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.

The shear shallow water model provides a higher order approximation for shallow water flows by including the effect of vertical shear in the model. This model can be derived from the depth averaging process by including the second order velocity fluctuations which are neglected in the classical shallow water approximation. The resulting model has a non-conservative structure which resembles the 10-moment equations from gas dynamics. This structure facilitates the development of path conservative schemes and we construct HLL, 3-wave and 5-wave HLLC-type solvers. An explicit and semi-implicit MUSCL-Hancock type second order scheme is proposed for the time integration. Several test cases including roll waves show the performance of the proposed modeling and numerical strategy. Figure 1: Shallow water approximation: The free surface is given by x 3 = ξ(x 1 , x 2 , t) and the bottom surface is given by(acoustic) and b-waves (shear), and develops an approximate Riemann solver for each one independently. Each of these sub-systems is also augmented with the energy conservation equation (3) which is used to derive some jump conditions required to develop the Riemann solvers. The approach in [4] also uses the same acoustic and shear sub-systems and develops fluctuation splitting schemes for each sub-system on unstructured grids, but does not make use of the total energy equation (3).In the present work, we cast the SSW equations in a particular non-conservative form which is similar to the 10-moment equations [15, 2] from gas dynamics. In this model, instead of equations for the stress P, we have equations for an energy tensor E, while the mass and momentum equations remain unchanged. This form of the equations naturally arises when we perform the depth averaging of the 3-D Euler equations and the derivation is given in Appendix A. In fact, the equation for the energy tensor appears in [19] but it has not been used by any of the researchers to develop a numerical approximation. We suggest that the form of the equations is important and hence we retain the equation structure arising from depth averaging to build a numerical approximation. The non-conservative terms in this form contain only derivatives of the water depth h unlike model (2) which has derivatives of v, P in the non-conservative terms. The presence of only the derivatives of h in the non-conservative terms facilitates the construction of path conservative schemes [8]. By using the generalized Rankine-Hugoniot (RH) jump conditions arising from taking a linear path in the state space, we build HLL-type Riemann solvers for the new system. We construct the HLL, a 3-wave HLLC and a 5-wave HLLC solver, with the last one including all the waves in the Riemann problem. Unlike previous works, we do not split the model in several sub-systems but instead we construct a unified Riemann solver for the full system. A higher order version of the scheme is constructed following the MUSCL-Hancock approach [21] where we make the source terms implicit. The resulting semi-implici...

Non-linear magnetohydrodynamic (MHD) simulations play an essential role in active research and understanding of tokamak plasmas for the realization of a fusion power plant. The development of MHD codes such as JOREK is a key aspect of this research effort. In this paper, we present an operational version of the full-MHD model implemented in JOREK, a significant advancement from the reduced-MHD model used for previous studies, where assumptions were made on the perpendicular dynamics and the toroidal magnetic field. The final model is presented in detail, and benchmarks are performed using both linear and non-linear simulations, including comparisons between the new full-MHD model of JOREK and the previously extensively studied reduced-MHD model, as well as results from the linear full-MHD code CASTOR3D. For the cases presented, this new JOREK full-MHD model is numerically and physically reliable, even without the use of numerical stabilization methods. Non-linear modeling results of typical tokamak instabilities are presented, including disruption and edge-localized-mode physics, most relevant to current open issues concerning future tokamaks such as ITER and DEMO.