Layer Formation in Stably and Unstably Stratified Flows

Abstract

This thesis investigates the dynamics of layered velocity and buoyancy structures in stably and unstably stratified flows. Geophysical examples of layers in stably stratified turbulence include the ocean's equatorial deep jets (EDJs) and small-scale horizontal density layers that are ubiquitous in both the ocean and atmosphere. These layers are visually striking and have implications for the broader climate system. Layered horizontal mean flows also form in the Rayleigh-Benard model of dry convection maintained by unstable background stratification, where they substantially modify the convective heat flux. We first analyze the dynamics of mean flow formation in two-dimensional (2D) Rayleigh-Benard convection. Horizontal mean shear flows form in this system via an instability of laminar roll convection known as the tilting instability. The tilting instability depends strongly on the horizontal lengthscale of the convection pattern so that mean flows are suppressed in domains with sufficiently large aspect ratio of length to height. We show that this suppression results from a perturbation-perturbation advection mechanism that intervenes at linear order in the tilting instability. We also analyze the dynamics of simplified models of convection that exhibit suppression of mean flows even though they do not retain the perturbation-perturbation nonlinear mechanism. We show that in such models, mean flow suppression operates by a different mechanism that is an artifact of low resolution, and that mean flow suppression ceases in such models when they are appropriately extended. We then analyze the dynamics of layered mean flow formation in stably stratified turbulence. Turbulent jets known as vertically sheared horizontal flows (VSHFs) form ubiquitously in simulations of stratified turbulence made using the stochastically maintained Boussinesq system. Jets form in both 2D and 3D with similar properties and have structure resembling that of the EDJs. The mechanisms of formation and maintenance of the jets have remained poorly understood. To analyze VSHFs we use fully nonlinear simulations as well as statistical state dynamics (SSD). In SSD, equations of motion are written directly for statistical variables of the turbulence rather than for the evolution of the detailed system state. We apply SSD in the form known as S3T, which is a second-order closure in which the SSD is expressed by coupling the equation for the horizontal mean structure with the equation for the ensemble mean perturbation covariance. To facilitate applying S3T we focus on the case of 2D stratified turbulence. We show that VSHFs form via a statistical instability of the state of homogeneous stratified turbulence that is associated with a bifurcation from the homogeneous state to the VSHF state as the strength of the stochastic excitation increases. We identify the mechanism of VSHF emergence as mean state-perturbation interaction and identify the physical processes that underlie this mechanism. Finally, we address the formation of buoyancy layers in stratified turbulence. A commonly invoked mechanism of buoyancy layer formation is the Phillips negative diffusion instability, by which layers form from an initially uniform stratification if the turbulent buoyancy flux weakens as stratification increases. Although attractive as an explanation for layering, it has remained unclear whether the Phillips mechanism operates in stratified turbulence. In particular, it is not well understood, in a given example of stratified turbulence, whether and by what mechanism the turbulent buoyancy flux weakens with increasing stratification. We apply S3T to analyze the Phillips instability in 2D uniformly sheared and stratified turbulence. We show that, in some circumstances, the buoyancy flux indeed weakens as stratification increases, and that uniformly stratified homogeneous turbulence undergoes a negative diffusion instability that we identify as the Phillips instability. This analysis provides a comprehensive theory of the Phillips instability, for an idealized example of turbulence, based directly on the fundamental equations of motion.This theory complements previous analyses of the Phillips instability based on phenomenological models of turbulence.