pith. sign in

arxiv: 2605.17138 · v1 · pith:AIBJJCQZnew · submitted 2026-05-16 · ⚛️ physics.plasm-ph

Shafranov shift and finite β effects on Alfv\'en Eigenmodes and microinstabilities in global electromagnetic gyrokinetic simulations

B. Rofman , G. Di Giannatale , A. Mishchenko , E. Lanti , A. Bottino , T. Hayward-Schneider , J.N. Sama , A. Biancalani
show 3 more authors
B.F. McMillan S. Brunner L. Villard
This is my paper

Pith reviewed 2026-05-20 14:30 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords Shafranov shiftToroidal Alfvén EigenmodesTAEgyrokinetic simulationsenergetic particlesITGKBMplasma beta
0
0 comments X

The pith

Shafranov shift from energetic particle pressure stabilizes toroidal Alfvén eigenmodes by up to 90 percent in growth rate.

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

The paper examines how the Shafranov shift, an outward displacement of the magnetic axis due to finite plasma pressure, influences instabilities in fusion plasmas containing energetic particles. Researchers begin with simple electrostatic models in ideal MHD equilibria and progressively add electromagnetic effects, kinetic electrons, and multi-species dynamics in global gyrokinetic simulations. They find that the shift depends on toroidal mode number and provides stronger stabilization at longer wavelengths, most notably reducing TAE growth rates by 90 percent when energetic particle pressure is included self-consistently in the equilibrium. This produces diminishing returns in TAE growth as the energetic particle fraction rises, while also shifting frequencies of ITG and TAE modes and damping KBMs.

Core claim

Shafranov shift effects are a function of the toroidal mode number, that they are mainly stabilizing, and stronger at longer wavelengths, impacting TAEs the most with a 90% reduction in growth rate for cases which consistently account for the EP pressure in the MHD equilibrium. Leading to a law of diminishing returns for the TAE growth rate as a function of EP fraction. With Shafranov shift asymptotically pushing the ITG frequency up and the TAE frequency down, while KBMs are strongly damped by both EPs and Shafranov shift effects.

What carries the argument

The Shafranov shift arising from finite plasma beta and energetic particle pressure in the MHD equilibrium, incorporated stepwise from ideal-MHD electrostatic models to full electromagnetic kinetic multi-species gyrokinetic simulations.

If this is right

  • TAE growth rates follow a law of diminishing returns as energetic particle fraction increases due to the stabilizing Shafranov shift.
  • Shafranov shift asymptotically increases ITG frequencies while decreasing TAE frequencies.
  • Kinetic ballooning modes experience strong damping from both energetic particles and the Shafranov shift.
  • Linear TAE stabilization by the shift does not change nonlinear saturation levels but reduces the associated heat and particle fluxes.
  • ITG saturation levels and fluxes remain largely unaffected by the inclusion of the Shafranov shift.

Where Pith is reading between the lines

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

  • High-beta reactor scenarios with substantial energetic particle populations may see lower TAE-driven transport than models without equilibrium shift predict.
  • The stronger effect at long wavelengths suggests that global mode stability in future devices will be particularly sensitive to accurate equilibrium modeling.
  • Extending the approach to self-consistent nonlinear equilibrium evolution could uncover additional feedback loops between instabilities and the Shafranov shift.

Load-bearing premise

A stable MHD equilibrium exists and the systematic increase in model realism from electrostatic adiabatic-electron beta-zero cases to full kinetic electromagnetic multi-species captures the dominant self-consistent contributions without missing important nonlinear or non-local couplings.

What would settle it

A side-by-side comparison of TAE linear growth rates in equilibria that include versus exclude energetic particle pressure in the Shafranov shift calculation, for a range of low toroidal mode numbers.

Figures

Figures reproduced from arXiv: 2605.17138 by A. Biancalani, A. Bottino, A. Mishchenko, B.F. McMillan, B. Rofman, E. Lanti, G. Di Giannatale, J.N. Sama, L. Villard, S. Brunner, T. Hayward-Schneider.

Figure 1
Figure 1. Figure 1: On the left: initial temperature and density (flux-surface-averaged) logarithmic gradients, for ions, electrons, and EPs. The profiles of the bulk species (ions and electrons) change between the standard (ST) and peaked (PK) cases, while the EP profiles remain the same. The q profile is indicated by the black dashed line. The vertical gray dashed lines mark the mode rational surfaces (nq = m + 1/2) of the … view at source ↗
Figure 2
Figure 2. Figure 2: ITG growth rate γ per toroidal mode number n for adiabatic and kinetic electrons in the electrostatic limit. To study the effect of Shafranov shift on the ITG growth rate, we compare in the left panel of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: ITG dispersion relation for electrostatic and electromagnetic cases with kinetic electrons. Cases with self-consistent MHD equilibria show very little electromagnetic effects on the growth rates and some effect on the frequency. ITG destabilization by Shafranov shift (standard profiles) In [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ITG dispersion relation driven by standard profiles in two magnetic geometries. Inconsistent βMHD = 0, and self-consistent βMHD = βST ideal MHD equilibria. To better understand this, we plot in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Poloidal cross-sections of ϕ the perturbed electrostatic potential by the n = 30 ITG mode in βMHD = 0 and βMHD = βST cases. On the right, contours of local shear and a black line to marks the border between ”good” and ”bad” curvature. Internal Kink mode stabilization by Shafranov shift (peaked profiles) In the following sections we show how the same peaked profiles can destabilize different modes de￾pendin… view at source ↗
Figure 6
Figure 6. Figure 6: Internal Kink mode dispersion relation for peaked profiles in an ad-hoc, i.e. circular concentric equilibrium. The most unstable, n = 5, mode structure in ϕ and A∥ . KBM stabilization by Shafranov shift (peaked profiles) In a βMHD = 0 MHD equilibrium, the small Shafranov shift stabilizes the internal kink mode, and the peaked profiles excite another mode instead. We identify the mode as KBM based on its fr… view at source ↗
Figure 7
Figure 7. Figure 7: ITG and KBM dispersion relations for peaked profiles. KBMs are excited in the inconsistent βMHD = 0 equilibrium, and are stabilized by Shafranov shift present in the self-consistent βMHD = βPK MHD equilibrium Part II - Energetic particles Energetic particles are a small and hot minority in the plasma, which as a result have a substantial kinetic pressure that should be accounted for in the MHD equilibrium … view at source ↗
Figure 8
Figure 8. Figure 8: KBM is excited in a βMHD = 0 MHD equilibrium and is stabilized by the inclusion of 1% − 3% of EPs as a third species [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: KBM is excited in a βMHD = 0 MHD equilibrium and is stabilized by the inclusion of 1% − 3% of EPs as a third species. KBM stabilization by kinetic effects We map the dispersion relation of the KBMs in βMHD = 0 MHD equilibrium in order to study the kinetic effect of EPs. We use an inconsistent magnetic equilibrium with βMHD = 0 because the Shafranov shift in the self-consistent case will stabilize the KBM b… view at source ↗
Figure 10
Figure 10. Figure 10: ITG dispersion relations for cases with the same MHD equilibrium, with and without EPs. There is little to no direct effect of the EPs on the ITG growth rate, and an up-shift on the ITG frequency with standard profiles. shift of the system without adding to the ITG drive - unlike the effect of increasing the profiles, which contribute to both. As a result, we see in [PITH_FULL_IMAGE:figures/full_fig_p010… view at source ↗
Figure 11
Figure 11. Figure 11: ITG dispersion relations for cases with and without EP pressure (βEP ) as part of the MHD equilibrium. The Shafranov shift effect on the growth rate is more pronounced in the longer wavelengths (lower n). The frequency spectrum shows a consistent downshift due to the additional Shafranov shift. Here ∆(βx) is the Shafranov shift due to βx component in βMHD 10 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: ITG mode structure - correct figures Part III - Alfv´en Eigenmodes In a fluid description, one finds that toroidal magnetic geometry couples the m and m + 1 poloidal harmonics, leading to the formation of so called gaps in the shear Alfv´en wave (SAW) continuum. In these gaps, Alfv´en Eigenmodes can resonate undamped, at radial locations where nq = (m + 1/2) with a typical frequency of ω0 ∼= ωA/2q. Where … view at source ↗
Figure 13
Figure 13. Figure 13: TAE dispersion relation in βMHD = 0 with peaked (PK) and low temperature (LT) thermal profiles. Shafranov shift and EP fraction effects on TAE stability Next we add Shafranov shift effects to the magnetic equilibrium, where in all cases we include 1% EPs. Thus we have two main sources of kinetic pressure - the thermal bulk plasma and the EPs. In 11 [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: TAE dispersion relations driven by 1% EPs, in magnetic equilibria with βMHD = [0; βth; βth + βEP ]. Here ∆(βx) is the Shafranov shift due to βx component in βMHD. We select the TAE with the highest growth rate (n = 2) to represent the TAE system and scan in EP fraction and βMHD. On the one hand, as we increase the EP fraction we increase the TAE drive and thus its growth rate. On the other hand, when we i… view at source ↗
Figure 15
Figure 15. Figure 15: Alv´en continuum (under the slow-sound approximation [44]) for the n = 2 mode with standard (left) and peaked (middle) thermal profiles. The measured frequencies of the = 2 modes are plotted in doted lines, which match the TAE gaps well [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: TAE mode structure for inconsistent case with βMHD = βST on the top row, and self￾consistent case with βMHD = βST + βEP on the bottom row. Full spectrum overview We summarize for brevity the combined effects of profiles and Shafranov shift on the plasma response, in scenarios with 1% EPs. Here, as before, we consider cases with MHD equilibria that are self￾consistent with the plasma profiles e.g. for the … view at source ↗
Figure 17
Figure 17. Figure 17: Toroidal mode number n dependence of Shafranov shift effects on the growth rates γ and frequencies ω of the unstable modes. Due to the separation in scales we separate between the TAE [1-5] and ITG [15-50] ranges. different with both curve intersecting (or getting close to) the 1/n line. Indicating that Shafranov shift has a stabilizing effect on the longer wavelengths and a destabilizing effect at the sh… view at source ↗
Figure 18
Figure 18. Figure 18: On the left: Electrostatic potential of the unstable TAE. On the : TAE saturation level vs. the linear growth rate as a function of EP fraction which increases from left to right: 0% ; 1% ; 3%. To expand our understanding of the nonlinear TAE dynamics we scan in EP fraction [0%, 1%, 3%], for 5 different MHD equilibria and two sets of profiles. Plotted in [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Heat and particles fluxes driven by the n = 2 TAE in a system with standard profiles.The fluxes are averaged over a time window ∆t =(111 − 172) [a/cs0 ] situated after the initial saturation phase. Note, the y-scale is different for each species and flux. The peaked cases which exhibit a similar behavior, are omitted for brevity. These results raise the question whether or not linear stabilization transla… view at source ↗
Figure 20
Figure 20. Figure 20: On the left: Electrostatic potential of the unstable ITG. On the right: Temperature gradient relaxation in a time-averaged window after the initial saturation phase, separated between the ion (left) and electron channels (right). right we plot the initial temperature profile for comparison to the time averaged profile taken after the nonlinear saturation phase. Although Shafranov shift changes the profile… view at source ↗
Figure 21
Figure 21. Figure 21: Heat and particles fluxes driven by the n = 25 ITG in a system with standard profiles. The fluxes are averaged over a time window ∆t =(89−144) [a/cs0 ] situated after the initial saturation phase. Note, the y-scale is different for each species and flux. Conclusions We perform first-principles gyrokinetic numerical simulations to study the effects of plasma pressure on the plasma stability. On our time sc… view at source ↗
read the original abstract

Future nuclear fusion reactors will have to confine plasma with strong kinetic gradients and small fractions of fusion-born energetic particles (EP) that are ~100 times hotter than the thermal ions. In our analysis, we assume the existence of a stable MHD equilibrium and study the unstable plasma perturbations. In this electromagnetic, kinetic, multi-scale, self-organizing system, all species contribute both to the Shafranov shift (equilibrium effect) and to the plasma $\beta$ (plasma response). Nonetheless, due to the high complexity of the problem, many works neglect these effects. We use the global, gyrokinetic code ORB5 to study the plasma stability. Starting from an electrostatic, thermal plasma with adiabatic electrons in a $\beta = 0$ ideal-MHD equilibrium, we systematically increase the realism of our models. And study the linear stability and nonlinear fluxes of Toroidal Alfv\'en Eigenmodes (TAE), and the Ion Temperature Gradient (ITG), and Kinetic Ballooning Modes (KBM) microinstabilities as they arise. Linearly, we find that Shafranov shift effects are a function of the toroidal mode number, that they are mainly stabilizing, and stronger at longer wavelengths, impacting TAEs the most with a 90% reduction in growth rate for cases which consistently account for the EP pressure in the MHD equilibrium. Leading to a law of diminishing returns for the TAE growth rate as a function of EP fraction. We find that with Shafranov shift asymptotically pushes the ITG frequency up and the TAE frequency down. Furthermore, we show that KBMs are strongly damped by both EPs (kinetic) and Shafranov shift (equilibrium) effects. Nonlinearly we find that the linear TAE stabilization does not effect the saturation levels. Nonetheless, the heat and particle fluxes carried by the TAE, are reduced by the Shafranov shift. While, the ITG fluxes and saturation levels are unaffected by the Shafranov shift.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript uses global electromagnetic gyrokinetic simulations with the ORB5 code to study Shafranov shift and finite-β effects on Toroidal Alfvén Eigenmodes (TAEs), Ion Temperature Gradient (ITG) modes, and Kinetic Ballooning Modes (KBMs) in the presence of energetic particles (EPs). Starting from an electrostatic adiabatic-electron β=0 ideal-MHD equilibrium and progressively incorporating electromagnetic, kinetic, and multi-species effects with consistent EP pressure in the MHD equilibrium, the authors report that Shafranov shift effects are mode-number dependent and primarily stabilizing, strongest at long wavelengths, yielding up to a 90% TAE growth-rate reduction and a law of diminishing returns with EP fraction. The shift also shifts frequencies (ITG up, TAE down), damps KBMs, and reduces nonlinear TAE fluxes without altering saturation levels or ITG fluxes.

Significance. If the central results hold after addressing isolation of effects, the work is significant for fusion plasma stability predictions. It demonstrates the importance of self-consistent EP contributions to equilibrium in global simulations, provides a systematic comparison of model realism levels, and identifies stabilizing mechanisms that could influence EP-driven instability thresholds and transport in reactor scenarios. The stepwise model escalation and nonlinear flux results add practical value.

major comments (2)
  1. [Abstract] Abstract: The 90% TAE growth-rate reduction and stabilization attributed specifically to Shafranov shift effects when EP pressure is included in the MHD equilibrium cannot be cleanly separated from concurrent changes in total plasma β. Adding EP pressure augments both the shift and β (as noted in the abstract), yet no control comparisons holding total β or other equilibrium quantities fixed are indicated. This leaves open the possibility that observed stabilization arises from β-driven or profile-driven effects rather than the shift alone.
  2. [Linear stability analysis] Linear stability results: The 'law of diminishing returns' for TAE growth rate versus EP fraction is presented as a key outcome but appears observational from the scanned cases. Without a quantitative fit, error bars, or theoretical derivation showing it is not a post-hoc description of the data, the claim risks being under-supported for the central conclusion.
minor comments (2)
  1. [Abstract] The abstract and summary statements would benefit from explicit mention of the number of toroidal modes scanned and the range of EP fractions used to support quantitative claims such as the 90% reduction.
  2. [Methods] Convergence tests, grid resolution, and time-step details for the global ORB5 runs are essential to substantiate the reported growth rates and flux changes but are not referenced in the provided summary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address the major comments point by point below, indicating planned revisions where appropriate to improve clarity and support for the claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The 90% TAE growth-rate reduction and stabilization attributed specifically to Shafranov shift effects when EP pressure is included in the MHD equilibrium cannot be cleanly separated from concurrent changes in total plasma β. Adding EP pressure augments both the shift and β (as noted in the abstract), yet no control comparisons holding total β or other equilibrium quantities fixed are indicated. This leaves open the possibility that observed stabilization arises from β-driven or profile-driven effects rather than the shift alone.

    Authors: We acknowledge that incorporating EP pressure into the MHD equilibrium simultaneously modifies the Shafranov shift and increases the total plasma β. Our methodology employs a systematic escalation of model realism, beginning from an electrostatic adiabatic-electron case in a β=0 ideal-MHD equilibrium and progressively including electromagnetic, kinetic, and multi-species effects. The additional stabilization is observed specifically upon consistent inclusion of EP pressure in the equilibrium. While dedicated control simulations holding total β fixed were not performed (as this would require non-standard adjustments to the equilibrium construction), we will revise the abstract and discussion sections to explicitly note the coupled nature of these equilibrium changes and clarify that the reported stabilization reflects the net effect of the self-consistent EP contribution rather than the shift in isolation. revision: yes

  2. Referee: [Linear stability analysis] Linear stability results: The 'law of diminishing returns' for TAE growth rate versus EP fraction is presented as a key outcome but appears observational from the scanned cases. Without a quantitative fit, error bars, or theoretical derivation showing it is not a post-hoc description of the data, the claim risks being under-supported for the central conclusion.

    Authors: The observed trend of diminishing returns in TAE growth rate with increasing EP fraction is drawn directly from the parameter scans presented in the linear stability analysis. To provide stronger quantitative support, we will add a fitted curve to the growth-rate data versus EP fraction in the revised manuscript, including available error estimates from the simulations. A complete theoretical derivation lies beyond the numerical focus of this study; however, we will include references to relevant analytic expectations for EP-driven Alfvén modes to contextualize the trend. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct outputs of forward gyrokinetic simulations

full rationale

The paper derives its claims on Shafranov shift stabilization and TAE growth-rate trends exclusively from numerical simulations performed with the external ORB5 code on an input MHD equilibrium. No equation or result reduces to its own inputs by construction, no parameter is fitted and then relabeled as a prediction, and no load-bearing step relies on a self-citation chain. The 'law of diminishing returns' is an observed trend across scanned EP fractions rather than a tautological fit, and the systematic increase in model realism is a controlled numerical experiment rather than a self-referential definition. The analysis is therefore self-contained against the simulation framework and external MHD equilibria.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions of gyrokinetic theory and ideal MHD equilibrium construction; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Existence of a stable MHD equilibrium
    Explicitly stated: 'we assume the existence of a stable MHD equilibrium and study the unstable plasma perturbations.'
  • domain assumption Adiabatic electrons and β=0 ideal-MHD starting equilibrium
    Described as the baseline from which realism is systematically increased.

pith-pipeline@v0.9.0 · 5963 in / 1500 out tokens · 65047 ms · 2026-05-20T14:30:53.323407+00:00 · methodology

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

45 extracted references · 45 canonical work pages

  1. [1]

    H., Grad, and H., Rubin, 1958,Conference on the Peaceful Uses of Atomic Energy2190-197

    work page 1958

  2. [2]

    D., 1958,Sov

    Shafranov, V. D., 1958,Sov. Phys.633

    work page 1958

  3. [3]

    Fusion,65043002

    Salewski, M., et al, 2025,Nucl. Fusion,65043002

    work page 2025

  4. [4]

    D., 1966,Rev

    Shafranov, V. D., 1966,Rev. Plasma Phys.2103

    work page 1966

  5. [5]

    K., and Ramos, J

    Coppi, B., Ferreira, A., Mark, J. K., and Ramos, J. J., 1979,Nucl. Fusion,19, 715

    work page 1979

  6. [6]

    Troyon, F., et al, 1984,Plasma Phys. Control. Fusion,26, 209

    work page 1984

  7. [7]

    J., 1991,Phys

    Ramos, J. J., 1991,Phys. Fluids B3, 8

    work page 1991

  8. [8]

    N., and Sagdeev, R

    Coppi, B., Rosenbluth, M. N., and Sagdeev, R. Z., 1967,Phys. Fluids,10, 582

    work page 1967

  9. [9]

    N., Chen, L., Tang, W

    Guzdar, P. N., Chen, L., Tang, W. M., and Rutherford, P. H, 1983,Phys. Fluids,26

    work page 1983

  10. [10]

    Y., Horton, W., Dong, J

    Kim, J. Y., Horton, W., Dong, J. Q., 1993,Phys. Fluids B: Plasma Phys.,54030–4039

    work page 1993

  11. [11]

    M., Connor, J

    Tang, W. M., Connor, J. W., and Hastie, R. J., 1980,Nucl. Fusion,20, 1439

    work page 1980

  12. [12]

    E., and Rosenbluth, M

    Drummond, W. E., and Rosenbluth, M. N., 1962,Phys. Fluids,5, 1507-1513

    work page 1962

  13. [13]

    Fusion,32151

    Weiland, J., and Hirose, A., 1992,Nucl. Fusion,32151

    work page 1992

  14. [14]

    J., & Jenko, F., 2010,Phys

    Pueschel, M. J., & Jenko, F., 2010,Phys. Plasmas17

    work page 2010

  15. [15]

    Ishizawa, A., et al 2019,Phys. Rev. Lett.,123, 025003

    work page 2019

  16. [16]

    Fluids,29, 3695-3701

    Cheng, C., and Chance, M., 1986,Phys. Fluids,29, 3695-3701

    work page 1986

  17. [17]

    Chen, L., and Zonca, F., 2016,Rev. Mod. Phys.88015008

    work page 2016

  18. [18]

    W., & Sadler, G

    Heidbrink, W. W., & Sadler, G. J., 1994,Nucl. fusion,34, 535

    work page 1994

  19. [19]

    Lanti, E., et al, 2020,Comput. Phys. Commun.251107072

    work page 2020

  20. [20]

    Ohana, N., et al, 2021,Comput. Phys. Commun.262107208

    work page 2021

  21. [21]

    Mishchenko, A., et al, 2023,Plasma Phys. Control. Fusion65064001

    work page 2023

  22. [22]

    N., et al, 2026,Phys

    Antlitz, F. N., et al, 2026,Phys. Plasmas33042101

    work page 2026

  23. [23]

    F., et al 2008Phys

    McMillan B. F., et al 2008Phys. Plasmas15052308

    work page

  24. [24]

    Di Giannatale, G., et al 2025,Plasma Phys. Control. Fusion67085003

    work page 2025

  25. [25]

    Mishchenko A et al 2019Comput. Phys. Commun.238194

    work page

  26. [26]

    Cole, M. D. J., Mishchenko, A., Bottino, A., & Chang, C. S. 2021,Phys. Plasmas,28

    work page 2021

  27. [27]

    Biancalani, A., et al 2021Plasma Phys. Control. Fusion63065009

    work page

  28. [28]

    Fusion62112007

    Hayward-Schneider, T., Lauber, P., Bottino, A., & Mishchenko, A., 2022,Nucl. Fusion62112007

    work page 2022

  29. [29]

    Plasmas31112503

    Sama, J.N., et al 2024,Phys. Plasmas31112503

    work page 2024

  30. [30]

    Ivanov, R.et al 2025, arXiv:2601.05886 (submitted)

    work page arXiv 2025

  31. [31]

    et al 2021Nucl

    Vlad G. et al 2021Nucl. Fusion61116026

    work page

  32. [32]

    Vlad G., et al 2025,Rev. Mod. Phys.9

    work page 2025

  33. [33]

    Dimits A. M. et al 2000Phys. Plasmas7969

    work page

  34. [34]

    L¨ utjens, H., Bondeson, A., and Sauter, O., 1996,Comput. Phys. Commun.97219

    work page 1996

  35. [35]

    M., 1990,Phys

    Rewoldt, G., and Tang, W. M., 1990,Phys. Fluids B2318–323

    work page 1990

  36. [36]

    Plasmas22062303

    Dominski, J., et al 2015,Phys. Plasmas22062303

    work page 2015

  37. [37]

    Niiro, M., et al 2023Plasma Phys. Control. Fusion65065004

    work page

  38. [38]

    Lewandowski, J. L. V., and Persson, M., 1995,Plasma Phys. Control. Fusion371199

    work page 1995

  39. [39]

    Marinoni A., Sauter, O., and Coda, S, 2021,Rev. Mod. Phys.56

    work page 2021

  40. [40]

    N., et al, 1975,Phys

    Bussac, M. N., et al, 1975,Phys. Rev. Lett.,35, 1638

    work page 1975

  41. [41]

    Porcelli, F., 1991,Phys. Rev. Lett.,66, 425

    work page 1991

  42. [42]

    Plasmas84745840602

    Aleynikova, K., et al 2018,Phys. Plasmas84745840602

    work page 2018

  43. [43]

    Plasmas23012108

    Biancalani, A., et al 2016,Phys. Plasmas23012108

    work page 2016

  44. [44]

    S., et al 1992,Phys

    Chu, M. S., et al 1992,Phys. Fluids B: Plasma Phys.,43713-3721

    work page 1992

  45. [45]

    W., et al 1999,Phys

    Heidbrink, W. W., et al 1999,Phys. Plasmas,6, 1147-1161. 18

    work page 1999