Tearing Mode Instability — Magnetic Island Formation in a Tokamak

Tearing Mode

Resonant mode (m/n)

Tearing stability index Δ′ 1.2

Edge safety factor q(a) 3.0

Live Readouts

Growing

Island width W

0.0% of a

Resonant surface

r_s /a = 0.72, q_s = 2.00

dW/dt

0.000 /s

How it works

In a tokamak the safety factor q(r) — toroidal turns per poloidal turn of a field line — increases outward. A tearing mode is a resistive instability that reconnects field lines at the "rational surface" r_s where q(r_s) = m/n, opening a chain of m magnetic islands around the poloidal cross-section.

q(r) = q0 + (qa − q0)(r/a)²         profile used here
q(r_s) = m/n                        rational (resonant) surface
ψ(x,θ) ≈ ½ψ0″x² + δψs·cos(mθ)       helical flux, x = r − r_s
W = 4·√(δψs / ψ0″)                  island full width

Growth follows a simplified, linearised form of the classic Rutherford equation, where Δ′ is the tearing stability index computed from the outer ideal-MHD solution:

τ_R · dW/dt = r_s · Δ′ · (1 − W / W_sat)

Δ′ > 0 — linearly unstable: the island grows exponentially at first, then saturates nonlinearly at W_sat once the destabilising Δ′ is balanced by the island's own flattened current/pressure profile.
Δ′ < 0 — linearly stable: any seed island decays back toward zero width.
q(a) slider — reshapes the whole q-profile, moving r_s and the local magnetic shear, which sets how "tight" the island looks.
Points outside the separatrix trace ordinary nested flux surfaces; points inside trace the island's own closed flux surfaces around its O-point(s) — this is the same Poincaré-plot picture used to diagnose real tokamak discharges. Large 2/1 islands are a well-documented precursor to major disruptions.

W(t)

Island flux surfaces Nested (unperturbed) surfaces Toroidal field-line flow

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