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The study of low-frequency turbulence in magnetized plasmas is a difficult problem due to both the enormous range of scales involved and the variety of physics encompassed over this range. Much of the progress that has been made in turbulence theory is based upon a result from incompressible magnetohydrodynamics (MHD), in which energy is only transferred from large scales to small via the collision of Alfv ́n waves propagating oppositely along the mean magnetic field. Improvements in laboratory devices and satellite measurements have demonstrated that, while theories based on this premise are useful over inertial ranges, describing turbulence at scales that approach particle gyroscales requires new theory. In this thesis, we examine the limits of incompressible MHD theory in describing collisions between pairs of Alfvén waves. This interaction represents the fundamental unit of plasma turbulence. To study this interaction, we develop an analytic theory describing the nonlinear evolution of interacting Alfv ́n waves and compare this theory to simulations performed using the gyrokinetic code AstroGK. Gyrokinetics captures a much richer set of physics than that described by incompressible MHD, and is well-suited to describing Alfvénic turbulence around the ion gyroscale. We demonstrate that AstroGK is well suited to the study of physical Alfvén waves by reproducing laboratory Alfvén dispersion data collected using the LAPD. Additionally, we have developed an initialization alogrithm for use with AstroGK that allows exact Alfvén eigenmodes to be initialized with user specified amplitudes and phases. We demonstrate that our analytic theory based upon incompressible MHD gives excellent agreement with gyrokinetic simulations for weakly turbulent collisions in the limit that k⊥ ρi << 1. In this limit, agreement is observed in the time evolution of nonlinear products, and in the strength of nonlinear interaction with respect to polarization and scale. We also examine the effect of wave amplitude upon the validity of our analytic solution, exploring the nature of strong turbulence. In the kinetic limit where k⊥ ρi ≥ 1 where incompressible MHD is no longer a valid description, we illustrate how the nonlinear evolution departs from our analytic expression. The analytic theory we develop provides a framework from which more sophisticated of weak and strong inertial-range turbulence theories may be developed. Characterization of the limits of this theory may provide guidance in the development of kinetic Alfvén wave turbulence.

No description available.

Rotating convection in the Earth's core produces columnar vortices of radius ~10 km or less near the inner core boundary. Small-scale motions in the core can travel as Alfvén waves in the face of Ohmic diffusion, provided the ratio of the magnetic diffusion time th to the Alfvén wave travel time tA (measured by the Lundquist number S0) is much greater than unity. These motions transfer angular momentum from the core to the mantle, a process that can help explain variations in length of day. Vortices subject to the combined influence of a magnetic field and background rotation give rise to fast and slow Magneto-Coriolis (MC) waves whose damping is not well understood. This thesis investigates the long-time evolution of magneto hydrodynamic (MHD) waves generated by an isolated, small-scale motion in an otherwise quiescent, electrically conducting fluid. The first part of the study focuses on the damping of small-scale Alfvén waves, which is independent of rotation. For a plausible magnetic field strength in the Earth's core, it is shown that flows of lengthscale ~ 5 km or larger can propagate across the core as damped Alfvén waves on sub-decadal timescales. The second part of the study looks at MC waves generated from an isolated blob under rotation and a uniform axial magnetic field. The decay laws for these waves are obtained by considering the decay of fast and slow waves individually. While the fast waves are subject to strongly anisotropic magnetic diffusion, the slow waves diffuse isotopically. New timescales are derived for the onset of damping and the transition from the wave-dominated to the diffusion-dominated (quasi-static) phase of decay. This study shows for the first time that MC waves originating from small-scale vortices of magnetic Reynolds number Rm ~ 1 can be long-lived. The results of this study are extendible to small-scale MHD turbulence under rotation, whose damped wave phase has not been adequately addressed in the literature. Furthermore, it is thought that this study would help place a lower bound on the poloidal magnetic field strength in the Earth’s core.