Topic/Type: 1.2 Fusion Plasmas (magnetic & inertial confinement), Poster

### Numerical Aspects of a code to solve the drift kinetic equations for electrons and ions in general toroidal geometries

J.M. Reynolds1, D. L?pez-Bruna2

1 BIFI (Instituto de F?sica de Sistemas Complejos), Zaragoza, Spain
2 Laboratio Nacional de Fusion-CIEMAT, Madrid, Spain

A new code has been developed for solving the drift-kinetic equation for ion and electron species in complex geometries [1,2]. Presently, the code is able to reach stationary states with fixed electric and magnetic fields. The magnetic geometries go from the simplicity of an axisymmetric tokamak to the complexity of the TJ-II Heliac [3]. Good agreement has been found with neoclassical theory in simple geometries. This code is one the very few [4] capable of studying neoclassical stationary states with full five-dimensional distribution function, which allows to look for non-averaging properties of transport in complex machines. Other capabilities are: the possible inclusion of magnetic islands (no magnetic nesting is assumed); and the design itself, which should permit the inclusion of self-consistent electric fields (electrostatic turbulence), complex interactions with the
energy sources etc, without essential changes in the numerical procedures.

In this work, from a computational point of view, some of the characteristics of the code are shown. The kinetic equation evolves in a five-dimensional space taking into account drift terms and collisions through a conservative linearized collision operator [5]. The expansion of the distribution function is split in spatial and velocity parts. In the spatial part, the Spectral Difference Method [6] is used because of its good parallelization properties, ease of implementation and conservation of convected quantities. The code includes a tool for generating meshes adapted to every magnetic toroidal configuration. In velocity space, the distribution function is expanded in modes using Laguerre-Legendre polynomials. The expansion requires a normalization velocity that should be near the thermal velocity to obtain an optimum projection over the polinomial basis. This normalization velocity does not evolve in time. In this way, if the electric and magnetic fields are stationary, the beneficial properties of the resulting linear kinetic equation are not lost due to the projection in the basis. If needed, after some time evolution, the basis can be adapted to the new temperature in every point. Obviously, this eliminates typical non linear convective terms like $\small \mathbf{v} \cdot \nabla \mathbf{v}$ and their related problems. Additionally, the method allows for discontinuous normalization velocity between elements of the spatial mesh: the discontinuities are taken into account through a modified Roe solver [7] to maintain the stability of the SDM and, moreover, to help in keeping strict conservation properties. The time stepping can be done explicit or implicit. The algebraic system in the implicit case is solved with an iterative parallelized Krilov method.

[1] J. M. Reynolds,a D. Lopez-Bruna, J. Guasp, J. L. Velasco, A. Tarancon, A new code for collisional drift kinetic equation solving, AIP Conf. Proc., 1071, 22, (2008).

[2] J. M. Reynolds, D. Lopez-Bruna, J. Guasp, J. L. Velasco, A. Tarancon, Simulating drift-kinetic electron-ion equation with collisions in complex geometry, 35th EPS Conference on Plasma Physics, ECA Vol. 32, P2.049. Hersonissos, Crete, Greece 9-13 june (2008).

[3] A. H. Boozer et al., Plasma Physics and Controlled Nuclear Fusion Research, Proc. 9th Int. Conf. Baltimore, 1982. Vol. III, p. 129, IAEA, Vienna (1983).

[4] E.A. Belli, J. Candy, Kinetik calculation of neoclassical transport including self-consistent electron and impurity dynamics Plasma Phys. Control. Fusion 50, 095010, (2008).

[5] J. Y. Ji, E. D. Held, Exact linearized Coulomb collision operator in the moment expansion, Phys. Plasmas 13, 102103 (2006).

[6] Y. Liu, M. Vinokur, Z. J. Wang, Discontinuous spectral difference method for conservation laws on unstructured grids, Proceedings of the 3rd International Conference on Computational Fluids Dynamics, Toronto, Canada, July 12-16, (2004).

[7] P. L. Roe, Approximate Riemann solvers, parameter vectors, and diference schemes, J. Comp. Phys. 43, 357 (1981).