The research presented in this thesis is part of the research area of controlled nuclear fusion, more specifically magnetic confinement fusion. It is motivated by observations in the experiment currently closest to an operating fusion reactor, the tokamak. In tokamaks large transport fluxes of particles and energy arise, which are thought to be the consequence of micro-instabilities, leading to a turbulent state. Although much progress in the understanding of turbulent tokamak transport has been made, the understanding is still incomplete. The gyrofluid model, which is further developed in this thesis, is one of the most commonly used models to describe the turbulent transport. It is a fluid like moment model able to treat finite-Larmor-radius effects to all orders. Up to now, the gyrofluid equations were derived by taking moments of the Gyrokinetic equation (the kinetic equation in which an average over the fast gyromotion has been performed), using the so called strong drift ordering to partly linearise the final equation of evolution, and closing the system such that only a finite number of moments is retained. This method leads to equations that are homogeneous in the plasma parameters. Furthermore, the associated conservation laws are results of the construction of the equations, and it is therefore not clear how to generalise them. In this thesis we present a method of deriving the gyrofluid equations that guarantees full non-linearity as well as an exact energy conservation law. Two sets of electrostatic gyrofluid equations are presented, in the first only the perpendicular temperature is kept, while the second retains both the parallel as well as the perpendicular temperature. In both cases a Lagrangian is used to derive the equations of evolution, such that the energy conservation law is automatically satisfied through Noether's theorem. The equations of evolution are calculated from the variation of the action integral with the help of constraints that in both cases represent conservation laws of the system. Two different variational methods are used, one in which the field variables are retained as dynamical variables (Lagrange multipliers method) and another in which the field variables are expressed as functions of only one dynamical variable (virtual displacement method). Both variational methods are equivalent, and yield the same equations of evolution. After the equations are obtained from the variation of the Lagrangian, diamagnetic cancellations are inserted manually in such a way that energy conservation is preserved. Comparison with previous models shows a very good agreement for the lowest order so called zero-Larmor-radius terms in both the sets of the gyrofluid equations. The final set of equations includes the evolution of the three or four first moments, as well as a Polarisation (Poisson) equation that determines the electric field, and an exact energy conservation law. The new features of the models, namely the full-nonlinearity and the exact energy conservation law are important for understanding tokamak turbulent transport. Fully nonlinear equations are needed to study the highly nonlinear phenomena that occur especially in the edge of the tokamak, and furthermore, turbulent transport phenomena in general cover a spatial range involving significant variations of the background parameters. Moreover, a lack of an energy conservation law exact in all levels of nonlinearity can give rise to unphysical drive or damping of the large scale fluxes and currents that participate in the turbulence. These considerations will help in the further study of turbulence in tokamaks.     «

The research presented in this thesis is part of the research area of controlled nuclear fusion, more specifically magnetic confinement fusion. It is motivated by observations in the experiment currently closest to an operating fusion reactor, the tokamak. In tokamaks large transport fluxes of particles and energy arise, which are thought to be the consequence of micro-instabilities, leading to a turbulent state. Although much progress in the understanding of turbulent tokamak transport has been...     »

Eines der meist benutzten Modelle, um turbulenten Transport in Tokamakplasmen zu beschreiben, ist das Gyrofluid-Modell, da es Phaenomene auf Skalen vergleichbar mit dem Ion-Gyroradius mit einschliesst. In vorliegender Arbeit wird eine Methode praesentiert, diese Gleichungen durch Entwicklung aus einer Lagrangefunktion mit Hilfe von Zwangsbedingungen herzuleiten. Diese Lagrange-Beschreibung garantiert, dass die Gleichungen einem Energieerhaltungssatz folgen, und dass sie alle Nichtlinearitaeten enthalten. Diese beiden Eigenschaften, die fuer Turbulenzuntersuchungen in Tokamaks unerlaesslich sind, wurden von bisherigen Gyrofluid-Modellen nicht erfuellt. Zwei Gyrofluid-Modelle werden praesentiert: eines, das nur Temperaturen senkrecht zum Hintergrundmagnetfeld beschreibt, und ein anderes, das sowohl parallele als auch senkrechte Temperaturen einschliet. Die Modelle bestehen aus Gleichungen fuer die Evolution der Momente, einer Poisson Gleichung und einem Energieerhaltungsgesetz.     «

Eines der meist benutzten Modelle, um turbulenten Transport in Tokamakplasmen zu beschreiben, ist das Gyrofluid-Modell, da es Phaenomene auf Skalen vergleichbar mit dem Ion-Gyroradius mit einschliesst. In vorliegender Arbeit wird eine Methode praesentiert, diese Gleichungen durch Entwicklung aus einer Lagrangefunktion mit Hilfe von Zwangsbedingungen herzuleiten. Diese Lagrange-Beschreibung garantiert, dass die Gleichungen einem Energieerhaltungssatz folgen, und dass sie alle Nichtlinearitaeten e...     »