Gyrokinetic studies of trapped electron mode turbulence in the HSX

stellaratorB. J. Faber,1, 2 M. J. Pueschel,2 J. H. E. Proll,3, 4 P. Xanthopoulos,3 P. W. Terry,2 C. C. Hegna,2, 5 G. M. Weir,1

K. M. Likin,1 and J. N. Talmadge11)HSX Plasma Lab, University of Wisconsin-Madison, Madison, WI 537062)Department of Physics, University of Wisconsin-Madison, Madison, WI 537063)Max Planck Institute for Plasma Physics, Wendelsteinstr. 1, 17491 Greifswald,Germany4)Max-Planck/Princeton Research Center for Plasma Physics, 17491 Greifswald,Germany5)Department of Engineering Physics, University of Wisconsin-Madison, Madison,WI 53706

Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented.Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed inthe ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electronmodes. For the most part, the heat transport exhibits properties associated with turbulence driven by thesetypes of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structuresnear the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels.Integrated heat fluxes are compatible with experimental observations in the corresponding density gradientrange. Despite low shearing rates, zonal flows are observed to regulate turbulence but can be overwhelmedat higher density gradients by the long-wavelength coherent structures.

I. INTRODUCTION

The development of quasi-symmetry1 has providedstellarators with a viable path to magnetic confinementfusion as an alternative to the tokamak. The HelicallySymmetric eXperiment (HSX) is a four period modularstellarator that possesses quasi-helical symmetry and hasbeen shown to reduce neoclassical transport levels to val-ues similar to those observed in tokamaks2. The residualtransport observed in HSX discharges is turbulent. Inthis work, we employ gyrokinetic simulation techniquesto study the properties of trapped electron mode turbu-lence in HSX geometry.The density-gradient-driven, collisionless trapped elec-

tron mode (TEM)3 has been shown to be a viablecandidate to explain the electron thermal fluxes ob-served in many tokamak experiments with peaked densityprofiles4,5 and is detrimental to achieving confinementin fusion plasmas. Fundamentally, the density-gradient-driven TEM exists for drift-wave frequencies ω < ωb,where ωb is the electron bounce frequency on a flux sur-face. It is destabilized by density perturbations of thetrapped particle population in the presence of bounce-averaged bad magnetic curvature. For tokamaks, thiscriterion is typically satisfied in a single poloidally local-ized region near the outboard midplane.Over the past decade, there has been significant work

and understanding gained in the gyrokinetic simulationof stellarator plasmas as it relates to the ion tempera-ture gradient (ITG) and electron temperature gradient(ETG) modes6–13 and subsequent turbulent transportoptimization14–19. However, only recently have compu-tational capabilities reached the point where TEMs instellarators can be simulated effectively. As such, anal-ysis of TEM-driven turbulence in stellarators is in the

early stages, with most prior work relating to the W7-Xstellarator12,20–23.The TEM destabilization condition is easily met in

HSX, as will be shown in Sec. II, due to the overlapof regions of bad curvature and magnetic wells, a prop-erty that generally occurs for quasi-symmetric stellaratorconfigurations8,24. One would then expect TEMs to belinearly unstable in HSX for some density gradient, whichhas been shown in the linear gyrokinetic growth rate cal-culations performed in Ref.25. The present work repre-sents the first examination of the nonlinear state of TEM-driven turbulence in HSX, and together with the workpublished in Ref.23 the first examination of TEM turbu-lence in neoclassical-transport-optimized stellarators.Previous work on the density-gradient-driven TEM in

tokamaks has shown that zonal flows are the primarynonlinear saturation mechanism26–28. Zonal flows canprovide a saturation mechanism through flow shear29,30

or by providing a route through which an instabilitycan couple and transfer energy to damped modes30–36.The nonlinear simulations of HSX presented in this pa-per show the development of zonal flows consistent withthe standard picture in tokamaks, and we show that thezonal flows are the nonlinear saturation mechanism.The paper is structured as follows: In Sec. II the de-

tails of HSX equilibria are introduced and important nu-merical considerations in applying gyrokinetics to HSXare discussed. In Sec. III, we present linear gyrokineticsimulation results, which show the existence of multi-ple, distinct ion-scale modes propagating in the electrondirection. The nonlinear simulation results in Sec. IVshow that the excitation of an ion-propagating directioncoherent structure not predicted by linear simulations isassociated with the development of zonal flows. Resultsare summarized and discussed in Sec. V.

2

II. NUMERICAL MODELING

We investigate trapped particle drift-waves throughthe gyrokinetic framework37,38. TheGene code39 is usedto solve the coupled gyrokinetic Vlasov-Maxwell systemof equations in both linear and nonlinear simulations.Simulations are performed in flux tube geometry40,41,where, by definition, the background equilibrium pres-sure and its gradients are constant. The GIST code42

is used to calculate the metric elements of the 3-D mag-netic field configuration of HSX in a flux tube, wherethe metric elements are now solely a function of the par-allel coordinate along the magnetic field. In a periodicmodular stellarator such as HSX, the flux tube is onlycomputed for one field period and quasi-periodic bound-ary conditions are applied in the parallel direction.In a tokamak, one flux tube is identical to all other flux

tubes on the same magnetic surface, and thus a single fluxtube can be used to represent a magnetic surface. This isnot the case in stellarators43, where different flux tubescan experience different curvatures and trapping regionsalong the field line, with resulting changes to drift-wavebehavior. The two representative flux tubes used in thiswork at the half-radius in the toroidal flux coordinateΨ/Ψ0 ≃ 0.5 (r/a ≃ 0.71) are identified by the shape ofthe plasma cross-section at zero poloidal angle. The nor-malized normal curvature and normalized magnetic fieldstrength of the flux tubes are given in Fig. 1, correspond-ing to a bean shaped cross-section, and Fig. 2, corre-sponding to a triangular cross-section. The flux tubeswill hereafter be identified as HSX-b and HSX-t, respec-tively, and are the up-down symmetry planes one-halffield period away from each other. As seen in Figs. 1,2, a feature of both flux tubes is the correlation of theregions of bad (negative) curvature and magnetic fieldminima (trapped particle regions), which is similar totokamak geometry. However, unlike a tokamak, the ma-jority of particles trapped in these wells are not toroidally,but rather helically trapped.The geometry of HSX is such that the averaged mag-

netic field shear is very small across the plasma radius. Influx tube simulations, periodicity at the parallel bound-ary requires41

Lx = N/(

|s|kminy ρs

)

(1)

where the background magnetic shear is defined as s =r0ι

dιdx

∣

∣

r0, where d/dx denotes the radial derivative of the

rotational transform profile ι and r0 is the minor radiusof the flux surface under investigation. Lx designatesthe radial box size, N is a positive integer and kmin

y ρsis the minimum binormal mode number simulated. Ifs becomes small, the box size in the radial direction canbecome quite large, in contrast to many tokamak applica-tions with s ∼ O(1). At the half-toroidal flux surface, thebackground magnetic shear is s = −0.045 and decreasesas r/a → 0, which leads to typical box sizes Lx & 200ρs,where ρs is the ion sound gyroradius. As the numericalresolution in the radial direction is set such that O(ρs)

-0.1

0

0.1

-1 -0.5 0 0.5 1

1.1

1.2

1.3

κ

|B|

z (π)

κ|B|

FIG. 1. Normalized normal curvature (red dotted line) andnormalized magnetic field strength (blue dashed line) in theparallel direction in HSX-b. Negative curvature indicates badcurvature.

-0.1

0

0.1

-1 -0.5 0 0.5 1

1.1

1.2

1.3

κ

|B|

z (π)

κ|B|

FIG. 2. Normalized normal curvature (red dotted line) andnormalized magnetic field strength (blue dashed line) in theparallel direction in HSX-t. Negative curvature indicates badcurvature

features are captured, the large radial box size providesa significant computational constraint for flux-tube sim-ulations.Two different TEM parameter sets will be investigated.

As HSX operates with primarily ECRH heating, the ionshave a flat temperature profile for a majority of the mi-nor radius. As such there will be no ion temperature gra-dient in the following simulations. The first parameterset will be called the “standard” TEM and was selectedto capture the essential physics of the density-gradient-driven TEM when both a density and electron temper-ature gradient are present but the density gradient islarger (ηe = dlnTe/dlnne < 1). It uses the normalizeddensity gradient a/Ln = 2 and normalized electron tem-perature gradient a/LTe = 1, where a is the averaged

minor radius, Lξ =1ξdξdx

is the inverse length scale for any

3

scalar ξ and Te0/Ti0 = 1 with mi/me = 1836, where Ts0,ms is the background temperature and mass of speciess. The standard TEM parameter set will be applied tosimulations in both flux tubes.The second parameter set was selected to examine the

sensitivity of the TEM to only the driving density gradi-ent. This parameter set uses four different density gradi-ent values, a/Ln = 1, 2, 3, 4 and no electron temperaturegradient. This parameter set will only be used in simula-tions done for HSX-b. The normalized electron pressureis β = 8πne0Te0/B

20 = 0.05%, where ne0 is the back-

ground electron density.

III. LINEAR INSTABILITY ANALYSIS

The linear growth rates, frequencies and mode struc-tures presented in this section have been calculated bytwo different methods. Initial value simulations are usedto obtain only the fastest growing mode at each kyρs.Eigenvalue simulations calculate both the dominant andsubdominant modes at each kyρs. The three-dimensionalnature of the magnetic geometry makes for a rich eigen-mode landscape characterized by many subdominant,but growing modes for each kyρs.The growth rates have been checked for convergence

with resolution, by performing a series of simulationswith increasingly fine resolution until the growth ratesshow no change. Two factors make these simulationssubstantially more expensive than tokamak simulations:(1) the magnetic field along the field line has more struc-ture, and as such, stellarator simulations may require be-tween 3 and 10 times as much resolution in the paralleldirection9,23; and (2) the low s allows linear modes at lowkyρs to become very extended along the field line and re-quire large numbers of radial connections to resolve, cor-responding to increasingly fine radial grids in real space.For linear simulations, we used a numerical grid size ofNz ×Nv‖ ×Nµ = 64× 36× 8 with 17 ≤ Nx ≤ 97, whereµ is the magnetic moment. The parallel hyper-diffusioncoefficient, see Ref.44, was set to ǫz = 4.

A. Linear dispersion relation

First we examine the linear behavior for both HSX-band HSX-t. The growth rates and real frequencies for thestandard TEM case in both flux tubes are presented inFig. 3. The magnitude of the maximum growth rates aredifferent (γ = 0.643 in HSX-b vs. γ = 0.443 in HSX-t),and the peak occurs at slightly different kyρs (kyρs = 2.1in HSX-b vs. kyρs = 1.6 in HSX-t). The difference ingrowth rate magnitudes can be explained by examiningFigs. 1 and 2. With z in Figs. 1, 2 as the parallel coor-dinate, one can see in Fig. 1 (HSX-b) a region centeredaround z = 0 corresponding to a magnetic field minimumand unfavorable curvature. The situation is reversed inHSX-t, where modes localized near z = 0 see favorable

0 0.1 0.2 0.3 0.4 0.5 0.6

γ (c

s/a

)

HSX-bHSX-t

-0.4

0

0.4

0.1 1.0

ω (

cs/a

)

kyρs

B1 B2 B3 B4T1 T2

FIG. 3. Linear growth rates (top) and real frequencies (bot-tom) in HSX flux tubes for canoncial TEM parameter set:a/Ln = 2, a/LTe = 1. The different dominant mode regionsare identified as the B modes for HSX-b and the T modes forHSX-t. The red dashed lines indicate the approximate bound-aries between different dominant B modes and the blue solidline is the approximate boundary between T modes. Neg-ative real frequencies indicate modes with electron-directiondrift frequencies.

curvature. The observed faster growth at high-kyρs inHSX-b is more strongly ballooning, i.e. its structure ismainly confined to kx = 0, whereas its HSX-t counter-part peaks at a finite |kx| = ±4, 6π.The peak growth rates in HSX are at larger

kyρs compared with density-gradient-driven TEM intokamaks26,28, where kyρs (γmax) . 0.7. They are com-parable to the kyρs of peak growth rates in W7-X23,where kyρs (γmax) ≈ 1.6. Applying a simple quasilinear

estimate for the heat flux, QQLky ∝ γk/ (kyρs)

2(assuming

kx = 0) suggests that scales at kyρs < 1, where bothHSX-b and HSX-t have similar linear growth rates, willlikely dominate transport. Thus despite there being asignificant difference in the maximum linear growth ratesbetween the two flux tubes, the nonlinear behavior of theflux tubes may be similar.The real frequencies show electron propagation direc-

tion modes for kyρs ≤ 2. This feature of the linear modesis robust against variations in β, where the low-kyρs lin-ear modes maintain electron propagation direction. Nokinetic ballooning modes are expected, as β = 0.05% iswell below marginal ideal MHD ballooning stability pointfor HSX45. The real frequencies in Fig. 3 show discon-tinuities in kyρs for kyρs < 1, which is indicative of thecoexistence of different modes. These modes will be clas-sified in Sec. III C. At the high-kyρs end of the spectrum,the real frequencies for both flux tubes cross the ωr = 0boundary shortly after the peak growth rate, switch-ing from electron to ion propagation direction. Thistype of mode has been observed in linear TEM simu-lations for a variety of configurations from stellarators23

to tokamaks26,28,46 and the reversed-field pinch47, and is

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

γ (c

s/a

)a/Ln = 1a/Ln = 2a/Ln = 3a/Ln = 4

-2-1.6-1.2-0.8-0.4

0 0.4

0.1 1.0

ω (

cs/a

)

kyρs

TEM1 TEM2 TTEM UTEM

FIG. 4. Linear growth rates (top) and real frequencies (bot-tom) in HSX-b for the density gradient scan: a/Ln = 1− 4,a/LTe = 0. Approximate mode boundaries are indicated bythe dashed vertical lines. The mode types are determined bythe analysis in Secs. III B and III C.

referred to as the “ubiquitous” mode48. Despite the highgrowth rates in these modes, the high kyρs modes con-tribute a relatively small amount to the overall transport,as is shown in Sec. IV, due to the quasilinear k−2

⊥ scaling.To isolate the role of the density gradient, we set the

electron temperature gradient to zero and only vary thedensity gradient. The growth rates and real frequenciesfor the density gradient scan in HSX-b are presented inFig. 4. Without an electron temperature gradient, thelinear growth rates and frequencies for the density gradi-ent scan are similar to that of the standard TEM, provid-ing evidence that the standard TEM is a density-gradientdriven TEM with little sensitivity to a/LTe, and is a gen-eral property of HSX plasmas, consistent with previouswork8,24,25. The high kyρs ubiquitous mode transitionsto a different mode at even higher kyρs as a/Ln is in-creased. The behavior of the different modes at low kyρsbecomes clearer, as the different mode boundaries aredelineated by discontinuities in the real frequencies. Ex-amination of the potential mode structures, shown forthe standard TEM parameters in Sec. III C, reveal thatthe dominant mode at each kyρs for each a/Ln are thesame as the modes at each kyρs for the standard TEM.At kyρs = 0.1, the growth rate for each a/Ln is small,which was also observed in the standard TEM simula-tions, indicating it may not be necessary to resolve lowerkyρs in nonlinear simulations.

B. Linear eigenmode structure

The eigenmodes in Figs. 3 and 4 are distinguished un-der the assumption that the real frequencies of a particu-lar mode branch should be a continuous function in kyρs.One can distinguish four different dominant linear modesin HSX-b but only two different dominant linear modes

-1-0.5

0 0.5

1

-40 -30 -20 -10 0 10 20 30 40

Φ

θ (π)

B1 kyρs = 0.1

Re(Φ)Im(Φ)

-1-0.5

0 0.5

1

-40 -30 -20 -10 0 10 20 30 40

Φ

θ (π)

B2 kyρs = 0.3

Re(Φ)Im(Φ)

-1-0.5

0 0.5

1

-10 -5 0 5 10

Φ

θ (π)

B3 kyρs = 0.8

Re(Φ)Im(Φ)

-1-0.5

0 0.5

1

-4 -3 -2 -1 0 1 2 3 4Φ

θ (π)

B4 kyρs = 1.5

Re(Φ)Im(Φ)

FIG. 5. The normalized real and imaginary parts of the ex-tended mode structure in the electrostatic potential Φ for thefour different dominant modes in Fig. 3 for HSX-b for stan-dard TEM parameters: a/Ln = 2, a/LTe = 1.

in HSX-t. The potential structure in the ballooning rep-resentation for the different mode branches of HSX-b andHSX-t are shown in Figs. 5 and 6, respectively, as a func-tion of the ballooning angle θ = z + 2πp, where p is aninteger49. For HSX-b, there is significant variation inthe structure between modes. At kyρs = 0.1, mode B1 inFig. 5 has a strongly extended envelope in θ with max-ima symmetrically located at roughly θ ∼ ±10π in theballooning angle. On top of the extended envelope, thereare spikes in the mode amplitude within each radial con-nection (every 2π in θ), which correspond to the localizedbad curvature at z = 0 in Fig. 1. These spikes are well-resolved. The broad envelope and narrow spikes indicatetwo disparate scales in the modes at low kyρs. The com-bination of low kyρs and small s leads to a slab-like modethat extends along the field line, where the larger scaleis related to the magnetic shear s and the small scale isrelated to the localization via bad curvature. This re-sult is consistent with what occurs in a tokamak, wherelower kyρs modes have broader ballooning structure50,particularly in the s ≪ 1 limit49,51.For 0.2 ≤ kyρs ≤ 0.6, corresponding to mode B2 of

Fig. 5, the mode still has an extended structure, but theenvelope peaks more narrowly at θ ≈ ±20π, which cor-responds to a finite-kx mode. Mode B3, dominant for0.7 ≤ kyρs ≤ 0.9, is considerably more localized but pos-

5

-1-0.5

0 0.5

1

-40 -30 -20 -10 0 10 20 30 40

Φ

θ (π)

T1a kyρs = 0.1

Re(Φ)Im(Φ)

-1-0.5

0 0.5

1

-40 -30 -20 -10 0 10 20 30 40

Φ

θ (π)

T1b kyρs = 0.3

Re(Φ)Im(Φ)

-1-0.5

0 0.5

1

-15 -10 -5 0 5 10 15

Φ

θ (π)

T2a kyρs = 0.8

Re(Φ)Im(Φ)

-1-0.5

0 0.5

1

-15 -10 -5 0 5 10 15

Φ

θ (π)

T2b kyρs = 1.5

Re(Φ)Im(Φ)

FIG. 6. The normalized real and imaginary parts of the ex-tended mode structure in the electrostatic potential Φ for thetwo different dominant modes in Fig. 3 in HSX-t for standardTEM parameters: a/Ln = 2, a/LTe = 1.

sesses tearing parity, Φ has odd parity, unlike the othermodes. The mode peaks in the first radial connection,±2π, indicating it is a finite-kx mode. For kyρs > 0.9,the ubiquitous mode is now highly localized to θ = 0.This mode is responsible for the maximum growth rateand the structure of the mode does not change as thereal frequency changes from electron to ion propagationdirection.Some of the modes in HSX-t show a markedly different

character than the modes at equivalent kyρs for HSX-b.The mode T1a at kyρs = 0.1 has two distinct, extendedlobes that peak at θ = ±9π. Unlike the mode B1 ofFig. 5, the amplitude in the central −π < θ < π range,corresponding to kx = 0, is small. The T2 modes ofFig. 6, dominant for kyρs ≥ 0.8 have the same characterand parity, with peak amplitudes on either side at θ =±5π. The T2 modes are identified with the ubiquitousmode for HSX-t and are distinct from the B3 mode dueto lack of tearing parity. There does not appear to be anequivalent mode with tearing parity in HSX-t.

C. Linear eigenmode classification

The sensitivity of these modes to the driving gradientswas examined by independently applying a 10% variation

TABLE I. Classification of the different linear modes in HSX-b and HSX-t based on ∆γ/γ0 change in growth rate for 10%increases in driving density and temperature gradient, stabi-lization due to β, negative (positive) sign denotes propagationin electron (ion) drift direction, and mode parity (ballooningor tearing). The acronyms are explained in the text.

Mode B1 B2 B3 B4 T1 T2a/Ln + 10% 0.5 0.076 0.044 0.058 0.068 0.076a/LTe + 10% 0.0 0.020 0.001 0.012 0.05 0.012β stab. No Yes Yes Yes Yes YesDrift Dir. − − − −/+ − −/+Parity Ball. Ball. Tear. Ball. Ball. Ball.Name TEM1 TEM2 TTEM UTEM TEM UTEM

to the density and temperature gradients of the standardTEM parameter set. For each of the different modes inFigs. 3 and 4, the growth rates have a positive corre-lation with increasing density and electron temperaturegradient, but consistently show a stronger dependenceon the density gradient. Increases in β generally resultin moderately decreasing growth rates, hence partial sta-bilization, for all modes except B1.With all of these considerations in mind, we have clas-

sified the B modes in HSX-b and the T modes in HSX-taccording to Table I. We will call the B1 mode TEM1and mode B2 TEM2. The B3 modes are called “tearing-parity” TEM (TTEM) and behave much like the ubiq-uitous mode B4. Finally, the B4 modes are ubiquitousmodes (UTEM). HSX-t shows only two distinct modes,with the T1 mode being a more traditional TEM andthe T2 mode the ubiquitous mode (UTEM) for that fluxtube.To examine the behavior of the dominant eigenmodes

in Figs. 3 and 4, we computed the first five most unstablelinear eigenmodes for the standard TEM at every kyρs.The results are shown in Fig. 7 with the five eigenvaluesdenoted as EV#1 to EV#5. An aspect of the eigenvaluedecomposition that is pervasive in HSX simulations andmarkedly different from tokamak results is the clusteringof linear eigenmodes. The first four most subdominantmodes in the range kyρs ≤ 0.6 all have growth ratesthat are only slightly smaller than that of the dominantmode and very similar, real frequencies. As such, theprecise ordering of the four subdominant modes was notinvestigated for kyρs ≤ 0.6.Fig. 7 shows that the TTEM is the dominant mode for

0.7 ≤ kyρs ≤ 0.9 (the red �) but becomes subdominantto the ubiquitous mode for kyρs ≥ 1. At no point in therange 0.7 ≤ kyρs ≤ 2.1 does the TTEM cease to existand all the subdominant modes at high kyρs exhibit thesame real frequency behavior as the ubiquitous mode.One feature of Fig. 7 is the pairing of two subdominant,unstable modes which are almost identical in both growthrate and frequency, but not mode structure. For example,in Fig. 7, for 1 ≤ kyρs, EV#2 has tearing parity (it isthe TTEM) and EV#3 has ballooning parity.This analysis shows that HSX is unstable to TEM tur-

6

0 0.1 0.2 0.3 0.4 0.5 0.6

γ (c

s/a

)EV #1EV #2EV #3EV #4EV #5

-0.6

-0.4

-0.2

0

0.2

0.2 0.5 1 2

ω (

cs/a

)

kyρs

TEM TTEM UTEM

FIG. 7. First 5 largest eigenmodes for the standard TEMparameter set in HSX-b. The eigenvalues at each kyρs are or-dered from largest to smallest linear growth rate. Continuityin real frequencies (bottom) shows the existence of at least 5distinct mode branches for kyρs ≥ 0.7. The different dom-inant modes (red �) identified by the designations in TableI.

bulence and there is a variety of linear modes that existin the transport-relevant regime in HSX. HSX-b exhibitsmultiple TEMs in the range kyρs ≤ 1 while HSX-t ex-hibits only two different modes. The linear growth ratescalculated in each flux tube are similar in this range de-spite differences in mode structure. A quasilinear esti-mate predicts that despite the large growth rates of theubiquitous mode, it will not contribute significantly totransport. The existence of a significant number of multi-ple, subdominant, unstable eigenmodes at each kyρs addscomplexity to subsequent nonlinear analysis as multiplemodes at the same kyρs can contribute to the turbu-lence. As a consequence, energy input to the nonlinearstate may be significantly greater than what is inferredfrom the dominant mode alone.

IV. NONLINEAR SIMULATIONS OF TURBULENCE IN

HSX

Using the insight gained from linear simulations, non-linear simulations were performed on the standard TEMand density gradient scan parameter sets. The defaultnonlinear resolution settings use Nky = 48 modes andkminy ρs = 0.1, which sets the radial box size according to

Eq. (1), where N = 1 was chosen to yield the smallestradial box size possible. The value of kmin

y ρs was checked

for convergence by halving kminy ρs and ensuring that the

saturated flux does not change. With kminy ρs = 0.1, the

radial box size is Lx = 222ρs and the radial resolution isset atNx = 192, but as we will show a posteriori, the non-linear structures are considerably smaller than the radialbox size. Doubling the Nx resolution does not change sat-urated flux values. The linear resolutions Nz ×Nv‖ ×Nµ

0

1

2

3

4

5

6

7

8

0 50 100 150 200 250 300 350

⟨Qe

s⟩ (

cs n

e T

e (

ρ s∗ )2 )

t (a/cs)

HSX-b, kymin = 0.1

HSX-t, kymin = 0.1

HSX-b, kymin = 0.05

FIG. 8. Time trace of the electron heat flux in HSX-bean(solid red) and HSX-triangle (dashed blue) flux tubes for stan-dard TEM parameters: a/Ln = 2, a/LTe = 1. Also shownis the time trace of the heat flux in a simulation for HSX-b where kmin

y ρs was halved (dotted green), showing that thesimulations are converged.

= 64× 36× 8 also result in nonlinearly converged fluxes.Quantities of interest, such as the heat flux, are deter-

mined, unless otherwise noted, by time averaging over thequasi-stationary state. All heat fluxes reported here havegyro-Bohm normalizaion, QgB = csne0Te0 (ρ

∗s)

2, where

ρ∗s = ρs/a with a the average minor radius, ne0 is thebackground electron density and Te0 is the backgroundelectron temperature. The fluctuating electrostatic po-tential Φ and electron density ne are reported as normal-ized values: Φ in terms of (Te0/e)ρ

∗s and n in terms of

ne0ρ∗s.

A. Standard TEM

The time traces of the electron heat flux for both HSX-b and HSX-t are shown in Fig. 8. From the time traces,it is apparent that the quasi-stationary electrostatic elec-tron heat fluxes in the two flux tubes are very similar,with 〈Qes〉 = 1.75 for HSX-b and 〈Qes〉 = 1.59 for HSX-t. This result is consistent with the arguments made inSec. III A, where the quasilinear estimate predicts thatthe kyρs < 1 modes, where both flux tubes have similargrowth rates, dominate the transport.The electron heat fluxes in Fig. 8 have been decom-

posed into spectral components in kyρs in Fig. 9, high-lighting some moderate differences between the flux tubeswhile confirming the relevance of the kyρs ≤ 1 region.The heat flux spectrum for HSX-t has only one peak atkyρs =0.5-0.6 and exhibits a quick decay toward highkyρs. The spectrum in HSX-b is similar, but the peakheat flux occurs at slightly higher kyρs = 0.7-0.8. Theimportant difference between the flux tubes, however, isthe presence of significant flux at kyρs = 0.1 for HSX-b,which is not predicted by the linear growth rate results

7

0

0.04

0.08

0.12

0.16

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

⟨Qe

s⟩ (

cs n

e T

e (

ρ s∗ )2 )

kyρs

HSX-b, kymin = 0.1

HSX-t, kymin = 0.1

2*HSX-b, kymin = 0.05

FIG. 9. Heat flux spectrum in HSX-b with kminy ρs = 0.1

(red solid), kminy ρs = 0.05 (green dotted) and HSX-t with

kminy ρs = 0.1 (blue dashed) for standard TEM parameters:

a/Ln = 2, a/LTe = 1. The simulation with kminy ρs = 0.05

has been scaled by a factor of 2 to illustrate more clearly itsnear-identical integrated flux level.

FIG. 10. Nonlinear frequencies ω in HSX-b for standard TEMparameters. The frequencies are obtained by performing aFourier transform in time over the quasi-stationary state. Thecolor scale is linear and independent for each kyρs. Linearfrequencies from Fig. 3 are overlaid with the white and blackdash line.

of Sec. III A.In a numerically resolved simulation, it is desired to

have only a small amount of flux in the kminy ρs wavenum-

ber of the simulation. Fig. 8 shows that HSX-b exhibitsno change in the integrated flux when kmin

y ρs is loweredto kyρs = 0.05. Fig. 9 demonstrates there is little flux inthe kyρs = 0.05 wavenumber but a peak in the heat fluxspectrum remains at kyρs = 0.1 (and a second peak atkyρs = 0.2). This indicates that the overall flux in thislow-kyρs feature is captured in either case. As such, ev-ery result hereafter that we will show uses kmin

y ρs = 0.1.The nonlinear real frequencies vs. kyρs are shown in

Fig. 10 for HSX-b. The linear frequencies have been over-laid and the magnitude of the nonlinear frequencies athigher kyρs are roughly a factor of 3 larger than the lin-

FIG. 11. Contours of Φ (normalized by Te0ρ∗s/e) and ne (nor-

malized by ne0ρ∗s) fluctuations at zero poloidal angle in (a)

HSX-b and (b) HSX-t. The x direction is radial and the ydirection is poloidal. Zonal flows are present in both fluxtubes, however a coherent structure centered at x = 0 is onlyobserved in HSX-b.

ear frequencies. The modes corresponding to dominanttransport peak (kyρs ≥ 0.5) have negative frequency,which indicates they are TEM. The difference in magni-tude between the linear and nonlinear frequencies couldbe the result of either subdominant modes with higherfrequencies or of three wave coupling between differentunstable modes. Without any ion temperature gradient,no ITG modes are present in these simulations, however,the modes at low kyρs, near to the low-kyρs transportpeak, all propagate in the ion direction. It is importantto reiterate that no linear subdominant mode with posi-tive frequency was found at these kyρs.The contours of the electrostatic potential and electron

density for HSX-b and HSX-t are shown in Figs. 11a and11b, respectively. These contours are a representativesnapshot taken at the final time step of the simulationand are not averaged over the saturated phase. Thereis zonal flow activity in both flux tubes, as evidencedby the well-defined, alternating vertical bands of positiveand negative potential. Zonal density structures are rel-atively weak in the present case, indicating that they donot play an important role in nonlinear saturation, con-sistent with corresponding density-gradient-driven toka-mak results27,28 . The density fluctuations do show thatthe turbulence has a somewhat different character be-tween the two flux tubes. The density fluctuations inHSX-b (Fig. 11a) away from x = 0 are anisotropic in the

8

x-y plane with radial elongation not exceeding the zonalflow width, which is consistent with one requirement foreddy shearing by the zonal flows29,30. The HSX-triangledensity fluctuations (Fig. 11b) are more isotropic but arestill appear sheared along the zonal flow boundaries.The appearance of the zonal flow corresponds with the

turnover of the heat flux at t ≈ 15a/cs in Fig. 8 andthe development of the quasi-stationary state. This in-dicates that the zonal flows may play a role as the non-linear saturation mechanism in these simulations, how-ever the time-averaged, fluctuation-driven E × B shear-ing rate ωE =

⟨

d2Φky=0/dx2⟩

≈ 0.35 in both flux tubesand is on the order of the respective linear growth rates.This shearing rate has not been corrected for the finite-frequency effects of Ref.52. Typically, uncorrected valuesof ωE as defined are consistent with the rule of thumb es-tablished in Refs.10,53, where ωE ≃ 10γmax should be sat-isfied if the zonal flows are to be an important nonlinearsaturation mechanism. A different definition of the shear-ing rate was used in Refs.28,54, where time scales shorterthan an eddy lifetime and spatial scales smaller than theradial correlation length were filtered from fluctuatingzonal potential, as a result of which their subsequentshearing rate has to be compared directly to the lineargrowth rate, without the application of finite-frequencycorrections or a factor of ≃ 10. To confirm equivalence ofboth approaches, our present formalism was applied tothe data of Ref.54, and our definition was found to yielda shearing rate value about eight times as large as theother definition, consistent with the aforementioned ruleof thumb.For the present case, the values of ωE reported here

indicate that the zonal flows may not play as big a roleas a saturation mechanism. A simulation performed withthe zonal flows artificially removed showed that the TEMturbulence still saturates in the absence of the zonal flowsbut with a quasi-stationary flux level 2.7 times larger.This result confirms that zonal flows are important tothe saturation of the TEM in HSX; this is consistentwith previous results of density-gradient driven TEMs intokamaks27,53, where the absence of zonal flows leads tolarger transport. As a consequence, it is concluded thatthe use of the shearing rate to determine saturation viazonal flows does not apply in the present case, perhapsdue to low magnetic shear or the specifics of the satura-tion.In the contours of HSX-b in Fig. 11, there is a large

scale coherent structure at x = 0 in the fluctuating po-tential and density that is not present in HSX-t. Thiscoherent structure is not static and drifts in the -y (ion)direction. Filtering out the 0.1 ≤ kyρs ≤ 0.3 componentsin Fig. 11 removes this structure and reveals an under-lying zonal flow smaller in amplitude than the coherentstructure and similar in amplitude to the zonal flows atother radial positions in the box. The large flux seen inthe flux spectrum of Fig. 9 at kyρs ≈ 0.1 is a product ofthis coherent structure.The coherent structure observed in Fig. 11a shows a

0

0.04

0.08

0.12

0.16

0.2

0 0.5 1 1.5 2 2.5 3

⟨Qe

s⟩ (

cs n

e T

e (

ρ s∗ )2 )

kyρs

HSX-b, β = 5 x 10-4

HSX-b, β = 5 x 10-5

FIG. 12. kyρs resolved electron heat flux spectrum for stan-dard TEM parameters in HSX-b for β = 5× 10−4 (solid red)and β = 5×10−5 (dashed blue). While these values appear tobe rather low, when normalizing to the ballooning threshold,they become more sizable.

significant dependence on several parameters: the pres-ence of zonal flows, β, and the background magneticshear. In the zonal-flow-free simulation, the coherentstructure was also absent, suggesting that the evolutionof the coherent structure may be moderated by the zonalflow. Fig. 12 shows that the low-kyρs peak in transportis eliminated when β is changed from β = 5 × 10−4 toβ = 5×10−5. The time trace (not shown) of heat flux hasslightly higher quasi-stationary flux levels for the lower-βcase, consistent with the linear result where growth ratesthroughout most of the kyρs range are slightly stabilizedwith increasing β. Simulations performed with an artifi-cially increased background magnetic shear (not shown)have reduced the low-kyρs flux and lack a nonlinear co-herent structure at x = 0, while the integrated flux levelsare not affected as much.The cross phases of Φ with ne, Te‖, and Te⊥, shown

in Fig. 13, show a clear difference between linear andnonlinear simulations at low kyρs in HSX-b. In all threephase relations, the potential is out of phase with theother quantities at kyρs ≤ 0.2, corresponding to the co-herent structure. As kyρs increases, the linear and non-linear phases agree reasonably well for kyρs > 0.3. TheΦ vs. ne and Φ vs. Te⊥ cross phases, Figs. 13a,c, fol-low the same trend as the trapped-particle population inRef.55 for temperature-gradient-driven turbulence, wherethe phase angle increases with increasing kyρs. This in-dicates that despite the linear-nonlinear frequency mis-match, the turbulence is of TEM type. Next, this inter-pretation shall be confirmed by comparing dependenciesin the driving density gradient.

9

FIG. 13. Nonlinear cross phases between Φ and ne, Φ andTe‖, and Φ and Te⊥ for the standard parameters in HSX-b. Overlaid are the linear cross phases (black line) for eachquantity. The color scale for the nonlinear cross phases islogarithmic. The linear and nonlinear phases differ for kyρs ≤

0.2 where the coherent structure lies, but generally agree forkyρs > 0.3, where linearly the TEM is dominant.

B. Impact of the density gradient

The nonlinear electrostatic electron heat fluxes for thedensity gradient scan parameters, where a/LTe = 0, arepresented in Fig. 14. These values are on the order of theobserved experimental fluxes obtained from calculationsof the power deposition profile56, where 3 . a/Ln . 4and 2 . 〈Qexp

es 〉 /QgB . 7 in the region 0.4 ≤ r/a ≤ 0.6.This suggests that TEM turbulence is responsible for theobserved heat fluxes in HSX.The scaling of the nonlinear saturated heat flux with

-2 0 2 4 6 8

10 12 14 16

0 50 100 150 200 250 300 350

⟨Qe

s⟩ (

cs n

e T

e (

ρ s∗ )2 )

t (a/cs)

a/Ln = 1a/Ln = 2a/Ln = 3a/Ln = 4

FIG. 14. Time trace of the electron electrostatic heat fluxes inHSX-bean for density gradient scan: a/Ln = 1−4, a/LTe = 0.Larger variability in the flux corresponds with the growth ofthe central coherent structure, much like in Fig. 11a.

0

1

2

3

4

5

6

7

0 1 2 3 4 5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

⟨Qe

s⟩ (

cs n

e T

e (

ρ s∗ )2 )

γ (cs /a)

a/Ln

γ⟨Qes⟩

FIG. 15. Density gradient scaling of the linear growth rates atkyρs = 0.7 (red �) and nonlinear electron heat flux (blue ⊙)in HSX-b. A square root fit predicts a linear critical gradientat a/Ln ≈ 0.2. A linear fit of the a/Ln ≤ 3 range of thenonlinear fluxes produces an upshift with a nonlinear criticalgradient predicted at a/Ln ≈ 0.8.

driving gradient is shown in Fig. 15 and compared withthe linear growth rates at kyρs = 0.7, approximatelywhere the nonlinear heat flux spectrum peaks in Fig. 16.A linear fit has been applied to the nonlinear flux datafor a/Ln ≤ 3. An upshift in critical density gradientis observed for the nonlinear saturated fluxes, consis-tent with the TEM nonlinear upshift discovered in toka-mak simulations26, with a critical density gradient ata/Ln ≈ 0.8 as compared with a critical density gradi-ent at a/Ln ≈ 0.2 predicted by the linear growth ratescaling. The fact that there is an upshift again providesevidence that zonal flows play a role in moderating theturbulence28.The electron heat flux spectrum presented in Fig. 16

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2

⟨Qe

s⟩ (

cs n

e T

e (

ρ s∗ )2 )

kyρs

a/Ln = 1a/Ln = 2a/Ln = 3a/Ln = 4

FIG. 16. Electron electrostatic heat flux spectrum for HSX-b for the density gradient scan: a/Ln = 1 − 4, a/LTe = 0.The flux at low-kyρs increases steadily with increasing a/Ln,however the TEM peak at kyρs = 0.7-0.8 still determines theoverall flux level.

shows that the fraction of the flux contributed by the low-kyρs wavenumbers increases with driving density gradi-ent. At a/Ln = 4, the fraction of flux for kyρs ≤ 0.3is approximately 22% of total integrated flux. Again,the nonlinear frequencies for the low-kyρs modes are ionfrequencies, consistent with Fig. 10.The increase in a/Ln leads to an increase both in the

variability of the heat flux and in the transport in thelowest-kyρs wavenumbers, the cause of which can be de-termined by examining the contours of electrostatic po-tential, shown for all four a/Ln in Fig. 17. The poten-tial structures in Figs. 17a and 17b are similar to bothFig. 11a and Fig. 11b for standard TEM parameters,showing clear zonal flows. However, there is no coher-ent structure observed for a/Ln = 1 and subsequentlyno low-kyρs transport peak in Fig. 16. For a/Ln = 2(Fig. 17b), the coherent structure is located near theradial boundary of the box and is responsible for thelow-kyρs transport peak in Fig. 16. For a/Ln = 3 anda/Ln = 4, Figs. 17c and 17d show a large amplitude po-tential structure centered at x = 0 that drifts in the iondirection and locally dominates the zonal flows.Decreasing kmin

y ρs to 0.05 (equivalent to doubling thebox size in the y direction) for the a/Ln = 4 simulationshows the coherent structure at x = 0 retains a similarscale as in Fig. 17d and that the quasi-stationary fluxlevel is similar to the kmin

y ρs = 0.1 simulation, indicatingthat the a/Ln = 4 simulations presented here are suf-ficiently resolved to capture the turbulent dynamics atlow kyρs. The bursts and fluctuations seen in Fig. 14 fora/Ln = 4 are also present in the higher resolution simu-lation due to the large size and amplitude of the coherentstructure at x = 0.The same observations about the zonal flow as the sat-

uration mechanism as for the standard TEM parametersapply here. For a/Ln ≤ 3, the shearing rate ωE is al-

FIG. 17. Contours of Φ fluctuations at zero poloidal anglein HSX-bean tube with a/LTe = 0 in all simulations. Zonalflows are present in all simulations and the coherent structuredevelops as a/Ln is increased.

ways smaller than the maximum linear growth rate andonly slightly larger than the maximum linear growth ratefor a/Ln = 4. Therefore, zonal-flow-based turbulent sat-uration may rely on energy transfer to damped modesrather than shearing. The associated density fluctua-tions to Fig. 17 are comparable to the potential fluctu-ations but zonal density as a saturation mechanism isagain precluded by the absence of zonal structures in thedensity.

V. CONCLUSIONS

We have performed an in-depth gyrokinetic study ofdensity-gradient-driven TEM turbulence in the HSX stel-larator. Linear simulations show that HSX profiles areunstable to TEMs across a large kyρs range, with theHSX flux tube corresponding to the “bean” shape havinglarger growth rates than the HSX flux tube correspond-ing to the triangular cross section. In the investigated

11

range 0.1 ≤ kyρs ≤ 6 we have found four distinct domi-nant TEM-type modes in HSX-b and two in HSX-t. TheTEMs at low kyρs have a two-scale ballooning structure,with an extended envelope whose width is influenced bythe low shear and localized spikes in amplitude due to lo-calized regions of overlapping bad curvature and particletrapping. At higher kyρs, the modes become substan-tially more localized, and an analysis of the five mostunstable modes for each kyρs in HSX-b shows that thereis a distinct tearing parity TEM that transitions fromdominant to subdominant as kyρs exceeds 0.9. HSX-tdoes not exhibit a dominant tearing parity TEM at anypoint.Nonlinear TEM simulations show that the two flux

tubes have similar quasi-stationary flux levels, in agree-ment with quasilinear estimates based on the similarity ofgrowths for kyρs < 1. The kyρs flux spectra between thetwo flux tubes only differ in that there is a peak in the fluxat kyρs = 0.1 in HSX-b, which does not occur for HSX-t and is not predicted by a quasilinear estimate. Thislow-kyρs transport peak is the consequence of a robustcoherent structure in HSX-b, which is further destabi-lized by a/Ln and β and can be eliminated by increasingthe background magnetic shear. The characteristic fre-quencies and phase relations associated with the coherentstructure have no equivalents in any of the performed lin-ear calculations. Examination of the potential contoursshows that the nonlinear coherent structures are radiallylocalized to the resonant magnetic surface at the centerof the simulation domain.Zonal flows were present in all nonlinear simula-

tions, consistent with the tokamak-based expectations fordensity-gradient-driven TEM. Artificially removing thezonal flows confirms that the TEM is saturated by zonalflow interactions, however, the E×B shearing rates aremuch smaller than what one would expect in turbulencethat saturates through zonal flow shearing. Instead, en-ergy transfer between different modes and moderated bythe zonal flows may be responsible for nonlinear satura-tion. This, however, has not been confirmed and will beleft for future investigation.Simulated turbulent heat fluxes are comparable with

experimental observations. While additional study alongthese lines will be necessary – e.g., investigating the im-pact of non-unity temperature ratio – it is concluded thatdensity-gradient-driven TEM microturbulence is an ex-cellent candidate in explaining heat fluxes in the HSXstellarator.

ACKNOWLEDGMENTS

The authors would like to acknowledge H. E. Mynickand D. R. Ernst for useful discussions. This work wassupported by US Department of Energy Grant No. DE-FG02-93ER54222 and No. DE-SC0006103. This researchused resources of the National Energy Research ScientificComputing Center, a DOE Office of Science User Facility

supported by the Office of Science of the U.S. Departmentof Energy under Contract No. DE-AC02-05CH11231.The authors are extremely grateful to D. R. Mikkelsenand F. Jenko for generously providing computing re-sources.

1J. Nuhrenberg and R. Zille, Phys. Lett. A 129, 113 (1988).2J. M. Canik, D. T. Anderson, F. S. B. Anderson, K. M. Likin,J. N. Talmadge, and K. Zhai, Phys. Rev. Lett. 98, 085002 (2007).

3B. B. Kadomtsev and O. P. Pogutse, Nucl. Fusion 67, 67 (1971).4P. C. Liewer, Nucl. Fusion 25, 543 (1985).5D. R. Ernst, K. H. Burrell, W. Guttenfelder, T. L. Rhodes,L. Schmitz, A. M. Dimits, E. J. Doyle, B. A. Grierson, M. J.Greenwald, C. Holland, A. Marinoni, G. R. McKee, R. Perkins,C. C. Petty, J. C. Rost, D. Truong, G. Wang, L. Zeng, andDIII-D and Alcator C-Mod Teams, in 2014 IAEA Fusion En-

ergy Conf. St. Petersburg, Russ. (2014) available as MIT PSFCJA-14-27.

6F. Jenko and W. Dorland, Phys. Rev. Lett. 89, 225001 (2002).7F. Jenko and a. Kendl, Phys. Plasmas 9, 4103 (2002).8G. Rewoldt, L. Ku, and W. M. Tang, Phys. Plasmas 12, 102512(2005).

9P. Xanthopoulos and F. Jenko, Phys. Plasmas 14, 042501 (2007).10P. Xanthopoulos, F. Merz, T. Gorler, and F. Jenko, Phys. Rev.Lett. 99, 035002 (2007).

11J. A. Baumgaertel, E. A. Belli, W. Dorland, W. Guttenfelder,G. W. Hammett, D. R. Mikkelsen, G. Rewoldt, W. M. Tang,and P. Xanthopoulos, Phys. Plasmas 18, 122301 (2011).

12J. A. Baumgaertel, G. W. Hammett, D. R. Mikkelsen,M. Nunami, and P. Xanthopoulos, Phys. Plasmas 19, 122306(2012).

13G. G. Plunk, P. Helander, P. Xanthopoulos, and J. W. Connor,Phys. Plasmas 21, 032112 (2014).

14H. E. Mynick, Phys. Plasmas 13, 058102 (2006).15H. E. Mynick, P. Xanthopoulos, and A. H. Boozer, Phys. Plas-mas 16, 110702 (2009).

16H. E. Mynick, N. Pomphrey, and P. Xanthopoulos, Phys. Rev.Lett. 105, 095004 (2010).

17H. E. Mynick, N. Pomphrey, and P. Xanthopoulos, Phys. Plas-mas 18, 056101 (2011).

18H. E. Mynick, P. Xanthopoulos, B. J. Faber, M. Lucia, M. Rorvig,and J. N. Talmadge, Plasma Phys. Control. Fusion 56, 094001(2014).

19P. Xanthopoulos, H. E. Mynick, P. Helander, Y. Turkin, G. G.Plunk, F. Jenko, T. Gorler, D. Told, T. Bird, and J. H. E. Proll,Phys. Rev. Lett. 113, 155001 (2014).

20J. H. E. Proll, P. Helander, J. W. Connor, and G. G. Plunk,Phys. Rev. Lett. 108, 245002 (2012).

21J. H. E. Proll, P. Helander, P. Xanthopoulos, and J. W. Connor,J. Phys. Conf. Ser. 401, 012021 (2012).

22P. Helander, J. H. E. Proll, and G. G. Plunk, Phys. Plasmas 20,122505 (2013).

23J. H. E. Proll, P. Xanthopoulos, and P. Helander, Phys. Plasmas20, 122506 (2013).

24T. Rafiq and C. C. Hegna, Phys. Plasmas 13, 062501 (2006).25W. Guttenfelder, J. Lore, D. T. Anderson, F. S. B. Anderson,J. M. Canik, W. Dorland, K. M. Likin, and J. N. Talmadge,Phys. Rev. Lett. 101, 215002 (2008).

26D. R. Ernst, P. T. Bonoli, P. J. Catto, W. Dorland, C. L. Fiore,R. S. Granetz, M. Greenwald, A. E. Hubbard, M. Porkolab, M. H.Redi, J. E. Rice, and K. Zhurovich, Phys. Plasmas 11, 2637(2004).

27J. Lang, S. E. Parker, and Y. Chen, Phys. Plasmas 15, 055907(2008).

28D. R. Ernst, J. Lang, W. M. Nevins, M. Hoffman, Y. Chen,W. Dorland, and S. Parker, Phys. Plasmas 16, 055906 (2009).

29P. Terry, Rev. Mod. Phys. 72, 109 (2000).30P. H. Diamond, S.-I. Itoh, K. Itoh, and T. S. Hahm, PlasmaPhys. Control. Fusion 47, R35 (2005).

12

31R. Gatto, P. W. Terry, and D. A. Baver, Phys. Plasmas 13,022306 (2006).

32P. W. Terry, D. A. Baver, and S. Gupta, Phys. Plasmas 13,022307 (2006).

33D. R. Hatch, P. W. Terry, W. M. Nevins, and W. Dorland, Phys.Plasmas 16, 022311 (2009).

34D. R. Hatch, P. W. Terry, F. Jenko, F. Merz, M. J. Pueschel,W. M. Nevins, and E. Wang, Phys. Plasmas 18, 055706 (2011).

35D. R. Hatch, P. W. Terry, F. Jenko, F. Merz, and W. M. Nevins,Phys. Rev. Lett. 106, 115003 (2011).

36K. D. Makwana, P. W. Terry, and J.-H. Kim, Phys. Plasmas 19,062310 (2012).

37P. H. Rutherford and E. A. Frieman, Phys. Fluids 11, 569 (1968).38E. A. Frieman and L. Chen, Phys. Fluids 25, 502 (1982).39F. Jenko, W. Dorland, M. Kotschenreuther, and B. N. Rogers,Phys. Plasmas 7, 1904 (2000).

40M. A. Beer, PhD Thesis (1995).41M. A. Beer, S. C. Cowley, and G. W. Hammett, Phys. Plasmas2, 2687 (1995).

42P. Xanthopoulos, W. Cooper, F. Jenko, Y. Turkin, A. Runov,and J. Geiger, Phys. Plasmas 16, 082303 (2009).

43R. Dewar and A. Glasser, Phys. Fluids 08544, 3038 (1983).44M. J. Pueschel, T. Dannert, and F. Jenko, Comput. Phys. Com-mun. 181, 1428 (2010).

45S. R. Hudson, C. C. Hegna, R. Torasso, and A. Ware, PlasmaPhys. Control. Fusion 46, 869 (2004).

46G. Rewoldt and W. M. Tang, Phys. Fluids B Plasma Phys. 2,318 (1990).

47D. Carmody, M. J. Pueschel, J. K. Anderson, and P. W. Terry,Phys. Plasmas 22, 012504 (2015).

48B. Coppi, S. Migliuolo, and Y.-K. Pu, Phys. Fluids B PlasmaPhys. 2, 2322 (1990).

49J. Candy, R. E. Waltz, and M. N. Rosenbluth, Phys. Plasmas11, 1879 (2004).

50W. M. Tang, Nucl. Fusion 18, 1089 (1978).51J. Citrin, C. Bourdelle, P. Cottier, D. F. Escande, O. D. Gurcan,D. R. Hatch, G. M. D. Hogeweij, F. Jenko, and M. J. Pueschel,Phys. Plasmas 19, 062305 (2012).

52T. S. Hahm, M. A. Beer, Z. Lin, G. W. Hammett, W. W. Lee,and W. M. Tang, Phys. Plasmas 6, 922 (1999).

53M. J. Pueschel and F. Jenko, Phys. Plasmas 17, 062307 (2010).54W. M. Nevins, J. Candy, S. Cowley, T. Dannert, A. Dimits,W. Dorland, C. Estrada-Mila, G. W. Hammett, F. Jenko, M. J.Pueschel, and D. E. Shumaker, Phys. Plasmas 13, 122306 (2006).

55T. Dannert and F. Jenko, Phys. Plasmas 12, 072309 (2005).56G. M. Weir, B. J. Faber, K. M. Likin, J. N. Talmadge, D. T.Anderson, and F. S. B. Anderson, Phys. Plasmas 22, 056107(2015).