Numerical strategies for MHD instabilities: context of magnetic confinement fusion

Abstract: The next step large-scale tokamak ITER device (International Thermonuclear Experimental Reactor), which is under construction in Cadarache in France, has the main goal to study physics of burning D-T plasma magnetically confined in tokamak device. ITER should prove that exploitation of controlled fusion energy as a resource of electrical power is conceivable and that fusion power plants could be achievable before the end of the 21st century. If ITER reaches its goals and demonstrates that fusion may be used to produce economically exploitable energy, then next step DEMO will be the first prototype of what could later be a fusion reactor. Plasma confinement in H-mode is considerably improved (“H” for high confinement) compared to a low confinement L-mode plasma typically observed below L/H transition power threshold. H-mode is one of the main baseline ITER scenarios, where in particular a fusion power gain Q=10 can be achieved. Note also that the maximum fusion plasma performance in terms of maximum plasma current, normalized plasma pressure (“beta”) and pressure gradient is limited in many cases by large-scale MHD instabilities. Hence the achievable fusion power gain in ITER (and future fusion reactor DEMO) even in H-mode will depend on these operational limits ruled by MHD activity. A large edge pressure gradient accompanied by a large edge current density (so-called “bootstrap” current) are the typical conditions met in an H-mode, and are particular prone to MHD activity. The pressure gradient and the current density are the two driving forces of the so-called Edge Localized Modes (ELMs), resulting from a coupling between kink (or peeling) modes driven by current and ballooning MHD modes driven by pressure gradient. During their non-linear evolution, ELMs evacuate very rapidly energy and particles through the separatrix, so that the pressure gradient decreases and ballooning/peeling modes become stable again.

With this respect, the non-linear MHD theory can provide further physical and numerical improvements to refine knowledge of basic ELM dynamics and related ELM control techniques. In view of particular complexity of the problem the numerical simulations provide a key component of this effort, since computations are performed at a tiny fraction of the cost of an experiment. To be effective, simulations are combined with a well-focused, well-diagnosed experimental physics program to guide model development, and to assure the validity of numerical results. Recent developments in plasma theory, computational physics, and computer science, along with anticipated advances in computer hardware performance, combine to make this simulation capability of non-linear MHD codes a very attractive option to study ELM physics.

We will focus on the modeling and numerical strategies capability for non-linear MHD simulations of ELMs. Numerical stabilization compatible with the physical properties is the main challenging issue. Some well established Finites elements codes as XTOR, NIMROD and M3D, use stabilization techniques based on Taylor-Galerkin methods. This is a family of high-order time-stepping schemes with stabilization of convective term by the means of intrinsic streamline diffusion. Taylor-Galerkin formulation is a class of more general sub grid scale concept. This strategy has been widely developed and recently applied for incompressible MHD. The main idea is to split the continuous solution of the problem in two components, the weak solution at a given scale and the subscales or subgrid scales, which are the part of the solution that cannot be captured by the discretization. In this situation, the problem is reduced to obtain a good approximation for the subscales. Many of these stabilizations depend on arbitrary algorithmic parameters, which need some tuning in order to achieve the best accuracy possible. We will present optimized strategies with applications to JET, ASDEC, DIID and ITER tokomaks geometries.