Explore the nature of density-stratified flow over and around topography, including applications to the flow of the atmosphere and ocean.

As the atmosphere and oceans feature density variations with depth, the flow of a density-stratified fluid over topography is central to various geophysical and meteorological applications and has been studied extensively. For reasons of convenience and mathematical tractability, the majority of theoretical treatments of stratified flow over a finite-amplitude obstacle assume idealized background flow conditions, namely constant free-stream velocity and either a homogeneous or two-layer buoyancy-frequency profile. In this work, a numerical model is developed that accounts for general variations in the buoyancy-frequency profile far upstream and the presence of unsteadiness in the free-stream velocity. The model employs a second-order projection method for solving the Euler equations for stratified flow over locally confined topography in a horizontally and vertically unbounded domain - the flow configuration most pertinent to atmospheric applications - combined with absorbing viscous layers at the upper and lateral boundaries of the computational domain. Using this model, a study is first made of the effect of variations in the buoyancy frequency on the generation of mountain gravity waves.

The dynamics of flows in density-stratified fluids has been and remains now an important topic for scientific enquiry. Such flows arise in many contexts, ranging from industrial settings to the oceanic and atmospheric environments. It is the latter topic which is the focus of this book. Both the ocean and atmosphere are characterised by the basic vertical density stratification, and this feature can affect the dynamics on all scales ranging from the micro-scale to the planetary scale. The aim of this book is to provide a “state-of-the-art” account of stratified flows as they are relevant to the ocean and atmosphere with a primary focus on meso-scale phenomena; that is, on phenomena whose time and space scales are such that the density stratification is a dominant effect, so that frictional and diffusive effects on the one hand and the effects of the earth’s rotation on the other hand can be regarded as of less importance. This in turn leads to an emphasis on internal waves.

Stratified flow over topography is examined in the context of its establishment from rest. A key element of numerical and steady state analytical solutions for large amplitude topographic flow is the splitting of streamlines which then enclose a trapped wedge of mixed fluid above the rapidly moving deeper layer. Measurements have been acquired which show that this wedge arises from small scale instabilities and mixing formed initially by the acceleration of subcritical stratified flow over the obstacle crest. The volume of trapped fluid progressively increases with time, permitting the primary flow to descend beneath it over the lee face of the obstacle. Throughout the evolution of this flow, small scale instability and consequent entrainment is the mechanism responsible for producing the weakly stratified wedge, thus allowing establishment of the downslope flow to take place. Velocity structure of instabilities within the entrainment zone is observed and the associated entrainment rate determined. The entrainment is sufficient to produce a slow downstream motion within the upper layer and a density step between the layers that decreases with downstream distance. The resulting internal hydraulic response is explained in terms of a theory which accommodates the spatially variable density difference across the sheared interface. The measurements described here were acquired in a coastal inlet subject to gradually changing tidal currents. It is proposed that the observed mechanism for flow establishment also has application to atmospheric flow over mountains.

This thesis investigates several aspects of stratified flow over isolated topography in ocean, lake, and atmospheric settings. Three major sub-topics are addressed: steady, inviscid internal waves trapped over topography in a pycnocline stratification, topographically generated internal waves and their interaction with the viscous bottom boundary layer, and flow over large-scale crater topography in the atmosphere. The first topic examines the conditions that lead to very large internal waves trapped over topography in a fluid with a pycnocline stratification. This type of stratification is connected to ocean or lake settings. The steady-state Euler equations of motion are used to derive a single partial differential equation for the isopycnal displacement in supercritical flows under two conditions: a vertically varying background current under the Boussinesq approximation and a constant background current under non-Boussinesq conditions. A numerical method is developed to solve these equations for an efficient exploration of parameter space. Very large waves are found over depression topography when the background flow speed is close to a limiting value. Variations in the background current are examined, as well as comparisons between Boussinesq and non-Boussinesq results. The second topic aims to extend the above subject by considering unsteady, viscous flows. Once again, supercritical flow over topography in a pycnocline stratification generates internal waves. These internal waves interact with the viscous bottom boundary layer to produce bottom boundary instabilities. The three-dimensional aspects of these instabilities are studied under changes in viscosity. The boundary layer instabilities have important implications for sediment transport in the coastal oceans or lakes. Lastly, the final topic is motivated by the connection between dust streaks on the Martian surface and crater topography. Flow over a large 100-km diameter crater is examined with numerical simulations conducted using the Weather Research and Forecasting model. Modifications to the stratification and topography are applied. It is found that a large hydraulic structure of amplitude comparable to the crater depth forms in many cases. This structure may have important implications for dust transport in the atmosphere. In addition, Martian atmospheric parameters are used to study the flow properties under Mars-like conditions.