Benchmarking the generalized Rutherford equation with reduced MHD simulations

Introduction The nonlinear growth of neoclassical tearing modes (NTMs) in tokamaks is commonly discussed in the framework of the generalized Rutherford equation (GRE) [1, 2]. We perform a theoretical / numerical validation of the GRE by means of numerical simulations implementing the set of 2D reduced MHD equations for the helical magnetic flux ψ and the potential φ [3]. The code uses finite differences in the radial direction and a Fourier decomposition in the periodic poloidal direction. This choice of numerical method allows radial boundary conditions for the flux to be set by the step in the logarithmic derivatives over the simulated radial domain [−L : +L] of each Fourier component k in accordance with the tearing stability parameters ∆k,BC: i.e. ψ ′ k(±L)/ψk(±L) = ±0.5∆ ′ k,BC. The corresponding boundary condition for the dominant Fourier harmonic of the potential is obtained in accordance with linear ideal MHD, which should be valid outside the island region. The code focusses on the nonlinear dynamics in the narrow layer in the poloidal plane of a tokamak around the resonant surface rs including the magnetic island. In this layer the dynamics are expected to be well approximated by the 2D reduced MHD equations (se Chapter 2.4 of [4]). The equilibrium helical flux is represented by its Taylor series around the resonant surface, ψeq(x) = ∑n≥2(x/n!)ψ (n) eq , where x = r− rs. When only the leading order n = 2 term is taken into account, the code reproduces both the linear and the nonlinear Rutherford phase in close correspondence to the theoretical expectations [3]. In this contribution we analyze the nonlinear saturation of a classical tearing mode, and the growth and suppression by electron cyclotron current drive (ECCD) of a neoclassical tearing mode (NTM).