Micro-Tearing Modes in Tokamaks

• Micro-tearing modes are electromagnetic instabilities characterized by tearing parity, driven by electron temperature gradients that enhance radial heat transport.
• They are identified by the formation of small magnetic islands and distinctive eigenmode structures, with simulations highlighting regime-dependent collisional and geometric effects.
• Recent high-fidelity gyrokinetic models and surrogate Gaussian-process methods reveal multiple MTM sub-regimes, guiding improved predictive simulations in fusion research.

Micro-tearing modes (MTMs) are a class of tearing-parity electromagnetic instabilities with perpendicular scales of order a few ion Larmor radii, $k_\perp \rho_i \sim 1$. In tokamak turbulence they are treated as the micro-scale analogue of classical tearing modes: they perturb the radial magnetic field, form small magnetic islands, can stochastize field lines, and thereby enhance radial electron heat transport. Across core, pedestal, and spherical-tokamak studies, the common free-energy source is the electron temperature gradient, but the relative importance of collisions, trapped-particle dynamics, magnetic drifts, and geometry is strongly regime-dependent rather than universal (Applegate et al., 2011, Moradi et al., 2013, Balestri et al., 27 Jul 2025).

1. Physical role and transport significance

MTMs matter because overlapping magnetic islands can create stochastic field-line regions, allowing electrons to move radially along perturbed field lines and thereby enhance electron heat transport. In spherical tokamaks this makes MTMs a plausible contributor to the high $T_i/T_e$ and anomalous electron transport often seen in devices such as MAST. In comparative gyrokinetic studies of NSTX, ASDEX-UG, and JET-like core plasmas, MTMs are reported as the dominant linear instability for experimentally relevant parameters, and the transport they drive is predominantly electron heat transport through $\delta A_\parallel$, while ion heat and particle transport are small (Applegate et al., 2011, Moradi et al., 2013).

The transport channel is fundamentally electromagnetic. In nonlinear collisional MTM simulations, magnetic flutter overwhelmingly dominates electron thermal transport, while $E\times B$ convection is negligible. A corresponding quasilinear-flavored interpretation is used in reduced-order modeling, where the linear growth rate $\gamma$, mode frequency $\omega$, and electron flutter heat flux are treated as the ingredients needed for a quasi-linear heat transport model (Fan et al., 2024, Hornsby et al., 2024).

The same transport fingerprint appears outside the spherical-tokamak core. In the DIII-D pedestal, the measured transport near the $n=3$ and $n=5$ modes has $D_e/\chi_e \sim 1/10$, indicating transport dominated by electron heat rather than particles. In W7-X, nonlinear GENE simulations identify an MTM-dominated state with $Q_e \gg Q_i$, electromagnetic heat flux dominating over electrostatic heat flux, and simulated heat and particle fluxes in reasonably close agreement with experiment (Nelson et al., 2021, Cu-Castillo et al., 22 Oct 2025).

2. Mode structure, parity, and identification

The defining structural property of an MTM is tearing parity. In local linear gyrokinetic calculations, the eigenmode of $T_i/T_e$0 has even parity across the resonant surface, while the electrostatic potential $T_i/T_e$1 is odd. Poincaré plots of perturbed field lines in MAST show chains of magnetic islands centered on rational surfaces, with the current perturbation localized in a narrow current layer around the resonant surface; the O-point is at a current minimum and the X-point at a current maximum, as expected for a tearing mode (Applegate et al., 2011, Moradi et al., 2013).

A useful geometric measure of tearing character is

$T_i/T_e$2

used in MTM filtering for GS2 surrogate datasets, with cases rejected when $T_i/T_e$3. In a high-$T_i/T_e$4 spherical tokamak equilibrium, an analogous field-line diagnostic gives $T_i/T_e$5 for the low-$T_i/T_e$6 MTM and $T_i/T_e$7 for the high-$T_i/T_e$8 MTM, confirming that both branches retain tearing character (Hornsby et al., 2023, Patel et al., 2021).

The current layer is a central object in MTM physics. In MAST, its width is defined by

$T_i/T_e$9

and the result $\delta A_\parallel$0 is used to argue that the instability is genuinely micro-scale rather than a macroscopic tearing instability. The eigenfunctions there show $\delta A_\parallel$1 localized narrowly in ballooning angle, while $\delta A_\parallel$2 and $\delta A_\parallel$3 are extended along the field line, consistent with tearing physics and the constant-$\delta A_\parallel$4 picture (Applegate et al., 2011).

Common linear and nonlinear signatures used to separate MTMs from electrostatic branches are summarized below from GENE-based triangularity studies, where $\delta A_\parallel$5 is a parity measure:

Mode	Parity and frequency	Transport signature
TEM	$\delta A_\parallel$6	$\delta A_\parallel$7
ITG	$\delta A_\parallel$8	$\delta A_\parallel$9
MTM	$E\times B$0	$E\times B$1

This identification strategy is complemented in surrogate-model datasets by requiring the mode frequency to be within $E\times B$2 of the electron diamagnetic frequency and the field structure to depart significantly from the equilibrium flux surface (Balestri et al., 27 Jul 2025, Hornsby et al., 2024).

3. Drive mechanisms and collisional regimes

Two classic MTM drive mechanisms are repeatedly cited in the literature summarized here: the thermal-force/current-layer mechanism and the nearly trapped electron boundary-layer mechanism. Both require finite electron temperature gradient and finite collisionality in their standard form. Core gyrokinetic studies across NSTX, ASDEX-UG, and JET-like plasmas support the statement that finite electron temperature gradient is the fundamental drive and that finite collisionality is needed for MTMs to become unstable in those cases; no unstable MTMs are found at $E\times B$3, and the growth-rate dependence on collisionality is generally non-monotonic, peaking at intermediate collisionality (Moradi et al., 2013).

The MAST core study modifies that picture in an important way. It finds that the instability is driven by the free energy in the electron temperature gradient as described in the literature, but is not substantially affected by either of the destabilising mechanisms proposed in previous theoretical models. Instead, the mode is strongly destabilised by magnetic drifts and by the electrostatic potential $E\times B$4. Removing $E\times B$5 weakens the instability greatly, and turning off both $E\times B$6 and magnetic drifts stabilizes it. The same study also finds that removing the energy dependence of the collision operator does not strongly change the instability, which argues against the classic thermal-force explanation as the main driver for that MAST regime (Applegate et al., 2011).

A further refinement comes from high-$E\times B$7 spherical-tokamak calculations in which the low-$E\times B$8 MTM remains unstable with adiabatic ions, survives with only $E\times B$9 perturbations, does not require trapped particles, and becomes stable if $\gamma$0 and curvature drifts are removed. In that equilibrium the mode is therefore not a slab MTM; it is a toroidally modified micro-tearing instability requiring toroidal geometry. The same work reports a critical gradient near $\gamma$1 at $\gamma$2, non-monotonic growth-rate dependence on $\gamma$3, and stabilization of the low-$\gamma$4 MTM by increasing $\gamma$5 beyond $\gamma$6 in a fixed-$\gamma$7 scan (Patel et al., 2021).

Pedestal-top MTMs introduce a genuine regime split. In MAST edge conditions just inside the top of the pedestal, the dominant edge MTM does not peak at a finite collision frequency. Its growth rate is maximized at $\gamma$8 and remains finite, or even rises slightly, as collisions are reduced to zero. That behavior is interpreted as a collisionless trapped-particle mechanism sensitive to magnetic drifts, enhanced by high magnetic shear and high trapped-particle fraction. The resonant velocity-space region lies roughly within $\gamma$9, and the mode is described as not well captured by existing core-oriented theories (Dickinson et al., 2012).

A collisional fluid-global treatment in BOUT++ adds another regime statement: in the strongly collisional regime used there, MTM growth rates decrease as collisionality increases, while increasing the temperature-gradient drive raises both growth rate and frequency. The same study derives a unified dispersion relation containing both MTM and drift-Alfvén-wave branches and emphasizes a spatial separation: MTM instability occurs near the rational surface where $\omega$0, whereas the drift-Alfvén instability appears away from the rational surface where $\omega$1 is finite (Fan et al., 2024).

Taken together, these results suggest that “MTM” denotes a family of tearing-parity electromagnetic instabilities sharing a common $\omega$2 free-energy source, but not a single universal collisional drive.

4. Geometry, shaping, and device dependence

The prevalence of MTMs in spherical tokamaks is not attributed to one geometric ingredient alone. In MAST, flux-surface shaping is not the primary reason the mode is unstable: the MTM remains unstable when the realistic numerical equilibrium is replaced by an analytic $\omega$3-$\omega$4 shifted-circle model. The large trapped-particle fraction is also not by itself strongly destabilizing in that case; trapped particles can be destabilizing at lower collisionality but become stabilizing at higher collisionality because they do not carry parallel current effectively. The explicit conclusion is that the strong occurrence of MTMs in spherical-tokamak plasmas is mainly due to higher plasma $\omega$5 and stronger magnetic drifts from smaller radius of curvature (Applegate et al., 2011).

Comparative core simulations reinforce that MTMs are not unique to spherical tokamaks. For experimentally relevant core plasma parameters, MTMs are found as the dominant linear instability not only in NSTX but also in ASDEX-UG and in a JET-like case. The mode numbers at maximum MTM growth differ across devices, but the JET-like case peaks at $\omega$6, comparable to NSTX at $\omega$7. The often-cited distinction between high-mode-number MTMs in spherical tokamaks and low-mode-number MTMs in conventional tokamaks is therefore reported as not universally valid (Moradi et al., 2013).

Triangularity changes the geometric balance substantially. Linear and nonlinear GENE flux-tube simulations across TCV, DIII-D, DEMO, SMART, and MAST-U show that negative triangularity is more susceptible to MTMs than positive triangularity. At sufficiently large $\omega$8, magnetic shear, and ratio of electron to ion temperature gradient, all the negative-triangularity scenarios become dominated by MTM turbulence, whereas the corresponding positive-triangularity scenarios remain dominated by electrostatic turbulence or have MTMs that are subdominant or stable. The stated threshold for MTM dominance in negative triangularity is

$\omega$9

The physical explanation given is that magnetic drifts are faster in the negative-triangularity geometry, and the MTM growth rate depends mainly on the poloidally averaged drift $n=3$0, not on the detailed poloidal variation of the drift profile (Balestri et al., 27 Jul 2025).

The triangularity study also rejects a simplistic aspect-ratio explanation. When the SMART aspect ratio is artificially changed toward a conventional value, MTMs remain dominant. The vulnerability of spherical tokamaks is instead traced to the parameter mix of higher $n=3$1, larger magnetic shear, and operation closer to the MTM boundary. Lowering magnetic shear can restore electrostatic turbulence and preserve the negative-triangularity benefit, which is why the study identifies lower $n=3$2 as the key mitigation route for spherical tokamaks (Balestri et al., 27 Jul 2025).

A distinct geometric route to MTM dominance appears in W7-X. There, large density gradients, moderate temperature gradients, low plasma beta, moderate collisionality, and very low magnetic shear are reported together with a quasi-omnigenous, nearly max-$n=3$3 magnetic configuration. The max-$n=3$4 property suppresses competing trapped-particle instabilities, leaving MTMs as the dominant branch. This is not the spherical-tokamak high-$n=3$5 pathway, but it shows that MTM dominance can also arise when competitors are removed by magnetic optimization (Cu-Castillo et al., 22 Oct 2025).

5. Edge and pedestal manifestations

MTMs are not confined to the core. In the shallow-gradient region just inside the top of the pedestal in MAST, local gyrokinetic calculations identify unstable MTMs that may play an important role in pedestal evolution. The reference case at $n=3$6 has a spectrum peaking at $n=3$7 in the full shaped equilibrium, and the dominant mode remains tearing parity with $n=3$8 even and $n=3$9 odd in $n=5$0. Compared with core MTMs, the edge mode has a more localized electrostatic response and an anomalous collisionality dependence, supporting the interpretation of an edge-specific trapped-particle drift-resonant drive (Dickinson et al., 2012).

The DIII-D pedestal study provides dynamic experimental evidence that MTMs are active there. The analysis uses

$n=5$1

for the electron diamagnetic frequency and

$n=5$2

for the Doppler shift, with the predicted observed MTM frequency given by $n=5$3. Time-resolved profile measurements and magnetic fluctuation data show an excellent match between the profile-based prediction and a chirping magnetic signal after an ELM. Fast vertical jogs then decouple the instability drive from the resonant location by moving rational $n=5$4-surfaces through the pedestal, and the observed frequency evolution follows the predicted change as the rational surface moves away from the peak drive. The stated conclusion is strong evidence of edge MTMs regulating electron heat flow in tokamaks (Nelson et al., 2021).

The collisional BOUT++ study reaches a closely related conclusion from global simulation: MTMs are strongest when the rational surface aligns with the peak of the electron diamagnetic frequency $n=5$5, while nonalignment stabilizes them. The stated instability criterion is $n=5$6, where $n=5$7 is the rational-surface offset from the $n=5$8 peak and $n=5$9 is the width of the $D_e/\chi_e \sim 1/10$0 profile. That result is consistent with the pedestal observation that resonant-surface location relative to the local electron-gradient drive is dynamically decisive (Fan et al., 2024).

These edge studies correct a common overgeneralization from core MTM theory. Finite collisionality is essential in some core plasmas, but pedestal-top MTMs can remain unstable in the collisionless limit, and resonant-surface alignment can be as important as the magnitude of the local gradients.

6. Numerical frameworks, reduced models, and unresolved structure

The MTM literature summarized here is methodologically diverse but numerically intensive. Core and spherical-tokamak studies employ local linear gyrokinetics with GS2, gyro, and GENE; the MAST study emphasizes a full numerical solution of the linear gyrokinetic-Maxwell equations with kinetic electrons and ions, collisions, electromagnetic effects, and equilibrium geometry; and the collisional global study uses a reduced fluid-electromagnetic model within BOUT++ (Applegate et al., 2011, Moradi et al., 2013, Fan et al., 2024).

Because MTMs have extremely fine radial structure, repeated high-fidelity simulation becomes a bottleneck for integrated plasma modelling. Two recent studies construct Gaussian-process surrogates for local linear GS2 calculations on a single flux surface at $D_e/\chi_e \sim 1/10$1, using a Miller equilibrium parameterization and a seven-dimensional input space: $D_e/\chi_e \sim 1/10$2 The outputs are the linear growth rate $D_e/\chi_e \sim 1/10$3, mode frequency $D_e/\chi_e \sim 1/10$4, and electron flutter heat flux, normalized to $D_e/\chi_e \sim 1/10$5. A GP classifier predicts whether a point is an unstable MTM, and GP regressors then predict $D_e/\chi_e \sim 1/10$6, $D_e/\chi_e \sim 1/10$7, and $D_e/\chi_e \sim 1/10$8 in the unstable region (Hornsby et al., 2023, Hornsby et al., 2024).

The active-learning workflow begins from a 300-point maximin Latin hypercube sample and adds new GS2 evaluations by classifier-guided rejection sampling. The reported hit rates rise from about $D_e/\chi_e \sim 1/10$9 MTM hits in the initial sampling to $Q_e \gg Q_i$0 and $Q_e \gg Q_i$1 in the first two expansion batches, and the broader 2023 study reports development with about 5000 data points and around 1 million CPU-hours. Five-fold cross-validation in the 2023 surrogate gives, for the Bernoulli GP with Matérn $Q_e \gg Q_i$2, accuracy $Q_e \gg Q_i$3, precision $Q_e \gg Q_i$4, recall $Q_e \gg Q_i$5, and F1 $Q_e \gg Q_i$6. The regressors are reported to be well calibrated, with $Q_e \gg Q_i$7 coverage approximately $Q_e \gg Q_i$8 to $Q_e \gg Q_i$9 across the three outputs (Hornsby et al., 2023).

The 2024 extension argues that a single smooth GP reaches a plateau in fidelity and is under-confident near marginal stability, especially because MTM physics may contain multiple sub-types. A mixture-of-experts construction therefore clusters the data into three submanifolds, uses a Dirichlet Gaussian process classifier to predict cluster membership, and combines expert posteriors through weighted total-mean and total-variance formulas. The identified branches are a low-$T_i/T_e$00 branch, a high-$T_i/T_e$01 branch, and, within the low-$T_i/T_e$02 branch, further separation by temperature gradient. The reported improvement is stronger in MSLL than in SMSE, indicating that the main gain is improved uncertainty calibration rather than only improved point prediction (Hornsby et al., 2024).

That surrogate-model outcome is physically notable. A plausible implication is that the literature’s differing reports on collisionality scaling, trapped-particle sensitivity, and magnetic-drift dependence do not merely reflect numerical disagreement; they may also indicate that MTMs occupy several nearby but distinct electromagnetic tearing sub-regimes that a single globally smooth reduced model does not represent well.