Global Effects on Drift Wave Microturbulence in Tokamak Plasmas

Hui Li(李慧)$^{1\hyperlink{s}{*}}$, Ji-Quan Li(李继全)$^{2}$, and Zheng-Xiong Wang(王正汹)$^{1\hyperlink{s}{*}}$

doi:10.1088/0256-307X/40/10/105201

⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 105201⇨ Global Effects on Drift Wave Microturbulence in Tokamak Plasmas Hui Li (李慧)1*, Ji-Quan Li (李继全)2, and Zheng-Xiong Wang (王正汹)1* Affiliations 1Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams (Ministry of Education), School of Physics, Dalian University of Technology, Dalian 116024, China 2Southwestern Institute of Physics, Chengdu 610041, China Received 2 July 2023; accepted manuscript online 18 September 2023; published online 26 September 2023 ^*Corresponding authors. Email: huilee@dlut.edu.cn; zxwang@dlut.edu.cn Citation Text: Li H, Li J Q, and Wang Z X 2023 Chin. Phys. Lett. 40 105201 Abstract Microturbulence excited by ion temperature gradient (ITG)-dominant and trapped electron mode (TEM)-dominant instabilities is investigated by employing an extended fluid code (ExFC) based on the so-called Landau fluid model, which includes the trapped electron dynamics. Firstly, the global effect is emphasized through direct comparison of ITG and TEM instability domains based on local and global simulations. The global effect makes differences in both linear instability and nonlinear transport, including the fluxes and the structure of zonal flow. The transitions among ITG, TEM, and ITG & TEM (ITG & TEM represents that ITG and TEM coexist with different wavelengths) instabilities/turbulence depend not only on the three key drive forces $({R/L_{\rm n}, R/L_{\rm Te}, R/L_{\rm Ti}})$ but also on their global (profile) effects. Secondly, a lot of electrostatic linear gyro-fluid simulations are concluded to obtain a distribution of the instability.

DOI:10.1088/0256-307X/40/10/105201 © 2023 Chinese Physics Society Article Text Tokamak plasma systems are well recognized for their extremely nonlinear and complicated characteristics. They involve kinds of magnetohydrodynamic modes and micro-instabilities. Furthermore, in magnetically confined fusion reactors, energy and particle losses are attributed to turbulent transport driven by micro-instabilities.[1-5] It is well known that turbulent transport significantly affects and restricts the performance of plasma confinement in magnetic fusion devices because of the tremendous radially outward particle and heat fluxes.[6] Much of the transport, especially in plasma cores, is caused by the turbulence induced by the ion temperature gradient (ITG) mode and trapped electron mode (TEM) in a tokamak.[7-9] For instance, to demonstrate the anomalous particle convection reversal generated by the ITG and TEM modes, a crucial physical model has been statistically constructed.[10] It has been generally recognized that ITG and TEM are responsible for the ion transport and electron transport respectively.[11,12] The coexistence of ITG and TEM turbulence may improve the anomalous transport in tokamaks. Furthermore, the theories,[13] simulations [14] and experiments[15] have indicated that both ITG and TEM may cause energetic particle transport as well. Recently, influence of micro-instability on anomalous fluxes has been illustrated in both L-mode (low confinement mode)[16] and H-mode (high confinement mode)[17-20] plasmas. In the L-mode, the core rotation is significantly complicated, which depended on the electron density, magnetic topology, magnetic field, and so on.[21-23] Furthermore, some physical effects have been considered, including the effect of finite $\beta$, collision effect, magnetic shear, and the finite Larmor radius (FLR) effect.[24-28] Generally, the micro-instabilities are driven by gradients of density, ion and electron temperatures.[29,30] The ITG turbulence is recognized as the dominant mode, in which the inverse scale length of the ion temperature gradient is larger than the critical threshold with the fixed inverse scale length of the electron temperature gradient. Otherwise, the TEM turbulence may dominate. Additionally, the type of micro-turbulence plays a crucial role in turbulent transport. The investigations have shown that the wave spectrum distributions of the ITG and TEM turbulence related to the wave-particle resonance and FLR have influences on the particle transport and heat diffusivity. In addition to the tokamak, which focuses on the transition between ITG and TEM, stellarators have also revealed the mechanism of ITG and TEM. The ‘stability valley’ has been displayed and analyzed.[29] It was demonstrated that the electrostatic instabilities are partly suppressed. Various efforts[31-34] have been made to helpfully understand the influence of the global effect on the plasma simulation. The essential importance of the global effect was found.[31] It was realized that for the outer core positions (i.e., $\rho_{tor} \approx 0.5$–0.7) nonlocal effects are important. In the simulations with less corrugation, the global stabilization was weaker than the local stabilization.[32] Furthermore, the importance of global effects has been addressed via the comparison between the global and local simulations with the observation of different behavior.[33] In Ref. [34], the authors illustrated that some physical effects which are not retained in local simulations are included in the global gyrokinetic simulations of turbulence. Here, global effect means that the physics of the entire cross section is considered, and the profile variations are included. The local simulation is set that just a thin annulus or flux tube is treated, which assumes constant profile gradients. In an effort to understand the influence of the global effect, we compare the global gradient-driven and local gyro-fluid simulations using the extended fluid code (ExFC). Then, the electrostatic instability maps are presented and analyzed. In the following work, nonlinear equations are numerically solved by applying a newly developed global fluid-type turbulence and transport code, i.e., ExFC. It is based on the three-dimensional finite difference method. Further details of the equations and the simulation code were shown in our recent work.[35,36] The ITG and TEM instabilities are involved in this multi-mode model, including plasma density $n_{\rm e}$, electron temperature $T_{\rm e}$, vorticity $\varOmega$, parallel ion velocity $\upsilon_{\parallel}$ and ion temperature $T_{\rm i}$: \begin{align} d_{\rm t} n_{\rm e} =\,&-\omega_{\rm dte} ({n_{0} \phi -T_{\rm e0} n_{\rm e} -n_{0} T_{\rm e}})+D_{\rm n} \nabla_{\bot }^{2} n_{\rm e}, \tag {1}\\ d_{\rm t} T_{\rm e}=\,&-T_{\rm e0} \omega_{\rm dte}[{({\varsigma -1})({\phi-\tau_{\rm en} n_{\rm e}})-({2\varsigma -1})T_{\rm e}}\notag\\ &-({\varsigma-1})\sqrt {8m_{\rm e} T_{\rm e0} /m_{\rm i}\pi}|{\nabla_{\parallel}}|T_{\rm e} +D_{\rm Te} \nabla_{\bot}^{2}T_{\rm e}, \tag {2}\\ d_{\rm t} \varOmega=\,&-T_{\rm i0}a({\partial_{r} {\rm ln}n_{0}+\partial_{r} {\rm ln}T_{\rm i0}})\partial_{\theta } \nabla_{\bot}^{2} \phi\notag\\ &+f_{\rm c} a\partial_{r} {\rm ln}n_{0} \partial_{\theta} \phi-\nabla_{\parallel } \upsilon_{\parallel}\notag \\ &+\omega_{\rm di}[{({1+f_{\rm c}/\tau})\phi +T_{\rm i} +f_{\rm t}\tau_{\rm in} n_{\rm e}}]\notag\\ &+f_{\rm t}\omega_{\rm dte}({\phi -T_{\rm e} -\tau_{\rm en} n_{\rm e}})+D_{U} \nabla_{\bot }^{2} \varOmega, \tag {3}\\ d_{\rm t} \upsilon_{\parallel}=\,&-\nabla_{\parallel } T_{\rm i} -f_{\rm t} \tau_{\rm in}\nabla_{\parallel}n_{\rm e}-({1+f_{\rm c}/\tau})\nabla_{\parallel } \phi\notag\\ &+D_{\upsilon } \nabla_{\bot }^{2} \upsilon_{\parallel}, \tag {4}\\ d_{\rm t}T_{\rm i}=\,&-T_{\rm i0}({\varsigma-1})\nabla_{\parallel}\upsilon_{\parallel} +T_{\rm i0}\omega_{\rm di}\{({\varsigma -1})[({1+f_{\rm c}/\tau})\phi\notag\\ &+f_{\rm t} \tau_{\rm in} n_{\rm e}]+({2\varsigma-1})T_{\rm i}\}\notag \\ &-({\varsigma-1})\sqrt {8T_{\rm i0}/\pi}|{\nabla_{\parallel}}|T_{\rm i} +D_{\rm Ti} \nabla_{\bot }^{2} T_{\rm i}. \tag {5} \end{align} The first two equations describe the non-adiabatic response of trapped electrons. Meanwhile, the latter three equations mainly govern the evolution of the well-known toroidal ITG mode under the quasi-neutrality condition through \begin{align} \nabla_{\bot}^{2} \phi=\frac{f_{\rm c}({\phi-\langle \phi\rangle})}{T_{\rm e0}} -\frac{\varOmega }{n_{0}} , \tag {6} \end{align} where $\langle \phi \rangle$ represents the magnetic surface average of the electrostatic potential.[37] Time and spatial coordinates are normalized by $a/c_{\rm s0}$ and $\rho_{\rm s0}$, respectively, with $\rho_{\rm s0} =m_{\rm i}c_{\rm s0}/e_{\rm i} B_{0}$ being the ion sound gyroradius and $c_{\rm s0} =({T_{0}/m_{\rm i}})^{1/2}$ being the sound speed; $\lambda_{\rm t} =\frac{1}{4}+\frac{2s}{3}$ characterizes the dependence between the precession frequency of trapped electron and magnetic shear $s=\rho dq/qd\rho$; $\varepsilon =a/R$ is the inverse aspect ratio with $a$ and $R$ being the minor and major radii; $f_{\rm t} =2[\pi({2r/R})^{1/2}]^{-1}$ and $f_{\rm c} =1-f_{\rm t} $ are the fraction of trapped and passing electron, respectively; $\omega_{\rm dte} =2\varepsilon (\frac{1}{4}+\frac{2s}{3})qr^{-1}\partial_{\varphi } $ is the trapped electron precession drift operator; and $\omega_{\rm di} =2\varepsilon ({\cos \theta r^{-1}\partial_{\theta}+\sin\theta\partial_{r}})$ is the ion curvature drift operator. The relations between instabilities and real frequency are defined. The mode rotates in the electron diamagnetic direction with $\omega >0$. However, when the real frequency is negative ($\omega < 0)$, the mode rotates in the ion diamagnetic direction. The other operators are defined as $d_{\rm t} =\partial_{\rm t} f+[{\phi ,f}]$, $[{f,g}]=r^{-1}({\partial_{r} f\partial_{\theta } g-\partial_{\theta } f\partial_{r} g})$. The adiabatic compression index is $\varsigma =5/3$. The following results are completed by applying Cyclone reference cases including the density, temperature and safety factor profiles: \begin{align} &n(r)=n_{0}\exp\{{-({\Delta_{r} /L_{\rm n}})\tanh[{({r-r_{0}})/\Delta_{r}}]}\}, \tag {7}\\ &T_{j}(r)=T_{0j}\exp\{{-({\Delta_{rj}/L_{Tj}})\tanh[{({r-r_{0}})/\Delta_{rj}}]}\}, \tag {8}\\ &q(r)=q_{0} +q_{a}({r/a})^{2}. \tag {9} \end{align} Here $j$ in subscripts stands for electrons and ions. Other parameters are $q_{0}=0.85$ and $q_{a} =2.18$. The computational domain is set as $[{L_{r}, L_{\theta}, L_{\varphi}}]\approx [{a,2\pi,2\pi}]$ in toroidal configuration $({r,\theta,\varphi})$. Otherwise, the ratios between the temperature and density are defined as \begin{align} \tau =T_{\rm e0} /T_{\rm i0},~~\tau_{\rm en} =T_{\rm e0} /n_{0},~~\tau_{\rm in} =T_{\rm i0} /n_{0}. \tag {10} \end{align}

Fig. 1. Diagrams for the parametric domain of dominant instability depending on three key drive forces $({R/L_{\rm n}, R/L_{\rm Te}, R/L_{\rm Ti}})$: (a) $R/L_{\rm Te} =1.1$, (b) $R/L_{\rm Te} = 4.4$, (c) $R/L_{\rm Te} = 8.8$. The instability boundary is labeled by the dashed (local) and solid (global) curves.

Firstly, simulations are performed to show the significance of global effects in determining unstable regimes of ITG and/or TEM modes and associated turbulent transport. The instability type depends sensitively on some key parameters so that a clear identification of the instability boundary in the parametric domain is essential. Hence, simulations are carried out first to identify the instability regime depending on key parameters $({R/L_{\rm n}, R/L_{\rm Te}, R/L_{\rm Ti}})$ with comparison between local and global models.

Fig. 2. Comparison of mode structures of dominant instability in linear (a) local and (b) global simulations.

Based on the above discussions, three types of instabilities are involved in the scan, depending on the dominant one in the initial value simulations with the ExFC code. Figure 1 plots the parametric domain of dominant instability. Here, three or four regions for the most unstable mode are partitioned including ITG, TEM, coexisting ITG and TEM mixture (ITG & TEM) as well as the stable one. The regional boundary is drawn by dashed or solid curves, which correspond to the results from local or global simulations. Distinct characteristics are discussed in the following. The region of small $R/L_{\rm Te}$ in Fig. 1(a) is the stable region. It indicates the existence of instability thresholds for three types of instabilities when all the three gradients are small. It also evidences the $\nabla n$-TEM instability when ion and electron temperature profiles are flatter. Here, $\nabla n$-TEM means that the TEM is driven by the density gradient $\nabla n$. Another region, unstable ITG & TEM is narrow for small $R/L_{\rm Te} $ roughly around $\eta_{\rm i} \sim 1$. ITG & TEM represents that ITG and TEM coexist with different wavelengths. Meanwhile, the global effect remarkably expands the ITG & TEM region toward the ITG side but slightly toward the TEM side. As $R/L_{\rm Te} $ increases, the stable region disappears due to the $\nabla T_{\rm e} $-TEM (electron temperature driven TEM) excitation; the ITG & TEM instability region is enlarged toward ITG side, especially in the local simulations, as shown in Fig. 1(b). For much steeper electron temperature, on the one hand, the ITG region is further shrunk and the TEM region is slightly expanded, as illustrated in Fig. 1(c). On the other hand, the global effect makes much more difference in the TEM region from the local prediction but no change in ITG region. Hence, the transition between ITG and TEM as well as the ITG & TEM instabilities depends not only on the three key drive forces $({R/L_{\rm n}, R/L_{\rm Te}, R/L_{\rm Ti}})$ but also on their global (profile) effects. For further comparison, mode structures are checked in Fig. 2. Both global and local cases with parameters $R/L_{\rm Te} =R/L_{\rm Ti} =R/L_{\rm n} =8.8$ are simulated. In the local case [Fig. 2(a)], the dominant instability is TEM. However, ITG is stronger in the global case as displayed in Fig. 2(b).

Fig. 3. Comparison of nonlinear global (a) and local (b) simulations of corresponding particle flux.

As shown in Fig. 3, the global effect also has influence on turbulent transport, not only due to the transition of dominant instability but also through the change of turbulence properties. This work is focused on turbulent particle transport, considering that the formation of particle transport structure, e.g., particle ITB, is extensively ascribed to the ITG and TEM dynamics. Similar to the above linear results (Fig. 2), nonlinear simulations with parameters $R/L_{\rm Te} =R/L_{\rm Ti} =R/L_{\rm n} =8.8$ are performed for both global and local cases, as shown in Fig. 3. Comparisons do not only confirm the dominant instability changes from TEM to ITG due to the global effect, but also exhibit that the radial width of turbulent fluctuations is limited by the profiles, showing a shrunken structure of turbulence and turbulent particle flux. Similar phenomena are observed for structures of heat fluxes. The corresponding characteristic of zonal flow in global ITG turbulence is more robust, so that the ITG turbulence is saturated and suppressed as shown in Fig. 4. For further comparison, the turbulent transports of local ITG and global ITG are compared. Figure 5 shows the ion heat fluxes caused by ITG turbulence. The global effect may drive lightly larger ion heat flux compared with the local case. Additionally, the radial structure is also different slightly.

Fig. 4. Comparison of time evolution of corresponding zonal flows in the nonlinear global (a) and local (b) simulations.

Fig. 5. Comparison of nonlinear global (a) and local (b) simulations of corresponding ion heat flux for ITG.

In summary, the global and local gradient-driven simulations are compared using the ExFC code. The main results of this study can be summarized as follows. For small $R/L_{\rm Te}$, unstable ITG & TEM domain is narrow, roughly around $\eta_{\rm i} \sim 1$. The global effect remarkably expands the ITG & TEM region toward the ITG side but slightly toward the TEM side. As $R/L_{\rm Te} $ increases, the ITG & TEM instability region is enlarged toward ITG side. For much steeper electron temperature, the global effect makes much difference in the TEM region from the local prediction but no change in ITG region. Therefore, with the comparison between the local and global simulation, the influence of global effects is illustrated, in which different behavior is observed. Furthermore, the nonlinear phenomena are also affected with the global effect. Acknowledgement. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12205035, 11925501, 12275071, and U1967206). References Drift waves and transportUnderstanding of the density profile shape, electron heat transport and internal transport barriers observed in ASDEX UpgradeCross-Scale Interactions between Electron and Ion Scale Turbulence in a Tokamak PlasmaStabilization of Short Wavelength Resistive Ballooning Modes by Ion-to-Electron Temperature and Gradient Ratios in Tokamak Edge PlasmasBasic features of the multiscale interaction between tearing modes and slab ion-temperature-gradient modes*Turbulent particle transport in Tore SupraInward transport of energy during off-axis heating on the DIII-D tokamakStudies of impurity mode and ion temperature gradient mode in toroidal plasmasComparison of ITG and TEM Microturbulence in DIII–D Tokamak^*Anomalous Convection Reversal due to Turbulence Transition in Tokamak PlasmasTurbulent Transport of Trapped-Electron Modes in Collisionless PlasmasComparisons and physics basis of tokamak transport models and turbulence simulationsTurbulent transport of ions with large Larmor radiiTransport of Energetic Particles by Microturbulence in Magnetized PlasmasEvidence for Fast-Ion Transport by MicroturbulenceIdentifying microturbulence regimes in a TCV discharge making use of physical constraints on particle and heat fluxesParticle transport in low-collisionality H-mode plasmas on DIII-DCollisionality driven turbulent particle transport changes in DIII-D H-mode plasmasHeat transport driven by the ion temperature gradient and electron temperature gradient instabilities in ASDEX Upgrade H-modesDensity gradient driven microinstabilities and turbulence in ASDEX Upgrade pellet fuelled plasmasSpontaneous L-mode plasma rotation scaling in the TCV tokamakObservations of core toroidal rotation reversals in Alcator C-Mod ohmic L-mode plasmasCore intrinsic rotation behaviour in ASDEX Upgrade ohmic L-mode plasmasBeta scaling of transport in microturbulence simulationsElectromagnetic effects on plasma microturbulence and transportThe effect of safety factor and magnetic shear on turbulent transport in nonlinear gyrokinetic simulationsGyrokinetic analysis of radial dependence and global effects on the zero particle flux condition in a TCV plasmaCollisional Effects on Drift Wave Microturbulence in Tokamak PlasmasSuppression of electrostatic micro-instabilities in maximum-J stellaratorsChapter 2: Plasma confinement and transportComparisons between global and local gyrokinetic simulations of an ASDEX Upgrade H-mode plasmaGyrofluid Simulation of Slab ITG Turbulence in Plasmas Including Pressure Profile CorrugationNonlocal effects in negative triangularity TCV plasmasNonlinear Energy Cascading in Turbulence during the Internal Reconnection Event at the Sino-United Spherical TokamakSimulation prediction of micro-instability transition and associated particle transport in tokamak plasmasMachine learning of turbulent transport in fusion plasmas with neural networkTurbulence suppression in the neighbourhood of a minimum- q surface due to zonal flow modification in reversed shear tokamaks

[1]	Horton W 1999 Rev. Mod. Phys. 71 735
[2]	Peeters A G, Angioni C, Apostoliceanu M, Pereverzev G V, Quigley E, Ryter F, Strintzi D, Jenko F, Fahrbach U, Fuchs C, Gehre O, Hobirk J, Kurzan B, Maggi C F, Manini A, McCarthy P J, Meister H, and Schweinzer J 2005 Nucl. Fusion 45 1140
[3]	Maeyama S, Idomura Y, Watanabe T H, Nakata M, and Nunami M 2015 Phys. Rev. Lett. 114 255002
[4]	Xu J Q, Peng X D, Qu H P, and Hao G 2020 Chin. Phys. Lett. 37 062801
[5]	Wei L, Wang Z X, Li J Q, Hu Z Q, and Kishimoto Y 2019 Chin. Phys. B 28 12
[6]	Hoang G T, Bourdelle C, Garbet X, Pegourie B, Artaud J F, Basiuk V, Bucalossi J, FenziBonizec C, Clairet F, Eriksson L G, Gil C, Guirlet R, Imbeaux F, Lasalle J, Lowry C, Schunke B, Segui J L, Travere J M, Tsitrone E, and Vermare L 2006 Nucl. Fusion 46 306
[7]	Petty C C and Luce T C 1994 Nucl. Fusion 34 121
[8]	Dong J Q and Horton W 1995 Phys. Plasmas 2 3412
[9]	Hu W, Feng H Y, and Zhang W L 2019 Chin. Phys. Lett. 36 085201
[10]	Sun T T, Chen S Y, Wang Z H, Peng X D, Huang J, Mou M L, and Tang C J 2015 Chin. Phys. Lett. 32 035201
[11]	Xiao Y and Lin Z 2009 Phys. Rev. Lett. 103 085004
[12]	Dimit A M, Bateman G, Beer M A, Cohen B I, Dorl W, Hammett G W, Kim C, Kinsey J E, Kotschenreuther M, Kritz A H, Lao L L, Mandrekas J, Nevins W M, Parker S E, Redd A J, Shumaker D E, Sydora R, and Weiland J 2000 Phys. Plasmas 7 969
[13]	Vlad M, Spineanu F, Itoh S I, Yagi M, and Itoh K 2005 Plasma Phys. Control. Fusion 47 1015
[14]	Zhang W, Lin Z, and Chen L 2008 Phys. Rev. Lett. 101 095001
[15]	Heidbrink W W, Park J M, Murakami M, Petty C C, Holcomb C, and van Zeeland M A 2009 Phys. Rev. Lett. 103 175001
[16]	Mariani A, Brunner S, Dominski J, Merle A, Merlo G, Sauter O, Görler T, Jenko F, and Told D 2018 Phys. Plasmas 25 012313
[17]	Mordijck S, Wang X, Doyle E J, Rhodes T L, Schmitz L, Zeng L, Staebler G M, Petty C C, Groebner R J, Ko W H, Grierson B A, Solomon W M, Tala T, Salmi A, Chrystal C, Diamond P, and McKee G R 2015 Nucl. Fusion 55 113025
[18]	Mordijck S, Rhodes T L, Zeng L, Salmi A, Tala T, Petty C C, McKee G R, Reksoatmodjo R, Eriksson F, Fransson E, and Nordman H 2020 Nucl. Fusion 60 066019
[19]	Ryter F, Angioni C, Dunne M, Fischer R, Kurzan B, Lebschy A, McDermott R M, Suttrop W, Tardini G, Viezzer E, Willensdorfer M, and the ASDEX Upgrade Team 2019 Nucl. Fusion 59 096052
[20]	Angioni C, Lang P T, Manas P 2017 Nucl. Fusion 57 116053
[21]	Duval B P, Bortolon A, Karpushov A, Pitts R A, Pochelon A, Sauter O, Scarabosio A, Turri G, and the TCV Team 2008 Phys. Plasmas 15 056113
[22]	Rice J E, Duval B P, Reinke M L, Podpaly Y A, Bortolon A, Churchill R M, Cziegler I, Diamond P H, Dominguez A, Ennever P C, Fiore C L, Granetz R S, Greenwald M J, Hubbard A E, Hughes J W, Irby J H, Ma Y, Marmar E S, McDermott R M, Porkolab M, Tsujii, and Wolfe S M 2011 Nucl. Fusion 51 083005
[23]	Mcdermott R M, Angioni C, Conway G D, Dux R, Fable E, Fischer R, Putterich T, Ryter F, Viezzer E, and the ASDEX Upgrade Team 2014 Nucl. Fusion 54 043009
[24]	Candy J 2005 Phys. Plasmas 12 072307
[25]	Snyder P B and Hammett G W 2001 Phys. Plasmas 8 744
[26]	Kinsey J E, Waltz R E, and Candy J 2006 Phys. Plasmas 13 022305
[27]	Mariani A, Brunner S, Merlo G, Sauter O 2019 Plasma Phys. Control. Fusion 61 064005
[28]	Hu W, Feng H Y, and Dong C 2018 Chin. Phys. Lett. 35 105201
[29]	Alcusón J A, Xanthopoulos P, Plunk G G, Helander P, Wilms F, Turkin Y, von Stechow A, and Grulke O 2020 Plasma Phys. Control. Fusion 62 035005
[30]	Doyle E J, Houlberg W A, Kamada Y, Mukhovatov V, Osborne T H, Polevoi A, Bateman G, Connor J W, Cordey J G, and Fujita T 2007 Nucl. Fusion 47 S18
[31]	Navarro A B, Told D, Jenko F, Grler T, Happel T, and ASDEX Upgrade Team 2016 Phys. Plasmas 23 042312
[32]	Miyata S, Li J, Imadera K, and Kishimoto Y 2011 J. Plasma Fusion Res. 6 2403113
[33]	Merlo G, Huang Z, Marini C, Brunner S, Coda S, Hatch D, Jarema D, Jenko F, Sauter O, and Villard L 2021 Plasma Phys. Control. Fusion 63 044001
[34]	Chai S, Xu Y H, Gao Z, Wang W H, Liu Y Q, and Tan Y 2017 Chin. Phys. Lett. 34 025201
[35]	Li H, Li J Q, Fu Y L, Wang Z X, and Jiang M 2022 Nucl. Fusion 62 036014
[36]	Li H, Fu Y, Li J, and Wang Z 2021 Plasma Sci. Technol. 23 115102
[37]	Miyato N, Kishimoto Y, and Li J Q 2007 Nucl. Fusion 47 929