The water in most reservoirs is density stratified with depth. This stratification leads to the inhibition of vertical movement, consequently, when water is withdrawn from the reservoir it tends to move in a jet-like layer called a withdrawal layer, towards the sink. By placing the sink at a certain depth, one is able to selectively withdrawal water from a limited range of depths and thus obtain water of a desired quality. Much work has been done in this field by considering a simplified boundary geometry, usually rectangular. However little attention has been given to the effects of accurate boundary geometry. For this thesis, five numerical experiments were conducted for the problem of a two-dimensional, viscous, incompressible, slightly-stratified flow towards a sink in a triangular reservoir. / Science, Faculty of / Mathematics, Department of / Graduate

Modal analysis for strongly stratified lakes was extended to obtain a better understanding of the dynamics of the basin-scale motions. By viewing the basin-scale motions as a superposition of modes, that have distinct periods and three-dimensional structures, the method provides a conceptual understanding for the excitation, evolution, and damping of the basin-scale motions. Once the motion has been decomposed into modes, their evolution and energetics may be extracted from hydrodynamic simulation results and field data. The method was applied to Lake Biwa, Japan, and Lake Kinneret, Israel, and used for a theoretical study. The real lake applications showed that winds excited basin-scale motions that had a surface layer velocity structure similar to the wind stress pattern. Three-dimensional hydrodynamics simulations of Lake Biwa indicated that most of the energy input from winds was partitioned into the internal waves that decayed within a few days. The gyres, on the other hand, received much less energy but dominated the dynamics during calm periods due to their slow damping. Analyses of field data from Lake Kinneret suggested that the internal waves, excited by the strong winds every afternoon, were damped over a few days primarily due to bottom friction. Theoretical investigations of damping mechanisms of internal waves revealed that bottom friction induced a velocity anomaly at the top of the boundary layer that drained energy from the nearly inviscid interior by a combination of internal wave cancelling and spin-down. These results indicate that gyres induce long-term horizontal transport near the surface and internal waves transfer energy from winds to near-bottom mixing. Modal structure of dominant basin-scale internal waves can induce large heterogeneity of nearbottom mass transfer processes. The method presented here provides a tool to determine how basin-scale motions impact on biogeochemical processes in stratified lakes.

The onset of summer stratification in temperate lakes and reservoirs forces a decoupling of the hypolimnion from the epilimnion that is sustained by strong density gradients in the metalimnion. These strong gradients act as a barrier to the vertical transport of mass and scalars leading to bottom anoxia and subsequent nutrient release from the sediments. The stratification is intermittently overcome by turbulent mixing events that redistribute mass, heat, dissolved parameters and particulates in the vertical. The redistribution of ecological parameters then exerts some control over the ecological response of the lake. This dissertation is focused on the physics of deep vertical mixing that occurs beneath the well-mixed surface layer in stratified lakes and reservoirs. The overall aim is to improve the ability of numerical models to reproduce deep vertical mixing, thus providing better tools for water quality prediction and management. In the first part of this research the framework of a one-dimensional mixed-layer hydrodynamic model was used to construct a pseudo two-dimensional model that computes vertical fluxes generated by deep mixing processes. The parameterisations developed for the model were based on the relationship found between lake-wide vertical buoyancy flux and the first-order internal wave response of the lake to surface wind forcing. The ability of the model to reproduce the observed thermal structure in a range of lakes and reservoirs was greatly improved by incorporating an explicit turbulent benthic boundary layer routine. Although laterally-integrated models reproduce the net effect of turbulent mixing in a vertical sense, they fail to resolve the transient distribution of turbulent mixing events triggered by local flow properties defined at far smaller scales. Importantly, the distribution of events may promote tertiary motions and ecological niches. In the second part of the study a large body of microstructure data collected in Lake Kinneret, Israel, was used to show that the nature of turbulent mixing events varied considerably between the epilimnion, metalimnion, hypolimnion and benthic boundary layer, yet the turbulent scales of the events and the buoyancy flux they produced collapsed into functions of the local gradient Richardson number. It was found that the most intense events in the metalimnion were triggered by high-frequency waves generated near the surface that grew and imparted a strain on the metalimnion density field, which led to secondary instabilities with low gradient Richardson numbers. The microstructure observations suggest that the local gradient Richardson number could be used to parameterise vertical mixing in coarse-grid numerical models of lakes and reservoirs. However, any effort to incorporate such parameterisations becomes meaningless without measures to reduce numerical diffusion, which often dominates over parameterised physical mixing. As a third part of the research, an explicit filtering tool was developed to negate numerical diffusion in a threedimensional hydrodynamic model. The adaptive filter ensured that temperature gradients in the metalimnion remained within bounds of the measured values and so the computation preserved the spectrum of internal wave motions that trigger diapycnal mixing events in the deeper reaches of a lake. The results showed that the ratio of physical to numerical diffusion is dictated by the character of the dominant internal wave motions.

A numerical model has been developed for the study of stratified tidal circulation in Indian Arm - a representative inlet on the southern coast of British Columbia. Equations for horizontal velocity, salt conservation, continuity, density (calculated as a linear function of salinity), and the hydrostatic approximation govern the dynamics. All equations have been laterally integrated under the assumption of negligible cross-inlet variability. The model is time dependent and includes nonlinear advective terms, horizontal and vertical turbulent diffusion of salt and momentum, and variations in width and depth. Provisions for surface wind stress and a flux of fresh water are also included, although neither was utilized in this study. An explicit finite difference scheme centred in both time and space was used to solve for the horizontal and vertical velocity components, salinity, and surface elevation on a staggered rectangular grid. A backward Euler scheme was used to suppress the computational mode. Tests using a semi-implicit scheme to solve the finite difference equations over realistic topography led to numerical instabilities at modest values of the time step - in spite of the unconditional stability criteria - suggesting that linear stability analysis may give misleading results for strongly nonlinear systems. Surface elevations calculated from tidal harmonic analysis and salinity timeseries derived from linearly interpolated CTD casts were prescribed at the open boundary. Initial and boundary conditions based on observations in Burrard Inlet and Indian Arm during the winter of 1974-75 were used to study the inlet's response to tidal forcing and to simulate the deep-water renewal that occurred during this period. Coefficients for the horizontal turbulent diffusion of momentum and salt were set equal to 10⁶ cm² s⁻¹. Reducing this value by a factor of two was found to have little impact on the solution. A further reduction to 10³ cm² s⁻¹ led to numerical instabilities under conditions of dense water inflow. The side friction term in the momentum balance was tuned to match calculated and observed dissipation rates in Burrard Inlet; leading to good agreement between the observed and calculated barotropic tide. Contour plots of tidal amplitudes and phases for model currents and salinities revealed a standing wave pattern for the K₁ and M₂ internal tides in Indian Arm; thus allowing for the possibility of resonance. A comparison of model results with vertical amplitude and phase profiles from harmonic analysis of Cyclesonde current meter timeseries at two locations in Indian Arm was consistent with this result. A least-squares fit was made of the vertical modal structure in the model to the complex tidal amplitudes. This led to calculations of the kinetic energy contained in each of the modes along the model inlet for the M₂ and K₁ constituents. Most of the energy was found to be contained in the barotropic and first baroclinic modes, with the latter dominating in the deep basin, and the former dominating near the sill. Second mode energy was significant for the K₁ constituent at some locations in Indian Arm. There are clear indications in the model of barotropic tidal energy being radiated into the inlet basin via the internal tide. Simulations of the influx of dense water into Indian Arm yielded exchange rates that are consistent with observed values and suggest the possibility of fine-tuning the model coefficients to allow prediction of future overturning events. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

[Truncated abstract] The lake boundaries are an important source of sediment, nutrients and chemicals. For life inside the lake, the exchange between the lake boundaries (littoral) and lake interior (pelagic) is of central importance to Limnology as the net flux of nutrients into the water column is both the driving force and limiting factor for most algae blooms found during the stratification period. Consequently, the understanding of the relevant processes defining such an exchange is a further step toward a sound basis for future decisions by lake managers in order to ensure high water quality. The objective of this research was to investigate the physical processes responsible for the exchange of water and particles between the lake boundaries and the lake interior. An integrated approach using field experiments and 3D modelling as applied to Lake Kinneret (Israel) is presented. The field data revealed large-scale metalimnion oscillations with amplitudes up to 10 m in response to westerly diurnal winds, the existence of a well-defined suspended particle intrusion into the metalimnion of the lake, characterized by high concentrations of organic matter, and a well-mixed benthic boundary layer (BBL). The changes in the thermal structure explained the observed vertical and horizontal movements of the suspended particle intrusion. The horizontal advective transport via the metalimnion, associated with the velocities induced by the basin-scale mode-two Poincare wave, controlled the exchange between the lake boundaries and lake interior on daily time scales. The observed BBL over the lake slope varied markedly with time and space. Detailed comparison of simulation results with field data revealed that the model captured the lake hydrodynamics for time scales from hours to days. The model could then be used to extract the residual motions in the various regions of the lake. The residual motions below the surface layer were predominantly forced by the basin-scale internal wave motions, but the residual motion in the surface layer was found to be very sensitive to the curl of the wind field. The residual circulation was responsible for redistributing mass throughout the lake basin on time scales from days to weeks. A clear connection of dynamics of the BBL with the large-scale features of the flow was addressed. The time history of the mixing in the BBL and the resulting cross-shore flux was shown to vary with the phase of the basin-scale internal waves.