PHYSICS OF PLASMAS 15, 072308 共2008兲
Collisionality dependence of the quasilinear particle flux due
to microinstabilities
T. Fülöp,1 I. Pusztai,1 and P. Helander2
1
Department of Radio and Space Science, Chalmers University of Technology
and Euratom-VR Association, Göteborg, Sweden
2
Max-Planck-Institut für Plasmaphysik, Greifswald, Germany
共Received 28 January 2008; accepted 22 May 2008; published online 30 July 2008兲
The collisionality dependence of the quasilinear particle flux due to the ion temperature gradient
共ITG兲 and trapped electron mode 共TEM兲 instabilities is studied by including electron collisions
modeled by a pitch-angle scattering collision operator in the gyrokinetic equation. The inward
transport due to ITG modes is caused mainly by magnetic curvature and thermodiffusion and can be
reversed as electron collisions are introduced, if the plasma is far from marginal stability. However,
if the plasma is close to marginal stability, collisions may even enhance the inward transport. The
sign and the magnitude of the transport are sensitive to the form of the collision operator, to the
magnetic drift normalized to the real frequency of the mode, and to the density and temperature
scale lengths. These analytical results are in agreement with previously published gyrokinetic
simulations. Unlike the ITG-driven flux, the TEM-driven flux is expected to be outwards for
conditions far from marginal stability and inwards otherwise. © 2008 American Institute of Physics.
关DOI: 10.1063/1.2946433兴
I. INTRODUCTION
Density peaking in tokamak plasmas has been shown to
decrease with increasing collisionality in ASDEX Upgrade1
and JET 共Joint European Torus兲2 H-modes.3–5 These experimental results suggest that the particle transport, which is
usually dominated by ITG- and TEM-driven turbulence, depends on the collisionality, although it has been suggested
that the source from ionization may also play an important
role.6,7
In the collisionless limit, numerical simulations of ITGmode driven turbulence give an inward particle flux in fluid,
gyrofluid, and gyrokinetic descriptions. The inward flow is
caused mainly by magnetic curvature and thermodiffusion.
However, nonlinear gyrokinetic calculations show that even
a small value of the collisionality affects strongly the magnitude and sign of the anomalous particle flows.8 The inward
particle flow obtained in the collisionless limit is rapidly converted to outward flow as electron-ion collisions are included. Linear gyrokinetic calculations with GS2 and a quasilinear model for the particle fluxes9 have confirmed the
strong collisionality dependence of the quasilinear particle
flux for small collisionalities and show a good agreement
with nonlinear gyrokinetic results from Ref. 8. Both gyrokinetic models find that the total particle flux becomes directed
outward for much smaller values of collisionality than the
lowest collisionality presently achieved in tokamaks. This
means that according to these simulations, for present tokamak experiments the particle flow should be outward.
The present paper addresses the collisionality dependence of the quasilinear flux due to ITG and TEM modes.
The aim is to derive analytical expressions for the quasilinear
flux to show explicitly the dependence on collisionality, density, and temperature gradients, so that the sign and magnitude of the flux can easily be estimated. We focus on the
1070-664X/2008/15共7兲/072308/9/$23.00
collisionality dependence of the direction and the magnitude
of the quasilinear flux, and give approximate analytical expressions for weakly collisional plasmas with large aspect
ratio and circular cross section.
The collisionality dependence has previously been studied in Refs. 10 and 11 by approximating the collision operator with an energy-dependent Krook operator. The main difference between this paper and the references above is the
form of the collision operator. Here we use a pitch-angle
scattering collision operator, but we include the results for
the Krook operator for comparison and completeness. As we
will show here, the form of the collision operator determines
the scaling with collisionality and therefore affects the collisionality threshold at which the particle flow reverses. The
eigenfrequency and growth rate of the modes are only
weakly dependent on the collisionality,8 and in this paper we
do not analyze the dispersion relation and the stability
boundaries, but instead focus on the quasilinear particle flux
driven by the mode. The collisionality dependence of the
quasilinear flux due to the TEM instability has been studied
in Ref. 12 using a pitch-angle scattering collision operator,
and here we generalize the expression presented there by
including the magnetic drift.
The structure of the paper is the following: In Sec. II the
general gyrokinetic formalism is presented. In Sec. III approximate solutions of the gyrokinetic equation are given in
the limit of high mode numbers and the perturbed electron
density is calculated. In Sec. IV the quasilinear flux is calculated and the effect of collisions is discussed. The possibility
of flux reversal, comparison with previous work and the validity of our approximations are discussed in Sec. V. Finally,
the results are summarized in Sec. VI.
15, 072308-1
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072308-2
Phys. Plasmas 15, 072308 共2008兲
Fülöp, Pusztai, and Helander
II. GYROKINETIC EQUATION
We consider an axisymmetric, large aspect ratio torus
with circular magnetic surfaces. The nonadiabatic part of the
perturbed distribution function is given by the linearized gyrokinetic equation10
⑀ = r / R. Performing the orbit average on the scattering
operator, Eq. 共4兲 becomes
共␻ − 具␻De典兲ge0 −
共6兲
v储 ⳵ga
− i共␻ − ␻Da兲ga − Ca共ga兲
qR ⳵␪
ea f a0
共␻ − ␻T a兲␾J0共za兲,
=−i
*
Ta
共1兲
where ␪ is the extended poloidal angle, ␾ is the perturbed
electrostatic potential, f a0 = na共ma / 2␲Ta兲3/2exp共−w / Ta兲 is the
equilibrium Maxwellian distribution function, w = mav2 / 2,
na, Ta, and ea are the density, temperature, and charge of
species a, respectively, ␻*a = −k␪Ta / eaBLna is the diamagnetic frequency, ␻T a = ␻*a关1 + 共w / Ta − 3 / 2兲␩a兴, ␩a = Lna / LTa,
*
Lna = −关⳵共ln na兲 / ⳵r兴−1, LTa = −关⳵共ln Ta兲 / ⳵r兴−1, are the density
and temperature scale lengths, respectively, k␪ is the poloidal
2
/ 2 + v2储 兲共cos ␪ + s␪ sin ␪兲 / ␻caR is
wave-number, ␻Da = −k␪共v⬜
the magnetic drift frequency, ␻ca = eaB / ma is the cyclotron
frequency, q is the safety factor, s = 共r / q兲共dq / dr兲 is the magnetic shear, r is the minor radius, R is the major radius, J0 is
the Bessel function of order zero, and za = k⬜v⬜ / ␻ca.
III. PERTURBED ELECTRON DENSITY RESPONSE
where Ĵ = E共␬兲 + 共␬ − 1兲K共␬兲 and ␶ˆ B = K共␬兲. We introduce
a parameter ␯ˆ ⬅ ␯e / ␻0⑀, where ␻0 = ␻ / y, y = ␴ + i␥ˆ ,
␴ = sign共Re兵␻其兲 denotes the sign of the real part of the eigenfrequency, and ␥ˆ = ␥ / ␻0 is the normalized growth rate. The
equation for ge0 is
⬙ + 共ln Ĵ兲⬘ge0
⬘ 兲+i
␯ˆ 共ge0
2␰ ⳵
⳵
␰␭ ⬅ ␯e共v兲L,
B ⳵␭ ⳵␭
ghom共␬兲 =
i共␻ − 具␻De典兲ge0 + 具Ce共ge0兲典 =
ie具␾典 T
共␻ e − ␻兲f e0 ,
*
Te
共3兲
where 具¯典 is the orbit average. The circulating electrons are
assumed to be adiabatic, while in the trapped region ge0 is
given by
共␻ − 具␻De典兲ge0 −
=−
2i␯e冑2⑀ ⳵
␭
␶ˆ BB ⳵␭
冉冕 冊
␰d␪
冑⍀Ĵ
冉
E共␬兲 1 2rq⬘ E共␬兲
− +
+␬−1
K共␬兲 2
q K共␬兲
y−
冊
SK
具␻De典
ge0 = i
,
␻0
␻0Ĵ
冋 冉 冕
c1 sinh ␯ˆ −1/2
冉 冕
+ c2 cosh ␯ˆ −1/2
共7兲
␬
␬
⍀共z兲dz
⍀共z兲dz
冊册
冊
共8兲
.
The solution of the inhomogeneous equation can then be
obtained with the method of variation of parameters, using
the boundary conditions at ␬ = 0 and ␬ = 1 to determine the
integration constants c1 and c2.
To make further progress analytically, we need to approximate the elliptic functions with their asymptotic limits
for small argument, as was done in Ref. 12, so that
K共␬兲 / Ĵ共␬兲 = 2 / ␬. The homogeneous solution becomes
ghom共␬兲 =
1
关c1 sinh共2冑␬u/␯ˆ 兲 + c2 cosh共2冑␬u/␯ˆ 兲兴,
共␬u兲1/4
共9兲
共4兲
where the orbit-averaged precession frequency for trapped
electrons is
冋
Ĵ
冉
ˆ D兲, and ␻
ˆ D = ␻D0 / ␻0 is the normalized
where u = −i共2y − ␻
magnetic drift frequency. The inhomogeneous part of the distribution is given by
⳵ge0
⳵␭
e具␾典
共␻ − ␻T e兲f e0 ,
*
Te
具␻De典 = ␻D0
1
共2兲
where ␯e共v兲 = ␯T / x3, x = v / vTe, ␰ = v储 / v, and ␭ = ␮ / w with
2
␮ = m av ⬜
/ 2B. If the electron distribution is expanded as
ge = ge0 + ge1 + ¯ in the smallness of the normalized collisionality ␯*e = ␯e / ⑀␻b Ⰶ 1 and ␻ / ␻b Ⰶ 1, where ␻b is the bounce
frequency, then in lowest order we have ⳵ge0 / ⳵␪ = 0. In next
order, we arrive at
K
where S = −共e具␾典 / Te兲共␻ − ␻T e兲f e0. The perturbed electrostatic
*
potential is approximated by ␾共␪兲 = ␾0共1 + cos ␪兲 / 2关H共␪
+ ␲兲 − H共␪ − ␲兲兴, where H is the Heaviside function and then
具␾典 = ␾0E共␬兲 / K共␬兲. Assuming weakly collisional plasmas
such that ␯ˆ Ⰶ 1, the WKB solution to the homogeneous
␯ˆ 共ge0
equation
⬙ + 共ln Ĵ兲⬘ge0
⬘ 兲 = ⍀2ge0, where ⍀2 = −i共y
− 具␻De典 / ␻0兲K / Ĵ, is
Turning to the electron kinetic equation, we retain collisions and use a pitch-angle scattering operator
C e = ␯ e共 v 兲
⳵ge0
i␯e ⳵
e具␾典
Ĵ共␬兲
=−
共␻ − ␻T e兲f e0 ,
*
⳵␬
Te
⑀␶ˆ B ⳵␬
冊册
,
共5兲
where ␻D0 = −k␪v2 / ␻ceR and E, K are the complete elliptic
functions with the argument ␬ = 共1 − ␭B0共1 − ⑀兲兲 / 2⑀␭B0,
where B0 is the flux-surface averaged magnetic field and
ginhom共␬兲 =
冋
册
冑␲ z2
− 2iŜ
2
关e Erf共z兲 + e−z Erfi共z兲兴 ,
1−
u␻0
4z
共10兲
where z = 冑2共␬u / ␯ˆ 兲1/4, Ŝ = SK共␬兲 / E共␬兲, Erf共z兲 is the error
function, and Erfi共z兲 = Erf共iz兲 / i is the imaginary error function. In the limit of z → ⬁ 共consistent with the assumption
␯ˆ Ⰶ 1兲, the error functions can be expanded, and the inhomogeneous solution simplifies to
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Phys. Plasmas 15, 072308 共2008兲
Collisionality dependence of the quasilinear particle…
ginhom共␬兲 =
− iŜ
2
共4 − 冑␲ez /z兲.
2u␻0
共11兲
Since ge0共␬ = 0兲 is regular, we choose c2 = 0, and the
boundary condition ge0共␬ = 1兲 = 0 共Ref. 12兲 gives
c1 = −ginhom共1兲u1/4 / sinh共2冑u / ␯ˆ 兲, so that the solution for the
perturbed trapped-electron distribution is
batic circulating electron response, the perturbed electron
density is
冉
⫻ 1 − 共1 + ␩e兲
− iŜ共4 − 冑␲ez /z兲
ge0 =
2u␻0
冋
2
冑
except in a narrow boundary layer close to ␬ = 0, that has a
negligible contribution to the velocity space integrals. The
nonadiabatic part of the perturbed trapped-electron density is
proportional to
冓冕 冔
3
ge0d v
= 4冑2⑀
冕 冕
冕
⬁
v dv
0
= 4冑2⑀
K共␬兲ge0d␬
⬁
共13兲
v 2d v I 1 .
Using Eq. 共12兲, the integral I1 becomes
冕
␲iŜ
共u − 冑u␯ˆ 兲,
u 2␻ 0
1
K共␬兲ge0d␬ = −
0
共14兲
where we retained terms only to the lowest order in ␯ˆ 1/2 and
we approximated K共␬兲 ⯝ ␲ / 2.
The above analysis is not valid in the boundary layer at
␬ ⯝ 1. The effect of the boundary layer reduces the collisional term, with a factor ␲ / 2冑ln ␯ˆ −1/2 共see Appendix A for
details兲. The reduction is less than 20% in the experimentally
relevant collisionality regime, and in the following analysis
this will be neglected.
ˆ D and keeping terms to
Expanding in the limit of small ␻
the first order, we have
ˆ D冑− iy ␯ˆ + 4y共␻
ˆ D − i冑− 2iy ␯ˆ 兲兲
2I1 Ŝ共8y 2 − 3i冑2␻
=
.
3
␲
8y ␻0
共15兲
ˆ Dt = ␻
ˆ D / x 2, ␻
ˆ e = ␻ e / ␻0 and ␯ˆ t = ␯ˆ x3, we obtain
Introducing ␻
*
*
冓冕 冔
ge0d3v
=
再冉 冊 冉
再
冉
冊冎
␲3/2S0
8
−i
1−
␻ˆ *e
y
+
␻ˆ *e
ˆ Dt
3␻
1 − 共1 + ␩e兲
4y
y
冊冎
␻ˆ *e 9␻ˆ
␲⌫共3/4兲S0冑− i2y ␯ˆ t
Dt
+
1 + 共3␩e − 4兲
8y
4y
64y
⫻ 4 − 共4 + ␩e兲
␻ˆ *e
y
,
⫻ 4 − 共4 + ␩e兲
冊册
␻ˆ *e
y
+
+
+
ˆ Dt
3␻
4y
⌫共3/4兲i冑− iy ⑀␯ˆ t
冑␲ y
ˆ Dt
9␻
64y
冊册冎
4y
␻ˆ *e
y
冊
共17兲
.
IV. QUASILINEAR PARTICLE FLUX
⌫e = Re共n̂evE*兲 =
0
0
I1 =
y
␻ˆ *e
The quasilinear particle flux is given by13
1
2
冉
共12兲
1−
␻ˆ *e
⫻ 1 + 共3␩e − 4兲
iŜ共4 − 冑␲ez / ␬␬1/4/z兲 sinh共z2兲
,
+
2u␬1/4␻0
sinh共z2/冑␬兲
2
再 冋冉
n̂e e␾0
=
1 − 冑2⑀
ne 2Te
共16兲
where S0 = −共e␾0 / Te兲ne4冑2⑀ / ␲3/2. Neglecting the nonadia-
冏 冏 冉
k␪ pe e␾0
2eB Te
2
Im
冊
n̂e/ne
,
e␾0/Te
共18兲
where the radial E ⫻ B velocity is vE ⬇ −ik␪␾0 / 2B. Taking
the imaginary part of the perturbed electron density from Eq.
共17兲, we obtain
冉
Im
n̂e/ne
e␾0/Te
冊
=−
␻ˆ *e 3␻ˆ
␻ˆ *e
⑀
Dt
Im −
+
1 − 共1 + ␩e兲
2
y
4y
y
冑
+
+
再
⌫共3/4兲冑⑀␯ˆ t
冑␲
冋
再 冋
Im
冉
冊
册冎
␻ˆ *e
i冑− iy
3
1 + ␩e − 1
4
y
y
冉 冉 冊 冊册冎
ˆ Dt
9␻
␩e ␻ˆ *e
1− 1+
16y
4 y
.
共19兲
The imaginary part of the perturbed density is sensitive to
the sign of the real part of the eigenfrequency ␴, and the
magnitude of the normalized growth rate ␥ˆ . In the following
analysis the quasilinear flux will be calculated for negative
共ITG兲 and positive 共TEM兲 signs.
A. ITG
ITG modes propagate in the ion diamagnetic direction,
so the real part of the eigenfrequency is negative. Figure 1
shows the quasilinear electron flux from Eqs. 共18兲 and 共19兲
normalized to pek␪ / 共2eB兲兩e␾0 / Te兩2冑⑀ as function of normalˆ Dt and ␩e for a case
ized collisionality for various values of ␻
where the plasma is far from marginal stability: ␥ˆ = 0.7.
In the absence of collisions, the flux is inwards if the
curvature and thermodiffusive fluxes 共the terms proportional
ˆ Dt and ␩e in the first row of Eq. 共19兲兲 dominate over
to ␻
diffusion. If collisions are included, the particle flux may be
reversed, if the part of the flux that is dependent on the
collisionality is positive. This reversal happens, for instance,
ˆ Dt = 0.2 and ␩e = 4.5 关see Fig. 1共d兲兴.
for ␻
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Fülöp, Pusztai, and Helander
ˆ e = 1. In the upper figures 共a and b兲: from
FIG. 1. Normalized quasilinear electron flux driven by ITG as function of normalized collisionality for ␥ˆ = 0.7 and ␻
*
ˆ Dt is 0 共solid兲, 0.1 共long-dashed兲, 0.2 共short-dashed兲, and 0.4 共dotted兲. 共a兲 ␩e = 3 and 共b兲 ␩e = 6.5. In the lower figures: ␩e is 3 共solid兲, 4.5 共long-dashed兲,
above ␻
ˆ Dt = 0 and 共d兲 ␻
ˆ Dt = 0.2.
6.5 共short-dashed兲, and 8.5 共dotted兲. 共c兲 ␻
However, if the ITG-instability growth rate is weak
共␥ˆ Ⰶ 1兲 and ␩e is large, the situation is completely different.
Figure 2 shows the normalized quasilinear electron flux for
the same parameters as in Fig. 1, but for ␥ˆ = 0.1, representing
a case close to marginal stability. The term proportional to
the 冑␯ˆ t will change sign and now this will also lead to an
inward flux. If the magnetic drift is high enough to give an
inward flux for zero collisionality, then collisions will
enhance this and the flux will therefore never be reversed. If
the magnetic drift is very small, the flux is outwards for ␯ˆ t
= 0. Then collisions may reverse the sign of the flux, but now
from outwards to inwards.
It is instructive to expand Eq. 共19兲 for small ␥ˆ , and show
explicitly the sign of the different terms in the expression for
the flux. If y = ␻ / ␻0 = −1 + i␥ˆ , then to lowest order in ␥ˆ , we
have
FIG. 2. Same as Fig. 1, but for ␥ˆ = 0.1.
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Phys. Plasmas 15, 072308 共2008兲
Collisionality dependence of the quasilinear particle…
ˆ e = 1. In the upper figures: from below
FIG. 3. Normalized quasilinear electron flux driven by TEM as function of normalized collisionality for ␥ˆ = 0.7 and ␻
*
␻ˆ Dt is 0 共solid兲, 0.2 共long-dashed兲, 0.4 共short-dashed兲, and 0.6 共dotted兲. 共a兲 ␩e = 3 and 共b兲 ␩e = 8.5. In the lower figures: from below ␩e is 3 共solid兲, 4.5
ˆ Dt = 0 and 共d兲 ␻
ˆ Dt = 0.2.
共long-dashed兲, 6.5 共short-dashed兲, and 8.5 共dotted兲. 共c兲 ␻
⌫ITG
=
e
冏 冏 再冑 冋
k␪ pe e␾0
2eB Te
−
冑
2
册
ˆ Dt
3␻
⑀
ˆ e
1−
共1 + ␩e兲 ␥ˆ ␻
*
2
2
再
冉
⌫TEM
=
e
3
⑀ 3␻ˆ Dt␥ˆ ⌫共3/4兲冑⑀␯ˆ t
␥ˆ
−
− 1 + + ␩e − 1
冑2␲
4
2 4
2
冉 冊
冋
冉 冊 册冎冎
冉 冊
⫻ 1−
ˆ Dt
9␻
3␥ˆ
3␥ˆ
␩e
+ 1+
␻ˆ *e +
1−
2
16
2
4
⫻ 1−
5␥ˆ
␻ˆ *e
2
冊
共20兲
.
If the plasma is close to marginal instability 共␥ˆ ⯝ 0兲, collisions 共represented by the term proportional to 冑␯ˆ t兲 lead to an
inward flux if ␩e ⬎ ␩crit:
␩crit =
ˆ e兲 − 9 ␻
ˆ Dt共1 + ␻
ˆ e兲兴
4关16共1 + ␻
*
*
ˆ Dt兲␻
ˆ e
3共16 + 3␻
*
.
共21兲
For typical experimental parameters, ␩e is expected to be
larger than ␩crit, and therefore the total flux is expected to be
inwards. However, if the plasma is further away from marginal instability, so that ␥ˆ ⬎ 2 / 3, the terms to 1 – 3␥ˆ / 2 and
1 – 5␥ˆ / 2 change sign, and then collisions will lead to an outward flux, as Fig. 1 shows. Note that the figures show the
quasilinear flux calculated from the unexpanded solution
共Eqs. 共14兲 and 共18兲兲 and they are valid even for ␥ˆ ⯝ 1.
冏 冏 再冑 冋
k␪ pe e␾0
2eB Te
−
冑
2
册
ˆ Dt
3␻
⑀
ˆ e
1+
共1 + ␩e兲 ␥ˆ ␻
*
2
2
再 冉 冊
3
⑀ 3␻ˆ Dt␥ˆ ⌫共3/4兲冑⑀␯ˆ t
␥ˆ
−
1 − + ␩e − 1
冑2␲
4
2 4
2
冉 冊
冋 冉 冊
冉 冊 册冎冎
⫻ 1−
ˆ Dt
9␻
3␥ˆ
␩e
␻ˆ *e +
1− 1+
2
16
4
⫻ 1−
5␥ˆ
␻ˆ *e
2
.
共22兲
There are two main differences compared with the ITGdriven flux. First, the part of the flux that is driven by the
curvature has opposite sign compared with ITG, and therefore contributes to the outward flux instead of driving an
inward pinch. Second, the part of the flux that arises due to
collisions is different and may have opposite sign compared
with the ITG case, depending on the parameters. Figure 3
shows the normalized quasilinear flux for different parameters if the plasma is far from marginal stability 共␥ˆ = 0.7兲 and
Fig. 4 shows the same for ␥ˆ = 0.1. Also here, the magnitude
of ␥ˆ changes the sign of the flux from outward to inward, and
collisions contribute to the inward flux.
B. TEM
C. Collisions modeled by a Krook operator
The real part of the eigenfrequency is positive, and this
means that y = ␻ / ␻0 = 1 + i␥ˆ and the electron flux to lowest
order is
Starting from the gyrokinetic equation for the electrons
but modeling the collision operator with an energydependent Krook operator, we have
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Phys. Plasmas 15, 072308 共2008兲
Fülöp, Pusztai, and Helander
FIG. 4. Same as in Fig. 3, but for ␥ˆ = 0.1.
i共␻ − 具␻De典兲ge0 − ␯effge0 = −
e␾0
共␻ − ␻T e兲f e0
*
Te
共23兲
so that
␻ − ␻T e
e␾0
*
ge0 = −
f e0 ,
Te ␻ − 具␻De典 + i␯eff
共24兲
where ␯eff = ␯T / ⑀x3.
The velocity-space integral of the perturbed electron distribution can be used to determine the imaginary part of the
perturbed electron density, and that gives the quasilinear flux
from Eq. 共18兲. If the plasma is far from marginal stability, the
results for the pitch-angle scattering and Krook operator are
qualitatively same, as shown in the upper figures of Fig. 5.
However, as the lower figures in Fig. 5 show, as we approach
marginal stability, the form of the collision operator matters
more and more, and both the sign and the magnitude of the
flux may be very different.
V. DISCUSSION
We have shown that in plasmas that are dominated by
ITG turbulence and are far from marginal stability the sign of
the electron flux will be changed from inward to outward if
the collisionality is increased. Figure 6 shows the threshold
in collisionality for which the flux reverses, i.e., ␯ˆ c, as a
function of ␩e for different values of the normalized magnetic drift frequency. The red curves correspond to the pitchangle scattering model-operator and the black curves correspond to the Krook model. It is interesting to see that the
pitch-angle scattering operator gives lower threshold for flux
reversal. If we compare the collisionality for zero flux in Fig.
4 in Ref. 8, we find that our result is of the same order
of magnitude. Figure 4 in Ref. 8 is computed for ␩e = 3,
R = 3 m, r = 0.5 m, a = 1 m, s = 1, q = 2. For these parameters
the trapped-electron flow changes sign for ␯e ⯝ 0.006cs / a,
where cs = 冑Te / mi is the ion sound speed. This collisionality
corresponds to ␯ˆ c ⯝ 0.1. This is in agreement with our threshˆ Dt = 0.7. Note that Fig. 4
old, shown in Fig. 6 for ␩e = 3 and ␻
in Ref. 8 is the result of a nonlinear gyrokinetic simulation
ˆ Dt is not constant and therefore exact comparison is not
so ␻
possible.
The analytical calculation presented in this paper is an
attempt to shed light on the numerical calculations mentioned above. For this purpose it is necessary to make a
number of simplifications, some of which may be justified
within a rigorous ordering scheme. However, it should be
noted that some approximations are more qualitative, in particular those having to do with the mode structure, which we
do not solve for. The approximation we use for the perturbed
electrostatic potential breaks down for low shear or near
marginal instability. It appears that the qualitative features of
the transport are captured by our calculations, but for quantitatively accurate results one of course has to resort to numerical simulations.
As we have seen, the effect of magnetic drift is important to understand the sign change of the quasilinear flux due
to the ITG modes. The magnetic drift gives an inward flux
for zero collisionality, but this is reversed when ␯ˆ t ⬎ ␯ˆ c.
VI. CONCLUSIONS
The collisionality dependence of the quasilinear particle
flux due to microinstabilities has been determined for large
aspect ratio, circular cross section plasmas. It has been
shown that if the plasma is far from marginal stability, the
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072308-7
Phys. Plasmas 15, 072308 共2008兲
Collisionality dependence of the quasilinear particle…
ˆ e = 1. The red curves are for the pitch-angle
FIG. 5. 共Color兲 Normalized quasilinear electron flux driven by ITG as function of normalized collisionality for ␻
*
ˆ Dt is 0 共solid兲, 0.2 共dashed兲, and 0.6 共dotted兲.
scattering operator and the black are for the Krook operator. In the left figures 共a and c兲 ␩e = 3 and from above: ␻
ˆ Dt = 0.2 and from above: ␩e is 3 共solid兲, 5.5 共dashed兲, and 8.5 共dotted兲. The upper figures 共a and b兲 are for ␥ˆ = 0.7, and the lower
On the right, figures 共b and d兲 ␻
共c and d兲 for ␥ˆ = 0.1.
inward transport due to ITG modes is reversed as electron
collisions are introduced, in agreement with nonlinear gyrokinetic simulations. However, if the plasma is close to marginal stability, collisions will lead to an additional inward
flux, and therefore the total flux is expected to be inwards.
The transport is therefore affected significantly by the parameter ␩e, both directly via the terms proportional to ␩e in the
expression for the flux, but also indirectly via the ITG
growth rate, which is important to determine the sign of the
flux.
If the electron collisions are modeled with a pitch-angle
scattering collision operator, the particle flux is proportional
to the square-root of the collisionality. The choice of the
model collision operator affects the collisionality threshold
for the reversal of the particle flux; i.e., ␯ˆ c. This is especially
important when the plasma is close to marginal stability. The
collisionality threshold ␯ˆ c depends on the magnitude of the
ˆ Dt and the ratio of density and
normalized magnetic drift ␻
temperature scale lengths; i.e., ␩e. For higher ␩e and higher
␻ˆ Dt, higher collisionality is needed to reverse the particle
flux.
The magnitude and the sign of the TEM-driven quasilinear flux has also been determined. The TEM-driven flux is
expected to be outwards if the plasma is far from marginal
stability and inwards otherwise, for typical experimental parameters, and the presence of collisions contributes to the
inward flow.
ACKNOWLEDGMENTS
This work was funded by the European Communities
under Association Contract between EURATOM, Germany
and Vetenskapsrådet. The authors would like to thank Dr. J.
Hastie and Dr. P. Catto for useful discussions of the paper.
The views and opinions expressed herein do not necessarily
reflect those of the European Commission.
FIG. 6. 共Color兲 Collisionality threshold as a function of ␩e for ␥ˆ = 0.7 and
␻ˆ e = 1. Above the lines, the transport is outwards, and below it is inwards.
*
ˆ Dt: from below ␻
ˆ Dt is 0.2
The different lines correspond different values of ␻
共solid兲, 0.4 共long-dashed兲, 0.6 共short-dashed兲, and 0.8 共dotted兲. The red lines
are for the pitch-angle scattering operator and the black are for the Krook
operator.
APPENDIX A: BOUNDARY LAYER ANALYSIS
FOR ␬ ¶ 1
In the outer region, far away from the trapped/passing
boundary we can neglect collisions, and the solution of Eq.
共7兲 is
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072308-8
Phys. Plasmas 15, 072308 共2008兲
Fülöp, Pusztai, and Helander
gouter =
ŜE共␬兲
.
共␻ − 具␻De典兲K共␬兲
共A1兲
The width of the boundary layer can be estimated by comparing 兩␯ˆ ge0
⬙ 兩 using the outer solution for ge0 and the term on
the right-hand-side of Eq. 共7兲, leading to 共1 − ␬兲 ⬃ ␯ˆ 1/2.
In the inner region we can approximate the elliptical integrals with their asymptotic forms for ␬ = 1,
giving E共␬兲 ⯝ 1, Ĵ共␬兲 ⯝ 1, K共␬兲 ⯝ ln共1 / 冑1 − ␬兲 ⯝ ln ␯ˆ −1/4 ⬅ K̂.
Changing variables in Eq. 共7兲 to t = 共1 − ␬兲 / 冑␯ˆ gives
冉
冊
Ŝ
⳵2ginner
具␻De典
+ iK̂ y −
ginner = i ,
2
␻0
⳵t
␻0
the expanded solution and the full WKB solution兲 and the
one published in Ref. 12. The perturbed electron density is
关Eq. 共28兲 of Ref. 12兴
再
8冑2⑀
n̂e e␾0
=
1 − 3/2
ne Te
␲
冋
⫻ 1−
␻ *e
␻
冉
共A2兲
Im
共␻ − 具␻De典兲K̂
冑
+ ĉ1 exp关− 共1 − ␬兲 ûK̂/␯ˆ 兴
冑
+ ĉ2 exp关共1 − ␬兲 ûK̂/␯ˆ 兴,
共A3兲
冑
兵
其
ŜE
1 − exp关− 共1 − ␬兲 ûK̂/␯ˆ 兴 .
共␻ − 具␻De典兲K
共A4兲
Using the global solution from Eq. 共A4兲, the collisional term
becomes
冕
1
K共␬兲
0
⯝−
冑
− ŜE共␬兲
exp关− 共1 − ␬兲 ûK̂/␯ˆ 兴d␬
共␻ − 具␻De典兲K共␬兲
Ŝ冑␯ˆ
共A5兲
冑
共␻ − ␻D0/2兲 ûK̂
since the dominant part of the integral comes from ␬ ⯝ 1.
Comparing with the corresponding term in Eq. 共14兲, we find
that the effect of the boundary layer reduces the collisional
term, with a factor 共␲ / 2冑ln ␯ˆ −1/2兲.
APPENDIX B: COMPARISON WITH THE SOLUTION
ˆ DT = 0 IN REF. 12
FOR ␻
In Ref. 12, the effect of the magnetic drift has been
ˆ Dt = 0兲, and the perturbed trapped electron
neglected 共that is, ␻
distribution has been calculated to be
ge0 = −
冋
册
2J1共a兲
e␾0
共1 − ␻T e/␻兲f e0 1 −
,
*
Te
aJ0共a兲
冋
0
册冎
2J1共a兲
aJ0共a兲
册
共B2兲
.
冊
再
2冑2⑀
n̂e/ne
=−
␻ˆ *e␥ˆ + ⌫共3/4兲
e␾0/Te
␲
冑
冎
␯ˆ t
ˆ e共1 − 3␩e/4兲兴 .
关1 + ␻
*
␲
共B3兲
ˆ Dt = 0.
This is in agreement with our results in the limit of ␻
where û = −i共y − 具␻De典 / ␻0兲. ĉ1 is determined by the boundary
condition ge0共␬ = 1兲 = 0 and ĉ2 = 0 to match the inner and
outer solutions. The global solution is then
ge0 =
2
x2e−x dx 1 −
If the real part of the frequency is negative 共␻ = −␻0
+ i␥兲, where ␻0 ⬎ 0, then for ar Ⰷ 1 and ␥ˆ Ⰶ 1, we have
⫻
Ŝ
⬁
关1 − ␩e共x2 − 3/2兲兴
and the solution is
ginner =
冕
共B1兲
where a = 共1 + i兲冑4␻⑀ / ␯e共v兲 ⬅ ar共1 + i兲. There is excellent
ˆ Dt = 0 共both
agreement between our results in the limit of ␻
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