Nonlinear asymmetric tearing mode evolution in cylindrical geometry

The growth of a tearing mode is described by reduced MHD equations. For a cylindrical equilibrium, tearing mode growth is governed by the modified Rutherford equation, i.e., the nonlinear Δ′(w). For a low beta plasma without external heating, Δ′(w) can be approximately described by two terms, Δ′ql(w), ΔA′(w) [White et al., Phys. Fluids 20, 800 (1977); Phys. Plasmas 22, 022514 (2015)]. In this work, we present a simple method to calculate the quasilinear stability index Δql′ rigorously, for poloidal mode number m≥2. Δql′ is derived by solving the outer equation through the Frobenius method. Δ′ql is composed of four terms proportional to: constant Δ′0, w, wlnw, and w². ΔA′ is proportional to the asymmetry of island that is roughly proportional to w. The sum of Δql′ and ΔA′ is consistent with the more accurate expression calculated perturbatively [Arcis et al., Phys. Plasmas 13, 052305 (2006)]. The reduced MHD equations are also solved numerically through a 3D MHD code M3D-C1 [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)]. The analytical expression of the perturbed helical flux and the saturated island width agree with the simulation results. It is also confirmed by the simulation that the ΔA′ has to be considered in calculating island saturation.