Plasma and Fusion Research: Rapid Communications

Volume 17, 1203077 (2022)

Impact of Geodesic Curvature on Zonal Flow Generation in
Magnetically Confined Plasmas
Motoki NAKATA1,2,3) and Seikichi MATSUOKA1,2)
1) National Institute for Fusion Science, National Institutes of Natural Sciences, Toki 509-5292, Japan2) The Graduate University for Advanced Studies, Toki 509-5292, Japan
3) PRESTO, Japan Science and Technology Agency, 418 Honcho, Kawaguchishi, Saitama 332-0012, Japan(Received 14 April 2022 / Accepted 8 May 2022)
The impact of magnetic geometry on zonal-flow generation in ion temperature gradient driven turbulence
is investigated by means of linear and nonlinear gyrokinetic simulations. The modulation of geodesic curvature
on various configurations has revealed amplification of the zonal-flow intensity in relatively smaller geodesiccurvature. Based on these findings, a nonlinear proxy model for explorations of novel magnetic geometry toactivate the zonal-flow dynamics is proposed.
© 2022 The Japan Society of Plasma Science and Nuclear Fusion Research

Keywords: magnetic geometry, geodesic curvature, zonal flow, plasma turbulence, gyrokinetic simulation
DOI: 10.1585/pfr.17.1203077
Magnetically confined plasma is regarded not only as
the medium of fusion reactions, but also as an attractive
non-equilibrium system with an extreme spatial inhomogeneity/gradient embedded in magnetic fields with the geometric flexibilities. One of the most prominent physicalprocesses is spontaneous emergence of mesoscopic coherent structures such as zonal flows, which are nonlinearly
excited in microscopic turbulence. The zonal-flow generation is now widely recognized as a key mechanism leading
to improved plasma confinement. An important earlier
finding by gyrokinetic simulations is the suppression of
turbulent transport by enhanced zonal flows [1, 2].
By fully utilizing the diversity of the three-dimensional
magnetic field structures, both axisymmetric and nonaxisymmetric plasmas have much capabilities to suppress/optimize the turbulence. In several pioneering earlier works, turbulence optimization scheme of “proxyapproach” has been developed, where the proxy model
proxyγk⊥ /k ⊥2 is constructed as a function of the geometricquantities to qualitatively reproduce the linear mixinglength diﬀusivity for the ion temperature gradient (ITG)
driven mode and the trapped electron mode (TEM) [3–5].
In this study, a key geometric dependence to amplify
the zonal-flow intensity is extracted by means of the linear
and nonlinear gyrokinetic Vlasov simulations of the ITGdriven turbulence in non-axisymmetric and axisymmetricplasmas. Particular focus is put on the geodesic curvature
Kg of the field line on the flux surface, where the radial drift
excursion for trapped particle is proportional to Kg . Then,the linear damping of zonal flows, which is attributed tothe shielding of the zonal potential by the radial orbit widthof trapped ions, can also aﬀect the zonal-flow intensity inauthor’s e-mail: nakata.motoki@nifs.ac.jp

the nonlinear state. In addition to the linear zonal-flow
response [6–9], the impact of the geodesic curvature on thenonlinearly generated zonal flows are examined here.
The governing equation solved in GKV, a multi-species
electromagnetic gyrokinetic Vlasov code [10], is briefly
shown as follows:()∂µb ·∇B ∂+ 3 ∥ b ·∇ +iωDs −δgsk⊥∂t
ms ∂3 ∥∑()()
c−b · k⊥′ × k⊥′′ δψsk⊥′ δgsk⊥′′ − Cs δgsk⊥
B ∆()µb · ∇Bes FMs ∂δψk⊥=+iω∗T s δψsk⊥ + 3 ∥δψsk⊥ ,Ts∂tTs(1)where δgsk⊥ = δgsk⊥ (t, z, 3 ∥ , µ) stands for the non-adiabatic(g)part of the perturbed gyrocenter distribution function δ fsk⊥
for the particle species “s”. The gyro-averaged potential
fluctuation is denoted by δψk⊥ := J0s [δϕk⊥ − (3 ∥ /c)δ A ∥k⊥ ],where the former and latter terms mean the electrostaticand electromagnetic parts, respectively. The magnetic drift
frequency ωDs is expressed by)(ck⊥ · b × µ∇B + ms 3 2∥ b ·∇b
ωDs =
es B
c(ms 3 2∥ + µB) ()=Kg k x + Kn k y ,(2)
es Baxwhere Kg and Kn represents the geodesic and normal curvature components, respectively. More detailed expressions
are available in e.g., Refs. [11] and [12].
The local fluxtube gyrokinetic simulations of the linear
zonal-flow response and the ITG-driven turbulence havebeen carried out for the several magnetic configurations,
i.e., Large Helical Device (LHD), NCSX-like [13], and the1203077-1© 2022 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Rapid Communications

Volume 17, 1203077 (2022)

Fig. 1 Geodesic curvature dependence of residual ZF level.

Fig. 2 Time evolution of (a) relative zonal-flow intensity Λzf and(b) turbulent transport diﬀusivity.

axisymmetric cases, where the adiabatic electron response
is assumed. The radial position of ρ = 0.5 with the safety
factor q(ρ = 0.5) = {1.99(LHD inward), 2.16(LHD standard), 2.14(NCSX-like), 1.41(Axisymmetric)} is considered, where Rax /Ln = 2, Rax /LTi = 8, and Ti /Te = 1. In thesimulations, the geodesic curvature Kg is artificially modulated in order to extract the geometric dependence, and theother geometrical quantities are fixed.
Figure⟨ 1 shows⟩ ⟨the residual⟩ zonal-flow (ZF) level
Rkzf := δϕkzf (t) z / δϕkzf (t = 0) z as a function of thegeodesic curvature Kg evaluated by the linear gyrokinetic
calculations for the zonal component at t = 50Rax /3ti ,where the quantities are normalized by the reference values corresponding to the original magnetic configuration
without any modulations to Kg . Such normalization enable us to compare the various magnetic configurations in
the context of relative amplification and attenuation. One
finds that the similar amplification/attenuation behavior of
the residual zonal-flow level appears in all configurations
considered here, where a power-decay function X −α withthe exponent of α = 1.5 well approximates Kg dependence.
Then the nonlinear gyrokinetic simulations of the ITGdriven turbulence have been carried out to examine thezonal-flow amplification for the case with a reduced Kg .
The LHD standard configuration, the LHD inward-shifted
configuration, and its modulated configuration with a reduced Kg /Kg(ref) = 0.5 are considered as representative
cases. As shown in Fig. 2 (a), the relative zonal-flow intensity defined by Λzf := Z/(T + Z) strongly depends on∑
the magnetic configuration, where Z := k x |δϕk x,k y=0 | 2∑ ∑and T = k x k y ,0 |δϕk x,k y | 2 . One can confirm the previous finding that the inward-shifted case shows stronger
zonal-flow generation in comparison to the standard case[1], where the turbulent transport is reduced, as shown inFig. 2 (b). It is also emphasized that a more significant amplification of the relative zonal-flow intensity (Λzf ∼ 0.75)
is observed in the case with reduced Kg . Note that, as indicated for t < 25Rax /3ti in Figs. 2, the modulation of Kg has
almost no impacts on the linear ITG growth because theoperator Kg k x in Eq. (2) does not act on the most unstableeigenmode with k x = 0. The zonal-flow amplification is,Fig. 3 Kg dependence of relative zonal-flow intensity Λzf .

thus, attributed from the improved response Rkzf inducedby the geodesic curvature Kg .
The results of systematic scans for Λzf with respect
to Kg are summarized in Fig. 3, where the three kinds ofconfigurations are used as the reference. Although we seea slightly diﬀerent geodesic curvature dependence of theresidual zonal-flow levels (see Fig. 1), a qualitatively similar power-like decay Λzf ∝ Kg−α is identified even for quite
diﬀerent three magnetic geometry, where the exponent has
some extent of 0.4 < α < 1.
One possible reason for the diﬀerence of Kg dependence in Λzf and Rkzf is that the zonal-flow intensity in the
nonlinear state depends not only on the linear response, butalso on the turbulence intensity (source of zonal flows) withvarious spectral shapes in the wavenumber space. Also, thegeodesic acoustic modes (GAM) are not explicitly distinguished in this study, and such individual modeling for
GAM and the steady zonal flows becomes important depending on the q-value. Further modeling studies including
them are remained as future work.
In conclusion, this paper has presented gyrokinetic
simulations to extract the geodesic curvature dependenceof nonlinearly generated zonal flows. The zonal-flow amplification by the magnetic geometry revealed here would
be one of the plausible strategies towards a new discovery

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Plasma and Fusion Research: Rapid Communications

Volume 17, 1203077 (2022)

of innovative confinement geometry and the optimizationsof magnetic configurations in the experiments. Indeed, utilizing the nonlinear turbulence modeling [14] and the proxyapproach for the linear ITG/TEM optimizations [3–5], we
can derive a nonlinear proxy function with Λzf ∝ Kg−α ,proxyχNL =C1(∑

proxy 2k⊥ γk⊥ /k ⊥1 + C2 Kg−α /(∑)β

proxyk⊥ γk⊥/k⊥2)δ ,(3)where (C1, C2, α, β, δ) are optimal parameters to be determined [15]. The nonlinear proxy model encourages theexplorations of novel magnetic geometry to activate zonalflow dynamics, as will be reported in future work.
The authors thank NGS (Next Generation Stellarator)
creation research team at NIFS for their fruitful discussions.
This work is supported by the MEXT Japan, Grant No.
19K03801, in part by the NIFS collaborative Research Programs (NIFS22KIST017, NIFS22KIST010), and in part byJST, PRESTO Grant Number JPMJPR21O7, and in part by

PLADyS, JSPS Core-to-Core Program. Numerical simulations were performed by JFRS-1 at IFERC-CSC, andPlasma Simulator at NIFS.
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