Physics Letters A 358 (2006) 154–158

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Observation and explanation of the JET n = 0 chirping mode

C.J. Boswell a,∗,1, H.L. Berk b, D.N. Borba c,d, T. Johnson e, S.D. Pinches f, S.E. Sharapov g

a Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USAb Institute for Fusion Studies, University of Texas at Austin, Austin, TX 78712-1060, USA

c Centro de Fusão Nuclear Associação Euratom-IST, Instituto Superior Técnico, 1049001 Lisboa, Portugald EFDA Close Support Unit, Culham Science Centre, OX14 3DB, UK

e Alfvén Laboratory, KTH, Euratom-VR Association, Swedenf Max-Planck Institute for Plasma Physics, EURATOM Association, D-85748 Garching, Germany

g Euratom-UKAEA Fusion Association, Culham Science Centre, Abingdon OX14 3DB, UK

Received 14 April 2006; accepted 4 May 2006

Available online 24 May 2006

Communicated by F. Porcelli

Abstract

Persistent rapid up and down frequency chirping modes with a toroidal mode number of zero (n = 0) have been observed in the JET tokamakwhen energetic ions, with a mean energy ∼ 500 keV, were created by high field side ion cyclotron resonance frequency heating. This heatingmethod enables the formation of an energetically inverted ion distribution function that allows ions to spontaneously excite the observed instability,identified as a global geodesic acoustic mode. The interpretation is that phase space structures form and interact with the fluid zonal flow to producethe pronounced frequency chirping.© 2006 Elsevier B.V. All rights reserved.

PACS: 52.35.Bj; 52.35.Mw

Keywords: Geodesic acoustic mode; Phase-space structures

Future burning plasma experiments, such as ITER, will dealwith plasmas which consist of a core component characterizedby some base temperature and an energetic particle componentwith a much higher mean energy. Such plasmas often give riseto nonlinear phenomena that are based on both MHD fluid prop-erties and intrinsic kinetic particle properties. In this Letter weinvestigate an unusual set of JET data [1,2] which exhibits non-linear oscillations based on these dual properties. Specifically,high clarity, up and down frequency chirping modes are ob-served at frequencies that are initiated at about 1/3 the TAEfrequency. Heeter [1] was the first to realize that these weren = 0 modes, where n is the toroidal mode number, but thephysics of the mode was undetermined. Here we infer that they

* Corresponding author.E-mail address: christopher.boswell@navy.mil (C.J. Boswell).

1 Present address: NSWC, Indian Head Div., Building 600, 101 Strauss Ave.,Indian Head, MD 20640, USA.

are due to the combined effects of MHD fluid behaviour thatis related to the geodesic acoustic mode (GAM) [3] and zonalflow (see Refs. [4–6] and references therein) and the kineticresponse arising from the kinetic resonant interaction of the en-ergetic particle component.

In Fig. 1 we show a specific case where n = 0 chirping oscil-lations were observed in JET. Frequency chirping, that persistsfor nearly a second, is seen and in some cases persists for theentire time the ion cyclotron resonance heating (ICRH) is ap-plied. These oscillations are initiated only when high field side(HFS) ICRH of minority protons is applied. The heating powervaries, but is typically ∼ 5 MW. It is found, through numeri-cal modeling with the SELFO code [7,8], that the protons forman energetic tail with a mean energy of ∼ 500 keV. The chirp-ing is quenched by the application of comparable neutral beaminjection (NBI) heating power or the termination of the ICRH.These modes are picked up by Mirnov coils, which measure themagnetic fluctuations at the plasma’s edge. These signals de-

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C.J. Boswell et al. / Physics Letters A 358 (2006) 154–158 155

Fig. 1. Spectrograms of JET Mirnov coil data. (a) Rapidly chirping n = 0 mode. (b) Zoom showing frequency chirping in detail. In both plots the white line isproportional to

√Te(r/a = 0.15). At 2.0 s the electron temperature is 0.85 keV and at 3.0 s it is 3.35 keV. Yellow curve in (a) indicates the applied ICRH power

where the power is off at 2.0 s and is 5 MW at 3.0 s.

termine the instantaneous mode frequency and toroidal modenumber. The initiating frequencies typically occur between 20and 60 kHz depending on equilibrium plasma conditions. Mostof these oscillations have been observed in discharges with a re-versed q profile deduced from the existence of Alfvén cascades[9] but they have also been observed in discharges with steadyfrequency TAEs and no Alfvén cascades, suggesting that thesemodes can be excited in either flat or monotonically increas-ing q profiles. Additional evidence for this mode arises fromreflectometer signals [10] and from soft X-ray detectors thatmeasure perturbations in the plasma core. Both showing strongfrequency correlation with the Mirnov coil signals measured atthe plasma edge.

The initiation frequency is found to scale as T1/2e (r/a =

0.15), as measured by the multi-channel ECE diagnostic. Thescaling of the initiating chirp frequencies with T 1/2 is seen inFig. 1 by noting correlation of the chirping frequencies withthe drawn in white curve, which is a plot of T

1/2e . Note that

Te � Ti as energetic protons (∼ 500 keV) primarily heat elec-trons rather than deuterium ions in the bulk plasma. When themode is excited the effect of Ti is observed by suddenly apply-ing neutral beams (the beam energy is a mix of 80 and 120 keVcomponents which directly heat core deuterium ions and elec-trons at comparable rates) with a combined power that exceeds5 MW. The initiation frequency of the mode first increases sub-stantially over a short time interval of ∼ 0.1 s and then thechirping mode terminates. This is interpreted as being due tothe sudden increase of Ti in accord with theoretical considera-tions discussed below.

The linear eigenmodeThe experimental data shown in Fig. 1, where Te � Ti ,

demonstrates that the basic mode frequency scales as T1/2e . This

scaling suggests that the n = 0 continuum geodesic acousticmode (GAM) [3] is the appropriate linear mode as its dispersionrelation in MHD theory is ω2 = γp

ρR2 (2+ 1q2 ) where γ is the adi-

abatic constant, ρ is the mass density, p = ρTe(1 + Ti/Te)/mi

is the plasma pressure, R is the major radius of the tokamak,

and mi is the mean mass per ion. Note that in kinetic theorywhen Ti/Te < 1 one finds a somewhat modified relationship,γp/ρ → Te(1 + 7Ti/4Te)/mi , with the same basic structureas MHD theory. As Ti/Te � 1, we obtain the observed scal-ing of frequency with electron temperature. However, the GAMmode is a continuum mode with no definite frequency associ-ated with it. In past literature the global property of the GAMhas been attributed to such physics mechanisms in the plasmaas the anisotropy of particle sources and turbulent fluxes [4–6]. In addition we have found that a global geodesic curvaturemode (GGAM) emerges from first principle MHD equations.The eigenfunction structure of the GGAM is found to havesimilar structure as the GAM, with the addition of a signifi-cant magnetic component that cannot be predicted from localcontinuum mode theory.

The properties of the GAM are obtained by examining thefollowing equations that have been derived in MHD theory fora continuum mode [11] on a magnetic flux surface,

(1)ρω2|∇ψ |2

B2Y + ∂

J∂θ

( |∇ψ |2B2J

∂Y

∂θ

)+ γpκsZ = 0,

(2)κsY +(

γp + B2

B2

)Z + γp

ρω2J

∂

∂θ

(1

B2J

∂Z

∂θ

)= 0.

We refer to reference [11] for the definition of the quantitiesrepresented here in their standard notation. Note that the geo-desic curvature is defined as κs = 2κ · B × ∇ψ/B2 with κthe magnetic curvature and the dependent variables defined as,Y(θ) = ξ · B ×∇ψ/|∇ψ |2 and Z(θ) = ∇ · ξ , which are surfacequantities.

To solve for a low frequency mode at low beta we takeY(θ) = 1 + δY (θ) (where δY (θ) is assumed � 1) and with theperiodicity conditions we find,

∂

∂θ

( |∇ψ |2B2J

∂δY

∂θ

)= −JγpκsZ − ρ

ω2J |∇ψ |2B2

⇒

(3)

π∫−π

dθ J

(γpκsZ + ρ

ω2|∇ψ |2B2

)= 0.

156 C.J. Boswell et al. / Physics Letters A 358 (2006) 154–158

Then using Eq. (2) to solve for Z(θ), we find

(4)Z + γp

ρω2J

∂

∂θ

(1

B2J

∂Z

∂θ

)= −κs.

The solution to Eq. (4) in the large aspect-ratio, circular cross-section limit is, Z(θ) = [2(sin θ)Bθ/B][1 − γp

ρω2(rB/Bθ )2 ]−1.Using this expression and the relation κs = 2(sin θ)Bθ/B , weevaluate the integral in Eq. (3) to find the desired eigenfre-quency.

By using the MHD normal mode analysis code CASTOR[12] for a model reversed shear equilibrium with circular crosssection, a global mode was found when the GAM contin-uum had a maximum frequency at an off-axis radial position.Such a maximum can occur when there is shear reversal. Theeigenmode structure is shown in Fig. 2 and the mode fre-quency is found to lie just above the maximum GAM fre-quency, see Fig. 3. Like the GAM, we see in Fig. 2(a) thatY , has a radially localized mode structure that is dominantlym = 0 while Z, which is proportional to the plasma densityperturbation, is also highly localized radially with primarily an|m| = 1 poloidal mode component. Local GAM theory predictsZm=1/Ym=0 = Bθ/B . For the parameters of the numerical cal-culation this ratio is reproduced at the peak of the mode withinan inverse aspect ratio. In Fig. 2(c) we plot the poloidal m-spectrum of the geodesic component of the perturbed magnetic

field, δBg ≡ δB·b×∇ψ|∇ψ | |m, as a function of radius. Though this

component, dominated by |m| = 2, is predominantly localizedin the same region as Y and Z, there is a smaller, but signif-icant, nonlocalized component of the perturbed magnetic fieldthat extends outside the region of localization. It is this “halo”

of the perturbed magnetic field that extends to the radial plasmaedge which allows detection with Mirnov coils. The measuredfrequencies are found to be synchronized with soft X-ray fre-quency signals arising from internal plasma perturbations thatare compatible with the internal flow eigenfunctions shown inFig. 2(a) and (b).

The GAM’s maximum frequency will be on the magneticaxis when the electron temperature monotonically decreases ra-dially (this is always the case in our experimental studies) andthe q-profile monotonically increases radially (this appears to

Fig. 2. Eigenmode structure for the global geodesic acoustic mode for areversed shear, circular JET-like equilibrium. (a) ReYm(r), (b) ImZm(r),(c) Im δB

gm(r). Note that ReYm(r) = ReY−m(r), ImZm(r) = − ImZ−m(r),

Im δBgm = − Im δB

g−m and that ImYm(r) = ReZm(r) = Re δB

gm = 0.

Fig. 3. Plot of the n = 0 GAM and acoustic continuum showing that the global mode (GGAM) is situated just above the GAM continuum. Note, the GGAMexperiences continuum damping well away from it’s region of localization.

C.J. Boswell et al. / Physics Letters A 358 (2006) 154–158 157

be the case in a few of the experimental observations). In thiscase a GGAM has not been found in our MHD studies. How-ever, as the GGAM eigenmode when found is very localized,we conjecture that the inclusion of additional finite Larmorradius terms into the theory will enable a global mode to beestablished when the GAM frequency has a maximum at themagnetic axis. Alternatively, it is possible that the GGAM isbeing established in these experiments via other mechanismssuch as from an anisotropic source or turbulence as is cited inthe literature [4–6] or even from the ICRH itself that induces anenergetic particle flux to leave the resonantly heated flux sur-face.

We also note that accounting for the ion temperature willimprove the comparison of the data with theory. In Fig. 1(a)the white curve is the inferred frequency of the mode whenone neglects the ion temperature. Accounting for finite Ti/Te

would increase the frequency of the theoretical curve, and thatshould make the correlation of experimental and theoretical fre-quencies closer to each other. Increasing ion temperature alsoexplains the increase in the frequency when more than 5 MWof 80/120 keV neutral beams are injected. These deuteriumbeams cause a direct heating of ions, which then increases bothTi and Te , as well as Ti/Te to enable an increase of the fre-quency of a mode associated with a GAM as ω is proportionalto (Te + Ti)

1/2. We also note that when neutral beam heating isimposed, additional dissipative processes arise that can preventthe GGAM from being spontaneously excited. Two additionalmechanisms are apparent. These are the ion Landau dampingthat arises from the increased ratio of Ti/Te that results from in-creased ion beam heating and the other is the resonance damp-ing that arises from the neutral beams themselves which forms amonotonically decreasing distribution function as a function ofenergy for constant magnetic moment, μ, and toroidal canoni-cal momentum, Pφ . We suggest that these increased dissipativeprocesses are the cause of the observed quenching of the spon-taneous excitation of the GGAM that takes place followingthe imposition of neutral beams, but only after a time delayof ∼ 0.1 s during which the above mentioned increase of theGGAM initiation frequency arises.

Free energy sourceThe source of instability to spontaneously excite n = 0

modes must come from an inverted energy distribution,∂F (E,μ,Pφ)/∂E|μ,Pφ > 0. To ascertain whether such a driveis formed in this experiment the Monte Carlo SELFO code wasused to determine the distribution function for HFS minorityproton ICRH. It was found that mirror trapped energetic pro-tons whose turning point magnetic fields are less than the ICRHresonance magnetic field, do indeed form such a distribution.

We attribute the pronounced frequency sweeping being ob-served to the formation of phase-space structures, whose gen-eral nonlinear theory has been discussed in Refs. [13,14]. Itis required that the instantaneous experimental frequency beequal to an orbit resonance frequency, which in our case is thebounce orbit frequency of a magnetically trapped particle. TheHAGIS code [15] was used to determine the bounce frequen-cies of particles with a fixed μ and Pφ and it was found that

the experimental frequency range of the chirp (32–42 kHz) ofthe observed n = 0 mode corresponded to the bounce frequencyrange of particles whose energy varied from 220 to 260 keV ina region where the SELFO code indicated that the distributionwas energy inverted. The highest energy corresponded to thebounce frequency of a deeply trapped particle and the lowestfrequency to a particle with a turning point at the ICRH res-onance layer. It should be noted that it is the nature of mirrorconfinement that higher (lower) energy particles correspond tolower (higher) bounce frequencies.

Non-linear evolutionThe changing mode frequency in the experiment is inter-

preted as a synchronism between the mode frequency and theresonance frequency as explained in Ref. [14]. At the phase-space region associated with a given mode frequency there areparticles trapped in the fields of the wave. The distributionfunction of these trapped particles are in the form of a clump(hole) where the wave trapped particle distribution has a larger(smaller) value than the ambient distribution. The clump (hole)cannot be stationary in time as the structure needs to compen-sate for the power being absorbed by dissipation present in thebackground plasma which causes the structure to move in phasespace in a manner that extracts energy from the distributionfunction. The mode frequency is synchronized to this structureand likewise shifts. Hence a clump (hole) must head to lower(higher) energy, and thus because of the bounce frequency en-ergy dependence of mirror trapped particles the frequency mustincrease (decrease). This description is compatible with the sig-nals in Fig. 1 where we see the highest frequency signals stag-nating in time. They can be interpreted as clumps that have beenconvected in phase space to the deeply trapped mirror orbits ofthe tokamak. Having reached the maximum bounce frequencyof the particles, there is no higher frequency to shift to. Notethat in this experiment the frequency increase (decrease), asso-ciated with the motion of clumps (holes), is opposite from thatarising in most other chirping events reported [14,16].

Possible implicationsThe steady repetitious GAM chirping oscillations observed

on JET can conceivably lead to significant applications in futureburning plasmas. First, we note that fusion born alpha particlesalways have degree of temperature anisotropy, which rangesfrom a modest variation in a monotonic q-profile of the form(r/R)3/2 T‖−T⊥

T‖ [17], to unity in a magnetically shear reversedplasma of “advanced” tokamaks [18] with a relatively high min-imum q (it can be shown that ion Landau damping of a GAMis proportional to exp[−q2(7/4 + Te/Ti)], so high q can haveweak ion Landau damping even when Te/Ti ≈ 1). In conditionscloser to the latter case, the alpha particles may then have thefree energy to spontaneously drive the chirping GAM mode.This instability only causes energy diffusion of alpha particlesbut no spatial diffusion as the canonical angular momentum isconserved. Thus this instability does not lead to a parasitic lossof the energetic alpha particles. Instead, this energy is beingtransferred to the background plasma through the plasma dissi-pation that is giving rise to the chirping frequencies. This newchannel for the absorption of alpha particle energy might then

158 C.J. Boswell et al. / Physics Letters A 358 (2006) 154–158

compete with the standard electron drag process and allow thepopulation of energetic alpha particles to be significantly low-ered for a given amount of fusion energy production, therebyreducing the finite-n universal instability drive that can inducethe parasitic loss of alpha particles due to Alfvénic instabilities.A second application is to note that there is strong couplingof the GAM mode with zonal flows [19]. If the GAM chirp-ing is established by applying ICRH in burning plasmas, it maythen be possible to have external control of the zonal flows andthrough this means, have an external “knob” to regulate turbu-lent diffusion.

Acknowledgements

The authors would like to thank Dr. M.F.F. Nave for ouruseful discussions. Also, we are indebted to Dr. G.D. Con-way who brought to our attention the properties of geodesicacoustic modes. This work has been conducted under the Euro-pean Fusion Development Agreement and was funded partlyby US DoE contract DE-FG02-99ER54563 and DE-FG02-04ER54742.

References

[1] R.F. Heeter, PhD thesis, Princeton University, 1999.[2] A.C.A. Figueiredo, M.F.F. Nave, IEEE Trans. Plasma Sci. 33 (2005) 468.[3] N. Winsor, J.L. Johnson, J.M. Dawson, Phys. Fluids 11 (1968) 2448.[4] P.H. Diamond, S.I. Ithoh, K. Itoh, T. Hahm, Plasma Phys. Controlled Fu-

sion 47 (2005) R35.[5] G.D. Conway, et al., Plasma Phys. Controlled Fusion 47 (2005) 1165.[6] B. Scott, New J. Phys. 7 (2005) 92.[7] T. Hellsten, T. Johnson, J. Carlsson, Nucl. Fusion 44 (2004) 892.[8] M.J. Mantsinen, et al., Nucl. Fusion 39 (1999) 459.[9] S.E. Sharapov, et al., Phys. Plasmas 9 (2002) 2027.

[10] S.E. Sharapov, et al., Phys. Rev. Lett. 93 (2004) 165001.[11] C.Z. Cheng, M.S. Chance, Phys. Fluids 11 (1986) 3695.[12] W. Kerner, et al., J. Comput. Phys. 142 (1998) 271.[13] P.W. Terry, P.H. Diamond, T.S. Hahm, Phys. Fluids B 2 (1990) 2048.[14] H.L. Berk, B.N. Breizman, N.V. Petviashvili, Phys. Lett. A 234 (1997)

213.[15] S.D. Pinches, et al., Comput. Phys. Commun. 111 (1998) 133.[16] S.D. Pinches, H.L. Berk, M.P. Gryazenvich, S.E. Sharapov, Plasma Phys.

Controlled Fusion 46 (2004) S47.[17] T.E. Stringer, Plasma Phys. 16 (1974) 651.[18] V. Yavorskij, et al., Nucl. Fusion 43 (2003) 1077.[19] H. Sugama, T.H. Wanatabe, Phys. Plasmas 13 (2006) 0120501.