Gyrokinetic study of electromagnetic effects on toroidal momentumtransport in tokamak plasmasT. Hein, C. Angioni, E. Fable, J. Candy, and A. G. Peeters Citation: Phys. Plasmas 18, 072503 (2011); doi: 10.1063/1.3609841 View online: http://dx.doi.org/10.1063/1.3609841 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v18/i7 Published by the American Institute of Physics. Related ArticlesObservation of plasma array dynamics in 110GHz millimeter-wave air breakdown Phys. Plasmas 18, 100704 (2011) Charged particle motion in electromagnetic fields varying moderately rapidly in space Phys. Plasmas 18, 104510 (2011) Extended estimations of neoclassical transport for the TJ-II stellarator: The bootstrap current Phys. Plasmas 18, 102507 (2011) A method for calculating plasma rotation velocity due to internal and external sources Phys. Plasmas 18, 102505 (2011) Plasma diffusion in self-consistent fluctuations Phys. Plasmas 18, 102310 (2011) Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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Gyrokinetic study of electromagnetic effects on toroidal momentumtransport in tokamak plasmas

T. Hein,1 C. Angioni,1,a) E. Fable,1 J. Candy,2 and A. G. Peeters3

1Max-Planck-Institut fur Plasmaphysik, IPP-EURATOM Association, D-85748 Garching bei Munchen,Germany2General Atomics, P.O. Box 85608, San Diego, California 92186-5608, USA3Department of Physics, University of Bayreuth, D-95440 Bayreuth, Germany

(Received 20 April 2011; accepted 31 May 2011; published online 26 July 2011)

The effect of a finite be¼ 8pneTe=B2 on the turbulent transport of toroidal momentum in tokamak

plasmas is discussed. From an analytical gyrokinetic model as well as local linear gyrokinetic

simulations, it is shown that the modification of the parallel mode structure due to the nonadiabatic

response of passing electrons, which changes the parallel wave vector kk with increasing be, leads

to a decrease in size of both the diagonal momentum transport as well as the Coriolis pinch under

ion temperature gradient turbulence conditions, while for trapped electron modes, practically no

modification is found. The decrease is particularly strong close to the onset of the kinetic

ballooning modes. There, the Coriolis pinch even reverses its direction. VC 2011 American Instituteof Physics. [doi:10.1063/1.3609841]

I. INTRODUCTION

The study of turbulent toroidal momentum transport cur-

rently receives much attention in the field of fusion plasma

physics. Because of the toroidal symmetry properties of toka-

maks, the toroidal angular rotation is undamped, in contrast to

the poloidal component. The shear of toroidal velocity goes

into both the E�B shear, which is perpendicular to the unper-

turbed magnetic field, and the shear of the parallel velocity. It

was investigated that the latter may enhance turbulent trans-

port due to the fact that a parallel velocity gradient adds a drive

to ion temperature gradient (ITG) modes.1–3 E�B shearing,

on the other hand, is known to have a stabilizing influence and

thus has a beneficial effect on energy confinement in fusion

devices, as was found for instance in Ref. 4. The relative con-

tribution of these two mechanisms is mainly determined by

the ratio of poloidal to toroidal magnetic field strength. In

addition, even without an external torque, a so-called intrinsic

rotation is observed.5,6 The latter is of particular interest for

plasma experiments going towards a reactor size scale since

only a very small external torque will be present there.

Despite the extensive research on this topic in the past

years, electromagnetic effects on turbulent toroidal momen-

tum transport have not received many dedicated investiga-

tions so far. Up to now, theoretical work has focussed on

mechanisms responsible for the generation of toroidal mo-

mentum transport. The transport is mainly given by the par-

allel velocity moment of the perturbed distribution function

times the fluctuating E�B velocity in the radial direction.

For local simulations in an up–down symmetric equilibrium

geometry and without toroidal rotation, the gyrokinetic equa-

tion is symmetric under certain symmetry transformations,

as is explained in Ref. 7. Since parallel velocity fluctuations

are an antisymmetric quantity, the momentum flux will be

exactly zero under the aforementioned conditions. A finite

momentum flux exists only when the symmetry in the

direction along the magnetic field line is broken. This can

be caused by a finite toroidal velocity gradient u0 ¼ �R2rX/=cs, where X/ is the angular rotation frequency of the

plasma. This symmetry breaking mechanism is the origin of

the diagonal part and it is usually expressed in terms of the

Prandtl number v/=vi defined as the ratio of viscosity to ther-

mal conductivity. The second mechanism originates from

the toroidal rotation itself and can be elegantly derived by

transforming to the reference frame, which moves with the

plasma.8 In such a frame, inertial forces, namely, the Coriolis

force and the centrifugal force, occur. While the latter plays

only a minor role in tokamak plasmas due to the quadratic

dependence on the Mach number u ¼ RX/=cs � uk=cs,

which is usually less than 0.5, the former is an important

contribution leading to an inward momentum flux, the so-

called Coriolis pinch. As a consequence, a radial gradient of

the toroidal velocity profile is generated even in the absence

of an external torque, but it requires a seed rotation. There

are additional leading order contributions to a finite momen-

tum flux, namely due to the particle flux effect, E�B shear-

ing, and up-down asymmetric plasma shape, but here the

focus is put on the modification of the diagonal and Coriolis

pinch contribution due to electromagnetic effects. These can

be expected to be the leading terms in the core of a low

q*¼qs=a, rapidly rotating plasma.7 Then, the radial flux C/

of momentum is given by RC/ ¼ v/u0 þ RV/u, where V/ is

the pinch velocity in the radial direction due to a finite back-

ground rotation u.

In Refs. 9–11, some aspects of the electromagnetic

effects on turbulent momentum transport have been investi-

gated under different approaches. In the present paper, we

focus on the identification of the physics mechanisms which

are responsible for electromagnetic effects on the Prandtl

number and the Coriolis pinch by means of a gyrokinetica)Corresponding author: clemente.angioni@ipp.mpg.de

1070-664X/2011/18(7)/072503/9/$30.00 VC 2011 American Institute of Physics18, 072503-1

PHYSICS OF PLASMAS 18, 072503 (2011)

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treatment. In this context, the influence of the self-consistent

parallel mode structure, whose importance was already

pointed out in Ref. 12, plays a crucial role in explaining the

behavior of both the pinch and the Prandtl number.

This paper is organized as follows. Section II discusses

the influence of electromagnetic effects and the self consist-

ent mode structure on both the Prandtl number and the Cori-

olis pinch. Section III discusses linear results obtained with

GYRO (Refs. 13 and 14) for the Prandtl number and the Cor-

iolis pinch, pointing out the role of the modification of the

averaged parallel wave vector with increasing be. These

results and their interpretation are supported by a semi-ana-

lytical gyrokinetic model in Sec. IV. A realistic case for typi-

cal high b hybrid scenario parameters in present day

tokamaks at mid-radius is considered in V. Finally, Sec. VI

draws the main conclusions.

II. INFLUENCE OF ELECTROMAGNETIC EFFECTSAND THE SELF CONSISTENT MODE STRUCTURE

The breaking of the symmetry in the parallel direction

has a strong influence on the parallel momentum flux.1,12 In

an electromagnetic description, extra terms proportional to

the vector potential Ak ¼ eAk=Te appear in the expression

for the momentum flux. The formula for the momentum flux

can be written as

C/¼��kyqsc

2s j/j

2=

�ð

d3vvk 1�2vjjXrþ v2jj X2

rþX2i

� �h i x�x�x�xgc

FMJ0

� ��;

(1)

where quantities with the hat are normalized (velocities to

the sound speed cs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiTe=mD

pand frequencies to cs=R),

h…i denotes the flux surface average, x is the (complex)

eigenfrequency, qs¼ csmD=(eB),

x� ¼ kyqs

R

Lnþ �� 3

2

R

LTþ vku

0� �

; (2)

includes the effect of a finite plasma rotation gradient with

� ¼ ðv2k þ v2

?Þ=2, and xgc ¼ k � vgc comprises both the paral-

lel and the perpendicular gyrocenter motion. The latter

includes the Coriolis drift in the presence of a finite toroidal

rotation of the plasma. Equation (1) has been derived in close

analogy to Eq. (6) in (Ref. 15). This electromagnetic expres-

sion is completely analogous to the electrostatic one with the

additional explicit electromagnetic terms proportional to

XðbeÞ, which is defined as the relation between the normal-

ized fluctuating electrostatic potential / ¼ e/=Te and the

parallel component of the fluctuating vector potential A,

A ¼ ðc=csÞX/ ¼ ðc=csÞ Xr þ iXi

h i/. This suggests that

through the inclusion and the symmetry properties of X new

contributions in the expression of the momentum flux may

account for a modification of the momentum flux when com-

pared to an electrostatic formulation. The ballooning struc-

ture of / shows a deviation when the toroidal rotation is

finite, as it was reported for instance in (Ref. 12). While this

behavior is not very strong for the real part of /(h), where h

denotes the ballooning angle, see Fig. 1(a), this is in particu-

lar visible for the imaginary part of /(h), as it is shown in

Fig. 1(b) for the GA-std case (see, e.g., Ref. 16) obtained

with GYRO (Refs. 13 and 14), with u ¼ 0:2 and u0 ¼ 0. The

imaginary part of the parallel component of the vector poten-

tial, however, does not exhibit a strong deviation from a per-

fect antisymmetry, see Fig. 1(d), while the real part does, as

shown in Fig. 1(c). Qualitatively, the same symmetry proper-

ties are also seen in cases with pure diagonal momentum

transport, for example, when taking u ¼ 0 and u0 ¼ 0:2. As a

consequence, X has to compensate the symmetry properties

through the dependence on kk and=or through additional

terms originating from the Coriolis drift. This implies that Xremains no longer fully antisymmetric, such that the addi-

tional terms modify the symmetry properties of Eq. (1) in

case of a finite toroidal rotation and=or its gradient.

However, the electromagnetic terms proportional to Ximpact the momentum flux only by a very small amount.

From the discussion in (Ref. 15), it follows that Ak, which is

mostly connected with the parallel electron dynamics, scales

asffiffiffilp

, where l ¼ m=mD. This factor is large for electrons

FIG. 1. (Color online) Ballooning dependence of the real (a) and imaginary

part (b) of the electrostatic potential / as well as the parallel component of

the vector potential Ak for u ¼ 0:2 [(c), (d)], respectively. The GA-std case

has been used.

072503-2 Hein et al. Phys. Plasmas 18, 072503 (2011)

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ffiffiffiffiffile

p� 60

�, while it is small for ions

ffiffiffiffili

p� 1

�. Since to-

roidal momentum transport is dominated by ion dynamics

due to the higher mass as compared to the electrons (momen-

tum transport isffiffiffiffiffile

psmaller for electrons), the aforemen-

tioned additional terms / X do neither influence the

magnitude of the “electrostatic” symmetry breaking nor do

they play a strong role through the additional phase shift,

since they are proportional toffiffiffiffili

pbe � be and therefore very

small for be values being reached in tokamaks. Thus, it is

expected that the influence of a finite be on momentum trans-

port is an indirect, “electrostatic-like” one.

The importance of the influence of the self-consistent

mode structure on momentum transport was discussed

recently in Ref. 12. Therein, a low field side gyrofluid model

was used in order to show that it is possible to formally com-

bine the Coriolis drift with the parallel dynamics. This is

done by combining parallel rotation velocity and parallel

wave number into one single term �u ¼ uþ kk, where

kk ¼ kkR=ð2kyqsÞ. This suggests that the two can actually be

interchanged. There is, however, an important difference

between the two. The Coriolis drift term is entirely deter-

mined by the prescribed background rotation velocity, while

the parallel wave vector has to be determined from the self-

consistent solution of the dispersion relation.

The following discussion is limited to the most unstable

mode corresponding to a particular choice of kk. This is a

common assumption made in the literature. It was shown in

Ref. 12, that for adiabatic electrons, the most unstable mode

satisfies �u ¼ 0 or

u ¼ �kk: (3)

According to this solution, the parallel dynamics is such that

it effectively eliminates the Coriolis drift from the equations.

This is referred to as the “compensation effect” in Ref. 12.

Such a compensation is indeed present, and gyrokinetic sim-

ulations with adiabatic electrons give zero momentum flux.

Simulations with kinetic electrons reveal that the compensa-

tion is not complete such that a finite momentum flux is

obtained. The mode structure along the field line adjusts and

generates a finite distortion, which can be described by a fi-

nite average parallel wave number hkki, defined as

hkki ¼ �i

Rq

Ðdh/�@h/Ðdh/�/

; (4)

where h is the angle-like coordinate, which measures the dis-

tance along the field line. The compensation effect is

(almost) perfect for adiabatic electrons, in agreement with

the simple gyrofluid model. This shows that the breaking of

symmetry is a necessary but not sufficient condition for the

generation of a finite momentum flux. Since electromagnetic

effects modify the parallel mode structure through a decrease

of the mobility of passing electrons, as it can be already seen

from Fig. 1, it has to be expected that also the momentum

pinch will change, since kk beð Þ. In the next two sections, it

will be demonstrated by gyrokinetic simulations and analyti-

cal derivations that the modification of kk due to electromag-

netic effects is indeed the leading order effect responsible for

the modification of momentum transport.

From the fluid equation12

xvþ 4vþ 2�unþ 2�uT ¼ u0 � 2�u½ �/ (5)

also the effect of a finite be on the Prandtl number can be

identified. The toroidal rotation gradient is removed from the

parallel force balance equation if u0 � 2hkki ¼ 0, which

implies that in this case hkki has to be a finite positive num-

ber for this compensation to take place, in contrast to the

Coriolis pinch case. Consequently, the specific impact of a fi-

nite hkki on the electromagnetic behavior of the Coriolis

pinch and the Prandtl number is discussed in the remainder

of the present paper in more detail using linear gyrokinetic

simulations as well as an analytical gyrokinetic model.

III. LINEAR SIMULATION RESULTS FOR CORIOLISPINCH AND PRANDTL NUMBER

The aim of this section is the investigation of the de-

pendence on be of both the Prandtl number v/=vi and the

Coriolis pinch number RV/=v/. In addition, the numerical

calculations allow us to explore the modifications in the av-

erage parallel wave number induced by the increase of be,

which can impact the momentum transport, as it was antici-

pated in the Sec. II. To this purpose, both Prandtl number

and Coriolis pinch are calculated with GYRO for the GA-std

case, as well as the corresponding average parallel wave vec-

tor hkk beð Þi for the mode distortion andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihk2kðbeÞi

qfor the

mode width, are computed. The discussion is focussed on the

behavior on be keeping fixed b0 ¼ 8p@rp=B2¼ 0. The param-

eters u ¼ 0 and u0 ¼ 0:2, or u ¼ 0:2 and u0 ¼ 0, have been

used in order to compute the momentum flux for both the

Prandtl number and the Coriolis pinch, respectively. The

results are shown for the Prandtl number in Figs. 2(a) and

2(c) and for the Coriolis pinch in Figs. 2(b) and 2(d), respec-

tively. Both of them decrease in size with increasing be in

the ITG branch (be � 0.6%), which is predominantly due to

the change in the mode distortion kk beð Þ with increasing be,

FIG. 2. (Color online) Behavior of the Prandtl number v/=vi (a) and the

Coriolis pinch number RV/=v/ (b) with the corresponding wave vectors

kk beð Þ (c) andffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihk2kðbeÞi

q(d), respectively, as a function of be.

072503-3 Gyrokinetic study of electromagnetic effects Phys. Plasmas 18, 072503 (2011)

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as it will be identified in Sec. IV. It is clear that in case of the

Prandtl number hkki is positive and for the Coriolis pinch

hkki is negative. The increase of hkki with be in absolute val-

ues means a stronger symmetry breaking of the electrostatic

potential /, which can be already deduced from Fig. 1(a).

Therefore, the main reason for the decrease of both the

Prandtl number and the Coriolis pinch with increasing be in

the ITG branch is the increase of the averaged hkki, thus

increasing the compensation of both the influence of the to-

roidal velocity shear and the toroidal velocity with raising

be, respectively. The different signs of hkki are fully consist-

ent with the simple fluid model discussed in Sec. II and

reflect the fact that the symmetry of the electrostatic poten-

tial in the case of a finite u0 and u ¼ 0 is broken in different

directions with respect to the ballooning angle as compared

to a finite u and u0 ¼ 0. Approaching the kinetic ballooning

mode (KBM) branch (be> 0.6%), the Prandtl number

increases again, and the Coriolis pinch number even reverses

sign with a subsequent decrease. The be dependence in the

KBM branch, however, cannot be described by the change in

the average parallel wave number, since hkki is practically

zero, which can be expected because for ballooning modes

Ak is the dominant fluctuating field. The latter is less affected

by a finite toroidal rotation because the electron motion,

which mainly determines Ak, is much faster than u. In Sec.

IV, it will be shown that this behavior is mainly due to a

change in the mode real eigenfrequency (see also Fig. 3(a)).

In order to identify the parameter domain in which the

dependences on be of the Prandtl number and the Coriolis

pinch are particularly strong, it is instructive to explore their

behavior as a function of be for different values of the local

safety factor q. This is done by keeping all other parameters

FIG. 3. (Color online) Electromagnetic behav-

ior of c (open symbols) and xr (full symbols (a),

parallel wavenumbers hkki for both (u ¼ 0,

u0 ¼ 0:2, open symbols) and (u ¼ 0:2, u0 ¼ 0,

full symbols (b), Prandtl number v/=vi (c) and

Coriolis pinch number RV/=v/ (d) for different

values of the safety factor q, respectively. In (e)

and (f), the Prandtl number and the Coriolis

pinch are shown versus q2be.

072503-4 Hein et al. Phys. Plasmas 18, 072503 (2011)

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fixed at their values in the GA-std case and by scanning

q¼ 1.1, 1.5, 2.0, and 2.5. The results are shown in Fig. 3.

From Fig. 3(a), it is apparent that with increasing safety factor

q, the KBM becomes the most unstable mode at progressively

lower values of be. Both the Prandtl number and the Coriolis

pinch numbers depend very weakly on be for the cases with

low safety factor, and their dependence on be becomes stron-

ger with increasing safety factor. We observe that the behavior

of the Prandtl number and the Coriolis pinch in the ITG

branch is mainly determined by the behavior of hkki, shown in

Fig. 3(b). Also, the quantitative agreement is noteworthy. For

instance, in the case of the Coriolis pinch at q¼ 2.5, the hkkiincreases compared to the electrostatic case in absolute values

up to approximately �0.15 for the last ITG dominated value

at be¼ 0.4%. This leads to the conclusion that the pinch

should be drastically reduced, since u ¼ 0:2, and is actually

demonstrated by the corresponding curve in Fig. 3(d).

Approaching the KBM branch, the Coriolis pinch num-

ber even reverses its sign as a consequence of the effects of

hkki, hk2ki

0:5and mainly xr. However, the drastic change of

the pinch in the KBM branch might not be relevant for

experiments, since it is unlikely that the plasma core of

tokamaks can operate in this regime. In the experimentally

relevant ITG regime, the reduction of the pinch is sizeable

for this case. The Prandtl number remains positive for both

the ITG and the KBM branch. These behaviors will be

explained using the gyrokinetic formula Eq. (17), defined in

Sec. IV.

It is of interest to plot the same results as a function of

q2be, the parameter which really determines the strength of

perpendicular magnetic field fluctuations, as shown for

instance in Ref. 17. This is done in Fig. 3(e), (f) for both the

Prandtl number and the Coriolis pinch, respectively. All the

curves tend to overlay, particularly in the KBM branch, and

the same value of q2be is found at the onset of the KBM. In

the ITG branch, the curves for the different values of q ex-

hibit small differences, already at their electrostatic values.

This can be directly attributed to the dependence on q of

hkki / 1=q, see Fig. 3(b), such that for high values of the

safety factor the compensation effect is weaker leading to

larger Prandtl and Coriolis pinch numbers.

Figs. 3(e) and 3(f) allows us to conclude that the

dependences of both the Prandtl and the Coriolis pinch num-

bers on be are particularly strong in proximity to the onset of

the KBM branch, whereas they remain rather shallow in con-

ditions which are far from it, like in the previous cases with

low values of the safety factor.

Finally, also the electromagnetic influence under trapped

electron mode (TEM) conditions has been investigated. It is

found that both the Prandtl number as well as the Coriolis

pinch are almost unaffected by a finite value of be. This is

attributed to the fact that TEMs produce smaller fluctuations

of the magnetic field lines. Consequently, the averaged paral-

lel wave vector hkki remains constant with increasing be in

the TEM branch. Moreover, it can be deduced from Fig. 3(a)

that since for TEM modes the linear growth rate remains

almost constant with increasing be, while ITG modes are sta-

bilized, the KBM becomes the dominant mode at higher val-

ues of be.

IV. ANALYTICAL MODEL

In this section, a general analytical gyrokinetic expres-

sion for the parallel momentum flux is derived for electro-

static fluctuations. We shall show, then, that by the

application of this electrostatic formula of the momentum

flux, the fully electromagnetic numerical results can be quan-

titatively reproduced provided that the actual values of the

parallel wave numbers and eigenfrequencies as a function of

be, as found in the numerical calculations, are used in the an-

alytical formula. This will allow us in particular to demon-

strate the leading role played by hkki in determining the

dependence on be of both the Prandtl and the Coriolis pinch

numbers in the presence of ITG instabilities.

The present analytical kinetic treatment consistently

introduces the effect of both a finite mode distortion as well

as a finite mode width due to u0 and=or u for the first time.

As a result, the electromagnetic behaviors of both v/=vi and

RV/=v/ are well recovered.

First, it is shown how the relevant physics can be

extracted from a generic kinetic equation by performing

appropriate averages and applying a parametrization of the

mode structure along the direction parallel to the field lines

h. In the following, the vk, v? representation of the distribu-

tion function is used. The mirror force term is neglected

here. As starting point a (complex-valued) linear eigenvalue

problem is considered, combining a kinetic equation for the

non-adiabatic part of the perturbed ion distribution function

h with the quasi-neutrality condition

�ixþ v@h þ iDðv2; kv; h2Þ� �

hðv; hÞ ¼ Lðv2; h2ÞUðhÞ; (6)

e

1

Teþ 1

Ti

UðhÞ ¼

ðhðv; hÞdv; (7)

where v¼ [�1, þ1], k 1, L is an arbitrary linear right

hand side operator (which will represent the “source” term

of the gyrokinetic equation). Notice that, if k¼ 0, this

equation admits two classes of independent solutions:

h(v, h) ¼6 h(� v, �h) and U(h) ¼6 U(�h), that is sym-

metric or antisymmetric with respect to the (v, h) ! (�v,

�h) transformation. In the case of typical ITG or TEM

gyrokinetic turbulence, the symmetric mode is the most

unstable one, while the antisymmetric one is stable, but it

has to be kept in mind that in the presence of a convective

type non-linearity @yU@xh, the equation now admits

only the antisymmetric solution, which means that the

actual mode structure will exhibit strong oscillations

between the two symmetry solutions. In the following, the

focus is put on the symmetric solution. On the other hand,

if k= 0, the solution will not have a defined symmetry.

However, two types of asymmetries can be distinguished:

first, an asymmetric distortion of the mode (k), and sec-

ond a net shift off the reference surface at h¼ 0, again

which scales as k. Moreover, the impact of k on both the

mode width and the average curvature drift operator Dgoes as k2. When representing the mode U(h) as

UðhÞ ¼ U0UeiF, where U0 is a complex number, while Uand F are real functions of h, one can show that the rele-

vant averages are given by

072503-5 Gyrokinetic study of electromagnetic effects Phys. Plasmas 18, 072503 (2011)

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hkki ¼ �i

ÐU�@hUdhÐjU2jdh

¼Ð

U2@hFdhÐU2dh

;

hk2ki ¼ �

ÐU�@2

hUdhÐjU2jdh

¼

Ð@hU �2þU2 @hFð Þ2h i

dhÐU2dh

;

hhi ¼ÐjUj2hdhÐjU2jdh

¼Ð

U2hdhÐU2dh

;

hf ðhÞi ¼ÐjUj2f ðhÞdhÐjU2jdh

¼Ð

U2f ðhÞdhÐU2dh

;

(8)

from which one sees that, while the average parallel wave

number is given only by the phase factor F of the mode, the

average shift hhi is given only by the shape factor U, and the

average parallel wave number squared is given by both the

shape and the phase factors first derivatives squared.

Let us assume first that the two operators D, L in Eq. (6)

do not depend on h. In this case, Eq. (6) admits plane waves

/ eikh as eigenvectors, and as such, the mode structure can

be parameterized as a two-point (in k space) function

UðhÞ ¼ U�eið�kþdÞh þ UþeiðkþdÞh; (9)

where d k kð Þ takes into account a preferential direction

of motion of the two plane waves. Notice that jUþj¼ jU�j,since the two plane waves have to carry the same energy con-

tent, or alternatively they have the same statistical weight,

while the two eigenvectors Uþ,� can differ by a phase factor

eir, i.e. Uþ¼ eirU�. One can compute the different averages

as in equations (8) to find, to first order in k: hkki ¼ d, k2k ¼ k2,

hhi � �=ðU�þU�Þ=ð2k Uþj j2Þ sin rð Þ= 2kð Þ r= 2kð Þ. Both

d and r will be given as self-consistent solution of the eigen-

value problem provided by Eq. (6) and Eq. (7). If D, L do

depend on h, as it is usually the case for the curvature drift

and the gyro-average operators, one can proceed in the same

way, but preliminarily one has to replace D(h) with an appro-

priate average D, such that one can then apply the plane wave

parametrization. This averaging is performed by means of a

shape function G(h), through hDi ¼ÐðG2DdhÞ=

ÐG2dh,

which should resemble the actual mode shape. To first order

in k, the best fit can be obtained with a Gaussian-like shape of

the type: G ¼ e�14ðkhÞ2 , where an average width through k2 has

been used. Note that this whole procedure reminds of the one

applied in the GLF23 model,18 where in our case, mode

asymmetries for the evaluation of momentum transport are

consistently taken into account to first order in uk. More

recently, a more general expansion procedure in terms of Her-

mite polynomials has been used in the new TGLF model,19

which takes into account mode asymmetries in a consistent

form. It can be shown that the plane wave approach and the

Hermite polynomials approach are equivalent in the limit of

small hhi and small uk (i.e., small distortion close to the reso-

nant surface). The justification for the choice of G becomes

clear as one compares the averages computed with a numeri-

cally resolved mode structure and this approximated choice.

Going back to the system described by Eqs. (6) and (7),

and considering the previous discussion, one may choose to

factorize the h dependence of h and U in the following form,

h(v, h)¼Hn(v)Bn(h)G(h), and U(h)¼U0nBn(h)G(h), where ncan have the two values 61. The function Bn(h) is expressed

in the terms of plane waves Bn(h)¼ ei(nkþ d)h, where n ¼6 1

and G(h)¼ e�(kh)2=4. The physical meaning of this splitting

of the solution in two separate branches (identified by n),

and the choice of the h-shape of the Bn functions and of G is

the following: the mode structure U(h) can be thought of as

composed by statistically independent waves that propagate

in the two opposite directions 6h, as evident from the re-

spective parallel phase velocity vx=(6ji@hj). Each solu-

tion carries a Gaussian envelope whose width is determined

by w 1=ffiffiffiffiffi@2

h

q, which explains why k appears in both Bn

and G. From the expression of Bn, one can evaluate

@hBn¼ i(nkþ d)Bn. First, an average of D is performed, then

the mode structure parametrization is employed. One gets

two equations for the two values of n, which read

�ixþ iðnk þ dÞvþ ihDiðv2; kvÞ� �

HnðvÞ ¼ hLiðv2ÞU0n;

(10)

where hDi ¼Ð

G2Ddh=Ð

G2dh, with no dependence on n in

the limit k 1.

This procedure is applied to approximate the gyrokinetic

equation, which for our purpose is taken in circular geome-

try, in order to explain the behavior of the Prandtl number

and the Coriolis pinch. Starting point is the gyrokinetic equa-

tion for the nonadiabatic part of the perturbed ion distribu-

tion function ~hi, which, replacing appropriate notation in Eq.

(10), is given by

�ixþ iðnk þ dÞvk þ i hxd;0i þ 2hxDivku �� �

~hin

¼ �i x� x�ð ÞFMJ0/n; (11)

where frequencies are normalized to cs=R, FM is the Max-

wellian and J0 ¼ J0 k?qð Þ, and

xd;0 ¼ xD v2k þ v2

?=2� �

: (12)

We redefine

nk þ d ¼ 2hxDi�kk;n ¼ 2hxDi nffiffiffiffiffiffiffiffihk2ki

qþ hkki

� �; (13)

where Eq. (4),

hk2ki ¼ �

1

ðRqÞ2

Ðdh/� @2

h/Ðdh/�/

¼ � 1

2ðRqÞ2Xn¼6

B�n @2hBn

� �(14)

and

xD ¼ kyqs cos hþ sh sin hð Þ (15)

have been used. The normalization of the wave vectors hkkiand hk2

ki is to cs= 2xDð Þ and c2s= 2xDð Þ2, respectively. Thus,

one can formally write the solution to Eq. (11) as

~hin ¼x� x�

x� xd;0 � 2hxDivk �kk;n þ u �FMJ0/n: (16)

072503-6 Hein et al. Phys. Plasmas 18, 072503 (2011)

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As discussed before, k ¼ffiffiffiffiffiffiffiffihk2ki

qrepresents the average width

of the eigenfunction, while the splitting n ¼6 1 gives the two

solutions of the eigenvalue problem (Eq. (11) coupled with the

Poisson equation for /n), which should provide the same com-

plex eigenfrequency x, or the same growth rate and real fre-

quency. Since we are solving an eigenvalue problem with

three equations (two signs of n plus Poisson equation), for one

unknown (eigenfrequency), it seems that there is an inconsis-

tency. That is only apparent, since one can show that, when

ðu ¼ u0 ¼ 0; d ¼ 0Þ, the two equations for n ¼6 1 are exactly

the same equation, re–mapping vk ! �vk. On the other hand,

if u 6¼ 0 or u0 6¼ 0, this is not possible, and as such one has to

solve an eigenvalue problem for two unknowns, i.e., the com-

plex frequency and dðu; u0Þ. From this discussions, it emerges

that the parameter d plays the role of a second eigenvalue,

which has a finite value when the parallel velocity, or its gradi-

ent, is finite. Since d ¼ hkki, it represents the average parallel

wavenumber, or an average asymmetric distortion of the mode

structure around h¼ 0 of the eigenfunction, which will be a

function of u, u0, linear in it for u, u0 1.

The expression for the generalized flux of parallel

momentum and energy can be written in the form

C/;Q� �

¼ � 1

2

Xn¼6

= kyqscs/�n

ðd3v csvk; c

2s �

� �~hin

� �

¼ � kyqsc2s j/j

2

2

Xn¼6

=ð

d3v csvk; c2s �g

��

� x� x�ð ÞFMJ0

x� xd;0 � 2hxDivk nffiffiffiffiffiffiffiffihk2ki

qþ hkki þ u

� �9>=>;:(17)

Here, the sum over the two signs of n takes into account that

both solutions of the eigenvalue problem contribute to the

total flux. This is necessary since one can easily see that, if

u ¼ u0 ¼ 0 (and so hkki ¼ 0), then C/¼ 0. In fact, putting

both the toroidal rotation u and its gradient u0 and conse-

quently the mode distortion hkki to zero, a tedious, but

straightforward analysis of Eq. (17) reveals that the momen-

tum flux is exactly zero. Moreover, it is evident from Eq. (16)

that hkki þ u can be combined to a single variable, such that

the aforementioned “compensation effect” can take place.

This may be seen in a more obvious way in the limit in

which the mode widthffiffiffiffiffiffiffiffihk2ki

qis neglected in the expression

for the momentum flux, Eq. (17), compared to the distortion

hkki. Then, the momentum flux is given by

C/ ¼� kyqsc2s j/j

2=

�ð

d3vvkx� x�

x� xd;0 � 2hxDivk hkki þ u �FMJ0

( ):

(18)

Like Eq. (17), this formula includes the result that the Corio-

lis pinch vanishes for perfect compensation, namely,

hkki þ u ¼ 0. It is also clear that it is impossible to fully bal-

ance the effect of a finite u0 through hkki, consistent with the

numerical results.

At this point, the analytical results of Eq. (17), which

are expressions for the electrostatic E�B fluxes, are com-

pared with the fully electromagnetic GYRO results for the

GA-std case. Inserting the parameters hkki, hxdi, hk2ki

0:5, xr,

and c, which are obtained from GYRO and all depend on be,

into the analytical expression of the flux, Eq. (17), the behav-

iors of the Prandtl number and the pinch with increasing be

are well recovered for the GA-std case, as it can be seen

from Figs. 4(a) and 4(b), where the corresponding simulation

results are shown in Figs. 4(c) and 4(d). Especially for the

Coriolis pinch in the ITG branch, the quantitative agreement

is very good, while for the Prandtl number, the effect of a fi-

nite be is slightly overestimated by the analytical model, still

remaining in a reasonable quantitative agreement. Thereby,

we can unambiguously identify through the analytical model

the key parameters governing the observed dependences in

the numerical results. The decrease of the Prandtl number

and the Coriolis pinch in the ITG branch with raising be is

found to be predominantly due to the increase of hkki in

absolute values. For be& 0:6%, it turns out that the pinch

changes its sign due to an interplay of a smaller hkki in nega-

tive direction in combination with a much higher mode real

frequency. When be is raised even further, the KBM growth

rate increases and the real eigenfrequency decreases, leading

to a decrease of the Coriolis pinch and an increase of the

Prandtl number. Therefore, since the influence of hkki is very

weak in the KBM branch, the physics mechanism for a finite

momentum flux has to be attributed to a finite uk, and by the

changes of xr and c. In conclusion, the electrostatic analyti-

cal model with the inclusion of hkki, hxdi, hk2ki

0:5, xr, and c

as functions of be works satisfactorily well in order to

explain the main physics mechanism for the electromagnetic

FIG. 4. (Color online) Theoretical model using the gyrokinetic formalism to

compute the be-behavior of the Prandtl number (a) and the Coriolis pinch (c)

for the GA-std case. The numerical values for hkki, hk2ki

0:5, D, c, and xr

obtained by GYRO have been used. In (b) and (d), the corresponding simu-

lation results for Prandtl number and Coriolis pinch are shown.

072503-7 Gyrokinetic study of electromagnetic effects Phys. Plasmas 18, 072503 (2011)

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behavior of the Prandtl number and the Coriolis pinch num-

ber. In particular, the dependences of both Prandtl and Corio-

lis pinch numbers on be in the presence of ITG instabilities

are encapsulated in the dependence of hkki on be.

Here below, the impact of the parallel dynamics on the

dependences of both the Prandtl and the Coriolis pinch num-

bers is further investigated. Since the main difference

between kinetic electrons and adiabatic electrons in the

description of the momentum flux is connected with the pres-

ence of trapped electrons, it has to be expected that the mo-

mentum flux depends on the trapped particle fraction, at least

for the Coriolis pinch part.12 Figure 5 shows both v/=vi as

well as RV/=vi as a function offfiffiffiffiffiffiffiffir=R

pbeing a measure for

the fraction of trapped particles, with other parameters kept at

their values of the GA-std case. The Coriolis pinch exhibits a

strong reduction with decreasingffiffiffiffiffiffiffiffir=R

p. This is obtained for

all the three values of be. In the electrostatic limit, the extrap-

olation of the GYRO results reveals that for purely passing

electronsffiffiffiffiffiffiffiffir=R

p� �, which are fully adiabatic due to be¼ 0,

the pinch is indeed at the null, consistent with.12 For finite be,

however, passing electrons are no longer fully adiabatic, such

that a reduction of the pinch is obtained (or even a reversal

when the trapped particle fraction is low). In this context, it

has to be underlined that, while the momentum flux is mainly

given by the ions due to their high inertia, the electrons play a

crucial role through their influence on the parallel mode

structure and therefore on hkki, which is clearly demonstrated

by Fig. 5. The results are consistent with those obtained in

Fig. 3. The Prandtl number, on the other side, does not go to

zero for electrostatic fully passing electrons (the limitffiffiffiffiffiffiffiffir=R

p! 0), which shows that the effect of a finite hkki does

not fully balance a finite u0. The decrease of v/=vi with

increasing be is nevertheless visible. The unphysical negative

values of the Prandtl number, obtained for be¼ 0.5% at smallffiffiffiffiffiffiffiffir=R

p. 0:15, are caused by the fact that while the trapped

particle fraction is reduced because of moving towards the

magnetic axis, the other parameters are held fixed. This leads

to large values of hkki with simultaneously low growth rate

and real eigenfrequency. From the analytic model Eq. (17), it

can be shown that the combination of the latter parameters

and in particular the decrease of the real eigenfrequency

indeed produces a negative Prandtl number.

V. EXPERIMENTAL HIGH BETA CASE

Lastly, also the effect of a finite be on Prandtl and Corio-

lis pinch numbers for a realistic set of plasma parameters is

shown, typical of high b H-mode plasmas in hybrid scenarios

at ASDEX Upgrade (e.g., Ref. 20). A local Miller equilib-

rium21 with r=R¼ 0.1667, q¼ 1.5, s¼ 0.8, elongation

j¼ 1.4, sj¼ (r=j)@j=@r¼ 0.05, triangularity d¼ 0.05 with

sd¼ r@d=@r¼ 0.07 and Shafranov shift D ¼� 0.17 is

assumed. The local parameters for the two species plasma

consisting of deuterons and electrons are R=Ln¼ 2,

R=LTi¼ 7, R=LTe¼ 6, and Ti=Te¼ 1. The collisionality is

(R=cs)�ei¼ 0.06. For this case, both the Prandtl number and

the Coriolis pinch are calculated by linear simulations at

kyqs¼ 0.3. The results are shown in Fig. 6. The grey shaded

area indicates values of be, which are achieved in high b(hybrid scenario) operation in present day tokamaks. Due to

the fact that this area can be close to the onset of KBMs, a

substantial decrease of the Coriolis pinch and a small

decrease of the Prandtl number is obtained. Similar results

are obtained for a wide range of the binormal wavenumber

spectrum (kyqs< 0.8). Also, preliminary results from nonlin-

ear GYRO simulations confirm these findings. In high boperational scenarios in present day devices, and therefore

usually in the presence of a substantial neutral beam injec-

tion (NBI) torque, the effects of a decreasing Prandtl number

and Coriolis pinch with increasing be could balance, since

the first implies an increase of the toroidal rotation velocity

gradient, while the second a decrease. In the absence of sig-

nificant NBI torque, which is expected to be the case in a

fusion reactor, the toroidal velocity gradient will be mainly

determined by the ratio between the off-diagonal contribu-

tions to the momentum flux and the diagonal viscosity. Pres-

ent investigations lead to the prediction of a decrease of the

toroidal velocity gradient in operational conditions close to

the onset of KBMs.

FIG. 5. (Color online) Prandtl number and Coriolis pinch as a function offfiffiffiffiffiffiffiffir=R

pfor different values of be in the GA-std case. Dashed and full lines

denote linear and quadratic (see text) fits to the numerical results on the

Prandtl number and the Coriolis pinch obtained with GYRO, respectively.

FIG. 6. (Color online) Prandtl number (a) and Coriolis pinch (b) as a func-

tion of be for linear simulations of an ASDEX-Upgrade hybrid scenario case

(parameters see text). The grey shaded area indicates values of be, which are

typical for H-mode plasmas in hybrid scenario operation in present day

tokamaks.

072503-8 Hein et al. Phys. Plasmas 18, 072503 (2011)

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For values of be between 1.5% and 2.2%, a different kind

of mode is found, namely, a micro-tearing mode (MTM).

These are characterized by the fact that magnetic field pertur-

bations form small scale magnetic islands such that for them

the electron heat transport by magnetic flutter is large while

the other transport channels go practically to zero. For this

reason, the MTM branch is not shown in Fig. 6.

VI. SUMMARY AND CONCLUSIONS

The effects of a finite be on the Prandtl number and the

Coriolis pinch have been investigated by means of a gyroki-

netic treatment. It has been pointed out that the additional

electromagnetic terms proportional to be provide only very

small corrections to the toroidal momentum flux. Thus, the

effects of a finite be on the Prandtl and the Coriolis pinch

numbers can be considered as “electrostatic-like,” and are

encapsulated in the dependence on be of other parameters,

like the average parallel wave number and the eigenfre-

quency. From linear gyrokinetic simulations, it is shown that

the modification of the parallel mode structure due to a nona-

diabatic response of passing electrons, which results in

changing average parallel wave vector kk with increasing be,

is the dominant effect responsible for the decrease of both

the diagonal momentum transport as well as the Coriolis

pinch in the presence of ITG instabilities. The decrease is

particularly strong in the vicinity to the onset of the KBM

branch, while far from it can be rather shallow. At the KBM

onset, the Coriolis pinch even reverses its direction, due to

the jump in eigenfrequency. In contrast, in the presence of

TEM instabilities, electromagnetic effects are practically

negligible. The dependences of the Prandtl and the pinch

number as a function of be are explained by an analytical

gyrokinetic analysis in the electrostatic limit (thus neglecting

the explicit electromagnetic terms in the momentum flux),

but allowing the parallel wave vector (and the eigenfre-

quency) to depend on be, showing a remarkable quantitative

agreement with the simulation results. A linear case with

typical parameters at mid-radius of present tokamaks in high

b (hybrid scenario) operation reveals that a small decrease of

the Prandtl number and a strong decrease of the Coriolis

pinch in the ITG branch can occur at the experimentally

achieved values of be, which can be close to the onset of

KBMs in hybrid scenario operation.20 The present linear

results have to be confirmed by nonlinear electromagnetic

gyrokinetic simulations, which for the study of momentum

transport become extremely expensive, since reliable time

averages of turbulent momentum fluxes require very long

time windows. At present, preliminary nonlinear simulations

confirm the Prandtl number and Coriolis pinch dependences

on be found in this linear study and will be subject of future

research.

ACKNOWLEDGMENTS

The authors thank A. Bottino and B.D. Scott for fruitful

discussions.

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