Tokamak equilibria with strong toroidal current density reversal

Abstract

The equilibrium of large magnetic islands in the core of a tokamak under conditions of strong toroidal current density reversal is investigated by a new method. The method uses distinct spectral representations to describe each simply connected region as well as the containing shell geometry. This ideal conducting shell may substitute for the plasma edge region or take a virtual character representing the external equilibrium field effect. The internal equilibrium of the islands is solved within the framework of the variational moment method. Equivalent surface current densities are defined on the boundaries of the islands and on the thin containing shell, giving a straightforward formulation to the interaction between regions. The equilibrium of the island–shell system is determined by matching moments of the Dirichlet boundary conditions. Finally, the macroscopic stability against a class of tilting displacements is examined by means of an energy principle. It is found out that the up–down symmetric islands are stable to this particular perturbation and geometry but the asymmetric system presents a bifurcation in the equilibrium.

A fast, accurate, and easy to implement Kapur-Rokhlin quadrature scheme for singular integrals in axisymmetric geometries

“…Layer potentials are fundamental tools in representing solutions to the partial differential equations that arise in magnetostatic and magnetohydrodynamic calculations for magnetic confinement fusion (Merkel 1986;Chance 1997;Ludwig et al 2006Ludwig et al , 2013Landreman & Boozer 2016;Drevlak et al 2018). Given a surface Γ and a free-space Green's function (x, y) → G(x, y) for a partial differential equation, the single layer operator S and the double layer operator D are defined by (Guenther & Lee 1996)…”

“…Integral formulations and integral equations are effective and popular tools for magnetostatic and magnetohydrodynamic problems in magnetic confinement fusion (Shafranov & Zakharov 1972;Zakharov 1973;Freidberg et al 1976;Merkel 1986;Hirshman & Neilson 1986;Chance 1997;Ludwig et al 2006Ludwig et al , 2013Lazerson et al 2013;Drevlak et al 2018;O'Neil & Cerfon 2018;Malhotra et al 2019a;Pustovitov & Chukashev 2021). They have intuitive physical interpretations (Shafranov & Zakharov 1972;Zakharov 1973;Hirshman & Neilson 1986;Lazerson et al 2013;Hanson 2015;Pustovitov & Chukashev 2021), provide geometric flexibility (Merkel 1986;Chance 1997;O'Neil & Cerfon 2018;Malhotra et al 2019a), and often reduce the dimension of the unknown quantities one solves for, thus reducing the number of unknowns (Merkel 1986;Chance 1997;O'Neil & Cerfon 2018;Malhotra et al 2019a).…”

“…However, there typically is a price to pay for these advantages. Integral formulations often involve singular integrands, which are subtle to handle numerically (Freidberg et al 1976;Merkel 1986;Atkinson 1997;Chance 1997;Ludwig et al 2006Ludwig et al , 2013Klöckner et al 2013;Kress 2014;Ricketson et al 2016;Malhotra et al 2019a;Landreman & Boozer 2016). The numerical difficulty of integrating these singular integrands depends on the nature of the singularity, the distribution of sources, and the relative location of the evaluation points (often known as target points or observation points) with respect to the sources.…”

Exact solutions of the Grad–Shafranov equation via similarity reduction and applications to magnetically confined plasmas

“…[15,16,17] or in the large aspect ratio limit [18]. Speaking of TCR equilibria we mention that an alternative method of spectral representation of the flux surfaces has been introduced in [19]. As we have already mentioned the methods exposed above are applicable in the case g(u) = f (u), hence the general linearized GS equation contains 3 instead of 4 free parameters,…”

Deep neural network Grad–Shafranov solver constrained with measured magnetic signals

“…Note that disputing on whether the off-line EFIT results we use to train the networks are accurate or not is beyond the scope of this work. If we find more accurate EFIT results, e.g., MSE(Motional Stark Effect)constrained EFIT or more sophisticated equilibrium reconstruction algorithms that can cope with current-hole configurations (current reversal in the core) [46][47][48], then we can always re-train the networks with new sets of data as long as the networks follow the trained EFIT data with larger similarity than the rt-EFIT results do. This is because supervised neural networks are limited to follow the training data.…”