Gyrokinetics is a rich and rewarding playground to study some of the mysteries of modern physics – such as turbulence, universality, self-organization and dynamic criticality – which are found in physical systems that are driven far from thermodynamic equilibrium. One such system is of particular importance, as it is central in the development of fusion energy – this system is the turbulent plasma found in magnetically confined fusion device. In this thesis I present work, motivated by the quest for fusion energy, which seeks to uncover some of the inner workings of turbulence in magnetized plasmas. I present three projects, based on the work of me and my collaborators, which take a tour of different aspects and approaches to the gyrokinetic turbulence problem.
I begin with the fundamental theory of gyrokinetics, and a novel formulation of its extension to the equations for mean-scale transport – the equations which must be solved to determine the performance of Magnetically confined fusion devices. The results of this work include (1) the equations of evolution for the mean scale (equilibrium) density, temperature and magnetic field of the plasma, (2) a detailed Poynting's theorem for the energy balance and (3) the entropy balance equations.
The second project presents gyrokinetic secondary instability theory as a mechanism to bring about saturation of the basic instabilities that drive gyrokinetic turbulence. Emphasis is put on the ability for this analytic theory to predict basic properties of the nonlinear state, which can be applied to a mixing length phenomenology of transport. The results of this work include (1) an integral equation for the calculation of the growth rate of the fully gyrokinetic secondary instability with finite Larmor radius (FLR) affects included exactly, (2) the demonstration of the robustness of the secondary instability at fine scales (kρi for ion temperature gradient (ITG) turbulence and kρe [special characters omitted]1 for electron temperature gradient (ETG)) which rules out the possibility that ultra-fine streamers could produce significant transport, (3) a demonstration that the variation in the phasing of the primary mode (which depend on the values, of the equilibrium scale lengths of the system) effects the strength of the secondary instability, distinguishing the gyrokinetic model from a previous gyrofluid model, (4) parameter scans for the mean-scale gradient lengths which suggest a possible role of secondary instabilities in the Dimits shift and the formation of electron internal transport barriers (ITB) in tokamaks, (5) a formulation of the theory for fully gyrokinetic ions and electrons in order to explore the transition between ETG and ITG scales and (6) demonstrate the existence of a mechanism for the saturation of long-wavelength ETG modes in this ETG-ITG transition range (modes which have been demonstrated in simulations not to saturate when employing the ETG Boltzmann-ion gyrokinetic system).
The final project is an application of the methods from inertial range understanding of fluid turbulence, to describe the stationary state of fully developed two-dimensional gyrokinetic turbulence. This work explores the relatively new idea of a phase-space cascade, whereby fine scales are nonlinearly generated in both position space and velocity space, and ultimately smoothed by collisional entropy production. This process constitutes the thermodynamic balance which occurs in the true steady state of a turbulent plasma, including those found in fusion devices. The results of this work include (1) exact third order relations (in analogy to Kolmogorov's four-fifths law), (2) phenomenological scaling theories for the forward and inverse cascades, (3) a detailed description of the relationship of the two-dimensional gyrokinetic cascade to the Charney-Hasegawa-Mima and two-dimensional Navier-Stokes cascades, (4) a Hankel transform formalism for treating velocity scales in the distribution function and (4) power law predictions for the phase-space free energy spectra.