Nonlinear Pattern Formation in Stratified Kolmogorov Flow

Abstract

This thesis explores stratified Kolmogorov flow (SKF), a fundamental model in geophysical fluid dynamics that combines a vertically-varying sinusoidal shear flow with vertical density stratification. This setup gives rise to linear instabilities, exact coherent structures (ECS), chaotic dynamics, and turbulent transport, all of which are crucial for understanding mixing processes in layered environments like oceans and atmospheres. A central theme through out this work is the examination of linear stability, ECS, and chaotic dynamics in stratified shear flows, drawing on tools from advance dynamical systems theory to uncover underlying patterns and behaviors. Chapter 1 outlines the two complementary scalings employed in this thesis: the anisotropic scaling in strongly stratified Kolmogorov flow (SSKF) at order one Prandtl number (Pr) and the isotropic scaling in standard stratified Kolmogorov flow at Pr≪1. These two flow regimes are highlighted as prime examples of forced-dissipative pattern forming systems, viewed through a dynamical systems tools, with direct relevance to oceanic and atmospheric flows. We also provide an overview quasilinear (QL) approximations for both systems, com- paring their performance against the full nonlinear (NL) model to assess how well simplified models capture complex behaviors. The chapter concludes with a comprehensive outline of the thesis structure, laying the groundwork for the subsequent analyses. Chapter 2 discusses the limit of strong stratification, in which a highly anisotropic, layer-like horizontal flow develops with emergent vertical shear sufficiently strong to trigger spatially localized instability regions. We employ tools from modern dynamical systems theory to better understand the spatial structure of these instabilities, their nonlinear saturation, and the diabatic mixing they drive. Specifically, we develop a Newton-Krylov iterative solver to compute ECS, i.e., fully nonlinear, invariant solutions, in the idealized and well-controlled setting of two-dimensional strongly stratified Kolmogorov flow. Unlike prior related studies, We investigate the physically relevant regime of small Froude number (Fr; strong stratification) and large buoyancy Reynolds number (Reb). We explore the dependence of the numerically computed ECS on these parameters, and compare our results with DNS. Chapter 3 examines SKF under low-P´eclet-number (LPN) conditions, where thermal diffusion dominates convective heat transport, a scenario pertinent in astrophysical settings including stellar interiors. This regime influences instability thresholds, pattern emergence, and chaotic dynamics by damping buoyancy effects. We adapt the Newton-Krylov solver described in Chapter 2 to search for ECS, comparing results between the standard governing equations and the LPN-reduced equations. Unlike prior related studies, we focus on practical parameter spaces with large Reynolds numbers (Re), small Richardson numbers (Ri), and small P´eclet numbers (Pe), exploring ECS dependencies on these factors and cross-checking with the DNS to highlight diffusion’s role in stability and structure formation. Chapter 4 examines the QL approximation applied to both the strongly stratified Kolmogorov flow at order one Pr and the stratified Kolmogorov flow at Pr ≪1 using the standard equations. In the first part, we compare the QL model against the full NL system for SSKF to evaluate its accuracy in capturing essential dynamics. This involves analyzing the structure of the flow fields, mean velocity and buoyancy profiles, and energy spectra, highlighting where QL successfully approximates self-organized patterns and transport in extreme stratification limits. In the second part, we perform similar comparisons of the SKF using the standard equations, assessing how well QL reproduces field structures, profiles, and spectra under dominant diffusion effects. These analyses reveal the strengths and limitations of QL as a simplified reduced model for exploring instability and mixing in stratified shear flows, particularly in parameter spaces challenging for full NL computations. In Chapter 5, we synthesize insights across stratification strengths and diffusion regimes, blending linear theory, ECS computations, and DNS to illuminate how structure arises in stratified shear flows. The QL method emerges as a powerful means of accessing challenging parameter regimes, with implications for improving geophysical models. Future directions for both systems could include three-dimensional extensions and integrations with observational data to further bridge theory and real-world applications.