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.

A benchmark exercise for the modeling of vertical displacement events (VDEs) is presented and applied to the 3D nonlinear magneto-hydrodynamic codes M3D-C 1 , JOREK and NIMROD. The simulations are based on a vertically unstable NSTX equilibrium enclosed by an axisymmetric resistive wall with rectangular cross section. A linear dependence of the linear VDE growth rates on the resistivity of the wall is recovered for sufficiently large wall conductivity and small temperatures in the open field line region. The benchmark results show good agreement between the VDE growth rates obtained from linear NIMROD and M3D-C 1 simulations as well as from the linear phase of axisymmetric nonlinear JOREK, NIMROD and M3D-C 1 simulations. Axisymmetric nonlinear simulations of a full VDE performed with the three codes are compared and excellent agreement is found regarding plasma location and plasma currents as well as eddy and halo currents in the wall.

In recent years, the nonlinear 3D magnetohydrodynamic codes JOREK, M3D-C1, and NIMROD developed the capability of modeling realistic 3D vertical displacement events (VDEs) including resistive walls. In this paper, a comprehensive 3D VDE benchmark is presented between these state-of-the-art codes. The simulated case is based on an experimental NSTX plasma but with a simplified rectangular wall. There are differences between the physics models and numerical methods, and the VDE evolution leads to sensitivities on the initial conditions that cannot be avoided as can be done in edge localized modes (ELM) and sawtooth simulations (due to the non-cyclical nature of VDEs). Nonetheless, the comparison serves to quantify the level of agreement in the relevant quantities used to characterize disruptions, such as the 3D wall forces and energy decay. The results bring confidence regarding the use of the mentioned codes for disruption studies, and they distinguish aspects that are specific to the models used (e.g., reduced vs full MHD models). The simulations show important 3D features for a NSTX plasma, such as the self-consistent evolution of the halo current and the origin of the wall forces. In contrast to other reduced MHD models based on an ordering in the aspect ratio, the ansatz-based JOREK reduced MHD model allows capturing many aspects of the 3D dynamics even in the spherical tokamak limit considered here.

For the simulation of disruptions in tokamak fusion plasmas, a fluid model describing the evolution of relativistic runaway electrons and their interaction with the background plasma is presented. The overall aim of the model is to self-consistently describe the non-linear coupled evolution of runaway electrons (REs) and plasma instabilities during disruptions. In this model, the runaway electrons are considered as a separate fluid species in which the initial seed is generated through the Dreicer source, that eventually grows by the avalanche mechanism (further relevant source mechanisms can easily be added). Advection of the runaway electrons is considered primarily along field lines, but also taking into account the E × B drift. The model is implemented in the non-linear magnetohydrodynamic (MHD) code JOREK based on Bezier finite-elements, with current-coupling to the thermal plasma. Benchmarking of the code with the one-dimensional runaway electron code GO is done using an artificial thermal quench on a circular plasma. As a first demonstration, the code is applied to the problem of an axisymmetric cold vertical displacement event in an ITER plasma, revealing significantly different dynamics between cases computed with and without runaway electrons. Though it is not yet feasible to achieve fully realistic runaway electron velocities close to the speed of light in complete simulations of slowly evolving plasma instabilities, the code is demostrated to be suitable to study various kinds of MHD-RE interaction in MHD-active and disruption relevant plasmas.

Experimentally it is observed that TAE modes are difficult to excite with an external antenna when the plasma is in x-point geometry. Here, the effect of the X-point geometry on the efficiency of the TAE excitation with the external antenna is investigated. In the first part of the paper the influence of the near-LCFS (Last Closed Flux Surface) layer from the core side on the damping of the TAE modes is calculated using the linear resistive eigenvalue MHD code CASTOR. The resistive damping is identified as the main cause of the TAE damping in the open gap in the Alfven continuum. It is shown that one aspect of the difficulty of excitation of the TAE modes in X-point geometry is an increased damping from the region inside the separatrix. However, the increased damping with the plasma boundary approaching the separatrix is not general, and depends on the density profile shape. The second part of the paper discusses the influence of the TAE behaviour in the limiter and X-point geometries including the scrape-off layer (SOL) in the reduced MHD code JOREK. It is shown that the dominant effect on the damping of the original TAE mode observed in the limiter configuration is caused by the additional damping in the region of open field lines, i.e. the SOL.

Magnetic triggering of Edge Localized Modes (ELMs) in ohmic H-mode plasmas was first reported in the TCV tokamak [1]. This method, showing reliable locking of the ELM frequency to an imposed axisymmetric vertical plasma oscillation, was also demonstrated in the ITER-relevant type-I ELM regime in ASDEX Upgrade [2] and JET [3]. However, the mechanisms of the ELM triggering due to a vertical motion has not been studied extensively. The non-linear reduced MHD code JOREK-STARWALL has been extended for 3D free-boundary computations [4], which has allowed us to simulate for the first time realistic vertical oscillations together with ELM simulations in a single consistent scheme. Our simulations demonstrate that stable plasmas can be destabilized by the application of a vertical oscillation. During the vertical motion, a toroidal current is induced in the pedestal. The origin of this current is analysed in detail with the use of simulations and a simple analytical model, revealing that it arises from the compression of the plasma cross section due to its motion through an inhomogeneous magnetic field. Initially lower pedestal currents require bigger vertical displacements to destabilize ELMs, which directly points towards the increased edge current as the ELM driving mechanism. Finally the ELM triggering shows a very weak dependence on the plasma velocity in agreement with experiments.

Edge localized modes (ELMs) are repetitive instabilities driven by the large pressure gradients and current densities in the edge of H‐mode plasmas. Type‐I ELMs lead to a fast collapse of the H‐mode pedestal within several hundred microseconds to a few milliseconds. Localized transient heat fluxes to divertor targets are expected to exceed tolerable limits for ITER, requiring advanced insights into ELM physics and applicable mitigation methods. This paper describes how non‐linear magneto‐hydrodynamic (MHD) simulations can contribute to this effort. The JOREK code is introduced, which allows the study of large‐scale plasma instabilities in tokamak X‐point plasmas covering the main plasma, the scrape‐off layer, and the divertor region with its finite element grid. We review key physics relevant for type‐I ELMs and show to what extent JOREK simulations agree with experiments and help reveal the underlying mechanisms. Simulations and experimental findings are compared in many respects for type‐I ELMs in ASDEX Upgrade. The role of plasma flows and non‐linear mode coupling for the spatial and temporal structure of ELMs is emphasized, and the loss mechanisms are discussed. An overview of recent ELM‐related research using JOREK is given, including ELM crashes, ELM‐free regimes, ELM pacing by pellets and magnetic kicks, and mitigation or suppression by resonant magnetic perturbation coils (RMPs). Simulations of ELMs and ELM control methods agree in many respects with experimental observations from various tokamak experiments. On this basis, predictive simulations become more and more feasible. A brief outlook is given, showing the main priorities for further research in the field of ELM physics and further developments necessary.

3D non-linear magnetohydrodynamic simulations of a disruption triggered by a massive injection of argon gas in JET are performed with the JOREK code. The key role of the thermal drive of the m = 2, n = 1 tearing mode (i.e. the drive from helical cooling inside the island) in the disruption process is highlighted by varying the amplitude and position of the argon source across simulations, and also during a simulation. In cases where this drive persists in spite of the development of magnetic stochasticity, which is favoured by moving the argon source in an ad hoc way from the plasma edge into the 2/1 island at some point in the simulation, a relaxation in the region q ⩽ 2 (roughly) takes place. This relaxation generates a plasma current spike comparable to the experimental one. Simulations are compared in detail to measurements via synthetic diagnostics, validating the model to some degree.