Fast and Slow Nonlinear Tearing Mode Reconnection

1. Introduction. The standard theory of the tearing-mode evolution identifies three stages. The first is the linear stage described by the Furth–Killeen–Rosenbluth (FKR) theory [1]. During this stage, the island width W grows exponentially in time until it reaches the width of the resistive dissipation layer, ℓ η ∝ η 2/5 ∆ ′1/5 , where η is the resistivity and ∆ ′ is the instability parameter [see Eq. (3)]. Once W ∼ ℓ η , nonlinear terms are sufficiently large to replace inertia as the force opposing the inflow pattern. A slow down of the growth ensues, from exponential to linear in time: dW/dt ∼ η∆ ′. This is the second stage of the tearing-mode evolution, known as the Rutherford regime [2]. Finally, the third, saturated, stage is reached when the island width becomes comparable to the equilibrium shear length [3]. In this paper, we find the tearing-mode evolution to be, in fact, a four-stage process: the FKR regime, the Rutherford regime, a regime of fast nonlinear island growth that we identify as Sweet–Parker (SP) reconnection, and saturation. We carry out a set of numerical simulations that demonstrate two main points. First, we show that, given sufficiently small η, the Rutherford regime always exists; larger values of ∆ ′ require smaller values of η. Rutherford's negligible-inertia assumption is validated and the asymptotically linear dependence of dW/dt on η and ∆ ′ is confirmed. Second, we find that, at large ∆ ′ , the Rutherford regime is followed by a nonlinear stage of fast growth linked to X-point collapse and formation of a current sheet. This causes the reconnection to become SP-like. The signature η