Resistive Pressure-Driven MHD Modes

• Resistive pressure-driven MHD modes are instabilities in toroidal plasmas driven by pressure gradients and finite resistivity, critical for defining operational β limits.
• They include resistive tearing, infernal, and wall modes, with eigenmode spectra shaped by magnetic shear, plasma shaping, and kinetic effects.
• The topic integrates analytic theory, computational modeling, and experiments to improve our understanding and control of plasma stability in fusion devices.

Resistive pressure-driven magnetohydrodynamic (MHD) modes are a fundamental class of instabilities in toroidal plasmas, arising from the interplay between finite plasma resistivity, equilibrium pressure gradients, and magnetic geometry. These modes, which include classic resistive tearing, resistive infernal, and resistive wall modes, are pivotal in setting operational stability limits (notably the so-called soft $\beta$ limits) in both tokamaks and stellarators. Their study integrates analytic theory, computational modeling, and experimental observation, particularly in regimes with low magnetic shear, reversed shear, and non-trivial shaping, where conventional ideal MHD stabilization by field-line bending is suppressed (Coste-Sarguet et al., 3 Sep 2025, Brennan et al., 2014, Halfmoon et al., 2017).

1. Fundamental Physics of Resistive Pressure-Driven Modes

Resistive pressure-driven MHD modes are governed by the reduced resistive MHD equations, supplemented by terms capturing pressure gradient effects (e.g., the Mercier parameter), resistivity $\eta$, and equilibrium properties such as current and $q$ profiles. These modes are destabilized by pressure gradients, with resistivity enabling reconnection and access to unstable branches not present in the ideal MHD limit.

The resistive tearing mode is characterized by island formation at rational $q=m/n$ surfaces, with a growth rate scaling as $\gamma \sim \eta^{3/5} (\beta \Delta')^{2/5}$ in the constant-$\psi$ case. In advanced tokamak and stellarator regimes with extended low or reversed shear, the reduced field-line bending stabilization enables a new spectrum of pressure-driven resistive modes—so-called "infernal" modes—with long radial wavelengths and complex mode structures (Coste-Sarguet et al., 3 Sep 2025).

Key controlling parameters include:

• Pressure gradient and shaping: Entering through the generalized Mercier parameter $DI(r)$, defined as

$DI(r) \simeq a(r)[2 - 1 + 2(K-1)(1-2\delta)],$

with $a(r)=-\frac{r dq}{dr} \frac{dP}{dr}/B^2$, $K$ elongation, and $\delta$ triangularity.

• Magnetic shear ($s$): Small $s$ suppresses field-line bending stabilization, promoting infernal mode growth.
• Resistivity ($\eta$): Allows magnetic reconnection in thin, resistive layers and sets characteristic growth rates.

2. Linear Theory and Matched Asymptotics

The canonical analytic framework is the outer-inner layer matching, which decouples the ideal MHD region (outer) from a thin resistive layer at the rational surface. The ideal region provides outer solutions, subject to jump conditions across pressure and current profile discontinuities. The resistive inner region is described by the tearing layer equations, yielding a relation between the discontinuity in the perturbed flux derivative ($\Delta'$) and the instability growth rate.

For resistive infernal modes, the dispersion relation is generalized to include pressure drive, toroidal coupling $A_q$, and shaping (Coste-Sarguet et al., 3 Sep 2025):

• In the ideal limit ($\eta\to 0$), the eigenmodes satisfy a Bessel-function dispersion relation with a discrete spectrum of roots, each giving rise to an ideal infernal mode.
• Including resistivity, the classical Glasser–Greene–Johnson formalism leads to the scaling

$$\gamma_{infernal} \propto \eta^{1/3} \beta^{1/2} |s|^{-2/3} \cdots,$$

contrasting with classic tearing scaling and marking a transition in dominant instability character as $\beta$ or shaping increase.

Tables of marginal $\beta$-thresholds and analytic scaling laws differentiate classical tearing, resistive wall, and infernal branches (see (Brennan et al., 2014) for explicit $\beta_{rp,iw}$, $\beta_{ip,iw}$, etc.).

3. Spectra, Mode Structure, and Parametric Dependence

In low-shear and/or shaped configurations, resistive infernal modes form a discrete ladder of instabilities for each $(m,n)$, indexed by radial node number $\ell$. Primary (lowest-$\ell$) modes are the most unstable, but subdominant modes exhibit increasingly oscillatory radial eigenfunctions, localized around the rational surface with characteristic width scaling as $S_L^{-1/3}$ ($S_L$ the Lundquist number) (Coste-Sarguet et al., 3 Sep 2025).

Spectral properties:

• Spacing: Eigenmode roots satisfy $x_\ell \simeq (\ell+m)\pi$, with spacing $\Delta r \simeq \pi r_1/x_\ell$.
• Sensitivity: Small changes in shaping (e.g., triangularity, elongation) or shear can switch the spectrum on or off.
• Compressibility: Finite sound speed introduces a mild reduction in growth rates but does not suppress the infernal spectrum.

Numerical approaches employ adaptive radial mesh refinement to resolve the thin resistive layers and multiple oscillatory subdominant modes up to high radial resolution ($N_r \sim 10^4$) (Coste-Sarguet et al., 3 Sep 2025).

4. Effects of Energetic Particles and Magnetic Shear

Energetic trapped ions affect resistive pressure-driven tearing and infernal modes via drift-kinetic interactions, modeled as a pressure perturbation added to the fluid pressure step in outer-region matching (Halfmoon et al., 2017). For positive magnetic shear, the precessional resonance leads to kinetic pressure perturbations that tend to damp and stabilize the mode, increasing the marginal $\beta$ threshold for instability. In contrast, in the presence of low or reversed shear, the resonance changes sign, leading to destabilization and reduction of $\beta_{crit}$.

This phenomenology explains empirical trends: energetic ion stabilization of 2/1 modes in monotonic shear tokamaks, and destabilization in reversed shear scenarios (Halfmoon et al., 2017).

5. Soft $\beta$ Limits in Stellarators and Tokamaks

Soft $\beta$ limits are associated with resistive MHD activity that limits achievable pressure at levels much lower than the ideal stability boundary. In stellarators (e.g., W7-AS), modeling demonstrates that low-$n$ resistive pressure-driven interchange or tearing modes set a soft $\beta$ limit: as resistivity approaches experimental values, mild MHD activity emerges, dominated by a nonlinearly significant (2,1) mode. The plasma typically remains separated into sub-volumes with ergodic field-line confinement, so that the enhanced perpendicular transport does not overwhelm heating power—allowing the soft limit to persist without macroscopic disruption (Ramasamy et al., 5 Feb 2024).

The resistive nature of observed MHD activity—in contrast to ideal mode predictions—emphasizes the criticality of resistive pressure-driven (rather than ideal) analysis in setting operational $\beta$ limits for reactor-relevant scenarios.

6. Implications for Control, Sawtooth Dynamics, and Scenario Design

Resistive pressure-driven infernal, tearing, and wall modes play a determining role in disruption dynamics and the design of stable high-performance scenarios:

• Sawtooth and reconnection phenomena: The existence of a discrete infernal-mode spectrum allows for a cascade of reconnection events, suggesting a mechanism for smoothing or mitigating classic sawtooth crashes by distributing reconnection over multiple, smaller-scale events (Coste-Sarguet et al., 3 Sep 2025).
• Feedback control: The stability of resistive wall modes (RWMs) is strongly affected by feedback mechanisms based on the normal and tangential magnetic field at the resistive wall, with complex gain structures (real and imaginary components) providing stabilization or destabilization, dependent on parameters such as plasma rotation and $\beta$ relative to analytic thresholds ($\beta_{rp,iw}$, etc.) (Brennan et al., 2014).
• Scenario optimization: Marginal stability boundaries are shifted by energetic particle content, pressure profile shaping, and shear profile, demanding precise tailoring of equilibrium and external heating/current-drive methods to maximize achievable $\beta$ while avoiding the rapid growth of resistive modes.

7. Contemporary Research and Outlook

Recent analytic and computational advances have produced robust models for resistive pressure-driven spectra in both axisymmetric and 3D geometries, capturing the influence of resistivity, magnetic geometry, pressure profile, toroidal coupling, and non-thermal populations (Coste-Sarguet et al., 3 Sep 2025, Halfmoon et al., 2017, Ramasamy et al., 5 Feb 2024). Research highlights the need for high-fidelity modeling, including full sideband coupling and accurate representation of resistive layers, as oversimplified treatments can artificially suppress or mischaracterize the spectrum.

A persisting challenge is the characterization and prediction of soft $\beta$ limits in non-axisymmetric stellarators, particularly given the mild and ergodic character of observed MHD activity at the soft limit. Future work is directed toward nonlinear extensions, global eigenmode analysis, and incorporation of kinetic effects and anisotropic transport (Ramasamy et al., 5 Feb 2024). A plausible implication is that proper exploitation of benign, low-$n$ resistive modes could be part of scenario control strategies to avoid major disruptions in next-generation reactor operation.