Linear and non-linear simulations using the EUTERPE gyrokinetic code Edilberto S´ anchez * , Ralf Kleiber † , Roman Hatzky ‡ , Alejandro Soba § , Xavier S´ aez § , Francisco Castej´ on * and Jose M. Cela § * Laboratorio Nacional de Fusi´ on. Avda Complutense 22, 28040 Madrid, Spain. Email: edi.sanchez@ciemat.es † Max-Planck Institut f¨ ur Plasmaphysik. EURATOM-Association, Wendelsteinstraße 1, 17491 Greifswald, Germany ‡ Max-Planck Institut f¨ ur Plasmaphysik. EURATOM-Association, Boltzmannstraße 2, 85748 Garching, Germany § Barcelona Supercomputing Center (BSC-CNS). Edificio C6-E201, Jordi Girona 1-3, 08034 Barcelona, Spain Abstract—In the present work we report on simulations recently carried out using the EUTERPE gyrokinetic code. The scaling of the code has been studied up to twenty thousand processing elements. Linear and non-linear simulations of Ion Temperature Gradient (ITG) instabilities have been carried out in screw-pinch geometry and the results are compared to those previously obtained using the TORB code finding a good agreement. The influence of a finite beta on the growth rates of instabilities and on the zonal flows in a screw-pinch has also been studied. The results are compared with previous ones. I. I NTRODUCTION The increase in the computational resources available in the last years has allowed the development of many codes based on the well established gyrokinetic framework [1], [2] for the simulation of turbulent fusion plasmas. There are codes that follow an Eulerian perspective [3]–[6], a Lagrangian one [7]–[15] and based on a semi-Lagrangian formalism [16]. The huge amount of computation resources required for a relevant simulation makes it only possible to carry out local simulations in some cases. An advantage of the Lagrangian codes is the ability to perform global simulations with moderate compu- tational resources, particularly in the case of the codes that use the delta-f approximation [17], that allows reducing the numerical noise originated in the discretization of the phase space by particles. Most of the gyrokinetic simulations have focused on the tokamak geometry, using either ad hoc simplified models or realistic tokamak equilibria. Use is made in many codes of the tokamak symmetry. An effort is needed for developing codes that are valid for stellarators, where the three dimensional effects can be crucial. The use of a finite number of coils in tokamaks, as well as the implementation of ergodic divertors, can also lead to important three dimensional effects that should be properly taken into account. The EUTERPE gyrokinetic code was created at the CRPP in Lausanne as a global linear particle in cell code for studying electrostatic plasma instabilities in arbitrary three-dimensional geometry [15]. It allows simulations using a plasma equilib- rium calculated with the VMEC code [18] as a starting point. The code was further developed at the Max Planck IPP and several linear calculations of ion temperature gradient (ITG) driven modes in tokamak and stellarator geometry have been carried out using it [19]–[21]. It has been afterwards heavily optimized and improved and the non-linear dynamics have been included. A kinetic treatment of the electron dynamics and of a third species have also been included in the recent times and work is in progress to include collisions. In what concerns the present work, only electrostatic fluctuations are taken into account and the electrons are considered to be adiabatic. The main purpose of this work is to benchmark EUTERPE against the TORB code [13] by comparing the results of linear and non-linear simulations with previous ones obtained with that code. The rest of the paper is distributed as follows. In section II the code, the equations that it solves and the basics of the numeric model are introduced. The results of the scaling studies are presented in section III. In section IV the results of linear simulations of ITGs are presented and compared to previous ones. In section V the results of the non-linear ITG simulations are presented and compared to previous ones. In sections V-A and V-B the influence of the radial resolution and finite beta on non-linear simulations of ITG are presented. Finally, in section VI some conclusions are drawn. II. THE EUTERPE CODE The EUTERPE code solves the gyroaveraged Vlasov equa- tion for the distribution function of ions ∂f ∂t + dv || dt ∂f ∂v || + d  R dt ∂f ∂  R =0 (1) assuming the equations of motion for the ions being given, in the electrostatic approximation, by: d  R dt = v ‖ e B + μB + v 2 ‖ B ∗ Ω i e B ×∇B + (2) + v 2 ‖ B ∗ Ω i (∇× B) ⊥ − ∇〈φ〉 B ∗ × e B dv ‖ dt = −μ  e B + v ‖ B ∗ Ω i (∇× B) ⊥  ∇B − (3) − q i m i  e B + v ‖ B ∗ Ω i [ e B ×∇B +(∇× B) ⊥ ]  ∇〈φ〉 dμ dt = 0 (4)