Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)585–588
000–000
J. Plasma Fusion

Saturation of Zonal Flow in Gyrofluid Simulations of Electrontemperature Gradient Driven Turbulence
LI Jiquan, KISHIMOTO Yasuaki1, IDOMURA Yasuhiro, MIYATO Naoaki andMATSUMOTO TaroNaka Fusion Research Establishment,JAERI, Ibaraki 311-0193, Japan
1Southwestern Institute of Physics, Chengdu, P. R. China(Received: 9 December 2003 / Accepted: 19 May 2004)

Abstract
An enhanced zonal flow is observed in gyrofluid simulations of electron temperature gradient (ETG) driven turbulence with weak magnetic shears. The Kelvin-Helmholtz (KH) instability is proposed as a primary damping mechanism of such flow. Some considerable evidences for the KH mode excitation are presented. Results seem to suggest apossibility of turbulence transition from the ETG-dominated one to the KH-dominated one due to the zonal flow
dynamics in weak shear plasmas.
Keywords:
KH instability, zonal flow saturation, spectral analysis, gyrofliud simulation, ETG turbulence

rate the growing flow [10-12]. In addition, the coupling ofzonal flows through the geodesic curvature to pressure fluctuations may also influence the level of zonal flows in a
toroidal plasma. In this work, the KH instability is proposed
as a primary mechanism to limit the level of enhanced zonal
flow in gyrofluid slab ETG turbulence. Some considerable
evidences for the KH mode excitation are presented based on
numerical analyses of the stability of zonal flow with regard
to KH instability, the spatial Fourier spectra and the time-frequency wavelet spectra of turbulent ETG fluctuations.

1. Introduction
In last decades, it is widely recognized that the anomalous transport plausibly originates from the drift wave fluctuations such as ion/electron temperature gradient (ITG/ETG)
driven turbulence in fusion plasmas [1]. The importance of
zonal flows in suppressing turbulent ion and electron heattransport has been intensively emphasized, especially in theITG turbulence [2]. The saturation level of zonal flow has
also received much attention [3]. This is because it is one ofkey factors to influence the flow-shearing rate, except for thespectral structure of zonal flow [4]. Recently, an enhanced
zonal flow generated by the slab ETG turbulence has been
observed in a regime with both weak magnetic shear andsteep electron temperature gradient by performing a longtime gyrofluid simulation, in which the electron heat transport is reduced much [5,6]. Similar phenomena were also
observed very recently in the drift-Alfven turbulence in theedge plasmas of stellarator and tokamak configurations [7].
Hence, a natural question arises, which refers to the zonal
flow saturation in such simulations.
It is generally known that zonal flows can be damped by collisional effects [8]. In low collisional plasmas, there exist
some other mechanisms. For example, first of all, the zonal
flow may be saturated by the spectral modulation of turbulence due to the flow, while it is also generated through amodulational instability [9]. On the other hand, a strongly
sheared flow may excite a secondary Kelvin-Helmholtz (KH)
instability, which can absorb the flow energy and, then, satu-

2. Basic model and simulation results
A three-field (normalized potential, φ , parallel electron
velocity, v|| , and electron pressure, p) gyrofluid model isemployed to describe the electrostatic slab ETG turbulence intokamak core plasmas, i.e., [9](1 − ∇2⊥)∂tφ − (1 + K∇ 2⊥)∂yφ −∇||v|| = ⎣φ , ∇ 2⊥φ ⎦ + µ⊥∇ 4⊥φ ,(1)∂tv|| − ∇||(φ + p) = − [φ , v||] − η⊥∇ 2⊥v||,(2)∂t p + K∂yφ + Γ ∇||v|| +2(Γ − 1) 8/π | k|| | (p + φ) = − [φ, p] + χ⊥∇⊥ p(3)Where Γ = 5/3 and K = 1 + ηe with ηe = dlnTe/dlnn. A 3dimensional (3D) initial value code is used to solve Eqs. (13) [9]. Lots of simulations are performed for different mag©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: lijq@swip.ac.cn
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working hypothesis that a KH (namely, flow-driven) instability may limit the zonal flow level. Two questions areaddressed: whether is the observed zonal flow unstable withregard to the KH mode? If yes, how is the KH fluctuation
identified in the spatio-temporal spectra of the mixed fluctuations?

3.1 Stability of KH mode
For a non-monotonously varied flow, it is not tractableto get an analytical derivation of the KH instability.
Alternatively, a numerical check can be performed to directly
examine the stability of KH mode in the enhanced zonal flow
observed in Fig. 1. This can be done by removing the nonlinear coupling, electron temperature and density gradients, andall turbulent fluctuations from the simulation. Note that asshown in Fig. 1, the growth of zonal flow becomes slow ataround t = 200, which may presage the excitation of KH
instability. Hence, the zonal flow at the normalized time t =200 are sampled as a constant energy source of KH fluctuations. Small random noise is chosen as the initial perturbation. With these settings, only the flow-driven KH instability
may occur if the zonal flow is unstable with regard to it. Fig.
2 displays the time evolution of fluctuating potential, whichshows that the KH mode is marginally unstable. Furthernumerical experiments by artificially increasing (or decreasing) the flow amplitude by 10% show that the KH fluctuations become unstable (or stable) as plotted by the dashed-dot
curves. These observations indicate that when the zonal flow
grows up to some high level, which approaches the amplitude
threshold of a KH instability, the flow itself may be damped
through exciting the KH mode. The growth of zonal flow isthen slowed down. It shows that the occurrence of KH instability is possible after around t = 200. To further confirm thispoint, the flow at t = 500, at which the zonal flow approaches
the quasi-steady state, is taken to analyze the KH stability. Itis found that the KH mode, which propagates along y direction, is weakly unstable in the spectral region of ky = 0.2 ~0.3. The unstable modes correspond well to the dominant turbulent fluctuations in the simulation of Fig. 1. Note that theweak magnetic shear also favors the KH instability. Hence,Fig. 1 Time evolution of the turbulent and zonal potential
energy as well as electron heat conductivity. ŝ = 0.1,
ηe = 6. The dark (light) shaded part marks the approximately exponentially fast (slow) growing phase ofzonal flows. Short straight line is for reference only.

netic shear and ηe. The artificial viscosities are nominated asµ⊥ = η⊥ = χ⊥ = 0.5 ; Space domain is set as Lx = 200ρe, Ly =20πρe and Lz = 2πLn. As a result, a high electron confinement
state, which is characterized by weak magnetic shear, steep
electron temperature gradient, enhanced zonal flow andreduced heat transport, is observed as typically shown in Fig.
1. It can be seen that the zonal flow component undergoes
two different growing phases after the initial saturation of
ETG fluctuations. The first fast growing phase clearly showsa zonal flow instability on average, in which the heat conductivity is higher. A slow growing phase follows accompanying
with decreasing turbulent fluctuations and heat transport.
Finally, the zonal flow is limited at some level, and turbulent
fluctuations, which are dominated by a few coherent structures with ky = 0.2 ~ 0.3, approach a quasi-steady state with alow transport level.
In the simulations above, the weak magnetic shear isfound to be a crucial factor to control the zonal flow dynamics and turbulent transport property. Analyses based on thewell-known Hasegawa-Mima turbulence, which is a simplified version of Eqs. (1-3), show that the weak shear favors
the zonal flow instability in ETG fluctuations, which is wellin agreement with the observations in ETG simulations. It isnoticed that the level of zonal flow determines the electrontransport property in weak shear ETG turbulence. Hence, thesaturation mechanism of zonal flows is an important issue.

3. Saturation of enhanced zonal flow
Lots of simulations similar to the above one have been
also performed with different viscosities and higher spatialresolution. A common feature is generally observed:
although initially saturated ETG fluctuations are highly turbulent, they slowly condense to some coherent structures as
the zonal flows grow up and saturate at some level. It seems
to show that enhanced zonal flows, suppressed ETG fluctuations and coherent vortices coexist in the quasi-steady state.
In what follows, numerical experiments are designed to test aFig. 2 Stability analyses of linear KH mode in the flowobserved at t = 200 in Fig. 1. It shows the KH mode ismarginally unstable.

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spectral distribution in the longer wavelength region with apeak at ky ≈ 0.3 and an isotropic distribution at shorter wavelengths in the kx − ky space, as shown in Fig. 3(a). The spectral distribution of dominant turbulent fluctuations at aroundky = 0.3 corresponds well to that of unstable KH modes. For
comparative purpose, a reference simulation is performed
with the same parameters as in Fig. 1, but the zonal flow
component is set to zero in time. An isotropic spectral distribution in both longer and shorter wavelength regions isobserved as shown in Fig. 3(b), which is of typical characteristics of the pure ETG turbulence. It may result from the nonlinear energy cascade or inverse energy cascade.
Furthermore, analyses on two-dimensional autocorrelation
functions of the turbulent fluctuations show that in the simulation with zonal flow, the turbulent fluctuation is characterized by a strongly inhomogeneous correlation in the x and ydirections. The ETG turbulence is strongly modulated by awave-like fluctuation propagating along the y direction sothat it seems to obviously possess a periodicity. The wave
number is estimated as ky ≈ 0.262, which also approaches thepeak spectrum of the KH mode.

Fig. 3 Spatial spectral density distributions of the turbulent
fluctuations 〈S{φ2}〉z, t(kx, ky) in kx − ky space in the simulations with (a) and without (b) zonal flows. The
dashed curves are for reference only.

these numerical experiments indicate that the role of magnetic shear in ETG turbulence seems to involve a complex interaction between zonal flows and the secondary KH mode,
which may significantly influence final saturation level ofETG turbulence.

3.3 Time-frequency wavelet analysis
To visually see the excitation of KH mode in time evolution, a new spectral method, termed wavelet transform, isused to analyze time series of the turbulent fluctuation. The
basic idea for the use of this technique is that: if the enhanced
zonal flow can drive a KH instability, which may back-react
to saturate the exponentially growing flow, the KH fluctuation should be diagnosed in the time-frequency sequence ofwavelet energy spectra. A software program of wavelet transform is employed and the Morlet wavelet is used in theimplementation [13]. Fig. 4 plots contours of turbulent potential energy [i.e., φ(ky)2/2] in the time-frequency plane in thesimulation of Fig. 1. It clearly reveals the excitation of a

3.2 Spatial fourier spectral analysis
Generally, any perturbation or fluctuation can be∼
described by a spatio-temporal spectrum under the ansatz f ∝→→exp[i( k⋅ x − ωt)]. The flow driven KH mode is characterized
by longer wavelength and lower frequency. To diagnose theKH fluctuation in ETG simulations, it is necessary to analyze
the turbulence spectra. Spatial Fourier spectral analyses showthat turbulent fluctuations are characterized by a shrinking kyFig. 4 Time-frequency wavelet energy spectra of the turbulent fluctuations in zonal flow dominated ETG turbulence. It shows theexcitation of a low frequency fluctuation (KH mode?) at the end of the fast growing phase of zonal flows in Fig. 1.

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lower frequency fluctuation from t ≈ 160 besides the ETG
fluctuation with higher frequency. It shows a visible frequency gap between these two fluctuations. Such phenomenon isnot observed in the reference simulation without zonal flows.
The latter displays a rather wider frequency spectral structure
after the saturation of ETG turbulence, which may be subject
to the nonlinear energy cascade or inverse cascade processes
like the spatial spectrum. Note that the time t ≈ 160 to excite
the low-frequency fluctuation is approximately in agreementwith the transition point t ≈ 200 of the evolution of zonal
flows in Fig. 1. Furthermore, the wavelet analysis of linear
KH instability in Fig. 2 shows that the frequency of KH
mode well corresponds to that of the low-frequency fluctuation observed in Fig. 4. Hence, from the correspondence ofthe spatio-temporal spectra of KH mode with the spectral distribution of the turbulent fluctuations in simulations, the lowfrequency fluctuation may be reasonably inferred as the secondary KH mode. Although the wavelet technique itself isstill suffered from some errors and noise problem, the observations from Fig. 4 may provide a significant evidence forthe excitation of KH mode in ETG turbulence due to theenhanced zonal flows.

slower than that of ETG turbulence. Additional calculation,in which the zonal flow at t = 500 in Fig. 1 is artificially
imposed in the reference simulation in Sec. 3.2, shows that
direct nonlinear coupling between ETG fluctuation and theKH mode does not seem to be strong [6]. Hence, the KH
instability in this work can be identified through the comparison of the spatio-temporal spectra of both linear KH modeand pure ETG fluctuation with the spectral distribution of
mixed fluctuations in ETG simulations. Other nonlinear
mechanisms, such as the spectral modulation of turbulent
fluctuations by the flow [9], energy cascade or inverse energy
cascade, may also occur in ETG turbulence, but they are notdominant ones. Furthermore, the zonal flow is also damped
by electron collision in ETG turbulence [14]. Although theviscosity damping has been applied to the zonal flow component in the simulations above, it is worthwhile to perform asimulation, which includes collisional damping effects, tocompare the contributions of different damping mechanisms
to the saturated level of zonal flow. The work is in progress.

Acknowledgements:
We would like to thank M. Azimi, M. Yagi and H. Sugamafor their supports and discussions.

4. Summary and discussion
References

In summary, based on 3D electrostatic slab ETG turbulence simulations, the saturation of zonal flow with regard toa KH instability is analyzed. All numerical analyses for theexcitation of KH mode in ETG simulations consistently support the working hypothesis that enhanced zonal flow may bedominantly limited by a flow-driven KH instability. It may
suggest a possibility of turbulence transition from the ETGdominated one to the KH-dominated one due to the enhanced
zonal flows in weak shear plasmas.
Although the dynamics of zonal flow involves complicated nonlinear interactions, some quasi-linear processes,
such as the modulation instability for flow generation and theKH instability for flow saturation, may be the most primarymechanisms in ETG simulations above. It depends on thetime scales and nonlinear coupling of different fluctuations.
In slab ETG simulations with weak shears, the generation ofzonal flow and the excitation of KH fluctuation are much

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