Gyrokinetic particle-in-cell global simulations of ion-temperature-gradient and collisionless-trapped-electron-mode turbulence in tokamaks

The goal of thermonuclear fusion research is to provide power plants, that will be able to produce one gigawatt of electricity. Among the different ways to achieve fusion, the tokamak, based on magnetic confinement, is the most promising one. A gas is heated up to hundreds of millions of degrees and becomes a plasma, which is maintained – or confined – in a toroidal vessel by helical magnetic field lines. Then, deuterium and tritium are injected and fuse to create an α particle and an energetic neutron. In order to have a favorable power balance, the power produced by fusion reactions must exceed the power needed to heat the plasma and the power losses. This can be cast in a very simple expression which stipulates that the product of the density, the temperature and the energy confinement time must exceed some given value. Unfortunately, present-days tokamaks are not able to reach this condition, mostly due to plasma turbulence. The latter phenomenon enhances the heat losses and degrades the energy confinement time, which cannot be predicted by analytical theories such as the so-called neoclassical theory in which the heat losses are caused by Coulomb collisions. Therefore, numerical simulations are being developed to model plasma turbulence, mainly caused by the Ion and Electron Temperature-Gradient and the Trapped-Electron-Mode instabilities. The plasma is described by a distribution function which evolves according to the Vlasov equation. The electromagnetic fields created by the particles are self-consistently obtained through Maxwell's equations. The resulting Vlasov-Maxwell system is greatly simplified by using the gyrokinetic theory, which consists, through an appropriate ordering, of eliminating the fast gyromotion (compared to the typical frequency of instabilities). Nevertheless, it is still extremely difficult to solve this system numerically due to the large range of time and spatial scales to be resolved. In this thesis, the Vlasov-Maxwell system is solved in the electrostatic and collisionless limit with the Particle-In-Cell (PIC) ORB5 code in global tokamak geometry. This Monte-Carlo approach suffers from statistical noise which unavoidably degrades the quality of the simulation. Consequently, the first part of this work has been devoted to the optimization of the code with a view to reduce the numerical noise. The code has been rewritten in a new coordinate system which takes advantage of the anisotropy of turbulence, which is mostly aligned with the magnetic field lines. The overall result of the optimization is that for a given accuracy, the CPU time has been decreased by a factor two thousand, the total memory has been decreased by a factor ten and the numerical noise has been reduced by a factor two hundred. In addition, the scaling of the code with respect to plasma size is presently optimal, suggesting that ORB5 could compute heat transport for future fusion devices such as ITER. The second part of this thesis presents the validation of the code with numerical convergence tests, linear (including dispersion relations) and nonlinear benchmarks. Furthermore, the code has been applied to important issues in gyrokinetic theory. It is shown for the first time that a 5D global delta-f PIC code can achieve a thermodynamic steady state on the condition that some dissipation is present. This is a fundamental result as the main criticism against delta-f PIC codes is their inability to deal with long time simulations. Next, the role of the parallel nonlinearity is studied and it is demonstrated in this work that this term has no real influence on turbulence, provided the numerical noise is sufficiently low. This result should put an end to the controversy that recently occurred, in which gyrokinetic simulations using different numerical approaches yielded contradictory results. Finally, thanks to the optimization of the code, the gyrokinetic model has been extended to include the kinetic response of trapped-electrons, in place to the usual adiabatic (Boltzmann) approximation. For the first time, global TEM nonlinear simulations are presented, and the role of the zonal flow on heat transport is analyzed. This study will help in acquiring some knowledge on the less-known TEM turbulence (as compared to ITG). In conclusion, this thesis is one of the main steps of the development of ORB5, which is now a state-of-the-art gyrokinetic code for collisionless ITG and TEM turbulence, and has brought several contributions to the understanding of these phenomena

IMPACT 2002+: A new life cycle impact assessment methodology

The new IMPACT 2002+ life cycle impact assessment methodology proposes a feasible implementation of a combined midpoint/damage approach, linking all types of life cycle inventory results (elementary flows and other interventions) via 14 midpoint categories to four damage categories. For IMPACT 2002+, new concepts and methods have been developed, especially for the comparative assessment of human toxicity and ecotoxicity. Human Damage Factors are calculated for carcinogens and non-carcinogens, employing intake fractions, best estimates of dose-response slope factors, as well as severities. The transfer of contaminants into the human food is no more based on consumption surveys, but accounts for agricultural and livestock production levels. Indoor and outdoor air emissions can be compared and the intermittent character of rainfall is considered. Both human toxicity and ecotoxicity effect factors are based on mean responses rather than on conservative assumptions. Other midpoint categories are adapted from existing characterizing methods (Eco-indicator 99 and CML 2002). All midpoint scores are expressed in units of a reference substance and related to the four damage categories human health, ecosystem quality, climate change, and resources. Normalization can be performed either at midpoint or at damage level. The IMPACT 2002+ method presently provides characterization factors for almost 1500 different LCI-results, which can be downloaded at http://www.epfl.ch/impac

Plasma shaping effects on tokamak scrape-off layer turbulence

The impact of plasma shaping on tokamak scrape-off layer (SOL) turbulence is investigated. The drift-reduced Braginskii equations are written for arbitrary magnetic geometries, and an analytical equilibrium model is used to introduce the dependence of turbulence equations on tokamak inverse aspect ratio (epsilon), Shafranov's shift (Delta), elongation (kappa), and triangularity (delta). A linear study of plasma shaping effects on the growth rate of resistive ballooning modes (RBMs) and resistive drift waves (RDWs) reveals that RBMs are strongly stabilized by elongation and negative triangularity, while RDWs are only slightly stabilized in non-circular magnetic geometries. Assuming that the linear instabilities saturate due to nonlinear local flattening of the plasma gradient, the equilibrium gradient pressure length Lp = -p(e)/del p(e) in the SOL is numerically computed and its dependence on epsilon, Delta, kappa and delta is analyzed, showing that stabilization of RBMs results in shorter Lp. An analytical estimate of L-p in the infinit aspect ratio limit and neglecting the Shafranov's shift is also derived. Nonlinear SOL turbulence simulations with non-circular magnetic geometries are carried out using the global, three-dimensional, flux-driven fluid code GBS (Ricci et al 2012 Plasma Phys. Control. Fusion 54 124047) and the results are compared with the findings obtained from the linear analysis of the SOL instabilities, showing good quantitative agreement

Global two-fluid simulations of tokamak Scape-Off-Layer turbulence

We present non-linear self-consistent 3D global fluid simulations of the SOL plasma dynamics using the Global Braginskii Solver (GBS) code. The code solves the drift-reduced Braginkii equations in a collisional plasma, with cold ions. The GBS code, originally developed for an electrostatic, 2D configuration has been recently upgraded to describe the SOL turbulence with the introduction of the variable curvature along the magnetic field lines, the magnetic shear, and the electromagnetic effects. The code peculiarity lies in the capability of evolving self-consistently equilibrium and 3D fluctuations as a results of the interplay among the sources, the turbulent transport and the plasma losses at the limiter plates. The non-linear simulations have been interpreted by means of linear analysis of the fluid equations modeling the system. This points out the presence of two main instabilities driving turbulence: the Drift Wave and the Resistive Balloning instabilities. The dependence of the instabilities growth rate and of their properties on the physical parameters of the system, for example the characteristic length of the plasma density, the magnetic shear and the beta ratio have been explained and the regions where each instability dominates have been identified

Low-frequency linear-mode regimes in the tokamak scrape-off layer

Motivated by the wide range of physical parameters characterizing the scrape-off layer (SOL) of existing tokamaks, the regimes of low-frequency linear instabilities in the SOL are identified by numerical and analytical calculations based on the linear, drift-reduced Braginskii equations, with cold ions. The focus is put on ballooning modes and drift wave instabilities, i.e., their resistive, inertial, and ideal branches. A systematic study of each instability is performed, and the parameter space region where they dominate is identified. It is found that the drift waves dominate at high R/L-n, while the ballooning modes at low R/L-n; the relative influence of resistive and inertial effects is discussed. Electromagnetic effects suppress the drift waves and, when the threshold for ideal stability is overcome, the ideal ballooning mode develops. Our analysis is a first stage tool for the understanding of turbulence in the tokamak SOL, necessary to interpret the results of non-linear simulations. [http://dx.doi.org/10.1063/1.4758809

Recent numerical developments in scrape-off-layer global fluid simulations using the GBS code

Turbulence in the scrape-off-layer (SOL) of magnetic fusion devices is one of the most outstanding issues in magnetic fusion. This open fied lines region determines the boundary condition of the core plasma and controls the plasma refueling, heat losses and impurity dynamics, therefore governing the fusion power output of the entire device. In this work, we present the global fluid code GBS [Ricci et al., Plasma Phys. Control. Fusion 54, 124047, 2012]. It employs a 5 field drift-reduced Braginskii model for both electrostatic and electromagnetic turbulence in a limited configuration. The model simulates a turbulent steady state resulting from plasma sources mimicking the plasma outflow from the core, turbulent perpendicular transport and parallel losses at the limiter sheaths. Unique features of the code are that gradients are a-priori unknown and there is no separation between the background gradient and the fluctuations. We will focus on recent advances to extend GBS from an infinite aspect ratio model to a general geometry model. One of the main features of SOL turbulence is its strong anisotropy characterized by k// /k⊥ << 1, It is therefore crucial to correctly describe the parallel gradient derivative, in particular at the limiter plates where the plasma is lost. In view of more complicated situations such as a diverted geometry, the GBS code does not employ field- aligned coordinates. The fluid fields are discretized on a toroidal and poloidal grid. Using alternative schemes that will be presented. These schemes are tested in a simplified model describing the propagation of shear-Alfven waves, which is the fastest wave propagating for this simulation model. Then, GBS nonlinear simulations using these new schemes will be presented and compared. Among those, we will also discuss some ot the simulation results, focusing on circular geometry with finite aspect ratio. In particular, it is shown that the characteristic pressure length can be well described by the gradient removal theory [Ricci et al., Phys. Plasmas 20, 010702, 2013] that uses the flattening of the gradient by the perturbation as a saturation mechanism

The role of the sheath in magnetized plasma fluid turbulence

In the sheath region at the interface between plasmas and solid surfaces, quasi-neutrality and, in the case of magnetized plasmas, drift-ordering are violated. These two assumptions are typically made in plasma fluid models; the presence of a plasma-wall transition region, typical of all bounded systems, hampers therefore their use. This problem can be overcome by introducing a set of boundary conditions (BCs) for fluid models that properly describe the physics of the plasma-wall transition region. While the classical Bohm-Chodura BCs for the ion and electron parallel velocities have been previously derived, no consistent BCs for the other fluid quantities existed up to date. Based on a recent theory [1,2], a complete set of analytical BCs for the density, temperature, potential, vorticity, and parallel ion and electron velocities, has been provided, which is fully consistent with kinetic simulations of the plasma-wall transition region [3]. These BCs have been implemented in a three-dimensional global fluid code, which is used to simulate turbulence in basic plasma physics devices and in the tokamak scrape-offlayer. It has been shown that BCs that faithfully supply the sheath physics to the fluid codes are crucial for the understanding of the equilibrium profiles, plasma recirculation, intrinsic plasma rotation, and blob propagation in basic plasma physics and fusion devices. References [1] J. Loizu, P. Ricci and C. Theiler, Physical Review E 83, 016406 (2011) [2] J. Loizu, J. Dominski, P. Ricci and C. Theiler, Phys. Plasmas 19, 083507 (2012) [3] J. Loizu, P. Ricci, F. Halpern and S. Jolliet, Phys. Plasmas 19, 122307, (2012

Turbulent regimes in the tokamak scrape-off layer

We identify the turbulent regimes in the tokamak scrape-off layer as a function of plasma resistivity, electron to ion mass ratio, safety factor, and magnetic shear. We find that two main instabilities drive turbulence in the SOL: Drift Waves, with their resistive and inertial branches, and the Ballooning instabilities, with their resistive, ideal, and inertial branches. First, we identify how the linear growth rate, and the properties of these instabilities depend on the system physical parameters. Then, according to the gradient removal saturation mechanism, we use the fact that the transport is dominated by the mode with the highest growth rate divided by the poloidal wave number and the non-linear saturated pressure scale length is proportional to this ratio. This allows us to evaluate the non-linear saturated pressure gradient and the poloidal wavenumber of the dominating mode as a function of the resistivity, the mass ratio, the safety factor, and the magnetic shear . We can therefore define the non-linear instability phase space, locating the regions in which each instability is influencing transport the most. In order to validate our calculations, we run non-linear SOL simulations and we compare the plasma pressure scale length and the mode characteristics to the prediction provided by the phase space description. The non-linear simulations are performed using the GBS code, which solves the drift-reduced Braginskii equations evolving self-consistently equilibrium and fluctuations in three-dimensional geometry. The non-linear simulations are interpreted in light of our non-linear analysis and confirm its validity

Global scrape-off layer electromagnetic fluid turbulence simulations

Electromagnetic effects play a key role in tokamak edge turbulence. It has been suggested that the density limit and the L to H mode transition may both be due to an interplay between electromagnetic effects, diamagnetic flows and collisionality. (See, e.g., Ref [1].) The present paper discusses the results of scrape-off layer (SOL) non-linear 3D fluid turbulence simulations including finite beta effects in the shear-less limit. These simulations were carried out using the GBS code [2], which evolves the drift-reduced Braginskii equations for a collisional plasma with cold ions in circular (s-α) geometry with a toroidal limiter in the high-field side midplane. The GBS code has been used to study turbulence in linear devices and in a simple magnetized torus configuration [2, 3]. We have recently adapted the code for tokamak edge geometry, and introduced s-α curvature operators as well as magnetic shear, finite aspect ratio, and finite beta effects. The objective of our work is to describe the phase-space relevant to the tokamak SOL turbulence. In this paper, in particular, the role played by finite beta effects upon the characteristic lengths of the profile gradients, turbulence saturation levels, and other basic turbulence properties, is assessed in the context of fully global non-linear turbulence simulations. The non-linear steady-state turbulent plasma profiles are obtained as the result of a balance between plasma density and heat sources, turbulent fluctuations, and parallel losses at the limiter plates. The turbulence drive is a priori unknown and there is no separation between fluctuations and background profiles. Linear analysis of the fluid equations has been carried out for SOL relevant parameters. In the presence of finite beta effects, we recover three instabilities: drift waves, resistive ballooning modes, and ideal ballooning modes. The onset of ideal ballooning modes is known to correspond to the instability threshold α_MHD = q^2 β R/Lp ~ 1. In the non-linear simulations, however, we observe the onset of catastrophic transport well below the ideal limit. The saturated states in this regime are characterized by large transport due to global ideal modes. These modes are linearly subdominant but non-linearly dominant due to the underlying turbulence saturation mechanism

Intrinsic toroidal plasma rotation in the scrape-off-layer

The origin and nature of intrinsic toroidal plasma rotation in the scrape-off-layer are theoretically investigated. We discuss and analytically estimate three mechanisms that give rise to SOL toroidal rotation: turbulent momentum transport associated with electrostatic instabilities, pressure gradients along the poloidal direction, and deviation of the plasma velocity at the sheath entrance with respect to the Bohm's value. The results of three-dimensional global fluid simulations of tokamak scrape-off-layer in a limiter configuration are shown and compared