Instability and Turbulence in Density-Stratified Mixing Layers

A vertically symmetric, stably stratified mixing layer may exhibit, depending upon the details of the initial profiles of horizontal velocity and density, either of two major classes of linear, supercritical instability. These are the familiar Kelvin-Helmholtz instability and its lesser -known relative, Holmboe instability. In this thesis, we employ a combination of linear and nonlinear mathematical techniques in order to (a) elucidate the dynamical characteristics of Holmboe instability and clarify its relationship with Kelvin-Holmholtz instability, and (b) investigate the processes through which flows exhibiting these instabilities may evolve towards the turbulent state. We begin by considering the evolution of small disturbances on an inviscid, Boussinesq, stably stratified shear layer. Conventional one-dimensional linear analysis is employed to study the temporal and spatial structures of these instabilities and the physical mechanisms which govern their evolution. Attention is focussed upon the manner in which Kelvin-Helmholtz instability is replaced by Holmboe instability for a sequence of background flows with successively larger values of the bulk Richardson number. Unstable normal modes that exist in the transition region between the Kelvin-Helmholtz and Holmboe regimes exhibit a distinctive spatial structure which causes kinetic energy that is transferred from the background flow to the disturbance to remain trapped in the vicinity of the steering level. These unstable structures are characterized by relatively low growth rates, and are shown to occur under conditions for which overreflection of neutrally propagating internal waves is impossible due to the fact that the gradient Richardson number at the steering level exceeds the value of 1/4. Detailed calculations of the propagation characteristics of internal waves in stratified shear layers reveal the extent to which resonant overreflection theory, based upon the reflection properties of temporally neutral waves, many predict (or fail to predict) the stability characteristics of such flows. We then generalize the stability problem to include dissipative effects, and demonstrate the existence of a class of initially parallel shear flows which, as a consequence of linear supercritical instability, evolve directly into three-dimensional flows without the requirement for an intermediate two-dimensional finite-amplitude state. This represents a counterexample to a common misinterpretation of Squire's theorem, namely that the fastest-growing unstable model of a dissipative parallel shear flow must be two -dimensional. The validity of the temporal scale assumption which must be made in order to assess the stability of dissipative flows using normal-mode analysis is tested by solving the linear initial-value problem and comparing the results with the results of the stability analyses. Next, we present results from a sequence of two -dimensional nonlinear numerical simulations of flows near the KH-Holmboe transition (i.e. having bulk Richardson numbers near 1/4), which clearly illustrate the structural relationship between Holmboe and Kelvin-Helmholtz waves at finite amplitude. Finally, the time-dependent nonlinear wave states delivered by these simulations are subjected to a three-dimensional normal-mode stability analysis in order to discover the physical processes that might drive the flows towards a chaotic state. Strong secondary instability is found to persist up to large spanwise wavenumbers, with no indication of a preferred length scale. These results enable us to advance theoretical explanations for two distinct mechanisms through which laboratory flows exhibiting Holmboe instability are observed to become turbulent.