Fluid Homogenization in Mixed Phase Spaces

The mixing dynamics of two dimensional incompressible systems can be broadly placed into three categories depending on the presence of stable structures in the advecting field. Integrable fields correspond to regular dynamics and are exactly solvable as eigenvalue problems, while uniformly chaotic fields are completely ergodic and can be well understood statistically. Mixed systems corresponding to partially chaotic advecting fields are not exactly solvable and due to their lack of uniformity, can not be trivially understood with statistical analysis. Mixed systems are especially interesting because they are ubiquitous in nature, and exhibit a resistance to homogenization. In this thesis we probe the scaling dynamics of persistent patterns in the homogenization of partially chaotic systems using both spectral analysis and the statistical Dirichlet quotient. We show that both methods provide equivalent insight into the diffusive scaling behavior. We find that the low diffusivity limit results in universal diffusive scaling and quantum-like spectral behavior. We provide a semi-analytic argument as to why this scaling behavior exists