Mean sheared flow and parallel ion motion effects on zonal flow generation in ion-temperature-gradient mode turbulence

0.19 MB·English

by  J. Anderson

Mean sheared flow and parallel ion motion ef-
fects on zonal flow generation in ion-temperature-
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0 gradient mode turbulence
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2

n
a J. Anderson1, Y. Kishimoto
J

4 Department of Fundamental Energy Science
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Graduate School of Energy Science, Kyoto University, Gokasho, Uji, Kyoto
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p 611-0011
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s Abstract
c
i
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y The present work investigates the direct interaction of sheared mean
h
p
flow with zonal flows (ZF) and the effect of parallel ion motion on ZF
[

1 generation inion-temperature-gradient (ITG)backgroundturbulence.
v
5 An analytical model for the direct interaction of sheared mean flows
1
0 with zonal flows is constructed. The model used for the toroidal ITG
2
.
1 driven modeis based on the equations for ion continuity, iontempera-
0
9 ture and parallel ion motion whereas the ZF evolution is described by
0
: the vorticity equation. The behavior of the ZF growth rate and real
v
i
X frequency is examined for typical tokamak parameters. It is shown
r
a that in general the zonal flow growth rate is suppressed by the pres-
ence of a sheared mean flow. In addition, with parallel ion motion
effects the ZFs become more oscillatory for increasing η (= L /L )
i n Ti
value.
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[e-mail]
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I Introduction
Thestudyofplasmaflowsforregulatingtheturbulenceandanomaloustrans-
port has in recent years attracted strong interest [1]. The regulation of drift
wave turbulence by sheared flows is due to shearing of the turbulent eddies
and thereby reducing the spatial scales of the eddies. Of particular inter-
est among the plasma flows are the so called zonal flows [2] which are self
generated from the background turbulence via Reynolds stress.
Unlike the sheared mean flows which have equilibrium scale size the zonal
flows are random E × B flows mainly in the poloidal direction with low
frequency and thus are almost stationary compared to the time scale of the
background turbulence. Since the zonal flows are quasistationary compared
to the background turbulence they may keep decorrelating turbulent eddies
for a relatively long time and thereby effectively suppressing the turbulent
transport.
The drift waves that are driven by gradients in the plasma density, tem-
perature,magneticfieldetcareresponsibleforcausingturbulenceandanoma-
lous transport. Candidates of such drift waves are the Ion-Temperature-
Gradient Mode (ITG) and the collisionless Trapped Electron Mode (TEM).
In the present paper the direct effects of a sheared mean flow on the
zonal flow instability are investigated analytically, however, the effects on the
background turbulence itself is not considered in this work. Moreover, the
effect of the parallel ion motion on the zonal flow growth rate and frequency
is studied.
The zonal flow generation from nonlinear interactions among drift waves
has been extensively investigated both analytically [3]- [15] and in computer
simulations using gyrokinetic [17]- [19] and advanced fluid models [20]- [23].
2
Furthermore, the effect of micro-scale electron temperature gradient (ETG)
driven turbulence driven zonal flow on semi-macro scale ITG turbulence was
studied by a fluid simulation [24]. However, the direct interaction of zonal
flowwithasheared meanflowandtheinteractionwiththeGAMhave mostly
been overlooked. There have been some previous studies on the interaction
of zonal flows and mean flows using simple drift wave models [13] and using
the coherent mode coupling method [16]. The present work extends the drift
wave modelto anadvanced fluidmodel forthe toroidalITG modeturbulence
while using the wave kinetic approach. The purpose of this study is to obtain
a qualitative estimate of the zonal flow growth rate and real frequency, and
their parametric dependence of the plasma under the influence of a sheared
mean flow and parallel ion motion. The effects of sheared mean flows and
parallel ion motion on zonal flows is not well known as well as how it is
influencing the turbulence and in the end how this effect may change the
overall transport.
Howeverthefullnon-lineareffectsoftheGeodesicAcousticMode(GAM)[25]
is out of the scope of the present study. The GAM is a perturbation where
the electrostatic potential (m = n = 0 mode) is coupled to the sideband
density perturbations (m = 1,n = 0 mode) by toroidal effects. The GAM
interacts with the zonal flow and act as a source or sink for the poloidal
flow leading to an oscillatory nature of the zonal flow. The complete role of
GAMs are still not well understood [26]- [29]. For these reasons it is of great
importance to study the four component system of drift waves, sheared mean
flow, GAM and zonal flow.
The methodology of the analytical model is based on the wave kinetic
approach [5], [13]- [15]. An algebraic equation which describes the zonal
flow growth rate and real frequency including the effects of a sheared mean
3
flow in the presence of maternal ITG turbulence including the effects of
parallel ion motion is derived and solved numerically. An advanced fluid
model including the ion continuity, ion temperature equations and the ion
momentum equation is used for the background ITG turbulence [30]. The
generation of zonal flows is described by the vorticity equation and the time
evolution of the ITG turbulence in the presence of the slowly growing zonal
flow is described by a wave kinetic equation.
It was found the generation of zonal flows was in general suppressed by
a sheared mean flow. By introducing collisional damping a modest suppres-
sion of the zonal flow growth rate is found. In addition, it is found that
the interaction with parallel iom momentum may reduce the ZF generation
significantly.
The paper is organized as follows. In Section II the analytical model for
the zonal flows generated from toroidal ITG modes including the effects of a
sheared mean flow is reviewed. The analytical model is extended to include
parallel ion motion in Section III. Section IV is dedicated to the results and
a discussion thereof. Finally there is a summary in section V.
II Analytical model for interaction of mean
flows and zonal flows
In this section the model excluding parallel ion motion is introduced. The
description used for toroidal ITG driven modes consists of the ion continuity
and ion temperature equations. For simplicity, effects of electron trapping
and finite beta are neglected in this work. Magnetic shear can, however,
modify the non-linear upshift as found in Ref. [10]- [11] and is accordingly
incorporated in the model including parallel ion motion. In this section
4
the model of how to construct the interaction of a sheared flow and the
zonal flows generated from ITG driven turbulence and the derivation of the
dispersion relation for zonal flows are summarized. The method has been
described in detail in Refs. [5], [13]- [15] (and References therein) and only a
brief summary is given here. The analysis of Ref. [5] is closely followed and
in the present work extended to be valid for an advanced fluid model. An
alternativestatisticalapproach, resultinginamodifiedwavekineticequation,
is presented in Ref. [12] which also contains an extensive discussion of and
comparisonwith theapproachused here. Indescribing thelargescale plasma
flow dynamics it isassumed that there is a sufficient spectral gapbetween the
small scale fluctuations and the large scale flow. The electrostatic potential
(φ = eϕ/T ) is represented as a sum of fluctuating and mean quantities
e
~ ~ ˜
φ(X,~x,T,t) = Φ(X,T)+φ(~x,t) (1)
˜
whereφ(~x,t)isthefluctuatingpotentialvaryingontheturbulent scalesx,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 coordinates X,T , (~x,t)
(cid:16) (cid:17)
are the spatial and time coordinates for the mean flows and small scale fluc-
tuations, respectively.
The wave kinetic equation see Refs. [5], [31] - [32] for the generalized
wave action N = 4γk2 |φ˜ |2 in the presence of a sheared mean plasma
k ∆2+γ2 k
k k
flow perturbing the other mean flow (the zonal flow in this case) due to the
interaction between mean flow and small scale fluctuations is
∂ ∂ ∂N (x,t) ∂ ∂N (x,t)
~ k ~ k
N (x,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,t)−∆ωN (x,t)2 (2)
k k k
Here the spectral difference (q << k , q ,k is zonal flow and drift wave
y y y y
wave numbers respectively) is used in solving the wave kinetic equation and
5
only the x direction is considered where similar spectrum of the background
turbulence and zonal flow is expected. Where ~v is the zonal flow part of the
0
E ×B velocity, and in the relation between small scale turbulence and the
generalized wave action density we have
k 4τ
y
∆ = 1−ǫ g + ǫ g (3)
k n n
2 3
(cid:18) (cid:19)
γ = k ǫ g(η −η ). (4)
k y n i ith
q
In the expression for the η the FLR effects are neglected,
ith
2 1 1 1 10
η ≈ − + +ǫ g + . (5)
ith n
3 2τ 4τǫ g 4τ 9τ
n (cid:18) (cid:19)
Here,andintheforthcomingequationsτ = T /T ,~v = ρ c ~y/L ,ρ = c /Ω
i e ⋆ s s n s s ci
where c = T /m , Ω = eB/m c. We also define L = −(dlnf/dr)−1,
s e i ci i f
q
η = L /L , ǫ = 2L /R where R is the major radius and α = τ (1+η ).
i n Ti n n i i
The perturbed variables are normalized with the additional definitions n˜ =
(L /ρ )δn/n , φ˜ = (L /ρ e)δφ/T , T˜ = (L /ρ )δT /T as the normalized
n s 0 n s e i n s i i0
ion particle density, the electrostatic potential and the ion temperature, re-
spectively. The perpendicular length scale and time are normalized to ρ
s
and L /c , respectively. The geometrical quantities are calculated in the
n s
strong ballooning limit (θ = 0, g(θ = 0,κ) = 1/κ (Ref. [33]) where g(θ)
is defined by ω (θ) = ω ǫ g(θ)). In this analysis it is assumed that the
D ⋆ n
RHS is approximately zero (stationary turbulence). The role of non-linear
interactions among the ITG fluctuations (here represented by a non-linear
frequency shift ∆ω) is to balance linear growth rate. In the case when
γ N (x,t) − ∆ωN (x,t)2 = 0, the expansion of the wave kinetic equation
k k k
is made under the assumption of small deviations from the equilibrium spec-
trum function; N = N0 +N˜ where N˜ evolves at the zonal flow time and
k k k k
space scale (Ω,q ,q = 0), as
x y
∂N˜ ∂N˜ ∂N0
k +iq v N˜ −k hV′i k +γ N˜ = iq k V˜ k (6)
∂t x gx k y E ∂k k k x y E ∂k
x x
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In this last expression the third term on the left hand side shows the explicit
interaction of a mean shear flow hV′i and N˜ . Here V˜ = iq Φ. This
E k E x
equationissolvedforN˜ assumingthathV′i2 issmall, byintegratingbyparts
k E
and obtaining an expansion in hV′i2 and introducing a total time derivative
E
D = ∂ −k hV′i ∂ . It is interesting to note that on the total time scale the
t ∂t y E ∂kx
shearing effect is explicit as D k = −k hV′i. The solution can be written
t x y E
as
N˜k = tdt′e−γ(t−t′)−iqx tt′′′dt′′vgxiqxkyV˜E∂Nk0
∂k
Zt0 R x
∂N0
= −iq2k ΦR (k ,q ,Ω,hV′i) k. (7)
x y 0 y x E ∂k
x
The evolution equations for thezonal flows isobtained afteraveraging the
ion-continuity equation over the magnetic flux surface and over fast scales
employing the quasineutrality and including a damping term [34]. The
evolution equation is obtained as
∂ ∂ ∂ ∂ ∂
∇2Φ−µ∇4Φ = (1+τ)∇2 φ˜ φ˜ +τ∇2 φ˜ T˜ (8)
∂t x x x*∂x k∂y k+ x*∂x k∂y ik+
where it is assumed that only the small scale self interactions are the im-
portant interactions in the RHS [35]. Using typical tokamak parameters
(T = T = 10kev, n = n = 1020m−3, r = 1m, R = 3m) µ = 0.78ν (r/R)
i e i e ii
q
and ν = 10−12n /T3/2 and ν is the ion-ion collision frequency, T is the ion
ii i i ii i
temperature in electron volts. Using typical tokamak parameters it is found
that µ ≈ 50. Expressing the Reynolds stress terms in Eq. 8 in N we obtain
k
−iΩ−µq2 Φ = (1+τ +τδ) d2kk k |φ˜ |2 (9)
x x y k
(cid:16) (cid:17) Z
where δ is a k independent factor
∆ k 2
k y
δ = η − (1+τ)ǫ g . (10)
∆2 +γ2 i 3 n
k k (cid:18) (cid:19)
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Utilize equations 7, 9 and 10 gives,
∆2 +γ2 ∂N0
−iΩ−µq2 = −q2(1+τ +τδ) k k d2kk2k kR . (11)
x x 4γ2 y x ∂k 0
(cid:16) (cid:17) k Z x
Here the response function (R ) is, considering only the first two even terms
0
in the expansion
1
R¯ = (12)
0
γ −i(Ω−v q )
k gx x
R = R¯ +R¯2D R¯ D . (13)
0 0 0 t 0 t
In contrast with Ref. [14] it is not assumed that the short scale turbulence
is close to marginal state (or stationary state, γ is small). Integrating by
k
parts in k and assuming a monochromatic wave packet N0 = N δ(k −k )
x k 0 0
gives
∆2 +γ2
Ω+iµq2 = −iq2(1+τ +τδ) k kk2R (k ,q ,Ω,hV′i)N (14)
x x 4γ2 y 0 y x E 0
(cid:16) (cid:17) k
Here, Ω is the zonal flow growth rate and real frequency, q is the zonal flow
x
wave number and k is the wave number for the ITG mode. The real part of
y
the sheared mean flow dependent term can now be written in the k << 1
⊥
limit as found in Ref. [13],
γ
k
Re(R ) ≈
0 γ2 +(Ω−q v )2
k x gx
5
γ
− 12q2(q2Φ )2k2 k . (15)
x f f y γk2 +(Ω−qxvgx)2!
Thisresultisvalidintheweakshear limitγ > hV′i. Inlaternumericalcal-
ZF E
culations the full expression is retained. In expressing the zonal flow growth
in dimensional form making use of the relation (∆2 +γ2)/(4γ2)N = |φ˜|2 it
k k k 0
is assumed that the mode coupling saturation level is reached [36]
γ 1
˜
φ = (16)
ω k L
⋆ y n
8
When calculating the group velocities the FLR effects in linear ITG mode
physics is important and given by the real frequency and growth rate as
follows from Ref [14]
k 10τ 5
ω = y 1− 1+ ǫ g −k2 α + τǫ g (17)
r 2(1+k2) 3 n ⊥ i 3 n
⊥ (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
k
y
γ = τǫ g(η −η ) (18)
1+k2 n i ith
⊥q
where ω = ω + iγ. The group velocities (v = ∂ω /∂k ) are in the long
r gj r j
wavelength limit (k2 << 1) given by,
⊥
5τ
v = −k k 1+(1+η )τ − 1+ ǫ g (19)
gx x y i n
3
(cid:18) (cid:18) (cid:19) (cid:19)
1 10τ
v = 1− 1+ ǫ g . (20)
gy n
2 3
(cid:18) (cid:18) (cid:19) (cid:19)
III Analytical model including parallel ion mo-
tion
It is known that parallel ion motion effects on the background turbulence
growth rate is only slightly modified, whereas, there is often a significant
effect on the real frequency (significant increase in |ω |). The zonal flow
r
dispersion equation is explicitly dependent of the background real frequency,
groupvelocity andgrowthrateandit isnowexpected that thereissignificant
effect on the zonal flow generation dependent on the parallel ion motion.
Accordingly, the previous model for the generation of zonal flows from ITG
background turbulence is extended to include the equation of motion for the
ions. The model for the drift waves consists of the following equations:
ion continuity equation
∂n˜ ∂ ∂ ∂φ˜ ∂ ∂v˜
− −α ∇2φ˜+ −ǫ g φ˜+τ n˜ +T˜ + i|| =
∂t ∂t i∂y! ⊥ ∂y n ∂y i ∂z
(cid:16) (cid:16) (cid:17)(cid:17)
−[φ,n]+ φ,∇2φ +τ φ,∇2 (n+T ) (21)
⊥ ⊥ i
h i h i
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ion energy equation
∂T˜ 5 ∂T˜ 2 ∂φ˜ 2∂n˜ 2
i i
− τǫ g + η − − = −[φ,T ]+ [φ,n] (22)
n i i
∂t 3 ∂y 3 ∂y 3 ∂t 3
(cid:18) (cid:19)
parallel ion momentum equation
∂v˜ ∂φ ∂
i|| = − +τ n˜ +T˜ − φ,v˜ . (23)
i i||
∂t ∂z ∂z !
(cid:16) (cid:17) h i
Here [A,B] = ∂A/∂x∂B/∂y − ∂A/∂y∂B/∂x is the Poisson bracket. The
quantitiesarenormalizedinthesamefashionasabovewithv˜ = (L /ρ )v /c .
i|| n s i|| s
The electrons are assumed to be Boltzmann distributed. Note that, for the
zonal flows k = 0 and Eq. 23 is identically zero and Boltzmann distributed
k
electrons cannot be used, instead the same model as earlier is employed c.f
Eq. 8. The dispersion relation for the ITG mode resulting from Eqs 21 - 23
is then
5τ 5τ
1+k2 1+ ω2 − 1−ǫ 1+ +α −k2τΓ
⊥ 3 n 3 r ⊥
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19)
ǫ s 5τ
n
−i 1+ ωk
y
2q (cid:18) 3 (cid:19)!#
5τ2 ǫ s
+ ǫ Γ−α + k2 (1+η ) +i n Γ k2 = 0 (24)
" n r 3 ⊥ i ! 2q !# y
Here α = 5τ, τ = T /T and
r 3 e i
2 5τ
Γ = τ η − + ǫ (1+τ) (25)
i n
3 3
(cid:18) (cid:19)
Here,thesolutionhavebeenfoundusinganapproximateeigen-modefunction
in the form of the lowest order Hermite polynomial (n = 0).
δφ ∝ e−z2/2σ2 (26)
where
iǫ
n
σ = (27)
k |s|qω
⊥
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