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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 1 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 2 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 3 contains a numerical study of vortex ring dynamics in a stratified environment, which represents my contribution to a research project conducted at the UNC Fluid Lab. Vortex rings made of fluid with density higher than the ambient fluid and propagating downward through a sharp density stratification are considered. Lab experiments have identified a critical phenomenon, which distinguishes the behavior of the falling vortex ring in either being fully trapped at the ambient density layer, or continuing through the layer in its downward motion. The numerical simulations presented are able to reproduce such behaviors and offer several details not visible in experiments.