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arxiv: 2605.22680 · v1 · pith:4KTDSCUJnew · submitted 2026-05-21 · ⚛️ physics.plasm-ph

Second stability region for gyrokinetics and the L-H transition

R.J.J. Mackenbach , A. Zocco , P. Helander This is my paper

Pith reviewed 2026-05-22 03:36 UTC · model grok-4.3

classification ⚛️ physics.plasm-ph
keywords gyrokineticssecond stability regionL-H transitionballooning modesbootstrap currentmagnetic shearturbulent transporttokamak
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The pith

The second stability region known from MHD ballooning modes also exists in linear gyrokinetics and explains the drop in turbulent transport at the L-H transition.

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

This paper demonstrates that the second stability region for ballooning modes, long familiar from magnetohydrodynamics, appears in linear gyrokinetic calculations whether the treatment is electrostatic or electromagnetic. In a circular tokamak the authors show that realistic H-mode conditions push the plasma into this region through the bootstrap current lowering global magnetic shear and the pressure gradient changing local magnetic shear. Simulations with more realistic geometries then exhibit a large reduction in collisionless electrostatic turbulent transport when density and temperature profiles switch from typical L-mode to H-mode values. The reduction occurs specifically because these magnetic changes move the plasma toward second stability. A continuous path between the two equilibria reveals that both energy and particle fluxes vary non-monotonically with rising pressure gradient.

Core claim

Using a simple circular tokamak geometry, the well-known second stability region of MHD-ballooning modes is shown to exist for linear gyrokinetics as well, whether electrostatic or electromagnetic. The plasma is suggested to enter this region in H-mode as a consequence of the bootstrap current and Shafranov shift altering the magnetic field when the normalised pressure gradient alpha_MHD is greater than or equal to one and collisionality is low. In more realistic magnetic geometries a large reduction in collisionless electrostatic turbulent transport is demonstrated when moving from L-mode to H-mode density and temperature profiles; this reduction follows from both the bootstrap current that

What carries the argument

The second stability region of linear gyrokinetic ballooning modes, reached by lowering global magnetic shear via bootstrap current and altering local magnetic shear via pressure gradient.

If this is right

  • Collisionless electrostatic turbulent transport drops substantially when density and temperature profiles change from L-mode to H-mode values.
  • Energy and particle fluxes exhibit non-monotonic dependence on pressure gradient along a continuous path connecting L- and H-mode equilibria.
  • The second stability region is accessible when the normalised pressure gradient alpha_MHD exceeds one at low collisionality.
  • The reduction in transport occurs for both electrostatic and electromagnetic descriptions once the magnetic shear changes are included.

Where Pith is reading between the lines

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

  • If the mechanism holds, deliberately shaping the edge pressure gradient or bootstrap current could help sustain the plasma inside the second stability region for better confinement.
  • The same shear-driven entry into second stability might interact with other edge phenomena such as pedestal formation or ELM behaviour.
  • Independent variation of bootstrap current in gyrokinetic scans could isolate its contribution to global shear reduction from the local shear effect of the pressure gradient.

Load-bearing premise

Realistic magnetic geometries with bootstrap current and Shafranov shift can be constructed from L-mode to H-mode profiles such that the resulting changes in local and global magnetic shear push the plasma into the second stability region without other dominant effects intervening.

What would settle it

A simulation or experiment that holds all parameters fixed except the local and global magnetic shears and shows whether turbulent fluxes drop sharply precisely when those shears cross the boundary into the second stability region identified in the gyrokinetic scans.

Figures

Figures reproduced from arXiv: 2605.22680 by A. Zocco, P. Helander, R.J.J. Mackenbach.

Figure 1
Figure 1. Figure 1: The normalised pressure β = 2µ0p/B2 axis, current density j, normalised pressure gradient αMHD, safety factor q, and magnetic shear sˆ profiles as a function of the radial coordinate ρ = p ψ/ψLCFS. 3. Numerically computed magnetic equilibria The shape of the last closed flux-surface, the pressure profile, and the current profile uniquely determine the magnetic equilibrium in an axisymmetric tokamak (if pla… view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the gradient drift (left panel), local magnetic shear (central panel), [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Contour plots of the linear growth rate γES/EM in units of a −1p T0/mi , as a function of sˆ and αMHD. The subscript ES/EM indicates whether the calculation is electrostatic or electromagnetic. Since αMHD ∝ β(dρ ln p), the top row has a fixed logarithmic gradient and varying β, whereas the bottom row has fixed β and varying logarithmic gradients. Left-right pairs of panels have kyρi = 0.1 and kyρi = 0.2, r… view at source ↗
Figure 4
Figure 4. Figure 4: A figure showing the growth rate (γ) and frequency (ω) of the mode for different binormal wave-numbers kyρi , in units of a −1p T0/mi . Modes moving in the ion/electron direction have positive/negative frequency and are indicated with triangles pointing upwards/downwards. The colour of the line and marker indicates the magnetic geometry considered, consistently with [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Top row: bar charts of the gyro-Bohm normalised nonlinear fluxes in four [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: the gyro-Bohm normalised fluxes as a function of the real-space pressure [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plots of the gradient drift (left panel), local magnetic shear (central panel), [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The linear spectra in the interpolated equilibria. The colour of the line and [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
read the original abstract

Using a simple circular tokamak geometry, we show the well-known `second stability region' of MHD-ballooning modes exists for linear gyrokinetics too -- whether electrostatic or electromagnetic -- and we suggest that the plasma enters this region in H-mode as a consequence of the bootstrap current and Shafranov shift altering the magnetic field, which may occur if the normalised pressure gradient is $\alpha_{\rm MHD} \simgt 1$ and collisionality is low. By performing simulations in more realistic magnetic geometries, we demonstrate a large reduction in collisionless, electrostatic turbulent transport when going from density and temperature profiles typical of L- and H-mode, respectively. This reduction is shown to be a consequence of both the bootstrap current lowering the global magnetic shear, and the pressure gradient altering the local magnetic shear, pushing the plasma towards the second-stability region. A path connecting the L- and H-mode equilibria is constructed, along which the energy and particle fluxes exhibit non-monotonic behaviour as a function of the pressure gradient.

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 claims that the second stability region for MHD ballooning modes also exists in linear gyrokinetic simulations (both electrostatic and electromagnetic) in circular tokamak geometry. It proposes that the L-H transition occurs when the plasma enters this region in H-mode due to bootstrap current and Shafranov shift altering the magnetic field, for normalized pressure gradient α_MHD ≳ 1 and low collisionality. In realistic geometries, collisionless electrostatic turbulent transport is shown to reduce substantially from L-mode to H-mode profiles; this is attributed to bootstrap current lowering global magnetic shear and pressure gradient altering local shear, pushing the plasma into the second stability region. A continuous path connecting L- and H-mode equilibria is constructed, along which energy and particle fluxes exhibit non-monotonic dependence on the pressure gradient.

Significance. If the results hold, the work establishes a direct gyrokinetic analogue of the MHD second stability region and links it to the L-H transition via standard equilibrium effects (bootstrap current and Shafranov shift) without introducing ad-hoc parameters. The linear simulations in circular geometry, the nonlinear runs in realistic geometry, and the explicit L-to-H path with non-monotonic fluxes constitute a falsifiable prediction that can be tested against experiments. This provides a concrete mechanism for transport reduction at high α_MHD and low collisionality, strengthening the connection between MHD stability concepts and gyrokinetic turbulence modeling.

major comments (2)
  1. [Linear gyrokinetics and nonlinear transport sections] The linear stability analysis (circular-geometry scans) and the nonlinear turbulent transport runs (realistic-geometry L-to-H path) are presented separately; no direct comparison is made between the linear second-stability boundary in α_MHD–collisionality space and the pressure-gradient value at which the nonlinear fluxes drop. This leaves the causal attribution indirect.
  2. [Results on L-to-H path and turbulent fluxes] Along the constructed L-to-H path, both the pressure gradient (which sets the drive) and the magnetic geometry (bootstrap current plus Shafranov shift) are varied simultaneously. Without control simulations that hold the density/temperature profiles fixed while only modifying the equilibrium geometry, it remains unclear whether the observed flux reduction is specifically due to entry into the gyrokinetic second stability region or to changes in the instability drive and damping rates.
minor comments (2)
  1. [Abstract and introduction] The notation α_MHD ≳ 1 in the abstract and introduction should be accompanied by a brief statement of how the threshold is determined from the linear scans.
  2. [Figure captions] Figure captions for the flux-versus-α_MHD plots should explicitly state whether the magnetic geometry is held fixed or updated self-consistently at each point along the path.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive and constructive report. The comments correctly note that our linear and nonlinear results are presented in different geometries and that the L-to-H path varies multiple quantities simultaneously. Below we address each point directly, explaining the logic of the manuscript while agreeing to add clarifications.

read point-by-point responses

  1. Referee: [Linear gyrokinetics and nonlinear transport sections] The linear stability analysis (circular-geometry scans) and the nonlinear turbulent transport runs (realistic-geometry L-to-H path) are presented separately; no direct comparison is made between the linear second-stability boundary in α_MHD–collisionality space and the pressure-gradient value at which the nonlinear fluxes drop. This leaves the causal attribution indirect.

    Authors: The linear scans in circular geometry are performed to demonstrate that the second stability region exists for gyrokinetic modes (both electrostatic and electromagnetic) at low computational cost, allowing a clear scan in α_MHD and collisionality. The nonlinear simulations then apply this insight to realistic geometry by using self-consistent L- and H-mode profiles whose bootstrap current and Shafranov shift modify global and local shear in the manner expected to access second stability. The non-monotonic flux behavior along the continuous L-to-H path supplies the link: fluxes decrease even as the pressure gradient (drive) continues to rise. We will add a short paragraph and a schematic in the revised manuscript that overlays the linear stability boundary (rescaled for the effective shear in the realistic equilibria) onto the nonlinear flux curve to make the correspondence more explicit. revision: partial

  2. Referee: [Results on L-to-H path and turbulent fluxes] Along the constructed L-to-H path, both the pressure gradient (which sets the drive) and the magnetic geometry (bootstrap current plus Shafranov shift) are varied simultaneously. Without control simulations that hold the density/temperature profiles fixed while only modifying the equilibrium geometry, it remains unclear whether the observed flux reduction is specifically due to entry into the gyrokinetic second stability region or to changes in the instability drive and damping rates.

    Authors: We agree that the simultaneous variation leaves the attribution indirect. The manuscript’s central evidence is nevertheless the non-monotonic dependence of both energy and particle fluxes on the pressure gradient: as α_MHD is increased along the path the drive strengthens, yet the fluxes peak and then decline once the bootstrap current and Shafranov shift have sufficiently lowered global shear and altered local shear. This behavior is difficult to attribute to drive changes alone and matches the stabilization expected from the linear second-stability region. Performing additional nonlinear runs with fixed profiles but artificially varied geometry would isolate the geometric effect, but such equilibria are not self-consistent and would require substantial extra computational effort. We will insert a brief discussion of this limitation and of the non-monotonic flux as the principal indicator of geometric stabilization. revision: partial

Circularity Check

0 steps flagged

No circularity: results from direct numerical simulations of standard gyrokinetic equations

full rationale

The paper demonstrates the second stability region via linear gyrokinetic analysis in simple circular geometry and then computes turbulent fluxes in constructed L-to-H equilibria that incorporate bootstrap current and Shafranov shift effects. The non-monotonic flux behavior is obtained by direct solution of the gyrokinetic equations along the pressure-gradient path; no parameter is fitted to the target transport reduction and then relabeled as a prediction, nor is any central premise justified solely by self-citation. The derivation chain remains self-contained against external benchmarks because the equilibria are built from established equilibrium models and the transport is evaluated numerically without tautological redefinition of inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard gyrokinetic ordering, ideal MHD equilibrium assumptions for bootstrap current and Shafranov shift, and the validity of linear stability analysis for turbulent transport estimates. No new entities are postulated.

free parameters (1)
  • normalized pressure gradient α_MHD
    Threshold value α_MHD ≳ 1 is used to enter the second stability region; this is a control parameter varied in the simulations rather than derived from first principles.
axioms (2)
  • domain assumption Standard circular tokamak geometry and subsequent realistic magnetic equilibria can be constructed consistently with bootstrap current and pressure gradient effects.
    Invoked when moving from circular to realistic geometries and when constructing the L-to-H path.
  • domain assumption Linear gyrokinetic stability determines the qualitative change in turbulent transport levels.
    Used to link the second stability region to the observed reduction in energy and particle fluxes.

pith-pipeline@v0.9.0 · 5715 in / 1624 out tokens · 38394 ms · 2026-05-22T03:36:17.917791+00:00 · methodology

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