Jörg Imberger

Natural convection in a shallow cavity with differentially heated end walls. Part 1. Asymptotic theory

The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes : a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. A solution valid at all orders in the aspect ratio A is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A → 0, are derived.

Structure of bubble plumes in linearly stratified environments

Bubble plumes in a linearly stratified ambient fluid are studied. Four well-defined flow regions were observed: an upward-moving bubble core, an inner plume consisting of a mixture of bubbles and relatively dense fluid, an annular downdraught and beyond that a horizontal intrusion flow. Depending on the gas flow rate with respect to the stratification, three types of intrusions were documented. At large gas flow rates a single intrusion was observed. As the gas flow rate was decreased, the buoyancy flux was insufficient to carry the lower fluid to the surface and a stack of intrusions were formed. At very low gas flow rates the intrusions became unsteady. The transition between these three regimes was observed to occur at critical values of the parameters N 3 H 4 /( Q B g ), Q B g / (4πα 2 u 3 s H ), and H / H T , where N is the buoyancy frequency, H is the water depth, H T is equal to H + H A , H A being the atmospheric pressure head, Q B is the gas flow rate at the bottom, g the acceleration due to gravity, α the entrainment coefficient and u s the differential between the bubble and the average water velocity commonly called the slip velocity. The height between intrusions was found to scale with the Ozmidov length ( Q B g / N 3 ) ¼ , the plunge point entrainment with the inner plume volume flux (

On the Nature of Turbulence in a Stratified Fluid. Part II: Application to Lakes

(Q_0 g)^{\frac{3}{4}} N^{-\frac{5}{4}}

The degeneration of large-scale interfacial gravity waves in lakes

and the radial distance to the plunge point with ( Q 0 g / N 3 ) ¾ , where Q 0 is the gas flow rate at the free surface. These results were used to construct a double annular plume model which was used to investigate the efficiency of conversion of the input bubble energy to potential energy of the stratification; the efficiency was found to first increase, reach a maximum, then decrease with decreasing gas flow rate. This agreed well with the results from the laboratory experiments.

The classification of Mixed-Layer Dynamics of Lakes of Small to Medium Size

Abstract A strong debate has continued for a number of years over the magnitude of the ratio of the buoyancy flux b to the rate of production of turbulent kinetic energy from the mean velocity sheer. This ratio has traditionally been called the flux Richardson number Rf. In part I of Ivey and Imberger this definition was generalized by broadening the denominator to include all sources and sinks of mechanical turbulent kinetic energy, the net being defined as m. It was shown that for mechanically energized turbulence (m > 0, b > 0) the magnitude of Rf was completely determined by the magnitude of the overturn Froude FrT and the Reynolds ReT numbers By contrast, for the penetrative convection case (b < 0) Rf was shown to be dependent only on the distance from the source of buoyancy. In the present contribution, scaling arguments are presented for the magnitudes of FrT and ReT. It is shown that these may vary widely and depend, in the first instance, on the physics of the underlying processes energizing the ...

The initial response of a stratified lake to a surface shear stress

Mechanisms for the degeneration of large-scale interfacial gravity waves are identified for lakes in which the effects of the Earths rotation can be neglected. By assuming a simple two-layer model and comparing the timescales over which each of these degeneration mechanisms act, regimes are defined in which particular processes are expected to dominate. The boundaries of these regimes are expressed in terms of two lengthscale ratios: the ratio of the amplitude of the initial wave to the depth of the thermocline, and the ratio of the depth of the thermocline to the overall depth of the lake. Comparison of the predictions of this timescale analysis with the results from both laboratory experiments and field observations confirms its applicability. The results suggest that, for small to medium sized lakes subject to a relatively uniform windstress, an important mechanism for the degeneration of large-scale internal waves is the generation of solitons by nonlinear steepening. Since solitons are likely to break at the sloping boundaries, leading to localized turbulent mixing and enhanced dissipation, the transfer of energy from an initial basin-scale seiche to shorter solitons has important implications for the lake ecology.

Mixing processes relevant to phytoplankton dynamics in lakes

Abstract An analysis of the time scales of processes relevant to wind mixing in lakes indicates that the response of a lake to wind may be classified into four regimes with respect to thermocline deepening behavior, depending on the relative sizes of the parameters describing wind strength, basin size and stratification. The dependence is described in terms of a mixed layer Richardson number and the aspect ratio of the mixed layer thickness to length. The classification is used to explain the diversity of phenomena reported in the literature for wind events in a number of different lakes and laboratory tanks which are either short enough or narrow enough for rotational effects to be unimportant. The classification is derived with reference to a two-layer, rectangular basin in the absence of Coriolis forces and surface heating. The classification is exended in a simple way to more realistic stratifications and basin shapes to predict the overall mixing features of a wind event. Response to wind varies from...

Pseudo two-dimensional simulations of internal and boundary fluxes in stratified lakes and reservoirs

Laboratory experiments are used to study the initial response of a stratified fluid to the action of a wind stress. The experiments are described in the context of a parameterization scheme that quantifies the strength of the applied stress relative to the bulk stability of the fluid and also the duration of the wind stress relative to the periods of the waves generated by the stress. This study concentrates on the first fundamental internal wave period in experiments where the fluid is considered to have upwelled, i.e. the stratified region of the fluid reaches the surface at the upwind endwall. The majority of the experiments use three-layer initial density profiles as an approximation to a continuously stratified water column. A linear model using normal modes proved successful prior to the commencement of upwelling and this enabled an estimate to be made of the time at which upwelling occurred. At this point the wave development ceased and the flows developed via entrainment mechanisms. Consideration of the energy budget showed that little of the input energy was stored in the system. The initial mixing efficiency, defined as the ratio of the mean potential energy gained to the energy imparted by the belt, never exceeded 30 %. Peak efficiency occurred when the surface stress was just sufficient to bring the interfacial region to the surface.

Matching Temperature and Conductivity Sensor Response Characteristics

Abstract Active turbulence in lakes is confined to the surface mixed layer, to boundary layers on the lake sides and bottom, and to turbulent patches in the interior. The density stratification present in most lakes fundamentally alters the pathways connecting external mechanical energy inputs, for example by wind, with its ultimate fate as dissipation to heat; the density stratification supports internal waves and intrusions that distribute the input energy throughout the lake. Intrusions may be viewed as internal waves with zero horizontal wavenumber and are formed each time localised mixing occurs in a stratified fluid. Intrusions are also formed in the epilimnion by differential heating or cooling and by differential deepening. The fraction of lake volume below the diurnal mixed layer that is subject to active turbulence is very small, probably of the order of 1% or less in small to medium‐sized lakes. By contrast, in the surface mixed layer, turbulence is less intermittent and maintains phytoplankton...

Measurements of diapycnal diffusivities in stratified fluids

Abstract New mixing algorithms to model the vertical mixing processes in stratified lakes have been developed for the Dynamic Reservoir Simulations Model, DYRESM, and have been validated using five lakes of different size, shape and wind forcing characteristics. An analysis of temperature profiles from Lake Kinneret, Canning Reservoir and Mundaring Reservoir, were used to develop a strong inverse relationship between the Lake number and lake‐wide average vertical eddy diffusion coefficient. Analysis of microstructure data collected in Lake Kinneret using the portable flux profiler suggests that the development of a turbulent benthic boundary layer (BBL) accounts for a large proportion of the lake‐wide average vertical flux. A pseudo two‐dimensional model with explicit BBL and internal fluxes was developed based on the Lake Kinneret field observations and similar investigations in the literature. A sensitivity analysis revealed that improvements in the ability of DYRESM to model the diverse range of lakes considered without user‐calibration was attributable to a wind‐sheltering algorithm and a process‐based description of BBL and internal fluxes.