA, we now adopt the fully nonlocal operator (cid:144)Ω 0 (cid:144)Ω 0.0 inferred from Eq. (A23): Re Im -0.2 -1 -0.4 Dˆ =ωˆ−2(ωˆ2−ωL2H)ωp2i/ωL2H. (28) -2 J=ΩLH -0.6 J=ΩLH 0 1 2 3 4 0 1 2 3 4 The corresponding ELEs are as follows: Γ0(cid:144)ΩLH Γ0(cid:144)ΩLH β¯(lh) δE(ebw) : ∂tE(ebw)∗ = DE(ωebw) eiϑt, (29) 2 HeL 1.0 HfL δ¯(lh)∗ : Dˆ ¯(lh) =iβ∗ (ebw)∗e−iϑt (30) 1 0.5 E E E H H L L (plusthetwoequationsadjointtothese,whichwedonot (cid:144)ΩΩ 0 (cid:144)ΩΩ 0.0 need to consider). Using Eq. (29), we get e m R I -1 -0.5 iβ E(ebw)∗e−iϑt = D(ωebw) (ωˆ−ϑ)−1E¯(lh). (31) -20 1 2J=23.5ΩLH4 -1.00 1 2J=23.5ΩLH4 Accordingly, Eq. (30) can be expressed as Γ0(cid:144)ΩLH Γ0(cid:144)ΩLH β 2 Dˆ ¯(lh) = | | (ωˆ ϑ)−1¯(lh), (32) FIG. 1: Solutions of Eq. (34) for ω versusγ , both measured E −D(ebw) − E 0 ω in units ω . The left column shows the real part of ω, and LH or, equivalently, the right column shows the imaginary part of ω. In (a) and (b), ϑ = 0.3ω . In (c) and (d), ϑ = ω . In (e) and (f), LH LH (ωˆ ϑ)(ωˆ2 ω2 )+2γ2ωˆ2 ¯(lh) =0, (33) ϑ=2.5ω . Ineachgivenplot,differentcurvesshowdifferent − − LH 0 E LH (cid:2) (cid:3) branches. The unstable branch is shown as a solid line. whereγ isgivenbyEq.(27). Thecorrespondingdisper- 0 sion relation is as follows: Specifically, for g 1 (strong pump) one gets (ω ϑ)(ω2 ω2 )+2γ2ω2 =0. (34) ≫ − − LH 0 ω /ω i/(g√2), ω /ω 2g2+1, (36) Equation (34), in combination with Eq. (27) for γ , is 1,2 LH 3 LH 0 ≈± ≈− our main result. It is a cubic equation for ω and has and for g 1 (weak pump) one gets an exact, albeit cumbersome, analytic solution. We do ≪ not present it here for brevity, but see numerical plots ω /ω 1 ig, ω /ω 1 g2/2. (37) in Fig. 1. Like in the case of resonant damping, tak- 1,2 LH ≈ ± 3 LH ≈− − ing ∆ω = 0 (which corresponds to ϑ = ω ) ensures LH The first two roots in Eq. (37) correspond to the two that there is no instability threshold within the adopted roots predicted by Eq. (24). [The additional real part of model. Inotherwords,evenanarbitrarilysmallγ causes 0 thefrequencythatispredictedbyEq.(37),namely,ω , LH one of the three roots of Eq. (34) to acquire a positive is due to the fact that here we derived the frequency of imaginary part. ¯(lh) rather than that of (lh).] Thus, the weak-pump One can also derive asymptotic expressions for ω at E E model discussed in Sec. IVA is successfully recovered as ∆ω =0 in terms of the dimensionless parameter anasymptoticlimitofthegeneraltheorypresentedhere. g =. γ0 ωpeωpi ωLH k(lh)Ex(x) . (35) lowIt-aismtpolibtuedneotliemditthianttohuartgitentearkaelstahneoardydditiffioenraslfrbormanicths ωLH ≈ |ΩeΩi|rωUH (cid:12)(cid:12)16πene(cid:12)(cid:12) of the dispersion relation into account. Although this 5 branch remains stable by itself [see ω in Eqs. (36) and where S, D, and P depend on the wavefrequency ω but 3 (37)], it still affects the stability of resonant branches. not on the wave vector. Specifically, This can be identified as a manifestation of polarization (“spin”)effectsthatwererecentlydiscussedforbothclas- S =1+ χs,xx, iD = χs,yx, P =1+ χs,zz, sical and quantum waves in Refs. [21, 22]. Xs Xs Xs and χ =χ∗ = χ , where s,xy s,yx − s,yx V. DISCUSSION ω2 ps χ = , (A2) s,xx −ω2 Ω2 For an estimate, let us adopt the parameters used in − s Ref. [2]. Namely, consider an electron-deuterium plasma iΩs ωp2s χ = , (A3) pweitrhatuerleectTron=de9n5s0iteyVn. e A=ls0o.,89B×0 1=0180m.2−51T,ankd(lht)em=- s,yx ω ωω22−Ω2s 2000m−1, and Ex(x) = 8 × 105V/m. The X wave is χs,zz =−ωp2s. (A4) launched at ω = 2π 10GHz, which is about 0.9ω , UH and ωUH = 6.9 101×0s−1. Also, ωLH = 7.9 108s−1, For waves propagating perpendicularly to B0, the lin- which is about 3×3.1Ωi. The corresponding LH×period is ear field equation Dˆ ·Ec =0 becomes 2π/ω 7.9ns. Consideringthat weneglectedthermal LH ≈ S iD 0 effects, which somewhat modify the plasma dispersion, x − E this is in reasonable agreement with the simulation re-  iD S N2 0  y =0, (A5) − E 0 0 P sultsinRef.[2],whereoscillationswereobservedwithpe- z   E  riod about 10ns. For the specified parameters, Eq. (27) . where N =ck/ω is the refraction index. Thus, givesγ 1.3 108s−1. Thiscorrespondstoγ 5.5Ω , 0 0 i ≈ × ≈ or g 0.17. Then, the instability rate is expected to be S =iD . (A6) γ ≈γ , which corresponds to γ−1 7.6ns. This result Ex Ey 0 ≈ ≈ is in the ballpark of the simulation results. Also, the extraordinarywavesthat are of interest in this In summary, we proposed a modified theory of the in- paper are defined as those with = 0, so Eq. (A5) is z E stability that is caused by the resonant scattering, or simplified down to parametricdecay,ofanintenseXwaveintoanEBWand LHW. Our theory extends to the high-intensity regime, S −iD Ex =0. (A7) wherethe instabilityrateγ ceasestobealinearfunction (cid:18)iD S N2 (cid:19)(cid:18) y (cid:19) − E of the incident-wave amplitude. We derived an explicit formula for γ and expressed it in terms of cold-plasma The corresponding dispersion relation is N2 = (S2 − parameters. Predictions of our theory are in reasonable D2)/S. The electrostatic limit corresponds to S = 0, agreement with the results of the PIC simulations pre- which equation defines the hybrid resonances. sented in Ref. [2]. The work was supported by the U.S. DOE through 2. UH waves Contract No. DE-AC02-09CH11466 and by the U.S. DOE-NNSA Cooperative Agreement No. de-na0002008. In the UH range,the ioncontributionto the dielectric tensor is negligible, so Appendix A: Basic properties of the relevant plasma waves ωp2e χ χ = , (A8) xx ≈ e,xx −ω2 Ω2 − e Here, we summarize some basic properties of the iΩ ω2 e pe plasmawavesrelevanttothediscussioninthemaintext. χyx ≈χe,yx = ω ω2 Ω2, (A9) − e ω2 pe χ χ = . (A10) 1. Basic equations zz ≈ e,zz −ω2 In particular, this implies Weconsidermonochromaticwavesinthemodelofcold stationary homogeneous plasma. The dc magnetic field S 1 iD 0 is adopted in the form B0 = ezB0. Then, the plasma χe ≈ i−D S−−1 0 , (A11) dielectric tensor can be expressed as follows [23]: 0 0 P 1  −  S iD 0 so, as seen easily, Eq. (A6) leads to − ǫ= iD S 0 , (A1) 0 0 P E(x)∗ χ (ω ) e = (x)∗. (A12)   · e UH · x −Ex 6 Since S 1 ω2 /(ω2 Ω2), the UH frequency is thermal corrections be taken into account. This leads to ω =(ω2≈+Ω−2)1/p2e. Th−is edetermines the frequency UH pe e ω2 ∞ n2ω2 2I (λ ) range for electrostatic EBW, although calculating the D(lh)(ω,k) 1+ pe pi n i e−λi, EBW dispersion relation requires taking thermal ef- ≈ Ω2e −nX=1ω2−(nΩi)2 λi fects into account. In particular, the dispersion rela- (A22) tion of the EBW in the electrostatic approximation is D(ebw)(ω,k)=0, where [23] whereλ =. (kv /Ω )2,andv istheionthermalspeed. i Ti i Ti In the regime of interest, λ 1. Nevertheless, the sum i ∞ n2ω2 2I (λ ) in Eq. (A22) can be approx≫imated with its cold limit, D(ebw)(ω,k)=1 pe n e e−λe, − ω2 (nΩ )2 λ except when ω is particularly close to one of cyclotron nX=1 − e e resonances (Appendix B). Hence, we adopt (A13) I the modified Bessel function of order n, λ =. D(lh)(ω,k) 1+ ωp2e ωp2i (knv /Ω )2, and v is the electron thermal speed.eFor ≈ Ω2e − ω2 Te e Te simplicity, we consider the interaction with the lowest- ω2 ω2 = 1+ pe 1 LH order mode only and adopt (cid:18) Ω2 (cid:19)(cid:18) − ω2 (cid:19) e D(ebw)(ω,k)≈1− ω2ωp2eΩ2 Θ(λe), (A14) = ωωL2p2Hi (cid:18)1− ωωL22H(cid:19). (A23) − e Θ(λ )=. 2In(λe)e−λe =1 λ + (λ2), (A15) e λe − e O e Appendix B: Applicability of the cold approximation for χxx usingthatλ issmall. Notethatω(ebw)(λ 0) ω . e e UH → → ForparameterslistedinSec.V,onehasλ 0.35.1,so e The numerical parameters adopted in this paper neglectingthermaleffects isareasonableap≈proximation. (Sec. V) correspond to λ =. (kv /Ω )2 317. This i Ti i ≈ number is far too largeto allowfor anasymptotic small- argument expansion of the modified Bessel functions in 3. LH waves Eq.(A22). Likewise,the large-argumentexpansionisin- . applicablebecauseofthe largevalueofa=ω/Ω 33.1, i ≈ In the LH range (Ωi ω Ωe), one has whichdeterminesthenumberofrelevantharmonics(n ≪ ≪ ∼ a). Thus, a different approach is needed to justify the ωp2i ωp2e cold-plasma approximation (A23). It is the purpose of χ , χ , (A16) i,xx ≈−ω2 e,xx ≈ Ω2 this appendix to provide such justification. e iΩ ω2 iΩ ω2 Westartbyadoptinganalternativeexpressionforχxx χ i pi, χ e pe. (A17) thatis equivalentto thatassumedinEq.(A22)but does i,yx ≈ ω ω2 e,yx ≈− ω Ω2 e not involve an infinite series [24]: Accordingly, S ≈1+ωp2e/Ω2e−ωp2i/ω2, and χ = ωp2 v ∂f0(v)T 2πv dv dv , (B1) ωpi xx ωΩZ ⊥ ∂v⊥ xx ⊥ ⊥ k ω = . (A18) LH a πa 1+ω2 /Ω2 T = J (z)J (z) 1 . (B2) q pe e xx z2 (cid:20)sin(πa) −a a − (cid:21) Assuming ωpe Ωe, this gives ωLH ωpi (The species index is henceforth dropped for brevity.) ∼ ∼ ∼ Ωe(ωpi/ωpe)∼|ΩiΩe|1/2. Hence, at the LHW frequency, Here, f0 is the unperturbed distribution that is assumed isotropic and is normalized such that χ χ χ i,xx 1, i,yx 1, e,xx 1, (A19) f0(v)2πv⊥dv⊥dvk = 1, v⊥ is the velocity per- χ ∼ χ ≪ χ ≪ e,xx e,yx e,yx Rpendicular to the magnetic field, vk i.s the velocity parallel to the magnetic field, and v =(v2 +v2)1/2. because ⊥ k Also, J are Bessel functions of order a. a χ Ω ω2 Ω2 m Suppose z is small in a sense that is yet to be defined. i,yx i pi e e, (A20) Then, T has the following asymptotic expansion [25]: χ ∼ Ω ω2 ω2 ∼ m xx e,yx e pe LH i χe,xx ωp2i ωLH Ω2e me 3/2. (A21) Txx = ∞ (−1)m+1a2√πΓ(3/2+m)z2m χe,yx ∼ Ω2e Ωe ωp2e ∼(cid:18)mi(cid:19) mX=0Γ(2+m)Γ(2−a+m)Γ(2+a+m)sin(πa) ∞ m+1/2 (2m)! az2m Like with EBWs, calculating the dispersion function = , ofelectrostaticLHW(ionBernsteinwaves)requiresthat mX=0 m+1 4m(m!)2 [a2−(m+1)2]...(a2−1) 7 where Γ is the gamma function. More explicitly, Here denotes the temporal average over the UH t h···i oscillations, denotes the spatial average over all x Txx = 2(a2a 1) + 8(a2 31a)(za22 22) oscAilslaatisolnows, afhun·n·dc·,itciolena,rlLy, hLemciatn,xc=onhtLaeinmioxn.lyevenpow- p t,x − −5az4 − ers of the UH field E(huh).i We neglect all powers higher + +... (B3) thanthesecondone,assumingthatwavesareclosetolin- 16(a2 1)(a2 22)(a2 32) − − − ear. (Forextensionsofthis approachtononlinearwaves, In our case, a 1, so the ratio of the neighboring terms see Refs. [27, 28].) Then, scales as z2/a≫2, provided that a is not too close to an 1 integer. Hence, the expansion is applicable roughly at L = L + . (C2) p t,x 0 x t,x z2 a2. [Thisislessrestrictivethanthevalidityrequire- h i h i 8π hUi ≪ ment for the small-argument expansion of Ja(z), which Here, L is the zeroth-order term independent of E(uh) is z2 ≪a.] By substituting Eq. (B3) into Eq. (B1) and and =0[Qˆ E(uh)]2, where Qˆ is some, generally nonlo- assuming the Maxwellian distribution, we get U · cal, operator. Using the complex notation, one can also ω2 ∞ ( 2)m+1a√πΓ(3/2+m)λm express t,x as follows: χ = p − hUi xx −Ω2 Γ(2 a+m)Γ(2+a+m)sin(πa) 1 mX=0 − = E(uh)∗ χˆ E(uh) , (C3) ω2 ∞ (2m+1)! λm hUit,x 2 D c · · c Ex =−Ωp2 mX=0 2mm! [a2−(m+1)2]...(a2−1), wmhiteiraenEo(cpuehr)a=t.oEr.(cx)L+etEu(cesbwd)ecaonmdpχôs=e. Qχˆˆ†i·nQtô itshae Hsloerw- or, more explicitly, part χˆ that is independent of the LH field and the 0 remaining part χˆ(int). Since is assumed small, we ωp2 1 3λ adopt that χˆ(int) is linear in thUe LH field. This implies χ = + xx −Ω2(cid:20)a2 1 (a2 1)(a2 22) χˆ(int) =Ξ( , ,E(lh))[29],whereΞisarealrank-3tensor − − − 15λ2 that is symmetric in its first two arguments. Hence, + +... . (B4) (a2 1)(a2 22)(a2 32) (cid:21) 1 . − − − χˆ(int) = χˆ(int)+χˆ(int) , χˆ(int) =Ξ( , ,E(lh)). 2 c c c c This expansion is somewhat known [26], but here we (cid:2) (cid:3) emphasize not the expansion per se but rather its va- This leads to lidity domain. The ratio of the neighboring terms away 1 fromresonancesscalesasλ/a2,sotheexpansionrequires, hUit,x = 2Re E(cx)∗·χˆ(cint)·E(cebw) . (C4) roughly, λ/a2 1. (This is less restrictive compared to (cid:2) (cid:3) the usual requ≪irement λ 1.) In particular, the cold- Likewise,we candecompose L0 x into terms that areof plasma limit ≪ the zeroth and second orderhin tihe LH field: 1 ω2 1 ω2 L =L + E(lh) χˆ(lh) E(lh) . (C5) χxx ≈−Ωp2a2 1 =−ω2 pΩ2 (B5) h 0ix p0 8π D · · Et,x − − Since L does not depend on any of the wave variables, is reproduced when the second term in the expansion p0 it does not contribute to ELEs for the wave fields (even (B4) is negligible compared to the first term. This im- though it contributes to ELEs for plasma particles) and plies 3λ/a2 1. For parameters adopted in this paper, 3λ/a2 0.87≪,soonecanexpectthecold-plasmaapprox- thus can be omitted. Hence, Eq. (C1) becomes ≈ imation to be applicable at least semiquantitatively. L=L(x)+L(ebw)+L(lh)+L(int), 1 L(q) = E(q) 2 B(q) 2+E(q)∗ χˆ(q) E(q) , Appendix C: Variational principle 16π n| c | −| c | c · · c o 1 L(int) = Re E(x)∗ χˆ(int) E(ebw) , In this appendix, we derive the Lagrangian density L 16π n c · c · c o introducedinSec.IIIB.ThisLconsistsoftheLagrangian where we introduced densityoftheelectromagneticfieldL =(E2 B2)/(8π) em − χˆ , q =X,EBW, (here and further, E denotes the wave electric field, and χˆ(q) = 0 (C6) B denotes the wave magnetic field) plus the Lagrangian (cid:26)χˆ(lh), q =LH. density of the plasma interaction with this field, L . p Using Faraday’s law and the notation defined in PartsofL=Lem+Lpthatoscillateatthehighfrequency Eq. (8), one gets B(q) =ωˆ−1kˆ E(q)/c. Also consider ω(x) ω(ebw) do not contribute to the action integral . c × c signifi≈cantly on scales of interest. Thus, we can replacSe ˆǫca0n=c1as+t Lχˆ(0x,) awnhderLe(e1bwi)satshefolulonwits:operator. Then, one L with its average: 1 L= Lem t,x+ Lp t,x. (C1) L(q) = 16π Ec(q)∗·Dˆ ·Ec(q), (C7) h i h i 8 where Dˆ is given by Eq. (7). By treating perturbation to the linear-susceptibility operator. Like- (E(x),E(x)∗,E(ebw),E(ebw)∗) as independent variables wise, we infer that L(lh) is the Lagrangian that deter- c c c c [18], one then arrives at the following ELEs: mines the linear propagation of the LHW. Thus, it too can be put in the form (C7). 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