## Section: New Results

### Full wave modeling of lower hybrid current drive in tokamaks

Participants : Pierre Bertrand, Simon Labrunie, Takashi Hattori, Jean Rodolphe Roche.

This work is performed in collaboration with Yves Peysson (DRFC, CEA Cadarrache).

The aim of this project is to develop a finite element numerical method for the full-wave simulation of electromagnetic wave propagation in aplasma. Full-wave calculations of the LH wave propagation is a challenging issue because of the short wave length with respect to the machine size. In the continuation of the works previously led in cylindrical geometry, a full toroidal description for an arbitrary poloidal cross-section of the plasma has been developed.

Since its wavelength $\lambda $ at the LH frequency is very small compared to the machine size $R$, a conventional full wave description represents a considerable numerical effort. Therefore, the problem is addressed by an appropriate mathematical finite element technique, which incorporates naturally parallel processing capabilities. This is particularly important aspect when simulations for plasmas of large size must be considered. It is based on a mixed augmented variational (weak) formulation taking account of the divergence constraint and essential boundary conditions, which provides an original and efficient scheme to describe in a global manner both propagation and absorption of electromagnetic waves in plasmas.

With such a description, usual limitations of the conventional ray tracing related to the approximation $\lambda <<{\phi}_{B}<<R$, where ${\phi}_{B}$ is the size of the beam transverse to the rf power flow direction, may be overcome. Since conditions are corresponding to $\lambda <<{\phi}_{B}\sim R$, the code under development may be considered as a WKB full wave, dielectric properties being local.

The domain considered is as near as possible of the cavity filled by a tokomak plasma. Toroidal coordinates are introduced. In our approach we consider Fourier decomposition in the angular coordinate to obtain stationary Maxwell equations in a cross-section of the tokamak cavity.

A finite element method is proposed for the simulation of time-harmonic electromagnetic waves in a plasma, which is an anisotropic medium. The approach chosen here is sometimes referred to as *full-wave modeling* in the literature: the original Maxwell's equations are used to obtain a second order equation for the time-harmonic electric field. These are written in a weak form using an augmented variational formulation (AVF), which takes into account the divergence. The variational formulation is then discretized using modified Taylor-Hood (nodal) elements.

During 2011 we introduced a new boundary condition in order to take account of the antenna and essential condition are considered in the code "FullWaveFEM" and new real case was considered.