Cyclotron waves in a non-neutral plasma column

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1 Cyloton waves in a non-neutal plasma olumn Daniel H. E. Dubin Citation: Phys. Plasmas, 41 (13); doi: 1.163/ View online: View Table of Contents: Published by the Ameian Institute of Physis. Additional infomation on Phys. Plasmas Jounal Homepage: Jounal Infomation: Top downloads: Infomation fo Authos:

2 PHYSICS OF PLASMAS, 41 (13) Cyloton waves in a non-neutal plasma olumn Daniel H. E. Dubin Depatment of Physis, Univesity of Califonia at San Diego, La Jolla, Califonia 993, USA (Reeived Febuay 13; aepted 4 Apil 13; published online 5 Apil 13) A kineti theoy of linea eletostati plasma waves with fequenies nea the yloton fequeny X s of a given plasma speies s is developed fo a multispeies non-neutal plasma olumn with geneal adial density and eleti field pofiles. Tems in the petubed distibution funtion up to Oð1=X s Þ ae kept, as ae the effets of finite yloton adius up to Oð Þ. At this ode, the equilibium distibution is not Maxwellian if the plasma tempeatue o otation fequeny is not unifom. Fo!, the theoy epodues old-fluid theoy and pedits sufae yloton waves popagating azimuthally. Fo finite, the wave equation pedits that the sufae wave ouples to adially and azimuthally popagating Benstein waves, at loations whee the wave fequeny equals the loal uppe hybid fequeny. The equation also pedits a seond set of Benstein waves that do not ouple to the sufae wave, and theefoe have no effet on the extenal potential. The wave equation is solved both numeially and analytially in the WKB appoximation, and analyti dispesion elations fo the waves ae obtained. The theoy pedits that both types of Benstein wave ae damped at esonanes, whih ae loations whee the Dopple-shifted wave fequeny mathes the loal yloton fequeny as seen in the otating fame. VC 13 AIP Publishing LLC. [ I. INTRODUCTION This pape onsides linea plasma waves nea the yloton fequenies of a multispeies ion plasma olumn with nea-maxwellian veloity distibutions. We fous on the z-independent omponent of the plasma esponse in the eletostati (non-elativisti) limit, in ode to simplify the analysis and make onnetions to expeimental systems that measue this omponent. A boad ange of devies use the eletial signal indued by this plasma esponse in ode to diagnose the hage to mass atio and/o the elative onentation of plasma speies 1 (via the tehnique of ionyloton mass spetomety ). While most of these devies wok in the low-density egime whee plasma effets ae small, yloton fequeny shifts univesally aise fom eleti fields that an oiginate eithe fom the plasma o potentials applied to eletodes, as will be disussed hee. Pevious papes,3 have desibed theoy of the eletostati plasma esponse nea the yloton fequeny, in the low tempeatue old-fluid limit, whee themal effets ae not inluded. It was found that the old fluid plasma esponse is peaked at fequenies assoiated with sufae yloton waves eletostati plasma waves that popagate azimuthally along the sufae of the plasma olumn as expði h ixtþ, with azimuthal mode numbe and fequeny x nea the yloton fequeny X s ¼ q s B=m s fo a given speies s. The diffeene between x and X s aises fom a Dopple shift and a Coiolis foe effet due to plasma otation, and fom plasma effets popotional to the density n s of speies s. The fequeny x an then be used to diagnose the plasma otation fequeny, the density of the speies, as well as the yloton fequeny (whih detemines the hage to mass atio of the speies, the main inteest in mass spetomety). In othe wok, 4 6 effets of finite tempeatue wee also onsideed. It was obseved that eletostati Benstein waves an be exited in addition to the old fluid sufae waves. These waves popagate both adially and azimuthally within the plasma, with fequenies that depend on the yloton fequeny, as well as the plasma density and tempeatuept, whih entes though the yloton adius ¼ ffiffiffiffiffiffiffiffiffiffi T=m s =X s. An appoximate dispesion elation fo the Benstein waves was deived in Refs. 5 and 6, based on WKB analysis. In this pape, we pesent a theoy of the plasma esponse nea the yloton fequeny, whih desibes both the sufae yloton waves and the Benstein waves in the egime x p =X s 1, whee x p is the plasma fequeny. A wave equation fo the petubed plasma potential is deived assuming =L1 and k 1 (whee k is the wavenumbe of the esponse and L is the adial sale length of the equilibium plasma), whih inludes both the sufae yloton and Benstein waves as solutions. This equation is solved numeially, as well as though the WKB appoximation, whih is valid povided kl 1, and we extend this WKB solution though to the egime k 1. The Benstein waves ae efleted at loations whee thei fequeny equals the plasma s uppe hybid fequeny and an then set up nomal modes inside the plasma olumn. We find that the dispesion elation fo the Benstein nomal modes is modified fom the qualitative esult of Ref. 6 due to linea mode oupling between these modes and the eletostati sufae yloton waves. This oupling also allows the Benstein modes to be obseved via thei effet on the extenal eletostati potential, whih an be piked up using eletodes. An expession is deived fo the eletode signal, whih an exhibit peaks at the Benstein mode fequenies, onsistent with expeiments. 5 (Pevious theoy ould not X/13/(4)/41/5/$3., 41-1 VC 13 AIP Publishing LLC

3 41- Daniel H. E. Dubin Phys. Plasmas, 41 (13) explain this phenomenon.) This effet is simila to the linea mode oupling between eletomagneti waves and Benstein waves that is known to ou at the uppe hybid esonane in neutal plasmas, of impotane to yloton heating and uent dive in magneti fusion appliations. 7 1 We also find a seond set of Benstein modes that do not ouple to the sufae yloton waves. These modes ae intenal to the plasma, having no effet on the extenal potential. To deive the wave equation, we must fist deive seveal new esults fo the ylindial plasma equilibium. Fist, in Se. II, we solve fo haged patile motion in the equilibium adial eleti field of the plasma, keeping finite yloton adius effets, whih ae neessay to desibe the finite-tempeatue Benstein modes. In so doing, we obtain finite eleti field and finite yloton adius oetions to the patile yloton fequeny and the dift otation fequeny. Next, in Se. III, we obtain a losed-fom expession fo the ollisional quasi-equilibium veloity distibution of the otating plasma, fo given density, tempeatue, and adial eleti field pofiles, keeping finite-yloton adius effets. The system evolves to this quasi-equilibium distibution due to ollisions between the plasma hages. The distibution deviates fom Maxwellian due to adial vaiations in the plasma otation fequeny and tempeatue; eventually on a longe tanspot timesale, these vaiations ae wiped out by visosity and themal ondution, but duing an intemediate timesale between the ollision time and the tanspot time, they ae pesent and affet the veloity distibution. Next, in Se. IV, we deive a geneal dispesion elation fo linea eletostati waves on this nea-maxwellian quasiequilibium by lineaizing and solving the Vlasov equation with an added Kooks ollision opeato. The solution is obtained fo geneal adial density, tempeatue, and eleti field pofiles. In Se. V, we fous on the plasma esponse fo z-independent petubations nea the yloton fequeny of a given speies, deiving the afoementioned wave equation, whih keeps tems to fist-ode in k. In Se. VI, we eview the old-fluid theoy of solutions to this equation (the! limit), disussing the sufae yloton waves that ae pedited to appea unde vaious senaios. In Se. VII, we add finite tempeatue and in Se. VIII, we onside WKB solutions to the wave equation. In Se. IX, we onside the behavio of the WKB solutions fo a few examples. II. PARTICLE ORBITS Conside the obit of a patile with hage 6q and mass m in a unifom magneti field 7B^z and a ylindially symmeti potential / ðþ. Hee, q and B ae positivedefinite quantities. Fo positive (negative) hages, we assume a magneti field in the ðþþz dietion, so that vaious fequenies (yloton, dift otation) ae positive fo eithe sign of hage; i.e., the esultant iula motions assoiated with eah fequeny ae ounte-lokwise when viewed fom a loation on the z axis above the obit. The Hamiltonian fo this patile, expessed in ylindial oodinates ð; h; zþ, is Hð; pþ ¼ p m þ p h þ qb þ p z m þ / ðþ; (1) m whee p ¼ m _; p h ¼ m ð h _ X =Þ; and p z ¼ m_z ae the momenta anonially onjugate to, h, andz, espetively, and X ¼ qb=m is the bae yloton fequeny of an isolated patile. Note that / has units of potential enegy; it is q times the eletostati potential. This potential an aise fom voltages applied to ylindially symmeti eletodes o fom a ylindially symmeti equilibium distibution of plasma hages, whih podues a mean-field equilibium potential. In this ase, H is the mean field Hamiltonian fo the motion of a hage in the stati field podued by the othe hages. Howeve, / ðþ an also aise fom the inteation of a hage with its own image in the ylindial eletodes of the tap, even in the absene of othe hages. 11 This image hage potential is typially weak ompaed to othe potentials and is often negleted, but it should be kept in highpeision wok. 1 Fo example, fo a point hage q at adius within a hollow ylindial onduto of adius w,thisimage potential is most easily expessed as an integal: / image ðþ ¼ q w X 1 ¼ 1 p ð1 dx I x K ðxþ w I ðxþ ; () whee I and K ae modified Bessel funtions. Expessions fo the image potential fo othe eletode geometies, both ylindially symmeti and asymmeti, an be found in Ref. 11. The Hamiltonian given in Eq. (1) is sepaable, with thee onstants of the motion p h, p z,andh? ¼ H p z =m. We will find Þ it useful to eplae the onstant H? by the ation l ¼ 1 p p d, whee the line integal is pefomed along the losed adial patile obit. Sine p an be expessed as a funtion of H? ; p h, and via Eq. (1), this implies l ¼ lðh? ; p h Þ. Inveting this elation yields H? ¼ H? ðl; p h Þ, the pependiula Hamiltonian witten in tems of the ation. When the magneti field is lage, this tansfomation an be aomplished petubatively in an expansion in 1/B via Hamiltonian petubation theoy. This expansion equies that the yloton fequeny assoiated with adial patile osillations is lage ompaed to the othe motional fequenies, in patiula the dift fequeny assoiated with h motion in the adial eleti field. As a oollay, this also equies that the spatial sale length of the eleti field, L, be lage ompaed to the yloton adius, so that we an pefom Taylo expansions of the field aound the guiding ente position. (This latte equiement is sometimes violated in yloton mass spetomety, whee lage amplitude yloton motion an be diven by extenal fields.) The esult, good to ode 1=B 4,is H? ðl; p h Þ¼lXð Þþ/ ð Þþ e E ð Þ mx 15 Eð Þ 3 15 E ð Þ þ e4 E 3 ð Þ m X 4 þ l e 4 8m X 5 E ð Þ 1 E ð Þ þoðe 6 Þ; (3)

4 41-3 Daniel H. E. Dubin Phys. Plasmas, 41 (13) whee e is an odeing paamete used to keep tak of the ode in 1/B of diffeent tems, pimes denote deivatives with espet to, EðÞ =@, and ðp h Þ is the adial loation of the effetive potential minimum, i.e., the minimum of / ðþþðp h þ qb =Þ =m, as given by the solution to R 4 ¼ 4 4Eð Þ 3 =mx ; (4) p whee R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p h =ðqbþ, a onstant of the motion (the lowest-ode guiding ente adius). The shifted yloton fequeny Xð Þ is given by the (exat) expession X ð Þ¼X 3Eð Þ E ð Þ m m : (5) Note that the ation l appeaing in Eq. (3) is of ode e. In fat, this equation implies that to lowest ode in e, l e mv? =X, the well-known expession fo the yloton ation. Also, Eq. (5) shows that the yloton fequeny is shifted by the adial eleti field. In expeiments with low plasma density o even single patiles, the shift aises pedominantly fom applied tap potentials and/o image hages. In this pape, we obtain fequeny shifts to olletive plasma modes (as well as single patile fequenies), inluding plasma effets, applied potentials, and image hages. The tansfomation fom ð; p ; h; p h Þ to the ation-angle vaiables ðw; l; h; p h Þ used in Eq. (3) an be aied out with geneating funtions W and W h whee 13 W h p h h; (6) ð W ð; l; p h Þ p ð; H? ; p h Þ; (7) and we have used H? ¼ H? ðl; p h Þ. These geneating funtions elate the new and old oodinates via Inveting, this yields ðw þ W ð; l; p hþ: (8) ¼ ðw; l; p h Þ þ dðw; l; p h Þ; (9) whee the seond fom is atually a definition of d, the deviation of fom due to finite yloton adius effets. Also, we have h ðw þ W h h ð; l; p h Þ; (1) whih we an eaange as whee h ¼ h þ dhðw; l; p h Þ; (11) dhðw; l; p h ððw; l; p h Þ; l; p h Þ: (1) Petubation analysis, desibed in Appendix A, povides us with expliit expessions fo d and dh: d ¼ X1 n¼ dh ¼ X1 n¼1 e n D n ðq; ; eþos nw; (13) e n Dh n ðq; ; eþsin nw: (14) Hee, D n and Dh n ae given as powe seies in e up to Oðe 4 Þ in Table I, and q l=mx. The paamete q is, to lowest ode in 1/B, the adius of the yloton obit. The oeffiient D is the adial hange in guiding ente position due to finite yloton adius effets. Note that fo n 1, both D n and Dh n ente d and dh at Oðe n Þ. The invese of these tansfomations an also be witten as powe seies in e. In patiula, lð; v ; v h Þ is given by l ¼ em v þ v h X e EðÞ v h X þ e 3 mv h ðeðþþ3e ðþþ þ mv ð3eðþþe ðþþ þ EðÞ 4mX þ Oðe 4 Þ: (15) The Hamiltonian of Eq. (3) implies that the angle vaiables h and w, and the oodinate z evolve in time aoding to dw ¼ Xð ; qþ; (16) d h h ¼ x ð ; pþ; (17) dz dt ¼ p z m ; (18) whee the fequenies X and x ae given by the seies expessions TABLE I. Obit oeffiients in Eqs. (13) and (14). n 3 4 e q e 4 q 4 þ e4 q 3 mx 1 q þ e q3 8 q þ e 3 q D n 3 EðÞ 8 þ 9 E ðþ 8 þ q ð 3 mx 4 e q ð 1 mx 8 EðÞ EðÞ þ 3 8 q 3 3 þ Oðe Þ q 4 64 þ Oðe Þ 3 n Dh n q 1 þ e 1 q þ e 1 q q 3 þ q 3 q 8 þ 1 4 E ð Þ þ Oðe 6 Þ þ 1 4 Eð ÞÞ þ Oðe 4 Þ E ðþ þ 1 1 E ð ÞÞ þ Oðe 4 Þ EðÞþ3E ðþ 4mX 3EðÞþ5E ðþþ E ðþ=3 mx q 3 þ Oðe Þ q4 þ Oðe Þ 4 4 þ Oðe 4 Þ þ Oðe 4 Þ

5 41-4 Daniel H. E. Dubin Phys. Plasmas, 41 (13) Xð ; qþ ¼ Xð Þ þ e3 q e 8mX 15 Eð Þ 3 15 E ð Þ 5E ð Þ 1 E ð Þ þ Oðe 5 Þ; (19) x ð ; qþ ¼ dp and whee l;pz ¼ ex E ð Þþ e3 x E ð Þ e3 q X 4mX 3Eð Þ 3E ð Þ E ð Þ þ Oðe 5 Þ; () x E ð Þ Eð Þ mx : (1) The dift otation fequeny x is, to lowest ode in 1/B, given by the E B dift otation fequeny x E in the adial eleti field. The seond tem in Eq. () is a oetion due to entifugal foe, whih ats as an exta adial foe that auses an F B dift in the h-dietion. The tems popotional to q ae finite yloton adius oetions to the otation ate. The yloton fequeny X also has finite yloton adius oetions. Howeve, when ompaing this expession to pevious expessions fo the yloton fequeny in the pesene of an eleti field, 14 it is impotant to emembe that hee the fequeny is deived as the ate of adial osillations, whih is not the same as the yloton otation ate with espet to fixed Catesian axes sine the dietion of the adial unit veto vaies in time as the patile moves in h. Thus, single patile esonanes an be shifted fom the yloton fequeny X by (multiples of) the otation fequeny x. In fat, when subjeted to extenal fields vaying in, h, and t as d/ðþe i h ixt, we will see that patiles an absob enegy esonantly when the applied fields ae at the fequenies x ¼ nx þ x fo any intege n. The esonant inteation with n ¼ 1 at a fequeny nea X is typially the stongest esonane and is the main effet obseved in ionyloton mass spetomety fo low density systems. Howeve, fo highe densities, thee ae olletive eletostati plasma waves that an be exited. These olletive exitations ae the subjet of the next setions. One type of system fo whih these fequeny fomulae simplify is the hamoni tap whee to a good appoximation (and negleting z dependene, valid fo patiles moving in the z ¼ plane), / ðþ /. Then, the finite yloton adius oetions to X and x vanish in Eqs. (19) and (), and these fequenies ae independent of adial position, whih simplifies the analysis. This is one eason why hamoni taps ae often pefeed in ion-yloton spetomety appliations. (Fo a hamoni tap, ou fequenies X and x ae elated to the standad hamoni tap fequenies x þ and x, 15 via the fomulae x ¼ x and X ¼ x þ x.) Of ouse, even in taps designed so that the vauum field is hamoni, effets suh as plasma spae hage and image hages an add anhamoni oetions to /, neessitating inlusion of the fequeny oetions desibed by Eqs. (19) and (). III. EVOLUTION OF THE DISTRIBUTION FUNCTION We assume that the patile distibution f ð; v; tþ fo a single speies plasma evolves aoding to the þ v f þ q m E þ v ¼ Cðf ; f Þ; whee C is the patile Boltzmann ollision opeato. We fist onside the ylindially symmeti quasiequilibium distibution pedited by Eq. (). Negleting ollisions, the ollisionless Boltzmann equation has timeindependent solutions of the geneal fom f ¼ f ðl; p h ; p z Þ (3) sine l, p h, and p z ae onstants of the ollisionless motion desibed by Eq. (1). Any funtion of these onstants of the motion is a ollisionless (Vlasov) equilibium. Howeve, when ollisions ae taken into aount, this distibution evolves on the timesale of the ollision ate to a quasiequilibium nea-maxwellian distibution whose dependene on the onstants of motion is detemined by the ollision opeato. 16 Howeve, the tempeatue, density, and otation ate of the equilibium an have abitay adial dependene. [This quasi-equilibium then poeeds to evolve in time on a slowe tanspot timesale due to adial fluxes diven by gadients in the plasma otation fequeny and the plasma tempeatue, towad a themal equilibium state with no suh gadients. We neglet this slow evolution hee.] The deivation of the quasi-equilibium distibution funtion is outlined in Appendix B. Assuming that the tempeatue gadient is of ode e while density and otation fequeny gadients ae of O(1), the quasi-equilibium is, to Oðe 4 Þ, NðRÞ f qe ðl; p h ; p z Þ¼ exp H=TðRÞ 3= ðptðrþ=mþ f lx ð1=t l ðrþ 1=TðRÞÞ þ Oðe 4 Þg; (4) p whee R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p h =mx ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v h =X is the lowestode guiding ente adius [see Eq. (4)], the funtion N(R) is elated to equilibium density n() and potential / ðþ, NðRÞ nðrþ expf½/ ðrþ 1=mR e x ðrþš=tðrþg; (5) the yloton tempeatue T l ðrþ is elated to the paallel tempeatue T(R) though T l ðrþ TðRÞ 1 þ e ; (6) and x ðrþ is the fluid otation fequeny, defined as

6 41-5 Daniel H. E. Dubin Phys. Plasmas, 41 (13) ð nðþx ðþ ¼ d 3 vv h f qe : (7) [Note that in these veloity integals, is held fixed, not R. The expession fo f qe given in Eq. (B5), while less elegant than Eq. (4), is easie to integate ove veloities.] Pefoming the veloity integals yields the following expession fo x up to Oðe 3 Þ: x ðþ ¼ e mx ðtðþnðþþ þ EðÞ þ e 3 " x T X mx x þ Oðe 5 Þ: (8) The lowest ode fluid otation fequeny in Eq. (8) is the familia expession fo diamagneti and E B difts. 17 The Oðe 3 Þ oetions ae due to entifugal foe (the fist tem) and themally aveaged finite yloton adius oetions due to shea in the fluid otation (the nd and 3d tems). Although x appeas on both sides of this expession, to Oðe 3 Þ one an use the lowest-ode dift expession fo x on the ight hand side to obtain an expliit expession fo x in tems of density, potential, and tempeatue. When T and x ae unifom (independent of ), Eq. (4) edues to the themal-equilibium fom N f qe ¼ ðpt=mþ 3= e H=Tþx p h =T ; (9) whee N is a onstant. This an be seen by noting that, when x and T ae onstant, Eqs. (5) and (8) imply that mx ¼ x ; (3) whih implies that in themal equilibium, NðRÞ ¼N e x p h =T. When applied to Eq. (4), this leads to Eq. (9). Howeve, when T and/o x ae not unifom, Eq. (4) is not a Maxwellian distibution due to the dependene of R on v h. Collisions dive the system as lose to a Maxwellian as possible when vaiations in x and T ae pesent. The non- Maxwellian natue of f qe is esponsible fo the diffeene between yloton tempeatue T l and paallel tempeatue T. With the definition of T l given by Eq. (6), the mean kineti enegy pe patile in eah degee of feedom, as detemined by veloity integation ove f qe, is given by hmv i¼tðþ 1 þ þ Oðe4 Þ ; (31) hmðv h x Þ i¼tðþ 1 þ Oðe4 Þ ; (3) and hmv z i¼tðþ: (33) Thus, the mean tansvese themal enegy h1=4 mðv þðv h x Þ Þi is equal to the mean paallel themal enegy hmv z =i. This is equied in quasi-equilibium; othewise thee will be equipatition of paallel and pependiula themal enegy on a ollisional timesale. (Note that the diffeenes in mean adial, axial, and azimuthal themal enegy ould, in piniple, be obseved using, say, lase dopple diagnostis.) IV. GENERAL DISPERSION RELATION FOR LINEAR WAVES We now onside small petubations df s ð; v; tþ of f s ð; v; tþ away fom the quasi-equilibium f qes given by Eq. (4). Hee, we e-intodue a speies label s fo speies with mass m s and hage q s. The petubations ae desibed by lineaization of the multispeies vesion of the Boltzmann equation, Eq. (). Fo simpliity, we use a simple Kook ollision opeato of the fom C ¼ ðf s f qes Þ. Substituting and into Eq. () and lineaizing, we obtain f s ¼ f qes þ df s ð; v; tþ (34) / ¼ / ðþþduð; tþ (35) d dt df s þ df s @w h ; (36) whee d þ v þ EðÞ m ^ v (37) is a deivative taken along the obit of the Hamiltonian of Eq. (1). The ight-hand side of Eq. (36) is witten in tems of ation-angle vaiables. Next, we Fouie analyze df and du in h, z, and t, witing these funtions as and df s ð; v; tþ ¼dF s ð; vþ e i hþik zz ixt (38) duð; tþ ¼d/ðÞ e i hþik zz ixt : (39) Applying Eqs. (9) and (11), we define d/ð þ dþ e i dh X1 n¼ 1 D/ n ðp h ; lþ e inw ; (4) whee the D/ n ae the Fouie oeffiients of the left-hand side, whih is peiodi in w [see Eqs. (13) and (14)]. We may then integate Eq. (36), obtaining e i hþikzz ixt df s ð; v ; v h ; v z Þ ¼ X1 D/ n ðp h ; lþ þ qe þ ik z n¼ 1 e t ð t 1 dt e inwðt Þþi hðt Þþik z zðt Þ iðxþiþt ; (41)

7 41-6 Daniel H. E. Dubin Phys. Plasmas, 41 (13) whee wðt Þ; hðt Þ, and zðt Þ ae detemined by the equations of motion (16) (19) wðt Þ¼w þ X s ð ; qþðt tþ; hðt Þ¼h þ x s ð ; qþðt tþ; zðt Þ¼z þ v z ðt tþ: (4a) (4b) (4) Hee, the onstants of integation w and h and the onstant of motion l ae detemined in tems of, h, v, and v h by the onditions h þ dhðw ; ; qþ ¼h; (43) þ dðw ; ; qþ ¼; (44) d _ðw ; ; qþ ¼v : (45) Equations (44) and (45) detemine w and l using Eqs. (4) and (13), and then Eq. (43) detemines h. Note that no equation fo v h is neessay sine v h is detemined in tems of and p h by p h ¼ mv h qb =. Of ouse, l is also detemined by,v, and v h via Eq. (15). Using Eqs. (4) in Eq. (41), the time integal an be pefomed, yielding df s ð; v ; v h ; v z Þ¼ X1 n¼ 1 e i dhðw ; ;qþþinw D/ n ðp h ; lþ qe qe qes þ k z nx s ð ; qþþ x s ð ; qþþk z v z x i : (46) The esonant denominato in Eq. (46) povides an expession fo the fequeny x at whih thee is a stong wave-patile esonant inteation, as we disussed at the end of Se. II. Finally, the dispesion elation fo d/ is obtained by substituting Eqs. (38) and (39) into Poisson s ¼ 4pe X s þ k z d/ðþ ð d 3 v df s ð; v ; v h ; v z Þ: (47) This intego-diffeential equation fo d/ an be solved subjet to the bounday onditions on d/. With egad to Eqs. (46) and (47), we note that the deivatives with espet to l and p h ae easiest to evaluate using the fom of f qes given in Eq. (4), but the veloity integals ae easiest to evaluate using the equivalent fom given in Eq. (B5). V. CYCLOTRON MODES FOR SPECIES s We fous on z-independent yloton modes, assuming k z ¼. Thee ae yloton modes nea multiples n of the yloton fequeny fo eah speies. In this pape, we onside only the modes fo whih n ¼ 1, nea the yloton fequeny X s of speies s, with x ¼ X s þ OðeÞ. Substituting Eq. (4) [o Eq. (B5)] fo f qes into Eq. (47), expanding the integand in e, and aying out the veloity integals, we keep enough tems in the seies expessions so that finite tempeatue oetions to the dispesion elation ae obtained. These oetions ente at Oðe Þ, so, noting that x ¼ OðeÞ and D/ n ¼ Oðe n Þ fo n 6¼, analysis of Eq. (47) implies that tems in the sum ove n in df s an be dopped only fo jnj > that f qes must be evaluated inluding tems up to Oðe 3 Þ and that D/ n must be evaluated up to Oðe 4 Þ. The petubed density fo speies s an then be evaluated by pefoming the veloity integal ove df s ð d 3 vdf s ¼ dn Fs ðþþdn Ts ðþ; (48) whee dn Fs ðþ is the T ¼ old fluid density esponse to the petubed potential d/, and dn Ts ðþ is the T > themal oetion. The old fluid density petubation an also be deived dietly to all odes in e fom fluid equations 6 and has the fom 4pq s dn F s ðþ ¼ x p s ðþ ðx vs x F s ðþþd/ ðþþ ð^x þ iþd/= ^x X vs ðx vs x F s ðþþ ð^x þ iþ " # þ x X vs d/= þð^x þ iþd/ ðþ s ðþ X vs ðx vs x F s ðþþ ð^x þ iþ ; (49) whee ^x x x Fs ðþ is the Dopple-shifted fequeny, X vs X s x Fs, x p s ðþ 4pq s n sðþ=m s is the squae of the speies s plasma fequeny, and x Fs ðþ is the old-fluid otation fequeny given by the solution to the equation x Fs ðx s x Fs Þ¼ P s x p s ðþ. Howeve, the veloity integal in Eq. (47) yields an expansion in e of this geneal expession, inluding tems of Oðe Þ: 4pq s dn F s ðþ ððd 1ÞuðÞÞ ðd 1Þ u þ Oðe Þ; (5) whee the field amplitude uðþ is defined as uðþ d/ ðþþ d/=; (51) DðÞ 1 bðþ aðþ is the dieleti onstant fo the speies s yloton modes, (5) aðþ x X s þð Þx E þ x EðÞ= þ i (53) is a fequeny offset, and bðþ x p s ðþ X s (54) epesents the loal speies density n s ðþ, expessed as the equivalent E B otation ate fo a unifom n s. We do not display the Oðe Þ tems in the old fluid density petubation as they ae faily omplex and will not be needed in what follows. The fequeny offset a is, in fat, (the negative of) the

8 41-7 Daniel H. E. Dubin Phys. Plasmas, 41 (13) esonant denominato that appeas in Eq. (46) fo X ¼ 1, expanded to lowest ode in e with the assumption that x ¼ X s þ OðeÞ. This implies that a is also of OðeÞ. Next, we onside the themal density oetion fo speies s, whih takes the fom 4pq s dn T s ðþ ¼ C 1 þ C a þ C 3 a 3 þ C 4 a 4 ; (55) whee C 1 ¼ b a d/ b u a ; (56) C ¼ x E ð ð 5Þ þ þa bu þ b u bu b u b u a b u þ bu ð Þ bu a bu; (57) C 3 ¼ a bu þ 3b u þð3 Þ bu þ a a x ð 1Þ E bu; (58) whee C 4 ¼ 3a 3 bu; : (6) Finally, we note that Eq. (49) implies that fo a diffeent speies s 6¼ s with X s X s ¼ Oð1=eÞ, dn Fs ¼ Oðe Þ. Also, we find the themal oetions to dn s ae even highe ode in e. We, theefoe, neglet dn s when solving Eq. (47), so only the density petubation fo speies s need be kept fo speies s yloton waves. Futhemoe, sine Eqs. (6) and (51) imply d/ ðuþ u ; (61) the themal density oetion fo speies s an be witten entiely in tems of u and its deivatives up to thid ode. Thus, Eq. (47) ombined with Eqs. (48), (5) and (55) onstitute a thid-ode odinay diffeential equation (ODE) fo u(), whih must be solved subjet to the bounday onditions fo d/. While the ODE is faily omplex, its fom an be tested in vaious ways. Fo instane, the themal oetions, popotional to, ente as expeted fom analysis fo a homogeneous system, i.e., dn Ts ¼ n s k 4 d/=ðm sx s aþ. 18 Also fo the ase ¼ 1 in a single speies plasma, a simple analyti solution exists 6 d/ðþ ¼A½x þ i X s þ x E ðþš; (6) due to the fat that this exitation is a ente-of-mass osillation in whih the entie olumn is displaed, and themal effets on the density petubation must vanish as a onsequene. Indeed, substitution of Eq. (51) and (6) into Eq. (55) fo ¼ 1, along with the Poisson equation elating total hage density q tot ¼ P s q sn s to E B otation fequeny, q tot ðþ ¼ ð x E Þ; (63) yields dn Ts ¼ if thee is only one speies, so that q tot ¼ q s n s. Futhemoe, the old-fluid density petubation satisfies d/ þ 4pq s dn F s ¼, showing that Eq. (6) is a solution of Eq. (47) fo ¼ 1 in a single speies plasma. Futhemoe, if x is hosen as x ¼ X s x E ð w Þ i, Eq. (6) implies that the petubed potential at w vanishes. This is the fequeny of the ¼ 1 ente of mass yloton eigenmode in a single speies plasma olumn [oet to OðeÞ]. The fequeny shift x E ð w Þ is aused by E B otation of the ente of mass in the plasma s image hage eleti field. VI. COLD FLUID THEORY OF SURFACE CYCLOTRON MODES In the zeo-tempeatue old-fluid limit, the yloton mode dispesion elation beomes d/ þ 4pq s dn F s ¼ ; (64) with dn Fs given by Eq. (5). Following Gould, 6 we will solve this equation fo d/ðþ using the elated field amplitude u(). Using Eqs. (5) and (61), Eq. (64) an be witten as a fistode ODE fo u(), 6 The solution of this equation ðduþ Du ¼ : (65) uðþ ¼A 1 = DðÞ; (66) whee the onstant A is detemined by bounday onditions on d/ðþ. We will onside the following bounday onditions d/ð w Þ¼/ w ; d/ð in Þ¼; in < w : (67) Fo the speial ase and in ¼, the nd bounday ondition must be modified to d/ðþ ¼finite: The fist bounday ondition oesponds to a potential of magnitude / w applied to an exteio eletode of adius w, osillating in time at fequeny x; and the seond bounday ondition oesponds to an inne onduto of adius in at fixed voltage (see Fig. 1). Taking in ¼ (by whih we mean no inteio onduto) is typial in many expeiments. The equilibium density of a given speies s, n s ðþ, an have abitay adial dependene, but in most expeiments, x E ðþ is monotonially deeasing; othewise, the plasma an be unstable. 6 The E B otation fequeny is detemined by the total hage density q tot though Eq. (63) (see Fig. 1). The density of speies s need not have the same pofile shape as q tot ðþ sine vaious foes at diffeently on diffeent speies and an even podue entifugal o hage

9 41-8 Daniel H. E. Dubin Phys. Plasmas, 41 (13) This is the expession fo the uppe hybid fequeny in a otating plasma olumn, in the low density limit x ps =X s 1. At UH, u() is undefined, but it is zeo eveywhee else. The potential oesponding to these uppe hybid osillations is, aoding to Eq. (69), d/ ¼ C þ ; < UH Bð UH =Þ (71) ; > UH ; FIG. 1. Shemati diagam of the hage densities and E B otation fequeny x E in a non-neutal plasma olumn onsisting of thee speies. Cylindial ondutos bound the plasma at in and w ; in many expeiments, the inne onduto is not pesent. Centifugal and/o hage sepaation 5 an ause the speies to sepaate adially, in ode of lagest to smallest hage to mass atio. sepaation of the speies densities at suffiiently low tempeatue and lage x E. 5 We will find that thee ae sufae yloton waves that popagate along the edge(s) of eah speies density pofile, poduing measuable eleti fields at the walls. When thee is no inne onduto, thee is also a seond set of intenal uppe hybid waves that do not affet the potential outside the plasma. With the bounday ondition d/ð in Þ¼, Eq. (51) implies d/ðþ ¼ ð in d uð Þ: (68) Fo the speial ase in ¼ (no inteio onduto), this must be modified to 3 ð d/ðþ ¼ 4C þ d uð Þ5: (69) Fo, the onstant C is undetemined, but fo >, we equie that C ¼ so that d/ emains finite at ¼. A. Uppe hybid utoff Fo any, a possible solution of Eq. (65) is Du ¼. This oesponds to a loalized uppe-hybid osillation with u ¼ Bdð UH Þ, at any loation UH fo whih Dð UH Þ¼, whih an be witten as að UH Þ¼bð UH Þ using Eq. (5). In the theoy of eletomagneti wave popagation, suh loations ae efeed to as uppe hybid esonanes, 18 but fo the eletostati Benstein waves disussed in this pape, these loations at as utoffs, efleting the waves. We, theefoe, efe to a loation whee a ¼ b as an uppe hybid utoff. Using Eqs. (53), (5), and(63), the fequeny at utoff an be witten as x þ i X s ¼ð 1Þx E ð UH Þ X s6¼s x ps ð UH Þ X s : (7) whee C ¼ if in > o >. In these instanes, the hoie B ¼ / w ð w = UH Þ mathes the bounday onditions at ¼ w, so the osillation amplitude is lamped by the value at the wall. But, if in ¼ and eithe B o C is undetemined, so any osillation amplitude is allowed; these ae singula uppe hybid eigenmodes. These modes make no potential outside the plasma; they ae intenal modes. Note that these ou only if thee is no inteio onduto ( in ¼ ). Finite tempeatue effets on these modes will be disussed in Ses. VII IX. B. Sufae yloton waves fo no inne onduto Tuning to the sufae yloton waves, we fist examine the ase whee thee is no inteio onduto. In this ase, the solution domain inludes ¼, and then Eq. (66) implies that nontivial solutions fo u() exist only fo 1. These waves have angula phase veloity x= X s = in the same dietion as the yloton motion. Then, Eq. (66) along with Eq. (69) implies that d/ðþ ¼A ð d ð 1Þ Dð ; > : (7) Þ The onstant A is detemined by the bounday ondition that d/ð w Þ¼d/ w, A ¼ ð w w d/ w d ð 1Þ =Dð Þ : (73) A dimensionless measue of the system esponse to the applied potential d/ w is the admittane funtion Y, whee Y w@d/=@ w d/ w : (74) This funtion is popotional to the sufae hage on the wall eletode fo a given wall potential. The out-of-phase (imaginay) omponent of Y is due only to the plasma and is a useful measue of the amplitude of the plasma esponse to the applied wall potential. Using Eq. (51), the admittane an be expessed in tems of d/ w and uð w Þ as Y ¼ þ w uð w Þ d/ w ; ¼ þ ð w w : (75) d 1 =Dð Þ

10 41-9 Daniel H. E. Dubin Phys. Plasmas, 41 (13) The imaginay pat of the admittane has peaks at fequenies fo whih the denominato in Eq. (75) is small, i.e., whee ð w d ð 1Þ =Dð Þ!: (76) Fo finite ollisional damping and eal x, this integal neve equals zeo, but fo weak damping, minima in its magnitude appoah zeo at one o moe fequenies oesponding to the fequenies of weakly damped yloton modes in the old fluid limit. Fo the ase of a unifom otation fequeny x E and a unifom density (possibly hollow) plasma olumn with oute adius and inne adius 1 (simila to the speies pofile shown in Fig. 1), the integal in Eq. (76) an be pefomed analytially. Thee is a single mode fequeny fo eah (positive) value of, given by x þ i X s ¼ð Þx E þ b 1 1 w ; : (77) The fequeny shift x E aises fom the Dopple effet due to plasma otation. The tem x E aises fom a shift in the yloton fequeny aused by the Coiolis foe. The tem popotional to b is the fequeny shift due to the selfonsistent plasma eleti field eated by the petubation. 1 Þ= w due to the effet of image hage eleti fields. Fo moe geneal density and otation pofiles, the integal in Eq. (76) must be pefomed numeially. In the limit of weak damping,!, thee is a singulaity in the integand at adial loations UH whee DðÞ!, oesponding to the afoementioned uppe hybid utoff. If thee is only one suh loation, at ¼ UH, appliation of the Plemelj fomula to Eq. (7) fo! þ allows one to wite the admittane as The shift is edued by the fato ð Y ¼ þ w, 4 ð w Pd ð 1Þ Dð Þ 5: jd ð UH Þj pi 1 UH 3 (78) This expession shows that if the density o otation fequeny gadient is lage (whih makes jd j lage), the amount of enhaned absoption due to the uppe hybid utoff will be small. (This is why no effet of the utoff appeas in Eq. (77) the utoff ous on the plasma edge, whih was taken to be abitaily naow.) An example with an edge of finite width is shown in Fig.. Hee, we plot the imaginay pat of Y vesus the applied fequeny fo a single-speies plasma with a density pofile of the fom nðþ ¼ n h tanh i þ 1 ; (79) D with assoiated equilibium potential (and hene otation fequeny) given by the solution to Poisson s equation, / ¼ 4pq nðþ. Hee, and thoughout the pape, we take ¼ w =. Fo this pofile, thee is a ange of FIG.. Imaginay pat of the admittane vesus fequeny fo an ¼ wall petubation, in the old fluid limit, at diffeent ollision fequenies (measued in units of b max ¼ Max ðbþ fo the density pofile of Eq. (79) with ¼ w =; D ¼ w =, whee w is the wall adius). fequenies fo whih thee is a single utoff. Fo finite edge width D, even if! the peak in Im Y has finite fequeny width aused by enegy absoption at the uppe hybid utoff. The width in Im Y deeases as D deeases (Fig. 3). Also, as D deeases, the loation of the peak in the plasma esponse appoahes the analyti esult given by Eq. (77), shown by the aow in the figue. We will see in Ses. VIII and IX that this absoption is due to the oupling of the sufae yloton wave to Benstein waves. The Benstein waves daw enegy fom the sufae yloton wave, ausing a boadened fequeny esponse. This damping mehanism is simila to the absoption of unmagnetized sufae plasma wave enegy that ous at bulk plasma esonanes in an inhomogeneous unmagnetized plasma. 19 C. Sufae yloton waves fo an inne onduto with adius in > Fo the bounday onditions d/ð in Þ¼; d/ð w Þ¼/ w, the oigin is not in the solution domain and, theefoe, Eq. (66) povides nontivial sufae yloton mode solutions fo all integes. Now the petubed potential is, fom Eqs. (68) and (66), d/ðþ ¼A ð w d ð 1Þ Dð Þ ; (8) in FIG. 3. Same as Fig. but fo fixed ¼ b max =1, at diffeent pofile widths D (in units of w ). The aow shows the fequeny fo a step pofile, Eq. (77).

11 41-1 Daniel H. E. Dubin Phys. Plasmas, 41 (13) and the onstant A is again given by the ondition that d/ð w Þ¼/ w. The admittane is then given by Eq. (74) Y ¼ ð w w in d ð 1Þ =Dð Þ : (81) Peaks in the admittane funtion again appea whee the denominato appoahes zeo, ð w d ð 1Þ Dð Þ in! : (8) Fo example, fo a unifom density hollow olumn with inne and oute adii 1 and and with unifom otation fequeny x E, thee is again one yloton mode pe value, at fequeny! x þ i X s ¼ð Þx E þ b 1 ð = w Þ ð 1 = w Þ 1 ð in = w Þ : (83) Fo > and in!, this fomula etuns to the pevious esult, Eq. (77). Howeve, thee ae now also modes fo. Fo example, fo ¼, Eq. (83) edues to ln w 1 x þ i X s ¼ x E þ b in ; ¼ : (84) lnð w = in Þ This is the fequeny of a yloton beathing mode, whee the olumn osillates adially without hanging its density. Fo moe geneal density pofiles with an inne onduto, fo whih Eq. (83) oughly applies, thee ae not neessaily any loations whee D() ¼ in the plasma (unlike the pevious example with no inne onduto), so in old fluid theoy, these modes ae then undamped disete eigenmodes when ¼. An example is shown in Fig. 4 fo ¼ and the same tanh density pofile as we used fo Figs. and 3, taking in ¼ 1 ¼ 1=1 w. The potential on the inne onduto is hosen so that x E is unifom inside the plasma fa fom the edges. Now, as deeases fo any fixed value of D, peaks in Im Y beome naowe without limit, signifying a disete undamped mode in the! limit. Fo small D, the peak in the plasma esponse ous at the fequeny pedited by Eq. (84) (the aow in Fig. 4). The modes ae undamped as! in this example beause, fo the ange of fequenies plotted in Fig. 4, thee is no longe an uppe hybid utoff, although D hanges sign fom a negative value inside the plasma to a positive value (unity) outside it. This is beause D hanges sign by passing though infinity, sine aðþ ¼ at a loation inside the plasma. This is the loation of a wave-patile esonane that has impotant impliations fo finite-tempeatue yloton wave popagation. Beause the fequeny of the sufae yloton mode depends on speies density, measuement of the mode FIG. 4. Imaginay pat of the admittane fo an ¼ old-fluid yloton mode in a plasma with an inne onduto of adius in ¼ w =1, fo two values of the pofile width D (in units of w ) and fo =b max ¼ 1=, 1/1 and 1/1 (in ode fom boadest to shapest admittane uves). The aow shows the fequeny fo a step pofile, Eq. (84). fequeny is a useful and nondestutive diagnosti fo the density of eah speies. The intenal uppe hybid modes have fequenies given by Eq. (7), whih also depend on density and otation fequeny, but these modes may be hade to detet expeimentally. Thee ae no peaks in the admittane funtion Y, indiating that, within the model of an infinitely long plasma olumn, these modes annot be obseved by thei effet on wall image hages. On the othe hand, it should be possible to detet these modes using a wall eletode loated at the end of a finite-length plasma olumn. VII. FINITE TEMPERATURE EFFECTS, BERNSTEIN WAVES When finite tempeatue tems ae added to the analysis of the sufae yloton waves, new waves appea, efeed to as Benstein waves. These waves wee analyzed qualitatively by Gould. 5,6 We will see that these waves ouple to the sufae yloton waves. Also, when thee is no inne onduto, the singula uppe hybid ontinuum given by Eq. (7) beaks into anothe set of finite tempeatue Benstein eigenmodes, whih do not ouple to the sufae yloton waves. The petubed potential now satisfies d/ þ 4pq ðdn Fs þ dn Ts Þ¼; (85) whee dn Fs and dn Ts ae given by Eqs. (5) and (55). Fo now, we onside only bounday onditions with no inne onduto whee ¼ is inluded in the domain, and d/ is speified on the oute wall. We also assume, fo now, that the speies s density pofile extends to the oigin, and that thee is a vauum egion between the plasma and the wall. Inside the plasma, Eq. (85) is a fouth ode homogeneous equation fo d/, o, altenately, a thid ode equation fo u when we apply Eq. (61), but outside the equation evets to Laplae s equation, seond ode in d/ (fist ode in u). The geneal solution of the thid ode equation fo u is a sum of thee independent solutions, u 1 ðþ; u ðþ; u 3 ðþ.

12 41-11 Daniel H. E. Dubin Phys. Plasmas, 41 (13) Fo >, it an be shown that one of these solutions (u 3, say) blows up at the oigin. The inteio solution within the plasma is then u in ðþ ¼B 1 u 1 ðþþb u ðþ: (86) This must be mathed onto the oute Laplae solution at the plasma edge. The oute Laplae solution is u out ¼ A 1 : (87) The mathing of inne and oute solutions is aomplished by setting u in ð out Þ¼u out ð out Þ; (88) whee out is a adius outside the plasma (typially hosen lose to the plasma edge), whee Eq. (85) is fist-ode. A WKB analysis (Se. VIII) shows that only one of the two inteio solutions emains finite outside the plasma (u 1, say); the othe blows up as nðþ!. Theefoe, we set B ¼, so Eqs. (86) (88) detemine B 1 in tems of A, B 1 ¼ A 1 out =u 1ð out Þ: (89) Finally the onstant A is detemined in tems of the applied wall potential via Eq. (69), / w ¼ B 1 w ð out u 1 ðþ d þ A w ð w out 1 d (9) (taking C ¼ sine > is assumed). Equation (9) detemines the amplitude of the plasma esponse to the applied wall potential / w. This amplitude is unbounded wheeve the hs of Eq. (9) equals zeo, i.e., ¼ 1 out ð out ð w u 1 ðþ d þ u 1 ð out Þ out 1 d: (91) These zeos oespond to a sequene of nomal modes the afoementioned Benstein modes. The behavio of these modes is analyzed in the next setions. Fo, one an show that two of the thee inteio solutions blow up at the oigin (u and u 3, say), so u in ðþ ¼B 1 u 1 ðþ: (9) This inteio solution is not neessaily finite outside the plasma; it geneally blows up as density n s!, so the only solution is A ¼ B 1 ¼, i.e., u ¼. Then, Eq. (69) implies that the solution fo d/ is a vauum potential: d/ ¼ d/ w ð= w Þ. Howeve, thee may be etain hoies of x fo whih the inteio solution does not blow up. These fequenies oespond to eigenfequenies of Benstein osillations. In the old-fluid theoy, the uppe hybid osillations wee exited ove a ontinuum of fequenies assoiated with an uppe hybid utoff and wee loalized to a given adius fo a given fequeny in the ontinuum. With finite tempeatue, these modes ae not loalized and may ou only at disete fequenies. VIII. WKB ANALYSIS OF FINITE TEMPERATURE CYCLOTRON MODES The pevious geneal ideas onening the solution of Eq. (85) an be illustated and expanded using a WKB solution of the poblem. We fist make some geneal obsevations about the Benstein wave solutions expeted fom this analysis. Fo a unifom plasma in the low-density egime x p =X 1, the finite-t dieleti onstant D T ðx; kþ nea the yloton fequeny fo speies s is 18, D T ðx; kþ ¼1 bx s I 1ðk xðx X s Þ e k Þ k ; (93) whee I 1 ðxþ is a modified Bessel funtion. The zeos of the dieleti onstant yield the Benstein mode dispesion elation, xðkþ ¼X s þ b e k I 1ðk Þ k þ O ¼ X s þ bð1 k þ Þ; k 1: (94) The time-aveaged enegy density in the waves is E w ¼ ðxd TÞ¼ jej X ; (95) 16p x X whee E ¼ d/ is the wave eleti field. The enegy flux is and 1 X! S ¼ v g E w ; (96) v ¼ 4b ½k ði 1 I ÞþI 1 Š k 3 3 e k ^k (97) is the goup veloity. Ou wave Eq. (85) should podue esults onsistent with the small k limit of these expessions. Outside the plasma, the solution fo u() is given by Eq. (87). Inside, we assume a WKB fom uðþ ¼e SðÞ ; (98) whee we expand the eikonal S() in a powe seies in the yloton adius, SðÞ ¼ S ðþ þ S 1 ðþþ S ðþþ: (99) This asymptoti expansion will be useful povided that =L1, whee L is the sale length of the equilibium. The only tem in dn T ðþ that entes the analysis to detemine S and S 1 is the fist tem in Eq. (56).

13 41-1 Daniel H. E. Dubin Phys. Plasmas, 41 (13) Substituting Eqs. (98) and (99) into Eq. (85) and keeping the lowest ode tems in yields the following equation fo S : DS b a S3 This implies S ¼ os ¼ 6ikðÞ, whee kðþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ad=ðb Þ ¼ ¼ : (1) p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 a=b = (11) is the loal adial wavenumbe of the solution. Fo the ase S ¼, the solution is slowly vaying and is entiely detemined at the next ode. Consideing the next-ode equation in yields the following expession fo S 1 when S ¼ : S 1 þ D ð 1Þ ¼ : (1) D The solution is the old fluid esult S 1 ¼ log D þð 1Þ log,o u 1 ðþ ¼ 1 DðÞ : (13) An ode oetion to this solution ould be found by woking to even highe ode, but we will not use that oetion hee. This old fluid solution is invalid if 1/D() vaies apidly, whih ous nea the uppe hybid utoff whee D() ¼. The amplitudes of the othe two apidly vaying solutions with S ¼ 6ikðÞ ae also obtained by onsideing the next-ode equation in, whih now yields S 1 ðþþ 1 þ 1 D D þ b a ¼ : (14) b a Thus, the seond and thid WKB solutions fo u() ae u ;3 ðþ ¼ 1 Ð 1 a 4e 6i kðþd pffiffi : (15) bd These solutions ae taveling waves moving adially inwad o outwad, depending on the sign of x and Re(k). Thei wavenumbe and amplitude (but not thei fequeny) vay in adius as the plasma density and/o otation ate vaies. The loal dispesion elation of these waves follows fom Eq. (11): v g ¼ bk : (17) The WKB amplitude fato fo the Benstein waves, u / 1 a 1=4 pffiffi ; (18) bd is onsistent with enegy onsevation. Fo WKB taveling waves, the onseved quantity is the total adial enegy flux S ^ given by Eq. (96). Identifying x X with a and identifying jej with u [using Eq. (5) and the WKB limit k ] implies the following geneal WKB amplitude fato: aðþ 1= u / : (19) X s v g ð; kþ To ompae this to Eq. (18), we substitute fo v g fom Eq. (17), noting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that the goup veloity an be witten as v g ¼ b 1 a=b using Eq. (16). Substituting this expession into Eq. (19) and using Eq. (5) yields Eq. (18), the k 1 fom of the amplitude fato. The two WKB solutions in Eq. (15) beak down nea ¼ ; nea any uppe hybid utoffs whee DðÞ!; and nea any esonanes whee aðþ ¼andDðÞ!1,oesponding to stong wave-patile esonant inteations. These esonanes, if they ou, ause damping of Benstein waves. Note that the slowly vaying old fluid solution u 1 also beaks down at the uppe hybid utoff, but not at ¼ oat the a ¼ esonane. Also, when bðþ! at the plasma edge, Eq. (11) implies that k! i1, so one Benstein solution blows up and the othe deays to zeo exponentially. Connetion fomulae 18 must be deived in ode to onnet WKB solutions on eithe side of a utoff o esonane. We fist onside the onnetion fomula fo a utoff. A. Connetion fomula fo an uppe hybid utoff Conside a situation whee ReðaðÞÞ > but Re(D()) hanges sign at ¼ UH, with ReðDÞ > fo > UH (see Fig. 5). Nea ¼ UH, DðÞ ð UH Þ=L, whee L 1 D ð UH Þ > by assumption. On the left side of the utoff, we wite the WKB solution as 3 u L ðþ ¼ A L 1 DðÞ þ p BL ffiffi a 1 4os ð 4 kd þ v5; (11) bd UH a ¼ bð1 k Þ: (16) Equation (16) mathes the long-wavelength limit of the Benstein-mode dispesion elation, Eq. (94), fo the sheafee ase; noting that fo a otating system, thee is a Dopple shift to x and a shift to X fom the Coiolis foe that appeas in Eq. (16) though a [see Eq. (53)]. The adial goup veloity follows fom the deivative with espet to k of Eq. (16): FIG. 5. Shemati of a plasma fo whih thee is a single uppe hybid utoff. Vaying x moves the aðþ pofile vetially and hanges the loation of the utoff.

14 41-13 Daniel H. E. Dubin Phys. Plasmas, 41 (13) whee A L and B L ae the amplitudes of the solution and v is the phase. [Thoughout the emainde of the pape, we assume a sign in the squae oot in Eq. (11) suh that Re k >. In this setion, we use the k 1 fom of the WKB amplitude fato, Eq. (18), sine nea utoff k!.] On the ight side of the utoff, the WKB solution is u R ðþ ¼ A R 1 DðÞ þ 1 Ð Ð 3 1 a 4 kd kd 6 pffiffi B R1 bd e UH 4 þ B R e UH 7 5: (111) To onnet these two solutions, we note that nea ¼ UH, Eq. (85) an be appoximated as u L u ¼ ; (11) whee x ¼ UH, we have used the elation b=aj ¼UH ¼ 1, and we have kept only the dominant balane in Eq. (85), assuming =L1. [This balane involves only the fist tem in dn F (see Eq. (5)) and the fist tem in Eq. (56).] The geneal solution an be witten in tems of Aiy funtions and integals of Aiy funtions: uðxþ ¼D 1 A i ðxþþd B i ðxþþd 3 C i ðxþ; (113) whee the funtion C i is defined as C i ðxþ pa i ðxþ ð x 1 ð1 dx B i ðx ÞþpB i ðxþ dx A i ðx Þ (114) and x x=ðl Þ1=3. Fo x 1 but jxj=l 1, we an onnet Eq. (11) to Eq. (113) using the asymptoti fom of Eq. (113). The asymptoti foms of the Aiy funtions A i and B i ae well known. 1 The asymptoti foms fo C i ðxþ ae C i ðxþ¼ 1 x þ x 4 þ 4 pffiffiffi x 7 þþ p ð xþ os 1=4 3 ð xþ3= þ p 4 1 þ x 3 þ 5 pffiffiffi p 48 ð xþ sin 7=4 3 ð xþ3= þ p 4 1 þ x 3 þ ; x 1 C i ðxþ¼ 1 x þ x 4 þ 4 ; x 1: (115) x 7 Using the x 1 asymptoti fom in Eq. (113) along with the oesponding foms fo the Aiy funtions yields the following lowest-ode fom fo u: lim uðxþ ¼ 1 pffiffiffi D x! 1 p ð xþ 1=4 1 sin 3 ð xþ3= þ p 4 þðd þ pd 3 Þ os 3 ð xþ3= þ p 4 x þ D 3 x : (116) Compaing Eq. (116) to Eq. (11), and noting that fo ð UH Þ=L1, we an appoximate 3 jxj3= ¼ Ð UH kd, DðÞ¼x=L, and ð a=ðbdþþ 1=4 ¼ ð L=xÞ 1=4, so we obtain A L ¼ D 3 UH 1 =L Þ 1=3 ; (117) ffiffiffiffiffiffiffi UH B L os v ¼ ð =LÞ 1=6 ½D 1 þ D þ pd 3 Š; p (118) ffiffiffiffiffiffiffi UH B L sin v ¼ ð =LÞ 1=6 ½D 1 D pd 3 Š: p (119) On the ight side of the utoff the lowest ode asymptoti fom of Eq. (113) is lim uðxþ ¼ 1 1 p ffiffiffi e x3= =3 D x!1 p x 1=4 1 þ D e x3= =3 þ D 3 =x: (1) Conneting to Eq. (111) fo x 1 but ð UH Þ=L 1 implies A R ¼ D 3 UH 1 ð =LÞ =3 ; (11) ffiffiffiffiffiffiffi UH ¼ ð =LÞ 1=6 D ; (1) p ffiffiffiffiffiffiffi UH ¼ ð =LÞ 1=6 D 1 p : (13) B R1 B R Eliminating D 1, D, and D 3 fom Eqs. (117) (119) and (11) (13) yields the onnetion fomulae at an uppe hybid utoff, A L ¼ A R ; (14) B L os v ¼ B R pffiffiffi 1 þ ffiffi sffiffiffiffiffiffi p pl BR þ 1= UH A R; (15) B L sin v ¼ B R pffiffi 1 þ ffiffi sffiffiffiffiffiffi p pl BR 1= UH A R: (16) These fomulae indiate that, at a utoff, the slowly vaying old fluid solution, esponsible fo sufae yloton waves, and popotional to oeffiients A R and A L, is mixed with the apidly vaying Benstein solutions, popotional to B R and B L. B. Behavio nea a esonane Thee may be loations in the plasma whee aðþ ¼, due to sheas in the E B otation fequeny. At suh loations, the WKB Benstein mode solutions, Eq. (15), ae invalid. This is beause Eq. (11) implies that k ¼ 1at a ¼, whih beaks the assumption used to deive Eq. (85) that k 1. Theefoe, we modify the Benstein WKB solutions by using the exat Benstein mode dispesion elation fo a unifom system, Eq. (94), witing the dispesion elation as aðþ ¼bðÞ e k I 1 ðk Þ k (17) in ode to aount fo Dopple and Coiolis foe shifts due to plasma otation. This WKB dispesion elation agees