Analysis of the nonlinear behavior of shear-Alfvén modes ... · Analysis of the nonlinear behavior of shear-Alfven modes in tokamaks based on Hamiltonian mapping techniques S. Briguglio,1,a)

Analysis of the nonlinear behavior of shear-Alfvén modes in tokamaks based onHamiltonian mapping techniquesS. Briguglio, X. Wang, F. Zonca, G. Vlad, G. Fogaccia, C. Di Troia, and V. Fusco Citation: Physics of Plasmas (1994-present) 21, 112301 (2014); doi: 10.1063/1.4901028 View online: http://dx.doi.org/10.1063/1.4901028 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/21/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Flow and shear behavior in the edge and scrape-off layer of L-mode plasmas in National Spherical TorusExperiment Phys. Plasmas 18, 012502 (2011); 10.1063/1.3533435 Double tearing mode induced by parallel electron viscosity in tokamak plasmas Phys. Plasmas 17, 112102 (2010); 10.1063/1.3503584 Nonlinear simulation of toroidal Alfvén eigenmode with source and sink Phys. Plasmas 17, 042309 (2010); 10.1063/1.3394702 Nonlinearly driven second harmonics of Alfvén cascades Phys. Plasmas 13, 042504 (2006); 10.1063/1.2192500 Hybrid magnetohydrodynamic-particle simulation of linear and nonlinear evolution of Alfvén modes in tokamaks Phys. Plasmas 5, 3287 (1998); 10.1063/1.872997

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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 fundamental

physics processes underlying the nonlinear dynamics of Alfv�enic modes resonantly excited by

energetic particles in tokamak plasmas and of the ensuing energetic particle transports. These

phenomena are investigated by following the evolution of a test particle population in the

electromagnetic fields computed in self-consistent MHD-particle simulation performed by the

HMGC code. Hamiltonian mapping techniques are used to extract and illustrate several features of

wave-particle dynamics. The universal structure of resonant particle phase space near an isolated

resonance is recovered and analyzed, showing that bounded orbits and untrapped trajectories,

divided by the instantaneous separatrix, form phase space zonal structures, whose characteristic

non-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 flattening

and/or gradient inversion of the fast ion density profile around the peak of the linear wave-particle

resonance. The connection of this phenomenon to the mode saturation is analyzed with reference to

two different cases: a Toroidal Alfv�en eigenmode in a low shear magnetic equilibrium and a

weakly unstable energetic particle mode for stronger magnetic shear. It is shown that, in the former

case, saturation is reached because of radial decoupling (resonant particle redistribution matching

the mode radial width) and is characterized by a weak dependence of the mode amplitude on the

growth 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 the

growth rate. [http://dx.doi.org/10.1063/1.4901028]

I. INTRODUCTION

In burning plasmas, shear-Alfv�en modes can be driven

unstable by resonant interactions with energetic ions pro-

duced by additional heating or nuclear fusion reactions. In

fact, these particles are characterized by velocities of the

order of the Alfv�en speed, the typical group velocity of shear

Alfv�en waves parallel to the equilibrium magnetic field. The

Alfv�enic fluctuation spectrum generated in this way can in

turn affect energetic ion confinement, preventing their ther-

malization in the plasma core and, possibly, increasing the

thermal load on the material wall. The assessment of plasma

operation 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 the

nonlinear saturation mechanisms (which saturation level is

expected for fluctuation amplitudes and what are the ensuing

effects on the fast ion confinement).

Due to the relevance of these issues, there exists a very

rich literature investigating linear stability of Alfv�en modes

and their nonlinear saturation, including theoretical and nu-

merical analyses, as well as comparisons of corresponding

predictions with experimental observations. Recent reviews

also exist, focusing on general aspects and applications to the

International Thermonuclear Experimental Reactor (ITER),1

on theoretical analyses,2–4 on basic physics properties and ex-

perimental evidences,5 on applications of model theoretical

descriptions to interpretation of experimental observations,6

and on numerical investigations and application to tokamak

experiments.7 Most available literature focuses on tokamak

plasmas; however, some works have addressed similarities

and differences of Alfv�en wave interactions with energetic

particles in tokamaks and stellarators.8,9

Nonlinear Alfv�en wave and energetic particle physics

can be understood and discussed along two main routes:3 (i)

nonlinear interactions of Alfv�enic fluctuations with energetic

particles; and (ii) wave-wave interactions and the resultant

spectral energy transfer. Although the second route is very

important3 and plays crucial roles in multi-scale dynamics of

burning plasmas,10 most nonlinear investigations address the

first route, as effects of the Alfv�enic fluctuation spectrum on

energetic 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 techniques

developed ad hoc for representing energetic particle phase

space; i.e., for isolating and understanding the fundamental

processes of resonant Alfv�en wave-energetic particle interac-

tions, nonlinear saturation of the Alfv�en fluctuation spectrum

and energetic particle transports.a)Electronic mail: [email protected]

1070-664X/2014/21(11)/112301/27/$30.00 21, 112301-1

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 nonlinear

interactions of Alfv�enic fluctuations with energetic particles.

The “bump-on-tail” paradigm4,11 applies only when the

plasma is sufficiently close to marginal stability such that

energetic particle orbit nonlinear distortions are small com-

pared to fluctuation wavelengths. It is based on the Ansatz

that resonant wave-particle interactions can be investigated

by analogy with Langmuir waves in a 1D Vlasov system in

the 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 are

generally important when resonant structures in the energetic

particle phase space (also dubbed “phase-space zonal

structures”16–18) coherently evolve with the underlying non-

linear mode structures due to phase-locking.11,16–20 Strictly

speaking, these two paradigms apply to limiting cases. For

example, the “bump-on-tail” paradigm can be adopted for

weak Toroidal Alfv�en Eigenmodes (TAE’s)21 close to mar-

ginal stability; and the “fishbone” paradigm well describes

fast growing Energetic Particle Modes (EPM’s)22 driven by

sufficiently strong fast ion pressure gradients. It has been

shown that weak TAEs saturate because of wave-particle

trapping,12–14,23–27 causing the resonant fast ion phase-space

gradients to vanish locally. As the fraction of particles

trapped within the wave grows with the mode amplitude, the

free energy source is progressively depleted. Saturation is

reached when a significant fraction of linearly resonant par-

ticles is trapped. Being this fraction proportional to the linear

growth 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 a

plasma wave excited by a cold electron beam in a 1D uni-

form plasma.28–31 Fast growing EPMs, meanwhile, saturate

because 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, increasing

continuum damping) quenches the process.4,11,16,17,32,33

The theoretical framework for bridging these two limit-

ing paradigms and discussing the fundamental processes

underlying nonlinear Alfv�en wave interactions with ener-

getic particles both in extreme and intermediate cases is laid

out in Refs. 4, 11, and 16–18. There, the concept of

“resonance detuning,” due to the nonlinear change in the

wave-particle phase, is introduced along with that of “radial

decoupling,” occurring when the nonlinear particle excursion

becomes comparable with the perpendicular wavelength.

Both mechanisms occur in nonuniform plasmas and are due

to resonant wave-particle interactions. However, only the

first one is accounted for by analyses adopting the “bump-

on-tail” paradigm discussed above; while both are generally

important for the “fishbone” paradigm. The relative role of

“resonance detuning” and “radial decoupling” depends on

the relative ordering of finite wave-particle radial interaction

length and radial mode width, which, in turn, depend on the

considered resonance (precession or precession bounce, for

magnetically trapped particles; and transit, for circulating

particles) and nonuniform plasma equilibrium (magnetic

shear, 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 case

of EPM32,33 and fishbones.34 A systematic and thorough

analysis of these process and, more generally, of the nonlin-

ear dynamics of “phase-space zonal structures”16–18 on the

basis of numerical simulation results is lacking; and it is the

main aim of the present paper. Our investigation is based on

numerical simulations performed by the HMGC code,36

which is capable of addressing the different processes self-

consistently, thereby shedding light on the various nonlinear

dynamics that will be important in burning plasma relevant

conditions.

The HMGC code, based on the hybrid MHD gyrokinetic

model35 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 modes

observed in existing devices (JT-60U,40 DIII-D41) or

expected in forthcoming burning plasmas (ITER32,42) and

proposed experiments (the Fusion Advanced Studies Torus

(FAST)43–45). It solves O(�3) reduced-MHD equations48 in

the zero pressure limit for the bulk plasma, with � being the

inverse aspect ratio. The reduced MHD equations are closed

by the energetic particle pressure gradient, computed by

solving the nonlinear Vlasov equation in the drift-kinetic

limit via particle-in-cell methods. Thus, fast ion finite

Larmor radius effects are neglected, but finite magnetic drift

orbit width effects are retained consistently.46,47 Energetic

particle dynamics are self-consistently retained and are

treated nonperturbatively: their pressure term contributes to

determine both structure and evolution of electromagnetic

fields. 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�enic

modes driven by energetic particles.49 This extended version

can also handle up to three independent particle populations

kinetically, assuming different equilibrium distribution func-

tions, as, e.g., bulk ions, energetic ions and/or electrons

accelerated by neutral beam injection (NBI), ion/electron cy-

clotron resonance heating (ICRH/ECRH), fusion generated

alpha particles, etc. In the present paper, however, these

extended physics simulation capabilities will not be used, as

they are not needed to analyze the cases discussed in this

work: namely, a n¼ 6 TAE (with n being the toroidal mode

number), in a low shear, large aspect ratio, circular magnetic

surface equilibrium; and a weakly unstable n¼ 2 EPM, in a

stronger magnetic shear equilibrium. Both modes fall in an

intermediate regime (c/x of order of few percent) between

the quasi-marginally stable “bump-on-tail” regime and the

strongly unstable “fishbone” one. Focusing on them will

allow to elucidate the role of “radial decoupling” and

“resonance detuning” without entering the extreme dynamic

regime of EPM avalanches.4,32,33

In order to investigate the nonlinear dynamics of shear-

Alfv�en modes and the corresponding behaviour of energetic

ions, we look at the evolution of a suited set of test

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


192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

particles50 in the perturbed electromagnetic fields obtained

by self-consistent HMGC simulations. Several features of

the wave-particle nonlinear interactions are evidenced by

resorting 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 the

times it crosses the equatorial plane. Drift motions induced

by the interaction with perturbed fields give rise to bounded

orbits in the (H, P/) phase-space for particles not too far

from 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. Such

density flattening involves an increasingly wider region as

the mode amplitude increases. At the same time, it causes an

increasing reduction of the free energy source for the system

instability. Mode saturation is reached when the region

affected 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 two

distinct regimes: in the low shear case, for not too little

mode growth rates, the potentially resonant region is limited

by the radial mode width rather than the finite wave-particle

radial interaction length introduced above4,16,18 (or

“resonance width”, for brevity, as it will be referred to in

what follows); in the larger shear case, for not too large

growth 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 obtained

in the latter case (“resonance detuning”). This corresponds to

the fact that the resonance width has a stronger dependence

on the growth rate than the mode width. This also shows the

important role of magnetic shear in determining mode satu-

ration, nonlinear dynamics and energetic particle redistribu-

tion. For the sake of simplicity, numerical simulations below

refer to fluctuations excited by transit resonance with circulat-

ing energetic particles. Precession and precession-bounce res-

onance with magnetically trapped particles deserve a separate

in depth analysis due to the implications they may have on

non-local behaviors.4,16,18 Furthermore, the case of a TAE

mode dominated by “radial decoupling” and of a EPM mode

with prevalent “resonance detuning” have been selected to

illustrate 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 describes

thermal plasma and energetic particle equilibrium profiles

adopted 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 linear

phase of mode evolution. The selection criteria of test-

particles suited for the investigation of the nonlinear

mode-particle dynamics are described in Sec. IV, and the

corresponding diagnostics based on Hamiltonian mapping

techniques are presented in Sec. V. The issue of the finite

radial 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 of

density gradient produced by the nonlinear particle flux; and

in Sec. IX, where the relative importance of the radial mode

and resonance width in determining the saturation level is

examined in the two considered reference cases. Conclusions

are drawn in Sec. X.

II. REFERENCE THERMAL PLASMA AND ENERGETICPARTICLE EQUILIBRIA

Our analysis will mainly focus on a specific case, which

has been recently adopted51 as a benchmark case for the

comparison of different simulation codes with respect to

the linear dynamics of energetic-particle driven TAEs in the

framework of the Energetic particle Topical Group of the

International Tokamak Physics Activity (ITPA). A circular

shifted magnetic-surface Tokamak equilibrium is consid-

ered, characterized by major radius R0¼ 10 m, minor radius

a¼ 1 m, safety factor q¼ q0þ (qa� q0) (r/a)2, q0¼ 1.71,

qa¼ 1.87 and on-axis magnetic field B0¼ 3T. The

bulk 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 ¼ mH

2pTH

� �32

nH sð Þe�E

TH ; (1)

with E being the kinetic energy of the energetic particle,

s � ð1� weq=weq0Þ1=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 yield

vH0=vA0’ 0:3 and qH0=a’ 0:03, with vH0�ðTH0=mHÞ1=2;vA0 being the on-axis Alfv�en velocity, qH0� vH0=XH0 and

XH0 being the energetic particle on-axis cyclotron frequency.

As in the ITPA benchmark,51 a single toroidal mode

number, n¼ 6, is retained: this corresponds to neglecting

fluid mode-mode coupling in the evolution of electromag-

netic fields, while fully taking into account wave-particle

nonlinearities. Poloidal harmonics are instead retained in the

range 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, with

the Lundquist number defined as S � 4pa2=ðgc2sA0Þ.The evolution of the perturbed magnetic energy is

shown, for this reference case, in Fig. 1, where energy is in

units of a3B20=8p. Saturation is reached at t ’ 1700 sA0: The

radial structure of the poloidal harmonics of the scalar poten-

tial (normalized to TH0/eH) and the parallel component of the

vector potential (normalized to B/0qH0) during the linear

phase are shown in Fig. 2 along with the corresponding

power spectrum (intensity contour plot in arbitrary units) of

the scalar potential in the (r, x) plane: the dominant poloidal

harmonics corresponds to m¼ 10 and 11, while the



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

following values are found for the real frequency and the

growth 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”), in

order to examine how the nonlinear dynamics depend on the

typical equilibrium and/or mode scale lengths: a weakly

unstable n¼ 2 EPM in an equilibrium characterized by larger

magnetic shear than the previous case; qa¼ 1.1, qa¼ 1.9 giv-

ing qðrÞ ¼ 1:1þ 0:8ðr=aÞ2: The bulk plasma density profile

is chosen in such a way to have aligned toroidal frequency

gaps in the Alfv�en continuum: ni ¼ ni0q20=q2; but still suffi-

ciently separate in radial location (due to choice of low

n¼ 2) that EPM nonlinear dynamics is local and not affected

by 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 perturbed

magnetic energy (normalized to a3B20=8p) and, respectively,

the linear mode structure and power spectrum for this second

reference 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 and

3, 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 units

of a3B20=8p.

FIG. 2. Radial structure of the poloidal

harmonics of the scalar potential (a)

and the parallel component of the vec-

tor potential (b) during the linear

phase; corresponding power spectrum

of the scalar potential (c).



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

In the following, unless stated explicitly, we will refer to

case 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 time

derivative

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 the

radial coordinate, h and / are the poloidal and toroidal angle,

respectively, M is the conserved magnetic momentum, U is

the parallel velocity and # the gyrophase) and Dzc!Z is the

Jacobian of the transformation from canonical zc to gyrocen-

ter coordinates.

The power transfer from energetic particles to the wave

is given by �dEH=dt: We can compute such quantity in the

following 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.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

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 by

the l-th macroparticle, is a constant of motion of the l–th

macro-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

� �1

þ FH0

dZi

dt

� �2

" #@

@Zi

mH

2U2 þMXH

� �

’ �2pX

l

�wdZi

dt

� �1

þ �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

� �1

þ �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 the

phase-space velocities dZi/dt, and taken into account that, as

the 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 power

transfer”. It corresponds to the power transfer weighted by

the distribution function and integrated in h and /; the over-

all cancellation, at each time, of the large contribution to the

power transfer yielded by the linear terms in the fluctuating

fields has been explicitly enforced. Note that such enforce-

ment would not be possible if we computed the particle

power transfer as pl ¼ 2p½�wl þ �DlFH0ðZlÞ�ðdE=dtÞl, through

a direct computation of the particle kinetic energy variation

(that is, without splitting the different contributions related

to linear or nonlinear terms in the perturbed fields).

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

following 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 toðdxSxðx� 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 the

interval ½xi�1; xiþ1�. The power transfer will be then approxi-

mated by

� dEH

dt’ 1

m3H

Xi;j;k

DriDMjDUkPi;j;k: (15)

In Fig. 5, the density of the wave-particle power exchange

(in arbitrary units) proportional toP

i2mode 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 is

limited to the region where the mode is localized. Positive

sign corresponds to destabilization of the mode.

IV. TEST PARTICLE SELECTION: A PROPERRESONANT PARTICLE SAMPLE

In order to get information about the resonant particle

behaviour and its connection with mode saturation, we



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

follow the evolution of a set of test particles (resonant particle

sample) in the fields computed from the self-consistent simu-

lation. In principle, the same information could be obtained

by the direct analysis of the behaviour of the simulation par-

ticles (that is, particles used in the self-consistent simulation

to sample the whole phase space). The advantage of a test

particle approach is that it allows us for a zoomed investiga-

tion of specific phase space regions (in particular, those where

the resonances are localized) by simply evolving the selected

test-particle set in the stored self-consistent fluctuating fields,

without requiring a new heavy self-consistent simulation per

each different choice of the phase space zoom.

At t¼ 0, we set h¼ 0 for all test particles. The other

coordinates are initialized in such a way that test particles

correspond to a set of resonant particles driving the mode

unstable 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 which

the linear drive (that is the particle-wave power transfer) is

significant. Following Ref. 50, we then define the quantity

C � xP/ � nE, with P/ � mHRU þ eHR0ðw� w0Þ=c being

the toroidal angular momentum. Such a quantity is a constant

of the (perturbed) motion, provided that the perturbed field is

characterized (as in the considered case) by a single toroidal

mode 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, characterized

by time and toroidal angle dependence in the form

H¼H(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 as

follows: 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 matching

the value M¼M0 and C¼C0; that is, U¼U(r, M0, C0). For

each 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 all

test particles corresponds to cutting the phase space into in-

finitesimal slices that do not mix together even during the

nonlinear evolution of the mode and looking at the dynamics

characterizing one of these slices (M¼M0 and C¼C0); that

is, the nonlinear evolution of an isolated linear resonance.

Note that this method remains valid for frequency chirping

modes, provided j _xj�jc2j, with c being the characteristic lin-

ear growth rate. In fact, in this limit, the resonance frequency

shift in one nonlinear time (�c�1) is jDxj�jcj, the typical

resonance width. Thus, resonant particle samples belonging

to separate linear resonances will not mix nonlinearly, even

for frequency chirping modes. In the present case, we fix

M 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 that

the test-particle set corresponds to a relevant resonant-particle

sample does not ensure that this set represents adequately the

whole resonant-particle population of the self-consistent sim-

ulation. In principle–as discussed in Sec. VI with reference to

case 2–, several different portions of the phase space (corre-

sponding to different C values) could play an important role

in the linear destabilization of the mode, with resonances

peaked around different radial positions; and the nonlinear

dynamics 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 profile

and time evolution of the power transfer integrated over the

test-particle set with that integrated over the whole particle

population of the self-consistent simulation. Fig. 6 compares

the radial profile of the power transfer for the two popula-

tions. We see that the two profiles exhibit both close peaks

and comparable width. Note that, if separated peaks were

found, we could improve the selection of the test-particle

population, by looking for a better choice of r0, M0 and U0

(that is, a better choice of C0). The radial width of the two

profiles cannot instead be controlled, as it depends on given

features of the dynamical system: namely, the spatial scales

characterizing 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 good

agreement of the two profiles, that the test-particle set

FIG. 5. Power exchange between the energetic particles and the mode in the

normalized (M, U) space, integrated over a radial shell around the mode

localization (0:448 � r=a � 0:552). Positive sign corresponds to power

transfer from particles to mode; that is, to mode destabilization. The solid

and dashed lines approximately indicate the loss cone for the inner and outer

radial surface of the considered shell, respectively. It is apparent that the

main contribution to the drive is given, here, by circulating particles.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

represents the whole resonant-particle population fairly well.

This is confirmed by comparing the time evolution of the

power transfer integrated over the two different populations,

(Fig. 7). We see that the agreement is quite satisfactory until

saturation 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 nonlinear

evolution of the self-consistent wave-particle interactions.

Two remarks have to be done concerning the results

shown in Fig. 6. First, power transfer has been computed re-

ferring each particle contribution to the radial coordinate

assumed by the same particle when it crosses the equatorial

plane in its outmost position. Let us call, conventionally,

such coordinate the “equatorial” radial coordinate to distin-

guish it from the “instantaneous” radial coordinate. In the

following, when reporting the radial dependence of quanti-

ties integrated over the particle population or the test-particle

set, r will represent the equatorial radial coordinate, unless

explicitly stated. The second remark is a technical one, con-

cerning the weight of each test particle contributing to the

power 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 in

the coordinates ZW� (r, h, /, M, C) as

�Dtest

l � DWlDzc!ZWðZWlÞ: (20)

The infinitesimal quantity

DWl � ½DrDhD/DMDC�l (21)

is the same for all test particles (as we are initializing test

particles with fixed M and C, h¼ 0 and several equispaced rand /). Then, we obtain, from Eqs. (18) and (19)

�Dtest

l / Dzc!ZWZWlð 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 are

considered (see Sec. VI), the element DC is not necessarily

the same for all particles. In our case, we will determine the

different values C0i by fixing several equispaced values of

parallel velocity U0i and computing C0i ¼ Cðr0; 0;M0;U0iÞ.With such choice, we will get

�Dtest

l / 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 outmost

R position. The wave phase seen by the particle at those

times 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 can

be 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 particle

population (blue) compared with the same quantity integrated over the test

particle population only (red). The agreement is fairly good until saturation

is 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.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

Here,4,16,18

xD �D/2p� r�q

� �xb (28)

is the precession frequency, D/ is the change in toroidal

angle 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 along

the particle orbit; and

xb �2psb

(31)

is the bounce frequency (in our case, transit frequency).

Equation (26) can be solved with respect to r, yielding

r ¼ rresðx;M0;C0; kÞ, or P/, yielding P/ ¼ P/ resðx;M0;C0; kÞ: In the present case, the relevant bounce harmonic is

k¼ 1, as shown in Fig. 8.

In the unperturbed motion, dP/=dt ¼ 0. Then, during

the linear phase of the mode evolution, the particle trajecto-

ries in the (H, P/) plane essentially reduce to fixed points for

P/ ¼ P/ res, while they correspond to drift along the H axis

in the positive (negative) direction, for P/ greater (less) than

P/ res. This is represented in Fig. 9 where P/ is given in units

of mHavH0 and H is reported to the interval ½0; 2p½. Here and

in the following, each marker is doubled by a twin marker

in the interval ½2p; 4p½, in order to yield a better visualization

of the relative dynamics. Moreover, for simplicity, we

shall indicate the quantity mod½DHj; 2p�=ðtjþ1 � tjÞ (with

mod½DHj; 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 particles

that 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 they

would properly close if the field amplitude were constant in

time. This is true for particles born close to the resonance,

while particles born with P/ far from the resonance maintain

drifting 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” by

the wave. Once the particle orbit becomes topologically

closed, we will describe the particle as “trapped” in the

wave, adopting the standard classification. As the fluctuating

field strength increases, the P/ drift increases and more and

more particles are captured and eventually trapped. This can

be 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 to

the last crossing of the equatorial plane of a test particle. We

consider two kinds of plots. A first kind, where the color (red

or blue) of each marker depends on the birth P/ value (less

or greater than P/ res). In the second kind of plots, marker

color 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 the

linear phase of the mode evolution and for two different

times of the nonlinear phase. We observe the formation of

wave-trapped particle structures, while, correspondingly, the

maximum-drive structures move outward (lower P/) and

FIG. 8. The quantity xresðr;M0;C0; kÞ is plotted for various values of k (red

dashed 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 adopted

in the present paper, has been considered here.

FIG. 9. Drift of test particles in the H direction, at constant P/ during the

linear phase of the mode evolution. Markers are coloured according to their

instantaneous 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 is

reported to the interval [0,2p]. Moreover, here and in many other following

figures, each marker is doubled by a twin marker in the interval [2p, 4p], in

order to yield a better visualization of the relative dynamics.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

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

decreasing 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 the

wave-particle power transfer profile. In Section V, we have

noted that, once the values of M and C are fixed, the reso-

nance condition x� xresðr;M;C; kÞ ¼ 0 identifies a radial

position r ¼ rresðx;M;C; kÞ, where the resonance takes

place. If the mode is characterized, in the linear phase, by a

wave-particle power transfer that we generally assume to be

of the order of the linear growth rate c, the resonance will be

anyway 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 the

resonant layer is shown.

Equation (32) is only a kinematic condition related to

the single particle motion; the effectiveness of resonant

wave-particle power exchange also depends on the structure

of the perturbed field and of the particle distribution function

in the resonant layer.4,16,18 This can be easily understood by

looking at the power transfer radial profile. We can write this

quantity, 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 curvature

and 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 of

motions, as all these contributions are typically small for

shear Alfv�en waves considered here. The quantity dFH obeys

to 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 during

the nonlinear phase of the mode evolution. Different colours have been

adopted here for different markers, in order to distinguish them.

FIG. 11. Test particle markers in the

(H, P/) plane during the linear phase

of the mode evolution. (a) Each marker

is coloured according to the birth P/

value of the particle (red for

P/ < P/ res, blue otherwise); (b) each

marker is coloured according to the

macroparticle power transfer (time-

varying color).



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

Taking into account that Dzc!Z ¼ m2HXHRr ¼ m2

HXH0R0r,

we can symbolically write

P rð Þ �ð

dMdU

�eHmHrR

xþ ic� xres

rFH b�r du� U

cdAk

� �

� eHM

mHr log Bþ eHU2

XHj

� � b�rdu

� ��h;/

:

(35)

It is apparent that the radial structures of perturbed

field and gradient of the distribution function will deter-

mine the radial shell where a significant power exchange

can occur. For fixed radial mode and distribution function

structures, P(r) will be predominantly given by particles

with 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 that

shell. Each (M, C) value, as noted in Section V, identifies

one individual wave-particle resonance condition in the

form of Eq. (26), extended over the resonant layer defined

by 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, at

t¼ 1640sA0.

FIG. 14. Radial localization (rres) and width (Drres) of the resonance for a

mode with frequency x and growth rate c.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

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, each

individual resonant layer will be larger than P(r). The two

limits are sketched in Fig. 15, which considers two situa-

tions corresponding, ceteris paribus, to small and large

growth rate, respectively. In the latter case, once fixed

M¼M0 and C ¼ C0, with M0 and C0 corresponding to the

resonance peak, the test particle sample population is repre-

sentative of the whole resonant particle population. Indeed,

the resonant layer corresponding to such test particle choice

essentially 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 of

the whole resonant particle population. In the present case,

Fig. 6 suggests that the individual resonance is wider than

or of the same order of the interaction region. This is con-

firmed by Fig. 16(a), comparing the power transfer radial

profile 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 to

six 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 to

resonances separated by more than the single-resonance

FIG. 15. Model comparison between

(potential) power transfer radial profile

and resonance width. The former quan-

tity (blue), integrated over the whole

velocity space, is significantly different

from zero in the radial shell where

power exchange can occur (in the pres-

ence of resonant particles), and is

determined by the radial structure of

the mode and the gradient of the ener-

getic particle distribution function. The

resonance width depends on the mode

growth rate and the radial variation of

xres. Here, we consider only the effect

of growth rate variation. (a) Small

growth rate limit; (b) large growth rate

limit.

FIG. 16. (a) Power transfer radial pro-

file for the selected test-particle popu-

lation compared with the resonance

width Drres (b) Power transfer radial

profiles 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 resonances

separated by more than the single-

resonance width from the peak one

yield negligible contribution to the

power transfer. (c) Normalized multi-Cset power transfer radial profile (black)

compared with that related to the sin-

gle-C set corresponding to the C value

adopted in our analysis (C ¼ �4:74,

close to the peak value) and with the

resonance width (dashed-blue).



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

width from the peak one yield negligible contribution to the

power transfer. We then expect that, in this case, a single-Csample, with C value close to resonance peak, is as adequate

as the multi-C set in representing the whole resonant particle

population 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 2

discussed in Sec. II. In such case, the single-C test-particle

sample cannot properly represent the whole resonant particle

population, and a multi-C approach is needed, consistently

with the larger value of dxres /dr, mainly due to larger shear.

Figure 17 refers to ten test-particle samples, corresponding

to equi-spaced parallel velocity values between U/vH0¼ 1.2

and 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 the

original (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 each

time, to the maximum power transfer with its birth value

P/0. In Fig. 18, this comparison is shown for ingoing and

outgoing resonant structures. We note that, in the nonlinear

phase, the mode is driven by different particles (born at dif-

ferent P/0 values) at different times. It is then clear that the

trajectory 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 be

seen from Fig. 19, which directly compares evolution of

maximum-drive structures and test particle markers. In the

upper, 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, respectively

In the left, central and right frames of each row test particles

are plotted in the (H, P/) positions they have at t¼ 500sA0,

t¼ 900sA0 and t¼ 1470sA0, respectively. In each frame, the

black markers contain the region characterized by a power

transfer greater than half of the maximum rate obtained at the

considered time t. In such way the frames on the “diagonal”

of the frame matrix show the drive structures at the three

times considered. The off-diagonal frames illustrate the evo-

lution of the set of the “row-time” most destabilizing particles

at earlier or later times (below and above diagonal, respec-

tively). It is apparent that the evolution of particles driving

the 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 shows

the trajectories in the (H, P/) plane of four different particles

FIG. 17. Same as Fig. 16, for case 2,

characterized by a larger magnetic

shear than case 1. Power transfer radial

profiles 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.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

(each represented by a different color), chosen as the most

destabilizing 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 is

marked by a black dot. So, the black boxes correspond to the

trajectory of a maximum-drive structure, while the colored

dots correspond to the individual particle trajectories, which

neither coincide with nor are tangent to those of maximum-

drive structures.

Other interesting features of wave-particle interactions

can 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 according

the instantaneous power transfer. First, as it also appears

from Fig. 18, macro-particles driving the mode during the

nonlinear phase (t¼ 1400, 1470sA0) were originally out of

resonance (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 resonance

with 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-particles

will be trapped into bounded orbits.

The second feature is that only a small portion of the

trapped particle orbit yields relevant contribution to the

mode 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 volume

represented by the test particle) becomes small or even nega-

tive, although its average contribution over the full orbit

remains positive.

The third interesting feature we can emphasize from the

analysis of test-particle dynamics is that the time needed to

complete 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 trapping

times. This is shown in Fig. 22 in a more quantitative way,

where the P/ range covered by the largest bounded orbit at

each time is plotted (red line). The blue line represents the

same quantity for orbits that have completed two bounces.

We see that saturation (t ’ 1700 sA0) occurs before any test

particle has completed two bounces.

Finally, it is interesting to note that, in the nonlinear

phase, most destabilizing particles exchange energy with rel-

atively large value of jP/ � P/ resj; that is, large values of

jx� xresðr;M;C; kÞj: In the framework of linear analysis,

we would then expect that such energy exchange takes place

at large values of dH/dt. However, this is not the case, as we

can 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 by

Eq. (11) and sums are taken only over test particles yielding

energy to the wave. The same quantities are plotted against

each 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 resonant

structures: the perturbed poloidal and toroidal velocities of

the particle, d _h and d _/, have to be retained on the same

footing.

VIII. SATURATION MECHANISM: DENSITYFLATTENING

In Fig. 7, we have noted that the energetic particle drive

(measured by the test particle sample) fast decreases in the

time interval t ¼ 1400� 1700 sA0. Below, we analyze the

reasons of such decrease and the consequent mode saturation.

In Section VII, we have seen that saturation occurs on a

time scale of the same order as the wave-particle trapping

time. 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.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

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 the

mode evolution (left frames, showing the unperturbed den-

sity profile and the linear-phase power transfer profile) along

with the corresponding plots of the markers in the (H, P/)

plane. We see that the formation of mutually penetrating

structures in such plane, corresponding to particles captured

and eventually trapped by the wave field, yields an exchange

of 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 linear

wave-particle interaction is strongest, and a consequent

steepening of the same profile at the boundaries of that

region. In particular, in the limit of negligible growth rate

and 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 (left

frames), 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 the

considered time.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

motion of particles captured in the wave field, we would get

density flattening after particle orbits have completed one

quarter bounce, while density profile inversion would occur

after half bounce. Time and radial dependence of the mode

amplitude cause quantitative, but not qualitative modification

of this simplified picture, on the time scale that characterizes

this phenomenon (shorter than of the order of x�1wp). In the

following, for the sake of simplicity, we will refer to such a

density 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 different

from density flattening that occurs on time scales much lon-

ger than x�1wp, as consequence of phase mixing, collisions

and/or quasilinear particle diffusion due to the presence of a

broad fluctuation spectrum with many overlapping resonan-

ces52,53 (cf. Ref. 18 for more in depth discussions).

Figure 25 compares the radial position of the density

gradient 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 the

wave 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 because

we 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 macroparticles

are considered, chosen as the most destabilizing ones, for t¼ 900sA0 (violet),

t¼ 1300sA0 (blue), t¼ 1400sA0 (green) and t¼ 1470sA0 (orange). The arrow

indicates the rotation direction of the trajectories during the nonlinear phase.

The black boxes correspond to the position of the different particles when

they become the most destabilizing particle: they indicate the trajectory of

the ingoing maximum-drive structure. It is apparent that the latter trajectory

neither coincides with nor is tangent to any of the former.

FIG. 21. Test particle orbits in the (H,

P/) plane. Three macroparticles are

considered, chosen as the most destabi-

lizing ones, for t¼ 900sA0 (a),

t¼ 1400sA0 (b) and t¼ 1470sA0 (c).

Markers are coloured according to

their instantaneous power transfer.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

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 distortion

of 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 examine

whether and how the density profile distortion is related to

resonant particles, we can characterize each test particle, at

each time, by its power transfer (normalized to the instanta-

neous maximum value among all the test particles). Defining

a threshold b< 1 in the normalized power transfer, we can

divide particles into two different groups: those whose nor-

malized power transfer never exceeds, in the course of the

whole mode evolution considered in the numerical simula-

tion, such a threshold, and those for which the threshold is

exceeded, at least within a certain time window. We can

then separate the whole test particle population into different

classes, according to the best drive performance reached by

particles during the mode evolution. Figure 26 shows, with

reference to case 1, the fraction of population belonging to

each class for a certain partition: namely, particles that never

drive the mode and particles whose best (and positive) drive

performance falls in left-open intervals of width Db¼ 0.1.

We see that there is no particle belonging to the former

group; 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 the

power transfer) versus time, for the

ingoing resonant structure (blue) and

the outgoing one (red). (b) Average ra-

dial position for the two resonant struc-

tures. (c) Average dH/dt versus

average 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 of

jdH=dtj much smaller than expected,

on the basis of the linear-phase rela-

tionship dH=dt ¼ x� xres, for the

observed radial drift.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

contribute to drive the mode, consistent to the fact that the

sample was chosen around a resonance in the phase space.

We also observe that there is a large population fraction

whose contribution to the drive never exceeds a normalized

value of b¼ 0.1. The fraction value falls down dramatically

in the next class (0:1 < b � 0:2), and then it continuously

increases for the upper classes, until reaching a relatively

large 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 that

such 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 that

exceed, in the course of mode evolution, this threshold; as

nonresonant particles, the others. Correspondingly, we shall

indicate by CresðtÞ and CnonresðtÞ the quantities, analogous to

that defined in Eq. (39), obtained when looking only at the

density of particles belonging to one or the other of these

two classes. Figure 27 compares the time evolution of the

density distortion rate, C, with Cres and Cnonres. Although, in

general, the absolute value that appears in Eq. (39) causes Cto be different from the sum of Cres and Cnonres, we see that

the 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 to

particles characterized by a power transfer greater than 0.2 at

that 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 rate

related to the density of particles affecting the drive at earlier

or 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 and

power transfer profiles are also reported (blue). Times are the same considered in Figs. 12 and 13 respectively: t¼ 1470sA0 (top) and t¼ 1640sA0 (bottom). The

corresponding 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 the

largest radial width of orbits that have completed, respectively, a quarter,

half, and a full bounce in the wave field.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

especially in the late nonlinear phase. Note, however, that

normalizing these two specific quantities to the number of

particles belonging to the corresponding particle subset, the

average instantaneously-resonant distortion rate per particle

is much larger than that computed over the remaining

resonant-particle subset (cf Fig. 29(a)); the two groups yield

values of the same order because the former is always much

less numerous than the latter (Fig. 29(b)).

It is also interesting to compare the time evolution of

dPtot=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 instantaneously

resonant particle rate (Fig. 30(a)), but not as well with the

full resonant particle rate (Fig. 30(a)). This shows that the

progressive reduction of the overall drive, along with the

consequent mode saturation, is primarily caused by the flat-

tening of the density profile of particles at the instantaneous

peak of power transfer to the wave. This is consistent with

the role played by the gradient of the distribution function

and the resonant denominator in Eq. (35), with the latter

quantity selecting the instantaneously resonant phase-space

region. The fact that the full resonant-particle distortion rate

remains relatively large even after the saturation has been

reached can be explained considering that the formation of

phase space zonal structures16–18–i.e., the (n¼ 0, m¼ 0)

structures made of particles captured by the resonance, with

the definition introduced in Sec. V–proceeds because of the

finite amplitude of the saturated mode; indeed, we have al-

ready observed in Sec. VII that particles strongly resonant in

the nonlinear phase will undergo further radial excursion

even 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 the

power transfer is reduced in the region where the density

profile is flattened, it is preserved and possibly increased

where 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 of

nonresonant ones being negligible.

FIG. 26. Fraction of particle population, corresponding to the test particle

set adopted for case 1, belonging to different classes, according to the best

drive performance reached during the mode evolution. No particles have

negative best performance: this means that all the considered particles some-

times contribute to drive the mode. A large population fraction is confined

in 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 increases

for the upper classes, until reaching a relatively large value (above 10%) for

the upmost one (0.9<b< 1).

FIG. 28. Density distortion rate for the “resonant” particles (Cres, shown in

Fig. 27), compared with the analogous quantities computed for the subsets

of particles whose drive performance exceeds b¼ 0.2 at the specific time

considered (“instantaneously resonant” particles) or at earlier or later times

(“other resonant” particles). Although the distortion rate computed for the

instantaneously resonant particles is large, the analogous quantity related to

the other resonant particles is not negligible at all, especially in the late non-

linear phase.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

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-density

region. 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 plotted

for 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 outgoing

structures, respectively. The linear-phase structure (the

same for both groups of test particles) is also plotted for

comparison, along with the actual density profile and the

unperturbed one.

If the radial dependence of the power transfer rate (cf.

the symbolic expression given in Eq. (35)) were due to the

density gradient only, the resonant structure splitting would

not 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 underlying

plasma 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 gradient

width), the density flattening, whose radial extension

increases with mode amplitude along with the wave-particle

trapping region, subtracts a finite and increasing fraction

FIG. 29. Average distortion rate per

particle (a) and relative numerosity (b),

for the two subsets (instantaneously

resonant particles and other resonant

particles) considered in Fig. 28. The

distortion rate per particle is much

higher for the instantaneously resonant

particles. The two subsets give rise, as

a whole, to comparable rates because

such particles always represent a rela-

tively small fraction of the whole reso-

nant particle group.

FIG. 30. Time derivative of the overall

particle power transfer compared with

the rate of density profile distortion for

the instantaneously resonant particle

set (a) and the full resonant particle set

(b).

FIG. 31. Radial profile of the power

transfer at the same times considered

in Fig. 24: t¼ 1470 sA0 (a) and

t¼ 1640sA0 (b). Two contribution are

computed separately, related respec-

tively to test particles having values of

H around the ingoing structure and the

outgoing one. The linear-phase profile

(the same for both groups of test par-

ticles) is also plotted for comparison,

along with the actual density profile

and the unperturbed one.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

from the effective drive. As soon as density flattening affects

the whole region where wave-particle power transfer can

take place, the drive vanishes, the mode saturation is reached

and 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 with

the radial profile of the linear power transfer and the time

evolution 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 radial

width of the region where wave-particle power transfer can

take place comes from the mode width, we dub the saturation

mechanism radial decoupling, indicating that potentially res-

onant particles get decoupled from the radial mode structure.

If such constraint comes from the resonance width, we refer

to the mode saturation mechanism as resonance detuning.

The two different limits correspond to different scaling

of the saturation field amplitude with the mode growth

rate.20,37 This can be qualitatively understood on the basis of

the following argument. The width of the density flattening

region scales with A1=2 (with A representing the field ampli-

tude), since this is the scaling expected for the separatrix

width of resonant particles trapped in the wave (with conse-

quent density flattening) and untrapped particles.54,55 We

can also assume, from Eq. (32), that the resonance width

scales as c, while the mode width is expected to exhibit a

much weaker dependence on the mode growth rate (let us

assume, for the sake of simplicity, no dependence at all).

Thus, resonance detuning would yield saturation when the

density flattening width matches the resonance width, or

A � c2;12–14,28–30 while radial decoupling would require the

matching with the mode width, or A � c020,37 (consistent

with the fact that increasing the growth rate and, hence, the

resonance width does not increase the mode capability of

extracting power from the particles). Such scalings are based

on a simplified picture of the saturation process, and they

should 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 to

radial decoupling takes place. More articulate saturation

models, following the same qualitative arguments, are dis-

cussed in Ref. 20 and explain that, when the different length

scales of radial mode structures become important in the

wave-particle power exchange, the transition between A �c2 (vanishing drive) to A � c0 (strong drive) can go through

A � cp (2< p< 4);4,20,37 and not be necessarily character-

ized by a progressive weakening of the power scaling from

p¼ 2 to p¼ 0.

In Sec. VI, we noted that in case 1 the mode width is of

the same order or even narrower than the linear resonance

width. 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, larger

growth 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 in

Figs. 16(a) and 32 for the reference case with those obtained

for a larger growth rate mode (xsA0 ’ 0:289, csA0 ’ 0:021,

saturation reached at t ’ 760sA0). If resonance detuning

were responsible for the limit imposed to the resonant-

structure splitting, we should get a proportional increase of

the maximum splitting. Instead, it is clear that splitting

depends very weakly on the linear growth rate (and the linear

resonance width). This can also be seen from Fig. 34, where

the maximum radial splitting is compared with the mode and

resonance widths for different values of the growth rate.

Consistently, we find that the saturation amplitude has a

weaker c scaling than the quadratic dependence expected for

dominant resonance detuning. This is shown in Fig. 35,

where the amplitude of the perturbed poloidal magnetic field

is reported for different values of c/x.

We now explore the conditions under which resonance

detuning 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 (that

is, stronger radial dependence of xres) or larger mode width,

the linear resonance is narrower than the mode. An example

of this condition has been shown, for case 2, at the end of

Sec. VI (Fig. 17). In these cases, we have seen that the linear

power transfer profile for a single-(M, C) test-particle set is

limited by resonance (rather than mode) width, so that a

multi-(M, C) approach can be required to properly represent

the whole resonant particle population (characterized by an

overall power transfer profile with mode-size width).

Consistent with these results, Fig. 36 shows that, in such a

situation, the linear resonance width also plays the main role

in 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, by

itself, for further splitting of the resonant structures. This is

confirmed by the results plotted in Fig. 37, which shows the

width of the relevant regions versus the growth rate of the

FIG. 32. Splitting of the power transfer peaks (red) compared with the radial

profile of the linear power transfer (dashed black) and the time evolution of

the mode growth rate (dashed blue). The finite width of the linear power

transfer profile puts a hard limit to the radial drift of the resonant structures.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

FIG. 33. Splitting of the power transfer

peaks (red) compared with the radial

profile (dashed black) of the linear

power transfer (left) and the linear res-

onance width (right). The reference

case (a) and a larger growth rate case

for 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 weakly

on linear growth rate (and linear reso-

nance width).

FIG. 34. Maximum radial splitting of the resonant structures (red) compared

with the mode (green) and resonance (blue) widths, for different values of

the growth rate. It is apparent that, in the large growth rate limit, the main

role in determining mode saturation is played by the radial decouplingmechanism.

FIG. 35. Amplitude of the perturbed poloidal magnetic field versus c/x. The

deviation from the quadratic scaling (shown for comparison) expected for a

purely resonance detuning saturation mechanism is clearly visible at large

growth rates, consistently with the prevalence, in this limit, of the radial

decoupling mechanism.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

mode. Correspondingly, a stronger scaling of the saturation

field amplitude with c/x is recovered, close to quadratic (cf.

Fig. 38). The dispersion of the obtained values around the

latter scaling is the natural consequence of the fact that, dif-

ferent from the ideal framework in which such scaling is

obtained, the 3D system investigated by our simulations is

anyway characterized by mode structures and plasma

nonuniformities.4

It is worthwhile noting, as discussed above in this

Section, that shear-Alfv�en fluctuations are generally charac-

terized by “internal” length scales finer than the mode

width2,4,10,17 due to the properties of the shear Alfv�en contin-

uous spectrum. Thus, the effect of plasma non-uniformities

and radial mode structures can become important even when

the linear resonance width is a fraction of the mode

width.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 effects

is, however, outside the scope of the present analysis.

The simple interpretation of a mode saturation reached

because the density flattening reduces the drive in the whole

region where the resonant particles can interact with the

perturbed field could underestimate important aspects of the

nonlinear mode-particle dynamics. In particular, radial

decoupling 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 to

the radial drift of the maximum density gradient and the

resonant structure.4,32,33,37–40 Resonance detuning, mean-

while, could lose effectiveness if mode structure and fre-

quency modifications consistent with resonant particle

nonlinear dynamics (phase-locking4,11,16–20,32–34) and/or

perturbed field contributions to the poloidal and toroidal

velocities of the particle (cf. Sec. VII) are able to keep

jdH=dtj small, in spite of the distance between the linear-

resonance and the location where the power transfer is

peaked because of the nonlinear evolution of phase space

zonal structures.16–18

FIG. 36. (a) splitting of the power

transfer peaks (red) corresponding to a

single-C test-particle set for case 2,

compared with the radial profile of the

linear power transfer for the same set

(black) and a multi-C one (green). (b)

linear resonance width. Saturation is

reached at t ’ 1230sA0 (indicated by a

vertical continuous black line). Both

the linear power transfer profile for the

single-C test-particle set and the

resonant-structure splitting are limited

by 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 for

case 2. A stronger scaling than that obtained for case 1 is observed, close to

the quadratic scaling (shown for comparison), expected for a purely reso-

nance detuning saturation mechanism.



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

In the first reference case considered in this paper (case

1), no significant frequency or mode structure modifications

are observed in the nonlinear phase (cf. Fig. 39 with Fig.

2). Concerning the nonlinear contribution to the resonance

condition, 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 of

the 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 resonance

are significant and they correspond to an increased power

transfer efficiency. We then expect that they contribute to

postponing mode saturation, at least in the low growth

rate 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 power

transfer with radial profile approximated by the following

expression:

Pmodel rð Þ �ð

dMdUhrFH eHmHrR

xþ ic� xres

b�r du� U

cdAk

� �" #(

� eHM

mHr log Bþ eHU2

XHj

� � b�rdu

� �0

�h;/

; (40)

with subscript 0 denoting quantities computed in the

linear-phase. Comparing Eq. (40) with Eq. (35), it can be

recognized that the model expression is obtained by

neglecting all nonlinear physics except those due to reso-

nant particle density gradient. That is, assuming prescribed

mode structure (justified on the basis of the comparison

between 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 integrated

power transfer corresponding to this model. We see that,

although the model properly includes the density profile

evolution, the actual nonlinear decrease of the integrated

power 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 evolution

and saturation.

FIG. 39. Mode structure during the

nonlinear phase: scalar potential (a),

parallel vector potential (b), power

spectrum of the scalar potential (c).



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

With reference to case 2, we observe analogous features

of the behaviour of the resonant denominator jdH=dtj in the

nonlinear phase (cf. Fig. 41): its absolute value is kept much

lower than expected from its linear-phase radial dependence

(jx� xresðrÞj), computed at the nonlinear radial position of

the 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 (linear

phase, 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) versus

average r for the two resonant structures; the quantity x� xresðr;M;C; kÞ is

also plotted (green). Also in this case, the energy exchange is characterized, in

the nonlinear phase, by values of jdH=dtj much smaller than expected, on

the basis of the linear-phase relationship dH=dt ¼ x� xres, for the observed

radial drift.

FIG. 40. Time evolution of the integrated power transfer obtained in case 1

(red), compared with the integrated power transfer corresponding to the

model power profile given by Eq. (40) (blue).

FIG. 42. Mode structure during the

nonlinear phase, for case 2, just before

the first saturation (t¼ 1200sA0): scalar

potential (a), parallel vector potential

(b), power spectrum of the scalar

potential (c).



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

saturation, at t¼ 1200sA0). Nevertheless, the first saturation

is followed, in this case, by complex nonlinear oscillations.

Different from case 1, the mode adjusts both its radial

localization and frequency in order to maximize resonant

wave-particle power exchange, typical of EPM nonlinear

evolution.4,10,17,32,33 The detailed investigation of this further

nonlinear evolution, however is beyond the scope of the

present work and will be the subject of a separate paper.

X. CONCLUSIONS

In this paper, the nonlinear evolution of shear-Alfv�en

modes, 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 electromagnetic

fields computed by self-consistent HMGC hybrid MHD-

particle simulations. The test particle set has been selected by

fixing 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. Two

different case have been analyzed: a n¼ 6 TAE, in a low

shear, 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 been

selected to illustrate the roles of resonance detuning vs. radial

decoupling in nonlinear mode dynamics and saturation, as

well as energetic particle transport. In particular, the fact that

the TAE case is dominated by radial decoupling and the weak

EPM case by resonance detuning shows that, depending on

mode growth rate and plasma equilibrium nonuniformity,

nonlinear dynamics may be different from that expected on

the basis of the theoretical paradigms (bump-on-tail and,

respectively, fishbone paradigms) currently adopted in the

investigation of Alfv�en wave-energetic particle interactions.

Hamiltonian mapping techniques have been used to

examine the evolution of resonant phase-space structures

and 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 the

equatorial-plane crossing times, the universal structure of

resonant particle phase space near an isolated resonance can

be recovered and illustrated, with bounded orbits and

untrapped trajectories divided by the instantaneous separatrix

in the time-evolving mode structure. Furthermore, the pres-

ent numerical simulation results allow us to demonstrate

that, in general, the characteristic time of phase space zonal

structures evolution is the same as the nonlinear time of the

underlying fluctuations. That is, wave-particle nonlinear dy-

namics in nonuniform plasmas are generally non-adiabatic.

Bounded orbits correspond to particles captured in the

potential 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 resonant

energetic particle density in a region around the linear reso-

nance peak. A detailed analysis shows that there is a strong

correlation between the decreasing rate of power transfer to

the fluctuation by resonant energetic particles and the

distortion of the density profile of the instantaneously reso-

nant particles. This fact can be easily understood considering

that flattening and/or inversion of the gradients of the distri-

bution function in the instantaneously resonant phase-space

region causes the subtraction of phase-space portions to the

drive; thus, drive decrease is associated to the widening of

the no longer destabilizing region. Saturation is reached

when the subtracted phase-space portion, whose radial width

increases with the mode amplitude, covers the whole poten-

tially resonant region. Such a correlation disappears if we

look at the overall density-profile distortion, as the latter

involves also particles that are no longer resonant, simply

because 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 resonant

region, depending on the relative size of finite interaction

length of resonant particles and characteristic length scale of

perpendicular mode structures. In the low shear case, this

region is radially limited by the mode width more than the

wave-particle resonance width (especially for relatively large

growth rates). In the larger shear case, the region is limited

by the resonance width. We refer to the former situation as

to radial decoupling saturation mechanism; to the latter, as to

resonance detuning. As the resonance width has typically a

stronger dependence on the mode growth rate than the mode

width, the mode amplitude at saturation exhibits a stronger

growth rate dependence in the resonance detuning than in

the radial decoupling regime. This is confirmed by the scal-

ings of the mode amplitude at saturation obtained in the two

considered cases.

The present work consists of a series of carefully diag-

nosed numerical simulation experiments set up to isolate and

illustrate the fundamental physics processes underlying the

nonlinear dynamics of Alfv�enic modes resonantly excited by

a sparse supra-thermal particle population in tokamak plas-

mas; and of the ensuing energetic particle transports. Further

to the saturation processes, several other phenomena, which

are undoubtedly of interest for the physics involved and have

possible practical impacts on energetic particle confinement

studies in fusion plasmas, have been enlightened by the pres-

ent investigation. In particular, we have demonstrated that, in

addition to nonlinear radial energetic particle excursions, non-

linear particle motions in poloidal and toroidal direction are

crucial 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 devoted

to a deep analysis of richer nonlinear dynamics, observed in

case 2 but not examined in the present paper, related to the

capability of the mode to adjust its spatial and frequency

structure, after the first saturation has been reached, in such a

way to maintain the power extraction from energetic particles.

ACKNOWLEDGMENTS

The authors are indebted to Liu Chen for suggesting the

analysis presented in this paper and improving it through



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

many valuable discussions. This work was supported by

Euratom Community under the contract of Association

between EURATOM/ENEA. It was also partly supported by

European Union Horizon 2020 research and innovation

program under Grant Agreement No. 633053 as Enabling

Research Project CfP-WP14-ER-01/ENEA_Frascati-01. The

computing resources and the related technical support used

for this work have been provided by CRESCO/ENEAGRID

High Performance Computing infrastructure and its staff.

1A. Fasoli, C. Gormenzano, H. L. Berk, B. N. Breizman, S. Briguglio, D. S.

Darrow, N. N. Gorelenkov, W. W. Heidbrink, A. Jaun, S. V. Konovalov,

R. Nazikian, J.-M. Noterdaeme, S. E. Sharapov, K. Shinohara, D. Testa,

K. Tobita, Y. Todo, G. Vlad, and F. Zonca, Nucl. Fusion 47, S264 (2007).2L. Chen and F. Zonca, Nucl. Fusion 47, S727 (2007).3L. Chen and F. Zonca, Phys. Plasmas 20, 055402 (2013).4L. Chen and F. Zonca, “Physics of Alfv�en waves and energetic particles in

burning plasma,” Rev. Mod. Phys. (submitted).5W. W. Heidbrink, Phys. Plasmas 15, 055501 (2008).6B. N. Breizman and S. E. Sharapov, Plasma Phys. Control. Fusion 53,

054001 (2011).7Ph. Lauber, Phys. Rep. 533, 33 (2013).8Ya. I. Kolesnichenko, V. V. Lutsenko, A. Weller, A. Werner, H. Wobig,

Yu. V. Yakovenko, J. Geiger, and S. Zegenhagen, Plasma Phys. Control.

Fusion 53, 024007 (2011).9K. Toi, K. Ogawa, M. Isobe, M. Osakabe, D. A. Spong, and Y. Todo,

Plasma Phys. Control. Fusion 53, 024008 (2011).10F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, A. V. Milovanov, Z. Qiu, G.

Vlad, and X. Wang, “Energetic particles and multi-scale dynamics in

fusion plasmas,” Plasma Phys. Control. Fusion (to be published).11L. Chen and F. Zonca, JPS Conf. Proc. 1, 011001 (2014).12H. L. Berk and B. N. Breizman, Phys. Fluids B 2, 2246 (1990).13H. L. Berk, B. N. Breizman, and M. Pekker, Phys. Rev. Lett. 76, 1256 (1996).14H. L. Berk, B. N. Breizman, and N. V. Petiashvili, Phys. Lett. A 234, 213

(1997).15B. N. Breizman, H. L. Berk, M. Pekker, F. Porcelli, G. V. Stupakov, and

K. L. Wong, Phys. Plasmas 4, 1559 (1997).16F. Zonca, S. Briguglio, L. Chen, G. Fogaccia, G. Vlad, and X. Wang,

“Nonlinear dynamics of phase-space zonal structures and energetic parti-

cle physics,” Proceedings of the 6th IAEA Technical Meeting on Theory

of Plasma Instabilities, Vienna, Austria, May 27–29 (2013).17F. Zonca and L. Chen, AIP Conf. Proc. 1580, 5 (2014).18F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, G. Vlad, and X. Wang,

“Nonlinear dynamics of phase-space zonal structures and energetic parti-

cle physics in fusion plasmas,” New J. Phys. (submitted).19R. B. White, R. J. Goldston, K. McGuire, A. H. Boozer, D. A. Monticello,

and W. Park, Phys. Fluids 26, 2958 (1983).20X. Wang, S. Briguglio, L. Chen, C. Di Troia, G. Fogaccia, G. Vlad, and F.

Zonca, Phys. Rev. E 86, 045401(R) (2012).21C. Z. Cheng, L. Chen, and M. S. Chance, Ann. Phys. (N.Y.) 161, 21 (1985).22L. Chen, Phys. Plasmas 1, 1519 (1994).23G. Y. Fu and W. Park, Phys. Rev. Lett. 74, 1594 (1995).24R. B. White, Y. Wu, Y. Chen, E. D. Fredrickson, D. S. Darrow, M. C.

Zarnstorff, J. R. Wilson, S. J. Zweben, K. W. Hill, G. Y. Fu, and M. N.

Rosenbluth, Nucl. Fusion 35, 1707 (1995).25Y. Wu, R. B. White, Y. Chen, and M. N. Rosenbluth, Phys. Plasmas 2,

4555 (1995).26Y. Todo, T. Sato, K. Watanabe, T. H. Watanabe, and R. Horiuchi, Phys.

Plasmas 2, 2711 (1995).27W. Park, E. V. Belova, G. Y. Fu, X. Z. Tang, H. R. Strauss, and L. E.

Sugiyama, Phys. Plasmas 6, 1796 (1999).

28I. N. Onishchenko, A. R. Linetskii, N. G. Matsiborko, V. D. Shapiro, and

V. I. Shevchenko, JETP Lett. 12, 281 (1970).29V. D. Shapiro and V. I. Shevchenko, Sov. Phys. JETP 33, 555 (1971), see

http://www.jetp.ac.ru/cgi-bin/dn/e_033_03_0555.pdf.30T. M. O’Neil, J. H. Winfrey, and J. H. Malmberg, Phys. Fluids 14, 1204

(1971).31M. B. Levin, M. G. Lyubarskii, I. N. Onishchenko, V. D. Shapiro, and V.

I. Shevchenko, Sov. Phys. JETP 35, 898 (1972), see http://www.jetp.ac.ru/

cgi-bin/dn/e_035_05_0898.pdf.32G. Vlad, S. Briguglio, G. Fogaccia, and F. Zonca, Plasma Phys. Control.

Fusion 46, S81–S93 (2004).33F. Zonca, S. Briguglio, L. Chen, G. Fogaccia, and G. Vlad, Nucl. Fusion

45, 477 (2005).34G. Vlad, S. Briguglio, G. Fogaccia, F. Zonca, V. Fusco, and X. Wang,

Nucl. Fusion 53, 083008 (2013).35W. Park et al., Phys. Fluids B 4, 2033 (1992).36S. Briguglio, G. Vlad, F. Zonca, and C. Kar, Phys. Plasmas 2, 3711

(1995).37S. Briguglio, F. Zonca, and G. Vlad, Phys. Plasmas 5, 3287 (1998).38S. Briguglio, G. Vlad, F. Zonca, and G. Fogaccia, Phys. Lett. A 302,

308–312 (2002).39F. Zonca, S. Briguglio, L. Chen, S. Dettrick, G. Fogaccia, D. Testa, and G.

Vlad, Phys. Plasmas 9, 4939–4956 (2002).40S. Briguglio, G. Fogaccia, G. Vlad, F. Zonca, K. Shinohara, M. Ishikawa,

and M. Takechi, Phys. Plasmas 14, 055904 (2007).41G. Vlad, S. Briguglio, G. Fogaccia, F. Zonca, C. Di Troia, W. W.

Heidbrink, M. A. Van Zeeland, A. Bierwage, and X. Wang, Nucl. Fusion

49, 075024 (2009).42G. Vlad, S. Briguglio, G. Fogaccia, F. Zonca, and M. Schneider, Nucl.

Fusion 46, 1–16 (2006).43A. Cardinali et al., “Energetic particle physics in FAST H-mode scenario

with combined NNBI and ICRH,” In Fusion Energy 2010, IAEA, Vienna,

2010, Paper no. THW/P7-04.44C. Di Troia, S. Briguglio, G. Fogaccia, G. Vlad, and F. Zonca,

“Simulation of EPM dynamics in FAST plasmas heated by ICRH and

NNBI,” In 24th IAEA FEC 2012, San Diego 8–13 Oct. 2012, Paper no.

TH/P6-21.45A. Pizzuto, F. Gnesotto, M. Lontano, R. Albanese, G. Ambrosino, M. L.

Apicella, M. Baruzzo, A. Bruschi, G. Calabr�o, A. Cardinali, R. Cesario, F.

Crisanti, V. Cocilovo, A. Coletti, R. Coletti, P. Costa, S. Briguglio, P.

Frosi, F. Crescenzi, V. Coccorese, A. Cucchiaro, C. Di Troia, B. Esposito,

G. Fogaccia, E. Giovannozzi, G. Granucci, G. Maddaluno, R. Maggiora,

M. Marinucci, D. Marocco, P. Martin, G. Mazzitelli, F. Mirizzi, S. Nowak,

R. Paccagnella, L. Panaccione, G. L. Ravera, F. Orsitto, V. Pericoli

Ridolfini, G. Ramogida, C. Rita, M. Santinelli, M. Schneider, A. A.

Tuccillo, R. Zag�orski, M. Valisa, R. Villari, G. Vlad, and F. Zonca, Nucl.

Fusion 50, 095005 (2010).46F. Zonca and L. Chen, Phys. Plasmas 21, 072120 (2014).47F. Zonca and L. Chen, Phys. Plasmas 21, 072121 (2014).48R. Izzo, D. A. Monticello, W. Park, J. Manickam, H. R. Strauss, R.

Grimm, and K. McGuire, Phys. Fluids 26, 2240 (1983).49X. Wang, S. Briguglio, L. Chen, G. Fogaccia, G. Vlad, and F. Zonca,

Phys. Plasmas 18, 052504 (2011).50R. B. White, “Modification of particle distributions by MHD instabilities

I,” Commun. Nonlinear Sci. Numer. Simulat. 17, 2200–2214 (2012).51A. K€onies et al., “Benchmark of gyrokinetic, kinetic MHD and gyrofluid

codes for the linear calculation of fast particle driven TAE dynamics,”

IAEA, ITR/P1-34.52R. Z. Sagdeev and A. A. Galeev, Nonlinear Plasma Theory (W. A.

Benjamin Inc., 1969).53W. E. Drummond and D. Pines, Nucl. Fusion Suppl. 3, 1049 (1962), see

http://www-naweb.iaea.org/napc/physics/FEC/1961.pdf.54T. M. O’Neil, Phys. Fluids 8, 2255 (1965).55R. K. Mazitov, Zh. Prikl. Mekh. Fiz. 1, 27 (1965).



192.107.52.30 On: Tue, 04 Nov 2014 15:04:38

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http://dx.doi.org/10.1063/1.4804628

http://dx.doi.org/10.1063/1.2838239

http://dx.doi.org/10.1088/0741-3335/53/5/054001

http://dx.doi.org/10.1016/j.physrep.2013.07.001

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http://dx.doi.org/10.1016/S0375-9601(97)00523-9

http://dx.doi.org/10.1063/1.872286

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http://dx.doi.org/10.1016/0003-4916(85)90335-5

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http://dx.doi.org/10.1063/1.872997

http://dx.doi.org/10.1016/S0375-9601(02)01136-2

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http://dx.doi.org/10.1063/1.3587080

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Analysis of the nonlinear behavior of shear-Alfvén modes ... · Analysis of the nonlinear behavior of shear-Alfven modes in tokamaks based on Hamiltonian mapping techniques S. Briguglio,1,a)

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