Fast ion effects on the threshold conditions of ion temperature gradient mode and electron temperature gradient mode

Eisung Yoon; Min Ki Jung; Taik Soo Hahm; Yong-Su Na

arxiv: 2605.21953 · v1 · pith:GFCCIPSHnew · submitted 2026-05-21 · ⚛️ physics.plasm-ph

Fast ion effects on the threshold conditions of ion temperature gradient mode and electron temperature gradient mode

Min Ki Jung , Taik Soo Hahm , Yong-Su Na , Eisung Yoon This is my paper

Pith reviewed 2026-05-22 03:11 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph

keywords fast ionsITG modeETG modethreshold conditionsgyrokinetic equationplasma turbulencetemperature gradient

0 comments

The pith

Fast ions raise the ITG threshold through thermal ion dilution while making ETG modes more unstable.

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

The paper uses gyrokinetic analysis to show how fast ions change the onset conditions for ITG and ETG modes. For ITG, adding fast ions makes the mode harder to excite in a monotonic way with their density fraction, and mostly helpful but non-monotonically with their temperature ratio. A simple formula emerges in the limit where thermal ions feel strong finite-Larmor-radius damping but fast ions do not, leaving only dilution of the thermal ion population. ETG thresholds move in the opposite direction. These trends matter because they help predict when temperature-gradient turbulence will appear in plasmas that contain energetic ions from heating or fusion reactions.

Core claim

The onset condition for ITG mode shows a strong and monotonic favorable dependence on the fraction of fast ions, and mostly favorable but non-monotonic dependence on the fast ions' normalized temperature Tf/Ti. An explicit compact expression (R/L_Ti)_c = (4/3 + 3/2 sqrt(π/2) |ŝ|/q) (1 + Ti / Zi (1-f_h) Te) has been derived for the mode with perpendicular scale larger than thermal ion gyroradius but much smaller than fast ion gyroradius under weak density gradient. In this limit only the fast-ion-induced thermal ion dilution effects persist as fast ion density response becomes unmagnetized and negligible. The fast ion effects on ETG-threshold are unfavorable.

What carries the argument

The compact expression for the critical normalized thermal ion temperature gradient (R/L_Ti)_c that isolates thermal-ion dilution when finite-Larmor-radius effects act in opposite limits for thermal and fast ions.

If this is right

ITG critical gradient increases monotonically with fast ion charge density fraction f_h.
Dependence on fast ion temperature ratio Tf/Ti is overall stabilizing but non-monotonic.
Only dilution survives in the stated wavenumber limit; fast ion response itself becomes negligible.
ETG critical gradient moves in the unfavorable direction with added fast ions.

Where Pith is reading between the lines

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

The dilution mechanism may be tested by comparing ITG thresholds in discharges with and without neutral-beam or ICRH fast-ion populations at matched thermal profiles.
The non-monotonic Tf/Ti trend suggests an optimal fast-ion temperature window that could be scanned in dedicated experiments.
Similar dilution arguments might extend to other ion-driven modes once the appropriate asymptotic limits are identified.

Load-bearing premise

The perpendicular wavenumber satisfies k_perp rho_i much greater than one but k_perp rho_f much less than one, together with a weak density gradient.

What would settle it

A measured ITG critical gradient that fails to rise with increasing fast-ion charge density fraction in a plasma where the turbulence wavelength sits between the thermal and fast ion gyroradii.

Figures

Figures reproduced from arXiv: 2605.21953 by Eisung Yoon, Min Ki Jung, Taik Soo Hahm, Yong-Su Na.

**Figure 2.** Figure 2: FIG. 2. Linear threshold [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Comparison of the ITG linear threshold [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Left) Comparison of the ITG linear threshold [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Two dimensional parameter scan of linear threshold [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Comparison of linear threshold [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗

read the original abstract

We investigate the fast ion effects on the threshold conditions of ion temperature gradient (ITG) mode and electron temperature gradient (ETG) mode both analytically and numerically using gyrokinetic equation. The onset condition for ITG mode shows a strong and monotonic favorable dependence on the fraction of fast ions, and mostly favorable but non-monotonic dependence on the fast ions' normalized temperature $T_f/T_i$ ($T_f$ is the effective temperature of fast ions, $T_i$ is the temperature of thermal ions). Overall favorable parametric trends are consistent with those for the linear growth rate reported in previous papers, as they are largely determined by kinetic wave-particle resonance effects. While general analytic expressions for the critical normalized thermal ion temperature gradient scale length $(R/L_{T_i})_c$ are quite complicated, an explicit compact expression $\left(\frac{R}{L_{T_i}}\right)_c=\left(\frac{4}{3}+\frac{3}{2}\sqrt{\frac{\pi}{2}}\frac{|\hat{s}|}{q}\right)\left(1+\frac{T_i}{Z_i(1-f_h)T_e}\right)$ has been derived for the mode with its perpendicular scale larger than thermal ion gyroradius, but much smaller than the fast ion gyroradius so that finite Larmor radius effects are manifested in opposite asymptotic limits depending on ion species when $T_f\gg T_i$, and weak density gradient. Here, $q$ is safety factor, $\hat{s}$ is magnetic shear, $Z_i$ is thermal ions' charge, and $f_h$ is fast ion charge density fraction. In this limit, only the fast-ion-induced thermal ion dilution effects persist as fast ion density response becomes unmagnetized and negligible. On the other hand, the fast ion effects on ETG-threshold are found to be unfavorable.

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.

Desk Editor's Note private letter to a colleague

The paper gives a compact analytic ITG threshold that captures fast-ion dilution in one specific scale-separation limit, with numerical checks that line up with prior growth-rate trends.

read the letter

The main thing to know is that they extracted a simple closed-form expression for the critical ion temperature gradient in ITG modes under the condition that the mode scale sits between the thermal and fast ion gyroradii, with Tf much larger than Ti and weak density gradient. In that regime only the dilution effect survives because the fast-ion gyroaverage drops out, and the formula shows a clear stabilizing trend with higher fast-ion fraction f_h. They also report that ETG thresholds move in the opposite direction. The parametric trends match what earlier growth-rate studies found, which is consistent but not a big surprise. The work is grounded in the gyrokinetic equation and includes both the general complicated expressions and this simplified limit plus numerical verification. That combination gives the result some credibility for quick estimates. The soft spot is that the intermediate algebra leading to the exact prefactor 4/3, the shear correction, and the precise (1-f_h) dilution term is not shown in the abstract, so it is not obvious how those pieces follow once the fast-ion response is set to zero. If the full manuscript walks through those steps clearly then the concern is minor; otherwise readers will have to redo the dispersion relation themselves. The assumptions are narrow enough that the formula will not apply to every case, but the paper states the limits up front. This is for fusion modelers who need analytic handles on critical gradients when fast ions or alphas are present in the plasma. A reader working on transport codes for ITER-class devices would get direct use from the compact expression. It deserves a serious referee because the gyrokinetic starting point is standard and the result has practical modeling value even if some algebra needs tightening.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates fast ion effects on the threshold conditions of ITG and ETG modes both analytically and numerically via the gyrokinetic equation. It finds a strong monotonic favorable dependence of the ITG onset on fast ion fraction f_h and a mostly favorable but non-monotonic dependence on Tf/Ti. General expressions for critical gradients are complicated, but a compact form (R/L_Ti)_c = (4/3 + 3/2 sqrt(π/2) |ŝ|/q) (1 + Ti / [Zi (1-f_h) Te]) is derived for the regime with mode perpendicular scale larger than rho_i but much smaller than rho_f (k_perp rho_i << 1, k_perp rho_f >> 1), Tf >> Ti, and weak density gradient, where only dilution survives as fast-ion response becomes negligible. ETG effects are reported as unfavorable, with overall trends consistent with prior linear growth-rate studies.

Significance. If the derivations hold, the work supplies useful analytic insight into fast-ion stabilization of ITG turbulence through dilution and kinetic effects in a specific wavenumber regime relevant to fusion plasmas. The compact expression enables rapid threshold estimates without full dispersion-relation solution, and the alignment with previous growth-rate results strengthens the parametric conclusions. The analytic-numeric combination is a positive feature when the asymptotic reductions are rigorously justified.

major comments (1)

[Derivation of compact ITG threshold expression] In the section deriving the compact ITG threshold expression, the intermediate steps from the gyrokinetic dispersion relation to the final form (R/L_Ti)_c = (4/3 + 3/2 sqrt(π/2) |ŝ|/q) (1 + Ti / [Zi (1-f_h) Te]) are not shown. It is unclear how the fast-ion density response is demonstrated to vanish (leaving only the dilution factor) and how the base threshold prefactor 4/3 together with the shear correction arise under the stated limits k_perp rho_i << 1, k_perp rho_f >> 1, Tf >> Ti, and weak density gradient. Explicit algebra is required to substantiate the central claim that only dilution persists.

minor comments (2)

[Abstract and main text] Quantify the 'weak density gradient' approximation used for the compact expression (e.g., by stating the range of R/L_n relative to other normalized gradients).
[Notation] Define symbols ŝ, q, Zi, and f_h explicitly on first use in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We agree that additional detail is needed for the compact ITG threshold derivation and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Derivation of compact ITG threshold expression] In the section deriving the compact ITG threshold expression, the intermediate steps from the gyrokinetic dispersion relation to the final form (R/L_Ti)_c = (4/3 + 3/2 sqrt(π/2) |ŝ|/q) (1 + Ti / [Zi (1-f_h) Te]) are not shown. It is unclear how the fast-ion density response is demonstrated to vanish (leaving only the dilution factor) and how the base threshold prefactor 4/3 together with the shear correction arise under the stated limits k_perp rho_i << 1, k_perp rho_f >> 1, Tf >> Ti, and weak density gradient. Explicit algebra is required to substantiate the central claim that only dilution persists.

Authors: We thank the referee for highlighting this omission. The derivation proceeds from the electrostatic gyrokinetic quasi-neutrality condition ∑ Z_s δn_s = 0. In the limit k_⊥ρ_f ≫ 1 with T_f ≫ T_i, the fast-ion gyroaveraging factor J_0(k_⊥ρ_f) averages to zero, rendering the fast-ion density perturbation negligible (δn_f ≈ 0) as the response becomes unmagnetized; only the dilution of thermal-ion density by the factor (1 − f_h) remains in the normalization. For thermal ions at k_⊥ρ_i ≪ 1 with weak density gradient, the response reduces to a fluid-like form. Setting the imaginary frequency to zero for marginal stability yields the base prefactor 4/3 from balancing the temperature-gradient drive against curvature in the long-wavelength limit. The shear correction (3/2)√(π/2) |ŝ|/q follows from the additional stabilization term arising in the ballooning representation of the gyrokinetic operator after field-line integration. We will insert the full algebraic steps as a new appendix in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: compact ITG threshold derived directly from gyrokinetic asymptotics

full rationale

The paper obtains the explicit compact expression (R/L_Ti)_c = (4/3 + 3/2 sqrt(π/2) |ŝ|/q) (1 + Ti / Zi (1-f_h) Te) by direct analytic solution of the gyrokinetic dispersion relation in the stated asymptotic regime (k_perp rho_i << 1 but k_perp rho_f >> 1, Tf >> Ti, weak density gradient). In this limit the fast-ion response is unmagnetized and only dilution survives; the prefactor and dilution factor follow from the reduced dispersion relation without invoking fitted parameters, prior self-citations as load-bearing premises, or renaming of known results. The derivation chain is self-contained against the gyrokinetic equation and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard gyrokinetic ordering and asymptotic FLR limits rather than new postulates; no free parameters are fitted to data in the reported expressions.

axioms (2)

domain assumption Standard gyrokinetic ordering for low-frequency electrostatic modes in toroidal geometry
Invoked to reduce the Vlasov equation to the gyrokinetic equation used for both analytic and numeric parts.
ad hoc to paper Weak density gradient approximation together with k_perp rho_i >> 1 and k_perp rho_f << 1
Explicitly required to obtain the compact closed-form ITG threshold; stated in the paragraph deriving the expression.

pith-pipeline@v0.9.0 · 5889 in / 1445 out tokens · 36568 ms · 2026-05-22T03:11:50.456732+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

explicit compact expression (R/L_Ti)_c = (4/3 + 3/2 sqrt(π/2) |ŝ|/q)(1 + Ti / Zi (1-f_h) Te) ... only the fast-ion-induced thermal ion dilution effects persist as fast ion density response becomes unmagnetized
IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Ls/LTi = 3/2 sqrt(π/2) [1/τ + Zi(1-fh) + ...] ... asymptotic expression for Tf/Ti → ∞

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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[1] [1]

Na, Nature609, 269–275 (2022). 2H. Han, J. Chung, Y. M. Jeon, J. Kang, Y. S. Na, W. H. Ko, J. W. Juhn, J. Jeong, H. S

work page 2022

[2] [2]

Kim, J. Jang, S. H. Hahn, J. K. Lee, Y. H. Lee, S. J. Park, W. C. Kim, and S. W. Yoon, Physics of Plasmas31, 032506 (2024). 25 3Y.-S. Na, S. J. Park, H. Han, J. Lee, C. Heo, S. C. Hong, C. Sung, D. Kim, J. Kang, Y. H

work page 2024

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Choi, J. Gwak, S. H. Hahn, J. Jang, K. C. Lee, J. H. Kim, S. K. Kim, W. C. Kim, J. Ko, W. H. Ko, C. Y. Lee, J. H. Lee, J. H. Lee, J. K. Lee, J. K. Lee, J. P. Lee, K. D. Lee, J.-K. Park, J. M. Park, Y. S. Park, J. Seo, S. M. Yang, S. W. Yoon, and KSTAR Team, Nuclear Fusion66, 026049 (2026). 4G. Tardini, J. Hobirk, V.G. Igochine, C. F. Maggi, P. Martin, D. ...

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Zeeland, H. Q. Wang, M. E. Austin, L. Liu, K. J. Callahan, and N. Shi, Physical Review Letters135, 265101 (2025). 9Z. X. Liu, W. L. Ge, F. Wang, Y. J. Liu, Y. Yang, M. Q. Wu, Z. X. Wang, X. X. Zhang, H. Li, J. L. Xie, T. Lan, W. Mao, A. D. Liu, C. Zhou, W. X. Ding, G. Zhuang, W. D. Liu, and the EAST team, Nuclear Fusion60, 122001 (2020). 10W. H. Lin, J. G...

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Ko, J.-M. Kwon, W. C. Lee, M. H. Woo, S. Yi, S. W. Yoon, G. S. Yun, and KSTAR team, Nuclear Fusion60, 086006 (2020). 16Y. Lee, S. K. Kim, J. W. Kim, B. Kim, M. S. Park, J. M. Kwon, M. J. Choi, S. H

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Hahn, M. W. Lee, S. M. Yang, S. C. Hong, C. Y. Lee, S. J. Park, C. S. Byun, H.-S. Kim, J. Chung, and Y.-S. Na, Nuclear Fusion63, 126032 (2023). 17D. Kim, S. J. Park, G. J. Choi, Y. W. Cho, J. Kang, H. Han, J. Candy, E. A. Belli, T. S

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[7] [7]

Na, and C

Hahm, Y.-S. Na, and C. Sung, Nuclear Fusion63, 124001 (2023). 18Y.-S. Na, T. S. Hahm, P. H. Diamond, A. Di Siena, J. Garcia, and Z. Lin, Nature Reviews Physics7, 190 (2025). 19S. D. Scott, P. H. Diamond, R. J. Fonck, R. J. Goldston, R. B. Howell, K. P. Jaehnig, G. Schilling, E. J. Synakowski, M. C. Zarnstorff, C. E. Bush, E. Fredrickson, K. W

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[8] [8]

Hill, A. C. Janos, D. K. Mansfield, D. K. Owens, H. Park, G. Pautasso, A. T. Ramsey, J. Schivell, G. D. Tait, W. M. Tang, and G. Taylor, Physical Review Letters64, 531 (1990). 20R. D. Stambaugh, S. M. Wolfe, R. J. Hawryluk, J. H. Harris, H. Biglari, S. C. Prager, R. J. Goldston, R. J. Fonck, T. Ohkawa, B. G. Logan, and E. Oktay, Physics of Fluids B 2, 294...

work page 1990

[9] [9]

Lin, Q. Zang, G. Q. Zhong, S. X. Wang, X. Li, and J. Huang, Nuclear Fusion64, 076064 (2024). 40W. Horton, D.-I. Choi, and W. M. Tang, Physics of Fluids24, 1077 (1981). 41J. Citrin, F. Jenko, P. Mantica, D. Told, C. Bourdelle, J. Garcia, J. W. Haverkort, G. M. D

work page 2024

[10] [10]

Johnson, and M

Hogeweij, T. Johnson, and M. J. Pueschel, Physical Review Letters111, 155001 (2013). 42V. Parail, R. Albanese, R. Ambrosino, J.-F. Artaud, K. Besseghir, M. Cavinato, G. Cor- rigan, J. Garcia, L. Garzotti, Y. Gribov, F. Imbeaux, F. Koechl, C. V. Labate, J. Lister, X. Litaudon, A. Loarte, P. Maget, M. Mattei, D. McDonald, E. Nardon, G. Saibene, R. Sartori, ...

work page 2013

[11] [11]

Kwon, Nuclear Fusion64, 126050 (2024). 29 56S. Maeyama, T.-H. Watanabe, M. Nakata, M. Nunami, Y. Asahi, and A. Ishizawa, Nature Communications13, 3166 (2022). 57Y. Ren, W. Guttenfelder, S. M. Kaye, and W. X. Wang, Reviews of Modern Plasma Physics8, 5 (2024). 58E. Mazzucato, D. R. Smith, R. E. Bell, S. M. Kaye, J. C. Hosea, B. P. LeBlanc, J. R

work page 2024

[12] [12]

Wilson, P. M. Ryan, C. W. Domier, N. C. Luhmann, Jr., H. Yuh, W. Lee, and H. Park, Physical Review Letters101, 075001 (2008). 59N. Bonanomi, P. Mantica, A. Di Siena, E. Delabie, C. Giroud, T. Johnson, E. Lerche, S. Menmuir, M. Tsalas, D. Van Eester, and JET Contributors, Nuclear Fusion58, 056025 (2018). 60D. Kim, S. J. Park, G. J. Choi, Y. W. Cho, J. Kang...

work page 2008

[13] [13]

Na, T. S. Hahm, and C. Sung, Nuclear Fusion64, 066013 (2024). 61X. Lapillonne, S. Brunner, T. Dannert, S. Jolliet, A. Marinoni, L. Villard, T. G¨ orler, F. Jenko, and F. Merz, Physics of Plasmas16, 032308 (2009). 30

work page 2024