An efficient treatment of the full Coulomb collision operator with applications

A formulation of full Coulomb (or Landau) collision operator is provided that allows for an efficient numerical implementation, both in unmagnetized and magnetized plasmas. The method is based on projecting the Boltzmann equations and the collision operator onto a Hermite-Laguerre velocity-space polynomial basis, obtaining a moment-hierarchy. This approach is implemented in a numerical simulation code and studies of systems of increasing complexity are being carried out, shedding light on the importance of retaining the full Coulomb collision operator, with respect to widely used simplified operators. First, the dynamics of electron-plasma waves is described at arbitrary collisionality for the first time by considering the full Coulomb collision operator. In particular, a purely damped entropy mode, characteristic of a plasma where pitch-angle scattering effects are dominant with respect to collisionless effects, is shown to emerge numerically, and its dispersion relation is analytically derived. This mode is absent when simplified collision operators are used, and like-particle collisions strongly influence its damping rate. Second, the linear properties of drift-waves are investigated, which allows for the comparison of the Coulomb collision operator to collision operators used in state-of-the-art turbulence simulation codes. Established collisional and collisionless limits are retrieved and an analysis on both the growth rate and eigenmode spectrum shows the need for retaining the Coulomb collision operator, specially at the intermediate levels of collisionality relevant for present and future magnetic confinement fusion devices.

Chauffage de plasma par ondes électromagnétiques à la troisième harmonique de la fréquence cyclotron des électrons dans le tokamak TCV

Gilles Arnoux

The Tokamak à Configuration Variable (TCV) programme is based on flexible plasma shaping capabilities together with a powerful electron cyclotron wave (ECW) additional heating for studies of stability, confinement, transport, control and power exhaust. In particular, ECW heating system allows an extended study of the β limit, defined as the ratio between the plasma kinetic pressure and the magnetic pressure, which is attributed to MHD (MagnetoHydroDynamic) instabilities. The ECW heating (ECH) is based on the resonant interaction between the electrons and an electromagnetic (EM) wave in a region of the plasma where the wave frequency is an harmonic of the electron cyclotron frequency. The TCV ECH system is composed of 6 gyrotrons (high power radio frequency sources), at the frequency of 82.7 GHz for second harmonic X-mode heating (X2) and 3 gyrotrons at the frequency of 118 GHz for third harmonic X-mode heating (X3), providing each a nominal power of 0.5 MW. In the moderate magnetic field of TCV (1.45 T), the X2 system is able to heat plasmas up to the X2 cutoff density (4.2 · 1019 m-3) above which the wave cannot propagate. The X3 system extends the accessible density range for ECH up to the X3 cutoff density (11.2 · 1019 m-3) and allows in particular the heating of plasmas in high confinement regime (H mode), most appropriate candidates to reach the β limit. The X2 wave is totally absorbed (plasma optically thick) being launched from the lateral side of TCV and crossing the vertical resonance layer. With this launching configuration, the X2 wave can also be used for non-inductive generation of the plasma current. Since the X3 absorption coefficient is weaker than the X2 absorption coefficient, the X3 wave is injected vertically in order to increase the beam path within the resonance layer, therefore maximizing the X3 optical depth. The present work, based on experiments and simulation, is the first detailed study of the X3 absorption properties in a top-launch configuration. The X3 absorption is shown to mainly depend on the wave injection conditions and the electron temperature. Full single-pass absorption is measured increasing nearly threefold the central electron temperature (1 keV –> 2.7 keV) when 1.35 MW of RF power is injected in low confinement regime plasmas (L-mode) with a central density of 4.0·1019 m-3. Experimental evidences show that a fraction of the power is absorbed on suprathermal electrons generated by the X3 wave itself. An absorption level of 85% is measured increasing threefold the central temperature by injecting 1.35 MW of X3 in H-mode plasmas with a central density of 8.2 · 1019 m-3. A new plasma dynamics is observed for the first time on TCV in these experiments. The X3 absorption is shown to depend strongly on the wave injection angle. In order to maximize the absorption during a plasma discharge by optimizing the injection angle, a real time feedback control has been developed and used. The system is based on the synchronous demodulation technique and uses a PI controller. In order to simulate the X3 wave propagation and absorption, the linear ray-tracing code TORAY-GA is used. These simulations predict an absorption dependence on the temperature and the injection conditions in agreement with the experimental results. Since TORAY-GA does not take into account the diffraction effects on the beam propagation, a comparison with the beam tracing code ECWGB which includes diffraction is discussed. The present results on X3 absorption properties demonstrate the efficiency of the X3 heating system on TCV, therefore extending the β limits study capabilities in elongated plasmas.

EPFL2005

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

Sébastien Jolliet

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.

EPFL2009

Pedestal Characteristics and MHD Stability of H-Mode Plasmas in TCV

Andreas Pitzschke

The tokamak à configuration variable (TCV) is unique in its ability to create a variety of plasma shapes and to heat the electron population in high density regimes using microwave power at the third harmonic of the electron cyclotron frequency. In the frame of this thesis, the impact of plasma shaping and heating on the properties of the edge transport barrier (ETB) in the high confinement mode (H-mode) was studied. This mode of operation is foreseen as one of the reference scenarios for ITER, the International Tokamak Experimental Reactor, which is being built to demonstrate the feasibility of thermonuclear fusion using magnetic confinement. A feature of H-mode regime operation are edge localized modes (ELMs), instabilities driven by the steep pressure gradients that form in the plasma edge region due to a transport barrier. During an ELM event, energy and particles are expelled from the plasma in a short burst. This will cause serious problems with respect to the heat load on plasma facing components in a tokamak of the size of ITER. Understanding of the phenomena associated with ELMs is thus required and dedicated investigations of their theory and experimental observations are carried out in many laboratories worldwide. This thesis presents several experimental and numerical investigations of tokamak behavior for configurations where the plasma edge plays an important role. From the experimental viewpoint, studies of transport barriers are challenging, as plasma parameters change strongly within a narrow spatial region. As part of the work presented here, the TCV Thomson scattering system was upgraded to meet the requirements for diagnosing electron temperature and density with high spatial resolution in the region of internal and external transport barriers. Simultaneously, the data analysis was significantly improved to cope with statistical uncertainties and alleviate eventual systematic errors. For measurements of the time evolution of density and temperature profile during the ELM cycle, the low repetition rate of the lasers used for Thomson scattering is a limiting. Although the system on TCV comprises 3 laser units that may be triggered in sequence with time separations down to 1 ms, time evolution over longer periods can only be reconstructed from repetitive events. In this context, an adjustment of the laser trigger to improve the synchronization with the ELM event is an advantage. A method was developed and implemented to generate a synchronizing trigger sequence, by a real-time monitoring of the D-alpha emission, which provides a marker for the ELM event. Recently, a "snowflake" (SF) divertor configuration, proposed as a possible solution to reduce the plasma-wall interaction by changing the divertor's poloidal magnetic field topology, was generated, for the first time, in TCV. A numerical code (KINX), based on a magnetohydrodynamic model (ideal MHD), was used to investigate the stability limits of this configuration under H-mode conditions and compare them with a similar standard single-null equilibrium. In a series of experiments, improved energy confinement was found and explained by improved stability of the edge region in the SF configuration. The influence of the pedestal structure in ELMy H-mode plasmas on the energy confinement and on ELM energy losses was investigated. The different ELM regimes found in TCV were analyzed, in particular the transition between type-III to type-I ELMs. The operational boundary of each ELM regime was characterized and verified by ideal MHD stability simulations for the ETB region. Recent studies on the scaling of the pedestal width with normalized poloidal pressure were confirmed. Using the capabilities of TCV, the influence of plasma shaping on pedestal parameters and MHD stability limits was investigated. In the past, models were developed to describe the onset of type-I ELMs, which are associated with modes in the ETB region arising from a coupling of pressure- and current-driven instabilities (coupled kink-ballooning modes). Experimental studies were performed to trace the temporal evolution of pedestal parameters characterizing the ETB during an ELM cycle. The results of these experiments were analyzed using information from MHD stability calculations. It is concluded that these models are capable of predicting limits as necessary conditions for ELM activity, but are not sufficient to fully explain ELM triggering.

EPFL2011