Linear gyrokinetic particle-in-cell simulations of Alfv!en instabilities intokamaks

A. Biancalani,1,a) A. Bottino,1 S. Briguglio,2 A. K€onies,3 Ph. Lauber,1 A. Mishchenko,3

E. Poli,1 B. D. Scott,1 and F. Zonca2,4

1Max-Planck-Institut f€ur Plasmaphysik, 85748 Garching, Germany2ENEA C. R. Frascati, Via E. Fermi 45, CP 65-00044 Frascati, Italy3Max-Planck-Institut f€ur Plasmaphysik, 17491 Greifswald, Germany4Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou, People’s Republic of China

(Received 7 October 2015; accepted 30 December 2015; published online 14 January 2016)

The linear dynamics of Alfv!en modes in tokamaks is investigated here by means of the globalgyrokinetic particle-in-cell code ORB5, within the NEMORB project. The model equations are shownand the local shear Alfv!en wave dispersion relation is derived, recovering the continuous spectrum inthe incompressible ideal MHD limit. A verification and benchmark analysis is performed forcontinuum modes in a cylinder and for toroidicity-induced Alfv!en Eigenmodes. Modes in a reversed-shear equilibrium are also investigated, and the dependence of the spatial structure in the poloidalplane on the equilibrium parameters is described. In particular, a phase-shift in the poloidal angle isfound to be present for modes whose frequency touches the continuum, whereas a radial symmetry isfound to be characteristic of modes in the continuum gap. [http://dx.doi.org/10.1063/1.4939803]

I. INTRODUCTION

Plasma heating is essential for reaching appropriatefusion temperatures in tokamak plasmas, but as a side effectglobal modes can become unstable by converting particle ki-netic energy into collective kinetic energy. The energeticparticle (EP) population produced in the process of heatingtogether with the alpha particles produced in fusion reactionsare important actors in this chain, driving plasma oscillationsunstable via resonant wave-particle interactions. On the otherhand, the plasma instabilities such as Alfv!en Eigenmodes(AEs) can redistribute the EP population making the plasmaheating less effective and leading to additional loads on thewalls.1–4 Typically, tokamak plasmas are turbulent plasmas.Thereby, turbulence adds one more level of difficulty to theEP-redistribution problem, where wave-wave nonlinearinteraction of turbulent and zonal modes with AEs competeswith wave-particle nonlinear saturation mechanisms of AEs.For these reasons, it is important to have a proper self-consistent theoretical framework to understand the AEs’instability threshold in present tokamaks and predict it infuture fusion reactors.

NEMORB5–7 is the project of development of an elec-tromagnetic, multi-species version of the nonlinear gyroki-netic particle-in-cell (PIC) code ORB5.8 The Lagrangianformulation that is used is based on the gyrokinetic GKVlasov-Maxwell equations of Sugama, Brizard, andHahm.9,10 Due to the method of derivation of the GKVlasov-Maxwell equations from a discretized Lagrangian,the symmetry properties of the starting Lagrangian arepassed to the Vlasov-Maxwell equations, and the conserva-tion theorem for the energy is automatically satisfied.6 As aconsequence, this model can be adopted in principle for rig-orous nonlinear electromagnetic simulations of global insta-bilities in the presence of EP and turbulence, where all

nonlinearities are treated on the same footing in a self-consistent way. Furthermore, a PIC formulation offers a finediscretization in v-space “for free” (because no grid in veloc-ity space is needed, but a sufficiently high number ofmarkers for the Monte Carlo method), which is crucial forstudying the wave-particle interaction in the narrow layersaround the resonances in phase-space.

When trying to solve the Vlasov-Maxwell set of equa-tions in a pk formulation with a df PIC method, one faces anumerical problem called “cancellation problem”.11,12 Thisarises in particular in the numerical resolution of theAmpere’s equation, i.e., in the equation for the vector poten-tial. One term of this equation, namely, the current integral,has to be calculated with a discretization in terms of macro-particles, i.e., markers, whereas the other terms are calculateddirectly as analytic integrals in phase-space (see Sec. II). Dueto fact that the statistical error affects only the term discretizedwith markers, the balance between these terms is not satisfied,and the result is a numerical error which can be orders of mag-nitude higher than the desired signal.

A brute-force solution to the cancellation problem is thedrastic increase in the number of markers, which in turnwould make electromagnetic PIC simulations unpractical. Asmart solution has been proposed as a split of the adiabaticand nonadiabatic parts of the electron distribution function,where only the physically relevant one, i.e., the nonadiabaticpart, is discretized with markers, whereas the adiabatic part(which is dominant in absolute value) can be calculateddirectly by means of an adjustable control variate.11,12 Thisscheme has been found to greatly mitigate the cancellationproblem, making electromagnetic PIC simulations feasible,with a reasonable number of markers. Recently, the control-variate scheme has been implemented also in NEMORB,5,7

making the investigation of the dynamics of shear Alfv!enwaves (SAW) possible, for linear and nonlinear simulations.Furthermore, a split of the vector potential into symplectica)URL: www2.ipp.mpg.de/!biancala

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PHYSICS OF PLASMAS 23, 012108 (2016)

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and Hamiltonian parts can be performed, and only theHamiltonian part has to be calculated self-consistently withthe Vlasov-Maxwell system of equations, whereas the sym-plectic part can be evaluated with alternative methods (forexample, by imposing the ideal MHD Ohm’s law, which isvalid for incompressible SAW to the leading order).13 Thisnew “pullback” scheme, which further helps in strongly miti-gating the cancellation problem, is also considered as one ofthe next numerical improvements to be done in NEMORB.

NEMORB has been recently verified and benchmarkedin electrostatic mode for linear dynamics of global instabil-ities driven by EP,14,15 but a detailed verification and investi-gation of the linear dynamics of Alfv!en instabilities has notyet been done with this code. This is a necessary step in thedirection of performing a trustable study of the richer nonlin-ear dynamics of Alfv!en modes in the presence of turbulenceand zonal flows. In this paper, we first describe a verificationand benchmark effort of NEMORB on Alfv!en instabilitiesdriven by EP. Second, we investigate the effect of the contin-uum on the radial structure of Alfv!en modes, finding that theradial symmetry is broken, when the mode frequency liesoutside the continuum gaps.

The structure of the paper is the following: in Section IIwe describe the model equations of NEMORB and derivethe local dispersion relation of SAW, which constitutes thecontinuous spectrum. The continuous spectrum is the fre-quency, varying continuously in space, where Alfv!en insta-bilities are damped due to continuum damping.16–19

Generally speaking, the continuous spectrum does not existin pure kinetic theory. But there exists a nearly dense spec-trum of modes that behaves as a continuum when one con-siders their cumulative impact on a given oscillation, withassigned frequency and wave-vector.20 In this paper, wefocus on the derivation of the continuum in the incompressi-ble ideal MHD limit, by neglecting the effect of the EP. Averification of NEMORB for the frequency of the continuousspectrum is performed in the limit of very small inverse as-pect ratio (i.e., in the cylindrical limit), and described inSection III, in the absence of EP. Toroidicity-induced AEs(TAEs) are investigated in the presence of EP in Sec. IV,and results compared with those obtained with the hybridMHD-gyrokinetic code HMGC21 and published in Ref. 22.Sec. V is devoted to a study of how EP drives continuummodes unstable, and in particular of the dependence of theirspatial structure on the continuum properties. Finally, Sec.VI is devoted to a summary of the conclusions and an outlineof the next steps.

II. THE MODEL AND THE LOCAL DISPERSIONRELATION OF SAW

A. Model equations

In this section, we describe the general model equationsof NEMORB, and we derive the local dispersion relation forthe SAW continuous spectrum, in the limit of a cold plasmaand neglecting the parallel equilibrium current Jk0.

The gyrokinetic equation in its general form is describedby the Liouville theorem, i.e., the property of incompressibility

of the distribution function in phase-space, in the absence ofcollisions

@f

@tþ 1

B#k

@

@Z$ B#k _Zf! "

¼ 0: (1)

The phase-space coordinates are Z ¼ ðR; pk; lÞ, i.e.,respectively, the gyrocenter position, canonical parallel mo-mentum pk ¼ mU þ ðe=cÞJ0Ak (e is considered negative forelectrons here) and magnetic momentum l¼mv2

?/(2B). TheJacobian is given by the parallel component ofB# ¼ Bþ ðc=eÞpk$( b, where B and b are the equilibriummagnetic field and magnetic unitary vector.

The properties of the system are described by the gyro-kinetic Lagrangian (see Ref. 6 and references therein)

L ¼Rsp

ðdVdW

e

cAþ pkb

$ %$ _R þ mc

el _h

$ %f

&

) H0 þH1ð Þf )H2fM* )ð

dVjr?Akj2

8p(2)

with the Hamiltonian divided into unperturbed, linear, andnonlinear part, H ¼H0 þH1 þH2, with

H0 ¼p2k

2mþ lB; (3)

H1 ¼ e J0/)pkmc

J0Ak

$ %; (4)

H2 ¼e2

2mc2J0Ak' (2 ) mc2

2B2jr/j2: (5)

Here f and fM are the total and equilibrium (i.e., independentof time) distribution functions, the integrals are over the phase-space volume, with dV being the real-space infinitesimal anddW ¼ ð2p=m2ÞB#kdpkdl the velocity-space infinitesimal. Thetime-dependent fields are the scalar potential / and the parallelcomponent of the vector potential Ak. Here A is the equilibriumvector potential. The summation is over all species present in theplasma (with e, m, and f depending on the species), and the gyro-average operator is labeled here by J0 (with J0 ¼ 1 for elec-trons). The gyroaverage operator reduces to the Bessel functionif we transform into Fourier space. The Lagrangian given in Eq.(2) is a function of the trajectories and of the fields and containsall information about the system we are interested in. In the fol-lowing, the particle trajectories and field equations are derivedfrom Eq. (2) (see also Refs. 9 and 23–25).

The particle gyrocenter trajectories are derived byimposing the minimal action principle with respect to thephase-space coordinates, which yields6 (the sign of the sec-ond term at the R.H.S. appears wrong in the original refer-ence, due to a typographical error)

_R ¼ @ H0 þH1ð Þ@pk

B#

B#kþ c

eB#kb(r H0 þH1ð Þ;

_pk ¼ )B#

B#k$r H0 þH1ð Þ:

Now by noting that rðH0 þH1Þ ¼ lrBþ erJ0

ð/) Akpk=mcÞ, we can write the trajectories in explicit form

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_R ¼ 1

mpk )

e

cJ0Ak

$ %B#

B#k

þ c

eB#kb( lrBþ erJ0 /)

pkmc

Ak

$ %& ); (6)

_pk ¼ )B#

B#k$ lrBþ erJ0 /)

pkmc

Ak

$ %& ): (7)

The Poisson equation is derived by imposing theminimal action principle with respect to the scalar potential,which yields6

)r $ n0mc2

B2r?/ ¼ Rsp

ðdWeJ0f (8)

with n0m being here the total plasma mass density.Similarly, the Ampere equation is derived by imposing

the minimal action principle with respect to the vector poten-tial, which yields6

Rsp

ðdW

epkmc

f ) e2

mc2AkfM

$ %þ 1

4pr2?Ak ¼ 0: (9)

Here the form with J0 ¼ 1 is given for simplicity. For morecomplicated models, see Ref. 6.

As mentioned in Sec. I, the resolution of Eq. (9) presentsa numerical problem, if one wants to solve the first term likeit is written here, with a particle-in-cell technique. This isbecause the total (equilibrium þ perturbed) electron distribu-tion function fe has to be integrated in the phase-space bymeans of a marker discretization, whereas the term involvingthe equilibrium distribution function is solved by direct inte-gration, yielding the equilibrium electron density. Due to thefact that each of these two terms is much bigger in amplitudethan their difference, and that the statistical error introducedby the marker discretization falls on the first term only, andnot on the second one, then we have that the resulting bal-ance (or “cancellation”) is not satisfied in the numerical solu-tion, which is dominated by the statistical noise.11,12 Asolution to this “cancellation problem” comes with a control-variate technique, which splits the perturbed distributionfunction df in an adiabatic part df ad ¼ )ðJ0/) pkJ0Ak=mcÞefM=kBT and a nonadiabatic part (i.e., the remaining part).With this technique, the integral to be performed with themarker discretization becomes in fact much smaller, andtherefore the resulting numerical noise is greatly miti-gated.11,12 This control-variate technique has been recentlyimplemented in NEMORB5,7 and allows us to perform thefirst numerical simulations of SAW.

Eqs. (6)–(9) are the constitutive equations of the model.The results of NEMORB simulations, where these equationsare solved numerically with particle-in-cell method, aredescribed in Secs. III–V.

B. Local dispersion relation

In the following, we take Eqs. (6)–(9) as starting pointto derive analytically the vorticity equation (see also Ref. 23and 24 for analogous derivations). We focus here on radially

localized perturbations in the incompressible ideal MHDlimit in a tokamak with large aspect-ratio and neglect theparallel equilibrium current. The shear-Alfv!en wave continu-ous spectrum is derived as a result, in tokamak geometry.

In order to derive the vorticity equation, we start by tak-ing the time derivative of Poisson equation for T¼ 0 andwriting it in a continuity form24

@w

@tþr $ JG ¼ 0; (10)

where the vorticity is defined by

w ¼ )r $ n0mc2

B2r?/ (11)

and where the gyrocenter current is

JG ¼ Rsp

ðdWef _R: (12)

Here the gyrokinetic equation, Eq. (1), has been used to givean explicit form to the time derivative of the distributionfunction (only the term with spatial derivatives survivesinside the phase-space integral, due to the divergencetheorem).

For the present derivation, we consider only the domi-nant term in the gyrocenter velocity. For a magnetizedplasma, this is the one along the equilibrium magnetic field

_R ’ 1

mpk )

e

cJ0Ak

$ %b: (13)

The linearized divergence of the gyrocenter current becomes

r $ JG ’ B $r 1

BRsp

ðdW

e

mpkdf )

eAkc

fM

$ %" #

;

¼ ) c

4pB $r 1

Br2?Ak

$ %; (14)

where we have used Ampere’s equation, Eq. (9), to evaluatethe integral in phase-space, in the limit of a cold plasma(J0ðk?qiÞ¼ 1).

We now consider a plasma where the parallel electricfield is zero, Ek ¼ )b $r/) ð1=cÞ@tAk ¼ 0, consistentlywith Ohm’s law in the ideal MHD regime. A Fourier trans-form is performed in time, so that @t ! )ix, and the Ohm’slaw can be written as iðx=cÞAk ¼ b $r/. A tokamak withcircular concentric flux surfaces is considered in the follow-ing. After performing a time derivative to the whole equa-tion, the vorticity equation for incompressible shear-Alfv!enwaves reads1,26

r $ x2

v2A

r?/þ b $rð Þr2? b $rð Þ/ ¼ 0; (15)

where v2AðRÞ ¼ B2=4pmn0 is the local Alfv!en velocity. The

singular solutions of this equation are the continuum modesand reflect the fact that a global mode approaching the posi-tion of the continuum, is damped by continuum damping.19

The importance of knowing the exact topology of the

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continuous spectrum is clear, for the existence of globalAlfv!en instabilities has a strong dependence on whether theirfrequency touches or not the continuum.

In the following, we derive the SAW continuous spec-trum formula in the tokamak geometry. The coordinatesused are the cylindrical (R;u; Z) and toroidal (r; f; h) coordi-nates linked by

R ¼ R0 þ r cos h; Z ¼ r sin h; u ¼ ) fR:

We consider a geometry with small inverse aspect ratio! ¼ a=R+ 1 for this derivation. The plasma nonuniformityis kept now only in the spatial dependence of the Alfv!envelocity: vAðr; hÞ ¼ vAðrÞ=ð1þ ! cos hÞ. The modes areFourier-decomposed in the poloidal and toroidal angles

/ðr; h;u; tÞ ¼X

m

/mðrÞ exp ið)mhþ nu) xtÞ:

Now we select one single toroidal mode, and the differentcomponents in m have

kkm ¼1

R0n) m

q rð Þ

$ %:

We write the vorticity equation, Eq. (15) as a matrix equa-tion Mijðr; n;xÞ/j ¼ 0, whose determinant is imposed to bezero. With the chosen decomposition in Fourier, it is clearthat the plasma nonuniformity in h of the Alfv!en velocitycouples the m and m61 modes. This reflects in the fact thatthe matrix has only diagonal terms and first off-diagonalterms nonzero, with value

Miiðr; n;xÞ ¼ ~x2 ) ~k2

km; (16)

Mijðr; n;xÞ ¼ ! ~x2 ðj ¼ i61Þ; (17)

Mijðr; n;xÞ ¼ 0 ðjj) ij > 1Þ; (18)

where ~x ¼ qRx=vA; ~kkm ¼ qRkkm and ! ¼ 3r=2R. When wefocus on a region of the plasma in the proximity of the cross-ing of an m and an mþ 1 continuum branches, the determi-nant reduces to a second-order algebraic equation, where thetwo solutions are

~x26 ¼

~k2

km þ ~k2

kmþ16

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~k

2

km ) ~k2

kmþ1

! "2

þ 4!2 ~k2

km~k

2

kmþ1

r

2 1) !2ð Þ :

(19)

This is the formula for the continuous spectrum of SAW ina tokamak geometry.1,26 In the cylindrical limit, !! 0,Eq. (19) reduces to the local SAW dispersion relationx2 ¼ v2

Ak2km. For a small but finite value of !, a gap is created

in the continuum at the position of the crossing of thetwo continuum branches with m and mþ 1 (where kkm¼ )kkmþ1). In this gap, global modes named TAEs can exist,essentially not damped by continuum damping.

In Secs. III–V, the model equations of NEMORB, Eqs.(6)–(9), are solved numerically with NEMORB, i.e., with a

particle-in-cell technique, and the results are compared withthe continuum formula, Eq. (19). In particular, in Sec. III thelimit of !! 0 is considered as a verification test, and theoscillation frequency of SAW is investigated in the absence ofenergetic particles. In Secs. IV and V, respectively, TAEs andcontinuum modes with zero shear are investigated in the pres-ence of energetic particles, and their dynamics is discussed inrelation to the continuum topology given by Eq. (19).

III. CONTINUOUS SPECTRUM IN A CYLINDER

A. Axisymmetric continuum

In this section, we show the results of the tests ofNEMORB for the simplest plasma confinement geometry,i.e., a cylindrical geometry. This is achieved by choosing an-alytical magnetic equilibria with very small inverse aspectratio (! ¼ 0:01). Numerical simulations with flat q profilesare performed, where an initial perturbation is let evolve intime (without EP) and the SAW oscillation frequency ismeasured, for different values of q, electron mass, and elec-tron beta be ¼ 8pneTe=B2

0 (where be regulates the densityin NEMORB). This frequency is the natural oscillationfrequency of the plasma and takes the name of continuousspectrum, or simply continuum.19

First, the continuum for axisymmetric perturbations(corresponding to a toroidal mode number n¼ 0) is consid-ered. We choose an analytical equilibrium with magneticfield on axis B¼ 2.4 T, major radius R¼ 1.667 m, minorradius a¼ 0.01667 m, q# ¼ qs=a ¼ 1=50 (with qs ¼ cs=Xi

being the sound gyroradius, and cs ¼ffiffiffiffiffiffiffiffiffiffiffiffiTe=mi

pbeing the

sound speed) and be ¼ 2( 10)4. Flat temperature anddensity profiles are chosen, with the ratio of electron to iontemperature seðqÞ ¼ TeðqÞ=TiðqÞ ¼ 1 (for all simulationsdescribed in this paper). An axisymmetric ion gyrocenterdensity perturbation is initialized and let evolve in time.Dirichlet boundary conditions and Neumann boundary con-ditions are imposed, respectively, to the external and internalboundaries for the potentials. The spatial grid of a typicalsimulation is (ns,nchi,nphi)¼ (64,32,4) and the time stepdt¼ 10 X)1

i , with 107 markers for ions and electrons.The signal is observed to oscillate and the oscillation fre-

quency is measured. Different simulations with different val-ues of q are performed (see Fig. 1), and the scan of thefrequency vs q is compared with the ideal MHD prediction forthe continuum given by Eq. (19), which reduces to xSAW

¼ vA=qR for axisymmetric geometries and axisymmetric oscil-lations. Different scans for different values of electron massare performed, and we note that for this value of q#, a goodconvergence is found for values of electrons mass 2000 timessmaller than the ion mass. A scan in be is also performed (seeFig. 2), for the case with q¼ 2, and me=mi ¼ 200. The code isfound to be stable (i.e., no signs of cancellation problem areobserved) also at this large values of be, where a good conver-gence with ideal MHD prediction is observed.

B. Non-axisymmetric continuum

Similar simulations like in Sec. III A are described here,but for non-axisymmetric perturbations. The same magnetic

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equilibrium profile as described in Sec. III A is considered,with the same value of plasma temperature and density.Several simulations are performed, each of them with a dif-ferent value of q, and each of them with flat q profile.

In this case, we consider the evolution of modes withtoroidal mode number n¼ 2 and poloidal mode numbersm¼ 4 and m¼ 5, corresponding, respectively, to khqs ¼m q# a=r ¼ 0.16 and 0.2 in the middle of the radial domain(r=a ¼ 0:5). Like in Sec. III A, no EP population is initial-ized; therefore, our SAW oscillations are stable. We measurethe frequency like in the axisymmetric simulations, by meas-uring the period of oscillation of the potential for simulationswhere each poloidal component is let evolve in time.

A good match is found, for both branches m¼ 4 andm¼ 5, with ideal MHD prediction, Eq. (19), which reducesin the cylinder limit to xSAW ¼ vAðm) nqÞ=qR (see Fig. 3).Like predicted, in this cylinder limit, no gap in the continu-ous spectrum due to toroidicity is found at the intersection ofthe two branches. A gap forms when toroidal curvature is

introduced, by increasing the value of !, and consequently afurther degree of nonuniformity breaks the symmetry in thepoloidal angle h giving rise to the modification of the twocylinder branches.

In Sec. IV, simulations with a non-negligible toroidalcurvature are performed, and the formation of a global modewith frequency lying in the toroidicity-induced gap is shown.

IV. TOROIDICITY-INDUCED ALFV!EN EIGENMODES

A. Equilibrium and numerical setup

In this section, we consider a more realistic value oftoroidal curvature, i.e., a larger value of inverse aspect ratio!. In this case, the continuous spectrum gap opens at theintersection of two neighbor cylinder branches m and mþ 1,and a global eigenmode is created with frequency lyingwithin the continuum gap. This mode takes the name ofTAE.1

The equilibrium for the simulations shown in thissection has been chosen consistently with the ITPA bench-mark.22 The equilibrium magnetic field is given by analyticaltoroidal flux surfaces without Grad-Shafranov shift, and withmagnetic field on axis B¼ 3 T, major radius R¼ 10 m, minorradius a¼ 1 m. The corresponding q-profile is parabolic,with minimum value of qð0Þ ¼ 1:72 and maximum value ofqð1Þ ¼ 1:84 (a slightly steeper profile was resulting from theVMEC equilibrium code, used for MHD equilibria anddescribed as the reference for gyrokinetic code EUTERPE inthe original ITPA publication22). Flat temperature anddensity profiles are considered for bulk ions and electrons,with electron temperature Te¼ 1 eV, corresponding to avalue of q# ¼ qs=a ¼ 1=927; se ¼ 1, and electron pressurecorresponding to a value of be ¼ 9:1( 10)4. This choice ofthe plasma temperature and density is made in order to haveTAEs with frequency well approximated by ideal MHD (theeffects of plasma compressibility turn out to be negligible).Electrons 200 times lighter than ions are considered.

FIG. 2. Scan of the SAW continuum frequency for different values of theelectron beta and me=mi ¼ 1=200. Good stability properties are found, up tovalues of be ! 0:1.

FIG. 1. Frequency dependence on q, obtained for different simulations withdifferent values of q (flat q, density, and temperature profiles are adopted foreach simulations here), and with only the m¼ 1, n¼ 0 poloidal component.Here the cylinder limit (! ¼ 0:01) is considered. Convergence with the pre-diction of ideal MHD theory is found for electrons 200 to 2000 times smallerthan ions, for this value of q# ¼ 1=50 and be ¼ 2( 10)4.

FIG. 3. Frequency vs q for different simulations with flat q profiles, andwith only one poloidal component (m¼ 4 or m¼ 5) evolved in time. Herethe cylinder limit (! ¼ 0:01) is considered. A good match with the predictionof ideal MHD theory is found for electrons 2000 times smaller than ions, forthis value of q# ¼ 1=50 and be ¼ 2( 10)4.

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A distribution function Maxwellian in vk is consideredfor the EP population. The EP averaged concentration ishnEPi=ne ¼ 0:00307 with radial profile given by

nEPðsÞ=nEPðs0Þ ¼ exp½D jntanhððs) s0Þ=) DÞ* (20)

with s0 ¼ 0:5; D ¼ 0:2, and jn ¼ 3:333, like shown in Figs.4 and 5. In the simulations shown in this paper, due to thevery small amount of fast particles, the bulk ion and electronprofiles are not corrected to satisfy quasi-neutrality to ahigher level of accuracy.

Linear simulations with 107 markers per species havebeen run, with a space resolution of (ns,nchi,nphi)¼ (256,256,64), and a time step of dt¼ 20 X)1

i . A filter inmode numbers is applied, which keeps only n¼ 6 mode, andm¼ 9, 10, 11, 12, corresponding to khqs ’ 0.02. A radialdomain going from s¼ 0.1 to s¼ 0.9 is considered for theevolution of the potentials. No finite-Larmor-radius effectsare studied in this paper, meaning that the gyroaverageoperators are set to 1 for all species in the code, for the simu-lations described here.

B. TAE frequency and structure

We initialize a perturbation in the density of the iongyrocenters at t¼ 0, with n¼ 6 and m¼ 10 and 11, and witha radial structure calculated in order to give a perturbationof potential radially localized around s¼ 0.5 at t¼ 0. Theperturbed vector potential is measured at different radii as afunction of time, at a given poloidal angle. The frequency ofthe mode is found to depend on the EP concentration (seeFig. 6, where a comparison with the continuum X-point isshown, and Fig. 7). The limit for vanishing EP concentrationmatches well with the center of the TAE gap calculated withEq. (19).

The mode is found to be peaked radially at the positionof the center of the SAW continuum gap, s¼ 0.5, and haspoloidal structure of the vector potential with characteristicballooning features (i.e., mixed m and mþ 1 features) at thehigh-field side (see Fig. 8). At the high-field side, around thes¼ 0.5 flux surface, the mode shows a phase shift in h with

respect to the value at s¼ 0.5. This creates a “boomerang”shape, which will be studied in details for continuum modesin Sec. V.

C. TAE growth rate

Due to the EP radial density gradient, energy is pumpedfrom the EP thermal energy to the macroscopic SAW kineticenergy. This leads to an exponential growing of the modeamplitude. The dependence of the growth rate on the EPconcentration is found to be linear (see Fig. 9). The dampingis found to be very small for this tokamak configuration,confirming that we are deep into the MHD regime. The drivedependence on the EP temperature is also compared withresults of the hybrid MHD-gyrokinetic code HMGC shownin Ref. 22, where the drive is estimated by adding the meas-ured growth rate and the damping rate. The damping rate isevaluated, for each EP temperature, as in Fig. 9, by takingthe limit of nEP=ne ! 0. It is negligible in the simulations ofNEMORB. This comparison can be seen in Fig. 10. A verygood agreement is found, except at very high temperatures,where a difference is present between the results of the twocodes.

FIG. 4. Energetic particle density profile, like in Eq. (20), normalized involume.

FIG. 5. Logarithmic gradient of nEP normalized with the minor radius a.

FIG. 6. Dependence of the TAE frequency on the EP averagedconcentration.

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V. ENERGETIC-PARTICLE DRIVEN CONTINUUMMODES

A. Equilibrium and numerical setup

In this section, we want to study how EP drives Alfv!enmodes of the continuum unstable. We focus on modes whereno continuum damping occurs. This is done by driving con-tinuum modes centered radially where the gradient of thecontinuous spectrum vanishes. We consider two equilibria,one where the modes are excited at the continuum accumula-tion point (CAP) created by an inversion of the magneticshear, and another one where the continuum frequency isconstant in radius. In the presence of a reversed shear, AEscan exist which are usually referred to as reversed-shear-induced AE (RSAEs).27 Nevertheless, in our case, due to thesmall value of !, the mode frequency in the limit of zeroEP concentration tends to the CAP (and not to a discretefrequency lying in the continuum gap, far from the CAP);therefore, we refer to them as EP driven continuum modes(EPMs).2

We consider the same analytical equilibrium as inSection IV (same minor and major radius, concentric fluxsurfaces with no Shafranov shift), with same magnetic fieldintensity at the axis (B¼ 3 T), but with a different poloidalcomponent, yielding a different safety factor profile. Thesafety factor has a value of 1.85 at the axis, it decreases fromq¼ 0 to q¼ 0.5, where the minimum value is located(qðq ¼ 0:5Þ¼ 1.78), and then it raises to the edge, where itreaches the maximum value (qðq ¼ 1Þ¼ 2.6), like shown inFig. 11. Ion and electron density and temperature profiles areflat. We consider two regimes with ion temperature, respec-tively, Ti¼ 3.44 keV (corresponding to q# ¼ 1=500) andTi¼ 55.1 keV (corresponding to q# ¼ 1=125). The electronto ion temperature ratio is always kept at 1 throughout thispaper, se ¼ 1, and the electron pressure is chosen for a valueof be ¼ 5( 10)4.

A similar distribution function as in Sec. IV is consideredhere for the EP, Maxwellian in vk and with a radial densitygradient peaked around s¼ 0.5. The EP concentration has a

FIG. 7. Fourier transform in time of the vector potential for the case withnEP=ne¼ 0.003, TEP¼ 500 keV.

FIG. 8. Structure of the vector potential at t¼ 70000 X)1i .

FIG. 9. Dependence of the TAE growth rate on the EP average concentra-tion, for TEP¼ 500 keV.

FIG. 10. Comparison of the dependence of the EP drives on the EP tempera-ture, for NEMORB and HMGC. HMGC data are taken from Ref. 22.

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radial profile given by Eq. (20), with s0 ¼ 0:5; D ¼ 0:16, andjn ¼ 10. The EP temperature is kept fixed for all simulationsof this section, with TEP ¼ 5510 keV. No finite-Larmor-ra-dius effects are considered here (like in the previous section);i.e., the gyroaverage operators are set to their value ofk?qi;e;f ¼ 0 (for the bulk-ion, electron and fast species).Modes with n¼ 6, m¼ 11 and modes with n¼ 6, m¼ 10 areconsidered, corresponding to khqs ’ 0.04 for the case withTi¼ 3.44 keV and khqs ’ 0.16 for the case withTi¼ 55.1 keV. The characteristic number of markers for bulkions, electrons, and fast ions is 106. Electrons 2000 timeslighter than ions are chosen (necessary for a better conver-gence for this case with larger q# with respect to the one con-sidered in the previous section). The characteristic space andtime resolutions are (ns,nchi,nphi)¼ (256,256,64), dt¼ 5X)1

i .

B. EPM frequency and growth rate

For the reversed-shear equilibrium, shear Alfv!en modesare initialized and observed to grow at the CAP location(q ¼ 0:5), where the EP density gradient is peaked, with apoloidal mode number selected with a filter. Modes withm¼ 10 and modes with m¼ 11 are studied separately. Thesignal of Ak is measured at different radial locations, andthe Fourier transform in time is performed to calculate thefrequency for each radial location. The mode frequency iscompared with the continuous spectrum formula, Eq. (19),which is crucial to characterize the nature of the unstablemode. As an example, in Fig. 12 a mode with filter at m¼ 11is shown, and the frequency is found indeed near the m¼ 11continuum branch. Note that the eigenmode structure isobserved (the frequency is independent of the radial posi-tion). Note also that the frequency lies below the CAP, andnot at the CAP frequency, for this EP concentration.

The dependence of the mode frequency and growth rateon the EP concentration is shown in Figs. 13 and 14 for them¼ 11 and m¼ 10 cases. In the same figures, the value ofthe CAP frequency calculated with Eq. (19) is reported. Theinclusion of plasma compressibility effects can be calculated

as in Ref. 34 and gives an upshift of about 0.005 vA=R forthe m¼ 11 mode, and negligible for the m¼ 10 mode. Wecan see that, for both m¼ 10 and m¼ 11 modes, the effect ofEP in this regime is to decrease the frequency of the mode.In both cases, the mode frequency tends to the CAP in thelimit of zero EP concentration. A comparison with the con-tinuous spectrum topology (Fig. 12) tells us that the m¼ 11mode enters the continuum in the presence of EP, whereasthe m¼ 10 mode leaves the continuum and stays into thegap. This difference reflects in a difference in the modestructure, as described in Sec. V C.

The growth rate has also been measured for the m¼ 11and m¼ 10 modes, for different EP average concentrations(see Figs. 13 and 14). Except for very high values of growthrates, a linear dependence is found in the EP density. Theinstability threshold is observed to be lower for the m¼ 10mode, consistently with the lower damping. In a simulationwhere the filter allows both the m¼ 10 and m¼ 11 mode tocoexist, we observe that the most unstable one, namely, them¼ 10 mode, grows faster, and the structure has a clearm¼ 10 poloidal component. This is due to the fact that, eventhough the geometry has toroidal curvature, nevertheless theradial location of the mode is far from the TAE gap, andtherefore, the EP population can drive a mode with a domi-nant poloidal number without relevant coupling to the m 6 1sidebands.

C. EPM structure

The poloidal structure of Ak has also been studied forthe modes with m¼ 11 and m¼ 10. The poloidal mode num-ber is defined for both cases by means of a filter. The radialstructure is centered near q ¼ 0:5 (corresponding tos¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw=wð1Þ

p¼ 0:54), where the peak of the EP gradient is

located, chosen at the SAW CAP. The external boundarycondition is set here at s¼ 0.8, in order to isolate the dynam-ics of interest around s¼ 0.5, from TAEs getting unstable atouter radii for the case where the Landau damping is lower(i.e., for low values of q#).

FIG. 11. Safety factor profile for the reversed-shear equilibrium.

FIG. 12. Fourier transform in time (in red) compared with the theoreticalcontinuum (in blue), for Ti¼ 55.1 keV and nEP=ne ¼ 0:03, for the reversed-shear equilibrium.

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The mode is observed to rotate in the poloidal anglewith amplitude increasing exponentially in time and welldefined growth rate (described in Sec. V B). The propagationdirection of the fluctuation has been measured to be in theion diamagnetic direction. For the mode with m¼ 11, astrong phase shift is observed at lower and bigger radii withrespect to the signal measured at the CAP flux surface, whichis visualized as a tilt of the poloidal structure at both sides,creating a “boomerang” shape (see also, e.g., Refs. 28–33).The tilt angle at both inner and outer radii is not observed todepend on the EP concentration, which seems to affect onlythe radial extension of the mode (see Fig. 15). We also com-pare the structure of the m¼ 11 mode for two cases with dif-ferent bulk-ion temperatures. Even in this case, the geometryis found not to change in the two cases (see Fig. 16).Similarly, no evident difference in the poloidal structure isfound by switching on/off the bulk-ion FLR (i.e., by actually

calculating the correct value of the gyroaverage operators forthe bulk-ions, for their finite value of k?qi).

The poloidal mode structure for the mode with m¼ 11is found to be modified, on the contrary, by the continuousspectrum topology. This is investigated by repeating thesame case as in Fig. 16 (for both bulk-ion temperatures) butfor an equilibrium where the poloidal magnetic field is cho-sen such as to have a flat-q profile (i.e., zero shear), by keep-ing all other parameters unchanged. The result is shown inFig. 17, for a case with Ti¼ 3.44 keV and a case withTi¼ 55.1 keV. In this case with flat-q profile, no tilt is foundin the poloidal mode structure. The difference with respectof the reversed-shear case is that in the flat-q case, the bulkplasma nonuniformity (reflected in the continuum slopeaway from the CAP) is pushed down to zero, and the modeis not affected by continuum damping at any radial position.Consistently, these m¼ 11 modes are found to have a higher

FIG. 13. Frequency (left) and growthrate (right) of the m¼ 11 mode for dif-ferent EP concentrations.

FIG. 14. Frequency (left) and growthrate (right) of the m¼ 10 mode, fordifferent EP concentrations.

FIG. 15. Poloidal structure of Akfor the m¼ 11 mode, for the casewith reversed-shear q profile shown inFig. 11, Ti¼ 55.1 keV (q# ¼ 1=125).Energetic particles are characterizedby TEP¼ 5510 keV and, respectively,nEP=ne ¼ 0:02 (left) and nEP=ne ¼0:05 (right). Radial mode extension isfound to depend on the EP concentra-tion, but no difference is found in thepoloidal mode section geometry.

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growth rate (e.g., c ’ 2( 10)2vA=R for nEP=ne ¼ 0:03) withrespect to the ones observed in the reversed-shear equilib-rium. This is due to the absence of continuum damping, forthis case where the mode is not touching a sloped continuum,at any radial location.

In order to complete the study of the dependence of thestructure on the position with respect to the continuum, weinvestigate also the poloidal structure of the m¼ 10 mode,for nEP=ne ¼ 0:01 and nEP=ne ¼ 0:03 (see Fig. 18). Nostrong tilt is found in any case (the mode is in the gap),although a slight upward tilt can be observed for the casewith lower EP concentration (the mode is closer to theCAP). In summary, we conclude that the tilt of the EP drivenmodes in this regime strongly depends on the slope of thecontinuum (which depends on the safety factor only, in thepresent paper, but can vary with other factors like the plasmadensity as well, in general), for those modes which strongly

interact with the continuum, being absent at the locationwhere the continuum slope goes to zero or where there is nointeraction with the continuum (i.e., for those cases wherethe continuum damping does not affect the modes).

The dependence of the poloidal mode structure ofAlfv!en instabilities on the features of the EP population (ra-dial location of the density gradient, value of the density gra-dient, temperature, etc.) and the comparison with analyticaltheory (see, e.g., Ref. 33) is outside the scope of this paperand will be investigated in details in a dedicated paper.

VI. SUMMARY AND CONCLUSIONS

The importance of understanding the SAW dynamics ismainly linked to their role in the redistribution of the EPpopulation. This is crucial for the achievement of a good the-oretical model of the plasma stability and heating. The linear

FIG. 16. Poloidal structure of Ak forthe m¼ 11 mode, for the reversed-shear q profile, and nEP=ne ¼ 0:03,TEP¼ 5510 keV. On the left, forTi¼ 3.44 keV (q# ¼ 1=500) and on theright for Ti¼ 55.1 keV (q# ¼ 1=125).

FIG. 17. Poloidal structure of Ak forthe m¼ 11 mode, for a case with flat qprofile (q¼ 1.78) (with nEP=ne ¼ 0:03,TEP¼ 5510 keV). On the left, forTi¼ 3.44 keV (q# ¼ 1=500) and on theright for Ti¼ 55.1 keV (q# ¼ 1=125).

FIG. 18. Poloidal structure of Ak forthe m¼ 10 mode, for the case withreversed-shear q profile shown in Fig.11, with Ti¼ 55.1 keV, TEP¼ 5510 keV,and, respectively, nEP=ne ¼ 0:01 (left)and nEP=ne ¼ 0:03 (right). Like for them¼ 11 mode, EP concentration seemsto influence only the radial modeextension.

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and nonlinear interaction of SAW instabilities with EP andwith other modes (e.g., with zonal flows) make their investi-gation not trivial, for many space and time scales becomeinvolved and the nonlinearities and the driving and dissipa-tion mechanisms must be treated as rigorously as possible.To face such a complicated system, a robust theoretical toolis required with a set of model nonlinear equationsconstructed in such a way to conserve the basic symmetriesof the system (energy and momentum).

We adopt here the code ORB5 within the NEMORBproject, which has been previously used for turbulence simu-lations and for the study of electrostatic global instabilitiesdriven by EP. NEMORB’s model equations are derived in agyrokinetic Lagrangian formulation, where the discretizationis performed at the Lagrangian level, so that the Vlasov-Maxwell governing equations satisfy the same symmetryproperties of the starting discretized Lagrangian.

In this paper, we have presented the results of the firstinvestigation performed with NEMORB on the linear colli-sionless dynamics of SAW instabilities. First, the modelequations have been presented and solved analytically forradially localized modes, in the incompressible ideal MHDlimit. This gives the frequency of the SAW continuous spec-trum, which is the local oscillation frequency of SAW in atokamak. The continuous spectrum also provides theposition of energy absorption of global SAW modes via con-tinuum damping, and its topology is therefore important toknow when studying the existence of global SAWinstabilities.

As a first test of NEMORB on local SAW dynamics, wehave performed numerical simulations in simplified geome-try with negligible inverse aspect ratio, in order to recoverthe cylindrical limit. The frequency of the SAW oscillationhas been measured and compared with the theoretical predic-tion for axisymmetric and non-axisymmetric perturbations.No EP population has been loaded at this stage.

The dynamics of global SAW instabilities has also beeninvestigated, e.g., for toroidicity induced Alfv!en Eigenmodesdriven by an EP population. The frequency of TAEs withrespect to the theoretical continuous spectrum has beeninvestigated, and the growth rate dependence with respect tothe EP temperature has been studied. Results have also beencompared with those of the hybrid gyrokinetic-MHD codeHMGC, giving a good match.

Finally, we have investigated the dynamics of globalSAW instabilities centered in a region with no magneticshear. For this tokamak equilibrium configuration, the SAWfrequency has been verified to tend to the prediction for thecontinuum, in the limit of vanishing EP concentration, and todecrease for increasing EP concentrations. The dependenceof the spatial structure in the poloidal plane on the equilib-rium parameters has been investigated in details. A phaseshift in the poloidal angle h at different radii has beenobserved, giving a characteristic “boomerang” shape. The ra-dial size of the mode has been observed to depend on the EPconcentration. The shape has been observed to be directlylinked to the position of the mode frequency with respect tothe continuous spectrum: modes with frequency in the con-tinuum gap have no phase shift in h, i.e., no boomerang

shape, whereas modes entering the continuum for increasingEP concentration have a well defined boomerang shape. Thiscan be placed in a broader context according to which thephase shift is directly caused by the radial dependence thelocal mode frequency, dominated by SAW continuum radialdispersion when the mode intersects the continuum. Theinvestigation of the dependence of the spatial structure onthe EP distribution function is outside the scope of this paperand will be investigated with NEMORB in a dedicated paperand the results will be compared with analytical theory.

ACKNOWLEDGMENTS

This work has been carried out within the framework ofthe EUROfusion Consortium and has received funding fromthe Euratom research and training program 2014–2018 underGrant Agreement No. 633053. The views and opinionsexpressed herein do not necessarily reflect those of theEuropean Commission. Simulations were performed on theIFERC-CSC Helios supercomputer within the framework ofthe ORBFAST and VERIGYRO projects. This work hasbeen done in the framework of the nonlinear energeticparticle dynamics (NLED) European Enabling ResearchProject (EUROFUSION WP15-ER-01/ENEA-03) andEuropean Enabling Research Project on verification anddevelopment of new algorithms for gyrokinetic codes(EUROFUSION WP15-ER-01/IPP-01). Interesting discussionswith L. Chen, M. Cole, Z. Qiu, C. Di Troia, G. Vlad, and A.Zocco are gratefully acknowledged. Part of this work wasdone when one of the authors, A. Biancalani, was visiting IPPat Greifswald, IFTS at Hangzhou, and ENEA at Frascati,whose teams are gratefully acknowledged for the hospitality.This paper has been written in Rochefort (France).

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