Cheng, C.Z.

Helander, P.

Hsu, C.T.

Sigmar, D.J.

White, R.

DSpace@MIT

PFC/JA-93-34Particle Dynamics in Chirped-Frequency FluctuationsC. T. Hsu, C. Z. Cheng,I P. Helander,2D. J. Sigmar, R. WhitelMIT Plasma Fusion CenterCambridge, Massachusetts 02139 USADecember 19931Permanent address: Princeton Plasma Physics Labatory, Princeton University, Princeton, NJ085432Permanent address: Institute for Electromagnetic Field Theory and Plasma Physics, ChalmersUniversity of Technology, S-412 96 G6teborg, SwedenThis work was supported by the US Department of Energy under contract DE-FG02-91ER-54109.Reproduction, translation, publication, use, and disposal, in whole or in part, by or for the USGovernment is permitted.Submitted for publication in: Physical Review LettersParticle Dynamics In Chirped-Frequency FluctuationsC. T. Hsu, C.Z. Cheng'), P. Helanderb),D.J. Sigmar, and R. Whitea)Massachusetts Institute of TechnologyPlasma Fusion Center167 Albany Street, NW16-260Cambridge, Massachusetts 02139Hamiltonian systems describing particle motion in a wave with time-dependent (chirped)frequency are studied. The wave is found to form a single-node separatrix (bucket) mov-ing in the phase space at a rate proportional to that of the frequency change. Particlestrapped inside the bucket undergo convection, while untrapped particles colliding with thebucket get a resonant kick, in phase space. In toroidal systems, these effects can result ina large radial convective flux roughly proportional to the size of bucket and the frequencychirping. Possible applications of this novel mechanism to tokamak plasmas are discussed.PACS numbers: 03.20.+i, 52.20.Dqa) Permanent address: Princeton Plasma Physics Laboratory, Princeton University, Prince-ton, New Jersey 08543.b) Permanent address: Institute for Electromagnetic Field Theory and Plasma Physics,Chalmers University of Technology, S-412 96 G6teborg, Sweden.1A common characteristic of an evolving nonlinear system is that the wave frequencyexcited in the system also evolves in time. Such behavior, referred to as frequency chirping[1], can be found in, nonlinear optics [1], developing turbulent systems, and unsaturatedsystems with nonlinear wave-wave and/or wave-particle interaction. In particular, beamdriven activities in tokamak plasmas, especially instabilities with bursting behavior suchas fishbone oscillations [2], usually exhibit strong frequency chirping. Yet, most particletransport studies carried out so far are based upon the interaction of particles with constantfrequency fluctuations. It is therefore worthwhile to study particle dynamics in a chirpedfrequency fluctuation, i.e., in an accelerating (or decelerating) wave.First, let us consider the one-dimensional (1-D) Hamiltonian systemH =p 2 /2 + A (1 - cos(q - 4(t))), (1)where 4 is the wave phase, so that 4 is wave frequency, 4 is the frequency chirping, andthe amplitude A is assumed to be constant. Note that this Hamiltonian system has beenextensively studied [3] in the limit of small frequency chirping 4 < 1, in which the adiabaticinvariant is preserved except for particles very close to the separatrix. In contrast, 4 isallowed to be finite in the present study.Particle motion in this Hamiltonian system is most easily described in the wave frame,in which Q = q - 4(t) and P = p - 4(t). The evolution of P and Q is governed by a newHamiltonian H = P 2/2 + V (Q) with the potential V(Q) = A(1 - cos Q) + 4Q equivalentto that of a pendulum subject to a drag force 4 along Q. We shall hereby assume that 4is constant so that both V(Q), as shown in Fig. la, and f become independent of time.This time-independent Hamiltonian is similar to the one governing electron motion nearthe synchronous energy in Free Electron Lasers (FEL) [4], where the average slope of V(Q)is induced by the synchronous phase instead of the frequency chirping.The particle trajectories in P, Q space, illustrated in Fig. 1b, coincide with contoursof constant H. With this in mind, it is straightforward to observe from Figs. la, lb that(i) A single-node separatrix is formed and divides the phase space into two regions: atrapped region inside and an untrapped region outside; this separatrix does not occupy2the whole Q space but leaves an open gap connecting the positive and negative Pspaces (see Fig. 1);(ii) The action f PdQ is invariant for trapped particles;(iii) Both and < Q > are constant for particles trapped in the potential well, where,< ... > indicates an average taken between two neighboring local maxima of P;(iv) - = - for passing particles far from the separatrix.Transforming to the lab frame, one finds that this single-node separatrix is movingin the p-direction at a rate p= q. Particles initially loaded inside the separatrix willalways be trapped in it and will be dragged along with it in phase space (see Fig. 2a). Itis therefore appropriate to refer this convecting separatrix as a "bucket" [4]. The size ofthe bucket is determined by the action along the separatrix in P, Q space which can bewritten asSb(A,4) = 8 Z dy Vsin y sin(y+ yi) - ay. (2)Here a = y, = arcsin a, and Y2 satisfies cos y 1 - cos y 2 = a(27r - y, - y2). It is evidentthat Sb decreases with a and vanishes when a > 1. For a < 1 Sb ~ 16VA (1 + M Ina).For passing orbits reasonably far away from the bucket, remains constant. Asshown in Fig. 2b however, when the bucket approaches a passing orbit, the p modulation ofthe orbit increases until it hits the resonance and decelerates (accelerates if 4 is negative)along the rim of bucket and "slides" over to the other side. Then, the bucket moves awayfrom it and becomes constant again. It should be noted here that the reason whythe passing orbits are allowed to slide through the resonant surface without undergoingseparatrix crossing is that the bucket does not occupy the entire Q space, as shown inFig. 1b, but leaves a gap open. This process causes to increase for particlesinteracting with a decelerating wave (q < 0) and to decrease when interacting with anaccelerating wave (q > 0). Since this change occurs within just a few bounce periods, theorbit effectively gets a resonant "kick" in momentum space.Upon noticing the symmetry property with respect to the P = 0 axis in P, Q space,one finds that the kick's magnitude is determined byAp = -2VAr,, Y1/ dy sin(2y yo) (3)3Here, G(y) = - sinysin(y + yo) - ay, o 0 , Po, Qo are the initial phase spacecoordinates, yo = ooQo, and y, satisfies G(yl) = 0. This kick decreases with increasinga and actually vanishes when a > 1. It was previously estimated for the case a < 1 byusing the adiabatic invariant [3,5,6] property of orbits outside the separatrix. In that limit,the particle is considered to undergo separatrix crossing [3] and gets a kick Ap ~ - vfA4.In summary, a chirped wave forms a moving bucket if the frequency chirping is smallerthan the wave amplitude. The bucket occupies the phase space of size Sb, given in Eq. (2),and transports trapped particles convectively in phase space at the rate of the frequencychirping q. In other words, the centers of the islands of trapped particle trajectoriesmove at the phase velocity of the wave and therefore stay in resonance. This results in atransport flux in velocity space proportional to 'Sb, which is roughly proportional &/A.Even for a non-adiabatic change of the frequency (i.e., a finite frequency chirping) thereexist adiabatic invariants for most of the phase space except for the passing region near thebucket, while passing particles colliding with the bucket get a resonant kick Ap quantifiedin Eq. (3). If the frequency chirping is larger than the wave amplitude, the bucket vanishesand no particles are trapped. In this case, most orbits remain unaffected by the wave untilthey collide with the moving resonance surface and get a momentum kick.The above conclusions are surprisingly well preserved even when is not constant, asshown in Fig. 2c, where a particle is trapped in a wave with the phase 0(t) = t exp(-t/20)and ~ - (1 - -)exp(-t/20) is shown to hold. However, it should be noted thateven though the amplitude is constant, the evolution of causes a change of the bucketsize according to Eq. (2). This results in separatrix crossings of particle orbits into orout of the bucket, according to Liouville's theorem, in the same way as in a system withnon-constant amplitude. If there exist two buckets moving at different speeds in the phasespace, a locally stochastic region forms when they collide. After the collision, they departfrom each other, and each one carries some particles with it. A particle originally in onebucket can then end up in the other bucket after the collision. A detailed disccusion ofthese more complex features of particle dynamics in chirped waves as well as more detailedcalculations of the phenomena presented here will be given in a separate paper [7].4In a system with many degrees of freedom, it is wellknown [6,8,9] that, as long as theperturbed Hamiltonian is sufficiently small so that the coupling between two neighboringresonant surfaces is negligible, the 1-D Hamiltonian faithfully reproduces the behaviorof the multi-dimensional system. Therefore, the effects of frequency chirping are at leastqualitatively similar for the two systems. In particular, they both can form buckets movingwith the resonant islands. In a toroidally confined axisymmetric plasma, the guiding centermotion of charged particles is governed by the following Hamiltonian [10]1 Ze B 2H - (PC + (,p - A()) - + pB + Ze4.2M c BCRHere, PC is the toroidal angular momentum, I = 1 MvI/B the magnetic moment, M theparticle mass, B the magnetic field strength, Z the particle charge, (AC, 4) the perturbed(vector, scalar) potentials, Op the poloidal magnetic flux, and R the major radius of thetorus.If q/$, Wb < w(, we, a particle in the (n, m) resonance bucket has a bounce-averagedguiding center motion, caused by the frequency chirping, in (Pc, E, tt) space satisfying:nwC - mue = q, motion of the bucket center; (4a)a constraint near the nth resonance; (4b)while p is constant since < Q. Here, Wb is the bounce frequency of particle trappedin the wave, and wc(PC, E, p) and we(PC, E, p) are bounce-averaged toroidal and poloidalangular rotational frequencies. More importantly, this motion in (PC, E) space leads to abounce-averaged radial motion governed by the toroidal force balanced= - 1 R)-PC , (4c)Ze dt Bwhere (- ) represents the bounce-average along the unperturbed orbit.An explicit relation between the radial motion and the frequency chirping requiresa calculation of wc(Pc, E, p) and we(PC, E, pt). For orbits with a finite width in generaltoroidal geometry, this is usually not possible, and a numerical study [11] is required.However, it is easy to see that a negative chirping, < 0, typically leads to decreasing5PC (thus decreasing energy) and consequently causes an outward radial drift, as shown inFig. 3c. In addition, as the particle moves radially outward, its pitch angle decreases. Foran originally well confined counter-circulating particle, the decrease of the pitch-angle cancause it to cross the trapped/passing boundary [8] and to turn into a fat banana, as shownin Fig. 3d. Note that Fig. 3 is obtained from deploying a chirped frequency wave and aD-shaped tokamak plasma equilibrium in the Hamiltonian guiding center orbit-followingcode [8,10,11]. For a more detailed studies of various types of chirped-frequency-inducedconvection and topological bifurcations, we refer the reader to Ref. [7]. Also note that therate of the radial drift is not sensitive to the wave amplitude. However, as indicated in the1-D model, to form a finite size bucket the ratio of wave amplitude to frequency changerate has to exceed a certain value.By assuming large aspect ratio and neglecting finite orbit width effects, Eqs. (4a)-(4c)yield the following analytic expressions for the radial drift induced by frequency chirping:- - / ( -- well passing orbits (5a)klr2 qR ) '2; deeply trapped orbit (5b)r (1 - S)-O + (2 - 3S)mnwe - M'where wo ~- . Here, q is the safety factor, S = q the magnetic shear, kl theparallel wave vector, and Q the gyro-frequency.When the poloidal bounce harmonic m = 0, as in fishbone oscillations, Eq. (5b)reduces to the simple relation . In other words, a frequency change of orderunity leads to a change of radial position of order unity. In particular, by assumingq = 1 + 2y, one finds 2 ()2, which indicates that in order to bring a deeplytrapped particle originally captured by the bucket at ro = a./yo to the plasma edge (r = a),one needs to change the frequency from wo to of = which has a minimum value atl+2VOyo = 0.5. Here, y (1) 2 , Yo = y(t = 0), and wo = w(t = 0) is the initial wave frequency.This also implies that if yo < 0.5, one needs to first decrease the frequency from wo toW1 = 21w to bring the particle to y1 = 0.5, and then to increase the frequency back towf to bring particle to the plasma edge.6Similarly, for a well-circulating particle close to the n = 1, m = 1 bounce resonance,with wo/l = 0.016 (appropriate for Shear Alfv6n modes induced by fusion alphas), Eq. (5a)yields (i - y -0.32 -. Thus, to bring the particle from the center to the plasmaedge requires ' < 4. Note that the higher order term in the denominator of Eq. (5a)Wfinvolving Sq/2 has been neglected here for simplicity. However, if there exists a rationalsurface where kil = 0, this term becomes dominant and results in t = 0. This is simplydue to the fact that the particle can not be resonant with the wave at this rational surfaceunless the wave frequency also vanishes. One way to smoothly get across this surface is tokeep decreasing the frequency from positive to negative.The radial particle convection induced by frequency chirping can be large. It can beestimated by N Z, where N is particle density, and bB the perturbed magnetic field.If B o 10 2 ~0-, and wo ~ 10's-1, the radial flux is equivalent to that of diffusionwith the diffusion coefficient D ~ 10m 2 /s. Still, for energetic particles with large orbitwidth, this is usually an underestimate. In particular, recall that a counter-circulatingorbit can get across the trapped/passing boundary and turn into a fat banana in a wavewith negative chirping, as shown in Fig. 3d. In addition, when untrapped particles areapproached by the bucket, the amplitude of orbit oscillations in phase space increases.This enhances the near-boundary loss [8].In summary, particle dynamics in chirped waves has been studied in both a 1-D systemand a toroidally confined plasma system. It is demonstrated that a convecting bucketis formed, which traps particles within it and gives resonant kicks to untrapped particleswhen it collides with them in phase space. In toroidal systems, this leads to a radialconvection. Analytic expressions for the radial flow induced by frequency chirping areobtained in the thin orbit width and large aspect ratio limit. A more detailed discussionswill be given elsewhere [7], and more complicated features such as bucket dynamics in thepresence of orbit stochasticity, are subject to future investigation. There are many potentialapplications of this mechanism since frequency chirping is a common feature of nonlinearsystems. In particular, a radial standing wave propagating in a magnetic. surface of atokamak inevitably undergoes nonlinear evolution accompanyied by a continuous frequencyshift before it saturates [12].7It apears worth while to investigate the additional anomalous transport fluxes of hotions induced by frequency chirping in toroidally confined plasmas. Previous such studieshave mostly relied on stochastic quasilinear diffusion [8] which requires the amplitude offluctuations to exceed the stochastic threshold. A recent experimental investigation [13]of energetic particle losses in tokamaks during beam driven instabilities such as fishbones[2] and/or Toroidal Alfv6n Eigenmodes [13,14] has shown that (1) there is no clear sign ofan amplitude threshold for the losses to occur, and (2) the loss flux appears dominated byconvection. The present mechanism can potentially lead to considerable anomalous radialconvection without an amplitude threshold, and may thus contribute importantly to theobserved transport fluxes.By launching controllable electromagnetic waves with chirped frequency, the presentmechanism may be utilized to selectively redistribute charged particles radially in tokamakplasmas. This mechamism may thus possibly be used (i) to pump out unwanted fusionashes or impurities to avoid fuel dilution; (ii) to inject energetic fuel ions into the plasmacore for continuous controlled fueling; and (iii) to move groups of locally produced par-ticles away from particular radial resonance location to avoid certain instabilities. Theapplicability and efficiency of these processes, which are obviously very sensitive to theproperties of wave to be excited in the plasma, are under investigation by the authors.AcknowledgementOne the authors (CTH) would like to thank Drs. C.P. Chen and S.C. Chen of MITfor many fruitful discussions and suggestions. This work was supported by the U.S. De-partment of Energy under Grant No. DEFG02-91ER-54109 and partially supported bythe Royal Swedish Academy of Sciences.8Figure Captions1 Potential and phase space trajectories in the wave frame with constant frequency chirping4: (a), Potential V(Q) vs Q; and (b), Contours of constant HI(P, Q).2 Momentum p vs time in a 1-D Hamiltonian system with finite frequency chirping 4:(a) for a particle trapped in the bucket with constant 4; (b) for an untrapped particlekicked by the bucket with constant 4; (c) for a particle trapped in the bucket with<0(t) = texp(-t/20).3 Orbit drift due to chirped-frequency wave in an axisymmtric toroidal plasma: (a) normal-ized frequency 4Ro/vo vs t vot/27rRo, where Ro the major radius, and vo the initialparticle velocity; (b) change of toroidal angular momentum PC(t) - PC(O) vs t; (c)trajectory of a co-going orbit trapped in the bucket; (d) trajectory of a counter-goingorbit which changes its pitch-angle in the bucket and finally turns into a fat banana.References[1] D. Marcuse, Applied Optics 20, 3573 (1981).[2] M.F.F Nave, et al., Nuclear Fusion 31, 697 (1991).[3] J.L. Tennyson, J.R. Cary, and D.F. Escande, Phys. Rev. Lett. 56, 2117 (1986);A.I. Neishtadt, Sov. J. Plasma Phys. 12, 568 (1986) [Fiz. Plazmy 12, 992 (1986)].[4] N.M. Kroll, P.L. Morton, and M.N. Rosenbluth, Physics of Quantum Electronics,(addison-Wesley, Reading Massachusetts, 1978), Vol. 7, Chap. 4.[5] P. Helander, M. Lisak, and V.E. Semenov, Phys. Rev. Lett. 68, 3659 (1992).[6] A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion, (Springer-Verlag, New York, 1983).[7] C.T. Hsu, P. Helander, and C.Z. Cheng, (in preparation).[8] C.T. Hsu, D.J. Sigmar, Phys. Fluids B4, 1492 (1992).[9] B.V. Chirikov, Phys. Reports 52, 265 (1979).9[10] R.B. White, M.S. Chance, Phys. Fluids 27, 2455 (1984).[11] C.T. Hsu, D.J. Sigmar, B.N. Breizman, # 3C37, Sherwood Theory Conference, New-port, Rhode Island, (1993); also in preparation for submission to Phys. Fluids.[12] F.Y. Gang, (private communication).[13] W.W. Heidbrink, et al., Phys. Fluids B5, 2176 (1993).[14] K.L. Wong, et al., Phys. Rev. Lett. 66, 1874 (1991); C.Z. Cheng, M.S. Chance, Phys.Fluids 29, 3695 (1985).101~1~1*1*1~ I I I I I I ITW6~ooo~II, ~0I c id (N)A11d6dI0. .&I I I I I I . I . IICd12'Kra~r:')K*000100800900100o00001008009oo-008013N%.001

Particle dynamics in chirped-frequency fluctations

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Particle dynamics in chirped-frequency fluctations

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