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

### On deformation-free geometry for turbulence computations

T.T. Ribeiro1, 2, B. Scott1

1 Max-Planck-Institut fuer Plasmaphysik, Garching, Germany
2 IPFN, Instituto Superior T?cnico, Lisboa, Portugal

Turbulence is a fundamental problem in magnetically confined plasmas. Physical understanding of the resulting transport properties relies upon large scale direct numerical simulation efforts. Turbulence in a magnetised plasma, with particle gyroradii small compared to the device size, consists of two dimensional, dynamically incompressible turbulence of $\small E\times B$ flows in the plane perpendicular to the magnetic field, and compressible, electromagnetic electron dynamics parallel to it. A space scale separation between perpendicular and parallel dynamics results. Natural coordinates are therefore those which follow the field lines in a Clebsch representation in either closed or open magnetic flux surfaces. Severe deformation of the coordinate metric results unless several countermeasures are taken. We describe the mathematics of restoring local orthogonality by describing a globally-valid field-aligned system which is orthogonal on each perpendicular plane. In simplified geometry (circular, high-aspect ratio torii) the problem has been solved simply by this re-mapping from plane to plane. However, in shaped geometry (non-circular, intermediate aspect-ratio torii) the Clebsch representation is necessarily problematic, as the perpendicular coordinate volume element is more slowly varying along a field line than the metric elements themselves (flux expansion: the variation of inter-surface distance with poloidal position). Turbulence, on the other hand, tends toward isotropicity at small scale. We describe a solution for this problem in terms of conformal coordinates in the poloidal plane. The ratio of the metric elements is made to vary slowly with position along the field line and barring exceptional cases is always within 50 percent of unity. The parallel direction is still treated with straight field line coordinates which can be cast in terms of a Clebsch representation. The ability to map between both coordinates using shifts in the angles in a one-to-one and onto way allows the perpendicular and parallel dynamics to interact in a most natural way, providing for maximally efficient computation. A side benefit is that the conformal property enables the simplest and fastest form of multigrid treatments in solving the elliptic field equations that are part of the model. We present the details of this treatment and its implementation together with some preliminary results of turbulence computations.