An improved hybrid electron model for global gyrokinetic simulations using the ORB5 PIC code

Gyrokinetic simulations are extensively used as a tool to describe various phenomena in magnetically confined plasmas. Among others, turbulent transport induced by drift-wave type instabilities such as Ion Temperature Gradient (ITG), Trapped Electron Mode (TEM) and Electron Temperature Gradient (ETG) plays a central role in fusion research. While the ITG mode in first approximations can be modeled using a reduced adiabatic electron model, the TEM and ETG modes require a kinetic description. The global delta-f PIC code ORB5 [1] currently implements three electron models: in the first, all the electrons are treated adiabatically, in the second they are all treated as kinetic and finally, in the so-called hybrid model the trapped electrons are treated as kinetic while the passing ones are considered to have an adiabatic response. Although the hybrid model is very useful to simulate electron driven modes such as the TEM at a lower numerical cost as compared to the fully kinetic electron model, in its standard form it does not ensure the ambipolarity condition and toroidal angular momentum conservation, which is clearly problematic for long nonlinear simulations. Recently, an improved hybrid electron model was proposed [2] to solve the ambipolarity problem while avoiding the electrostatic limit of the shear Alfvén mode (so-called ωH mode) that constrains the time step to small enough values so as to resolve this high frequency mode. This new model treats the trapped electrons as kinetic but separates the passing contribution into a zonal contribution that is treated kinetically and a non-zonal contribution that has an adiabatic response. Finally, all the zonal modes with a poloidal mode number m ≠ 0 are filtered out. Although this model clearly satisfies the ambipolarity condition, it affects the GAM frequency, which is problematic, for example for comparisons with the experiment [3]. In this work, we propose a further improved hybrid electron model based on [2]. This version considers the zonal contribution of the passing electrons as kinetic but contrarily to the reference, it also includes the zonal modes with m ≠ 0 as an adiabatic response. In addition to verifying the ambipolarity condition, it affects less the GAM frequency. Zonal flows damping (Rosenbluth-Hinton) tests are used to validate the linear regime of this improved hybrid electron model and comparison with other models and theory will be shown to assess the GAM properties. Nonlinear simulations confirm the ambipolarity condition and show a richer physics that will be discussed. In particular, the zonal flow dynamics is affected by the proximity of mode rational surfaces.

Development and application of Eulerian and Particle-in-Cell gyrokinetic codes for studying the effect of non-adiabatic passing electron dynamics on microturbulence

Julien Stanislas Pierre Dominski

In tokamak fusion plasmas, micro-turbulence transport is known to be the cause of large losses of heat and particles. The present work deals with the study of electrostatic micro-turbulence transport driven by instabilities of essentially two types: the ion temperature gradient (ITG) modes and the trapped electron modes (TEM). The plasma is described within the gyrokinetic framework, which permits to save computational resources compared to the classical Vlasov kinetic description. In gyrokinetic simulations of fusion plasmas, the passing electrons are often assumed fast enough so that they respond instantaneously to the electrostatic perturbations. In this case, their response is computed adiabatically instead of kinetically. The main advantage is that this simplified model for the electron response is less demanding in computational resources. This assumption is nonetheless incorrect, in particular near mode rational surfaces where the non-adiabatic response of passing electrons cannot be neglected. This thesis work focuses on the study of this passing electron non-adiabatic response, whose influence on microturbulence is studied by means of numerical simulations carried out with the gyrokinetic codes GENE and ORB5. In the first part of this thesis work, the response of passing electrons in ITG and TEM microturbulence regimes is studied by making use of the flux-tube version of the GENE code. Results are obtained using two different electron models, fully kinetic and hybrid. In the hybrid model, passing particles are forced to respond adiabatically while trapped are handled kinetically. Comparing linear eigenmodes obtained with these two models enables one to systematically isolate fine radial structures located at corresponding mode rational surfaces, clearly resulting from the non-adiabatic passing-electron response. Nonlinear simulations show that these fine structures on the non-axisymmetric modes survive in the turbulent phase. Furthermore, through nonlinear coupling to axisymmetric modes, they induce radial modulations in the effective profiles of density, ion and electron temperature and zonal flows

E \times B

shearing rate. Finally, the passing-electron channel is shown to significantly contribute to the transport levels, at least in our ITG case. Also shown is that the passing electrons significantly influence the

E \times B

saturation mechanism of turbulent fluxes. Following this study in flux tube geometry, the influence of the non-adiabatic passing electron response near mode rational surfaces is further studied in global geometry with the global gyrokinetic code ORB5, in which a new field solver is implemented for the gyrokinetic quasi-neutrality equation valid at arbitrary wavelength, overcoming the former long wavelength approximation made in the original version of the code. A benchmark is conducted against the global version of the gyrokinetic code GENE, showing very good agreement. Nonlinear simulations are carried out with the new solver in conditions relevant to the TCV tokamak, with the physical deuterium to electron mass ratio (

m_i/m_e=3672

) and are compared to simulations carried out with heavy electrons (

m_i/m_e=400

). The particular spectral organization of the passing electron turbulent flux and its dependence on the radial profile of the safety factor are revealed. In particular, the formation of short-scale transport barriers is studied near low-order mode rational surfaces. Results show that quantitatively correct nonlinear fully-kinetic simulations of tokamak transport must be carried out in a full torus and with the physical mass ratio.

EPFL2016

Collisions in Global Gyrokinetic Simulations of Tokamak Plasmas using the Delta-f Particle-In-Cell Approach

Thibaut Vernay

The present work takes place within the general context of research related to the development of nuclear fusion energy. More specifically, this thesis is mainly a numerical and physical contribution to the understanding of turbulence and associated transport phenomena occuring in tokamak plasmas, the most advanced and promising form of magnetically confined plasmas. The complexity of tokamak plasma phenomena and related physical models, either fluid or kinetic, requires the development of numerical codes to perform simulations of the plasma behaviour under given conditions defined by the magnetic geometry as well as density and temperature profiles. The studies presented in this work are based on electrostatic kinetic simulations, taking advantage of a reduced kinetic model (the gyrokinetic model) which is particularly suitable for studying turbulent transport in magnetically confined plasmas, in effect solving an approximate form of the Vlasov equation for the distribution function of each species (electrons, ions) along with a reduced form of the Poisson equation providing the self-consistent electric fields. The main tool of this work, the gyrokinetic ORB5 code making use of numerical particles according to the Particle-In-Cell (PIC) method, has been upgraded during this thesis with different linearized collision operators related to both ions and electrons. The BIRDIE code, enabling to study collisional effects on the evolution of Langmuir waves in an unmagnetized plasma, has been written in order to serve as a test-bed for the collision operators ultimately implemented in ORB5. Some essential algorithms related to collisional simulations have been jointly implemented, such as the two-weight scheme which is extensively described in this work. The collision operators in ORB5 have been further carefully tested through neoclassical simu- lations and benchmarked against other codes, providing reliable levels of collisional transport. Together with different procedures controlling the numerical noise, the collision operators have then been applied to the study of collisional turbulent transport in two different regimes, the Ion-Temperature-Gradient (ITG) regime and the Trapped-Electron-Mode (TEM) regime re- quiring a trapped electron kinetic response. Although not dominant in core tokamak plasmas, collisional effects nevertheless lead to interesting modifications in the turbulence behaviour which are not captured by the often considered collisionless gyrokinetic models. The so-called coarse-graining procedure, a noise-control algorithm which is suitable for collisional gyrokinetic simulations with particles, is shown to enable carrying out relevant simulations over many col- lision times. Consequently, reliable conclusions regarding turbulent transport in the presence of collisions could be drawn in this thesis. Namely, the turbulent transport in the ITG regime is found to be enhanced by ion collisions through interactions with so-called zonal flows as- sociated to axisymmetric modes, while it is reduced by electron collisions in the TEM regime through electron detrapping processes. The zonal flow dynamics in collisionless and collisional ITG turbulence simulations is studied, emphasizing the limitation of the zonal flow level due to Kelvin-Helmoltz-type instabilities. Additionally, some purely collisionless issues related to tokamak physics are discussed, such as the finite plasma size effects in TEM-dominated regime which are found to be important in non-linear simulations but unimportant in linear simu- lations. The role of zonal flows in temperature-gradient-driven TEM turbulence saturation is confirmed to be weak, in agreement with previous studies. Finally, a realistic global gy- rokinetic simulation, accounting for a proper TCV tokamak magnetic equilibrium and related experimental profiles, has been successfully carried out thus demonstrating the relevance of the ORB5 code for predictions related to physics of real tokamaks. A good agreement with GAM experimental measurements is indeed obtained.

EPFL2013

A gyrokinetic model for the plasma periphery of fusion devices

Rogério Manuel Cabete De Jesus Jorge, Baptiste Jimmy Frei, Paolo Ricci

Understanding the turbulent dynamics of the plasma in the periphery of fusion devices - the region extending from the external part of the closed flux surface region to the scrape-off layer - is of crucial importance on the way to fusion energy. Indeed, this region largely controls the plasma heat exhaust, the plasma refueling, the level of fusion ashes and, also, the low-to-high (L-H) mode transition. In the present talk, a gyrokinetic model is presented that can properly consider the main elements at play in the periphery. These include the presence of strong flows, large and small amplitude electromagnetic fluctuations on scale lengths ranging from the electron Larmor radius to the equilibrium perpendicular pressure gradient scale length, and large deviations from thermal equilibrium. Due to the large range of collisionality present across the periphery, the model retains a nonlinear full Coulomb collision operator cast in a form suitable for implementation in a gyrokinetic code. The gyrokinetic equation and the associated equations for the electromagnetic fields are valid for arbitrary deviations from thermal equilibrium and are accurate at arbitrary perpendicular wavenumber values. The formulation of the gyrokinetic model is based on the single gyrocenter dynamics obtained from fully nonlinear second-order accurate electromagnetic gyrokinetic equations of motion, derived from Lieperturbation theory where the fast particle gyromotion is decoupled from the slow drifts. Then, the collective behavior is obtained by the gyrokinetic equation including the collision operator. The gyrokinetic model takes the form of a set of coupled fluid equations referred to as the gyrokinetic moment (gyro-moment) hierarchy equation. To obtain the gyro-moment hierarchy, the gyro-averaged distribution function is expanded onto a velocity-space Hermite-Laguerre polynomial basis. By projecting the gyrokinetic equation, the gyro-moment hierarchy provides the spatial and temporal evolution of the Hermite-Laguerre expansion coefficients, which are velocity moments of the gyro-averaged distribution function referred to as gyro-moments. The Hermite-Laguerre projection of the full nonlinear Coulomb collision operator is performed using a multipole expansion of the Rosenbluth potentials, allowing us to derive a closed form of the Coulomb collision operator in terms of products between gyro-moments. Finally, the self-consistent evolution of the fields is described by a set of gyrokinetic Maxwell’s equations derived from a variational principle. Linear results of the present model are illustrated and show that the Hermite-Laguerre decomposition provides an efficient framework for a numerical implementation with reasonable cost for the study of the turbulent dynamics in the plasma periphery.

2019