UC Berkeley

Rotating shear flows are fundamental to the dynamics of numerous natural and engineered systems, from protoplanetary accretion disks to aircraft wake vortices. This dissertation investigates the multi-faceted dynamics of hydrodynamic stability in these systems, bridging the fundamental theory of wave-wave resonances in incompressible environments with the global wave-mean flow interactions inherent to rotating, stratified fluids. Part I investigates the weakly nonlinear stability of incompressible columnar vortices, demonstrating that the triadic resonance of wave modes is governed by a set of hydrodynamic selection rules. Employing a multi-scale perturbation analysis, we prove that resonant interactions between smooth neutral modes are strictly conservative and confined to the Manley—Rowe relations. Using wave pseudoenergy within a large-k WKBJ framework, we show that the selection rules topologically prohibit intrinsic instability. Consequently, the breakdown of a columnar vortex requires a specific symmetry-breaking mechanism. We identify and analyze two distinct pathways for this destabilization: (1) Parametric instability, where external forcing sustains a pumping wave and acts as an energy source. Following the pseudoenergy criterion, we identify the discrete critical layer mode as the sole neutral mode that carries negative pseudoenergy, and thus required for the onset of parametric instability. Using non-degenerate perturbation theory, we generalize classical elliptical instability to arbitrary driving frequencies and geometries, and identify continuous instability configurations involving discrete critical layer modes. (2) Active critical layers, where an embedded singularity breaks the Hermitian symmetry of the linear operator, enabling the extraction of mean-flow energy via a wave-mean resonance. Most importantly, Part I establishes that the critical layer is the fundamental engine of instability in columnar vortices, and that the presence of a critical layer is essential to the breakdown of an isolated vortex.Part II transitions to the global wave-mean resonances of stratified fluids, focusing on the Zombie Vortex Instability (ZVI). To overcome the numerical stiffness imposed by restorative inertio-gravity waves and rapid advective shear in the three-dimensional cylindrical Boussinesq equations, we develop a parallelized Exponential Time Differencing (ETD) pseudo-spectral solver. By analytically diagonalizing the linear operators, the ETD scheme enables efficient simulations in global cylindrical domains, while natively embedding the physical resonance characteristics into the discrete integration operators. Utilizing these global simulations, we address the geometric limitations of traditional local shearing box approximations, demonstrating that the uniform spacing of zombie vortices observed in prior literature is a result of the uniform background shear assumed within the local models. Instead, we propose the frequency-resetting chain reaction hypothesis: self-replication is driven by localized wave-mean flow resonance, where inertio-gravity wave packets are barotropically emitted, trapped at baroclinic critical layers, and undergo nonlinear roll-up to re-emit new waves at reset frequencies dictated strictly by the local background shear. Ultimately, this dissertation provides a unified physical framework for wave resonances in rotating shear flows through a synthesis of weakly-nonlinear theories and global numerical studies. Across both incompressible and stratified regimes, the critical layer emerges as the fundamental engine of flow destabilization. Whether acting as the requisite symmetry-breaking mechanism to extract energy in isolated columnar vortices, or functioning dually as the barotropic amplifier and baroclinic spatial trap that drives multi-generational vortex replication, critical layer dynamics dictate the transfer of energy. Together, these theoretical and computational contributions elucidate the complex mechanisms of momentum transport and energy exchange in rotating shear flows, linking the fundamental fluid dynamics of engineered aircraft wakes to the macroscopic behavior of astrophysical accretion disks.