Analysis of the nonlinear behavior of shear-Alfven modes ...cation of the soliton-like EPM mode structure, which contin-ues until plasma nonuniformity (in general, increasing continuum

Analysis of the nonlinear behavior of shear-Alfv!en modes in tokamaks basedon Hamiltonian mapping techniques

S. Briguglio,1,a) X. Wang,2 F. Zonca,1,3 G. Vlad,1 G. Fogaccia,1 C. Di Troia,1 and V. Fusco1

1ENEA C. R. Frascati, Via E. Fermi 45, C.P. 65-00044 Frascati, Rome, Italy2Max-Planck-Institut f€ur Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany3Institute for Fusion Theory and Simulation and Department of Physics, Zhejiang University,Hangzhou 310027, People’s Republic of China

(Received 31 July 2014; accepted 14 October 2014; published online 4 November 2014)

We present a series of numerical simulation experiments set up to illustrate the fundamentalphysics processes underlying the nonlinear dynamics of Alfv!enic modes resonantly excited byenergetic particles in tokamak plasmas and of the ensuing energetic particle transports. Thesephenomena are investigated by following the evolution of a test particle population in theelectromagnetic fields computed in self-consistent MHD-particle simulation performed by theHMGC code. Hamiltonian mapping techniques are used to extract and illustrate several features ofwave-particle dynamics. The universal structure of resonant particle phase space near an isolatedresonance is recovered and analyzed, showing that bounded orbits and untrapped trajectories,divided by the instantaneous separatrix, form phase space zonal structures, whose characteristicnon-adiabatic evolution time is the same as the nonlinear time of the underlying fluctuations.Bounded orbits correspond to a net outward resonant particle flux, which produces a flatteningand/or gradient inversion of the fast ion density profile around the peak of the linear wave-particleresonance. The connection of this phenomenon to the mode saturation is analyzed with reference totwo different cases: a Toroidal Alfv!en eigenmode in a low shear magnetic equilibrium and aweakly unstable energetic particle mode for stronger magnetic shear. It is shown that, in the formercase, saturation is reached because of radial decoupling (resonant particle redistribution matchingthe mode radial width) and is characterized by a weak dependence of the mode amplitude on thegrowth rate. In the latter case, saturation is due to resonance detuning (resonant particle redistribu-tion matching the resonance width) with a stronger dependence of the mode amplitude on thegrowth rate. [http://dx.doi.org/10.1063/1.4901028]

I. INTRODUCTION

In burning plasmas, shear-Alfv!en modes can be drivenunstable by resonant interactions with energetic ions pro-duced by additional heating or nuclear fusion reactions. Infact, these particles are characterized by velocities of theorder of the Alfv!en speed, the typical group velocity of shearAlfv!en waves parallel to the equilibrium magnetic field. TheAlfv!enic fluctuation spectrum generated in this way can inturn affect energetic ion confinement, preventing their ther-malization in the plasma core and, possibly, increasing thethermal load on the material wall. The assessment of plasmaoperation in next generation Tokamak experiments, thus,strongly relies on the comprehension of Alfv!en mode dy-namics, with regard to both the linear stability properties(which modes are expected to be unstable) as well as thenonlinear saturation mechanisms (which saturation level isexpected for fluctuation amplitudes and what are the ensuingeffects on the fast ion confinement).

Due to the relevance of these issues, there exists a veryrich literature investigating linear stability of Alfv!en modesand their nonlinear saturation, including theoretical and nu-merical analyses, as well as comparisons of correspondingpredictions with experimental observations. Recent reviews

also exist, focusing on general aspects and applications to theInternational Thermonuclear Experimental Reactor (ITER),1

on theoretical analyses,2–4 on basic physics properties and ex-perimental evidences,5 on applications of model theoreticaldescriptions to interpretation of experimental observations,6

and on numerical investigations and application to tokamakexperiments.7 Most available literature focuses on tokamakplasmas; however, some works have addressed similaritiesand differences of Alfv!en wave interactions with energeticparticles in tokamaks and stellarators.8,9

Nonlinear Alfv!en wave and energetic particle physicscan be understood and discussed along two main routes:3 (i)nonlinear interactions of Alfv!enic fluctuations with energeticparticles; and (ii) wave-wave interactions and the resultantspectral energy transfer. Although the second route is veryimportant3 and plays crucial roles in multi-scale dynamics ofburning plasmas,10 most nonlinear investigations address thefirst route, as effects of the Alfv!enic fluctuation spectrum onenergetic particles in fusion devices are of significant con-cern. The first route is also the subject of the present work,where we introduce novel numerical diagnostics techniquesdeveloped ad hoc for representing energetic particle phasespace; i.e., for isolating and understanding the fundamentalprocesses of resonant Alfv!en wave-energetic particle interac-tions, nonlinear saturation of the Alfv!en fluctuation spectrumand energetic particle transports.a)Electronic mail: [email protected]

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PHYSICS OF PLASMAS 21, 112301 (2014)

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As discussed in Refs. 4 and 11, there are two“paradigms” currently adopted for investigating nonlinearinteractions of Alfv!enic fluctuations with energetic particles.The “bump-on-tail” paradigm4,11 applies only when theplasma is sufficiently close to marginal stability such thatenergetic particle orbit nonlinear distortions are small com-pared to fluctuation wavelengths. It is based on the Ansatzthat resonant wave-particle interactions can be investigatedby analogy with Langmuir waves in a 1D Vlasov system inthe presence of sources and collisions.12–15 The “fishbone”paradigm,4,11 meanwhile, takes into account the self-consistent interplay of nonlinear dynamic evolutions of fluc-tuation structures and energetic particle transports, which aregenerally important when resonant structures in the energeticparticle phase space (also dubbed “phase-space zonalstructures”16–18) coherently evolve with the underlying non-linear mode structures due to phase-locking.11,16–20 Strictlyspeaking, these two paradigms apply to limiting cases. Forexample, the “bump-on-tail” paradigm can be adopted forweak Toroidal Alfv!en Eigenmodes (TAE’s)21 close to mar-ginal stability; and the “fishbone” paradigm well describesfast growing Energetic Particle Modes (EPM’s)22 driven bysufficiently strong fast ion pressure gradients. It has beenshown that weak TAEs saturate because of wave-particletrapping,12–14,23–27 causing the resonant fast ion phase-spacegradients to vanish locally. As the fraction of particlestrapped within the wave grows with the mode amplitude, thefree energy source is progressively depleted. Saturation isreached when a significant fraction of linearly resonant par-ticles is trapped. Being this fraction proportional to the lineargrowth rate of the mode, a direct dependence of the satura-tion amplitude on the growth rate itself is established, corre-sponding to dB=B ! ðc=xÞ2, consistent with the case of aplasma wave excited by a cold electron beam in a 1D uni-form plasma.28–31 Fast growing EPMs, meanwhile, saturatebecause of secular radial motion of energetic particles,locked in phase (avalanche32,33) with the convective amplifi-cation of the soliton-like EPM mode structure, which contin-ues until plasma nonuniformity (in general, increasingcontinuum damping) quenches the process.4,11,16,17,32,33

The theoretical framework for bridging these two limit-ing paradigms and discussing the fundamental processesunderlying nonlinear Alfv!en wave interactions with ener-getic particles both in extreme and intermediate cases is laidout in Refs. 4, 11, and 16–18. There, the concept of“resonance detuning,” due to the nonlinear change in thewave-particle phase, is introduced along with that of “radialdecoupling,” occurring when the nonlinear particle excursionbecomes comparable with the perpendicular wavelength.Both mechanisms occur in nonuniform plasmas and are dueto resonant wave-particle interactions. However, only thefirst one is accounted for by analyses adopting the “bump-on-tail” paradigm discussed above; while both are generallyimportant for the “fishbone” paradigm. The relative role of“resonance detuning” and “radial decoupling” depends onthe relative ordering of finite wave-particle radial interactionlength and radial mode width, which, in turn, depend on theconsidered resonance (precession or precession bounce, formagnetically trapped particles; and transit, for circulating

particles) and nonuniform plasma equilibrium (magneticshear, in particular).4,16,18 Meanwhile, depending on the dis-persive properties of the considered fluctuation, phase-locking may also modify the relative importance of“resonance detuning” and “radial decoupling” as in the caseof EPM32,33 and fishbones.34 A systematic and thoroughanalysis of these process and, more generally, of the nonlin-ear dynamics of “phase-space zonal structures”16–18 on thebasis of numerical simulation results is lacking; and it is themain aim of the present paper. Our investigation is based onnumerical simulations performed by the HMGC code,36

which is capable of addressing the different processes self-consistently, thereby shedding light on the various nonlineardynamics that will be important in burning plasma relevantconditions.

The HMGC code, based on the hybrid MHD gyrokineticmodel35 and originally developed at the Frascati laboratories,has been used to investigate energetic particle driven modes(such as TAEs and EPMs37–39), as well as to analyze modesobserved in existing devices (JT-60U,40 DIII-D41) orexpected in forthcoming burning plasmas (ITER32,42) andproposed experiments (the Fusion Advanced Studies Torus(FAST)43–45). It solves O(!3) reduced-MHD equations48 inthe zero pressure limit for the bulk plasma, with ! being theinverse aspect ratio. The reduced MHD equations are closedby the energetic particle pressure gradient, computed bysolving the nonlinear Vlasov equation in the drift-kineticlimit via particle-in-cell methods. Thus, fast ion finiteLarmor radius effects are neglected, but finite magnetic driftorbit width effects are retained consistently.46,47 Energeticparticle dynamics are self-consistently retained and aretreated nonperturbatively: their pressure term contributes todetermine both structure and evolution of electromagneticfields. Recently, an extended version of HMGC (XHMGC)has been developed to include kinetic thermal ion compressi-bility and diamagnetic effects, in order to account for ther-mal ion collisionless response to low-frequency Alfv!enicmodes driven by energetic particles.49 This extended versioncan also handle up to three independent particle populationskinetically, assuming different equilibrium distribution func-tions, as, e.g., bulk ions, energetic ions and/or electronsaccelerated by neutral beam injection (NBI), ion/electron cy-clotron resonance heating (ICRH/ECRH), fusion generatedalpha particles, etc. In the present paper, however, theseextended physics simulation capabilities will not be used, asthey are not needed to analyze the cases discussed in thiswork: namely, a n¼ 6 TAE (with n being the toroidal modenumber), in a low shear, large aspect ratio, circular magneticsurface equilibrium; and a weakly unstable n¼ 2 EPM, in astronger magnetic shear equilibrium. Both modes fall in anintermediate regime (c/x of order of few percent) betweenthe quasi-marginally stable “bump-on-tail” regime and thestrongly unstable “fishbone” one. Focusing on them willallow to elucidate the role of “radial decoupling” and“resonance detuning” without entering the extreme dynamicregime of EPM avalanches.4,32,33

In order to investigate the nonlinear dynamics of shear-Alfv!en modes and the corresponding behaviour of energeticions, we look at the evolution of a suited set of test

112301-2 Briguglio et al. Phys. Plasmas 21, 112301 (2014)


particles50 in the perturbed electromagnetic fields obtainedby self-consistent HMGC simulations. Several features ofthe wave-particle nonlinear interactions are evidenced byresorting to Hamiltonian mapping techniques: each test parti-cle is represented by a marker in the plane (H, P/), where His the wave-particle phase and P/ is the toroidal angular mo-mentum. Coordinates of each particle are computed at thetimes it crosses the equatorial plane. Drift motions inducedby the interaction with perturbed fields give rise to boundedorbits in the (H, P/) phase-space for particles not too farfrom the resonant P/. It is shown that bounded orbits corre-spond to a net outward particle flux, which produces a flat-tening of the fast ion density profile near the resonance. Suchdensity flattening involves an increasingly wider region asthe mode amplitude increases. At the same time, it causes anincreasing reduction of the free energy source for the systeminstability. Mode saturation is reached when the regionaffected by density flattening extends to the whole poten-tially resonant phase-space region. We show that the two dif-ferent cases considered in this paper are paradigmatic of twodistinct regimes: in the low shear case, for not too littlemode growth rates, the potentially resonant region is limitedby the radial mode width rather than the finite wave-particleradial interaction length introduced above4,16,18 (or“resonance width”, for brevity, as it will be referred to inwhat follows); in the larger shear case, for not too largegrowth rates, the region is limited by the resonance width.We find that in the former case the saturation mechanism(“radial decoupling”) yields a weaker scaling of the satura-tion mode amplitude with the growth rate than that obtainedin the latter case (“resonance detuning”). This corresponds tothe fact that the resonance width has a stronger dependenceon the growth rate than the mode width. This also shows theimportant role of magnetic shear in determining mode satu-ration, nonlinear dynamics and energetic particle redistribu-tion. For the sake of simplicity, numerical simulations belowrefer to fluctuations excited by transit resonance with circulat-ing energetic particles. Precession and precession-bounce res-onance with magnetically trapped particles deserve a separatein depth analysis due to the implications they may have onnon-local behaviors.4,16,18 Furthermore, the case of a TAEmode dominated by “radial decoupling” and of a EPM modewith prevalent “resonance detuning” have been selected toillustrate that, depending on mode drive and plasma equilib-rium nonuniformity, nonlinear dynamics may be more com-plicated than one may expect on the basis of “bump-on-tail”and “fishbone” paradigms taken as limiting cases.

The paper is organized as follows. Section II describesthermal plasma and energetic particle equilibrium profilesadopted for the reference cases considered for our study.Section III analyzes the structure of wave-particle resonan-ces in energetic particle velocity space during the linearphase of mode evolution. The selection criteria of test-particles suited for the investigation of the nonlinearmode-particle dynamics are described in Sec. IV, and thecorresponding diagnostics based on Hamiltonian mappingtechniques are presented in Sec. V. The issue of the finiteradial mode and resonance widths is discussed in Sec. VI.Section VII analyzes test particle numerical results, giving

evidence to several aspects of linear and nonlinear mode-particle interactions. The mechanisms that cause mode satu-ration are examined in Sec. VIII, focused on the flattening ofdensity gradient produced by the nonlinear particle flux; andin Sec. IX, where the relative importance of the radial modeand resonance width in determining the saturation level isexamined in the two considered reference cases. Conclusionsare drawn in Sec. X.

II. REFERENCE THERMAL PLASMA AND ENERGETICPARTICLE EQUILIBRIA

Our analysis will mainly focus on a specific case, whichhas been recently adopted51 as a benchmark case for thecomparison of different simulation codes with respect tothe linear dynamics of energetic-particle driven TAEs in theframework of the Energetic particle Topical Group of theInternational Tokamak Physics Activity (ITPA). A circularshifted magnetic-surface Tokamak equilibrium is consid-ered, characterized by major radius R0¼ 10 m, minor radiusa¼ 1 m, safety factor q¼ q0þ (qa& q0) (r/a)2, q0¼ 1.71,qa¼ 1.87 and on-axis magnetic field B0¼ 3T. Thebulk plasma is characterized by flat ion (Hydrogen) and elec-tron densities, ni¼ ne¼ 2' 1019 m&3, and temperatures,Ti¼Te¼ 1 keV. The fast (“Hot”) particle population (deuter-ons) has a Maxwellian initial distribution function

FH0 s;Eð Þ ¼mH

2pTH

! "32

nH sð Þe&E

TH ; (1)

with E being the kinetic energy of the energetic particle,s ( ð1& weq=weq0Þ

1=2, B ( R0B/0r/þ R0rweq 'r/, TH

¼TH0¼ 400 keV, nHðsÞ¼ nH0 c3 expf&ðc2=c1Þtanh½ðs& c0Þ=c2*g, c0¼0.49123, c1¼0.298228, c2¼0.198739, and c3

¼0.521298. Moreover, we fix nH0¼ 0:721'1017m&3, corre-sponding to nH0=ni0¼ 3:62'10&3. These parameters yieldvH0=vA0’ 0:3 and qH0=a’ 0:03, with vH0(ðTH0=mHÞ1=2;vA0 being the on-axis Alfv!en velocity, qH0( vH0=XH0 andXH0 being the energetic particle on-axis cyclotron frequency.

As in the ITPA benchmark,51 a single toroidal modenumber, n¼ 6, is retained: this corresponds to neglectingfluid mode-mode coupling in the evolution of electromag-netic fields, while fully taking into account wave-particlenonlinearities. Poloidal harmonics are instead retained in therange m ¼ 8+ 13.

HMGC evolves visco-resistive MHD equations; viscos-ity and resistivity are fixed as "sA0=a2 ¼ 10&8 (with sA0 ¼R0=vA0 being the on-axis Alfv!en time) and S&1 ¼ 10&6, withthe Lundquist number defined as S ( 4pa2=ðgc2sA0Þ.

The evolution of the perturbed magnetic energy isshown, for this reference case, in Fig. 1, where energy is inunits of a3B2

0=8p. Saturation is reached at t ’ 1700 sA0: Theradial structure of the poloidal harmonics of the scalar poten-tial (normalized to TH0/eH) and the parallel component of thevector potential (normalized to B/0qH0) during the linearphase are shown in Fig. 2 along with the correspondingpower spectrum (intensity contour plot in arbitrary units) ofthe scalar potential in the (r, x) plane: the dominant poloidalharmonics corresponds to m¼ 10 and 11, while the



following values are found for the real frequency and thegrowth rate during the linear phase:

xsA0 ’ 0:2886;

csA0 ’ 0:0087;

with c=x ’ 0:03.We shall refer to this case as “case 1.”

We will also consider a different case (“case 2”), inorder to examine how the nonlinear dynamics depend on thetypical equilibrium and/or mode scale lengths: a weaklyunstable n¼ 2 EPM in an equilibrium characterized by largermagnetic shear than the previous case; qa¼ 1.1, qa¼ 1.9 giv-ing qðrÞ ¼ 1:1þ 0:8ðr=aÞ2: The bulk plasma density profileis chosen in such a way to have aligned toroidal frequencygaps in the Alfv!en continuum: ni ¼ ni0q2

0=q2; but still suffi-ciently separate in radial location (due to choice of lown¼ 2) that EPM nonlinear dynamics is local and not affectedby avalanches.4,16,32,33 The energetic particle initial distribu-tion function is assumed to be Maxwellian, with flat temper-ature and density profile given by nH ¼ nH0 expð&19:53 s4Þ:The other relevant dimensionless parameters are the follow-ing: R0=a ¼ 10; vH0=vA0 ¼ 1, qH0=a ¼ 0:01, nH0=ni0 ¼1:75 10&3, mH/mi¼ 2, "sA0=a2 ¼ 0, and S&1 ¼ 10&6:

Figures 3 and 4 show the time evolution of perturbedmagnetic energy (normalized to a3B2

0=8p) and, respectively,the linear mode structure and power spectrum for this secondreference case. A first saturation is reached at t ’ 1225 sA0,followed by a rich nonlinear activity exhibiting damping,further growth and saturation phases, with nonlinear oscilla-tions. The dominant poloidal harmonics are now m¼ 2 and3, and the observed frequency and growth rate are

xsA0 ’ 0:524;

csA0 ’ 0:0125;

with c/x ’ 0.024.

FIG. 1. Time evolution of the magnetic energy for case 1. Energy is in unitsof a3B2

0=8p.

FIG. 2. Radial structure of the poloidalharmonics of the scalar potential (a)and the parallel component of the vec-tor potential (b) during the linearphase; corresponding power spectrumof the scalar potential (c).



In the following, unless stated explicitly, we will refer tocase 1.

III. WAVE-PARTICLE RESONANCE STRUCTURE

We are interested in identifying the phase-space struc-ture of wave-particle resonances in the linear phase. To this

aim, we compute the total fast particle energy and its timederivative

EH ¼1

m3H

ðd6ZDzc!ZFH

mH

2U2 þMXH

! "; (2)

dEH

dt¼ 1

m3H

ðd6Z

@

@tDzc!ZFHð Þ

mH

2U2 þMXH

! "

¼ & 1

m3H

ðd6Z

@

@ZiDzc!ZFH

dZi

dt

! "mH

2U2 þMXH

! "

¼ 1

m3H

ðd6ZDzc!ZFH

dZi

dt

@

@Zi

mH

2U2 þMXH

! "

(3)

Here, EH is the total energy of fast particles, Z (ðr; h;/;M;U; #Þ are the gyrocenter coordinates (r is theradial coordinate, h and / are the poloidal and toroidal angle,respectively, M is the conserved magnetic momentum, U isthe parallel velocity and # the gyrophase) and Dzc!Z is theJacobian of the transformation from canonical zc to gyrocen-ter coordinates.

The power transfer from energetic particles to the waveis given by &dEH=dt: We can compute such quantity in thefollowing way. Let us discretize the distribution function(times Dzc!Z) according to

Dzc!ZFH ’X

l

½#wl þ #DlFH0ðZlÞ*dð5ÞðZ & ZlÞ; (4)

Fig. 3. Same as Fig. 1 for case 2.

FIG. 4. Same as Fig. 2 for case 2.



with Zl¼ Zl(t) being the gyrocenter coordinates of the l-thmacroparticle, FH¼FH0þ dFH, and

dð5ÞðZ&ZlÞ(dðr&rlÞdðh&hlÞdð/&/lÞdðM&MlÞdðU&UlÞ;(5)

as Dzc!ZFH does not depend on the gyrophase #. Here

#Dl ( DlðtÞDzc!ZðZlðtÞÞ; (6)

with

DlðtÞ ( ½DrDhD/DMDU*l; (7)

and

#wl ¼ #wlðtÞ ( #DldFHðt; ZlðtÞÞ: (8)

Note that #Dl, the phase-space volume element occupied bythe l-th macroparticle, is a constant of motion of the l–thmacro-particle (Liouville’s theorem). Then, we define

& dEH

dt( 1

m3

ðdrdMdUP r;M;Uð Þ; (9)

with P(r, M, U) obtained from Eqs. (3) and (4)

P r;M;Uð Þ ( &2pð

dhd/Dzc!ZFHdZi

dt

@

@Zi

mH

2U2 þMXH

! "

¼ &2pð

dhd/Dzc!Z dFHdZi

dt

! "

1þ FH0

dZi

dt

! "

2

" #@

@Zi

mH

2U2 þMXH

! "

’ &2pX

l

#wdZi

dt

! "

1þ #DFH0 Zð Þ dZi

dt

! "

2

" #@

@Zi

mH

2U2 þMXH

! "$ %( )

l

' d r & rlð Þd M &Mlð Þd U & Ulð Þ

(X

l

pl d r & rlð Þd M &Mlð Þd U & Ulð Þ; (10)

with

pl ( &2p #wdZi

dt

! "

1þ #DFH0 Zð Þ dZi

dt

! "

2

" #@

@Zi

mH

2U2 þMXH

! "$ %( )

l

: (11)

Here, we have used the subscripts 1 and 2 to indicate the lin-ear and nonlinear contributions in the perturbed fields to thephase-space velocities dZi/dt, and taken into account that, asthe unperturbed distribution function does not depend on /,it contributes to the /–averaged quantity only when multi-plied by the nonlinear terms. The quantity pl will be indi-cated, in the following, as the “l–th macroparticle powertransfer”. It corresponds to the power transfer weighted bythe distribution function and integrated in h and /; the over-all cancellation, at each time, of the large contribution to thepower transfer yielded by the linear terms in the fluctuatingfields has been explicitly enforced. Note that such enforce-ment would not be possible if we computed the particlepower transfer as pl ¼ 2p½#wl þ #DlFH0ðZlÞ*ðdE=dtÞl, througha direct computation of the particle kinetic energy variation(that is, without splitting the different contributions relatedto linear or nonlinear terms in the perturbed fields).

We can compute P on a discrete grid ri, Mj, Uk in thefollowing way

Pi;j;k (ð

drdMdUSrðri& rÞSMðMj&MÞSUðUk&UÞPðr;M;UÞ

(12)

’X

l

pl Srðri & rlÞSMðMj &MlÞSUðUk & UlÞ (13)

In the above expressions, Sr, SM and SU are smoothing func-tion normalized according to

ðdxSxðx& xiÞ ¼ 1 8i; (14)

needed to collect the singular contributions of the macropar-ticles on the discrete grid. In HMGC, they are triangle func-tions centered, for each i, at x¼ xi and null outside theinterval ½xi&1; xiþ1*. The power transfer will be then approxi-mated by

& dEH

dt’ 1

m3H

X

i;j;k

DriDMjDUkPi;j;k: (15)

In Fig. 5, the density of the wave-particle power exchange(in arbitrary units) proportional to

Pi2mode DriPi;j;k, is shown

on the velocity-space grid ðMj; UkÞ, with M ( MXH0=TH0

and U ( U=vH0: The integration over the minor radius r islimited to the region where the mode is localized. Positivesign corresponds to destabilization of the mode.

IV. TEST PARTICLE SELECTION: A PROPERRESONANT PARTICLE SAMPLE

In order to get information about the resonant particlebehaviour and its connection with mode saturation, we



follow the evolution of a set of test particles (resonant particlesample) in the fields computed from the self-consistent simu-lation. In principle, the same information could be obtainedby the direct analysis of the behaviour of the simulation par-ticles (that is, particles used in the self-consistent simulationto sample the whole phase space). The advantage of a testparticle approach is that it allows us for a zoomed investiga-tion of specific phase space regions (in particular, those wherethe resonances are localized) by simply evolving the selectedtest-particle set in the stored self-consistent fluctuating fields,without requiring a new heavy self-consistent simulation pereach different choice of the phase space zoom.

At t¼ 0, we set h¼ 0 for all test particles. The othercoordinates are initialized in such a way that test particlescorrespond to a set of resonant particles driving the modeunstable during the linear phase (resonant particle sample).To this aim, we identify, from plots like that shown in Fig. 5,a set of coordinates r¼ r0, M¼M0 and U¼U0 around whichthe linear drive (that is the particle-wave power transfer) issignificant. Following Ref. 50, we then define the quantityC ( xP/ & nE, with P/ ( mHRU þ eHR0ðw& w0Þ=c beingthe toroidal angular momentum. Such a quantity is a constantof the (perturbed) motion, provided that the perturbed field ischaracterized (as in the considered case) by a single toroidalmode number and constant frequency, as it can be easily recov-ered from the equations of motion in the Hamiltonian form

dP/=dt ¼ &@H=@/ (16)

and

dE=dt ¼ @H=@t; (17)

where H is the single particle Hamiltonian, characterizedby time and toroidal angle dependence in the formH¼H(xt& n/). At the leading order, we can approximate

P/ ’ mHRU þ eHR0ðweq & weq0Þ=c ( P/ðr; h;UÞ (18)

and

C ’ xP/ & nðmHU2=2þMXHÞ ( Cðr; h;M;UÞ: (19)

We can then compute the value C0(C(r0, 0, M0, U0), corre-sponding to the considered resonance. Selection of test parti-cle coordinates for the resonant particle sample proceeds asfollows: several different values of the radial coordinate r(around the mode localization) are chosen. Once r is fixed,the value of the parallel velocity U is obtained by matchingthe value M¼M0 and C¼C0; that is, U¼U(r, M0, C0). Foreach set of these coordinates, several equispaced values of/ are chosen in the open interval ½0; 2p½.

Fixing two constants of motions in the same way for alltest particles corresponds to cutting the phase space into in-finitesimal slices that do not mix together even during thenonlinear evolution of the mode and looking at the dynamicscharacterizing one of these slices (M¼M0 and C¼C0); thatis, the nonlinear evolution of an isolated linear resonance.Note that this method remains valid for frequency chirpingmodes, provided j _xj!jc2j, with c being the characteristic lin-ear growth rate. In fact, in this limit, the resonance frequencyshift in one nonlinear time (!c&1) is jDxj!jcj, the typicalresonance width. Thus, resonant particle samples belongingto separate linear resonances will not mix nonlinearly, evenfor frequency chirping modes. In the present case, we fixM ¼ 0:2 and C ( C=TH0 ¼ &4:74 (corresponding to U ¼1:24 at r ( r=a ¼ 0:52: a resonance peak yielded by co-passing particles), and distribute particles in the radial inter-val 0:25 , r , 0:8.

In general, fixing constants of motion in such a way thatthe test-particle set corresponds to a relevant resonant-particlesample does not ensure that this set represents adequately thewhole resonant-particle population of the self-consistent sim-ulation. In principle–as discussed in Sec. VI with reference tocase 2–, several different portions of the phase space (corre-sponding to different C values) could play an important rolein the linear destabilization of the mode, with resonancespeaked around different radial positions; and the nonlineardynamics of each of such portions could evolve, nonlinearly,in a different way from the others. We can quantify the repre-sentativeness of test particles by comparing both radial profileand time evolution of the power transfer integrated over thetest-particle set with that integrated over the whole particlepopulation of the self-consistent simulation. Fig. 6 comparesthe radial profile of the power transfer for the two popula-tions. We see that the two profiles exhibit both close peaksand comparable width. Note that, if separated peaks werefound, we could improve the selection of the test-particlepopulation, by looking for a better choice of r0, M0 and U0

(that is, a better choice of C0). The radial width of the twoprofiles cannot instead be controlled, as it depends on givenfeatures of the dynamical system: namely, the spatial scalescharacterizing equilibrium gradients, mode and resonances.We will discuss this issue further in Sec. VI.

In the present case, we expect, on the basis of the goodagreement of the two profiles, that the test-particle set

FIG. 5. Power exchange between the energetic particles and the mode in thenormalized (M, U) space, integrated over a radial shell around the modelocalization (0:448 , r=a , 0:552). Positive sign corresponds to powertransfer from particles to mode; that is, to mode destabilization. The solidand dashed lines approximately indicate the loss cone for the inner and outerradial surface of the considered shell, respectively. It is apparent that themain contribution to the drive is given, here, by circulating particles.



represents the whole resonant-particle population fairly well.This is confirmed by comparing the time evolution of thepower transfer integrated over the two different populations,(Fig. 7). We see that the agreement is quite satisfactory untilsaturation is reached (t ’ 1700 sA0). We can then be confi-dent that investigation of the dynamics of the resonant parti-cle sample yields relevant information of the nonlinearevolution of the self-consistent wave-particle interactions.

Two remarks have to be done concerning the resultsshown in Fig. 6. First, power transfer has been computed re-ferring each particle contribution to the radial coordinateassumed by the same particle when it crosses the equatorialplane in its outmost position. Let us call, conventionally,

such coordinate the “equatorial” radial coordinate to distin-guish it from the “instantaneous” radial coordinate. In thefollowing, when reporting the radial dependence of quanti-ties integrated over the particle population or the test-particleset, r will represent the equatorial radial coordinate, unlessexplicitly stated. The second remark is a technical one, con-cerning the weight of each test particle contributing to thepower transfer (as well as, in the following, to other inte-grated quantities). We can compute the (constant) phase-space volume element corresponding to each test particle inthe coordinates ZW( (r, h, /, M, C) as

#Dtestl ( DWlDzc!ZW ðZWlÞ: (20)

The infinitesimal quantity

DWl ( ½DrDhD/DMDC*l (21)

is the same for all test particles (as we are initializing testparticles with fixed M and C, h¼ 0 and several equispaced rand /). Then, we obtain, from Eqs. (18) and (19)

#Dtestl / Dzc!ZW ZWlð Þ ¼ m2

HXHlRlrl

&&&&@Z

@ZW

&&&&l

/ rl

&&&&@U

@C

&&&&l

’ rlmHjxR rl; 0ð Þ & nUlj&1; (22)

where j@Z=@ZW j is the Jacobian of the coordinate transfor-mation Z ! ZW : Note that if multi-C test particle sets areconsidered (see Sec. VI), the element DC is not necessarilythe same for all particles. In our case, we will determine thedifferent values C0i by fixing several equispaced values ofparallel velocity U0i and computing C0i ¼ Cðr0; 0;M0;U0iÞ.With such choice, we will get

#Dtestl / rl

&&&&@U

@C

&&&&l

&&&&@C0

@U0

&&&&l

’ rlmHjxR r0; 0ð Þ & nU0jljxR rl; 0ð Þ & nUlj

: (23)

V. HAMILTONIAN MAPPING TECHNIQUES FORPHASE-SPACE NUMERICAL DIAGNOSTICS

Test particle coordinates are collected every time (t¼ tj)the particle crosses the equatorial plane (h¼ 0) at its outmostR position. The wave phase seen by the particle at thosetimes is

Hj ¼ xtj & n/j þ 2pjmr; (24)

where (m, n) are poloidal and toroidal mode numbers,respectively, and r( sign(U). The resonance condition is

DHj ( Hjþ1 &Hj ¼ 2pk; (25)

with the integer k denoting the “bounce harmonics”. It canbe written in the form

x& xresðr;M0;C0; kÞ ¼ 0; (26)

with, for circulating particles,

xresðr;M0;C0; kÞ ( nxD þ ½ðn#q & mÞrþ kÞ*xb: (27)

FIG. 7. Power transfer rate integrated over the whole simulation particlepopulation (blue) compared with the same quantity integrated over the testparticle population only (red). The agreement is fairly good until saturationis reached (t ’ 1700 sA0).

FIG. 6. Radial profiles of the power transfer rate integrated over test-particle(red) and selfconsistent-simulation (blue) populations. The two profiles ex-hibit both close peaks and comparable width. Note that the “instantaneous”radial coordinate has been considered here.



Here,4,16,18

xD (D/2p& r#q

$ %xb (28)

is the precession frequency, D/ is the change in toroidalangle over the bounce time sb (i.e., the time needed to com-plete a closed orbit in the poloidal plane), defined as

sb (þ

dh_h; (29)

#q ( r2p

þqdh; (30)

with q being the safety factor and the integral taken alongthe particle orbit; and

xb (2psb

(31)

is the bounce frequency (in our case, transit frequency).Equation (26) can be solved with respect to r, yieldingr ¼ rresðx;M0;C0; kÞ, or P/, yielding P/ ¼ P/ resðx;M0;C0; kÞ: In the present case, the relevant bounce harmonic isk¼ 1, as shown in Fig. 8.

In the unperturbed motion, dP/=dt ¼ 0. Then, duringthe linear phase of the mode evolution, the particle trajecto-ries in the (H, P/) plane essentially reduce to fixed points forP/ ¼ P/ res, while they correspond to drift along the H axisin the positive (negative) direction, for P/ greater (less) thanP/ res. This is represented in Fig. 9 where P/ is given in unitsof mHavH0 and H is reported to the interval ½0; 2p½. Here andin the following, each marker is doubled by a twin markerin the interval ½2p; 4p½, in order to yield a better visualizationof the relative dynamics. Moreover, for simplicity, we

shall indicate the quantity mod½DHj; 2p*=ðtjþ1 & tjÞ (withmod½DHj; 2p*= defined as DHj modulo 2p) as dH/dt.

In the nonlinear phase, P/ varies because of the mode-particle interaction (e.g., radial E' B drift). Even particlesthat were initially resonant are brought out of resonance, get-ting nonzero dH/dt and drifting in phase until the drift in P/

is inverted. Particles that cross the P/ ¼ P/ res line revert val-ues of dH/dt as well. Thus, their orbits are bounded and theywould properly close if the field amplitude were constant intime. This is true for particles born close to the resonance,while particles born with P/ far from the resonance maintaindrifting orbits, as they do not cross P/ ¼ P/ res. In the fol-lowing, we will refer to particles that cross P/ ¼ P/res (and,then, change the sign of dH/dt) as particles “captured” bythe wave. Once the particle orbit becomes topologicallyclosed, we will describe the particle as “trapped” in thewave, adopting the standard classification. As the fluctuatingfield strength increases, the P/ drift increases and more andmore particles are captured and eventually trapped. This canbe seen from Fig. 10.

Let us look, now, at the whole resonant-particle sample.At each time step, each marker in the plane (H, P/) refers tothe last crossing of the equatorial plane of a test particle. Weconsider two kinds of plots. A first kind, where the color (redor blue) of each marker depends on the birth P/ value (lessor greater than P/ res). In the second kind of plots, markercolor depends on the macroparticle power transfer (normal-ized to the instantaneous field energy) and evolves in time.In Figs. 11–13 plots of these kinds are represented for thelinear phase of the mode evolution and for two differenttimes of the nonlinear phase. We observe the formation ofwave-trapped particle structures, while, correspondingly, themaximum-drive structures move outward (lower P/) and

FIG. 8. The quantity xresðr;M0;C0; kÞ is plotted for various values of k (reddashed lines) and compared with the mode frequency x (black solid line).We see that the relevant resonance order, in the present case, is k¼ 1.Different from Fig. 6, the “equatorial” radial coordinate, generally adoptedin the present paper, has been considered here.

FIG. 9. Drift of test particles in the H direction, at constant P/ during thelinear phase of the mode evolution. Markers are coloured according to theirinstantaneous power exchange with the wave (see Fig. 11(b), below). The to-roidal angular momentum P/ is given in units of mHavH0 and the phase H isreported to the interval [0,2p]. Moreover, here and in many other followingfigures, each marker is doubled by a twin marker in the interval [2p, 4p], inorder to yield a better visualization of the relative dynamics.



inward (higher P/) with respect to the resonance, withdecreasing intensity.

VI. MODE WIDTH VS. RESONANCE WIDTH

Before analyzing the nonlinear wave-particle dynamics,let us further consider the issue of the radial width of thewave-particle power transfer profile. In Section V, we havenoted that, once the values of M and C are fixed, the reso-nance condition x& xresðr;M;C; kÞ ¼ 0 identifies a radialposition r ¼ rresðx;M;C; kÞ, where the resonance takesplace. If the mode is characterized, in the linear phase, by awave-particle power transfer that we generally assume to beof the order of the linear growth rate c, the resonance will beanyway significant in a radial layer, around rres, approxi-mately defined by the following condition:

jx& xresðr;M;C; kÞj!c: (32)

This is represented in Fig. 14, where the width Drres of theresonant layer is shown.

Equation (32) is only a kinematic condition related tothe single particle motion; the effectiveness of resonantwave-particle power exchange also depends on the structureof the perturbed field and of the particle distribution functionin the resonant layer.4,16,18 This can be easily understood bylooking at the power transfer radial profile. We can write thisquantity, at the lowest order, in the following way:

P rð Þ (ð

dMdUP r;M;Uð Þ

’ 2pð

dMdU

(Dzc!ZdFH

eHM

mHr log Bþ eHU2

XHj

! "- b'rdu

$ %)

h;/; (33)

where j ( b -rb is the equilibrium magnetic field curvatureand we have used the explicit form of the gyrocenter equa-tions of motion. We have also neglected the term due to par-allel electric field fluctuation as well as the nonlinear terms(in the fluctuating fields) appearing in the equations ofmotions, as all these contributions are typically small for

shear Alfv!en waves considered here. The quantity dFH obeysto the Vlasov equation in the form

@dFH

@tþ dZi

dt

@dFH

@Zi¼ & dZi

dt

@FH0

@Zi: (34)

FIG. 10. Trapped and passing test particle orbits in the (H, P/) plane duringthe nonlinear phase of the mode evolution. Different colours have beenadopted here for different markers, in order to distinguish them.

FIG. 11. Test particle markers in the(H, P/) plane during the linear phaseof the mode evolution. (a) Each markeris coloured according to the birth P/

value of the particle (red forP/ < P/ res, blue otherwise); (b) eachmarker is coloured according to themacroparticle power transfer (time-varying color).



Taking into account that Dzc!Z ¼ m2HXHRr ¼ m2

HXH0R0r,we can symbolically write

P rð Þ !ð

dMdU

(eHmHrR

xþ ic& xresrFH - b'r du& U

cdAk

! "

' eHM

mHr log Bþ eHU2

XHj

! "- b'rdu

$ %)

h;/:

(35)

It is apparent that the radial structures of perturbedfield and gradient of the distribution function will deter-mine the radial shell where a significant power exchangecan occur. For fixed radial mode and distribution functionstructures, P(r) will be predominantly given by particleswith M and U values (that is, M and C values) correspond-ing to a resonant layer (associated to the resonant denomi-nator) characterized by a nonnegligible overlap with thatshell. Each (M, C) value, as noted in Section V, identifiesone individual wave-particle resonance condition in theform of Eq. (26), extended over the resonant layer definedby Eq. (32). In the limit of small growth rates and/or large

FIG. 12. Same as Fig. 11, for the non-linear phase of the mode evolution(t¼ 1470sA0).

FIG. 13. Same as Fig. 12, att¼ 1640sA0.

FIG. 14. Radial localization (rres) and width (Drres) of the resonance for amode with frequency x and growth rate c.



dxres/dr, the resonant layer corresponding to each individ-ual resonance will be narrower than the overall P(r); and,generally, P(r) will be the envelope of a continuous distri-bution of narrow resonances. In the opposite limit, eachindividual resonant layer will be larger than P(r). The twolimits are sketched in Fig. 15, which considers two situa-tions corresponding, ceteris paribus, to small and largegrowth rate, respectively. In the latter case, once fixedM¼M0 and C ¼ C0, with M0 and C0 corresponding to theresonance peak, the test particle sample population is repre-sentative of the whole resonant particle population. Indeed,the resonant layer corresponding to such test particle choiceessentially includes the entire mode-particle interaction

region. In the opposite limit, a multi-C (and/or multi-M)test particle sample could be needed to be representative ofthe whole resonant particle population. In the present case,Fig. 6 suggests that the individual resonance is wider thanor of the same order of the interaction region. This is con-firmed by Fig. 16(a), comparing the power transfer radialprofile for the selected test-particle sample with the reso-nance width Drres Figure 16(b) shows the power transfer ra-dial profiles for six test-particle samples, corresponding tosix equi-spaced parallel velocity values between U/vH0¼ 1.10 and U/vH0¼ 1.40 at r /a¼ 0.52 (C ranging from&3.90 to &5.85). We see that samples corresponding toresonances separated by more than the single-resonance

FIG. 15. Model comparison between(potential) power transfer radial profileand resonance width. The former quan-tity (blue), integrated over the wholevelocity space, is significantly differentfrom zero in the radial shell wherepower exchange can occur (in the pres-ence of resonant particles), and isdetermined by the radial structure ofthe mode and the gradient of the ener-getic particle distribution function. Theresonance width depends on the modegrowth rate and the radial variation ofxres. Here, we consider only the effectof growth rate variation. (a) Smallgrowth rate limit; (b) large growth ratelimit.

FIG. 16. (a) Power transfer radial pro-file for the selected test-particle popu-lation compared with the resonancewidth Drres (b) Power transfer radialprofiles for six test-particle samples,corresponding to six equi-spaced paral-lel velocity values between U/vH0

¼ 1.10 and U/vH0¼ 1.40 at r/a¼ 0.52(C ranging from &3.90 to &5.85).Samples corresponding to resonancesseparated by more than the single-resonance width from the peak oneyield negligible contribution to thepower transfer. (c) Normalized multi-Cset power transfer radial profile (black)compared with that related to the sin-gle-C set corresponding to the C valueadopted in our analysis (C ¼ &4:74,close to the peak value) and with theresonance width (dashed-blue).



width from the peak one yield negligible contribution to thepower transfer. We then expect that, in this case, a single-Csample, with C value close to resonance peak, is as adequateas the multi-C set in representing the whole resonant particlepopulation power transfer. This is confirmed by Fig. 16(c),where the multi-C set power transfer radial profile is com-pared with the single-C one adopted in our analysis(C ¼ &4:74, corresponding to U/vH0¼ 1.24 at r/a¼ 0.52).

For different choices of plasma equilibrium and/or con-sidered fluctuations, the situation could be quite different.Figure 17 shows the same plots shown in Fig. 16 for case 2discussed in Sec. II. In such case, the single-C test-particlesample cannot properly represent the whole resonant particlepopulation, and a multi-C approach is needed, consistentlywith the larger value of dxres /dr, mainly due to larger shear.Figure 17 refers to ten test-particle samples, correspondingto equi-spaced parallel velocity values between U/vH0¼ 1.2and U/vH0¼ 2.0 at r/a¼ 0.38.

VII. SPATIOTEMPORAL STRUCTURESOF WAVE-PARTICLE RESONANCES

We have already noted (Figs. 11–13) that, in the nonlin-ear phase, the instantaneous P/ corresponding to the reso-nant structures is shifted away from P/ res, associated to theoriginal (linear phase) resonance; that is, we observe increas-ing values of jP/ & P/ resj for such structures. Let us com-pare the P/ coordinate of the marker corresponding, at eachtime, to the maximum power transfer with its birth value

P/0. In Fig. 18, this comparison is shown for ingoing andoutgoing resonant structures. We note that, in the nonlinearphase, the mode is driven by different particles (born at dif-ferent P/0 values) at different times. It is then clear that thetrajectory of the maximum-drive structures in the (H, P/)plane should not be identified with that of the markers that,at a certain time, belong to those structures. This can also beseen from Fig. 19, which directly compares evolution ofmaximum-drive structures and test particle markers. In theupper, middle and lower frames test-particle markers are col-ored according to their power transfer computed at t¼ tp,with tp¼ 500sA0, tp¼ 900sA0 and tp¼ 1470sA0, respectivelyIn the left, central and right frames of each row test particlesare plotted in the (H, P/) positions they have at t¼ 500sA0,t¼ 900sA0 and t¼ 1470sA0, respectively. In each frame, theblack markers contain the region characterized by a powertransfer greater than half of the maximum rate obtained at theconsidered time t. In such way the frames on the “diagonal”of the frame matrix show the drive structures at the threetimes considered. The off-diagonal frames illustrate the evo-lution of the set of the “row-time” most destabilizing particlesat earlier or later times (below and above diagonal, respec-tively). It is apparent that the evolution of particles drivingthe mode is different from that of maximum-drive structures.Note that this is true both in the linear (t¼ 500sA0,t¼ 900sA0) and in the nonlinear phase (t¼ 1470sA0).

We can note the same features in Fig. 20, which showsthe trajectories in the (H, P/) plane of four different particles

FIG. 17. Same as Fig. 16, for case 2,characterized by a larger magneticshear than case 1. Power transfer radialprofiles refer to ten test-particle sam-ples, corresponding to equi-spaced par-allel velocity values between U/vH0¼ 1.2 and U/vH0¼ 2.0 at r/a¼ 0.38.



(each represented by a different color), chosen as the mostdestabilizing ones at four different times: t¼ 900, 1300,1400, and 1470sA0, respectively. The position of each parti-cle when it becomes the most destabilizing particle ismarked by a black dot. So, the black boxes correspond to thetrajectory of a maximum-drive structure, while the coloreddots correspond to the individual particle trajectories, whichneither coincide with nor are tangent to those of maximum-drive structures.

Other interesting features of wave-particle interactionscan be illustrated by the analysis of test-particle behaviors.Figure 21 reports the orbits of three of the particles consid-ered in Fig. 20 (namely, those corresponding to the maxi-mum drive at t¼ 900, 1400, and 1470sA0), colored accordingthe instantaneous power transfer. First, as it also appearsfrom Fig. 18, macro-particles driving the mode during thenonlinear phase (t¼ 1400, 1470sA0) were originally out ofresonance (P/0 “far” from P/ res, with corresponding large Hdrift during the linear phase). Furthermore, they yield signifi-cant drive to the mode on the opposite side of the resonancewith respect to their initial position; that is, adopting the ter-minology introduced in Sec. V, only after being “captured”by the fluctuating field. Eventually, these macro-particleswill be trapped into bounded orbits.

The second feature is that only a small portion of thetrapped particle orbit yields relevant contribution to themode drive. Except for that portion, the macroparticle contri-bution (that is, the contribution of the set of physical par-ticles belonging to the infinitesimal phase-space volumerepresented by the test particle) becomes small or even nega-tive, although its average contribution over the full orbitremains positive.

The third interesting feature we can emphasize from theanalysis of test-particle dynamics is that the time needed tocomplete a bounded orbit is of the same order of the nonlin-ear saturation time scale: the mode evolution does not cover,in the considered case, multiple wave-particle trappingtimes. This is shown in Fig. 22 in a more quantitative way,where the P/ range covered by the largest bounded orbit ateach time is plotted (red line). The blue line represents thesame quantity for orbits that have completed two bounces.We see that saturation (t ’ 1700 sA0) occurs before any testparticle has completed two bounces.

Finally, it is interesting to note that, in the nonlinearphase, most destabilizing particles exchange energy with rel-atively large value of jP/ & P/ resj; that is, large values ofjx& xresðr;M;C; kÞj: In the framework of linear analysis,we would then expect that such energy exchange takes placeat large values of dH/dt. However, this is not the case, as wecan see from Fig. 23, where the average quantities

(dHdt

)

t

(X

p l tð Þ>0

dHdt

tð Þ&&&&l

pl tð Þ (36)

and

hrit (X

p lðtÞ>0

rðtÞjlplðtÞ; (37)

with plðtÞ ( plðtÞ=P

pjðtÞ>0 pjðtÞ, are plotted versus time(Figs. 23(a) and 23(b), respectively); here, pl is defined byEq. (11) and sums are taken only over test particles yieldingenergy to the wave. The same quantities are plotted againsteach other, along with x& xresðr;M;C; kÞ, in Fig. 23(c).Note that, in the linear limit

dHdt¼ x& xres: (38)

Thus, it is apparent that the nonlinear evolution of dH/dtcannot be computed only in terms of the xres (r, M, C, k)evolution due to the nonlinear radial drift of the resonantstructures: the perturbed poloidal and toroidal velocities ofthe particle, d _h and d _/, have to be retained on the samefooting.

VIII. SATURATION MECHANISM: DENSITYFLATTENING

In Fig. 7, we have noted that the energetic particle drive(measured by the test particle sample) fast decreases in thetime interval t ¼ 1400+ 1700 sA0. Below, we analyze thereasons of such decrease and the consequent mode saturation.

In Section VII, we have seen that saturation occurs on atime scale of the same order as the wave-particle trappingtime. Saturation cannot then be explained, in the present case,in terms of models assuming a clear time scale separation,

FIG. 18. Instantaneous (blue) and birth(red) P/ for the ingoing (a) and out-going (b) resonant structures.



sNL . x&1wp, with sNL being the nonlinear (saturation) time

scale and xwp the wave-particle trapping frequency.In Fig. 24 the density profile of test particle markers is

reported for two times during the nonlinear phase of themode evolution (left frames, showing the unperturbed den-sity profile and the linear-phase power transfer profile) alongwith the corresponding plots of the markers in the (H, P/)plane. We see that the formation of mutually penetratingstructures in such plane, corresponding to particles captured

and eventually trapped by the wave field, yields an exchangeof low-density (outer) resonant macroparticles with high-density (inner) ones: there are more particles moving out-ward than inward. This causes a density profile flattening(and, eventually, its inversion) in the region where the linearwave-particle interaction is strongest, and a consequentsteepening of the same profile at the boundaries of thatregion. In particular, in the limit of negligible growth rateand uniform field amplitude, with consequent isochronous

FIG. 19. Comparison between the evolution of the maximum-drive structures and that of the particle set belonging to such structures at different times.Test-particle markers are coloured according to their power transfer computed at fixed times t¼ tp, with tp¼ 500sA0 in the upper frames, tp¼ 900sA0 in the mid-dle frames and tp¼ 1470sA0 in the lower frames, respectively. In each row, test particles are plotted according to their (H, P/) positions at t¼ 500sA0 (leftframes), t¼ 900sA0 (central frames) and t¼ 1470sA0, (right frames). Then, the frames along the diagonal show the drive structures at the three times consid-ered. The off-diagonal frames show instead the evolution of the “row-time” most destabilizing particles at earlier or later times (below and above diagonal,respectively). In each frame, the black markers contain the region characterized by a power transfer rate greater than half of the maximum rate obtained at theconsidered time.



motion of particles captured in the wave field, we would getdensity flattening after particle orbits have completed onequarter bounce, while density profile inversion would occurafter half bounce. Time and radial dependence of the modeamplitude cause quantitative, but not qualitative modificationof this simplified picture, on the time scale that characterizesthis phenomenon (shorter than of the order of x&1

wp). In thefollowing, for the sake of simplicity, we will refer to such adensity profile distortion as “density flattening”, without dis-criminating, unless needed, between flattening and inversion.It is, however, worthwhile emphasizing that the resonant par-ticle redistribution analyzed here is qualitatively differentfrom density flattening that occurs on time scales much lon-ger than x&1

wp, as consequence of phase mixing, collisionsand/or quasilinear particle diffusion due to the presence of abroad fluctuation spectrum with many overlapping resonan-ces52,53 (cf. Ref. 18 for more in depth discussions).

Figure 25 compares the radial position of the densitygradient maxima created by the local density profile flatten-ing with the largest radial width of orbits that have com-pleted, respectively, a quarter, half and a full bounce in thewave field. We see that the largest gradient splitting essen-tially follows the completion of a quarter closed orbit.

Note that the density profile distortion is apparent becausewe consider a single-M and single-C test-particle sample,corresponding to a single resonance radial position rres.

FIG. 20. Test particle trajectories in the (H, P/) plane. Four macroparticlesare considered, chosen as the most destabilizing ones, for t¼ 900sA0 (violet),t¼ 1300sA0 (blue), t¼ 1400sA0 (green) and t¼ 1470sA0 (orange). The arrowindicates the rotation direction of the trajectories during the nonlinear phase.The black boxes correspond to the position of the different particles whenthey become the most destabilizing particle: they indicate the trajectory ofthe ingoing maximum-drive structure. It is apparent that the latter trajectoryneither coincides with nor is tangent to any of the former.

FIG. 21. Test particle orbits in the (H,P/) plane. Three macroparticles areconsidered, chosen as the most destabi-lizing ones, for t¼ 900sA0 (a),t¼ 1400sA0 (b) and t¼ 1470sA0 (c).Markers are coloured according totheir instantaneous power transfer.



Integrating over a broader (M, C) domain could involve popu-lations not resonating at all or, in cases different form case 1,resonating at different radial positions, then hiding the distor-tion effect.

In order to investigate in a more quantitative way the rela-tionship between density flattening, nonlinear wave-particle

resonance and mode drive, we measure the rate of distortionof the radial density profile n(r, t) by the following quantity:

C tð Þ (ða

0

dr

&&&&@

@t" r; tð Þ

&&&&; (39)

where " ( rnðr; tÞ.We have already observed in Sec. VII that, in general,

different particles drive the mode at different times (cf. Figs.18 and 19 with the following comments). To examinewhether and how the density profile distortion is related toresonant particles, we can characterize each test particle, ateach time, by its power transfer (normalized to the instanta-neous maximum value among all the test particles). Defininga threshold b< 1 in the normalized power transfer, we candivide particles into two different groups: those whose nor-malized power transfer never exceeds, in the course of thewhole mode evolution considered in the numerical simula-tion, such a threshold, and those for which the threshold isexceeded, at least within a certain time window. We canthen separate the whole test particle population into differentclasses, according to the best drive performance reached byparticles during the mode evolution. Figure 26 shows, withreference to case 1, the fraction of population belonging toeach class for a certain partition: namely, particles that neverdrive the mode and particles whose best (and positive) driveperformance falls in left-open intervals of width Db¼ 0.1.We see that there is no particle belonging to the formergroup; this means that all the considered particles sometimes

FIG. 22. P/ extension of the largest closed orbit that, at each time, has com-pleted one (red) or two (blue) bounces. Note that mode saturation occurs at t’ 1700 sA0, before any test particle has completed two bounces.

FIG. 23. (a) dH/dt (in units of s&A01)averaged over the destabilizing par-ticles (with weight proportional to thepower transfer) versus time, for theingoing resonant structure (blue) andthe outgoing one (red). (b) Average ra-dial position for the two resonant struc-tures. (c) Average dH/dt versusaverage r for the two resonant struc-tures; the quantity x& xresðr;M;C; kÞis also plotted for comparison (green).The energy exchange is characterized,in the nonlinear phase, by values ofjdH=dtj much smaller than expected,on the basis of the linear-phase rela-tionship dH=dt ¼ x& xres, for theobserved radial drift.



contribute to drive the mode, consistent to the fact that thesample was chosen around a resonance in the phase space.We also observe that there is a large population fractionwhose contribution to the drive never exceeds a normalized

value of b¼ 0.1. The fraction value falls down dramaticallyin the next class (0:1 < b , 0:2), and then it continuouslyincreases for the upper classes, until reaching a relativelylarge value (above 10%) for the upmost one (0:9 < b , 1):this means that if a particle yields a nonnegligible contribu-tion to the drive during its motion, it is relatively likely thatsuch contribution is large. On the basis of such distribution,in the following we will look at a single threshold, b¼ 0.2,and conventionally indicate as resonant particles those thatexceed, in the course of mode evolution, this threshold; asnonresonant particles, the others. Correspondingly, we shallindicate by CresðtÞ and CnonresðtÞ the quantities, analogous tothat defined in Eq. (39), obtained when looking only at thedensity of particles belonging to one or the other of thesetwo classes. Figure 27 compares the time evolution of thedensity distortion rate, C, with Cres and Cnonres. Although, ingeneral, the absolute value that appears in Eq. (39) causes Cto be different from the sum of Cres and Cnonres, we see thatthe total-density distortion is essentially due to resonant par-ticles, the relevance of noresonant ones being negligible.

It is also worthwhile investigating whether, at each time,the resonant-particle density distortion rate is mainly due toparticles characterized by a power transfer greater than 0.2 atthat specific time. Figure 28 indicates that the answer is neg-ative: although the relative weight of “instantaneously”resonant-particle rate is generally large, the analogous raterelated to the density of particles affecting the drive at earlieror later times of the mode evolution is not negligible at all,

FIG. 24. Radial density of the test-particle set at two times during the nonlinear phase of the mode evolution (left frames, red). The unperturbed density andpower transfer profiles are also reported (blue). Times are the same considered in Figs. 12 and 13 respectively: t¼ 1470sA0 (top) and t¼ 1640sA0 (bottom). Thecorresponding plots of the markers in the (H, P/) plane are reproduced in the central and right frames.

FIG. 25. Time evolution of the radial position of the density gradient max-ima created by the local density profile flattening (red), compared with thelargest radial width of orbits that have completed, respectively, a quarter,half, and a full bounce in the wave field.



especially in the late nonlinear phase. Note, however, thatnormalizing these two specific quantities to the number ofparticles belonging to the corresponding particle subset, theaverage instantaneously-resonant distortion rate per particleis much larger than that computed over the remainingresonant-particle subset (cf Fig. 29(a)); the two groups yieldvalues of the same order because the former is always muchless numerous than the latter (Fig. 29(b)).

It is also interesting to compare the time evolution ofdPtot=dt (with Ptot being the overall particle power transfer)with that of the density distortion rate. We find that&dPtot=dt is very well correlated with the instantaneouslyresonant particle rate (Fig. 30(a)), but not as well with thefull resonant particle rate (Fig. 30(a)). This shows that theprogressive reduction of the overall drive, along with theconsequent mode saturation, is primarily caused by the flat-tening of the density profile of particles at the instantaneouspeak of power transfer to the wave. This is consistent withthe role played by the gradient of the distribution functionand the resonant denominator in Eq. (35), with the latterquantity selecting the instantaneously resonant phase-spaceregion. The fact that the full resonant-particle distortion rateremains relatively large even after the saturation has beenreached can be explained considering that the formation ofphase space zonal structures16–18–i.e., the (n¼ 0, m¼ 0)structures made of particles captured by the resonance, withthe definition introduced in Sec. V–proceeds because of thefinite amplitude of the saturated mode; indeed, we have al-ready observed in Sec. VII that particles strongly resonant inthe nonlinear phase will undergo further radial excursioneven after decreasing their wave-power transfer (Fig. 21(b)and Fig. 21(c)).

IX. SATURATION MECHANISM: RADIAL DECOUPLINGVS. RESONANCE DETUNING

The density flattening is a local phenomenon, with a ra-dial extension limited by the phase space region character-ized by bounded resonant particle trajectories. While thepower transfer is reduced in the region where the densityprofile is flattened, it is preserved and possibly increasedwhere the density gradient maintains a significant amplitude.

FIG. 27. Time evolution of the density distortion rate C, defined in Eq. (39),compared with that of Cres and Cnonres (as defined in the text). The total-density distortion is essentially due to resonant particles, the relevance ofnonresonant ones being negligible.

FIG. 26. Fraction of particle population, corresponding to the test particleset adopted for case 1, belonging to different classes, according to the bestdrive performance reached during the mode evolution. No particles havenegative best performance: this means that all the considered particles some-times contribute to drive the mode. A large population fraction is confinedin the very weak drive class 0<b< 0.1. The fraction value falls down dra-matically in the next class (0.1<b< 0.2). Then, it continuously increasesfor the upper classes, until reaching a relatively large value (above 10%) forthe upmost one (0.9<b< 1).

FIG. 28. Density distortion rate for the “resonant” particles (Cres, shown inFig. 27), compared with the analogous quantities computed for the subsetsof particles whose drive performance exceeds b¼ 0.2 at the specific timeconsidered (“instantaneously resonant” particles) or at earlier or later times(“other resonant” particles). Although the distortion rate computed for theinstantaneously resonant particles is large, the analogous quantity related tothe other resonant particles is not negligible at all, especially in the late non-linear phase.



This is the case of the region not yet involved by the den-sity flattening, but also, and in a special way, of the large-gradient zones that form on both sides of the flat-densityregion. Correspondingly, a splitting of the resonant struc-tures is observed, as discussed in Sec. V and shown in Fig.31, where the radial profile of the power transfer is plottedfor the same two times considered in Fig. 24. Two contribu-tions are computed separately, related to test particles hav-ing values of H around the ingoing and the outgoingstructures, respectively. The linear-phase structure (thesame for both groups of test particles) is also plotted forcomparison, along with the actual density profile and theunperturbed one.

If the radial dependence of the power transfer rate (cf.the symbolic expression given in Eq. (35)) were due to thedensity gradient only, the resonant structure splitting wouldnot be necessarily accompanied by a decrease of the effec-tive drive nor by the consequent mode saturation.Correspondingly, the flattening/splitting process could pro-ceed indefinitely, until linear instability of the underlyingplasma equilibrium and energetic particle distribution func-tion prevents it. On the contrary, for finite mode and/or reso-nance widths (smaller than the equilibrium density gradientwidth), the density flattening, whose radial extensionincreases with mode amplitude along with the wave-particletrapping region, subtracts a finite and increasing fraction

FIG. 29. Average distortion rate perparticle (a) and relative numerosity (b),for the two subsets (instantaneouslyresonant particles and other resonantparticles) considered in Fig. 28. Thedistortion rate per particle is muchhigher for the instantaneously resonantparticles. The two subsets give rise, asa whole, to comparable rates becausesuch particles always represent a rela-tively small fraction of the whole reso-nant particle group.

FIG. 30. Time derivative of the overallparticle power transfer compared withthe rate of density profile distortion forthe instantaneously resonant particleset (a) and the full resonant particle set(b).

FIG. 31. Radial profile of the powertransfer at the same times consideredin Fig. 24: t¼ 1470 sA0 (a) andt¼ 1640sA0 (b). Two contribution arecomputed separately, related respec-tively to test particles having values ofH around the ingoing structure and theoutgoing one. The linear-phase profile(the same for both groups of test par-ticles) is also plotted for comparison,along with the actual density profileand the unperturbed one.



from the effective drive. As soon as density flattening affectsthe whole region where wave-particle power transfer cantake place, the drive vanishes, the mode saturation is reachedand the splitting process itself stops. This is shown in Fig.32, where the radial position of the peaks of the separate res-onant structures is reported versus time and compared withthe radial profile of the linear power transfer and the timeevolution of the mode growth rate.

Consistent with the definitions introduced in Sec. I,4,11,16–18,20 when the most stringent constraint over the radialwidth of the region where wave-particle power transfer cantake place comes from the mode width, we dub the saturationmechanism radial decoupling, indicating that potentially res-onant particles get decoupled from the radial mode structure.If such constraint comes from the resonance width, we referto the mode saturation mechanism as resonance detuning.

The two different limits correspond to different scalingof the saturation field amplitude with the mode growthrate.20,37 This can be qualitatively understood on the basis ofthe following argument. The width of the density flatteningregion scales with A1=2 (with A representing the field ampli-tude), since this is the scaling expected for the separatrixwidth of resonant particles trapped in the wave (with conse-quent density flattening) and untrapped particles.54,55 Wecan also assume, from Eq. (32), that the resonance widthscales as c, while the mode width is expected to exhibit amuch weaker dependence on the mode growth rate (let usassume, for the sake of simplicity, no dependence at all).Thus, resonance detuning would yield saturation when thedensity flattening width matches the resonance width, orA ! c2;12–14,28–30 while radial decoupling would require thematching with the mode width, or A ! c020,37 (consistentwith the fact that increasing the growth rate and, hence, the

resonance width does not increase the mode capability ofextracting power from the particles). Such scalings are basedon a simplified picture of the saturation process, and theyshould be considered as qualitative behaviors; nevertheless,we can expect a weakening of the c dependence of the satu-ration amplitude as the transition from resonance detuning toradial decoupling takes place. More articulate saturationmodels, following the same qualitative arguments, are dis-cussed in Ref. 20 and explain that, when the different lengthscales of radial mode structures become important in thewave-particle power exchange, the transition between A !c2 (vanishing drive) to A ! c0 (strong drive) can go throughA ! cp (2< p< 4);4,20,37 and not be necessarily character-ized by a progressive weakening of the power scaling fromp¼ 2 to p¼ 0.

In Sec. VI, we noted that in case 1 the mode width is ofthe same order or even narrower than the linear resonancewidth. In such a case, the two saturation mechanisms gener-ally play equivalent roles, but we can expect that consider-ing, for the same equilibrium, stronger modes (that is, largergrowth rates and resonance widths) the prevalence of the ra-dial decoupling mechanism becomes apparent. This is con-firmed by Fig. 33, which compares the results reported inFigs. 16(a) and 32 for the reference case with those obtainedfor a larger growth rate mode (xsA0 ’ 0:289, csA0 ’ 0:021,saturation reached at t ’ 760sA0). If resonance detuningwere responsible for the limit imposed to the resonant-structure splitting, we should get a proportional increase ofthe maximum splitting. Instead, it is clear that splittingdepends very weakly on the linear growth rate (and the linearresonance width). This can also be seen from Fig. 34, wherethe maximum radial splitting is compared with the mode andresonance widths for different values of the growth rate.Consistently, we find that the saturation amplitude has aweaker c scaling than the quadratic dependence expected fordominant resonance detuning. This is shown in Fig. 35,where the amplitude of the perturbed poloidal magnetic fieldis reported for different values of c/x.

We now explore the conditions under which resonancedetuning is more relevant than radial decoupling as satura-tion mechanism. We have observed how in different cases,characterized by lower growth rates and/or higher shear (thatis, stronger radial dependence of xres) or larger mode width,the linear resonance is narrower than the mode. An exampleof this condition has been shown, for case 2, at the end ofSec. VI (Fig. 17). In these cases, we have seen that the linearpower transfer profile for a single-(M, C) test-particle set islimited by resonance (rather than mode) width, so that amulti-(M, C) approach can be required to properly representthe whole resonant particle population (characterized by anoverall power transfer profile with mode-size width).Consistent with these results, Fig. 36 shows that, in such asituation, the linear resonance width also plays the main rolein controlling the resonant-structure splitting for the single-(M, C) test-particle set. In fact, the mode width (represented,here, by the multi-C power transfer profile) would allow, byitself, for further splitting of the resonant structures. This isconfirmed by the results plotted in Fig. 37, which shows thewidth of the relevant regions versus the growth rate of the

FIG. 32. Splitting of the power transfer peaks (red) compared with the radialprofile of the linear power transfer (dashed black) and the time evolution ofthe mode growth rate (dashed blue). The finite width of the linear powertransfer profile puts a hard limit to the radial drift of the resonant structures.



FIG. 33. Splitting of the power transferpeaks (red) compared with the radialprofile (dashed black) of the linearpower transfer (left) and the linear res-onance width (right). The referencecase (a) and a larger growth rate casefor the same equilibrium (b) are con-sidered. Saturation is reached at t ’1700 sA0 in the former case, at t ’760sA0 in the latter. The resonant-structure splitting depends very weaklyon linear growth rate (and linear reso-nance width).

FIG. 34. Maximum radial splitting of the resonant structures (red) comparedwith the mode (green) and resonance (blue) widths, for different values ofthe growth rate. It is apparent that, in the large growth rate limit, the mainrole in determining mode saturation is played by the radial decouplingmechanism.

FIG. 35. Amplitude of the perturbed poloidal magnetic field versus c/x. Thedeviation from the quadratic scaling (shown for comparison) expected for apurely resonance detuning saturation mechanism is clearly visible at largegrowth rates, consistently with the prevalence, in this limit, of the radialdecoupling mechanism.



mode. Correspondingly, a stronger scaling of the saturationfield amplitude with c/x is recovered, close to quadratic (cf.Fig. 38). The dispersion of the obtained values around thelatter scaling is the natural consequence of the fact that, dif-ferent from the ideal framework in which such scaling isobtained, the 3D system investigated by our simulations isanyway characterized by mode structures and plasmanonuniformities.4

It is worthwhile noting, as discussed above in thisSection, that shear-Alfv!en fluctuations are generally charac-terized by “internal” length scales finer than the modewidth2,4,10,17 due to the properties of the shear Alfv!en contin-uous spectrum. Thus, the effect of plasma non-uniformitiesand radial mode structures can become important even whenthe linear resonance width is a fraction of the modewidth.4,16 Radial decoupling and resonance detuning may,thus, play comparable roles even in the saturation of rela-tively weak modes.4,20,37 The investigation of these effectsis, however, outside the scope of the present analysis.

The simple interpretation of a mode saturation reachedbecause the density flattening reduces the drive in the wholeregion where the resonant particles can interact with theperturbed field could underestimate important aspects of thenonlinear mode-particle dynamics. In particular, radialdecoupling could be contrasted (and the consequent satura-tion delayed, and reached at a higher field amplitude level)if the mode structure can evolve, nonlinearly, to adapt tothe radial drift of the maximum density gradient and theresonant structure.4,32,33,37–40 Resonance detuning, mean-while, could lose effectiveness if mode structure and fre-quency modifications consistent with resonant particlenonlinear dynamics (phase-locking4,11,16–20,32–34) and/orperturbed field contributions to the poloidal and toroidalvelocities of the particle (cf. Sec. VII) are able to keepjdH=dtj small, in spite of the distance between the linear-resonance and the location where the power transfer ispeaked because of the nonlinear evolution of phase spacezonal structures.16–18

FIG. 36. (a) splitting of the powertransfer peaks (red) corresponding to asingle-C test-particle set for case 2,compared with the radial profile of thelinear power transfer for the same set(black) and a multi-C one (green). (b)linear resonance width. Saturation isreached at t ’ 1230sA0 (indicated by avertical continuous black line). Boththe linear power transfer profile for thesingle-C test-particle set and theresonant-structure splitting are limitedby resonance width.

FIG. 37. Same as Fig. 34, for case 2. In this case, the main role in determin-ing mode saturation is played by the resonance detuning mechanism.

FIG. 38. Amplitude of the perturbed poloidal magnetic field versus c/x forcase 2. A stronger scaling than that obtained for case 1 is observed, close tothe quadratic scaling (shown for comparison), expected for a purely reso-nance detuning saturation mechanism.



In the first reference case considered in this paper (case1), no significant frequency or mode structure modificationsare observed in the nonlinear phase (cf. Fig. 39 with Fig.2). Concerning the nonlinear contribution to the resonancecondition, we have already noted (Fig. 23) that the nonlin-ear wave-particle power exchange is characterized by val-ues of jdH=dtj much smaller than expected on the basis ofthe linear-phase relationship dH=dt ¼ x& xresðrÞ, com-puted at the radial position of maximum power transfer. In

other words, the nonlinear modifications of the resonanceare significant and they correspond to an increased powertransfer efficiency. We then expect that they contribute topostponing mode saturation, at least in the low growthrate limit, in which resonance detuning and radial decou-pling play comparable roles. We can prove this by an acontrario argument. Let us introduce a model powertransfer with radial profile approximated by the followingexpression:

Pmodel rð Þ !ð

dMdUhrFH -eHmHrR

xþ ic& xresb'r du& U

cdAk

! "" #(

' eHM

mHr log Bþ eHU2

XHj

! "- b'rdu

$ %*

0

)

h;/; (40)

with subscript 0 denoting quantities computed in thelinear-phase. Comparing Eq. (40) with Eq. (35), it can berecognized that the model expression is obtained byneglecting all nonlinear physics except those due to reso-nant particle density gradient. That is, assuming prescribedmode structure (justified on the basis of the comparisonbetween Fig. 2 and Fig. 39) as well as wave-particle reso-nance. Figure 40 compares the time evolution of the

integrated wave-particle power transfer with the integratedpower transfer corresponding to this model. We see that,although the model properly includes the density profileevolution, the actual nonlinear decrease of the integratedpower transfer is slower than the predicted one. This sug-gests that nonlinear modifications of the resonance condi-tion play an important role in the nonlinear mode evolutionand saturation.

FIG. 39. Mode structure during thenonlinear phase: scalar potential (a),parallel vector potential (b), powerspectrum of the scalar potential (c).



With reference to case 2, we observe analogous featuresof the behaviour of the resonant denominator jdH=dtj in thenonlinear phase (cf. Fig. 41): its absolute value is kept muchlower than expected from its linear-phase radial dependence(jx& xresðrÞj), computed at the nonlinear radial position ofthe most destabilizing particles.

As to mode structure and frequency, no significant mod-ification occurs until the first saturation is reached, at t ’1225sA0 (cf. Fig. 3), as can be seen comparing Fig. 4 (linearphase, at t¼ 702sA0) and Fig. 42 (just before the first

FIG. 41. Same as Fig. 23(c) for case 2: average dH/dt (in units of s&A01) versusaverage r for the two resonant structures; the quantity x& xresðr;M;C; kÞ isalso plotted (green). Also in this case, the energy exchange is characterized, inthe nonlinear phase, by values of jdH=dtj much smaller than expected, onthe basis of the linear-phase relationship dH=dt ¼ x& xres, for the observedradial drift.

FIG. 40. Time evolution of the integrated power transfer obtained in case 1(red), compared with the integrated power transfer corresponding to themodel power profile given by Eq. (40) (blue).

FIG. 42. Mode structure during thenonlinear phase, for case 2, just beforethe first saturation (t¼ 1200sA0): scalarpotential (a), parallel vector potential(b), power spectrum of the scalarpotential (c).



saturation, at t¼ 1200sA0). Nevertheless, the first saturationis followed, in this case, by complex nonlinear oscillations.Different from case 1, the mode adjusts both its radiallocalization and frequency in order to maximize resonantwave-particle power exchange, typical of EPM nonlinearevolution.4,10,17,32,33 The detailed investigation of this furthernonlinear evolution, however is beyond the scope of thepresent work and will be the subject of a separate paper.

X. CONCLUSIONS

In this paper, the nonlinear evolution of shear-Alfv!enmodes, driven unstable by energetic ions in Tokamak mag-netic equilibria, has been investigated by looking at the dy-namics of a test particle population in the electromagneticfields computed by self-consistent HMGC hybrid MHD-particle simulations. The test particle set has been selected byfixing the values of two constants of the perturbed motion:namely, the magnetic momentum M and the quantity C ¼xP/ & nE; the latter quantity is a constant of motion, pro-vided that the perturbed field is characterized by a single to-roidal mode number n and a constant frequency x. Twodifferent case have been analyzed: a n¼ 6 TAE, in a lowshear, large aspect ratio, circular magnetic surface equilib-rium; and a weakly unstable n¼ 2 EPM, in a stronger mag-netic shear equilibrium. These reference cases have beenselected to illustrate the roles of resonance detuning vs. radialdecoupling in nonlinear mode dynamics and saturation, aswell as energetic particle transport. In particular, the fact thatthe TAE case is dominated by radial decoupling and the weakEPM case by resonance detuning shows that, depending onmode growth rate and plasma equilibrium nonuniformity,nonlinear dynamics may be different from that expected onthe basis of the theoretical paradigms (bump-on-tail and,respectively, fishbone paradigms) currently adopted in theinvestigation of Alfv!en wave-energetic particle interactions.

Hamiltonian mapping techniques have been used toexamine the evolution of resonant phase-space structuresand its connection with the mode saturation. In particular,representing each test particle by a marker in the (H, P/)space, with coordinates computed, for each particle, at theequatorial-plane crossing times, the universal structure ofresonant particle phase space near an isolated resonance canbe recovered and illustrated, with bounded orbits anduntrapped trajectories divided by the instantaneous separatrixin the time-evolving mode structure. Furthermore, the pres-ent numerical simulation results allow us to demonstratethat, in general, the characteristic time of phase space zonalstructures evolution is the same as the nonlinear time of theunderlying fluctuations. That is, wave-particle nonlinear dy-namics in nonuniform plasmas are generally non-adiabatic.

Bounded orbits correspond to particles captured in thepotential well of the Alfv!en wave and give rise to a net out-ward particle flux. Such flux produces a flattening (and even-tually gradient inversion) of the radial profile of the resonantenergetic particle density in a region around the linear reso-nance peak. A detailed analysis shows that there is a strongcorrelation between the decreasing rate of power transfer tothe fluctuation by resonant energetic particles and the

distortion of the density profile of the instantaneously reso-nant particles. This fact can be easily understood consideringthat flattening and/or inversion of the gradients of the distri-bution function in the instantaneously resonant phase-spaceregion causes the subtraction of phase-space portions to thedrive; thus, drive decrease is associated to the widening ofthe no longer destabilizing region. Saturation is reachedwhen the subtracted phase-space portion, whose radial widthincreases with the mode amplitude, covers the whole poten-tially resonant region. Such a correlation disappears if welook at the overall density-profile distortion, as the latterinvolves also particles that are no longer resonant, simplybecause of the finite mode amplitude, and proceeds even af-ter the power transfer has been strongly reduced and satura-tion has been reached.

The two different cases examined in this paper corre-spond to different characteristics of the potentially resonantregion, depending on the relative size of finite interactionlength of resonant particles and characteristic length scale ofperpendicular mode structures. In the low shear case, thisregion is radially limited by the mode width more than thewave-particle resonance width (especially for relatively largegrowth rates). In the larger shear case, the region is limitedby the resonance width. We refer to the former situation asto radial decoupling saturation mechanism; to the latter, as toresonance detuning. As the resonance width has typically astronger dependence on the mode growth rate than the modewidth, the mode amplitude at saturation exhibits a strongergrowth rate dependence in the resonance detuning than inthe radial decoupling regime. This is confirmed by the scal-ings of the mode amplitude at saturation obtained in the twoconsidered cases.

The present work consists of a series of carefully diag-nosed numerical simulation experiments set up to isolate andillustrate the fundamental physics processes underlying thenonlinear dynamics of Alfv!enic modes resonantly excited bya sparse supra-thermal particle population in tokamak plas-mas; and of the ensuing energetic particle transports. Furtherto the saturation processes, several other phenomena, whichare undoubtedly of interest for the physics involved and havepossible practical impacts on energetic particle confinementstudies in fusion plasmas, have been enlightened by the pres-ent investigation. In particular, we have demonstrated that, inaddition to nonlinear radial energetic particle excursions, non-linear particle motions in poloidal and toroidal direction arecrucial for properly describing the nonlinear resonance condi-tion. This effect clearly impacts on resonance detuning.Further investigation and ad hoc numerical simulation experi-ments in the spirit adopted in the present work will be devotedto a deep analysis of richer nonlinear dynamics, observed incase 2 but not examined in the present paper, related to thecapability of the mode to adjust its spatial and frequencystructure, after the first saturation has been reached, in such away to maintain the power extraction from energetic particles.

ACKNOWLEDGMENTS

The authors are indebted to Liu Chen for suggesting theanalysis presented in this paper and improving it through



many valuable discussions. This work was supported byEuratom Community under the contract of Associationbetween EURATOM/ENEA. It was also partly supported byEuropean Union Horizon 2020 research and innovationprogram under Grant Agreement No. 633053 as EnablingResearch Project CfP-WP14-ER-01/ENEA_Frascati-01. Thecomputing resources and the related technical support usedfor this work have been provided by CRESCO/ENEAGRIDHigh Performance Computing infrastructure and its staff.

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Documents

Analysis of the nonlinear behavior of shear-Alfven modes ...cation of the soliton-like EPM mode structure, which contin-ues until plasma nonuniformity (in general, increasing continuum