Gregory Lawrence

Heat transport is one of the most important phenomena in environmental fluid mechanics. Of the heat transport processes, convection is by and large the most important. Convective processes in the environment, however, are rarely straightforward, necessitating investigation into the various complications that can exist for a better understanding of heat transport in real world flows. This dissertation investigates three topical problems in convective processes of environmental fluids, 1) differential cooling of a freshwater lake below the temperature of maximum density, 2) natural convection of a fluid with a temperature dependent viscosity, and 3) analysis methods of observational turbulence data. In the second chapter of this dissertation, a simple box model which describes the cooling of a freshwater lake below the temperature of maximum density is presented. Considering only the quadratic equation of state for (fresh) water and the heat fluxes due to a constant surface heat loss and a time-dependent buoyancy-driven exchange flow, reasonable predictions of the timing of ice-on (freezing) for Base Mine Lake in Alberta, Canada are obtained. In the third chapter of this dissertation, the effect of a temperature dependent viscosity on the dynamics of natural convection is explored. While an assumption of constant viscosity is appropriate for many flows involving air and water, it becomes problematic when considering the convection of fluids like silicate melts and magma which have highly temperature-dependent viscosities. Using asymptotic and numerical analysis, the effect that a temperature dependent viscosity has on buoyancy-driven convection along heated and cooled boundaries is described and quantified. In particular, the rate of heat transfer between the boundary and the fluid is solved for. Finally, in the fourth chapter of this dissertation, one of the most widely used approximations in interpreting scales from turbulence experiments is explored: Taylor's frozen eddy hypothesis (FTH). Using a novel representation of how space and time can be correlated in turbulent flows in application to temperature observations made during up-slope flow events in the roughness sublayer of the atmospheric boundary layer, a way forward for the field with regards to analysing turbulence datasets and the application of FTH is clarified.

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In the atmospheric boundary-layer (ABL), both the turbulent flow dynamics of the air and the structure of the surface below are complex and multi-scale. This dissertation aims to strengthen the theoretical and computational foundation for models that attempt to account for an increasing amount of this complexity. First, the relationship between one-dimensional profiles and the three-dimensional flow field is examined for urban-like geometry. The thesis makes the case that mean profiles, which generally involve explicit or implicit spatial averaging, are best understood as a special case of the volume averaging methodology that was developed in the context of flow through porous media. The two natural definitions of the volume average both have some convenient and some inconvenient properties. The thesis discusses these properties and how they affect the analysis and modelling of urban flow profiles, potentially resulting in leading-order errors if not handled correctly. Next, the computation of momentum budgets is discussed for simulations relying on the immersed-boundary method, where a non-smooth flow field encompasses both fluid and solid regions. The thesis argues that budgets should nonetheless be free of residuals if their computation closely follows the methods used during the simulation. Using a concrete example, it is shown how small numerical details can produce non-negligible contributions, how the handling of forcing terms and boundary nodes can be resolved by careful analysis, and how some ambiguity might have to be tolerated for terms that are derived with mathematical identities that do not hold exactly for discrete data. Finally, a new code for ABL flow simulations is presented. The code is designed for improved adaptability since simulation methods often have to be tailored to the surface geometry and other problem properties. To achieve this goal without sacrificing performance or correctness, the code is written in the high-level, high-performance language Julia, makes use of extensive automated tests, and relies on experience with earlier implementations to clean up the mathematical formulation and implementation of the methods. The code is shown to match numerical results and performance properties of existing codes, and early experience has shown first benefits of the enhanced adaptability.

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Methane ebullition (bubbling) from lake sediments is an important source of atmospheric methane. Rising bubbles can also negatively impact water quality. However, most ebullition surveys measure ebullition at intervals that are much greater than the duration of ebullition events (typically 2-4 days), introducing uncertainties in our understanding of temporal variations in ebullition. In addition, most lakes are seasonally ice-covered, but this period is often ignored.In this dissertation, I provide high-frequency (minute to hourly) ebullition data from Base Mine Lake, a boreal pit lake in Alberta, Canada. The high-frequency data reveal that during ice cover ebullition almost exclusively occurs when atmospheric pressure decreases below a threshold, which is approximately the average pressure. During open-water season, ebullition is still regulated by pressure variations. Over the 4-month open-water period, 24 ebullition peaks were observed. 22 of them occurred when atmospheric pressure was at its local trough. Semi-empirical equations are provided that can reproduce the time-series of ebullition events well. A physics-based ebullition model is also developed. This model only utilizes a minimum number of physical principles; yet its results compare well with field observations. The analyses suggest that there exist three different regimes, namely the saturation-controlled regime, the pressure-controlled regime and the undersaturation regime. Inside the pressure-controlled regime, ebullition is regulated by pressure variations and changes in atmospheric pressure can trigger and stop ebullition events.Due to the opaque nature of sediments, the rise of bubbles inside sediments is not readily observed. In laboratory experiments, transparent Carbopol is used as a surrogate for sediments. A layer of Carbopol is capped with a layer of water and air bubbles are injected at the base of the Carbopol layer to mimic ebullition. Many novel behaviors are observed. For example, a gas-venting conduit develops inside the Carbopol layer. Inside this conduit, rising bubbles resemble Taylor bubbles, but have velocities that are 5-7 times that of the standard Taylor bubble. Rising bubbles also expel water out of the conduit and overlying water flows back after the bubbles escape. This process can enhance the exchange of heat and contamination between a lake and its sediments.

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Turbidity from glacial meltwater limits light penetration in lakes and reservoirs with potential ecological consequences. Field observations, theory and hydrodynamic modelling were used to investigate changes in turbidity in response to changes in reservoir operation (e.g. water level, inflows and withdrawals), and to natural processes (e.g. particle settling, dispersion and upwelling) in Carpenter Reservoir, a long and narrow, hydroelectric reservoir situated in a glaciated catchment in British Columbia, Canada. Profiles of temperature, conductivity and turbidity, combined with meteorological measurements revealed that, during summer the relatively dense inflows into Carpenter Reservoir plunged into the hypolimnion, and despite the high glacial load entering the reservoir, the turbidity of the epilimnion declined due to particle settling. Occasionally, down-valley winds were strong enough to upwell turbid, metalimnetic water to the free surface. The upwelled water was blown down-valley, setting up a longitudinal turbidity gradient in the epilimnion. The dominant drivers of epilimnetic turbidity variations were particle settling out of the epilimnion, longitudinal dispersion along the length of the epilimnion, and wind-driven upwelling into the epilimnion. The relative importance of these drivers was investigated using scaling arguments and a mechanistic model based on the one-dimensional diffusion equation. Two nondimensional parameters were obtained: the epilimnetic inflow parameter, I, a measure of the turbidity flux into the epilimnion; and the dispersion parameter, D, a measure of longitudinal dispersion. In the case of Carpenter Reservoir, I
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Flows in the oceans and the atmosphere often involve the horizontal shearing of stably stratified density layers. These shear layers are subject to hydrodynamic instabilities that cause the transition from laminar to turbulent flow. When a density interface is sharper than the velocity interface, the Holmboe instability can arise. In this thesis, the dynamics and mixing of the Holmboe instability are investigated theoretically, numerically and in the laboratory. First, the spatial evolution of the Holmboe instability along an arrested salt wedge is investigated both in laboratory experiments and using linear stability analysis. The spatial evolution is dependent on the variation in the mean velocity and density profiles along the length of the salt wedge. The linear stability analysis incorporates this variation and predicts the growth rate, wavelength, phase speed and wave steepness of the Holmboe instability along the length of the salt wedge. These predictions are consistent with the laboratory measurements.The Reynolds stress ellipses associated with Holmboe instabilities are investigated using linear stability analysis, single wavelength simulations, multiple wavelength simulations, and a laboratory experiment. Conventionally, only the statistics of horizontal and vertical velocity perturbation pairs, (u',w'), are used to show the degree of anisotropy in perturbation fields. Here, a theory-based approach is used to investigate this anisotropy. Linear stability analysis predicts the aspect ratio of the major and minor axes and the orientation angle of the Reynolds stress ellipses, and thus predicts the resultant Reynolds stresses. Finally, the influence of initial perturbations on the mixing generated by Holmboe instabilities is examined. Initial perturbations are commonly used in direct numerical simulations to stimulate the shear instability of stratified fluids. The amplitudes of the primary and subharmonic Holmboe modes determine the development of the Holmboe instabilities. The excitation of the subharmonic mode induces the merging of the primary Holmboe mode, and further increases the amplitude of Holmboe waves and the overall mixing.

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Due to a number of factors including energy security and climate change, there is an urgent need to transition the global energy supply to renewables. Two potential sources are tidal-stream and hydrokinetic power, utilizing free-stream water turbines as generating devices. Much of the interest in tidal-stream power comes from resource assessments that suggest that significant amounts of electricity could be produced from tidal currents flowing through straits. These assessments inventoried the kinetic energy flux and do not account for flow reduction due to turbine resistance. As such, they do not present a realistic picture of the resource. An analytical model for flow reduction in tidal straits demonstrates that only 38% of the natural fluid power is theoretically extractible. This model does not capture the behaviour of bays, lagoons, or the open ocean. Maximum power production requires flows to be reduced to 58% of natural and if the flow is kept above 95% of nominal (due to environmental regulations) less than 10% of the total power is available. A large laboratory experiment was built to test the analytical model and the results agree with the analytical model. Predicted future levelized cost of energy from tidal generation in straits is an interplay of reduced production due flow reduction competing with decreasing technology costs. This is modelled, indicating levelized costs of energy will drop initially, then rise due to flow reduction. Considering hydrokinetic power near hydropower stations, a 1D model used Seton Canal data to simulate the installation of turbines. The results show that the installation of hydrokinetic turbines would decrease the output of the existing powerhouse. Furthermore, the decrease in hydroelectric production is greater than the hydrokinetic production. Thus, installing hydrokinetic turbines would cause a net energy loss. In conclusion, there are three key recommendations:1. Policy makers are cautioned in embracing tidal resource assessments that are based solely on kinetic energy flux. 2. Project proponents and regulators are advised to study far-field effects of any proposed free-stream turbine installation.3. Developers, investors, and policy makers are cautioned towards assuming that the long-term cost of energy from tidal power will decrease.

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This dissertation illustrates how the vertical distribution of temperature in lakes can be affected by the fact that the attenuation coefficient of light is often strongly dependent on wavelength. The potential importance of this spectral effect is first examined by considering the solar radiation in isolation, and then by including all non–penetrative heat fluxes using a modified version of the numerical model, DYRESM. Comparing the subsurface spectral irradiance of different lakes reveals that the spectral variability of the attenuation coefficient is more significant when calculating the light intensity in relatively clear lakes than in turbid lakes. Comparisons made between field measurements and theoretical predictions of hypolimnetic heating show the importance of accounting for the spectral irradiance for two relatively clear lakes: Pavilion Lake and Crater Lake. A new parameterization that better describes the spectral attenuation coefficient and the distribution of subsurface irradiance is added to DYRESM. The results obtained, when running the original and the modified DYRESM on Pavilion Lake, show a significant improvement in predicting the thermal structure of the lake with the modified version.The effects of the variation in solar angle and the seasonal variation in water quality on the attenuation coefficient are also examined for Pavilion Lake using DYRESM modified to accept a time varying attenuation coefficient. Simulations were performed for Pavilion Lake using the original and modified versions of DYRESM on diurnal and seasonal scales. Results show no significant improvement in the thermal evolution of the lake when considering the diurnal variations, while slight improvement was shown on a seasonal scale.

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Hydrodynamic instabilities occurring in two dimensional shear flows have been investigated. First, the process of resonant interaction between two progressive interfacial waves is studied. Such interaction produces exponentially growing instabilities in idealized, homogeneous or density stratified, inviscid shear layers. It is shown that two oppositely propagating interfacial waves, having arbitrary initial amplitudes and phases, eventually phase-lock, provided they satisfy a particular condition. Three types of shear instabilities - Kelvin Helmholtz, Holmboe and Taylor have been studied. The above-mentioned condition provides a range of unstable wavenumbers for each instability type, and this range matches the predictions of the canonical normal-mode based linear stability theory. The non-linear evolution of Kelvin-Helmholtz (KH) instability has been studied. The commonly known manifestation of KH is in the form of spiral billows. However, KH evolving from a piecewise linear shear layer is remarkably different; it is characterized by elliptical vortices of constant vorticity connected via thin braids. Using direct numerical simulation and contour dynamics, it is shown that the interaction between two counter-propagating vorticity waves is solely responsible for this KH formation. The oscillation of the vorticity wave amplitude, the rotation and nutation of the elliptical vortex, and straining of the braids have been investigated.Finally, the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow is studied. The base flow is characterized by the cross flow Reynolds number, Reinj and the dimensionless wall velocity, k. Corresponding to each dimensionless wall velocity, k ∈ [0,1], two ranges of Reinj exist where unconditional stability is observed. In the lower range of Reinj , for modest k we have a stabilization of long wavelengths leading to a cut-off Reinj. As Reinj is increased, we see first destabilization and then stabilization at very large Reinj. Analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien-Schlichting waves.

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Kelvin-Helmholtz instabilities are the most commonly studied type of instability in sheared density stratified flows. Turbulence caused by these instabilities is an important mechanism for mixing in geophysical flows. The primary objectives of this study are the evolution of these instabilities and quantifying the mixing they generate using direct numerical simulations. The results are presented in three chapters. First, the evolution of primary Kelvin-Helmhlotz instabilities in two dimensions is studied for a wide range of Reynolds and Prandtl numbers, representing real oceanic and atmospheric flows. The results suggest that some properties of KH billows are predictable by a semi-analytical model. It is shown that a new Corcos-Sherman scale is a useful guide when simulating turbulent KH flow fields. The details of the mixing process generated by the evolution of Kelvin-Helmholtz instabilities as it goes through different stages, is analyzed. As the Reynolds number increases a transition in the overall amount of mixing is found, which is in agreement with previous experimental studies. This transition is explained quantitatively by the entrainment and mixing caused by three-dimensional motions, in addition to those resulted from the two-dimensional growth of the instability.The effect of Prandtl number on mixing is studied to understand the characteristics of high Prandtl number mixing events in the ocean; these cases have usually been approximated by low Prandtl number simulations. The increase in the Pradtl number has some significant implications for the evolution of the billow, the time variation of mixing properties, and the overall mixing.

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The instabilities that occur at a sheared density interface are investigatedin the laboratory, the Fraser River estuary and with Direct Numerical Simulations (DNS).In the laboratory, symmetric Holmboe instabilities are observed duringsteady, maximal two-layer exchange flow in a long channel of rectangular cross section. Internal hydraulic controls at each end of the channel isolate the subcritical region within the channel from disturbances in the reservoirs. Inside the channel, the instabilities form cusp-like waves that propagate in both directions. The phase speed of the instabilities is consistent with linear theory, and increases along the length of the channel as a result of thegradual acceleration of each layer. This acceleration causes the wavelength of any given instability to increase in the direction of flow. As the instabilities are elongated new instabilities form, and as a consequence, the average wavelength is almost constant along the length of the channel. In the Fraser River estuary, a detailed stability analysis is conductedbased on the Taylor-Goldstein (TG) equation, and compared to direct observations in the estuary. We find that each set of instabilities observed coincides with an unstable mode predicted by the TG equation. Each of these instabilities occurs in a region where the gradient Richardson number is less than the critical value of 1/4. Both the TG predictions and echosoundingsindicate the instabilities are concentrated either above or below the density interface. These ‘one-sided’ instabilities are closer in structure to the Holmboe instability than to the Kelvin-Helmholtz instability. Although the dominant source of mixing in the estuary appears to be caused by shearinstability, there is also evidence of small-scale overturning due to boundarylayer turbulence when the tide produces strong near-bed velocities.Many features of the numerical simulations are consistent with lineartheory and the laboratory experiments. However, inherent differences between the DNS and the experiments are responsible for variations in thedominant wavenumber and amplitude of the wave field. The simulations exhibit a nonlinear ‘wave coarsening’ effect, whereby the energy is shifted to lower wavenumber in discrete jumps. This process is, in part, related to the occurrence of ejections of mixed fluid away from the density interface. In the case of the laboratory experiment, energy is transferred to lower wavenumber by the ‘stretching’ of the wave field by a gradually varying mean velocity. This stretching of the waves results in a reduction in amplitude compared to the DNS. The results of the comparison show the dependence of the nonlinear evolution of a Holmboe wave field on temporal and spatial variations of the mean flow.

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The Holmboe instability is known to occur in stratified shear layers that exhibit a relatively thin density interface compared to the shear layer thickness. At finite amplitude the instability appears as cusp-like propagating internal waves. The evolution and identification of these unstable waves is the subject of this thesis. The results are presented in four parts.First, the basic wave field resulting from Holmboe's instability is studied both numerically and experimentally. In comparing basic descriptors of the wave fields, a number of processes are identified that are responsible for differences between the simulations and experiments. These are related to variations in the mean flow that arise due to the different boundary conditions.Holmboe waves are known to produce vertical ejections of interfacial fluid from the wave crests. This `ejection process,' in which stratified fluid is transported against buoyancy forces, is caused by the formation of a vortex couple (i.e. two vorticies of opposite sign that travel as a pair). Results obtained by means of direct numerical simulations also show that the process is primarily two-dimensional and does not require the presence of both upper and lower Holmboe modes.Shear instability is then studied in the highly stratified Fraser River estuary. The observations are found to be in good agreement with the predictions of linear theory. When instability occurs, it is largely as a result of asymmetry between regions of strong shear and density stratification. The structure of the salinity intrusion is found to depend on the strength of the freshwater discharge, in addition to the phase of the tidal cycle. This has implications for whether estuarine mixing takes place through shear instability or boundary layer turbulence.Finally, the asymmetric stratified shear layer, which exhibits a vertical shift between the density interface and the shear layer centre, is examined by the formulation of a diagnostic that is based on the `wave interaction' mechanism of instability growth. This allows for a quantitative assessment of Kelvin-Helmhotz and Holmboe-type growth mechanisms in stratified shear layers. The predictions of the diagnostic are compared to results of nonlinear simulations and observations in the Fraser River estuary.

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