Zonal flow generation in collisionless trapped electron mode turbulence

0.15 MB·English

by  J. Anderson

Zonal flow generation in collisionless trapped
electron mode turbulence
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J. Anderson1, H. Nordman, R. Singh*, J. Weiland

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a Department of Radio and Space Science, EURATOM-VR Association
J

5 Chalmers University of Technology, SE-41296 G¨oteborg, Sweden
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*Institute for Plasma Research
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p Bhat, Gandhinagar
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m Gujarat, India 382428
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s Abstract
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Inthepresentworkthegenerationofzonalflowsincollisionlesstrapped
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1 electron mode (TEM) turbulence is studied analytically. A reduced
v
7 model for TEM turbulence is utilized based on an advanced fluid
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2 model for reactive drift waves. An analytical expression for the zonal
2
.
1 flowgrowthrateisderivedandcomparedwiththelinearTEMgrowth,
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9 and its scaling with plasma parameters is examined for typical toka-
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: mak parameter values.
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[email protected]
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1 Introduction
The study of the generation and suppression of turbulence and transport in
tokamak plasmas is still a high priority topic in theoretical and experimental
magneticfusionresearch. Inrecent studies, theimportantroleplayedbynon-
linearly self-generated zonal flows for the regulation of turbulent transport
has been emphasized [1]- [3]. These are radially localized flows (wave-vector
q = (q ,0,0)), propagating mainly in the poloidal direction, which can re-
x
duce the radial transport by shearing the eddies of the driving background
turbulence.
The turbulence and anomalous transport observed in tokamak plasmas
is generally attributed to short-wavelength drift-type instabilities, driven by
gradients in the plasma density, temperature, magnetic field, etc. For the
hot core region of a tokamak plasma, the two main drift-wave candidates
are the toroidal Ion Temperature Gradient (ITG) Mode and the collisionless
Trapped Electron Mode (TEM).
Accordingly,thegenerationofzonalflowsbynon-linearinteractionsamong
drift waves has recently been studied both analytically [4]- [11] and numeri-
cally [12]- [23]. While a substantial effort has been devoted to the study of
zonal flow generation by ITG modes (see e.g. [5], [7] and [9] for an analyt-
ical treatment), very little work has been published on zonal flows driven by
pure TEM [16]- [17]. In tokamak plasma experiments, TEM are expected
to play a dominant role in the hot electron regime (T > T ), relevant for ex-
e i
periments with dominant central electron heating [24]- [25] and in advanced
confinement regimes with electron transport barriers (η > η ). The study of
e i
zonal flows and turbulence driven by TEM is therefore crucial for the assess-
ment of these regimes. In addition, an estimate of the zonal flow generation
2
close to marginal stability is essential in order to discriminate between the
TEM and the potentially important Electron Temperature Gradient (ETG-)
mode [26]- [27] in comparison of experimental profiles against linear thresh-
olds. The experimental temperature gradients (or inverse scale lengths) are
typically found to be about a factor of 2 above the linear thresholds, both
for ITG and TEM. In the Cyclone study [23], this factor was 1.7 for the
ITG mode. However, the nonlinear upshift, due to zonal flows, increased
the effective ITG threshold by a factor 1.5, thus bringing it much closer to
the experimental gradient. Although a comprehensive and quantitative in-
vestigation of zonal flow generation would require a nonlinear gyrokinetic
treatment, a qualitative analytical study, based on a reduced set of fluid
equations, is feasible and also more transparent in terms of physics interpre-
tation.
In the present paper, the generation of zonal flows by pure TEM is stud-
ied analytically in the limit η = 0 where the ITG mode is suppressed, using
i
a reduced fluid model for the trapped electron dynamics. A system of equa-
tions is derived which describes the coupling between the background TEM
turbulence, described by a wave-kinetic equation, and the zonal flow modes
generated by Reynolds stress forces. The qualitative analytical technique
used here follows closely the WKB analysis employed in [7] and [9] for
zonal flow generation by ITG turbulence. The purpose of the study is to
obtain a qualitative estimate of the zonal flow growth rate driven by TEM
and to compare with an ITG driven case, and in addition examine its scaling
with plasma parameters for typical tokamak parameter values.
The paper is organized as follows. In Section II the model equations for
the ITG/TEM system is presented and a reduced model for TEM turbulence
is presented. The equations describing the coupling between the background
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TEM turbulence and the zonal flows are presented in Section III and most
explicit derivations are put in the Appendix A. In Section IV the results are
discussed and finally there is a summary in Section V.
2 Reduced Model for Trapped Electron Modes
The description used for the coupled toroidal ITG and collisionless TEM
system is based on the continuity and temperature equations for the ions
and the trapped electrons [28]:
∂n
j
+∇·(n ~v +n ~v )+∇·(n ~v +n ~v ) = 0 (1)
j E j ⋆j j Pj j πj
∂t
3 dT
j
n +n T ∇·~v +∇·~q = 0 (2)
j j j j j
2 dt
5 p
j
q = e ×∇T (3)
j k j
2m Ω
j cj
(cid:16) (cid:17)
where n , T are the density and temperature perturbations (j = i and
j j
j = et represents ions and trapped electrons) and ~v = ~v +~v +~v +~v ,
j E ⋆ Pj πj
~v is the E~ × B~ velocity, ~v is the diamagnetic drift velocity, ~v is the
E ⋆ Pj
polarization drift velocity, ~v is the stress tensor drift velocity and ~q is the
πj j
˜
heat flux. The derivative is defined as d/dt = ∂/∂t + ρ c ~z × ∇φ · ∇ and
s s
φ is the electrostatic potential. In the forthcoming equations τ = T /T ,
e i
~v = ρ c ~y/L , ρ = c /Ω where c = T /m , Ω = eB/m c. We also
⋆ s s n s s ce s e i ce e
q
defineL = −(dlnf/dr)−1,η = L /L ,ω /ω = ǫ g = 2Lng ,whereRis
f j n Tj Dj ∗j n j R j
the major radius, α = (1+ηi) and g represents the variation of ω along the
i τ j Dj
field line. The geometrical quantities are calculated in the strong ballooning
˜
limit (θ = 0, g = 1). The perturbed field variables are normalized as φ =
j
(L /ρ )eδφ/T , n˜ = (L /ρ )δn/n , T˜ = (L /ρ )δT /T . The perpendicular
n s e n s 0 j n s j e0
lengthscaleandtimearenormalizedtoρ andL /c , respectively. Equations
s n s
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(1) and (2) can now be simplified to
˜
∂n˜ ∂φ ∂
et ˜ ˜
+f +ǫ g −f φ+n˜ +f T =
t n e t et t et
∂t ∂y ∂y
(cid:16) (cid:17)
−[φ,n ] (4)
et
∂T˜ 5 ∂T˜ 2 ∂φ˜ 2 ∂n˜
et et et
+ ǫ g + η − − =
n e e
∂t 3 ∂y 3 ∂y 3f ∂t
(cid:18) (cid:19) t
2
−[φ,T ]+ [φ,n ] (5)
et et
3f
t
˜
∂n˜ ∂ ∂ ∂φ ∂ 1
i − −α ∇2φ˜+ −ǫ g φ˜+ n˜ +T˜ =
i n i i i
∂t ∂t ∂y! ∂y ∂y (cid:18) τ (cid:16) (cid:17)(cid:19)
1
− φ˜,n˜ + φ˜,∇2φ˜ + φ˜,∇2 n˜ +T˜ (6)
i ⊥ τ ⊥ i i
h i h i h (cid:16) (cid:17)i
∂T˜ 5 ∂T˜ 2 ∂φ˜ 2∂n˜
i i i
− ǫ g + η − − =
n i i
∂t 3τ ∂y 3 ∂y 3 ∂t
(cid:18) (cid:19)
2
− φ˜,T˜ + φ˜,n˜ . (7)
i i
3
h i h i
Here f = n /n is the fraction of trapped electrons. The Poisson bracket
t et 0
is [A,B] = ∂A/∂x∂B/∂y − ∂A/∂y∂B/∂x. The system is closed using the
quasineutrality condition
δn = δn = δn +δn . (8)
i e et ef
where a Boltzmann distribution has been assumed for the free electrons.
After linearizing equations (4)-(7), the dispersion relation for the coupled
ITG/TEM system is obtained as
ω 7 5
⋆
0 = ω(1−ǫ g )− −η − ǫ g ω
n i i n i Di
N 3 3
i (cid:20) (cid:18) (cid:19)
ω 5
− k2ρ2(ω −ω (1+η )) + ǫ g
y s ⋆i i ω 3τ n i
(cid:18) ⋆e (cid:19)(cid:21)
ω 7 5
⋆e
− f ω(1−ǫ g )− −η − ǫ g ω −1+f (9)
t n e e n e De t
N 3 3
e (cid:20) (cid:18) (cid:19) (cid:21)
where
10 5
N = ω2− ωω + ω2 (10)
j 3 Dj 3 Dj
5
Dependingontheplasma parameters, thedispersion relation(9)maycontain
0, 1 or 2 unstable modes. For modes propagating in the ion drift direction
(usually the ITG mode), N < N , while for modes propagating in the elec-
i e
tron drift direction (TEM), N < N . The modes become de-coupled when
e i
the inequalities are strong. Thus, for N >> N we obtain a pure ITG mode.
e i
For pure ITG mode physics, the fluid model used here has been found to be
in good qualitative agreement with a number of gyrokinetic treatments. For
example, both the η -scaling of the ion heat transport [23] and the nonlinear
i
upshift of the linear ITG threshold due to zonal flows [23] has been recovered
by the fluid model [29]. A more comprehensive version of the model, based
on the full ITG and TE system (Equations 4-7), has been heavily used in
predictive transport code simulations [30]- [32] of tokamak discharges. The
simulation results indicate that the model is able to reproduce experimental
profiles of temperatures and density, inside the edge region, with good ac-
curacy over a wide range of plasma parameters. In the limit N >> N , we
i e
obtain a pure TEM. In tokamak plasmas the TEM is expected to dominate
in the hot electron regime (T >> T ) and/or in regimes with η >> η .
e i e i
However, for peaked density profiles (small ǫ ), the ion and trapped electron
n
responses are strongly coupled (N ≈ N ) and a density gradient driven TEM
i e
appears for weak temperature gradients. In the following we will neglect the
effects of ion perturbations on the TEM and hence only consider electron
temperature gradient driven TEM (not including the electron temperature
gradient mode (ETG)). The dispersion relation then takes the form
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0 = ω2 +ωk ξ(1−ǫ g)− ǫ g
y n n
3
(cid:18) (cid:19)
2 7 5
+ k2ǫ g ξ η − ǫ g − (1−ǫ g) + ǫ g (11)
y n e 3 n 3 n 3 n
(cid:18) (cid:18) (cid:19) (cid:19)
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f
t
ξ = (12)
1−f
t
Equation 11 describes TEMs driven by R/L and suppressed by R/L lead-
Te n
ing to a linear TEM stability threshold in the parameter η = L /L . In the
e n Te
considered limit the TEM is fairly symmetrical to the toroidal ITG mode,
except that effects of finite-Larmor-radius and parallel electron dynamics do
not appear in the TEM dispersion relation. The solution to Equation 11 is
given by
k 10
y
ω = − ξ(1−ǫ g)− ǫ g (13)
r n n
2 3
(cid:18) (cid:19)
γ = k ξǫ g(η −η ) (14)
y n e eth
q
where ω = ω +iγ and the linear stability threshold is given by
r
2 ξ 10 ξǫ g ξ
n
η = − + ǫ g + + . (15)
eth n
3 2 9ξ 4 4ǫ g
n
In this regime we can define a reduced model for electron temperature gra-
dient driven TEM turbulence by retaining equations (4)-(5) for the trapped
electron fluid while neglecting ion dynamics (6)-(7) (see Appendix A). The
effect of the neglected ion dynamics on the linear physics will in the following
be quantified by comparing the results of the reduced (equation (11)) with
the complete (equation (9)) dispersion relation.
3 Zonal Flow Generation
In describing the large scale plasma flow dynamics it is assumed that there
is a sufficient spectral gap between the small scale TEM fluctuations and the
large scale flow. The electrostatic potential is represented by
˜
φ(X,x,y,T,t) = Φ(X,T)+φ(x,y,t) (16)
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˜
where φ(x,y,t) is the fluctuating potential varying on the turbulent scales
x,y,t and Φ(X,T) is the zonal flow potential varying on the slow scale X,T
(the zonal flow potential is independent on Y).
The evolution of the TEM turbulence in the background of the slowly
varyingzonalflowΦ(X,T)canbedescribedbythewave-kineticequation[5],[33]
and [34] for the adiabatic invariant N = C |φ˜ |2 (see Appendix A for a
k k k
derivation of N and C ).
k k
∂ ∂ ∂N (x,y,t) ∂ ∂N (x,y,t)
~ k ~ k
N (x,y,t) + ω +k ·~v − k ·~v
k k 0 0
∂t ∂k ∂x ∂x ∂k
x x
(cid:16) (cid:17) (cid:16) (cid:17)
= γ N (x,y,t)−∆ωN (x,y,t)2 (17)
k k k
Here, ~v is the zonal flow part of the ExB drift. In this analysis it is assumed
0
that the RHSis approximately zero (stationary turbulence). The role of non-
linear interactions among the TEM fluctuations (here represented by a non-
linear frequency shift ∆ω) isto balancelinear growthrate, i.e. γ N (x,y,t)−
k k
∆ωN (x,y,t)2 ≈ 0. The TEM turbulence is assumed to be adiabatically
k
modulated by the slowly growing potential Φ(X,T). Equation (17) is then
expanded under the assumption of small deviations from the equilibrium
spectrum function; N = N0 +N˜ where N˜ evolves at the zonal flow time
k k k k
and space scale (Ω,q ,q = 0), as
x y
∂2 ∂N0
−i(Ω−q v +iγ )N˜ = k Φ k (18)
x gx k k y
∂x2 ∂k
x
∂N0 i
N˜ = −q2k k Φ (19)
k x y ∂k Ω−q v +iγ
x x gx k
Here v = ∂ω/∂k ≈ 0, since the effects of electron FLR is neglected.
gx x
The evolution equations for the zonal flow is obtained after averaging the
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ion-continuity equation over the magnetic flux surface and over fast scales
and employing quasi-neutrality (equation 8)
∂ ∂ ∂
∇2Φ = ∇2 φ˜ φ˜ (20)
∂t x x*∂x k∂y k+
Here we have assumed that the turbulence is dominated by the TEM (n˜ <<
i
n˜ ) and hence only the small scale self interactions among the TEM are
et
contributing to the Reynolds stress in the RHS of (20) [35]. Expressing the
Reynolds stress terms in equation (20) in N we obtain
k
−iΩΦ = d2kk k C−1N (21)
x y k k
Z
The factorC defines therelationship between small scale turbulence andthe
k
wave action density, see Appendix equation A9 for details. Integrating by
parts in k and assuming a monochromatic wave packet N0 = N δ(k −k )
x k 0 0
and using equations (19) and (21) gives
Ω2 = −q2C−1k2N (22)
x k y 0
The dispersion relation for zonal flow Ω reduces to
Ω = iq k C−1N . (23)
x y k 0
q
Hence, the zonal flow growth rate scales as Ω ∝ |φ| . In expressing the zonal
k
flow growth in dimensional form making use of equation (23), it is assumed
that the background turbulence (in the absence of zonal flows) reach the
mixing length level for temperature gradient driven modes corresponding to
T˜ = 1 . We then obtain (see Appendix A)
e kxLTe
η
e
Ω = iq k F (24)
x y
k L
x n
(∆2 +γ2)
k k
F = (25)
|η −q2 ǫ g(1+ξ)|
e 3ξ n
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where q is the zonal flow wave number, k is the TEM wave number, ∆ =
x y k
−ky ξ −ǫ gξ + 4ǫ g andγ isthelinear TEMgrowthrate. ThefunctionF
2 n 3 n k
(cid:16) (cid:17)
is usually large in regions close to marginal stability due to the denominator
in equation (25). This is a result of the quasilinear treatment of the TEM
perturbations φ (appearing in equation 23) and T˜ = 1/(k L ).
k e x Te
4 Results and discussion
An algebraic equation (24) describing the zonal flow growth rate driven by
short-wavelength TEM turbulence has been derived. The zonal flow growth
rates will in the following be calculated and compared to the linear TEM
growth rates. First, the linear TEM descriptions are compared. In Figure
1, the solutions to the full system of 4 equations describing the coupled
ITG/TE system (squares, equation (9)) is compared to the reduced model
with 2 equations (asterisks, equation (11)) for the pure TEM. The results
are shown for η =0, τ = 1, ǫ = 1.0, kρ = 0.3 and f = 0.5. The η scalings
i n t e
for the TEM eigenvalues are found to be in good qualitative agreement in
this regime (for fairly flat density profiles); the growth rates are within 20%
except close to the linear threshold. The influence of the ion dynamics on the
TEM stability would be reduced further in the limit of cold ions (τ >> 1).
Next, the zonal flow growth rate is studied. In Figure 2 the effects of η
e
and ǫ on zonal flow growth rate (normalized to the TEM growth rate) are
n
displayed. The other parameters are f = 0.5, η = 0, τ = 1 and k ρ = k ρ =
t i x y
q ρ = 0.3. The results for the zonal flow growth rate are shown for ǫ = 0.5
x n
(with η = 1.38, boxes), ǫ = 0.7 (with η = 1.48, rings). There is a
eth n eth
significant increment in the zonal flow growth rate (normalized to the linear
TEM growth rate) just above marginal stability. Part of this increment is
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