Pith. sign in

REVIEW 4 major objections 4 minor 46 references

First direct observation of fishbone-driven zonal flows with a fine-scale, radially reversed structure inside the q=1 surface of a tokamak, produced by beat-driven self-coupling rather than energetic-particle expulsion.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

T0 review · deepseek-v4-flash

2026-08-01 05:02 UTC pith:3YUWULIL

load-bearing objection A genuinely new experimental observation with a plausible mechanism, but the central reversed-flow profile needs uncertainty quantification before it carries the claim. the 4 major comments →

arxiv 2607.22344 v1 pith:3YUWULIL submitted 2026-07-24 physics.plasm-ph

First Observation of Fishbone-Driven Zonal Flows with Fine Reversed Structure in Tokamak Plasmas

classification physics.plasm-ph
keywords fishbone instabilityzonal flowstokamakEASTDoppler reflectometrygyrokinetic simulationenergetic-particle-driven modesplasma confinement
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reports the first direct observation in the EAST tokamak of zonal flows driven by fishbone bursts, detected with multi-channel Doppler reflectometry. The flows show a fine-scale, radially reversed structure inside the q=1 surface — small, alternating directions — rather than the macroscopic well-like profile predicted by earlier models based on energetic-particle expulsion. Temporal analysis shows the flows grow faster and saturate earlier than the fishbone itself, indicating a beat-driven nonlinear process (self-coupling of the fishbone) rather than expulsion. Global nonlinear gyrokinetic simulations reproduce the measured radial profile and attribute the reversal to cancellation between comparable, opposite contributions from thermal ions and electrons, with energetic ions contributing only a modest long-wavelength component. If correct, this establishes fishbones as a source of sheared zonal flows with a distinct radial topology, potentially influencing turbulence and core confinement.

Core claim

In the core of the EAST tokamak, low-frequency E×B flows measured by Doppler reflectometry intensify in synchrony with fishbone bursts. Bispectral analysis shows a bicoherence peak at (fishbone frequency, minus fishbone frequency), demonstrating that the flows are generated by nonlinear self-coupling of the fishbone. The flows have sub-km/s amplitudes and, inside the q=1 surface, reverse sign radially on a fine scale — a structure not predicted by the standard energetic-particle-expulsion picture, which gives a macroscopic, non-reversing well. A global nonlinear gyrokinetic simulation, retaining only the n=1 mode and the zonal component, reproduces the measured radial flow profile quantitati

What carries the argument

The central mechanism is the beat-driven self-coupling of the fishbone: the n=1 mode couples with its complex conjugate (f + (-f) = 0, k + (-k) = 0) to produce a zonal (toroidally and poloidally symmetric, low-frequency) electric field. In the simulations, the fine reversed radial structure is explained by solving the flux-surface-averaged gyrokinetic Poisson equation with separate species responses; the zonal electron density response cancels part of the zonal ion density, and the residual field from thermal ions and electrons nearly cancels, yielding a small net field whose shape is set by the thermal species, not the energetic ions. The key diagnostic enabling the observation is multi-cha

Load-bearing premise

The claim rests on the Doppler reflectometer resolving a velocity reversal of less than 1 km/s with uncertainties smaller than the signal; the figures do not show error bars, so a comparable noise floor would make the reversed rings an artifact.

What would settle it

Take the same tokamak discharge (or a repeat) and measure the radial perpendicular-velocity profile with an independent diagnostic that resolves sub-km/s flows, such as charge-exchange recombination spectroscopy; if the radial sign reversal is not reproduced, the fine reversed structure is likely instrumental rather than physical.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Fishbones can generate sheared zonal flows without relying on energetic-particle loss, so the beneficial turbulence regulation might be achieved while preserving fast-ion confinement.
  • Predictive models of fishbone saturation and internal transport barriers must include the zonal electron density response and thermal-species kinetics; the energetic-particle-expulsion picture alone yields the wrong radial structure.
  • The two-stage growth (beat-driven flows before expulsion-driven flows) reconciles earlier observations linking fishbone bursts to transport barriers in several tokamaks, where both mechanisms could act together.
  • The measured flow profile and matched temperature-perturbation envelope provide a quantitative benchmark for global gyrokinetic simulations of fast-ion-driven instabilities in the core of burning plasmas.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the reversal is robust, the same fine-scale reversed topology may be generic for energetic-particle-driven modes whenever the electron kinetic response is retained; simulations of other modes (e.g., toroidal Alfvén eigenmodes) could be examined for the same cancellation.
  • Because the net flow comes from near-cancellation of large opposite thermal-ion and electron contributions, modest changes in the ion-to-electron temperature ratio or density gradient could flip the sign of the flow again, offering a potential experimental control of flow structure.
  • The two-stage picture suggests a reduced, source-term model for fishbone-driven transport — beat-driven early, expulsion-driven late — that could be included in integrated tokamak modeling codes.
  • The success of the Doppler reflectometry technique hints that it could be deployed to hunt for zonal flows driven by other chirping fast-ion instabilities, broadening the evidence base beyond a single mode.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper reports what it claims to be the first direct experimental observation of fishbone-driven zonal flows in a tokamak core, using multichannel Doppler reflectometry on EAST. The central observations are: (i) low-frequency perpendicular flows appear during fishbone bursts; (ii) bispectral analysis shows a peak at (f_FB, −f_FB), indicating self-coupling of the fishbone; (iii) the ensemble-averaged flow rises faster and saturates earlier than the fishbone, suggesting a beat-driven process; and (iv) the radial flow profile inside q=1 exhibits a fine-scale, direction-reversed structure. Nonlinear GTC simulations retaining n=1 and n=m=0 modes reproduce the radial electric field profile, and a species decomposition attributes the reversal to cancellation of comparable, opposite thermal-ion and thermal-electron contributions. The paper argues this structure is not captured by the energetic-particle-expulsion framework.

Significance. If the experimental observations withstand scrutiny, this would be a genuinely important result: direct evidence of fishbone-driven zonal flows with a fine-scale radial topology that is absent from existing EP-expulsion models, as well as a new mechanistic picture involving beat-driven excitation and thermal-species cancellation. The paper has real strengths: the bispectral evidence is a standard, appropriate tool; the temporal ensemble averaging over ten bursts is a good practice; the GTC simulations are global, nonlinear, and constrained by the experimental equilibrium and profiles, with no free parameters fitted to the measured Er profile; and the species decomposition is a useful diagnostic. The main weakness is that the paper's central novelty — the sub-km/s, radially reversed flow structure — is presented without uncertainty quantification, and the simulation mode truncation makes the mechanistic conclusion less independent than it appears.

major comments (4)
  1. [§Experimental setup; Fig. 3(b); Fig. 5(a)] The core claim of a fine, radially reversed flow structure rests on Doppler reflectometry measurements of u⊥ with mean amplitudes below 1 km/s, yet no error bars, calibration uncertainty, radial channel spacing, or statistical significance test are provided for the radial profile. Fig. 3(a) shows burst-to-burst scatter for the temporal evolution, but there is no equivalent scatter or uncertainty band for the radial profile in Fig. 3(b) or for the comparison with GTC in Fig. 5(a). At sub-km/s levels, systematic uncertainties from the DR k⊥ calibration, Doppler-shift extraction, beam refraction, and density-fluctuation contamination can be comparable to the signal. The authors should provide random and systematic uncertainties, demonstrate that the reversal between adjacent channels is significant above the measurement noise, and state the spatial resolution/channel spacing. Without this,
  2. [§Zonal electric field driven by Fishbone; Fig. 4(a)] The simulation is presented as evidence for a beat-driven mechanism because γ_n,m=0 ≈ 2γ_n=1. But the GTC runs retain only the n=1 toroidal mode together with n=m=0 zonal flows. Under this mode truncation, the n=m=0 flow can only be generated by self-coupling of n=1, so γ_n,m=0 ≈ 2γ_n=1 is a structural consequence of the model, not an independent confirmation of the beat-driven process over EP expulsion. To support the mechanistic claim, the authors need either a fuller toroidal-mode spectrum (including n>1 and the associated zonal-flow damping) or a controlled test isolating the beat-driven contribution, such as artificially suppressing the zonal electron response or comparing with a run that includes energetic-particle redistribution. As written, the simulation cannot independently arbitrate the mechanism.
  3. [§Temporal and radial characteristics; Fig. 3(a)] The claim that the flow 'rises faster and saturates earlier' than the fishbone is load-bearing for the two-stage, beat-driven picture. This is supported only by an unshown exponential fit to the ensemble-averaged traces; the fit rates, their uncertainties, and the criterion for 'faster'/'earlier' are not reported. Given the visible burst-to-burst scatter in Fig. 3(a), a statistical comparison (e.g., confidence intervals on the rise-rate ratio and the time-to-peak difference) is needed before this temporal-ordering evidence can be considered quantitative.
  4. [§Zonal electric field driven by Fishbone; Fig. 5(b)] The sentence 'The difference is attributed to the self-consistent inclusion of the zonal electron density response' is presented as a causal conclusion, but no controlled numerical experiment is shown. The species decomposition in Fig. 5(b) demonstrates comparable opposite ion and electron contributions; it does not by itself show that omitting the zonal electron response would produce a macroscopic, well-like field without reversal. A sensitivity run with the electron zonal response artificially removed, or a comparison against a model treating electrons as a passive background, should be included to substantiate the claimed causal role.
minor comments (4)
  1. [Introduction] Typographical error: 'sufficient' should be 'sufficient'.
  2. [Fig. 2(b) caption and §Flows driven by fishbone] The notation '⟨ eTe eTeeu⊥⟩' is confusing and unexplained. Define the cross-bispectrum in the text; also define f_flow, which is used to draw the line in Fig. 2(c) but not defined in the main text.
  3. [Fig. 4 and §Zonal electric field driven by Fishbone] The normalization 'δϕ (normalized by electron temperature Te and charge e)' is ambiguous. Clarify whether δϕ is divided by (Te/e) or by Te with e absorbed elsewhere.
  4. [References] Reference [45] is incomplete: 'Physics of Plasmas 31 (2024)' lacks an article number or DOI. Also, the GTC code reference [8] could be accompanied by the specific GTC version or link used in this work.

Circularity Check

0 steps flagged

No significant circularity; the simulation is not fitted to the observed flow profile and the central experimental claim stands independently.

full rationale

The paper's central claim is an experimental observation of fishbone-driven zonal flows from Doppler reflectometry, supported by bispectral coupling analysis and temporal evolution. The GTC simulation is an independent first-principles calculation initialized from kinetic-EFIT, TRANSP, and NUBEAM profiles; the measured Er/u⊥ radial profile is not used as an input or fitting target, so the 'quantitative reproduction' is not equivalent to its input by construction. The beat-driven interpretation is supported by the experimental observation that the flow rises faster and saturates earlier than the fishbone, not solely by the simulation. The simulation retaining only n=1 plus n=m=0 zonal flows means self-beating is the only non-zonal drive in the model, so it cannot independently arbitrate between beat-driven and EP-expulsion mechanisms, but the authors explicitly acknowledge that the two mechanisms are not mutually exclusive and that EP expulsion may contribute at later stages, which mitigates rather than creates circularity. Self-citations [24,38,39] appear only as background motivation or diagnostic references and are not load-bearing for the derivation. The absence of error bars on the DR radial profile is a validation and uncertainty-quantification concern, not a circularity. No fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is imported from self-authored prior work to force the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No fitted free parameters are used to force the central result: simulation inputs come from experimental equilibrium and profile reconstruction rather than from tuning to the measured flow profile. The main assumptions are model choices (gyrokinetic closure, single-n truncation, representative time slice) and the diagnostic interpretation of u⊥. No new physical entities are introduced.

axioms (5)
  • domain assumption Gyrokinetic δf description with fluid-kinetic electrons (GTC) captures fishbone nonlinear saturation and zonal flow generation.
    Invoked throughout the Simulation section; convergence is asserted but the model's validity for this discharge is not independently established.
  • ad hoc to paper Retaining only n=1 and n=m=0 modes is sufficient to reproduce the observed flow structure and mechanism.
    Simulation section states 'retain only the n = 1 toroidal mode together with the n = m = 0 ZFs'; this truncation forces zonal flow generation through n=1 self-beat and excludes turbulence and n>1 harmonics.
  • domain assumption Equilibrium and profiles at t=3.72 s (kinetic-EFIT, TRANSP, NUBEAM) are representative of all ensemble-averaged bursts.
    Simulation uses a single time slice; burst-to-burst equilibrium variation is not quantified.
  • domain assumption u⊥ measured by Doppler reflectometry is directly linked to the E×B flow and low-pass filtering isolates the zonal component.
    Based on ref [40] and standard DR practice; no in-situ absolute velocity calibration is given in this paper.
  • standard math The Kim-Powers bispectral criterion validly identifies nonlinear three-wave coupling from the 250-window ensemble.
    Statistically standard technique; the bicoherence noise background is asserted but details of the significance threshold are minimal.

pith-pipeline@v1.3.0-alltime-deepseek · 8441 in / 13809 out tokens · 148234 ms · 2026-08-01T05:02:24.377265+00:00 · methodology

0 comments
read the original abstract

We present the first direct experimental observation of fishbone-driven zonal flows in the core of the EAST tokamak. In contrast to the global pattern predicted by previous models and simulations based on the energetic-particle-expulsion mechanism, the observed flows exhibit a fine-scale, radially reversed structure inside the q = 1 rational surface. The flow rises faster and saturates earlier than the fishbone within a single burst, indicating that a beat-driven nonlinear process dominates the early stage rather than the energetic-particle-expulsion mechanism. Global nonlinear gyrokinetic simulations quantitatively reproduce the observed radial profile and reveal that this structure arises from the cancellation of comparable but opposite contributions from thermal ions and electrons. This cancellation mechanism is not captured in previous theoretical frameworks. These findings establish that the fishbone can generate sheared flows with a distinct radial topology, offering a promising pathway for regulating turbulence and improving core confinement.

Figures

Figures reproduced from arXiv: 2607.22344 by Adi Liu, Bin Zhang, Chu Zhou, Feifei Long, Ge Zhuang, Haiqing Liu, Huishan Cai, Jinlin Xie, Liqing Xu, Liutian Gao, Ming Xu, Mingyuan Wang, the EAST Team, Xiaoming Zhong, Yuehao Ma.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

46 extracted references

  1. [1]

    C. C. Porco, R. A. West, A. McEwen, A. D. Del Ge- nio, A. P. Ingersoll, P. Thomas, S. Squyres, L. Dones, C. D. Murray, T. V. Johnson, J. A. Burns, A. Brahic, G. Neukum, J. Veverka, J. M. Barbara, T. Denk, M. Evans, J. J. Ferrier, P. Geissler, P. Helfenstein, T. Roatsch, H. Throop, M. Tiscareno, and A. R. Vasavada, Science 299, 1541 (2003)

  2. [2]

    S. A. Balbus and J. F. Hawley, Rev. Mod. Phys. 70 (1998)

  3. [3]

    Terry, Reviews of Modern Physics 72, 109 (2000)

    P. Terry, Reviews of Modern Physics 72, 109 (2000)

  4. [4]

    P. H. Diamond, S.-I. Itoh, K. Itoh, and T. S. Hahm, Plasma Physics and Controlled Fusion 47, R35 (2005)

  5. [5]

    Biglari, P

    H. Biglari, P. H. Diamond, and P. W. Terry, Physics of Fluids B: Plasma Physics 2, 1 (1990)

  6. [6]

    Diamond and Y.-B

    P. Diamond and Y.-B. Kim, Physics of Fluids B: Plasma Physics 3, 1626 (1991)

  7. [7]

    K. H. Burrell, Physics of Plasmas 4, 1499 (1997)

  8. [8]

    Z. Lin, T. S. Hahm, W. Lee, W. M. Tang, and R. B. White, Science 281, 1835 (1998)

  9. [9]

    Connor, T

    J. Connor, T. Fukuda, X. Garbet, C. Gormezano, V. Mukhovatov, M. Wakatani, et al. , Nuclear Fusion 44, R1 (2004)

  10. [10]

    Fujisawa, Nuclear Fusion 49, 013001 (2009)

    A. Fujisawa, Nuclear Fusion 49, 013001 (2009)

  11. [11]

    Ida and T

    K. Ida and T. Fujita, Plasma Physics and Controlled Fu- sion 60, 033001 (2018)

  12. [12]

    Yoshida, R

    M. Yoshida, R. McDermott, C. Angioni, Y. Came- nen, J. Citrin, M. Jakubowski, J. Hughes, Y. Idomura, P. Mantica, A. Mariani, et al., Nuclear Fusion 65, 033001 (2025)

  13. [13]

    P. H. Diamond, Y.-M. Liang, B. A. Carreras, and P. W. Terry, Physical Review Letters 72, 2565 (1994)

  14. [14]

    Y. H. Xu, C. X. Yu, J. R. Luo, J. S. Mao, B. H. Liu, J. G. Li, B. N. Wan, and Y. X. Wan, Physical Review Letters 84, 3867 (2000)

  15. [15]

    Kim and P

    E.-j. Kim and P. H. Diamond, Physical Review Letters 90, 185006 (2003)

  16. [16]

    Schmitz, L

    L. Schmitz, L. Zeng, T. L. Rhodes, J. C. Hillesheim, E. J. Doyle, R. J. Groebner, W. A. Peebles, K. H. Burrell, and G. Wang, Phys. Rev. Lett. 108, 155002 (2012)

  17. [17]

    Z. Qiu, L. Chen, and F. Zonca, Physics of Plasmas 23, 090702 (2016)

  18. [18]

    Z. Qiu, L. Chen, and F. Zonca, Nuclear Fusion 56, 106013 (2016)

  19. [19]

    Z. Qiu, L. Chen, and F. Zonca, Nuclear Fusion 57, 056017 (2017)

  20. [20]

    Y. Chen, G. Y. Fu, C. Collins, S. Taimourzadeh, and S. E. Parker, Physics of Plasmas 25, 032304 (2018)

  21. [21]

    Brochard, C

    G. Brochard, C. Liu, X. Wei, W. Heidbrink, Z. Lin, N. Gorelenkov, C. Chrystal, X. Du, J. Bao, A. R. Polevoi, M. Schneider, S. H. Kim, S. D. Pinches, P. Liu, J. H. Nicolau, and H. Lütjens, Physical Review Letters 132, 075101 (2024)

  22. [22]

    P. Liu, X. Wei, Z. Lin, G. Brochard, G. J. Choi, W. W. Heidbrink, J. H. Nicolau, and G. R. McKee, Physical Review Letters 128, 185001 (2022)

  23. [23]

    P. Liu, X. Wei, Z. Lin, W. Heidbrink, G. Brochard, G. Choi, J. Nicolau, and W. Zhang, Nuclear Fusion 64, 076007 (2024)

  24. [24]

    Y. Ma, B. Zhang, L. Gao, P. Liu, J. Bao, Z. Lin, H. Cai, A. Liu, H. Zhao, and T. Zhang, Phys. Rev. Res. 8, 033092 (2026)

  25. [25]

    McGuire, R

    K. McGuire, R. Goldston, M. Bell, M. Bitter, K. Bol, K. Brau, D. Buchenauer, T. Crowley, S. Davis, F. Dylla, et al. , Physical Review Letters 50, 891 (1983)

  26. [26]

    L. Chen, R. White, and M. Rosenbluth, Physical Review Letters 52, 1122 (1984)

  27. [27]

    Coppi and F

    B. Coppi and F. Porcelli, Physical review letters 57, 2272 (1986)

  28. [28]

    Salewski, D

    M. Salewski, D. Spong, P. Aleynikov, R. Bilato, B. Breiz- man, S. Briguglio, H. Cai, L. Chen, W. Chen, V. Duarte, R. Dumont, M. Falessi, M. Fitzgerald, E. Fredrick- son, M. García-Muñoz, N. Gorelenkov, T. Hayward- Schneider, W. Heidbrink, M. Hole, Ye.O. Kazakov, V. Kiptily, A. Könies, T. Kurki-Suonio, Ph. Lauber, S. Lazerson, Z. Lin, A. Mishchenko, D. Mos...

  29. [29]

    Günter, A

    S. Günter, A. Gude, J. Hobirk, M. Maraschek, S. Saarelma, S. Schade, R. Wolf, A. U. Team, et al. , Nuclear fusion 41, 1283 (2001)

  30. [30]

    W. Chen, Y. Xu, X. Ding, Z. Shi, M. Jiang, W. Zhong, X. Ji, et al. , Nuclear Fusion 56, 044001 (2016)

  31. [31]

    W. Deng, Y. Liu, W. Ge, M. Jiang, Z. Shi, D. Li, X. Ji, Y. Dong, F. Wang, J. Cao, et al. , Physics of Plasmas 29 6 (2022)

  32. [32]

    Gao et al

    X. Gao et al. , Physics Letters A 382, 1242 (2018)

  33. [33]

    Z. Liu, W. Ge, F. Wang, Y. Liu, Y. Yang, M. Wu, Z. Wang, X. Zhang, H. Li, J. Xie, et al. , Nuclear Fusion 60, 122001 (2020)

  34. [34]

    Zhang, X

    B. Zhang, X. Gong, J. Qian, L. Zeng, L. Xu, Y. Duan, J. Zhang, Y. Hu, T. Jia, P. Li, et al. , Nuclear Fusion 62, 126064 (2022)

  35. [35]

    Brochard, C

    G. Brochard, C. Liu, X. Wei, W. Heidbrink, Z. Lin, M. Falessi, F. Zonca, Z. Qiu, N. Gorelenkov, C. Chrys- tal, X. Du, J. Bao, A. Polevoi, M. Schneider, S. Kim, S. Pinches, P. Liu, J. Nicolau, H. Lütjens, and the ISEP group, Nuclear Fusion 65, 016052 (2024)

  36. [36]

    Pinches, S

    S. Pinches, S. Günter, A. Peeters, A. U. Team, et al. , in 28th EPS Conference on Controlled Fusion and Plasma Physics (European Physical Society, 2001) pp. 57–60

  37. [37]

    Liu and G

    Z. Liu and G. Fu, Journal of Plasma Physics 89, 905890612 (2023)

  38. [38]

    L. Gao, X. Feng, A. Liu, C. Zhou, W. Ding, Z. Liu, G. Zhuang, J. Xie, X. Zhong, H. Liu, S. Wang, B. Zhang, Y. Zhang, J. Zhang, S. Wang, W. Shi, S. Qiu, L. Li, X. Chen, Y. Zhang, H. Li, T. Lan, W. Mao, Z. Liu, W. Liu, and the EAST team, Plasma Physics and Con- trolled Fusion 67, 055047 (2025)

  39. [39]

    L. Gao, A. Liu, W. Ding, Z. Liu, G. Zhuang, M. Xu, C. Zhou, X. Feng, L. Xu, H. Liu, B. Zhang, Y. Zhang, Y. Chu, Y. Jin, Y. Duan, S. Wang, L. Wang, X. Zhong, M. Wang, F. Long, J. Zhang, S. Wang, W. Shi, S. Qiu, L. Li, Y. Feng, X. Chen, Y. Zhang, and the EAST Team, Nuclear Fusion 65, 116019 (2025)

  40. [40]

    G. D. Conway, J. Schirmer, S. Klenge, W. Suttrop, E. Holzhauer, and T. A. U. Team, Plasma Physics and Controlled Fusion 46, 951 (2004)

  41. [41]

    Y. C. Kim and E. J. Powers, The Physics of Fluids 21, 1452 (1978)

  42. [42]

    W. W. Heidbrink, M. E. Austin, R. K. Fisher, M. García- Muñoz, G. Matsunaga, G. R. McKee, R. A. Moyer, C. M. Muscatello, M. Okabayashi, D. C. Pace, K. Shinohara, W. M. Solomon, E. J. Strait, M. A. Van Zeeland, and Y. B. Zhu, Plasma Physics and Controlled Fusion 53, 085028 (2011)

  43. [43]

    X. Zhu, L. Zeng, Z. Qiu, B. Hao, W. Shen, X. Gu, M. Wu, T. Tang, J. Qian, H. Liu, D. Jiang, L. Xu, J. Zhang, Y. Liu, Q. Zang, Y. Jie, X. Gao, and X. Lin, Journal of Plasma Physics 86, 905860610 (2020)

  44. [44]

    H. R. Lewis and P. M. Bellan, Journal of Mathematical Physics 31, 2592 (1990)

  45. [45]

    L. Chen, Z. Qiu, and F. Zonca, Physics of Plasmas 31 (2024)

  46. [46]

    Chen and F

    L. Chen and F. Zonca, Reviews of Modern Physics 88, 015008 (2016)