Fluctuation characteristics and transport properties of collisionlesstrapped electron mode turbulence

Yong Xiao,1 Ihor Holod,1 Wenlu Zhang,1 Scott Klasky,2 and Zhihong Lin1

1Department of Physics and Astronomy, University of California, Irvine, California 92697, USA2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

�Received 27 August 2009; accepted 4 January 2010; published online 12 February 2010�

The collisionless trapped electron mode turbulence is investigated by global gyrokinetic particlesimulation. The zonal flow dominated by low frequency and short wavelength acts as a veryimportant saturation mechanism. The turbulent eddies are mostly microscopic, but with a significantportion in the mesoscale. The ion heat transport is found to be diffusive and follows the local radialprofile of the turbulence intensity. However, the electron heat transport demonstrates somenondiffusive features and only follows the global profile of the turbulence intensity. Thenondiffusive features of the electron heat transport is further confirmed by nonlognormal statisticsof the flux-surface-averaged electron heat flux. The radial and time correlation functions arecalculated to obtain the radial correlation length and autocorrelation time. Characteristic time scaleanalysis shows that the zonal flow shearing time and eddy turnover time are very close to theeffective decorrelation time, which suggests that the trapped electrons move with the fluid eddies.The fluidlike behaviors of the trapped electrons and the persistence of the mesoscale eddiescontribute to the transition of the electron turbulent transport from gyro-Bohm scaling to Bohmscaling when the device size decreases. © 2010 American Institute of Physics.�doi:10.1063/1.3302504�

I. INTRODUCTION

Anomalous transport in tokamak plasma is generally be-lieved due to the microturbulence excited by drift waveinstabilities.1 At present the ion turbulent transport driven byion temperature gradient �ITG� turbulence becomes rela-tively clear after intensive research of the last several de-cades. For example, in the ITG turbulence, the zonal flowshearing is found to regulate the ion heat transport by break-ing the radial streamers2–5 and the ion transport scaling hasbeen found to change from the Bohm to the gyro-Bohmwhen the device size increases.6 However, the electron tur-bulent transport is much less understood. The electron trans-port is becoming an important issue in the burning plasmas,such as ITER,7 because the energetic fusion products��-particles� will mostly heat the electrons. As a prominentcandidate for the electron transport, the collisionless trappedelectron mode �CTEM� has gained a renewed interest withthe advance of gyrokinetic simulation. Driven by the mag-netically trapped electrons,8 the CTEM instability has twotypes of excitation mechanisms in the collisionless limit: thekinetic instability driven by the toroidal precessional reso-nance between the trapped electrons and the electrostaticdrift waves,9 and the reactive fluid instability driven by theelectron magnetic drift when the instability drive is verystrong.10

In the late 1970s, the linear CTEM instability was exten-sively studied by the analytic theory,11,12 when only verylimited attention was paid to the nonlinear physics of theCTEM turbulence13 due to its extreme complexity. Later,some nonlinear theory began to deal with thiscomplexity.14,15 With the advance of the gyrokinetictheory16–18 and numerical algorithm,19–23 as well as the fast-

increasing computational power, the massively parallel gyro-kinetic simulations have emerged as a major tool to investi-gate the nonlinear physics of the microturbulence. Here weapply the gyrokinetic toroidal code �GTC� �Ref. 2� to studythe CTEM turbulence physics, particularly, nonlinear satura-tion mechanism and transport mechanism.

The zonal flow shearing has been discovered in the ITGturbulence as the dominant saturation mechanism.2,24–26

However, its role in the CTEM turbulence has been an on-going debate between different gyrokinetic simulations, ei-ther particle or continuum.27–30 For example, the zonal flowshearing rates between two gyrokinetic codes do not agreewith each other for �e�3, at least quantitatively. The find-ings of these two codes,29,30 that the zonal flow is not impor-tant for Te�3Ti, may not be relevant since most tokamakexperiments operate at Te�Ti. However, even this agree-ment is soon challenged.31 Therefore, until a clear physicspicture is provided, such as that for the ITG turbulence,25,26

the effect of zonal flow in the CTEM turbulence remains anopen issue. In our simulation, the zonal flow is still veryimportant in saturating the CTEM turbulence. The temporaland spatial structures of the zonal flow are then carefullyexamined in our study. It is found that the low frequencyzonal flow dominates the frequency spectrum.

To understand the transport mechanism it is important toexamine the detailed turbulence properties, such as wavespectrum, frequency spectrum, correlation length and corre-lation time.32 These detailed properties can provide impor-tant information to validate the simulation withexperiments,33,31 as well. We calculated two-point correlationfunction and two-time-two-point correlation function34 forthe CTEM turbulence. These two-dimensional �2D� correla-

PHYSICS OF PLASMAS 17, 022302 �2010�

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tion functions are used to calculate the radial correlationlength, turbulence autocorrelation time and toroidal wavepropagation speed. The radial correlation length of the tur-bulence eddies has a two-scale structure: a microscopiclength of several ion gyroradii and a mesoscale length ofseveral tens of ion gyroradii. This two-scale structure persistsfor different device sizes. The mesoscale eddies are formedin a competing process between the breaking of the macro-scopic streamers by the zonal flow and the merging of themicroscopic eddies.35

The electron heat transport in the CTEM turbulence ex-hibits some nondiffusive features and smooths out the localradial profile of the turbulence intensity that the ion heattransport closely tracks. Our statistical analysis support thenondiffusive feature of the electron heat flux. A comprehen-sive set of characteristic time scales is calculated to investi-gate the transport mechanism. The fluid time scales, espe-cially the zonal flow shearing time �s and the eddy turnovertime �eddy, are found to be very close to the effective decor-relation time. This confirms the importance of the zonal flowin the turbulence saturation process. All the kinetic timescales are much longer than the fluid time scales. Therefore,the trapped electrons in the CTEM turbulence are believed tomove with the turbulence eddies as fluid elements. As a com-parison, the characteristic time analysis for the ions in theITG turbulence with adiabatic electrons shows that the par-allel wave-particle decorrelation is the dominant transportmechanism for the ITG turbulence. Therefore, the electrontransport in the CTEM turbulence is a one-dimensional �1D��radial� process, while the ion transport is a 2D �parallel+radial� process.

The resonant trapped electrons can move ballistically inthe mesoscale eddies, which leads to a nondiffusive compo-nent in the electron heat flux. The ballistic electron heat fluxmay be responsible for the experimental observations of theresidual electron transport inside the internal transport barrier�ITB�, where the ion transport is mostly neoclassical.36 Thepersistence of the mesoscale eddies in the CTEM turbulencefor different device sizes, together with the possible turbu-lence spreading effects,32 leads to the observed transition ofthe electron transport scaling from gyro-Bohm to Bohmwhen the device size decreases.35

II. NONLINEAR SATURATION

In this section, we introduce the physical parametersapplied in the simulation and study the heat transport, aswell as the nonlinear saturation mechanism of the CTEMinstability.

A. Physical parameters

The GTC code is a well-benchmarked gyrokinetic par-ticle simulation code for studying microturbulence intokamaks.2 The code applies a global field-aligned mesh thatchange the computation scaling on device size from �a /�i�3

to �a /�i�2 and reduces the number of toroidal grids to 100times fewer, where a is the tokamak minor radius and �i isthe ion gyroradius. For the ions, the GTC code solves thegyrokinetic equation, where the gyroaverage is implemented

by the four point or eight point average in the real configu-ration space.19,37 For the electrons, the GTC code employsthe drift kinetic equation because of the small electron gyro-radius. Since the electrons move �mi /me times faster thanthe ions, where mi is the ion mass and me is the electronmass, a much smaller time step is required to satisfy theparallel Courant condition and solve the kinetic electron mo-tion accurately. Apparently this brute force method is com-putationally challenging. Instead, the GTC code employs aelectrostatic fluid-kinetic hybrid model for the electrons inthe CTEM simulation.38,39 In this model, the electrostaticpotential is decomposed into a zonal part and a nonzonalpart, �̃= ���+��. Based on the fact that the electron’s re-sponse to the zonal part ��� is negligible40 and the responseto the nonzonal part �� is mainly adiabatic for passingelectrons,9 we can expand the electron drift kinetic equationusing the smallness parameter, �me /mi. As a matter of fact,we always keep the realistic mass ratio in the simulation,whereas most other gyrokinetic simulation codes have to usethe reduced mass ratio to relieve the computational burden.The GTC code uses the Adaptable I/O System,41 ADIOS, toprovide ultrafast parallel file I/O support for the large data inthe turbulence fluctuation analysis. The GTC code has beensuccessfully benchmarked with other simulation codes in-cluding the FULL code and the GT3D code on the linearCTEM instability.42

For the nonlinear simulation of the CTEM turbulence,we employ the following typical DIII-D parameters:R0 /LTe=6.9, R0 /LTi=2.2, R0 /Ln=2.2, Te /Ti=1, mi /me

=1837, q=0.58+1.09r /a+1.09�r /a�2, with q=1.4 ands�r /q��dq /dr�=0.78 at r=0.5a. The circular cross sectionmodel is used in the simulation for the equilibrium magneticfield. The simulation is carried out in an annulus between0.1a and 0.9a. From the linear benchmark,42 we learned thatwith this set of parameters the CTEM is the only electrostaticinstability on the �i scale. The field mesh in the simulationconsists of 32 toroidal grids and a unstructured perpendicularmesh with the perpendicular grid size �0.5�i to capture theshort wavelength mode of the CTEM turbulence. Accordingto the linear simulation,42 we note that this set of parametersfalls in the category of the kinetic CTEM instability,9 mainlydriven by the toroidal precessional resonance. In addition, asa comparison we carried out a simulation for the ITG turbu-lence by increasing the ion temperature gradient, which rep-resents ITG driven turbulence with adiabatic electrons. Forthis ITG case the ITG is increased to R0 /LTi=6.9, whilekeeping other parameters the same as the CTEM case. Theions in the ITG turbulence are the proactive species thatdrives the instability, which can be used to compare to theactive species in CTEM, i.e., the trapped electrons. The de-fault system size in this paper is set as a /�i=500.

In the current GTC code, the �f method is used to reducethe numerical noise with the one particle distribution func-tion f =F0+�f and F0 is set as a local Maxwellian in thesimulation.20 However, the numerical particle noise intrinsi-cally associated with the �f algorithm cannot be completelyannihilated.43 It has been known that the noise driven trans-port increases with the entropy of the system, i.e., the sum of

022302-2 Xiao et al. Phys. Plasmas 17, 022302 �2010�

the square of particle weights.44,45 Hence, the noise driventransport can be estimated as �N�t�=�N�0��w2�t�� / �w2�0��,where �N�t� is the noise driven heat conductivity at time tand w�t�=�f�t� / f is the particle weight and �w2�t�� is theensemble average of w2�t�, which represents the entropy ofthe system. As shown in Fig. 1, the noise driven heat trans-port is only around 1% of the total transport for both theelectrons and ions after the nonlinear saturation by using 100particles per cell. Therefore, by setting sufficient number ofparticles per cell, the particle noise cannot affect the physicsresults significantly and is no longer a major concern for ourCTEM simulation. In the simulation, the heat conductivity � j

is defined through qj =nj� j�Tj, j= i ,e. Note the current defi-nition of �e is traditional and different from our earlier defi-nition in Ref. 35, where the trapped electron density is usedto calculate the time scale of the heat transport, which mainlycomes from the trapped electrons. The heat flux qj is calcu-lated in the simulation by qj =d3v� 1

2v2− 32Tj��vr�f j, where v

is the particle velocity, �vr is the radial component of thegyroaveraged EB drift, and �f j is the perturbed distribu-tion function. The temperature Tj is related to thermalspeed v j by v j =�Tj /mj and the gyroradius � j =v j / j with j =ejBj /mjc the gyrofrequency.

B. Zonal flow regulation

As shown in Fig. 1, in the CTEM simulation, the par-ticles are initially randomly distributed and produces no heattransport; then the heat transport begins to grow exponen-tially due to the linear instability caused by the toroidal pre-cessional resonance; finally the heat transport saturates andreaches a stochastic equilibrium. It is known that zonal flow,the axisymmetric EB flow, excited by microturbulencethrough nonlinear mode coupling, are the dominant satura-tion mechanism in the ITG turbulence.2 The simulation ex-tends over several tens of maximum growth time, which ismuch longer than the characteristic time scale of the physics

we are studying, the nonlinear physics that leads to the tur-bulence transport. To evaluate the zonal flow effect in theCTEM turbulence, here we compare the two simulationcases, one with the zonal flow self-consistently generatedand the other one with zonal flow artificially removed, asshown in Fig. 2. The CTEM turbulence saturation level with-out zonal flow is shown to be much higher than the one withzonal flow. This clearly shows that zonal flow are very im-portant in saturating the CTEM turbulence for the simulationparameters. The formation of macroscopic size eddies orstreamers leads to the higher transport level in the case with-out the zonal flow regulation. With the zonal flow regulation,the streamers are broken into mostly small isotropic eddiesand a significant portion of mesoscale eddies, which substan-tially reduces the transport level.35 The result, that the zonalflow is important in regulating the CTEM turbulence, is ac-tually consistent with the existing literature for the parameterregime studied. In Fig. 2, we compare two cases of devicesize a /�i=250 to relieve the computational burden withoutsignificantly influencing the zonal flow physics.

The time-radial 2D structure of the zonal flow potentialis shown in Fig. 3. After nonlinear saturation �t�10 /�max

with �max=0.25Lne /vi �Ref. 42��, the zonal flow potential hasa strong short wavelength component with �kr�i��0.7, asshown in the upper panel of Fig. 4, since the short wave-length zonal flow are not easily screened by the neoclassicaleffects.46–49 The lower panel of Fig. 4 shows the radial wavespectrum at a typical time moment after saturation. To cal-culate �kr�i� from the lower panel, we exclude those verylong wavelengths that usually represents mean flow, sincethese zonal flows have very weak shearing effect50 andwhether they are treated in a sufficiently satisfactory way is atopic of current interest.51–53 There is a downshift of zonalflow wavelength during the nonlinear saturation. This down-shift is caused by the downshift of the perpendicular spec-trum of the drift wave turbulence in Fig. 6�a�, which is ex-

0 10 20 30 40

10−3

10−2

10−1

100

t(1/γmax

)

χ(ρ i2 v T

i/a)

χe

χi

χeno

χino

D

FIG. 1. �Color online� Time history of electron heat conductivity �e �solidline�, ion heat conductivity �i �circle�, particle diffusivity�dashed line�, andnoise induced transport �square for electrons and diamond for ions� in thenonlinear CTEM simulation, where �max represents the maximum lineargrowth rate.

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−7

t(1/γmax

)

χ e(a.u

.)

zonal flowno zonal

FIG. 2. �Color online� The time history of heat transport for a /�i=250. Thesolid line has the zonal flow self-consistently generated, while the dottedline has the zonal flow artificially removed.

022302-3 Fluctuation characteristics and transport properties… Phys. Plasmas 17, 022302 �2010�

cited by the linear CTEM instability and believed to be theenergy source of the zonal flow.25,26 Fixing on one radiallocation r�0.5a in Fig. 3, we can see that the zonal flowvaries with time. This time oscillation of zonal flow is thegeodesic acoustic mode �GAM�. For each radial location, thefrequency response of the zonal flow can be calculated andthen the GAM frequency spectrum can be averaged overdifferent radii. The average GAM frequency is found to be�GAM�2.6vi /R0, which is very close to the theoretical value2.68vi /R0 from the formula54,55

�GAM =� 2

mi�Te +

7

4Ti 1 +

46

49q2��� R0.

The GAM frequency is higher than the turbulence autocor-relation frequency �au�2 /�au�1.3vi /R0, which can befound in Table I. This indicates that the GAM mode couldonly have a weak regulation effect on the CTEM turbulence.Since the low frequency dominates the zonal flow spectrum,as shown in Fig. 3, it is still the low frequency zonal flowrather than the GAM that mainly regulates the CTEM turbu-lence. The GAM oscillation is stronger in the outside of theradial domain where the linear damping of GAM isweaker due to the larger q value since that the linear GAMdamping rate �GAM decreases with the safety factor q, i.e.,�GAM�q5 exp�−7q2 /4�.54,56 In addition, the GAM oscilla-tion occurs after the zonal flow is excited. This indicates thatfirst the turbulence excites the zonal flow and then the zonalflow excites the GAM mode by a geodesic energy transferprocess.57

III. TURBULENCE CHARACTERISTICS

In this section, we proceed to discuss the characteristicsof the CTEM turbulence and compare it with that of the ITGturbulence. In the GTC simulation, the 3D�r ,� ,�� turbulencedata is recorded at each time step for further quantitativeanalysis.

A. Ballooning structure

In a tokamak, the particles feel larger magnetic drift ��Band curvature drift� in the outer side ��=0� on each fluxsurface than the inner side ��= �. The anisotropy of themagnetic drift effect within a bounce period leads to a uni-versal ballooning mode structure of the turbulencefluctuations.58 Our simulation is able to demonstrate this bal-looning feature, as a verification of the global code. Theturbulence potential on the r=0.5a flux surface, plotted inFig. 5, shows a typical �� ,�� 2D turbulence structure at thenonlinear stage, where � is the toroidal angle and �=�−� /q is the field line label. In this figure the turbulence fluc-tuations are elongated along each field line, one of the keyfeatures of the microturbulence in a magnetized plasma.Moreover, the maximum potential point on each field lineforms a straight line with slope approximately −1 /q. Bytransforming the field-aligned coordinate �� ,�� to the mag-netic coordinates �� ,��, this feature actually shows that theturbulence should have a maximum value at the poloidalangle �=0. We can define a flux-surface-averaged parallelangle ��2�= �−

d�����2�2 /− d�����2� by integrating along

each field line, which represents the mean parallel wave-

TABLE I. Characteristic time scales for trapped electrons in the CTEMturbulences and for ions in the ITG turbulence.

�Lne /vi� �decor �� �� �eddy �au �s 1 / �max

CTEM e 0.61 � � 1.6 11 0.66 4.0

ITG i 1.7 1.8 2.0 4.9 7.2 1.4 9.1

FIG. 3. �Color� Zonal flow structures in both time and radial directions.

10 15 20 25 30 35 400

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t(1/γmax

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<krρ

i> for zonal flow

0 0.5 1 1.5 20

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krspectrum for zonal flow at t=29/γ

max

krρ

i

FIG. 4. �Color online� Upper panel shows the time history of the averageradial wave vector of the zonal flow. Lower panel shows that radial wavevector spectrum of the zonal flow at t=29 /�max.

022302-4 Xiao et al. Phys. Plasmas 17, 022302 �2010�

length. For a more precise representation of the parallelangle, we shall consider the statistical fluctuation of ��2� inthe nonlinear stage, i.e., averaging ��2� over many time steps.The time averaged parallel angle is found to be ��2�=1.14 forthe CTEM turbulence and ��2�=2.54 for the ITG turbulence.So the CTEM turbulence has a smaller mean parallel anglethan the ITG turbulence, which suggests that the kinetic elec-tron response reduces the mean parallel angle ��2�, i.e., theCTEM turbulence forms a stronger ballooning structure, andthe mode coupling between neighboring poloidal harmonicsis tighter in the CTEM turbulence. This is essentially afeature of the linear instability.42 With the mean parallelangle ��2�, we can calculate the mean radial wavelength��kr

2�=s2��2���k�2�, where s is the magnetic shear defined as

s=d ln q /d ln r. This value of ��kr2� is actually the range of

radial wavelengths that one particle could see within abounce motion, and will be used to calculate the particlediffusion time among different poloidal harmonics.34

B. Turbulence spectrum evolution

The 3D+time turbulence data can also be used tocalculate the parallel and perpendicular spectrum,e.g., �k��=�k���k�2�k�� /�k���k�2, �k� =��k� − �k���2, �k�

2 �=�k���k�2�k�� /�k���k�2, and �k�=��k�− �k���2, as shown inFig. 6. In the linear stage �t�8 /�max�, the perpendicularspectrum of the CTEM mode has a shorter wavelength thanthat of the ITG turbulence because the short wavelengthmodes have a larger linear growth rate than the long wave-length modes.42 Then the radial spectrum of zonal flow isshifted to the short wavelength regime by the mode-modecoupling of the ambient short wavelength turbulence�8 /�max� t�9 /�max�, as seen in Fig. 4. Later during thenonlinear saturation, the perpendicular spectrum of theCTEM turbulence shifts to the longer wavelength,29 so doesthe zonal flow. However, the parallel spectrum of the CTEMturbulence keeps roughly constant in this transition. While inthe ITG case, the perpendicular spectrum downshifts slightlyand the parallel spectrum upshifts slightly toward the shorter

wavelength, as shown in Fig. 6. Therefore, the spectrum evo-lution indicates that the CTEM turbulence is more related tothe perpendicular wave-particle dynamics, while ITG turbu-lence is more related to the parallel wave-particle dynamics.This will be further confirmed by our detailed time scaleanalysis.

C. Radial correlation length

The radial correlation length is an important measure ofthe turbulence structure, which can be clearly seen by theradial-toroidal 2D turbulence contour from the CTEM simu-lation in Fig. 7. This figure is dominated by turbulence ed-dies with radial length of several �i. From the data in Fig. 7,we can calculate the two-point correlation function,

Cr���r,��� =����r + �r,� + ������r,���

����2�r + �r,� + �������2�r,���. �1�

Then for each �r, the maximum of Cr���r ,��� can be cal-culated to give the radial correlation function Cr��r� for dif-ferent device sizes, as shown in Fig. 8. Note that all thesecorrelation curves are calculated from the data at one timestep. So there are some statistical fluctuations, especially forthe mesoscale structures of the a /�i=125 case. These radialcorrelation functions all drops exponentially for the smallradial separation with a characteristic length of 5�i, whichfits the exponential decay model exp�−�r /Lr� and the radial

FIG. 5. �Color� The contour plot of the electrostatic potential �� on a fluxsurface at t=29 /�max for r=0.5a with � the toroidal angle and � the fieldline variable.

10 20 30 400

0.4

0.8

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1.6

2

8 12 16 20 240

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2

t(1/γmax

)

<k||R

0>

<∆k||R

0>

<k⊥ρi>

<∆k⊥ρi>

<k||R

0>

<∆k||R

0>

<k⊥ρi>

<∆k⊥ρi>

FIG. 6. �Color online� Time history of the average parallel and perpendicu-lar wave vector and their respective spectrum width for CTEM turbulence inthe upper panel �a� and ITG�a� turbulence in the lower panel �b�.

022302-5 Fluctuation characteristics and transport properties… Phys. Plasmas 17, 022302 �2010�

correlation length Lr represents the microscopic eddy size. Inaddition, there is a significant tail in the correlation function,notably persisting for all device sizes, which represents themesoscale eddies of several tens of �i. On the contrary, thereexist mostly microscopic eddies in the ITG turbulence fordifferent device sizes.32 To reduce the statistical error, theradial correlation length can be time-averaged over multipletime steps to obtain the mean microscopic eddy size,Lr=5.6�i for the CTEM turbulence. As a comparison, wefind Lr=8.0�i for the ITG turbulence. So the mean micro-scopic eddy size in the CTEM turbulence is shorter than thatin ITG.

D. Wave propagation and autocorrelation time

An important characteristic time for the turbulence is theautocorrelation time �au, which describes how long the tur-bulence eddy can exist. In order to calculate �au, the two-time-two-point correlation function Ct���t ,��� defined by

Ct���t,��� =����t + �t,� + ������t,���

����2�t + �t,� + �������2�t,���, �2�

needs to be computed. From the turbulence data, the poten-tial value at line ��=0, r=0.5a� can be extracted at each timestep to obtain a 2D potential data ��t ,��. The two-time-two-point correlation function Ct���t ,��� can then be generated,as shown in Fig. 9. In this figure, the CTEM turbulencepropagates in the electron diamagnetic direction �−��. Thisdirection is counter to the direction of the ITG turbulencepropagation, which is in the ion diamagnetic direction �+��.From Fig. 9, the toroidal phase velocity for the CTEM tur-

bulence can be calculated by �̇=0.02vi /Ln, which is the tor-oidal precession frequency.

Following the maxima ridge in Fig. 9, the time correla-tion function Ct��t� can be obtained. The exponential decaymodel exp�−�t /�au� is found to very well suit the correlation

FIG. 7. �Color� The contour plot of the electrostatic potential �� on the�r ,�� plane at t=29 /�max for the poloidal angle �=0.

0 10 20 30 40 50

0.2

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0.6

0.8

∆r(ρi)

Cr(∆

r)

a=500ρi

a=250ρi

a=125ρi

FIG. 8. �Color online� Radial correlation function Cr��r� of the electrostaticpotential �� for different device sizes.

FIG. 9. �Color� Contour plot for the two-time-two-point correlation functionCt���t ,��� of the electronstatic potential ��, in which the unit of �� isradian.

0 10 20 300

0.2

0.4

0.6

0.8

1

∆t(Lne

/vi)

Ct(∆

t)

originalfitted

Exponential fit gives τau

= 11.1 Lne

/vi

FIG. 10. �Color online� Time correlation function Ct��t� of the electrostaticpotential �� and its exponential fit, which gives the autocorrelation time�au=11.1LTi /vTi.

022302-6 Xiao et al. Phys. Plasmas 17, 022302 �2010�

function Ct��t�, as shown in Fig. 10, which gives the auto-correlation time �au=11.1LTi /vi for the CTEM turbulence.

IV. NONLINEAR HEAT TRANSPORT MECHANISM

After exploring all the requisite turbulence characteris-tics, we proceed to investigate the nonlinear heat transportmechanism for the CTEM turbulence.

A. Features of CTEM transport

It is of great interest to investigate the electron heattransport mechanism because recent researches indicate thatthe CTEM turbulence can be a prominent candidate for theelectron heat loss in a tokamak.33,59,60 In the ITG turbulence,the ion heat transport can be described by a quasilinear dif-fusion model.61,62 An important consequence of the quasilin-ear diffusion model is that the local heat conductivity ��r� isproportional to the local EB drift intensity ��vr

2�r�� with � �denoting flux-surface average. For example, in the ITG tur-bulence, our simulation finds that the local heat conductivity�i�r� is indeed proportional to ��vr

2�r�� for a simulation withdevice size a /�i=500, as shown in Fig. 11�b�. For the CTEMturbulence, the local ion heat conductivity �i�r� is still pro-portional to ��vr

2�r��, as shown in Fig. 11�a�, which suggeststhat the ion heat transport is still driven by the local EBdrift intensity and the quasilinear diffusion model is suitablefor the ion heat transport in the CTEM turbulence. However,from Fig. 11�a�, the electron heat conductivity only tracks theglobal profile of the turbulence intensity ��vr

2�r�� and

smooths out its local oscillatory feature. This suggests thatthe electron heat transport follows a transport mechanismdifferent from that in the ion heat transport. The electron heattransport in the CTEM turbulence can contain a nondiffusivecomponent so that quasilinear diffusion model is insufficientto describe the electron dynamics. In addition, the 2D radial-time contour of the electron heat flux in Fig. 3�c� of Ref. 35shows that the electron heat flux has a radial ballistic propa-gation which may comes from this nondiffusive component.

The nondiffusive feature of the electron heat transportcan be further confirmed by the probability distribution func-tion �PDF� of the flux-surface-averaged electron heat flux, asshown by the circles in Fig. 12�a�. If the heat transport ispurely a diffusive process, the flux-surface-averaged heatflux should follow the lognormal distribution,63 which is truefor the ion heat fluxes and small electron heat fluxes in Fig.12�a�. However, neither lognormal nor Gaussian distributioncan fit well the simulation PDF in the whole range of theelectron heat flux. There is a significant large heat flux com-ponent that is above the lognormal distribution, which indi-cates a nondiffusive/superdiffusive component. The large ionheat fluxes are below the lognormal distribution, which may

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qi

<δvr2>

FIG. 11. �Color online� Radial profile of the heat flux qi ,qe and the EBdrift intensity ��vr

2� for a /�i=500 with the upper panel �a� for the CTEMturbulence and lower panel �b� for the ITG�a� turbulence.

1 2 3 4 5 6 710

−2

10−1

100

qe

1 2 3 4 5 6 710

−2

10−1

100

qi

lognormalGaussian

qi

FIG. 12. �Color online� The probability density function �PDF� of the flux-surface-averaged electron heat flux qe and ion heat flux qi in the CTEMturbulence, shown in the upper panel �a�, and ion heat flux qi in the ITGturbulence, shown in the lower panel �b� with the electron heat flux normal-ized by ntr�Te and ion heat flux normalized by n0�Ti, where ntr is thetrapped electron density. The solid line is a lognormal distribution fit and thedashed line is a Gaussian distribution fit.

022302-7 Fluctuation characteristics and transport properties… Phys. Plasmas 17, 022302 �2010�

indicate a subdiffusive component. On the contrary, the PDFof the flux-surface-averaged ion heat flux in the ITG turbu-lence is consistent with the lognormal distribution, as shownin Fig. 12�b�, which suggests that the ion heat transport in theITG turbulence is a diffusive process.32 The PDF is calcu-lated over a time window of 50Lne /vi for the electron heatflux in the CTEM turbulence and a time window of100Lne /vi for the ion heat flux in the ITG turbulence, respec-tively. As shown in Sec. IV B, these time windows are muchlarger than all the known characteristic time scales, espe-cially the effective decorrelation time, in their respectivetype of turbulence.

B. Characteristic time scales

Although the electron transport in the CTEM turbulencedoes not follow the local structure of the turbulence intensity,it still follows the global profile of the turbulence intensity,as shown in Fig. 11�a�. This global proportionality enables usto define an effective decorrelation time �decor=2D / ��vr

2� anda test particle diffusivity D can be related to the electronthermal conductivity �e by D=2�e /3. From Fig. 11�a�,���vr

2�=4.710−3vi. It is then calculated that �decor

=4�e / �3��vr�2��0.61Ln /vi for the trapped electrons. Simi-larly, �decor=1.7Ln /vi for the ions in the ITG turbulence. Thischaracteristic time scale may reflect the physical process rel-evant to the transport mechanism,34 which could be eitherkinetic wave-particle decorrelation or fluid eddy mixing.Through the following comprehensive analysis of the kineticand fluid time scales, we can identify the physical processresponsible for the transport.

The two kinetic time scales related to the CTEM eigen-modes are the parallel and perpendicular wave-particle deco-rrelation time ��� and ��� for the trapped electrons to crossthe turbulence eddies in the parallel and perpendicular direc-tions. Because of the fast bounce motion, which averages outthe parallel electric field, the trapped electrons cannot de-correlate from the wave in the parallel direction, i.e., �� =�.In the spectral range of interest, the CTEM frequency isroughly proportional to the toroidal mode number �i.e., non-dispersive�. Thus the resonant electrons cannot decorrelatefrom the wave in the toroidal direction. Moreover, althoughthe resonant electrons can decorrelate from the wave in theradial direction due to the radial dependence of the preces-sional frequency, this dependence is very weak �on the equi-librium spatial scale�. Therefore, ��=�. The trapped elec-trons thus remain resonant with the wave until the eddiesdisappear or the electrons jump from one eddy to another, i.e,the resonant electrons behave like fluid eddies in the trans-port process. On the contrary, the ions in the ITG turbulencecan experience parallel decorrelation with �� =1 / ��k�vi�=1.8Ln /vi, since the parallel spectral width can be measuredin Fig. 6�b�. The ions can also be scattered radially by theradial structure of the poloidal harmonics of the ballooningmodes, ��=3 / �4�is

2��2��k�2��=2.0Ln /vi.

The fluid time scales also include eddy turnover time�eddy, zonal flow shearing time �s, and eddy auto-correlation time �au. The eddy turnover time �eddy=Lr / ��vr�,describes how fast the eddy rotates due to the EB drift

without the zonal flow shearing. For microscopic eddies,�eddy�1.6Ln /vi by using the mean microscopic eddy sizecalculated in Sec. III C. For mesoscale eddies, the eddy turn-over time is much longer. Another fluid time scale relevant tothe dynamics of the turbulence eddies is the zonal flowshearing time,

�s = � Lr

L�

�

�r qVE

r��−1

,

which is calculated to be �s�0.66Ln /vi. As calculated inSec. V, the eddy autocorrelation time is �au�11Ln /vi.

We list all the characteristic time scales in Table. I. Com-pared with �decor, the zonal flow shearing time �s and eddyturnover time �eddy are the two closest time scales in theCTEM turbulence. This suggests that the decorrelation pro-cess should be mostly the eddy mixing regulated by thezonal flow. All the kinetic time scales are much larger than�decor, �s, and �eddy. Therefore, the electron heat transport inthe CTEM turbulence is mainly a fluid process although thelinear instability is driven by the kinetic process of the tor-oidal precessional resonance. The radially random distribu-tion of the microscopic and mesoscale eddies enables theelectrons to average out the local structure of the turbulenceintensity. The Kubo number is useful concept for the fluidturbulence. It is found that CTEM turbulence has a largeKubo number with K=�auto /�eddy�7 for the microscopic ed-dies, which could affect the transport scaling.64 This numberonly sets up a upper limit for the Kubo number because thereexist also mesoscale eddies. Since the mesoscale eddieswould have a smaller K value, the quasilinear estimate of theeffective decorrelation time �decor may be relevant to thetransport on the global scale, which is in the transitionalregime between the small and large K value.

For comparison, the ion characteristic time scales in theITG turbulence are also listed in Table. I. The kinetic timescales, such as �� and ��, are close to �decor. Therefore, thetransport in the ITG turbulence is mainly a kinetic processwith parallel decorrelation or perpendicular decorrelation bythe radial structure of the poloidal harmonics of the balloon-ing modes.34 The zonal flow time �s is also found to be closeto �decor, which suggests the importance of the zonal flowregulation on the ITG turbulence. For this kinetic decorrela-tion case, the Kubo number can be redefined as K=�� /�eddy,which leads to K=0.37�1. In this small Kubo number limit,the quasilinear theory should be valid.64

In the CTEM simulation, the transition from the Bohmscaling to gyro-Bohm scaling is observed when increasingthe device size and keeping other tokamak parameters fixed.The significant long tail of the radial correlation function inFig. 8 confirms the existence of a large number of mesoscaleeddies in Fig. 7. The resonant trapped electrons can be con-vected by the EB drift across the mesoscale eddies. Thismesoscale ballistic motion then drives the electron heattransport that contains a nondiffusive component on the me-soscale and smooth out the small radial structure of the tur-bulence intensity, as shown in Fig. 11�a�. The mesoscale bal-listic electron heat flux, together with the turbulencespreading,6 leads to the deviation from the gyro-Bohm scal-

022302-8 Xiao et al. Phys. Plasmas 17, 022302 �2010�

ing for the small devices. The transition from gyro-Bohmscaling to Bohm scaling when the device size decreases, isalso observed for ion heat transport in the ITG turbulence,6

which is mainly due to turbulence spreading.32

V. DISCUSSION AND CONCLUSION

In this paper, we carried out the nonlinear CTEM simu-lation using the global gyrokinetic code GTC with low par-ticle noise. Our simulation finds that zonal flow is importantin saturating the CTEM instability. The importance is furtherconfirmed by the characteristic time analysis in which thezonal flow shearing time �s and the eddy turnover time �eddy

are very close to the effective decorrelation time �decor thatcharacterizes the transport process. The CTEM turbulencezonal flow, unlike the ITG turbulence, has prominent com-ponents in the short wavelength range, �kr�i��0.7. TheGAM oscillation can be found on the outside of the radius,albeit the low frequency zonal flow still dominates the satu-ration process.

The turbulence properties found by our CTEM simula-tion is consistent with the ballooning mode theory. Even inthe nonlinear regime, the CTEM turbulence still exhibits aprominent ballooning structure. We find that the perpendicu-lar wavelength has a significant downshift to the longerwavelength during the nonlinear saturation. However, theparallel wavelength keeps roughly the same.

The two-point correlation function is calculated to derivethe radial correlation function, which shows that the CTEMturbulence contains both microscopic scale eddies of about5�i and mesoscale eddies of several tens of �i. This two-scalestructure of the turbulence eddies persists in different devicesizes. The two-time-two-point correlation function is calcu-lated to find the autocorrelation time of the turbulence. Com-paring to the ITG turbulence, the CTEM turbulence has ashorter microscopic eddy size and parallel wavelength. Thesedetailed properties of the turbulence fluctuations can provideuseful information for both theoretical studies and experi-mental validations.

We also investigated the nonlinear heat transport mecha-nism. It is found that in the CTEM turbulence, the ion heattransport tracks the local profile of the turbulence intensity,which suggests that the ion transport is driven by the localEB drift intensity. However, the electron heat transportonly tracks the global profile of the turbulence intensity in-stead of the local profile. This suggests that electron transportin the CTEM turbulence could be locally nondiffusive. Inaddition, the probability density function of the flux-surface-averaged electron heat flux demonstrates some nonlognormalfeatures, which further confirms the nondiffusiveness of theelectron heat transport. The characteristic time scale analysisfor the trapped electrons shows that the fluid time scales areclose to the effective decorrelation time while the kinetictime scales are much larger, which suggests that the electrontransport in the CTEM turbulence is a fluid process wherethe trapped electrons move with the turbulence eddies asfluid elements. For example, the resonant trapped electronscan be convected by the EB drift across the mesoscaleeddies. This mesoscale ballistic motion then drives the elec-

tron heat transport that contains a nondiffusive component onthe mesoscale and smooths out the small radial structure ofthe turbulence intensity. The electron transport in the CTEMturbulence is a 1D process with the decorrelation and trans-port processes both occurring in the radial direction. In ad-dition, it is found that the Kubo number K�1. Due to thelarge Kubo number and nonlocal feature of the electrontransport, the quasilinear theory may no longer be valid lo-cally for the electron transport in the CTEM turbulence. Thepersistence of mesoscale eddies changes the gyro-Bohmscaling to the Bohm scaling since the correlation length iscomparable to the device size for small devices. It is alsolikely for the turbulence spreading32,65 to play a role in thisgyro-Bohm to Bohm transition. The ballistic electron heatflux may also account for the residual electron heat transportexperimentally observed inside the ITB.

On the contrary, the ion transport in the ITG turbulenceis driven by the local turbulence intensity and the transportmechanism is found to be parallel decorrelation due to theguiding center parallel motion and the parallel and radialstructure of the poloidal harmonics of the ballooning modes.It is interesting to note that, the ion transport in the ITGturbulence, even the electron transport in the electron tem-perature gradient �ETG� turbulence,34 is a 2D process withthe decorrelation process occurring in the parallel directionand the transport process occurring the radial direction.Therefore, the Kubo number from the fluid turbulence needsto be modified to be relevant to the ITG turbulence and ETGturbulence,34 and it is found the modified Kubo numberK�1. In this limit, the quasilinear theory is suitable to de-scribe the ion transport in the ITG and CTEM turbulence, aswell as the electron transport in the ETG turbulence. In ad-dition, the transition from gyro-Bohm to Bohm scaling in theITG turbulence is mainly due to the turbulence spreadingsince no significant mesoscale eddies are observed.32

ACKNOWLEDGMENTS

This work was supported by U.S. DOE SciDAC GPS-TTBP and GSEP centers. Simulations used ORNL andNERSC supercomputers.

We acknowledge L. Chen, P. H. Diamond, and T. S.Hahm for useful discussions.

1W. Horton, Rev. Mod. Phys. 71, 735 �1999�.2Z. Lin, T. Hahm, W. Lee, W. Tang, and R. White, Science 281, 1835�1998�.

3T. Hahm and K. Burrell, Phys. Plasmas 2, 1648 �1995�.4M. Beer, Ph.D. thesis, Princeton University, 1995.5H. Biglari, P. Diamond, and P. Terry, Phys. Fluids B 2, 1 �1990�.6Z. Lin, S. Ethier, T. Hahm, and W. Tang, Phys. Rev. Lett. 88, 195004�2002�.

7For more information, see http://www.iter.org.8B. Kadomstev and O. Pogutse, Sov. Phys. Dokl. 14, 470 �1969�.9J. Adam, W. Tang, and P. Rutherford, Phys. Fluids 19, 561 �1976�.

10B. Coppi and G. Rewoldt, Phys. Rev. Lett. 33, 1329 �1974�.11P. Catto and K. Tsang, Phys. Fluids 21, 1381 �1978�.12W. Tang, Nucl. Fusion 18, 1089 �1978�.13L. Chen, R. L. Berger, J. G. Lominadze, M. N. Rosenbluth, and P. H.

Rutherford, Phys. Rev. Lett. 39, 754 �1977�.14T. S. Hahm and W. M. Tang, Phys. Fluids B 3, 989 �1991�.15P. W. Terry, Phys. Rev. Lett. 93, 235004 �2004�.16P. Catto, W. Tang, and D. Baldwin, Plasma Phys. 23, 639 �1981�.

022302-9 Fluctuation characteristics and transport properties… Phys. Plasmas 17, 022302 �2010�

17E. Frieman and L. Chen, Phys. Fluids 25, 502 �1982�.18W. Lee, Phys. Fluids 26, 556 �1983�.19W. Lee, J. Comput. Phys. 72, 243 �1987�.20A. Dimits and W. Lee, J. Comput. Phys. 107, 309 �1993�.21S. E. Parker and W. W. Lee, Phys. Fluids B 5, 77 �1993�.22R. D. Sydora, V. K. Decyk, and J. M. Dawson, Plasma Phys. Controlled

Fusion 38, A281 �1996�.23M. Kotschenreuther, G. Rewoldt, and W. Tang, Comput. Phys. Commun.

88, 128 �1995�.24A. M. Dimits, G. Bateman, M. A. Beer, B. I. Cohen, W. Dorland, G. W.

Hammett, C. Kim, J. E. Kinsey, M. Kotschenreuther, A. H. Kritz, L. L.Lao, J. Mandrekas, W. M. Nevins, S. E. Parker, A. J. Redd, D. E.Shumaker, R. Sydora, and J. Weiland, Phys. Plasmas 7, 969 �2000�.

25L. Chen, Z. Lin, and R. White, Phys. Plasmas 7, 3129 �2000�.26P. Diamond, S.-I. Itoh, K. Itoh, and T. Hahm, Plasma Phys. Controlled

Fusion 47, R35 �2005�.27D. 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, K. Zhurovich, and Alcator C-Mod Group, Phys. Plasmas 11, 2637�2004�.

28F. Merz and F. Jenko, Phys. Rev. Lett. 100, 035005 �2008�.29J. Lang, S. Parker, and Y. Chen, Phys. Plasmas 15, 055907 �2008�.30D. Ernst, J. Lang, W. Nevins, M. Hoffman, Y. Chen, W. Dorland, and S.

Parker, Phys. Plasmas 16, 055906 �2009�.31R. Waltz and C. Holland, Phys. Plasmas 15, 122503 �2008�.32Z. Lin and T. Hahm, Phys. Plasmas 11, 1099 �2004�.33L. Lin, M. Porkolab, E. M. Edlund, J. C. Rost, M. Greenwald, N. Tsujii, J.

Candy, R. E. Waltz, and D. R. Mikkelsen, Plasma Phys. Controlled Fusion51, 065006 �2009�.

34Z. Lin, I. Holod, L. Chen, P. H. Diamond, T. S. Hahm, and S. Ethier, Phys.Rev. Lett. 99, 265003 �2007�.

35Y. Xiao and Z. Lin, Phys. Rev. Lett. 103, 085004 �2009�.36M. E. Austin, K. H. Burrell, R. E. Waltz, K. W. Gentle, P. Gohil, C. M.

Greenfield, R. J. Groebner, W. W. Heidbrink, Y. Luo, J. E. Kinsey, M. A.Makowski, G. R. McKee, R. Nazikian, C. C. Petty, R. Prater, T. L.Rhodes, M. W. Shafer, and M. A. V. Zeeland, Phys. Plasmas 13, 082502�2006�.

37Z. Lin and W. Lee, Phys. Rev. E 52, 5646 �1995�.38Z. Lin and L. Chen, Phys. Plasmas 8, 1447 �2001�.39Z. Lin, Y. Nishimura, Y. Xiao, I. Holod, W. Zhang, and L. Chen, Plasma

Phys. Controlled Fusion 49, B163 �2007�.40Y. Xiao, P. J. Catto, and W. Dorland, Phys. Plasmas 14, 055910 �2007�.41J. Lofstead, F. Zheng, S. Klasky, and K. Schwan, Proceedings of the 24th

International Parallel and Distributed Processing Symposium, Rome, 2009�unpublished�.

42G. Rewoldt, Z. Lin, and Y. Idomura, Comput. Phys. Commun. 177, 775�2007�.

43A. Aydemir, Phys. Plasmas 1, 822 �1994�.44A. Bottino, Phys. Plasmas 14, 010701 �2007�.45I. Holod and Z. Lin, Phys. Plasmas 14, 032306 �2007�.46L. Wang and T. S. Hahm, Phys. Plasmas 16, 062309 �2009�.47G. Kagan and P. J. Catto, Plasma Phys. Controlled Fusion 50, 085010

�2008�.48Y. Xiao and P. Catto, Phys. Plasmas 13, 102311 �2006�.49F. Jenko, W. Dorland, M. Kotschenreuther, and B. Rogers, Phys. Plasmas

7, 1904 �2000�.50T. Hahm, M. Beer, Z. Lin, G. Hammett, W. Lee, and W. Tang, Phys.

Plasmas 6, 922 �1999�.51F. I. Parra and P. J. Catto, Plasma Phys. Controlled Fusion 51, 065002

�2009�.52F. I. Parra and P. J. Catto, Plasma Phys. Controlled Fusion 50, 065014

�2008�.53W. W. Lee and R. A. Kolesnikov, Phys. Plasmas 16, 044506 �2009�.54Z. Gao, K. Itoh, H. Sanuki, and J. Dong, Phys. Plasmas 13, 100702

�2006�.55T. Watari, Y. Hamada, T. Notake, N. Takeuchi, and K. Itoh, Phys. Plasmas

13, 062504 �2006�.56X. Xu, Z. Xiong, Z. Gao, W. Nevins, and G. Mckee, Phys. Rev. Lett. 100,

215001 �2008�.57V. Naulin, A. Kendl, O. E. Garcia, A. H. Nielsen, and J. J. Rasmussen,

Phys. Plasmas 12, 052515 �2005�.58J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Rev. Lett. 40, 396

�1978�.59J. DeBoo, S. Cirant, T. Luce, A. Manini, C. Petty, F. Ryter, M. Austin, D.

Baker, K. Gentle, C. Greenfield, J. Kinsey, and G. Staebler, Nucl. Fusion45, 494 �2005�.

60F. Ryter, C. Angioni, A. G. Peeters, F. Leuterer, H.-U. Fahrbach, and W.Suttrop, Phys. Rev. Lett. 95, 085001 �2005�.

61I. Holod and Z. Lin, Phys. Plasmas 15, 092302 �2008�.62W. Zhang, Z. Lin, and L. Chen, Phys. Rev. Lett. 101, 095001 �2008�.63R. Basu, T. Jessen, V. Naulin, and J. Rasmussen, Phys. Plasmas 10, 2696

�2003�.64M. Vlad, F. Spineanu, J. Misguich, J.-D. Reuss, R. Balescu, K. Itoh, and

S.-I. Itoh, Plasma Phys. Controlled Fusion 46, 1051 �2004�.65W. Deng and Z. Lin, Phys. Plasmas 16, 102503 �2009�.

022302-10 Xiao et al. Phys. Plasmas 17, 022302 �2010�