Microtearing Modes in Fusion Plasmas

• Microtearing modes are electromagnetic microinstabilities driven by electron temperature gradients that generate small-scale magnetic islands and enhance cross-field electron heat transport.
• They are characterized by tearing-parity fluctuations of Aₚₐᵣₐₗₗₑₗ and require specific collisionality regimes, with thresholds in ηₑ and βₑ setting their instability domains.
• Nonlinear saturation occurs via flattening of the temperature profile and overlapping islands, which imposes limits on temperature gradient steepness in fusion plasmas.

Microtearing modes (MTMs) are electromagnetic microinstabilities in magnetically confined plasmas, characterized by tearing-parity fluctuations of the parallel vector potential and driven primarily by the electron temperature gradient. MTMs are distinguished from macroscopic tearing modes by their small spatial scale, typically on the order of the ion or electron Larmor radius, and their role in stochastizing magnetic field lines at rational surfaces. This stochastization leads to enhanced cross-field electron heat transport, predominantly mediated by magnetic flutter. MTMs are most prominently observed in the core and edge of spherical tokamaks, conventional tokamaks, and reversed field pinches, and are increasingly recognized as a critical channel for regulating electron heat flux and limiting pedestal gradients in high-performance regimes.

1. Physical Properties, Drive Mechanisms, and Theoretical Models

MTMs are electromagnetic instabilities with even parity in the parallel vector potential $A_{\parallel}$ and odd parity in the electrostatic potential $\phi$ about the resonant (rational) surface. Their eigenstructure gives rise to chains of microscopic magnetic islands at rational $q = m/n$ surfaces; island overlap results in stochastic layers capable of rapidly transporting heat along newly reconnected field-lines. The essential free-energy source for MTMs is the electron temperature gradient, quantified by the parameter

$$\eta_e \equiv \frac{L_n}{L_{T_e}} = \frac{d\ln T_e}{d\ln n_e}$$

The classical “slab” or semi-collisional theory describes the instability as driven by a parallel ohmic current responding to $\nabla T_e$, with collisions providing both the required parallel conductivity $\sigma_{\parallel}$ and the width for the resonant layer. This is encoded in the typical slab dispersion relation (Drake–Lee, Hazeltine–Rogers),

$$\left[ \frac{d^2}{dx^2} - k_y^2 \right] A_{\parallel}(x) = - \mu_0 e^2 n_e / T_e \int d^3 v\, (v_{\parallel}/\omega - k \cdot v) g_e(x,v)$$

or, in compact form,

$$D(\omega, \gamma) \equiv \Delta'(k_y) - C_1 k_y^2 \delta_0^{-1} F_{\rm coll}(\omega/\nu_{ei}) = 0$$

where $\delta_0$ is the semi-collisional layer width and $F_{\rm coll} \to 0$ as $\omega/\nu_{ei} \to 0$ or $\infty$.

MTMs in toroidal geometry require additional physics, such as magnetic curvature, trapped-particle effects, and electromagnetic interactions. For example, toroidal trapped-particle theories predict a distinct collisional drive channel: $$D_{tp}(\omega) \sim 1 + i \sqrt{\pi} \left(\frac{\omega_{*e}}{|\omega|}\right) \epsilon^{1/2} \left( \frac{\omega}{\nu_{ei}} \right)^{1/2} - i \Delta' d(\omega, \epsilon) = 0$$ where $\epsilon = r/R$, the inverse aspect ratio, controls the trapped-particle fraction.

At low collisionality, inertia and non-ambipolar $\nabla B$ or curvature drifts can provide the necessary cross-field current even in the absence of resistivity, leading to robust MTM instability (the “collisionless microtearing mode”).

2. Dispersion Relations, Thresholds, and Scaling

Universal features of MTM dispersion relations observed across devices include:

• Growth rate $\gamma$ and real frequency $\omega_r$ are found by solving for complex $\omega$ in the system of coupled Ampère’s law and quasi-neutrality equations. In the slab and toroidal limits, and for relevant local parameters, one obtains:

$$\gamma \propto \eta_e\, \beta_e\, \frac{\nu_{ei}}{\omega_{*e}} F(\hat{s}, \mu, \ldots)$$

$$\omega_r \approx \omega_{*e} (1 + O(\beta_e, \nu/\omega_{*e}))$$

with $\beta_e = 8\pi n_e T_e / B^2$, $\omega_{*e}$ the electron diamagnetic frequency, $\hat{s}$ the magnetic shear, and $\mu$ measuring rational-surface offset from drive localization.

• Thresholds: Instability typically sets in for $\eta_e > \eta_e^{\rm crit} \sim 1.2$–$2.0$, $\beta_e > \beta_{thresh} \sim 0.01$–$0.02$ (in edge or RFP), and in the semi-collisional regime $0.3 \lesssim \nu_{ei}/\omega_{*e} \lesssim 3$. Farther from these thresholds, the drive vanishes and MTMs stabilize.
• Mode structure: The binormal (poloidal) wavenumber at maximal drive $k_y \rho_i$ is typically $0.2$–$1.0$, with considerably narrower radial structure ($k_x \rho_e \sim 1$ for collisionless modes).

Critical factors for instability are alignment of rational surfaces with the peak of the $\omega_{*e}(r)$ drive, and the local configuration of $\hat{s}$ and $q$. Global simulations confirm that only those rational surfaces coinciding with steep $T_e$ gradients or strong drive are observed as discrete bands in magnetic spectrogram data (Hatch et al., 2020).

3. MTM Nonlinear Saturation, Profile Relaxation, and Magnetic Flutter Transport

Nonlinear gyrokinetic simulations consistently show MTMs saturate via flattening of the local electron temperature gradient at rational surfaces—either by creating a stochastic layer through overlapping magnetic islands or by local relaxation of $\nabla T_e$ using zonal perturbations. The resulting heat flux is dominated by the magnetic flutter component,

$$Q_e^{\rm mag} \sim \langle T_e\, u_{\parallel e}\, \delta B_r \rangle$$

with $Q_e^{\rm mag} \gg Q_e^{E \times B}$ in all high-$\beta$ regimes (Giacomin et al., 2023, Fan et al., 11 Apr 2024).

When islands overlap, magnetic field lines become globally stochastic, resulting in rapid parallel transport and a Rechester–Rosenbluth-type electron heat flux: $$\chi_e \sim v_{the} \langle (\delta B/B_0)^2 \rangle L_c$$ with $L_c$ the parallel correlation length. Saturation occurs when the local $T_e$ gradient is flattened to marginality, and the electron thermal transport can reach values $\chi_e \sim 5$–$20$ m$^2$/s as measured in RFP and spherical tokamak regimes, and up to $\chi_e \sim 10\,D_{gB}$ in global simulations (Predebon et al., 2010, Fan et al., 11 Apr 2024). The cross-field electron heat transport severely limits achievable $T_e$ gradients in transport barriers and influences empirical $\tau_E$–collisionality scaling (Giacomin et al., 2023).

Magnetic shear $\hat{s}$ and the density of accessible rational surfaces play a key role in setting the radial extent of stochastic layers and, consequently, the saturated flux level. In regions with low shear, isolated islands lead to weak stochasticity and negligible MTM transport, while in high-shear regions, overlapping islands generate broad stochastic layers and substantial electron heat flux.

4. Collisional and Collisionless MTMs: Unified View and Instability Domains

MTMs exhibit distinct but overlapping regimes depending on collisionality:

• Collisional (semi-collisional) MTMs are well-described by the slab theory and require both a finite temperature gradient and finite $\nu_{ei}$. The drive is maximized for $\nu_{ei}/\omega_{*e} \sim 1$, with growth vanishing at both very low and very high $\nu_{ei}$ (Yagyu et al., 2022, Fan et al., 11 Apr 2024).
• Collisionless MTMs persist at $\nu_{ei} \to 0$ provided sufficient electron inertia, non-ambipolar drifts, and high $\eta_e$ are present. In this regime, the drive is provided by electron FLR, drift-resonant curvature, and inertia rather than collisional resistivity (Geng et al., 2020, Predebon et al., 2013). Collisionless slabs support tearing-parity ETG modes, which can dominate over the usual ETG “twisting parity” branch under high gradient and appropriate shear.

Trapped-particle effects can supply additional collisionless drive in the edge region of spherical and conventional tokamaks, particularly where both magnetic shear and the trapped fraction are large; here, resonance between the radial drift of trapped electrons and the mode frequency is the critical mechanism (Dickinson et al., 2012).

The table below summarizes the key MTM regimes.

Regime	Key Drive/Physics	Collisionality	Localization
Semi-collisional	Thermal-force, resistivity	$\nu_{ei} \sim \omega_{*e}$	Core/edge, wide pedestals
Collisionless	FLR, inertia, drift-resonant, trapped-particle	$\nu_{ei} \ll \omega_{*e}$	Edge, high-$\epsilon$

5. Experimental Signatures, Device-Specific Features, and Quantitative Validation

MTMs manifest experimentally as narrow-band magnetic fluctuations in Mirnov coil spectrograms or polarimeter measurements, producing discrete frequency bands with low toroidal mode number ($n \leq 10$) in H-mode pedestals of spherical tokamaks (e.g., NSTX, MAST) and in reversed-field pinches (RFX-mod, MST) (Curie et al., 2023, Predebon et al., 2010). The fluctuation frequencies track local $\omega_{*e}$ and match theoretical predictions to within 10–15% when equilibrium is tuned to align rational surfaces with the region of peak $T_e$ gradient.

Integrated workflows using reduced MTM models (e.g., SLiM) accelerated by neural networks demonstrate high-accuracy, rapid assessment of MTM mode frequencies across large equilibrium parameter spaces, achieving $98\%$ classifier accuracy and frequency predictions within $1$–$2\%$ of $\omega_{*e}$ at a computational cost $\sim 0.05$s/mode (Curie et al., 2023). Such tools enable systematic profile reconstruction and identification of MTM-driven transport channels in real discharges.

Global gyrokinetic and fluid simulations confirm that only low-$n$ bands with rational surfaces coinciding with peaks in $\omega_{*e}$ are robustly unstable (Hatch et al., 2020, Fan et al., 11 Apr 2024). Saturated flux levels predicted in simulation match experimental power-balance estimates of the electron heat diffusivity and fluctuation amplitudes in both RFP and advanced tokamak regimes.

Device-specific characteristics, such as the short connection length and high $\beta$ of RFPs, enhance the robustness and amplitude of MTMs, with profile stiffness and $\beta$ thresholds up to four times higher than in conventional tokamaks (Carmody et al., 2013). In modern stellarators (W7-X), max-J configurations with low magnetic shear suppress dangerous TEMs and ITGs, leading to a regime where MTMs dominate transport, set electron heat flux floor, and leave “ion-clamping” signatures matching experimental observations (Cu-Castillo et al., 22 Oct 2025).

6. Impact, Mitigation Strategies, and Limitations

MTMs fundamentally limit the steepness of the electron temperature gradient throughout the plasma, both in internal transport barriers and in the edge pedestal. This establishes a "profile stiffness" effect, whereby attempts to steepen $T_e$ are countered by strong, self-regulated, magnetic-flutter-driven transport.

Mitigation and control strategies identified include:

• Profile tailoring to reduce $\eta_e$ at rational surfaces (flatten $T_e$, broaden pedestal),
• Control of safety factor and magnetic shear profiles to dislocate rational surfaces away from gradient peaks,
• Application of $E \times B$ shear: nonlinearly, equilibrium flow shear can advect ballooning angle $\theta_0$ and suppress MTM transport, primarily effective at high magnetic shear (Patel et al., 12 Sep 2024).
• Increased collisionality by impurity seeding to move away from the collisional peak, though this simultaneously affects other instability channels and may be inconsistent with burning-plasma regimes,
• External magnetic perturbations or helical boundary fields in RFPs to suppress small-scale magnetic islands.

Current models are limited by geometric simplifications (slab limit, neglect of shaping), the lack of complete MHD consistency in equilibrium variation workflows, and incomplete inclusion of multi-scale or multi-niumbral MTM–ETG coupling. Full non-local conductivity models and nonlinear, multi-$n$ simulations are required for a predictive, device-scale understanding (Fan et al., 11 Apr 2024).

The implications for reactor scenarios are substantial: as device size increases (and $\rho_* \to 0$), the "flattened" fraction of radius due to MTM saturation decreases, leading to worse-than-gyro-Bohm scaling for electron heat flux (J. et al., 2022). In high-$\beta$ devices without active mitigation, this threatens to cap performance and degrade overall confinement.

7. Future Directions and Open Challenges

A comprehensive, quantitatively accurate theory of MTMs in complex device geometry remains under active development. Immediate research frontiers include:

• Development of global, fully nonlinear gyrokinetic tools capable of simultaneously resolving overlapping rational surfaces and their mutual interactions,
• Incorporation of zonal-field and multi-scale coupling physics into MTM saturation models,
• High-fidelity experimental validation using targeted diagnostics (magnetic fluctuation amplitude, $A_{\parallel}$ parity, electron heat flux footprint),
• Systematic paper of cross-device scaling, especially in transition regimes as reactors move toward low collisionality and high $\beta$.

Improved closure of electron conductivity (e.g., non-local and kinetic formulations) and self-consistent equilibrium reconstruction (integration with Grad–Shafranov solvers) are required for orbit-resolved transport modeling and for robust, real-time prediction of MTM-dominated operating points (Curie et al., 2023, Fan et al., 11 Apr 2024).

The overarching challenge is predicting and controlling MTM-driven heat transport in high-performance, reactor-scale plasmas, where these modes are poised to set hard constraints on achievable gradients and global energy confinement. Successful suppression or optimization of MTMs is essential for unlocking next-generation fusion scenarios.