Section: New Results
Application of Vlasov codes to magnetic fusion
Participants : Pierre Bertrand, Nicolas Besse, Jean-Philippe Braeunig, Nicolas Crouseilles, Daniele DelSarto, Nicolas Dubuit, M Elmounden, Etienne Gravier, Rudy Klein, Guillaume Latu, Jean Roche, Eric Sonnendrücker.
The computation of turbulent thermal diffusivity in fusion plasmas is of prime importance since the energy confinement is determined by these transport coefficients. Fusion plasmas are thus prompt to instabilities and require a self-consistent analysis of the plasmas and the electromagnetic fields. Since collisions between particles play a negligible role, the kinetic plasma behavior is described by the well-known Vlasov model. A key issue is the resonant interaction between particles and waves, which has to be accurately described. The self consistent coupling of the electromagnetic fields then exhibits resonant (Landau) interactions between charged particles and electromagnetic waves, a feature that cannot be addressed in the fluid limit.
Development of a 5D gyrokinetic code
The collaboration around the optimization of the GYSELA code used for gyrokinetic simulations of turbulence in tokamaks went on. The upgrade from four to five dimensions of the phase space is now efficient on several thousands of processors. Last year, several developments of the code enabled to achieve this task.
In the frame of gyrokinetic models in Magnetic Confinement Fusion, we are interested in simulation of plasma motion with a strong anisotropy in the velocity field. This situation occurs in tokamaks and ITER in particular, where the magnetic field is so strong that the particles are mostly following magnetic field lines. Parallel (to the magnetic field lines) velocity and perpendicular velocities are then of different magnitudes and should be treated specifically. Therefore, considering that the magnetic field is constant, a gyrokinetic model written in curvilinear coordinates is currently investigated, which should allow to make simulations using a mesh aligned to the magnetic field lines. Such a mesh should reduce significantly numerical diffusion in magnetic lines direction. Early tests using a simple model show that numerical diffusion is reduced even for not strictly aligned meshes. However, this technique introduces some difficulties on boundary conditions. In fact, the physical location of domain boundaries could be complex curves and for periodic boundary conditions, one has to find values on one side of the domain to imposes equal values on the other side. If a curvilinear cartesian mesh is not aligned with the periodic direction, one has to find values at boundaries taking into account the lag between mesh lines and periodic direction. Curvilinear coordinates impose also to work on data post-treatment, adapting visualization tools to this kind of meshes.
We investigate a modification of the scheme in order to have field aligned coordinates. This feature, used by many other gyrokinetic codes, will improve the numerical quality of the simulations. Nevertheless the numerical method and the parallelization should be deeply revised.
In the Gyro-Water-Bag (GWB) approach, a discrete distribution function taking the form of a multi-step like function is used in place of the continuous distribution function along the velocity direction. According to Liouville's phase-space conservation property the distribution function remains constant in time between the bag contours. The time evolution of the system is completely described by the knowledge of the contours. We get a set of hydrodynamic equations, where the system behaves as N fluids coupled together by the electromagnetic fields (in our case the quasi-neutrality). As a matter of fact a small bag number (not more than 10) has been shown to be sufficient to correctly describe the Ion Temperature Gradient (ITG) instability observed in fusion plasmas (see  ). Thus the water-bag offers an exact description of the plasma dynamics even with a small bag number, allowing more analytical studies and bringing the link between the hydrodynamic description and the full Vlasov one. First very encouraging results have been obtained with a 3-D code in cylindrical geometry and for electrons following the adiabatic law (see  ) based on discontinuous-Galerkin type methods.
Trapped-ion driven turbulence (4D model)
The work focuses on trapped-ion driven modes (TIM), which belong to the family of ion gradient driven modes. Theses instabilities are characterized by frequencies of the order of the trapped precession frequency and radial scales of the order of several banana orbits. Trapped-ion driven modes are a prototype of kinetic instability since they are driven through the resonant interaction between a wave and trapped ions via their precession motion. Averaging the kinetic equation over both cyclotron and bounce motions allows the number of independent variables in phase space to be reduced. The new Vlasov equation function depends on only two variables (precession angle and poloidal flux), the final problem is 2D, parametrized by the particle energy and trapping parameter, allowing for an efficient parallelization of the code. The goal is here to take into account the effects of the magnetic shear on the stabilization of the instability for an ensemble of values of the trapping parameter. The development (and optimization) of a 4D Vlasov code on a parallel computer is thus in progress. The step is necessary to check the validity and the accuracy of this method of reduction based on action-angle formalism. First numerical results of the code implemented on a parallel IBM-Power4 supercomputer show a very good stability of the numerical scheme taking into account a marginally stable initial condition. This very good stability of the Vlasov method (which is noiseless in comparison with the standard Particles-In-Cell codes) allows us to control the start up of the ITG instability (which grows from the round-off errors of the computer).
A key element in the understanding of the role played by trapped ions in the turbulence may be obtained by making numerical simulations using the full-gyrokinetic GYSELA code (which describes of course trapped and passing particles) and the reduced action-angle code which indeed takes into account only TIM. A major goal in the future is to implement the closed-loop strategy combining the implementation of reduced models (which are faster and cheaper than the full gyrokinetic treatment) with a series of numerical comparisons using the GYSELA 5D code. The existing cooperation with teams of CEA makes this a realistic objective on a long time (more than one year) with potentially large overall impact.
Full wave modeling of lower hybrid current drive in tokamaks
This work is performed in collaboration with Yves Peysson (DRFC, CEA Cadarache). The goal of this work is to develop a full wave method to describe the dynamics of lower hybrid current drive problem in tokamaks. The wave dynamics may be accurately described in the cold plasma approximation, which supports two independent modes of propagation, the slow wave which corresponds to a cold electrostatic plasma wave, and the fast wave, namely the whistler mode. Because of the simultaneous presence of the slow and fast propagation branches a vectorial wave equation must be solved. The wave equation is obtained from the Maxwell equations with a time harmonic approximation. We consider a toroidal formulation of the Maxwell equations.
We have developed a P1 finite element method (FEM) in the spirit of F. Assous et al. method, which is based on a mixed augmented variational formulation (MAVF) of the problem. We have written a Matlab code for the method, which gives correct results in academic examples. We develop a new version introducing domain decomposition techniques and source terms.