Kinetic theory of low-frequency Alfvén modes in tokamaks

Transcription

1 Plasma Phys. Control. Fuson Prnted n the UK Knetc theory of low-frequency Alfvén modes n tokamaks Fulvo Zonca, Lu Chen and Robert A Santoro Department of Physcs and Astronomy, Unversty of Calforna, Irvne, CA 9-455, USA Receved 9 February 996, n fnal form June 996 Abstract. The knetc theory of low-frequency Alfvén modes n tokamaks s presented. The ncluson of both damagnetc effects and fnte core-plasma on compressblty generalzes prevous theoretcal analyses Tsa S T and Chen L 993 Phys. Fluds B of knetc balloonng modes and clarfes ther strong connecton to beta-nduced Alfvén egenmodes. The dervaton of an analytc mode dsperson relaton allows us to study the lnear stablty of both types of modes as a functon of the parameters characterzng the local plasma equlbrum and to demonstrate that the most unstable regme corresponds to a strong couplng between the two branches due to the fnte thermal on temperature gradent. In addton, we also show that, under certan crcumstances, non-collectve modes may be present n the plasma, formed as a superposton of local oscllatons whch are quas-exponentally growng n tme.. Introducton The expermental observaton [] of large energetc on losses due to Alfvén waves wth frequences lower than that of the torodal Alfvén egenmode TAE [] has recently demonstrated that low-frequency Alfvén waves can be as deleterous as TAE modes to energetc partcle confnement. Expermentally, these modes have the predomnant polarzaton of shear Alfvén waves [] and they have been gven the name of beta-nduced Alfvén egenmodes BAE [3] snce ther frequency s located n the low-frequency betanduced gap n the shear Alfvén contnuous spectrum [4], whch s caused by fnte plasma compressblty. Ideal magneto-hydrodynamc MHD theores predct the beta-nduced frequency gap at [3] </ A γβq, where s the mode frequency, γ s the rato of specfc heats, β the rato of knetc and magnetc pressures, q the safety factor, A = v A /qr the Alfvén frequency, R the major radus of the torodal plasma column, v A = B/ 4πϱ the Alfvén speed and ϱ the plasma mass densty. Ths fact, along wth the expermental observaton that BAEs are shear Alfvén waves wth frequency wthn or near the betanduced gap, ndcates that these modes have long parallel to the equlbrum magnetc feld B wavelengths,.e. k v A γβv A /R k γβ/r k beng the parallel wavevector, and that the relevant BAE frequency range s ordered as the thermal on transt frequency, t = T /m /qr T s the on temperature n energy unts and m the on mass. Furthermore, there s clear expermental evdence [] that damagnetc effects are mportant for the BAE dynamcs, snce typcally p = ct /e B k B ln P, the core-plasma on damagnetc frequency. Here, e s the on electrc charge, P the on pressure and k the wavevector. Permanent Address: Assocazone EURATOM-ENEA sulla Fusone, CP Frascat, Rome, Italy /96/+8$9.5 c 996 IOP Publshng Ltd

2 F Zonca et al From the prevous dscusson, t s evdent that deal MHD s nadequate to construct a realstc theory of BAE modes, for whch t p, snce fnte core-plasma on compressblty s expected to strongly affect the mode dynamcs va resonant nteractons wth the on transt moton along magnetc feld lnes. Moreover, t s mportant to clarfy any relatonshp of BAEs wth knetc balloonng modes [5, 6] KBM, whch are expected to occur n the same frequency range. Prevous theores of resonant exctatons of KBM by energetc partcles [5, 6] have shown that these modes, lke BAEs, belong to the shear Alfvén branch and have p. However, these theores assumed ncompressble oscllatons, thereby neglectng t wave partcle resonances wth coreplasma ons. In the present paper we develop a unfed theory for Alfvén waves belongng to the BAE/KBM branches by accountng for fnte core-plasma compressblty and damagnetc effects on the same footng. In ths respect, we present the knetc theory of hgh torodal mode number [5, 6] low-frequency Alfvén modes n a hgh-β plasma β = Oɛ; ɛ = a/r, a beng the plasma mnor radus, whch we may refer to as drft Alfvén knetc balloonng modes. As a relevant and novel result we show that the most unstable scenaro corresponds to the stuaton n whch BAE and KBM are strongly coupled due to the presence of a fnte temperature gradent of the thermal ons. The valdty of the deal MHD assumpton of neglgble parallel electrc feld perturbatons δe s also dscussed, snce, n general, the couplng between shear Alfvén and acoustc branches s not neglgble at t p. More specfcally, we show that, for long wavelength modes k β / /R, the δe assumpton holds for waves propagatng n the on damagnetc drecton, whereas t may break down for modes propagatng n the electron damagnetc drecton and/or modes wth p / = O/β, whch are strongly coupled to the slab-lke on temperature gradent ITG drven wave [ 9]. Snce our goal s to study the BAE/KBM modes whch may be resonantly excted by energetc ons, only the branches propagatng n the on damagnetc drecton are consdered here. Nevertheless, n the present analyss we neglect the resonant exctaton of the BAE/KBM branch by energetc partcles. The prmary reason for ths choce s that of smplcty, whch allows us to focus on the relevant features of the knetc Alfvén spectrum due to the wave resonances wth thermal ons. A second reason s that wave partcle resonances wth core-plasma ons are mportant only n a narrow boundary layer the nertal layer centred at the mode ratonal surface, where the dynamcs of energetc partcles may be neglected [6] because of ther large orbts compared to the layer wdth. In ths sense, the ssue of the resonant exctaton of BAE/KBM by energetc partcles can be addressed by smply addng the energetc partcle dynamcs to the present theory [6]. Ths problem wll be analysed n a separate work. The plan of the paper s as follows. In secton the theoretcal model s presented and the relevant egenmode equatons are derved. Secton 3 s devoted to a dscusson of the characterstc two-scalelength mode structures of the Alfvén waves we wsh to analyse. The knowledge of mode structures s used n secton 4 to derve an analytc dsperson relaton for BAE/KBM modes. The general features of BAE/KBM spectra are dscussed n secton 5, whereas detaled numercal studes of the analytc dsperson relaton are presented n secton 6. Secton gves fnal dscussons and conclusons. An analyss of the δe deal MHD assumpton s presented n appendx A. Fnally, appendx B provdes an elementary dervaton of the shear Alfvén contnuous spectrum, wth ts modfcatons due to damagnetc effects and core-plasma on compressblty. There, a bref dscusson of the relatonshp between sngular mode structures and contnuous spectra s also gven.

3 Knetc theory of Alfvén modes 3. Theoretcal model and egenmode equatons We consder a large aspect-rato axsymmetrc torodal plasma equlbrum wth shfted crcular magnetc flux surfaces and wth major and mnor rad gven by R and a. For the sake of smplcty, we assume a hgh-β β = 8πP/B ɛ = a/r, P beng the total core-plasma pressure and B the equlbrum magnetc feld s, α model equlbrum [], whch s entrely determned by the local equlbrum parameters s, the magnetc shear, and α = R q β. We also concentrate on waves wth hgh torodal mode numbers, such that k ϑ ρ L ɛ k ϑ beng the polodal component of the wavevector k and ρ L the on Larmor radus. Ths assumpton does not cause any loss of generalty, snce ths s the range of most unstable mode numbers [6]. As usual [, ], we wll descrbe the plasma oscllatons n terms of three fluctuatng scalar felds: the scalar potental perturbaton δφ; the parallel to b = B/B magnetc feld perturbaton δb ; and the perturbed feld δψ, related to the parallel vector potental fluctuaton δa by c δa b δψ. Wth ths representaton, the parallel electrc feld fluctuaton s δe = b δφ δψ, and the deal magneto-hydrodynamc MHD lmt, δe =, s obtaned for δψ = δφ. Isolatng adabatc and convectve partcle responses to the wave, the perturbed partcle dstrbuton functon can be expressed as [, ] δf s = e m s [ F E δφ J k ρ L QF δψ el k ] s + δk s e L ks where s s the speces ndex, e s the speces electrc charge, m s the mass, F s the equlbrum dstrbuton functon, E = v / the energy per unt mass, J the Bessel functon of zero ndex, k the perpendcular to b wavevector, ρ Ls = m s cv /e s B the Larmor radus, QF s = E + ˆ s F s, ˆ s F s = m s c/e s Bk b F s, and L ks = m s c/e s Bk b v. Adoptng the balloonng mode representaton [] n the space of the extended polodal angle varable θ, the partcle dstrbuton functon δk s s derved from the gyroknetc equaton [] e [ tr θ d ] s δk s = QF s m s [ d J k ρ Ls δφ δψ + J k ρ Ls δψ + v ] s k c J k ρ Ls δb where tr = v /qr s the transt frequency, k = k ϑ [ + sθ α sn θ ] and ds s the magnetc drft frequency ds θ = gθk ϑ m s cv / + v /e sbr, gθ = cos θ + [sθ α sn θ] sn θ. In the followng, we wll assume the electron response to be adabatc,.e. δk e =. Furthermore, t may be shown that, for the Alfvén modes we are nterested n [], δb = 4π c B k b P δψ. 3 If we multply both sdes of equaton by 4πe s J k ρ Ls /k ϑ c and then sum over the speces ndex and ntegrate over the velocty space, t s well known that the followng vortcty equaton s obtaned [5, 6]

4 4 F Zonca et al [ k Bb Bkϑ b δψ ]+ va p k kϑ δφ + α q R gθδψ = 4πe s s kϑ J k ρ Ls ds δk s 4 c where... = dv..., ps = ns + Ts, ns = T s c/e s Bk b n s /n s, Ts = T s c/e s Bk b T s /T s, n s s the speces partcle densty, T s the temperature n energy unts, and use has been made of the parallel Ampére s law k kϑ b δψ= 4π kϑ c e s v δf s. s s e sδf s =, form Equatons 4, along wth the quas-neutralty condton, a closed set of ntegro-dfferental equatons for the modes we are nterested n,.e. drft Alfvén knetc balloonng modes. The quas-neutralty equaton can be put nto the followng form + δφ δψ + τ p b δψ = T ne J k ρ L δk 5 where τ = T e /T, b = k m c T /e B, n = n = n e and core-plasma ons wth unt electrc charge have been assumed. 3. Two-scale mode structures Equatons 5 descrbe a varety of drft Alfvén balloonng modes. They are n a complcated ntegro-dfferental form and lttle can be gleaned drectly from these equatons concernng the general propertes of those waves. However, some analytc progress can be made and further nsght can be ganed when we recall the characterstc frequency and wavelength orderngs assumed here;.e. p t Oβ / A and k ϑ ρ L Oβ. 3.. Inertal layer physcs: the large θ soluton It can be recognzed that, at large θ =Oβ /, equatons 5 always have a twoscale structure: n fact, the fluctuatng felds vary on the short scale θ and on the long scale θ β /. We consder ths statement as an ansatz, to be self-consstently verfed a-posteror. Furthermore, for convenence, we work wth new feld quanttes defned as follows: δ = k /k ϑ δφ, δ = k /k ϑ δψ and δ ˆB = k /k ϑ δb. Each feld s thought to be expressed n terms of an asymptotc seres n powers of β / ; e.g., δ = δ + δ + δ +, where δ = Oβ /, δ = Oβ, etc. It s readly recognzed that large θ values correspond, n real space, to a narrow torodal layer centred around the mode ratonal surface, n deal MHD usually referred to as the nertal layer. At large θ =Oβ /, we have k ρ L b d / / A β. Equaton, thus, gves e δk QF k ϑ = δ δ k m whch, substtuted nto the quas-neutralty condton, equaton 5, yelds + δ δ = n δ δ τ

5 Knetc theory of Alfvén modes 5.e. δ = δ to the lowest order. Thus, to the lowest order, equaton 4 predcts that δ = δ θ. To the next Oβ / order, equaton 4 gves θ δ = θ θ δ =.e. δ =, snce the θ dependence of δ can be ncorporated nto δ. Therefore, equaton reads tr θ δk = tr θ δk + k [ ϑ e QF δ + ] d k m δ whch yelds δk = tr θ δk + k ϑ e/m QF k tr {[ + sθ tr ] d δ + δ c + tr δ s cos θ [ + sθ tr ] } d δ + δ s tr δ c sn θ 6 where we have assumed δ = δ c θ cos θ + δ s θ sn θ. When substtuted nto the quas-neutralty condton, equaton 5, equaton 6 yelds + τ + T QF nm tr δ T d = QF nm tr δ 8 whch gves δ c = ct k ϑ N/ t eb R D/ t δ δ s = sθ δ c. 9 Here, t = T /m /qr and the functons Nx = Dx = n + x τ [x +/+x Zx] T [x/ + x + /4 + x 4 Zx] + n Zx T [x + x /Zx] have been ntroduced, where Zx = π / /y xdy s the plasma dsperson e y functon. From equaton 9, t s evdent that our asymptotc expanson s consstent as long as D/ t > Oβ /. We assume that ths s the case. Proceedng further to the next Oβ order, the vortcty equaton, equaton 4, becomes θ δ + θ δ + A p δ = k ϑ k 4πe kϑ q R c dδk. In order to avod seculartes of δ on the short θ scale, equaton becomes θ δ + A p δ + q t A [ n F/ t T G/ t N ] / t δ =. D/ t

6 6 F Zonca et al Here, the functons Fx = xx x4 + x + Zx Gx = xx 4 + x + + x 6 + x 4 / + x + 3 Zx 3 4 have been defned and use has been made of the fact that δ = δ to lowest order. The determnaton of δk and of δ δ at Oβ s not necessary for our present purposes. However, t s gven n appendx A for completeness. Here, we just recall the result for the Oβ quas-neutralty equaton, whch s t p k [ ] kϑ k ϑ θ δ δ + k τ + n δ δ { + [ p b + q t b n F/ t T G/ t N / t D/ t ]}δ = b/ α t k ϑ 3/ q k p δ. Equaton 4 allows us to consder δe = for long wavelength modes propagatng n the on damagnetc drecton, such as those we are analysng n the present paper cf ntroducton and appendx A. Therefore, equaton s the relevant egenmode equaton n the large θ =Oβ / regon. Incdentally, we note that a smlar analyss of the nertal layer s presented n [3], where t was appled to the theory of resstve nterchange balloonng modes. 3.. Ideal regon: the moderate θ soluton For moderate θ < Oβ / values, equaton 4 does not exhbt a two-scale structure any longer. The contrbuton of core-plasma nerta and core-plasma compressblty.e. the core-on contrbuton to the angular brackets on the rght-hand sde can be neglected, whch s why ths s usually referred to as the deal regon. The vortcty equaton, thus, becomes θ δ s α cos θ [ + sθ α sn θ ] α cos θ δ + [ + sθ α sn θ ]δ =. 5 Equaton 5 contnuously matches onto equaton at large θ, and, hence, these two equatons defne a well posed egenvalue problem for the modes we wsh to analyse. However, before proceedng further, t s worthwhle notng that t was possble to drop the core-on nerta term n equaton 5 snce / A = Oβ and δ = δ was assumed. As explaned n appendx A, the latter assumpton whch s the crtcal one for modes wth long parallel wavelength k = Oβ / /qr and p / = O holds as long as /τ + n / = O, e.g. for waves propagatng n the on damagnetc drecton. In the followng, we assume that ths s the case, so that consderng δ = δ s reasonable Dsperson relaton In the prevous secton, we have shown that, n the deal regon θ θ O, the vortcty equaton s gven by equaton 5. Multplyng both ts members by δ ID here,

7 Knetc theory of Alfvén modes the subscrpt ID stands for deal soluton, we may construct the followng quadratc form [6] δ ID θδ ID + δw f = 6 where δw f s the deal MHD contrbuton to the potental energy perturbaton δw f = dθ [ θ δ ID + s α cos θ [ + sθ α sn θ ] αcos θ [ + sθ α sn θ ] ] δ ID. In the large θ nertal regon, equaton s readly solved and gves δ IN = exp θ 8 where the subscrpt IN stands for nertal regon soluton and s gven by { = A p +q [ t A n F/ t T G/ t N ]} / / t. D/ t 9 Here, the square root n the expresson for s taken such that the causalty constrant, Im >, s satsfed. The asymptotc matchng condton between δ IN and δ ID reads δ ID θδ ID + =. Thus, equaton 6 s equvalent to the followng dsperson relaton for drft Alfvén balloonng modes: = δw f. In equaton, δw f s the same used n the MHD theory of deal balloonng modes [] and t may be evaluated by one of the well known numercal methods. Incdentally, we note that the ncluson of energetc partcle dynamcs n the present theory would lead to the dsperson relaton of equaton, wth a contrbuton, δw K, of energetc ons to the potental energy perturbaton added on the rght-hand sde [6]. 5. Relevant lmts of the dsperson relaton A varety of Alfvén spectra are descrbed by the dsperson relaton equaton, derved n the prevous secton. Specfcally, the causalty constrant Im > reduces to δw f <,.e. to the condton for deal MHD nstablty. Snce δw f s purely real, s purely magnary and the correspondng dscrete spectrum can be dentfed wth that of a gap mode [6],.e. of a mode whose frequency falls wthn the gaps n the shear Alfvén contnuous spectrum. Ths fact can be clearly seen by takng the / t lmt, for whch = p / A. Then, the gap mode would be nsde the, p damagnetc gap [6]. In the present case, the frequency gap structure s complcated by the ncluson of core on compressblty effects. However, t s conceptually the same. The contnuous spectrum s obtaned for purely real [6]. In fact, n ths case the mode egenfuncton n θ space has a purely oscllatory asymptotc behavour and cannot be normalzed: the correspondng egenfuncton n real space has logarthmc sngulartes for qr k =±, as t may be readly verfed cf appendx B. The modes of the Alfvén

8 8 F Zonca et al contnuum result n ncoherent plasma oscllatons, obtaned as superposton of local n real space perturbatons of the type [4] exp rt t where r s a radal-lke flux varable and r s obtaned from the local dsperson relaton qrr k r =± r. 3 Note that r, as obtaned from equaton 3, s generally complex snce s a transcendental functon. Ths fact s remarkable snce t predcts the exstence of unstable contnua for Imr >, whch s mpossble n deal MHD. In fact, ths s also mpossble n knetc MHD when only damagnetc effects are ncluded. The novel feature s entrely due to the ncluson of core-plasma on compressblty n the theoretcal analyss. In order to study the varous modes descrbed by the dsperson relaton, equaton, t s useful to classfy them accordng to the accumulaton ponts of the contnuous spectrum, whch they merge nto when δw f. The accumulaton ponts are obtaned for =, and snce s a transcendental functon there are nfntely many of them. Thus, we wll lmt ourselves to consder the most unstable least stable ones. For smplcty, let us assume / t. In ths case, t may be shown that β = 4 q p + 4 q T q p / /τ + n /. 4 + πq e n n + p/ /τ + n / Here, = ln T / ln n, = / t, p = p / t and the other symbols are analogously defned. An explct expresson for the accumulaton ponts of the contnuous spectrum can be found for =, yeldng where = n and = π q 4 e 5 { = 4 + τq for n q for n. Thus, we see that, n the / t lmt, three accumulaton ponts of the contnuous Alfvén spectrum may be found close to the real frequency axs. Two drectly related to the beta-nduced gap and one assocated wth the on damagnetc gap. Hence, n the followng, we wll call KBM those modes mergng nto the p accumulaton pont when δw f. Smlarly, the modes mergng nto the 4 + τq or 3 4 q accumulaton ponts as δw f wll be referred to as BAE. More precsely, only the branch wth Re > wll be called BAE, snce we are nterested only n modes propagatng n the on damagnetc drecton cf sectons, 3 and appendx A. Approxmate expressons for the contnuum accumulaton ponts can also be found n the general case. For p 4 + τq, the KBM accumulaton pont turns out to be [ = p q + ] πq 5 p 4 + +η e p p

9 Knetc theory of Alfvén modes 9 whereas the BAE accumulaton ponts are gven by = + q + + η q + η /4 + η π n τ p + q 6 + η e = 3 q [ + 4 ] η. 8 Note that the approxmatons leadng to equaton 8 requre.e. large and real. Analogously, for p 4 + τq,wefnd = + 5 π + τ n n n n [ + + τ + 4 ] / τ+ +τ 9 [ = n 4 + 4τ/ τ++ + τ + + +τ ] + 4 / τ+ +τ for the KBM accumulaton pont, whle those related to the beta-nduced gap are gven by = n π q 4 + τ + + τ q 4 e η n 3 = 4 + τ q. Equatons and 8 refer to the stuaton n whch the core-plasma dynamcs s domnated by on damagnetc effects and core-plasma on compressblty may be gnored. Hence, t s not surprsng that the hghest frequency accumulaton pont s close to that of the well known on damagnetc frequency gap,, p. Equatons 9 and 3, meanwhle, correspond to the case where damagnetc effects are small wth respect to core-plasma compressblty [3, 4]. The hghest frequency accumulaton ponts of the Alfvén contnuum, 4 + τq, are now those related to the beta-nduced Alfvén gap. Ths s the lmtng case we must refer to n order to establsh a brdge between the present knetc theory and prevous theoretcal analyses [3, 4], based on deal MHD, whch predct the accumulaton ponts of the beta-nduced Alfvén gap at = γq, where γ s the rato of specfc heats. The next secton s devoted to numercal studes of the dsperson relaton, equaton, to pont out the peculartes of both BAE and KBM modes and to clarfy the strong relaton exstng between these two branches. However, before proceedng further, t s worthwhle analysng the condtons under whch the accumulaton ponts of the contnuous spectrum, mentoned so far, may be located n the upper half complex plane,.e. may become unstable. From a drect check of equatons 9 and 3 t s readly verfed that the KBM accumulaton pont s always stable for p /4 + τq. Ths s not the case for the BAE accumulaton pont that wth Re >. In fact, t may become unstable for larger than a crtcal value c, gven by t c n c τ q n Equaton 3 can be nterpreted as the threshold condton for the onset of an unstable contnuous spectrum. Note that the threshold s only an estmate, although t should gve the correct scalng wth equlbrum parameters. For p 4 + τq, equaton predcts the KBM accumulaton pont to be unstable, although Im s expected to

10 F Zonca et al be exponentally small compared wth that obtaned from equaton 3 for > c. Meanwhle, equaton 8 shows that the BAE accumulaton pont s always stable for p 4 + τq. From the prevous analyss we get a qualtatve pcture of the Alfvén spectra descrbed by the dsperson relaton equaton. When p 4 + τq, only the BAE accumulaton pont may be unstable for >c, wth Im ncreasng lnearly wth and n. If p s further ncreased, the unstable BAE accumulaton pont s expected to smoothly connect to an unstable KBM accumulaton pont, wth exponentally small Im when p 4 + τq. Ths fact allows us to antcpate a result of the next secton;.e. that the most unstable BAE/KBM accumulaton pont occurs at p 4 + τq, when BAE and KBM branches are strongly coupled. 6. Numercal studes of the dsperson relaton In the numercal studes of the dsperson relaton, equaton, we focus our attenton on the gap mode for both BAE and KBM branches, and on the local frequences assocated wth the contnuous spectrum. We beleve that these analyses are suffcent to exhaustvely llustrate the relevant aspects of the low-frequency Alfvén spectrum. Theoretcal studes of the energetc partcle contnuum modes [6] for the BAE and KBM branches wll be presented n a future work. Detaled numercal smulatons of energetc partcle exctatons of KBMs n tokamak plasmas are presented n [5]. The numercal results, presented n the followng, assume β =., τ = T e /T = and q =.5. These parameters are kept fxed and are consdered to be representatve of a typcal tokamak local plasma equlbrum. In fgure, the BAE spectrum s shown for n =. The three curves whch are shown are characterzed by dfferent values of. The branches marked wth open squares correspond to δw f rangng n the nterval.,,.e. to the BAE gap mode. Open crcles refer to the soluton of equaton 3, wth qrr k r varyng between,.,.e. to the contnuous spectrum. The accumulaton ponts of the Alfvén contnuum are vsble as the postons where the gap mode merges nto the contnuous spectrum,.e. where open squares and open crcles overlap. Fgure shows the same parametrc studes reported n fgure, but focused on the KBM branch. For completeness, n fgure 3 we also report the analyss of the branch propagatng n the electron damagnetc drecton, whch confrms that t does not exhbt nterestng features. For ths reason and for those dscussed n appendx A, we shall neglect t n the followng. A drect comparson of fgure and fgure shows how the frequency spectra qualtatvely change wth. More specfcally, the =.5 case dffers from the others, snce t s characterzed by unstable BAE and stable KBM gap modes. Furthermore, the Alfvén contnuum assocated wth the BAE accumulaton pont s clearly unstable. Ths fact confrms the exstence of unstable contnua above a crtcal c and ndcates that ths phenomenon s deeply connected wth a strong couplng between BAE and KBM modes. Fgure 4 shows ths pont more clearly. The four curves are all obtaned for δw f rangng n the nterval.4,.6. Open squares refer to the KBM branch, whereas open crcles ndcate BAE gap modes. The couplng between the two branches s evdent and t ndcates a value of c n the nterval.5,.8; consstent wth the predcton of equaton 3. Fgures 5 and 6 show the same parametrc studes and use the same conventons of fgures and, except that here n =.5. The same consderatons whch were made n the prevous case hold here. In contrast, fgures and 8, where n = 3, exhbt new qualtatve features of the BAE/KBM spectra. Frst of all, the BAE branch never becomes

11 Knetc theory of Alfvén modes.5 =.5 ImΩ.5 = = ReΩ Fgure. The BAE branch s shown for β =., τ =, q =.5 and n =. The BAE spectrum s reported for three values of =,.5,.5. Open squares refer to the gap mode δw f., ; open crcles ndcate the contnuum qrr k r,...5 ImΩ.5 = = ReΩ =.5 Fgure. The KBM branch s shown for the same parametrc studes reported n fgure. unstable; second, even f a value for can stll be dentfed between.5 and, above whch BAE and KBM strongly couple, ths value can no longer be consdered as a threshold for the contnuous spectrum to be unstable. In fact, Im at the most unstable that of KBM accumulaton pont s exponentally small, as predcted by equaton. The dfference wth respect to the prevous cases s entrely due to the value of n. For p < + τq, the 4 features of the BAE/KBM spectrum are those of fgures, and 5, 6. The domnant modes for >c are those of the BAE branch, and n ths case part of the contnuous Alfvén spectrum s unstable. In the p > + τqcase, however, the features of the BAE/KBM 4 spectrum are those of fgures and 8. The domnant modes are of the KBM branch and the Alfvén contnuum s always stable t concdes wth that predcted n deal MHD wth damagnetc effects ncluded. The present dscusson s consstent wth the statement, made n the prevous secton, that the KBM accumulaton pont has an exponentally small Im for p > 4 + τq.

12 F Zonca et al -.5 = ImΩ -. = = ReΩ Fgure 3. The branch propagatng n the electron damagnetc drft drecton s shown for the same parametrc studes reported n fgure. ImΩ.5 =.5 = =.8 = ReΩ Fgure 4. Two spectra wth =.5 and =.8 llustrate the couplng of BAE open crcles and KBM open squares branches. δw f.4,.6 n all cases. Equlbrum parameters are those of fgure. For p + τq, BAE and KBM branches are strongly coupled. In the prevous 4 secton, t was antcpated that n ths parameter range Im of the unstable BAE/KBM accumulaton pont s expected to be peaked. Ths aspect s examned n fgures 9 and, where magnary and real parts of the BAE/KBM accumulaton pont are respectvely shown aganst n and for dfferent values of. Open crcles refer to the =.5 case, open squares to =. and open trangles to =.5. For p + τq, the 4 numercal results are well descrbed by the analytcal estmate, equaton 3, ndcatng that the unstable accumulaton pont s of BAE type note that >c. In fact, Im ncreases lnearly wth both n and, whereas Re lnearly decreases. When p 4 + τq, the strong couplng of BAE wth KBM results n the break down of equaton 3: Re begns to ncrease and the behavour of Im s no longer lnear. Fnally, the dependence of the unstable accumulaton pont on n and becomes of KBM type for p 4 + τq, as predcted by equaton. Thus, fgures 9 and llustrate the smooth transton from

13 Knetc theory of Alfvén modes 3.5 = ImΩ.5 =.5 = ReΩ Fgure 5. The BAE branch s shown for β =., τ =, q =.5 and n =.5. The BAE spectrum s reported for three values of =,.5,. Open squares refer to the gap mode δw f., ; open crcles ndcate the contnuum qrr k r,...5 ImΩ = =.5 = ReΩ Fgure 6. The KBM branch s shown for the same parametrc studes reported n fgure 5. an unstable BAE to an unstable KBM branch, passng through a stuaton characterzed by a spectrum of modes wth a mxed BAE/KBM nature. Ths stuaton, occurrng for p + τq, corresponds to the most unstable regme, as antcpated n the prevous 4 secton.. Conclusons and dscussons In the present work, we have dscussed a comprehensve knetc theory of hgh torodal mode number low-frequency p t Alfvén waves n a hgh-β tokamak equlbrum. By ncludng damagnetc effects and fnte core-plasma on compressblty on the same footng, we have generalzed the KBM theory of Tsa and Chen [6], and have dscussed the mportant relatonshp whch generally exsts between BAE and KBM spectra. BAE and KBM are separate branches of the low-frequency shear Alfvén spectrum, whch are ndependent only at =. In fact, they are coupled n the general case.

14 4 F Zonca et al ImΩ = =.5 = ReΩ Fgure. The BAE branch s shown for β =., τ =, q =.5 and n = 3. The BAE spectrum s reported for three values of =,.5,. Open squares refer to the gap mode δw f., ; open crcles ndcate the contnuum qrr k r,...5 = =.5 ImΩ.5 = ReΩ Fgure 8. The KBM branch s shown for the same parametrc studes reported n fgure. It has been shown that the theory of [6] apples for p 4 + τq t, as expected, a condton under whch KBM are the most unstable modes of those consdered here. More mportantly, we have demonstrated that, for p < 4 + τq t, a crtcal value c exsts, above whch the BAE mode s the most unstable branch, thereby showng that BAEs are not always Landau damped as smple consderatons based on ts frequency would suggest [6]. It has also been shown that, for >c, part of the contnuous Alfvén spectrum may be unstable. Ths mples that non-collectve modes may be present n the plasma, formed as a superposton of local oscllatons whch are quas-exponentally growng n tme. Ths new feature s entrely due to the ncluson of fnte core-plasma on compressblty n the theoretcal analyss. Fnally, t has been shown that the most unstable low-frequency Alfvén modes occur at p 4 + τq t, correspondng to a parameter range n whch BAE and KBM are strongly coupled. The present theory of BAE/KBM lnear stablty yelds an analytc expresson of the mode dsperson relaton, whch may be generalzed to nclude resonant wave exctatons

15 Knetc theory of Alfvén modes 5. ImΩ =.5 =.. = Ω *n Fgure 9. Im of the most unstable BAE/KBM accumulaton pont s shown aganst n for β =., τ =, q =.5, δw f = and dfferent values of : =.5 open crcles, =. open squares and =.5 open trangles =. ReΩ =.5.5 = Ω *n Fgure. Re of the most unstable BAE/KBM accumulaton pont s shown aganst n for the same parameters of fgure 9. by energetc partcles wth fnte orbt wdths [6]. It has been shown that, n general, two types of spectra may exst: a dscrete spectrum of gap modes, whch s present only when the tokamak plasma s deal MHD unstable; and the Alfvén contnuous spectrum, whch may be characterzed, under certan condtons, by unstable accumulaton ponts. The ssue of energetc partcle contnuum modes [6], whch may also exst for δw f >, but requre that the energetc partcle drve be strong enough to overcome contnuum dampng, s not analysed here. The numercal studes of the analytc dsperson relaton deal only wth BAE/KBM gap modes and wth the Alfvén contnuous spectrum. Fnally, t s clear that the present theoretcal results may have mportant mplcatons for the stablty and transport propertes of tokamak experments wth and wthout energetc partcles. Further theoretcal delneatons and comparsons wth expermental results are, however, beyond the scope ntended for ths work and wll be dscussed n future publcatons.

16 6 F Zonca et al Acknowledgments Ths work was supported by the US Department of Energy, DOE DE-FG3-94ER54, the Natonal Scence Foundaton, NSF grant ATM , and the Unversty of Calforna Energy Insttute, UCEI-594. One author, FZ, s grateful to the Department of Physcs and Astronomy, Unversty of Calforna at Irvne for the hosptalty and support receved durng the perod n whch part of ths work was done. Appendx A. Valdty of the deal MHD lmt δe = At order Oβ, the quas-neutralty equaton, equaton 5, becomes τ + n δ δ + p b δ = T k k ρ L δk + δk. nek ϑ 4 A Here, δk s obtaned from tr θ δk = tr θ + d δk ekϑ + m k [ QF δ δ k ρ L δ δ + m v ] 4 eb δ ˆB. A In order to get from equaton A an expresson for δ δ vald up to Oβ, we need to solve equaton A just for δk, where [...] = /π... dθ. It s possble to show that [ δk ekϑ QF k = ρl δ δ m v ] m k 4 eb δ ˆB tr θ δk + d δk. A3 When substtuted nto equaton A, equaton A3 yelds the Oβ quas-neutralty condton, reported n secton 3: t p k [ kϑ k ϑ { + θ p ] δ δ + k τ + n b + q [ t b n ]} δ T G/ t N / t D/ t = b/ α t k ϑ 3/ q k F/ t δ δ p δ. A4 Equaton A4 demonstrates that δ δ = Oβ,.e. δe, for long wavelength modes θ = Oβ / wth p / = O and whch propagate n the on damagnetc drecton /τ + n / = O. In the present paper, we refer to ths type of mode. For waves whch propagate n the electron damagnetc drecton, equaton A4 predcts that an apprecable parallel electrc feld can be orgnated for /τ + n / = Ob. Another possblty for the δe assumpton to break down s p / = O/β. These modes would nteract strongly wth the long wavelength slab-lke on temperature gradent ITG drven mode [ 9] and are out of the scope of the present analyss.

17 Knetc theory of Alfvén modes Equaton A4 serves also to the scope of dscussng the valdty of the δe = assumpton at moderate θ values. In fact, for θ < Oβ / fnte b and d / effects may be neglected, ndcatng that the scale of varaton of δ δ s a constant. Ths can be also seen from a drect examnaton of equatons and 5. Moreover, snce the large θ structure of the modes we are analysng s characterzed by a long scale of varaton, we must also have θ = constant = Oβ / when actng on δ δ for θ < Oβ /. Recallng that /τ + n / and p / are O n our analyss, the present argument and equaton A4 ndcate that δe = to lowest order also for moderate θ values,.e. δ δ =. The fact that θ = Oβ / when actng on δ δ does not ndcate that the functons δ and δ vary on the long scale θ only, but smply that any short-scale varaton of δ s balanced by the same varaton of δ. The prevous dscusson demonstrates that the δ δ = assumpton also holds n the moderate θ regon, where δ does not have a two-scale structure, as ponted out n secton 3., and, n general, θ = O due to fnte s, α effects, as t emerges from equaton 5. Appendx B. Logarthmc sngulartes and contnuous spectrum A smple dervaton of equaton 3 can be obtaned transformng the balloonng egenfuncton δψθ = [k ϑ /k θ]δ θ back to real space [] + δ θ exp[ nq mθ] δψnq m = dθ. B [ + sθ α sn θ ] / From equaton B, t s easly shown that s δψnq m nq m = + sθδ θ exp[ nq mθ] [ + sθ α sn θ ] / dθ. B Defne, now, a value such that and β /. In ths way, the ntegraton nterval n equaton B can be subdvded nto [, ], where the deal soluton δ ID can be used, and, ] [,+, where the nertal layer soluton δ IN s approprate. Wth the explct use of equaton 8, t s possble to show s δψnq m nq m = sθδ ID θ exp[ nq mθ] [ + sθ α sn θ ] / dθ + [exp m nqθ exp nq mθ]e θ dθ α +O + O + O. B3 s s Snce the contrbuton from the [, ] ntegral s typcally O, whereas that from the [, nterval s Oβ / for nq m =Oβ /, equaton B3 mples [ ] s δψnq m exp m nq exp nq m = e + O nq m nq m + nq m s α +O + O + Oβ /. B4 s Equaton B4 s readly ntegrated and yelds δψnq m = C s ln[ nq m ] B5

18 8 F Zonca et al for nq m =Oβ /, where C s an ntegraton constant and Oβ / terms have been consstently neglected. Equaton B5 ndcates that the wave feld n real space has a logarthmc sngularty when nq m =±,.e. when qrr k r =± r whch s equaton 3. In an ntal-value analyss, the just derved logarthmc sngularty becomes a logarthmc branch pont n the complex omega plane. When the tme asymptotc behavour s computed, the fnte jump of the wave feld across the branch cut, orgnatng from the logarthmc branch pont, gves the contrbuton t exp rt of equaton. A detaled analyss, yeldng to ths result, can be found n [4]. Here, we just wsh to note that the concepts of contnuous spectrum, satsfyng equaton 3, and of sngular local plasma oscllatons are closely related. References [] Hedbrnk W W, Strat E J, Chu M S and Turnbull A D 993 Phys. Rev. Lett. 855 [] Cheng C Z, Chen L, Chance M S 985 Ann. Phys. 6 [3] Turnbull A D, Strat E J, Hedbrnk W W, Chu M S, Greene J M, Lao L L, Taylor T S and Thompson S J 993 Phys. Fluds B [4] Chu M S, Greene J M, Lao L L, Turnbull A D and Chance M S 99 Phys. Fluds B 4 33 [5] Bglar H and Chen L 99 Phys. Rev. Lett [6] Tsa S T and Chen L 993 Phys. Fluds B [] Chen L, Brguglo S and Romanell F 99 Phys. Fluds B 3 6 [8] Romanell F, Chen L and Brguglo S 99 Phys. Fluds B [9] Guo S C and Romanell F 994 Phys. Plasmas [] Connor J W, Haste R J and Taylor J B 98 Phys. Rev. Lett [] Antonsen T M and Lane B 98 Phys. Fluds 3 5 [] Chen L and Hasegawa A 99 J. Geophys. Res [3] Romanell F and Chen L 99 Phys. Fluds B 3 39 [4] Sedlà cek Z 9 J. Plas. Phys [5] Santoro R A and Chen L 996 Phys. Plasmas [6] Bett R, Berk H L and Fredberg J P 994 Bull. Am. Phys. Soc. 39 [] Zonca F and Chen L 993 Phys. Fluds B