Proceedings of ITC/ISHW2007

Eﬀects of multi-helicity confinement fields on zonal flows and ion
temperature gradient turbulenceTomo-Hiko WATANABE1,2) , Sergi FERRANDO-MARGALET1) , and Hideo SUGAMA1,2)
1)2)

National Institute for Fusion Science, Toki 509-5292, JapanThe Graduate University for Advanced Studies (Sokendai), Toki 509-5292, Japan
October 17, 2007

Eﬀects of multi-helicity confinement magnetic fields on zonal flows and ion temperature gradient (ITG)
turbulence are investigated by means of the gyrokinetic-Vlasov simulation code (GKV code). Detailed magnetic field parameters for the standard and inward-shifted configurations of the Large Helical Device (LHD)experiments are introduced in the simulation model. By the linear GKV simulations, it has been shown that, in
the inward-shifted configuration, the maximum ITG mode growth rate slightly increases while the zonal flow
is maintained for a longer time [S. Ferrando-Margalet, H. Sugama, and T.-H. Watanabe, “Zonal flows and ion
temperature gradient instabilities in multiple-helicity magnetic fields” Phys. Plasmas (2007) in press]. Correspondingly, the nonlinear GKV simulations of the ITG turbulence in simple model configurations are extended
to the more elaborate ones in the present study. The obtained results show eﬀective regulation of the ITG turbulence by the zonal flows in the inward-shifted case. The zonal flows generated with three-times larger averaged
amplitudes than those in the standard configuration lead to a lower ion heat transport level. The obtained results
confirm the theoretical prediction that the neoclassical optimization of helical systems contributes to reduction of
the anomalous transport by enhancing the zonal-flow level.
Keywords: zonal flow, ITG turbulence, gyrokinetic simulation, helical system, LHD

1Introduction

Sheared E × B plasma flows with toroidal and poloidal
symmetries, namely the zonal flows, have widely been believed as one of key ingredients for regulating the turbulent
transport in magnetic confinement fusion [1]. In toroidalsystems, the zonal flow is coupled to the geodesic acoustic mode (GAM) oscillation [2]. A linear response of thezonal flow driven by the ion temperature gradient (ITG)
turbulence in a tokamak was derived from the gyrokinetic
theory by Rosenbluth and Hinton [3]. It is considered thatthe residual zonal flow remaining constant after Landau
damping of the GAM plays an important role in reductionof the tokamak ITG turbulent transport.
The gyrokinetic theory of the zonal flow has been extended to helical systems [4, 5], and the idea of the zonal
flow optimization for eﬀective reduction of the turbulent
transport has come out [4, 5, 6]. The zonal flow was first
identified in the Compact Helical System (CHS) experiments [7], and is recognized as a mutual important subject
in the anomalous transport studies for tokamak and helicalsystems.
The role of zonal flows in regulating the turbulenceis, thus, investigated for understanding the transport property observed in the Large Helical Device (LHD) [8] experiments. From the LHD experiments it is found that not
only the neoclassical but also the anomalous transport isreduced in the inward-shifted configuration [9]. Here, it
author’s e-mail: watanabe.tomohiko@nifs.ac.jp

is noteworthy that the radial drift motion of helical-rippletrapped particles is decreased by shifting the magnetic axis
inward while the unfavorable magnetic curvature destabilizing the ballooning-type modes such as the toroidal ITG
mode is increased. The gyrokinetic theory of zonal flows
in helical systems has shown that the slower decay of thezonal flow expected in the inward-shifted LHD configuration lead to lower turbulent transport than that in thestandard one [4, 5]. The theoretical estimate of the linear
response of the zonal flow is verified by the gyrokineticVlasov simulations [4, 5, 10, 11] by means of the GKV
code [12]. The nonlinear GKV simulations for the simplified models of the LHD have demonstrated that theITG turbulent transport in the inward-shifted configuration, which has 60% larger growth rates of the ITG stability, is regulated by the stronger zonal flows to a levelcomparable to the standard case [10].
In the present study, realistic parameter sets for theLHD experimental conditions are employed in the GKV
simulations of the ITG turbulence, where diﬀerences in
the linear stability between the standard and inward-shiftedcases become smaller [11, 13] while slower decay of thezonal flow in the inward-shifted case. Then, it is expectedthat the lower anomalous transport will be found in theinward-shifted case, as actually observed in the LHD experiments. This paper is organized as follows. After a brief
introduction of our simulation model in section 2, the GKV
simulation results of the ITG turbulence and zonal flows
are shown in section 3. A conclusion is given in section 4.

Proceedings of ITC/ISHW2007
Table 1 Parameters used in GKV simulations of ITG turbulenceand zonal flows in LHD model configurations.

Standard

Inward-shifted

0.099
1.9-0.850.087
0.91
-0.28
00.22
1.02
1.96
-0.63
00.114
1.7
-0.96
0.082
1.20
-0.74
-0.24
0.71
1.00
2.44
-0.36
-0.61

r/R0q0ŝϵt
ϵL /ϵtϵL−1 /ϵt
ϵL+1 /ϵt′(r/ϵt )ϵ00′(r/ϵt )ϵt(r/ϵt )ϵL′′(r/ϵt )ϵL−1′(r/ϵt )ϵL+12

Simulation Model

The nonlinear gyrokinetic equation [14] for the perturbed
ion gyrocenter distribution function, δ f , in the low-β electrostatic limit is numerically solved by the GKV code[12] as a partial diﬀerential equation defined on the fivedimensional phase-space. We introduce a model collision term given by the gyrophase average of the LenardBernstein collision operator [15]. The quasi-neutrality
condition with the adiabatic electron response (for T e = T i )is used for calculation of the electrostatic potential, ϕkx ,ky .
In the GKV code, we employ the toroidal flux tube model[16] with the field-aligned coordinates, and also assume
constant volume-averaged density and temperature gradients with scale-lengths of Ln and LT as well as the constantmagnetic shear parameter, ŝ = (r0 /q0 )dq/dr. Here, q(r)stands for the safety factor, and q0 = q(r0 ). See Ref. [12]for more details.
In the helical version of the GKV code, the averaged
minor radius, r0 , is defined by Ψt = πB0 r02 where Ψt means
the toroidal flux. The toroidal and helical eﬀects of theconfinement field are introduced by the change of magnetic
field strength,
B = B0 {1 − ϵ00 (r) − ϵt (r) cos z−l=L+1∑
l=L−1, (1)ϵl (r) cos[(l − Mq0 )z − Mα]where L and M denote the poloidal and toroidal periodicities of the main component of the helical field. For theLHD, L = 2 and M = 10. The main helical field is represented by ϵL . The side-band components and the averaged normal curvature are also given by ϵL−1 , ϵL+1 , and′ϵ00= dϵ00 /dr, respectively. We also set the field-line label
α to be constant (α = 0) because of weak α-dependenceof the linear ITG instability. For the ITG turbulence simulations shown below, we employ a huge number of gridFig. 1 Color contours of the electrostatic potential of the zonal
flow and the ITG turbulence obtained by the GKV simulation for the standard model configurations at t = 25(upper) and t = 100Ln /vti (lower).

points over 50 billions [that is, 128 × 128 × 512 × 128 × 48
in the fluxtube coordinates of (x, y, z, v∥ , µ) of the fivedimensional phase-space]. Further details of the GKV simulation model for helical systems are found in Ref. [10].

3Gyrokinetic Simulation of ITG
Turbulence and Zonal Flows

Nonlinear gyrokinetic simulations of the ITG turbulenceand zonal flows for helical systems [10] are extended so as
to incorporate the realistic parameter sets for the LHD experiments as has been done in the linear GKV simulationsof the ITG instability and zonal flows [11]. Diﬀerences in
the linear growth rates of the ITG instability between thestandard and inward-shifted model configurations, whichwere about 60% in Ref. [10], are largely decreased by using the new parameter sets [11]. Moreover, the collisionless decay of the zonal flow in the realistic model for theinward-shifted configuration takes longer time than thosefor the standard and the previous cases [11]. These results
support the scenario derived from the theoretical analysisof the zonal flow response in helical systems [4, 5], suchthat the stronger zonal flow driven by the ITG turbulencein the inward-shifted case may lead to a lower level of theturbulent transport than that in the standard configuration.
The GKV code employs the toroidal flux tube domain
with the same local plasma parameters of ηi ≡ Ln /LT i = 3,Ln /R0 = 0.3, T e /T i = 1, and α = 0 for the standard
and inward-shifted cases. Other parameters are summarized in Table 1. We set the toroidal periodicity of thesimulation domain Nα = 6, and the half-widths of the ra-Proceedings of ITC/ISHW2007

Spectrum ∑kx〈|φkx,ky|2〉/∆ky40
Standard
Inward-Shifted

35
30
25
20
15
10
500.010.1
kyρi1Fig. 2 Color contours of the electrostatic potential of the zonal
flow and the ITG turbulence obtained by the GKV simulation for inward-shifted model configurations at t = 25(upper) and t = 100Ln /vti (lower).

dial and toroidal (field-line-label) box size are given byr0 ∆q/q0 ŝ and πr0 /q0 Nα , where the change of the safety
factor ∆q = −1/6.
Color contours of the electrostatic potential ϕ obtained from the GKV simulations of the ITG turbulence areshown in Figs. 1 and 2 for the standard and inward-shiftedmodel configurations, respectively. Here, the potential
fluctuations are mapped on the innermost flux surface
and the elliptic poloidal cross-section. Radially-elongatededdy patterns of potential (streamers) are first driven by thetoroidal ITG instability, and propagate in the direction of
the ion diamagnetic drift. The ballooning-type mode structures are clearly found in the linear growth phase of the instability. The growth of the ITG instability is saturated by
the self-generated E × B zonal flows, and the streamers aredestroyed into small eddies in the later turbulent state.
Power spectra of the potential fluctuations of the ITG
turbulence are shown in Fig. 3, where the data is timeaveraged from t = 60 to 100Ln /vti . We see the same peak∑amplitude of the spectrum, kx ⟨|ϕkx ,ky |2 ⟩/∆ky between theinward-shifted and standard cases, while the spectrum inthe former broadens into the low-ky side.
It is expected from the linear GKV simulations of
the ITG mode [11] that, because of the higher maximum
growth rate, ion thermal diﬀusivity, χi , grows faster for
the inward-shifted configuration than for the standard one.
In the nonlinear simulations, we have observed that thepeak value of χi ≃ 3.8ρ2ti vti /Ln for the former case isabout 50% larger than that of χi ≃ 2.6ρ2ti vti /Ln for thelatter case. In the statistically steady states of the turbulent transport, however, χi averaged from t = 60 tot = 250Ln /vti for the inward-shifted case is about 20%

Time-Average of 〈φky=0(x)〉Fig. 3 Power spectra of potential fluctuations of the ITG turbulence obtained from the GKV simulations for the standard (black) and the inward-shifted (red) model configurations. The spectrum is time-averaged from t = 60 to
100Ln /vti .

Standard
Inward-Shifted420-2-4-30-20-100x/ρi102030

Fig. 4 Radial profiles of the zonal flow potential averaged fromt = 60 to 250Ln /vti . Black and red curves represent
the results obtained from the standard and inward-shiftedcases, respectively.

smaller than that from the result for the standard configurations, that is, χi ∼ 1.45ρ2ti vti /Ln for the former case andχi ∼ 1.78ρ2ti vti /Ln for the latter one. See also Ref. [17].
The lower ion heat transport found in the inwardshifted case is attributed to a larger amplitude of thezonal flows generated by turbulence. Radial profiles ofthe zonal flow potential are shown in Fig. 4, where theflux-surface average of the ky = 0 components of thepotential, ⟨ϕky =0 (x)⟩, are time-averaged from t = 60 tot = 250Ln /vti . The zonal-flow potential spectrum also
shows about a three-times larger amplitude in the inwardshifted case than that in the standard configuration [17]. It
should be noted that the stronger zonal-flow generation in
the inward-shifted model configuration is consistent withlarger values of the zonal-flow response function as discussed in our previous works [4, 5, 10, 11].
The transport reduction associated with the zonalflow generation is also found in Lissajous plots of timehistories of the simulation data in Fig. 5, where the vertical axis measures the squared potential of the zonal flow,∑⟨|ϕkx ,0 |2 ⟩. The horizontal axes in upper and lower panels

Proceedings of ITC/ISHW2007

Zonal Flow ∑〈|φkx,0|2〉

10

prediction derived from the gyrokinetic analysis of thezonal flow response in helical systems, that is, the neoclassical optimization results in reduction of the anomaloustransport through enhancement of the zonal flow. This provides ones a physical understanding on a possible mechanism of the confinement improvement found in the inwardshifted configurations of the LHD experiments.

Standard
Inward-Shifted86
42000.51Zonal Flow ∑〈|φkx,0|2〉

10

1.5 2 2.5
χi/(ρ2i vti/Ln)

3

3.5

4

Standard
Inward-Shifted

86
4
2005
1015
Turbulence ∑〈|φkx,ky|2〉

20Fig. 5 Lissajous plots of the simulation results shown in the∑∑∑space of χi - ⟨|ϕkx ,0 |2 ⟩ (upper) and ⟨|ϕkx ,ky |2 ⟩- ⟨|ϕkx ,0 |2 ⟩(lower). Black and red curves represent the results obtained from the standard and inward-shifted cases, respectively.

of Fig. 5 stand for the transport coeﬃcient, χi /(ρ2ti vti /Ln ),∑and the turbulent fluctuations, ⟨|ϕkx ,ky |2 ⟩, respectively. In
the upper panel, starting from the initial condition near theorigin of (0, 0), the data point moves along the horizontal axis as the instability grows. One can see that χi isgreatly reduced when the zonal flow is excited, where theorbit of the data point turns to the upper left corner of thefigure. In the nonlinear saturation stage of the instability (t > 60Ln /vti ), the data points for the two configurations fluctuate only inside two diﬀerent regions, namely,
the region with high transport and weak zonal flows for thestandard configuration and the region with low transportand strong zonal flows for the inward-shifted configuration. Contrarily, the data points in the lower panel movearound in a common horizontal range with a similar turbulent fluctuation level, but with diﬀerent zonal flow amplitudes.

4Conclusion

The nonlinear gyrokinetic-Vlasov simulations present theresults that the ion thermal diﬀusivity χi driven by theion temperature gradient turbulence for the inward-shifted
LHD plasma takes a lower time-averaged value in thesteady turbulent state because of the stronger zonal-flow
generation. The obtained results confirm the theoreticalAcknowledgments
This work is supported in part by grants-in-aid of
the Ministry of Education, Culture, Sports, Science and Technology (No.
16560727, 17360445,and 17·05373), and in part by the National Institute for Fusion Science (NIFS) Collaborative Research Program (NIFS05KKMT001, NIFS06KTAT038,
NIFS06KDAD006, and NIFS06KNXN060). Numericalsimulations are carried out by use of the Earth Simulator
under the support by Japan Agency for Marine-Earth Science and Technology, and by use of the Plasma Simulator
and the LHD numerical analysis system at National Institute for Fusion Science.
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