Quasi-linear theory of forced magnetic reconnection for the transition from linear to Rutherford regime

Abstract

Using the in-viscid two-field reduced MHD model, a new analytical theory is developed to unify the Hahm-Kulsrud-Taylor (HKT) linear solution and the Rutherford quasi-linear regime. Adopting a quasi-linear approach, we obtain a closed system of equations for plasma response in a static plasma in slab geometry. An integral form of analytical solution is obtained for the forced magnetic reconnection, uniformly valid throughout the entire regimes from the HKT linear solution to the Rutherford quasi-linear solution. In particular, the quasi-linear effect can be described by a single coefficient $K_s\propto S^{8/5} \psi_c^2$, where $S=\frac{\tau_R}{\tau_A}$ and $\psi_c$ are the Lunquist number and amplitude of external magnetic perturbation, respectively. The HKT linear solution for response can be recovered when the index $K_s\rightarrow 0$. On the other hand, the quasi-linear effect plays a key role in the island growth when $K_s\sim1$. Our new analytical solution has also been compared with reduced MHD simulations with agreement.