# EUTERPE

The EUTERPE gyrokinetic code was created at the CRPP in Lausanne as a global linear particle in cell code for studying electrostatic plasma instabilities [1]. It allows three-dimensional turbulence simulations using a plasma equilibrium calculated with the VMEC code as a starting point. EUTERPE was further developed at the Max Planck IPP and several linear calculations of ion temperature gradient (ITG) driven turbulence in tokamak and stellarator geometry have been carried out using it [2] [3] [4] [5]. Afterwards, the code has been heavily optimized and improved. The perturbation to the magnetic field, a third species (in adition to electrons and ions) and the non-linear dynamics have been included.

The EUTERPE code solves the gyroaveraged Vlasov equation for the distribution function of ions

$\frac{\partial f}{\partial t} + \frac{\rm{d}v_{||}}{\rm{d}t} \frac{\partial f}{\partial v_{||}} + \frac{\rm{d}\vec{R}}{\rm{d}t} \frac{\partial f}{\partial \vec{R}} = 0$

The code is based on the particle-in-cell (PIC) scheme, where the distribution function is discretized using markers. The δf approximation is used, so that the distribution function is decomposed in an equilibrium part (Maxwellian) and a time-dependent perturbation.

$f(\vec R, v_{||}, \mu, t) = f_{0}(\vec R, v_{||}, v_{\perp})+ \delta f(\vec R, v_{||}, \mu, t)$

Each marker along with its weight is evolved following the particle trayectories and contributes a part to the distribution function, so that

$\delta f = \sum_{p=1} ^{N} w_p \delta ^{3}(\vec R - \vec R_p)\delta(v_{||} - v_{||p})\delta(\mu - \mu_p) /(2 \pi B),$

where the $w_p$ are the weights (contribution to the distribution function) associated to each marker.

The electric potential is represented on a spatial grid, the electric charge being carried by the markers. Two coordinate systems are used in the code: a system of magnetic coordinates (PEST) $(s, \theta,\phi )$ is used for the electrostatic potential and cylindrical coordinates $(r, z,\phi )$ are used for pushing the particles, where $s=\Psi / \Psi_0$ is the normalized toroidal flux. The change between coordinate systems, which is facilitated by the existence of the common coordinate $(\phi)$, is done in a continuous way. The equation for the field is discretized using finite elements (B-splines) and the PETSc library is used for solving it. The integration of the motion is done using a fourth order Runge-Kutta scheme. In linear simulations a phase factor transformation can be used and the equations can be integrated using a predictor-corrector scheme.

An equilibrium state calculated with the code VMEC is used as a starting point. The equilibrium quantities computed by VMEC are mapped onto the spatial grid using an intermediate program.

EUTERPE features several techniques for the noise control: the filtering of Fourier modes (square and diagonal filters can be used) and the optimized loading [6].

## References

1. G. Jost, T. M. Tran, K. Appert, W. A. Cooper, and L. Villard, in Theory of Fusion Plasmas, International Workshop, Varenna, September 1998 (Editrice Compositori, Società Italiana di Fisica, Bologna, 1999), p. 419.
2. G. Jost, T. M. Tran, W. Cooper, and K. Appert. Phys. Plasmas 8: 3321 (2001)
3. V. Kornilov, R. Kleiber, R. Hatzky, L. Villard, and G. Jost. Phys. Plasmas 11: 3196 (2004)
4. V. Kornilov, R. Kleiber, and R. Hatzky, Nucl. Fusion 45: 238 (2005)
5. R. Kleiber, Global linear gyrokinetic simulations for stellarator and axisymmetric equilibria, Joint Varenna-Lausanne International Workshop. AIP Conference Proceedings, 871, p. 136, 2006
6. Hatzky, R Tran, TM Konies, A Kleiber, R Allfrey, SJ .Energy conservation in a nonlinear gyrokinetic particle-in-cell code for ion-temperature-gradient-driven modes in theta-pinch geometry. Phys. Plasmas, 9- 3, p. 898, 2002.