A robust solution for the resistive MHD toroidal Δ ′ matrix in near real-time

We introduce a new near real-time solution for the tokamak resistive MHD Δ′ matrix. By extending state transition matrix methods introduced in [Glasser et al., Phys. Plasmas 25(3), 032507 (2017)] and leveraging the asymptotic methods of [A. H. Glasser, Phys. Plasmas 23, 072505 (2016)], we have developed STRIDE - State Transition Rapid Integration with DCON (Asymptotic) Expansions - a code that solves for Δ′ in <500 ms. The resistive MHD stability remains a foremost challenge in successful tokamak operation, and its numerically demanding analysis has received attention for many years. Our code substantially improves upon the speed and robustness of earlier Δ′ calculation methods, affording solutions for previously intractable equilibria and helping enable the real-time control of ideal and resistive MHD tokamak stability. In this paper, we pedagogically review tearing stability analysis and motivate and define Δ′ in the slab, cylindrical, and toroidal geometries. We also benchmark STRIDE against the calculations of [Nishimura et al., Phys. Plasmas 5, 4292-4299 (1998)] and Furth et al. [Phys. Fluids 16, 1054 (1973)] for Δ′ in a cylindrical geometry, and the Δ′ matrix calculations of [A. H. Glasser, Phys. Plasmas 23, 112506 (2016)] in the full toroidal geometry.