The Tearing-Mode Equation

The resistive-MHD instability that reconnects field lines at rational surfaces, forming magnetic islands.

Physical Picture

A tearing mode is a resistive instability that occurs at a rational surface where the safety factor q equals m/n for integers m and n. There the perturbation is resonant, the field line closes on itself after m toroidal and n poloidal transits, and finite resistivity lets field lines break and reconnect. The reconnected topology forms magnetic islands that short-circuit the pressure gradient across their width, degrading confinement.

The Delta-Prime Criterion

Outside the thin resistive layer the plasma behaves ideally and the perturbed flux psi satisfies Newcomb's equation. The stability is set by the tearing stability index Delta', the jump in the logarithmic derivative of psi across the rational surface: Delta' = [psi'/psi] evaluated just outside minus just inside. If Delta' > 0 the mode grows; if Delta' < 0 it is stable. The classical linear growth rate scales with resistivity as eta^{3/5}, intermediate between ideal and fully resistive rates.

Neoclassical Tearing Modes

In high-pressure plasmas a more dangerous variant appears. Once a seed island forms, the loss of bootstrap current inside the flattened-pressure island removes a helical current that would otherwise be stabilizing, so the island grows even when the classical Delta' is negative. This neoclassical tearing mode is a leading limit on achievable beta in tokamaks and is typically controlled by driving replacement current at the island with localized electron-cyclotron waves.

Relevance

Because neoclassical tearing modes cap sustainable pressure and can trigger disruptions, they are a central stability concern for any tokamak breeder. For the Hyperion spherical-tokamak concept, the design-stage analysis assesses rational-surface placement and localized current-drive options; these are simulation studies, not results from operating hardware.