The Rutherford Equation

The nonlinear equation governing the growth and saturation of magnetic islands from tearing modes.

Island evolution

When a tearing mode reconnects field lines it forms a magnetic island whose width W grows nonlinearly. Rutherford derived the evolution equation for the island width in the regime where the island is wider than the linear resistive layer:

(tau_R / r) dW/dt = Delta' + island drive terms

Here tau_R is the resistive diffusion time and Delta' is the classical tearing stability index, the jump in the logarithmic derivative of the perturbed flux across the rational surface. Positive Delta' drives island growth; negative Delta' means the classical mode is stable.

Neoclassical tearing modes

The modern form adds neoclassical terms. Inside an island the pressure gradient flattens, removing the local bootstrap current; that missing current reinforces the island. This bootstrap term can drive an island unstable even when Delta' is negative, giving the neoclassical tearing mode a metastable threshold behavior.

dW/dt ~ Delta' + a*(bootstrap term)/W - b*(curvature term)/W - polarization

How it is solved numerically

• Compute Delta' from an outer-region ideal-MHD solve (the Newcomb equation)
• Evaluate bootstrap, curvature, and polarization terms from local profiles
• Integrate the ordinary differential equation for W(t) to predict saturation or growth

The threshold and saturated width determine whether a mode is benign or leads to confinement loss and possible disruption.

Why it matters

Neoclassical tearing modes limit the achievable pressure in long-pulse tokamaks and can trigger disruptions. Predicting their thresholds with the Rutherford equation is part of qualifying high-performance scenarios for devices like the Hyperion breeder, where island control informs the operating point.