<p> Tokamaks use magnetic fields to confine plasmas to achieve fusion; they are the leading approach proposed for the widespread production of fusion energy. The sawtooth crash in tokamaks limits the core temperature, adversely impacts confinement, and seeds disruptions. Adequate knowledge of the physics governing the sawtooth crash and a predictive capability of its ramifications has been elusive, including an understanding of incomplete reconnection, i.e., why sawteeth often cease prematurely before processing all available magnetic flux. In this dissertation, we introduce a model for incomplete reconnection in sawtooth crashes resulting from increasing diamagnetic effects in the nonlinear phase of magnetic reconnection. Physically, the reconnection inflow self-consistently convects the high pressure core of a tokamak toward the <i>q</i>=1 rational surface, thereby increasing the pressure gradient at the reconnection site. If the pressure gradient at the rational surface becomes large enough due to the self-consistent evolution, incomplete reconnection will occur due to diamagnetic effects becoming large enough to suppress reconnection. Predictions of this model are borne out in large-scale proof-of-principle two-fluid simulations of reconnection in a 2D slab geometry and are also consistent with data from the Mega Ampere Spherical Tokamak (MAST). Additionally, we present simulations from the 3D extended-MHD code M3D-C¹ used to study the sawtooth crash in a 3D toroidal geometry for resistive-MHD and two-fluid models. This is the first study in a 3D tokamak geometry to show that the inclusion of two-fluid physics in the model equations is essential for recovering timescales more closely in line with experimental results compared to resistive-MHD and contrast the dynamics in the two models. We use a novel approach to sample the data in the plane of reconnection perpendicular to the <i>(m,n)</i>=(1,1) mode to carefully assess the reconnection physics. Using local measures of reconnection, we find that it is much faster in the two-fluid simulations, consistent with expectations based on global measures. By sampling data in the reconnection plane, we present the first observation of the quadrupole out-of-plane magnetic field appearing during sawtooth reconnection with the Hall term. We also explore how reconnection as viewed in the reconnection plane varies toroidally, which affects the symmetry of the reconnection geometry and the local diamagnetic effects. We expect our results to be useful for transport modeling in tokamaks, predicting energetic alpha-particle confinement, and assessing how sawteeth trigger disruptions. Since the model only depends on local diamagnetic and reconnection physics, it is machine independent, and should apply both to existing tokamaks and future ones such as ITER.</p>

This thesis presents a complete theoretical framework for turbulence and transport in tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio of the gyroradius to the equilibrium scale length. Proceeding order-by- order in this expansion, a framework for plasma turbulence is developed. It comprises an instantaneous equilibrium, the fluctuations driven by gra- dients in the equilibrium quantities, and the transport-timescale evolu- tion of mean profiles of these quantities driven by the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the high-flow gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local cascade of free energy. Transport equations for the evolution of the mean density, temperature and flow ve- locity profiles are derived. These transport equations show how the neo- classical corrections and the fluctuations act back upon the mean profiles through fluxes and heating. This framework is further developed by exploiting the scale separation between ions and the electrons. The gyrokinetic equation is expanded in powers of the electron to ion mass ratio, which provides a rigorous method for deriving the electron response to ion-scale turbulence. We prove that such turbulence cannot change the magnetic topology, and ar- gue that, therefore, the magnetic field lies on fluctuating flux surfaces. These flux surfaces are used to construct magnetic coordinates, and in these coordinates a closed system of equations for the electron response is derived. All fast electron timescales have been eliminated from these equations. Simplified transport equations for electrons in this limit are also derived.