A non-linear MHD code, named JOREK, is under development with the aim of studying the non-linear evolution of the MHD instabilities thought to be responsible for edge localized modes (ELMs): external kink (peeling) and medium-n ballooning modes. The full toroidal X-point geometry is taken into account including the separatrix, open and closed field lines. Analysis of the influence of the separatrix shows a strong stabilization of the ideal and resistive MHD external kink/peeling modes. One instability remains unstable in the presence of the X-point, characterized by a combination of a tearing and a peeling mode. The so-called peeling–tearing mode shows a much weaker dependence on the edge q. Non-linearly the n = 1 peeling–tearing mode saturates at a constant amplitude yielding a mostly kink-like perturbation of the boundary with an island-like structure close to the X-point. The non-linear evolution of a medium-n ballooning mode shows the formation of density filaments. The density filaments are sheared off from the main plasma by an n = 0 flow non-linearly induced by the Maxwell stress. The amplitude of the ballooning mode is limited by this n = 0 flow and multiple (in time) density filaments can develop to bring the plasma below the stability boundary.

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JOREK is a massively parallel fully implicit non-linear extended magneto-hydrodynamic (MHD) code for realistic tokamak X-point plasmas. It has become a widely used versatile simulation code for studying large-scale plasma instabilities and their control and is continuously developed in an international community with strong involvements in the European fusion research programme and ITER organization. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and physics studies performed with the code. A dedicated section highlights some of the verification work done for the code. A hierarchy of different physics models is available including a free boundary and resistive wall extension and hybrid kinetic-fluid models. The code allows for flux-surface aligned iso-parametric finite element grids in single and double X-point plasmas which can be extended to the true physical walls and uses a robust fully implicit time stepping. Particular focus is laid on plasma edge and scrape-off layer (SOL) physics as well as disruption related phenomena. Among the key results obtained with JOREK regarding plasma edge and SOL, are deep insights into the dynamics of edge localized modes (ELMs), ELM cycles, and ELM control by resonant magnetic perturbations, pellet injection, as well as by vertical magnetic kicks. Also ELM free regimes, detachment physics, the generation and transport of impurities during an ELM, and electrostatic turbulence in the pedestal region are investigated. Regarding disruptions, the focus is on the dynamics of the thermal quench (TQ) and current quench triggered by massive gas injection and shattered pellet injection, runaway electron (RE) dynamics as well as the RE interaction with MHD modes, and vertical displacement events. Also the seeding and suppression of tearing modes (TMs), the dynamics of naturally occurring TQs triggered by locked modes, and radiative collapses are being studied.

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