## Section: New Results

### Plasma physics

#### Analysis of the drift approximation

Participants : Hervé Guillard, Afeintou Sangam, Philippe Ghendrih [ IRFM, CEA Cadarache ] , Yanick Sarazin [ IRFM, CEA Cadarache ] .

Drift approximation consider the slow evolution of the fields in the vicinity of a tokamak equilibrium. These models are typically used to study the micro-instabilities that are believed to be responsible of turbulent transport in tokamaks. Since the drift asymptotic use a “slow” scaling of the velocity field, the resulting models are significantly different from the MDH models. This is particularly true with respect to the computation of the electric field that is given by an Ohm's law in MDH models whereas it is computed by a vorticity-like evolution equation in drift approximations. Drift asymptotics models are extremely interesting from a computational point of view since they save substancial CPU time and computer memory. However, the mathematical and numerical properties of these models are essentially unknown. We have begun a detailed study of the derivation of these models from two-fluid Braginskii-type models in order to establish the range of applicability of these asymptotic models, understand their mathematical properties and design appropriate numerical methods for them.

#### Anisotropic heat Diffusion

Participants : Audrey Bonnement, Hervé Guillard, Richard Pasquetti.

Magnetized plasmas are characterized by extremely anisotropic properties relative to the direction of the magnetic field. Perpendicular motions of charged particles are constrained by the Lorentz force, while relatively unrestrained parallel motions lead to rapid transport along magnetic field lines. Heat transport models (e.g [34] are therefore characterized by an extreme anisotropy of the transport coefficient that differ by several order of magnitude in the parallel and perpendicular directions. The use of field aligned coordinates that essentially reduce the problem to one-dimension is one way to overcome this difficulty. However, for complex or unsteady magnetic configurations or for problems requiring a high degree of geometrical realism, this approach leads to serious numerical difficulties. An alternative is to use a numerical representation that has a high rate of spatial convergence like high order finite element methods or spectral elements. We have begun a numerical study to compare and evaluate the two approaches.

#### Stationary constraints in MHD

Participants : Hervé Guillard, Boniface Nkonga.

Although this is an old problem [33] , there are still unresolved arguments about how one should maintain the divergence-free property of the magnetic field in multidimensional MHD calculations. Numerically, in many computations, the divergence of the magnetic field is not exactly zero and terms proportional to this term acting as extra non-physical forces are not damped. This leads to inconsistent results and even eventually to the algorithm breakdown. Many different remedies have been proposed. However at the present time, there is no definite winner and the question is still open. In this direction, we have studied two different possibilities : first a variant of Powell's method that adds a non-conservative term proportional to div B to control the divergence and second in the framework of a M2 master internship, two different Hodge projection schemes. At present, from a computational point of view, the Hodge projection schemes are too costly (they require at each time step the solution of an elliptic equation).This study has to be completed by additional work to find more efficient linear solvers in this context.