UCLA

This thesis addresses several limitations in the current state of the art fractional models, and applies fractional transport concepts to transport in confined plasmas. One limitation that is addressed is the discrepancy between the strict mathematical properties of the Levy distributions that underlie fractional diffusion models and the distributions observed in experiment and numerical simulation. Specifically, the second and higher moments of the Levy distribution are infinite, while the moments of observed distributions are finite. This issue is dealt with using tempered, or truncated Levy statistics in which the moments of the Levy distributions are regularized by including exponential damping, with damping constant λ, into the tails of the distribution. In this thesis these tempered Levy processes are studied in the context of a spatially tempered fractional Fokker-Planck (TFFP) equation and in the context of fluctuation driven transport.

Other limitations that this thesis addresses are related to the dimensionality of the fractional derivatives commonly used to model transport, and the application of fractional derivatives to transport in bounded finite-sized domains. Many fractional models of radial transport in a cylindrical system are based on one-dimensional Cartesian fractional operators. In this thesis a fully two-dimensional fractional Laplacian operator is developed from a generalized random walk model. This 2D operator can be azimuthally averaged to produce a radial fractional operator that correctly incorporates, near the origin, all the geometric effects of a 2D circular system. There are also subtleties in applying fractional derivative operators, which are often formulated on an unbounded domain, to a finite domain. In this this thesis a bounded domain model of radial fractional transport is formed using mask functions that modify the kernels of the fractional operators and that go to zero in a boundary layer to remove mathematical singularities associated with limiting the range of integration.

This thesis also presents an analytic solution to the one-dimensional fractional thermal wave equation for the unbounded domain. To the author's knowledge, no such thermal wave solution is present in the literature.

Finally this thesis also compares the results from the radial fractional model to experimental results taken from several different tokamak fusion devices. The devices considered include the General Atomic's machine DIII-D and the major European devices: the Joint European Torus (JET), the Axially Symmetric Divertor Experiment (ASDEX-Upgrade), and the Rijnhuizen Tokamak Project (RTP). The survey focuses on two types of experiment: steady off-axis ECH experiments from RTP and power modulation experiments from JET, ASDEX-Upgrade, and DIII-D. In the steady heating experiments from RTP, hollow temperature profiles, or profiles that peak away from the center, are observed. Hollow profiles are an intrinsic feature of the radial fractional model, and it is found that the radial model is robust in describing the observed RTP profiles. In the power modulation experiments, the fractional model only achieves good agreement with the measurements for the high-frequency modulation in the purely Ohmic discharge # 10589 in ASDEX-Upgrade for α = 1.75. This disagreement suggests that fractional transport does not play a major part in determining the thermal waves excited in power modulation experiments.