Normal mode spectrum of Multi-region relaxed Magnetohydrodynamics *

It is well known, the nature of the ideal-MHD equilibria in three-dimensional toroidal plasma is profoundly affected by resonant surfaces, which give rise to a non-analytic dependence of the equilibrium. Thus, using ideal MHD to model a hot, near collisionless 3D plasmas cannot be formally justified. As a result, the study of MHD stability and Alfvén eigen-mode spectra of (m,n) field harmonics are limited, and remains an open problem in 3D equilibrium fields which are typically a blend of nested magnetic surfaces, islands, and chaotic regions. 

Throughout the last decade, the Multi-Region relaxed MHD [1] has been developed and advances as equilibrium [2], ideal and tearing stability theory [3,4], to examine 3D fields where the magnetic islands and stochastic fields co-exist. Theoretically, MRxMHD employs a generalization of Taylor relaxation model by using a series of sharp boundaries for which discontinuities in the magnetic field and pressure are present, allowing relaxation and “tearing” at rational surfaces. In this work, we present the Lagrangian formalism to compute the normal-mode spectra of MRxMHD for low-amplitude short-wavelength perturbation. Numerical solutions are constructed by upgrading the well-established Stepped Pressure Equilibrium Code (SPEC) [2] which uses the mixed spectral-Galerkin representation for the vector potential to normal modes of Stepped-Pressure Equilibrium Code, SPECN. Well-posed matrix perturbation theories are used to evaluate perturbed quantities. To explore the impact of relaxation principle of MRxMHD, a schematic verification study of frequency domain code, SPECN with CSCAS code [5] has been conducted for the spectrum of TAE, GAE in a axisymmetric equilibria. In addition, we are in the process to apply our approach to the tokamaks with imposed or self-generated 3D structure.