Tearing Mode Instability in Tokamaks

• Tearing mode instability is a resistive MHD phenomenon that drives magnetic reconnection, altering current sheet topology in magnetized plasmas.
• It is characterized by a unified dispersion relation linking drift-tearing, kinetic Alfvén, and internal kink modes with critical thresholds for instability.
• This instability critically influences sawtooth oscillations in tokamaks, where evolving parameters like Δ′ and β determine stabilization or disruption.

The tearing mode instability is a resistive magnetohydrodynamic (MHD) instability that mediates magnetic reconnection in current sheets, critically impacting the topology and transport properties of magnetized plasmas. In hot, large tokamak devices such as JET and ITER, the semi-collisional regime—where the electron reconnecting layer width is much less than the ion Larmor radius and semi-collisional electron dynamics prevail—defines a physically rich environment where the interplay among kinetic ions, electron diamagnetic effects, and temperature gradients governs stability. A unified dispersion relation for this regime links classic drift-tearing, kinetic Alfvén wave (KAW), and internal kink modes, permitting analytic delineation of tearing mode thresholds and stabilization mechanisms, and providing a quantitative basis for sawtooth modelling in fusion experiments.

1. Unified Dispersion Relation in the Semi-Collisional Regime

In the semi-collisional regime typical of large hot tokamaks, the reconnecting layer (width $\delta_0$) is narrow compared to the ion Larmor radius ($\rho_i$), requiring a two-region asymptotic approach:

• The “ion region” ($x \sim \rho_i$) demands a nonlocal gyro-kinetic ion response, capturing finite Larmor radius effects and high-$k$ tails in the ion distribution.
• The “electron region” ($x \sim \delta_0$) treats electrons using a semi-collisional fluid model, incorporating finite resistivity, temperature perturbations, and diamagnetic drifts.

Matching the asymptotic solutions across these layers and transforming to Fourier space yields a general dispersion relation of the symbolic form: $$e^{-i\pi/4} \frac{\delta_0}{\rho_i} A(\hat{\omega})B(\hat{\omega}) + \hat{\omega} \sqrt{1+\hat{\omega}\tau} C(\hat{\omega}) D(\hat{\omega}) = 0,$$ with:

• $\hat{\omega} = \omega/\omega_{*e}$ (normalized frequency),
• $A$, $B$, $C$, $D$: functions explicitly dependent on kinetic and diamagnetic parameters (see the paper for their construction),
• $\tau = T_e/T_i$,
• normalized pressure $\hat{\beta} = \beta_e (L_s/L_n)^2/2$.

Key branches in different parameter limits:

• Drift-tearing: $\hat{\omega} \simeq 1 + 0.74 \eta_e$ (with $\eta_e = L_n/L_{T_e}$), corresponding to the classic electron-drift-modified tearing mode.
• Kinetic Alfvén waves (KAW): Roots of $C(\hat{\omega}) = 0$; KAWs propagate predominantly in the electron diamagnetic direction.
• Modified internal kink mode: Recovers for $m=n=1$ by appropriate parameter selection; transitions to the dissipative internal kink in the low-frequency, finite-$\beta$ limit.

2. Ion Orbit Stabilization and Critical Tearing Parameter

The nonlocal kinetic treatment of ions reveals a strong stabilizing influence—ions “smear” the current perturbation over their orbits, diluting magnetic reconnection drive. The stabilizing effect is most prominent at high $k$, where the ion response's $1/k$ asymptotic tail (in the function $F(k)$) is significant. The analytic requirement for instability is that the tearing parameter $\Delta'$ must exceed a critical threshold,

$$\Delta'_{\text{crit}} \sim \rho_i^{-1} \hat{\beta} \eta_e^2 \ln\left(\frac{\rho_i}{\delta_0}\right),$$

demanding substantial current profile “sharpness” (large $\Delta'$) for instability onset. This regime reflects the original result of Cowley et al. and underscores that large orbit width and high temperature enhance stability, not susceptibility, to tearing.

3. High-$\beta$ Shielding by Electron Temperature Gradients

At high normalized pressure ($\hat{\beta}$), the dynamics fundamentally shift:

• With $\Delta' \rho_i \hat{\beta} \sim 1$, the drift-tearing mode couples with a KAW branch. At a crossover value—explicitly determined by the dispersion function and frequency—the initially unstable drift-tearing mode becomes stabilized, while a KAW branch may become unstable.
• For sufficiently large $\hat{\beta}$, strong electron temperature gradients produce intense shielding around the reconnection layer. This shielding effect stabilizes tearing for all $\Delta'$, generalizing the cold-ion electron-temperature-gradient shielding discovered by Drake et al. (1983) to regimes relevant for hot tokamaks.

The continuous transition—quantified in the dispersion relation—tracks the evolution from standard ion-orbit-dominated stabilization at low $\beta$ to high-$\beta$ electron-temperature-gradient shielding. Mode frequency and growth rate “bend” accordingly in $(\Delta'\rho_i, \hat{\beta})$ parameter space.

4. Stability Diagram and Trajectory During Sawtooth Cycles

A central result is the derived $(\Delta'\rho_i, \hat{\beta})$ stability diagram, with analytic boundaries demarcating stable and unstable regions for the drift-tearing, KAW, and dissipative internal kink modes:

• Regions I, IV, V: stable with respect to both tearing and internal kink.
• Regions II, III: unstable to drift-tearing/KAW or internal kink, respectively.

During a sawtooth period in a tokamak, plasma parameters (especially $\Delta'$ and $\hat{\beta}$) evolve: After a sawtooth crash, rapid recovery of density and temperature reduces $\hat{\beta}$, while resistive evolution of the current profile slowly increases $\Delta'$. The trajectory in the diagram tracks this evolution, and a sawtooth crash is triggered as the boundary for the dissipative internal kink or drift-tearing instability is encountered.

Key Equations:

• Dispersion relation (low $\hat{\beta}$, drift-tearing branch):

$$e^{-i\pi/4} \frac{\delta_0}{\rho_i} A(\hat{\omega})B(\hat{\omega}) + \hat{\omega} \sqrt{1+\hat{\omega}\tau} C(\hat{\omega}) D(\hat{\omega}) = 0.$$

• Drift-tearing frequency:

$\hat{\omega} \simeq 1 + 0.74\,\eta_e$

• Growth rate for $\Delta' > \Delta'_{\text{crit}}$:

$$\hat{\gamma} \sim \frac{\delta_0}{\rho_i} \frac{(\Delta' - \Delta'_{\rm crit})}{\pi\,\hat{\beta}}$$

• Crossover to KAW instability:

$$(\Delta'\rho_i \hat{\beta})_c = \frac{\pi}{\hat{\omega}_0 (\hat{\omega}_0-1) I(\hat{\omega}_0, \ldots)}$$

5. Implications for Sawtooth Phenomena and Tokamak Operation

The unified semi-collisional theory directly constrains sawtooth triggering conditions in devices such as JET and ITER:

• Immediately post-crash, high $\hat{\beta}$ (and fast electron temperature/density recovery) places the plasma in a region where drift-tearing/KAW may be unstable (or at least weakly damped).
• As the $q$-profile and thus $\Delta'$ evolve slowly, the trajectory passes into a stable window (between the drift-tearing/KAW and dissipative kink instability boundaries).
• Instability resumes (triggering a sawtooth crash) when $\Delta'$ grows sufficiently for the dissipative internal kink criterion to be met.

The model resolves ambiguity in earlier heuristic criteria for sawtooth onset, providing a firm framework for predictive or control-oriented transport codes: By tracking $\Delta'$, $\hat{\beta}$, $\eta_e$, and other physical parameters, one can forecast sawtooth periods and design external actuators to avoid deleterious instabilities or disruptions.

6. Theoretical Significance and Research Applications

The unified approach developed in this work:

• Bridges drift-tearing, KAW physics, and the (m = n = 1) internal kink mode within a single analytic formalism, capturing key stabilization mechanisms relevant to high-performance plasmas.
• Validates, generalizes, and quantifies previous ion-orbit stabilization and electron-temperature shielding results, with explicit scalings and thresholds.
• Enables construction of a stability map that tracks the evolution of inhomogeneous toroidal plasmas through sawtooth cycles, encoding the interplay among mode drive, pressure gradients, temperature perturbations, and kinetic effects.

This analytic structure is particularly amenable to integration into advanced transport and integrated modelling frameworks, allowing robust connections between micro-scale physics and macroscopic performance in fusion reactors.

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Summary Table: Physical Control Parameters

Symbol	Physical Meaning	Regime/Subtleties
$\delta_0/\rho_i$	Ratio: electron layer to ion Larmor radius	$\ll 1$ in semi-collisional
$\hat{\beta}$	Normalized electron pressure	High $\hat{\beta}$: shielding
$\Delta'$	Tearing mode stability parameter	Must exceed $\Delta'_{\text{crit}}$ for instability; increases through sawtooth cycle
$\eta_e$	Electron temperature gradient parameter	Drives drift-tearing regime
$\omega_{*e}$	Electron diamagnetic drift frequency	Sets mode frequencies
$\tau$	$T_e/T_i$ ratio	Entry to kinetic regime

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These analytic criteria, dispersion relations, and the stability diagram synthesized here form an essential basis for rigorous, physics-informed control of tearing instability, magnetic reconnection, and associated nonlinear dynamics in technologically relevant fusion plasmas.