Analytical and Numerical Studies on Stratified Fluid Flows

CLAUDIO VIOTTI: ANALYTICAL AND NUMERICAL STUDIES ON STRATIFIED FLUID FLOWS. (Under the direction of Roberto Camassa and Richard M. McLaughlin.) The mathematical modeling of stratified fluid flows is the overall subject of this work, which spans a range of more specific topics: internal gravity waves, shear instability, anomalous diffusion of passive scalars, vortex rings dynamics in stratified environments. This dissertation is organized in three parts. Part I focuses on shear-induced instability in large amplitude internal waves. Previous studies have shown that an instability of the Kelvin–Helmholtz type can occur within internal wave-induced shear layers, leading to wave breaking and production of turbulence. It was also recognized that such wave breaking resembles the instability of parallel shear flows only superficially. This study aims to refine the understanding of shear-induced instability in this specific context, and to identify the mechanisms that eventually make the onset of shear instability for such flows a significantly more subtle mechanism than its parallel counterpart. Part II is concerned with anomalous diffusive regimes for passive scalars advected in shear flows. The problem is studied from the standpoint of spectral analysis, which provides a natural way to classify diffusive regimes ensuing from different asymptotic limits in the governing parameters. The analysis identifies separate classes of eigenmodes, whose features dominate the evolution at different time and space scales in initial value problems with multiscale initial conditions. Part III contains a numerical study of vortex ring dynamics in a stratified environment, which represents my contribution to a research project conducted at the UNC