Theory of perturbed equilibria for solving the Grad–Shafranov equation

Abstract

The theory of perturbed magnetohydrodynamic equilibria is presented for different formulations of the tokamak equilibrium problem. For numerical codes, it gives an explicit Newton scheme for solving the Grad–Shafranov equation subject to different constraints. The problem of stability of axisymmetric modes is shown to be a particular case of the equilibrium perturbation theory.

Predictive simulations of ITER including neutral beam driven toroidal rotation

“…III C. The magnetic equilibrium is computed using the ESC code. 23 Sawtooth crashes are assumed to occur with a prescribed period of 5 s. The plasma profiles are mixed after every sawtooth crash, using a density matching method that is similar to the typical Kadomtsev mixing criterion, 24 resulting in flat plasma profiles within the sawtooth mixing region. Profile mixing is carried out for the following profiles: electron temperature T e , ion temperature T i , toroidal rotation frequency , electron density n e , ion density n k ͑including impurities, minority ions, fast ions, and helium ash͒, and current density.…”

Contour dynamics method for solving the Grad–Shafranov equation with applications to high beta equilibria

“…9 As the flux lines get squeezed on the low field side, linear system inversion becomes difficult and successive errors tend to increase. The method presented does not include such mathematical techniques and is designed to diminish successive errors.…”

“…The first sort is based on an Eulerian scheme, relying on a twodimensional (2D) mesh without any direct link to plasma shape or properties. 4,5 The second one is based on a Lagrangian scheme using curvilinear flux coordinates to map plasma geometry, 6 involving adaptive grid, 7 variational, 8 or perturbative approaches, 9 or inverse coordinates 10 methods. Nevertheless, many of these excellent methods cannot compute asymptotic high beta equilibria such as the one presented at the end of this paper.…”

Fusion-Fission Research Facility (FFRF) as a Practical Step Toward Hybrids

“…All presented calculations have been made with ASTRA-ESC code system (IPP, Garching [13], PPPL, USA [14]). Plasma global stability margins for free-boundary magneto-hydrodynamic modes (with the toroidal wave numbers n = 1, 2, 3) have been calculated using KINX code [15] (CRPP, Lausanne, Switzerland, Keldysh Inst., Moscow, RF) and all presented results correspond to the plasma parameters within these margins.…”