Resistive Pressure-Driven MHD Modes

• Resistive pressure-driven MHD modes are instabilities triggered by steep pressure gradients combined with finite resistivity, destabilizing magnetic confinement.
• Analytic and numerical studies reveal scaling laws, such as growth rates proportional to η^(1/3) for kink modes and η^(3/5) for tearing modes, which are critical for reactor design.
• Understanding these modes aids in mitigating instabilities through feedback control in fusion devices, astrophysical disks, and advanced plasma configurations.

A resistive pressure-driven magnetohydrodynamic (MHD) mode is an instability in conducting plasmas where free energy stored in pressure gradients drives plasma flows, reconnection, or large-scale changes in the magnetic field, and where resistive effects—arising from finite conductivity—modify the growth, structure, and nonlinear development of these instabilities. In the presence of resistivity, otherwise ideal-MHD-stable configurations may become unstable, facilitating phenomena such as magnetic reconnection, global relaxations, or transport-destabilizing behaviors, crucial in fusion devices (tokamaks, stellarators), astrophysical disks, and reversed field pinches. The characteristics, analytic descriptions, and control mechanisms for these modes depend sensitively on geometry, plasma profiles, the inclusion of kinetic effects, and the particular physical configuration.

1. Fundamental Physics and Linear Theory

Resistive pressure-driven MHD modes are initiated when pressure gradients are sufficiently steep to overcome stabilizing forces (field line tension, curvature), and resistivity enables magnetic topology changes not accessible in the ideal limit. The general form of the resistive MHD induction equation is:

$$\frac{\partial\mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B}) - \nabla \times (\eta \nabla \times \mathbf{B})$$

where $\mathbf{u}$ is the fluid velocity and $\eta$ the resistivity. The momentum equation includes pressure gradients and Lorentz forces:

$$\rho \left( \frac{\partial\mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mathbf{J} \times \mathbf{B} + \dots$$

Instabilities such as tearing modes, resistive wall modes (RWMs), and resistive ballooning or interchange modes fall into this class. Near rational surfaces, resistivity introduces singular layers where the otherwise stabilizing field line bending is reduced, allowing growth of instabilities that can be described by extended dispersion relations (e.g., Glasser–Greene–Johnson (GGJ), Mercier-like criteria, etc.) (Ham et al., 2013, Coste-Sarguet et al., 3 Sep 2025). In advanced scenarios with low or reversed shear, resistivity further amplifies long-wavelength “infernal” modes by weakening field line coupling, leading to discrete spectra of fast-growing modes (Coste-Sarguet et al., 3 Sep 2025).

2. Nonlinear Evolution and Phenomenology

Nonlinear simulations reveal the complexity of resistive, pressure-driven MHD modes. In reversed field pinch or tokamak geometries, once the mode saturates and significant magnetic flux is displaced or compressed (e.g., after helical core (HC) formation (Adulsiriswad et al., 15 Sep 2025)), steep local pressure gradients develop, giving rise to secondary resistive instabilities characterized by a broad spectrum of short-wavelength Fourier components with nearly identical linear growth rates. For instance, after the nonlinear formation of an HC state in ITER-scale scenarios with $\beta_\alpha \lesssim 1\%$, simulations show that a resistive pressure-driven mode can become unstable, localized where the bulk pressure gradient is maximized along compressed flux surfaces. The resulting mode grows as a coherent bundle and facilitates magnetic chaos, evidenced by the breakdown of nested flux surfaces and increased particle transport (Adulsiriswad et al., 15 Sep 2025).

Growth rates in the resistive regime typically follow analytic scalings:

• For resistive kink/ballooning modes: $\gamma \propto \hat{\eta}^{1/3}$
• For resistive tearing modes: $\gamma \propto \hat{\eta}^{3/5}$

with $\hat{\eta}$ the normalized resistivity. These features were observed both in core-localized secondary instabilities after HC formation and in reconnection-dominated relaxation events (Adulsiriswad et al., 15 Sep 2025, Coste-Sarguet et al., 3 Sep 2025).

3. Impact of Kinetic Effects and Energetic Particles

Kinetic resonances with thermal and suprathermal particle populations modify resistive, pressure-driven mode stability. In hybrid kinetic-MHD models, the inclusion of the kinetic tensor (with parallel and perpendicular pressure perturbations) introduces velocity-space resonances (e.g., drift, bounce, and precession frequencies) which can transfer energy between the mode and particle motions (Yadykin et al., 2011). The kinetic resonance operator formalism demonstrates that such resonances can result in stabilization (via energy extraction) at selected rotation frequencies, or even drive instabilities for specific velocity-space domains.

Energetic ions, particularly trapped populations with slowing-down distributions, exert a strong influence on tearing and interchange mode stability, with the sign and amplitude of their effect highly sensitive to local magnetic shear. For the classic $m/n=2/1$ tearing mode, energetic ions are stabilizing when orbiting in strong positive shear, and destabilizing in regions of weakly reversed shear—a result confirmed both numerically and experimentally (see JET and DIII-D/JT-60U comparisons) (Halfmoon et al., 2017).

At moderate $\beta_\alpha$ (fusion-born alpha pressure), the alpha population acts as an additive scalar pressure and increases the saturated HC displacement without generating significant flattening of profile gradients or omnigenity loss; at higher $\beta_\alpha \gtrsim 1\%$, both bulk and alpha pressure profiles are partially flattened, decreasing omnigenity and increasing susceptibility to resistive instabilities and degraded confinement (Adulsiriswad et al., 15 Sep 2025).

4. Role of Geometry, Shear, and Shaping

Extended low- or reversed-shear $q$ profiles in advanced tokamaks make the plasma particularly susceptible to resistive infernal modes: stabilizing field line bending is diminished, and the toroidal coupling of harmonics is enhanced, producing a discrete and often overlapping spectrum of pressure-driven instabilities (Coste-Sarguet et al., 3 Sep 2025). Shaping parameters—elongation ($\kappa$), triangularity ($\delta$)—and the configuration of the safety factor profile determine the dispersion relation’s stability threshold via modified Mercier or GGJ conditions. For example, terms such as $D_I \sim (\kappa-1)(1-2\delta)$ enter directly into growth rates and thresholds.

The analytic and numerical solutions demonstrate that the marginal stability threshold can depend simultaneously on shaping, local $q$ profile, and pressure-drive, with critical inequalities revealing parameter domains where entire spectra of unstable modes appear. For stellarator regimes (e.g., W7-AS), nonlinear resistive MHD modeling shows that soft $\beta$ limits emerge when plasma resistivity nears experimental values: mild $(2,1)$ tearing/interchange activity produces a fragmented, yet ergodic, but not disruptive confinement structure (Ramasamy et al., 5 Feb 2024).

5. Practical Control, Stability Margins, and Applications

In fusion devices, the control and suppression of resistive, pressure-driven MHD modes is essential for stable operation at elevated pressure (or elongation) and for the avoidance of disruptions. Active feedback techniques—using sensors and actuators with real and complex gain components—can stabilize resistive wall modes or extend the stability margin beyond various ideal/resistive thresholds ($\beta_{rp,rw}$, $\beta_{rp,iw}$, $\beta_{ip,rw}$, $\beta_{ip,iw}$) (Brennan et al., 2014). Optimization of complex gain (including rotation equivalence via imaginary gain terms for normal field sensors) is shown to maximize the stable region, but can also become destabilizing above certain thresholds due to inhibited feedback flux penetration (Brennan et al., 2014).

Analytic and numerical scaling relations have been derived for vertical instability (n=0 resistive wall mode) thresholds as a function of plasma shape, wall distance, current profile, and feedback system parameters, enabling optimization of high-confinement scenarios (Lee et al., 2017). In simulations and experiments, mode mitigation also depends critically on the global configuration (plasma-wall gap, shaping), magnetic Prandtl number, and resistivity profiles.

6. Global Astrophysical and Laboratory Implications

Resistive pressure-driven MHD modes are not limited to laboratory fusion contexts. In accretion disks, resistive MHD simulations of thin $\alpha$-discs reveal robust midplane backflow sustained by the interplay of pressure gradients and non-ideal MHD effects, with the character (steady vs. quasiperiodic) of the backflow strongly determined by the magnetic Prandtl number (Mishra et al., 2022). The overall angular momentum and mass transport in such systems is fundamentally influenced by these resistive MHD processes.

In high-energy and relativistic systems, simulations confirm that explicit resistivity—rather than numerical dissipation—accurately captures the scaling and onset of magnetic reconnection and pressure-driven instability, a requirement for reliable modeling of global reconnection and energy release (Grehan et al., 25 Mar 2025).

7. Summary Table: Typical Physical Regimes and Characterizations

Mode Type	Physical Driver	Key Instability Features
Resistive Tearing / Pressure-Driven	Pressure gradient	Growth rate $\propto \eta^{1/3}$ or $\eta^{3/5}$
Resistive Wall Mode (RWM)	Plasma pressure/current	Sensitivity to rotation, wall resistivity, feedback
Infernal Modes (Reversed Shear)	Low/reversed shear, shaping	Discrete unstable spectra, global reconnection
Drift-wave Instability (Resistive 2F-MHD)	Density/pressure gradient	Edge-localization, non-monotonic dispersion
Nonlinear Saturated State (e.g., HC)	Ideal kink/interchange sat.	Secondary resistive mode: magnetic chaos, confinement loss

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In summary, resistive pressure-driven MHD modes sit at the intersection of plasma pressure, magnetic topology, resistivity, and geometry, underpinning many of the critical stability and transport phenomena in both laboratory and astrophysical systems. Their analytic characterization, nonlinear simulation, and experimental validation remain active areas of research, especially given their importance for reactor performance and stability in advanced fusion regimes and stellarators (Coste-Sarguet et al., 3 Sep 2025, Adulsiriswad et al., 15 Sep 2025, Ramasamy et al., 5 Feb 2024, Brennan et al., 2014, Ham et al., 2013).