Nonlinear magnetohydrodynamic modeling of ideal ballooning modes in high-$β$ Wendelstein 7-X plasmas

Abstract: We present nonlinear magnetohydrodynamic (MHD) simulations of high-$β$ Wendelstein 7-X plasmas using the stellarator extension of the M3D-$C<sup>1$ code, building on the recent work that shows benign saturation of ideal ballooning modes above the designed $β$ limit in the standard configuration [Y. Zhou et al, Phys. Rev. Lett. 133, 135102 (2024)]. First, we examine the results' sensitivity to the parallel thermal conductivity. It is found that while an increased parallel conductivity reduces the linear growth rate, the saturated pressure profile is barely affected. Second, we consider the dependence on the profile shape. It is shown that an equilibrium with a peaked pressure profile and lower $β$ is subject to more significant change than a broad profile with higher $β$ and a larger growth rate, suggesting that benign saturation, or nonlinear stability, is not guaranteed and not dictated by linear growth. Third, we study the influence of the magnetic configuration, with the equilibrium rotational transform varied by adjusting the planar coil current. With similar growth rates, similar magnitudes of profile change are found regardless of the presence of a low-order resonance, which implies that the saturation mechanism is not specific to a resonant or non-resonant mode. These results indicate that MHD stability should still be treated seriously in stellarator operation and design, for which nonlinear modeling using tools like M3D-$C<sup>1$ can play an instrumental role.

• The paper demonstrates that nonlinear ballooning saturation remains robust across parallel conductivity and rotational-transform changes, but is strongly dependent on pressure-profile shape.
• Peaked profiles at 3.88–4.04% β experience more severe, core-focused pressure degradation than broad profiles at 4.9–5.4% β, despite weaker linear growth rates.
• The results show that resonance topology and linear stability margins do not reliably predict nonlinear outcomes, underscoring the need for full-MHD modeling and reduced saturation theories for W7-X operations.

This paper extends prior nonlinear MHD simulations of ideal ballooning modes in high-$\beta$ Wendelstein 7-X (W7-X) plasmas (2603.00869), building on the earlier finding that such modes saturate benignly above the designed $\beta$ limit of about 5% in the standard configuration. The authors, using the stellarator extension of M3D-$C^1$, systematically test the robustness of that conclusion along three axes: the parallel thermal conductivity $\kappa_\parallel$, the pressure profile shape, and the magnetic configuration as parameterized by the rotational transform profile. The central result is a cautionary one: benign saturation is real but not universal — it is not guaranteed by linear stability margins, not dictated by linear growth rates, and not specific to resonant or non-resonant mode dynamics.

Simulation framework

The simulations solve single-fluid extended MHD equations for density, velocity, pressure, and magnetic field in full stellarator geometry, initialized from VMEC fixed-boundary equilibria of the W7-X "EIM" standard configuration. The numerical setup uses 3807 reduced quintic $C^1$ elements in the poloidal plane and 160 Hermite cubic elements toroidally (full torus), with a time step of 0.573 μs for a hydrogen plasma at core density $1.5\times10^{20}\,\mathrm{m^{-3}}$ and roughly 5 keV core temperature. Resistivity is enhanced by a factor of 100 over Spitzer values, and equilibrium fields are subtracted in dissipative terms to act as effective sources. Boundary conditions are ideal on the magnetic field, no-slip on velocity, and fixed on density and pressure.

A notable limitation is acknowledged directly: two-fluid and other extended-MHD effects available in M3D-$C^1$ have not been fully verified in stellarator geometry, so only the single-fluid model is exercised here. Additionally, the coordinate mapping from VMEC does not evolve during the simulation, so flux-surface labels remain fixed even as actual surfaces deform.

Sensitivity to parallel thermal conductivity

Heat transport in magnetized plasmas is strongly anisotropic, with measured $\kappa_\parallel/\kappa_\perp$ ratios up to $\sim10^8$. Because the pressure evolution matters for ballooning dynamics, the authors vary $\kappa_\parallel/\kappa_0$ in single-field-period simulations of a broad-profile, $\beta$0 EIM equilibrium. For $\beta$1, the linear growth rate decreases substantially with increasing $\beta$2.

The origin of this reduction is diagnosed through two control experiments. Raising $\beta$3 by a factor of ten makes growth rates nearly insensitive to $\beta$4, indicating that part of the stabilizing effect is numerical pollution — spurious perpendicular transport arising from discretization error in parallel diffusion. However, increasing toroidal resolution from 32 to 48 elements per field period raises growth rates while preserving the same dependence on $\beta$5, demonstrating that $\beta$6 also has a genuine direct stabilizing effect on the mode.

Crucially, the nonlinearly saturated states are far more robust than the linear growth rates: full-torus simulations with $\beta$7 and $\beta$8 yield nearly identical saturated pressure profiles despite different linear growth. This validates the choice of $\beta$9 in the original study and establishes the saturated pressure-profile change as a more meaningful observable than the growth rate itself. The implication is practical: conclusions about benign saturation do not hinge on an arbitrarily chosen transport coefficient, even though accurately resolving the linear phase remains numerically formidable.

Dependence on pressure profile shape

The broad, parabolic-like profile used previously is favorable for high volume-averaged $C^1$0 but difficult to achieve experimentally; recent high-performance W7-X discharges exhibit more peaked profiles. The authors therefore consider a peaked profile $C^1$1 at $C^1$2 and $C^1$3, whose rotational transform profiles cross the low-order $C^1$4 resonance.

The key finding contradicts any simple extrapolation from linear theory: the peaked-profile equilibria, despite lower $C^1$5 and lower linear growth rates than the broad-profile case, suffer more pronounced ballooning-induced pressure degradation. Moreover, the degradation worsens sharply as $C^1$6 increases from 3.88% to 4.04%, whereas the broad-profile case degraded only modestly between $C^1$7 and $C^1$8. The authors conclude that with a peaked pressure profile in the standard configuration, the effective $C^1$9-limit is both lower and more rigid. This carries a direct operational implication: in future higher-$\kappa_\parallel$0 W7-X experiments, core MHD activity may naturally broaden the pressure profile, analogous to temperature flattening observed in NSTX.

Structural features of the nonlinear evolution reinforce that saturation is not simple relaxation toward linear marginality. The saturated pressure changes extend radially inward relative to the linear mode structures, which localize where the pressure gradient is largest (mid-radius for peaked profiles, periphery for broad ones); substantial pressure gradient persists where the linear modes reside. For the peaked cases, the degradation concentrates in the core, qualitatively resembling ballooning-induced core collapses observed and simulated in the Large Helical Device.

Poincaré plots add a further nuance: in the broad-profile case a large fraction of the field remains non-integrable in the periphery, while in the peaked cases most flux surfaces heal, with $\kappa_\parallel$1 islands forming at the $\kappa_\parallel$2 resonance. More integrable fields therefore do not correspond to softer $\kappa_\parallel$3-limits, corroborating the earlier finding that convective transport dominates over conduction in the degradation.

Influence of rotational transform

Exploiting W7-X's planar coil system, the authors vary $\kappa_\parallel$4 to generate vacuum configurations with shifted rotational transform profiles while preserving profile shape and shear, keeping core field and plasma volume roughly constant. Positive planar-coil current places an $\kappa_\parallel$5 resonance in the profile; negative current removes it. Finite-$\kappa_\parallel$6 equilibria use the peaked profile with $\kappa_\parallel$7 adjusted so all cases share comparable linear growth rates ($\kappa_\parallel$8).

Despite these controlled differences, the saturated pressure profiles show similar levels of degradation across all configurations, whether or not the low-order resonance is present. Linear-phase mode structures are indistinguishable, saturated states are largely similar, and only the island topology in the Poincaré plots differs. The authors infer that the saturation mechanism is not specific to a particular resonant or non-resonant mode, and suggest this invariance may permit a reduced, energetics-based theory of the saturation amplitude that is agnostic to detailed mode dynamics. They note explicitly that magnetic shear and other configuration parameters were not varied, leaving their influence open.

Limitations and open questions

Several caveats bound the results. The single-fluid MHD model excludes two-fluid and kinetic effects whose implementation in stellarator geometry remains unverified. Linear growth rates retain sensitivity to resolution and transport coefficients, so quantitative linear predictions carry uncertainty even though saturated states appear robust. The profile-shape study covers only two idealized shapes; experimental profiles will differ in detail. The rotational-transform scan holds shear approximately fixed, so the role of shear in the saturation mechanism is unresolved. Finally, each full-torus simulation costs hundreds of thousands of CPU hours, precluding iterative use in scenario optimization or device design; developing a validated reduced model for the saturation amplitude is identified as the necessary next step, with the present invariance results offering constraints for such a model.

Conclusion

By testing sensitivity to parallel conductivity, profile shape, and rotational transform, this work converts a single encouraging prediction into a more nuanced assessment of nonlinear ballooning stability in W7-X. Benign saturation survives variation of $\kappa_\parallel$9 and of the magnetic configuration, but fails to generalize across pressure profile shapes, where lower-$C^1$0, slower-growing modes produce worse degradation. The overall message for stellarator design and operation is that MHD stability must still be treated seriously, and that nonlinear simulation tools such as M3D-$C^1$1 — complemented by future reduced models — are instrumental for assessing it.