ALBERT

Description: A laboratory study was carried out to directly measure the turbulence properties in a benthic boundary layer (BBL) above a uniformly sloping bottom where the BBL is energized by internal waves. The ambient fluid was continuously stratified and the steadily forced incoming wave field consisted of a confined beam, restricting the turbulent activity to a finite region along the bottom slope. Measurements of dissipation showed some variation over the wave phase, but cycle-averaged values indicated that the dissipation was nearly constant with height within the BBL. Dissipation levels were up to three orders of magnitude larger than background laminar values and the thickness of the BBL could be defined in terms of the observed dissipation variation with height. Assuming that most of the incoming wave energy was dissipated within the BBL, predicted levels of dissipation were in good agreement with the observations. Measurements were also made of density and two orthogonal components of the velocity fluctuations at discrete heights above the bottom. Cospectral estimates of density and velocity fluctuations showed that the major contributions to both the vertical density flux and the momentum flux resulted from frequencies near the wave forcing frequency, rather than super-buoyancy frequencies, suggesting a strong nonlinear interaction between the incident and reflected waves close to the bottom. Within the turbulent BBL, time-averaged density fluxes were significant and negative near the wave frequencies but negligible at frequencies greater than the buoyancy frequency N. While dissipation rates were high compared to background laminar values, they were low compared to the value of ε(tr) ~ 15 ν N2, the transition value often used to assess the capacity of a stratified flow to produce mixing. Existing models relating mixing to dissipation rate rely on the existence of a positive-definite density flux at frequencies greater than N as a signature of fluid mixing and therefore cannot apply to these experiments. We therefore introduce a simple model, based on the concept of diascalar fluxes, to interpret the mixing in the stratified fluid in the BBL and suggest that this may have wider application than to the particular configuration studied here.

Description: Laboratory experiments are used to investigate the processes governing steady convectively driven circulation in a basin that communicates with a large external reservoir over a shallow sill. The motion is maintained by a steady loss of buoyancy distributed over the surface of the basin. Turbulent convection associated with the forcing produces a horizontal buoyancy gradient across the sill and the resulting mean flow consists of a layer directed into the basin near the surface with a dense counterflow below. To first order, the magnitude of the exchange flow over the sill is determined by the horizontal momentum balance within the basin. Measurements of the mean and turbulent flow fields are used to show that inertia, buoyancy and friction may each contribute significantly to the balance. The interior flow produces a horizontal pressure gradient near the surface which must also contribute to the momentum balance. The density of the lower layer at the sill reflects the cumulative effect of interior processes, such as mixing, and these in turn influence the hydraulically controlled exchange flow over the sill. The basin dynamics are therefore coupled in a nonlinear fashion with the submaximal sill exchange. This coupling is investigated first by showing how interior processes are affected by changes in the magnitude of the forcing, and then by observing the associated variation of the flow state at the sill. The flow state is defined in terms of its relative proximity to the theoretical maximal exchange limit. Results show that the exchange flows are submaximal with flow rate approximately 85% of the maximal limit. This state appears to change very little in response to increasing forcing. For a stratified basin, which exhibits a deep stagnant layer under the convectively driven near-surface exchange flow, the possibility of basin ventilation or erosion of deep fluid exists in the long term. This process and its dependence on external parameters is also explored.

Description: A laboratory investigation of exchange flows near the two-layer hydraulic limit is used to examine the generation of shear instability at the interface dividing the two layers. The present experiments differ from many previous investigations into shear instability, in that the instabilities are an active part of a quasi-steady flow regime rather than the product of a controlled initial state. Regimes characterized by either Kelvin-Helmholtz or Holmboe's instability are found to be separated by a well-defined transition. Observations of the transition from Kelvin-Helmholtz to Holmboe's instability are compared to predictions from scaling arguments that draw on elements of both two-layer hydraulic theory and linear stability theory. The characteristics of unstable modes near the transition, and the structure of both classes of instability are examined in detail.

Description: Internal hydraulic theory is often used to describe idealized bi-directional exchange flow through a constricted channel. This approach is formally applicable to layered flows in which velocity and density are represented by discontinuous functions that are constant within discrete layers. The theory relies on the determination of flow conditions at points of hydraulic control, where long interfacial waves have zero phase speed. In this paper, we consider hydraulic control in continuously stratified exchange flows. Such flows occur, for example, in channels connecting stratified reservoirs and between homogeneous basins when interfacial mixing is significant. Our focus here is on the propagation characteristics of the gravest vertical-mode internal waves within a laterally contracting channel. Two approaches are used to determine the behaviour of waves propagating through a steady, continuously sheared and stratified exchange flow. In the first, waves are mechanically excited at discrete locations within a numerically simulated bi-directional exchange flow and allowed to evolve under linear dynamics. These waves are then tracked in space and time to determine propagation speeds. A second approach, based on the stability theory of parallel shear flows and examination of solutions to a sixth-order eigenvalue problem, is used to interpret the direct excitation experiments. Two types of gravest mode eigensolutions are identified: vorticity modes, with eigenfunction maxima centred above and below the region of maximum density gradient, and density modes with maxima centred on the strongly stratified layer. Density modes have phase speeds that change sign within the channel and are analogous to the interfacial waves in hydraulic theory. Vorticity modes have finite propagation speed throughout the channel but undergo a transition in form: upwind of the transition point the vorticity mode is trapped in one layer. It is argued that modes trapped in one layer are not capable of communicating interfacial information, and therefore that the transition points are analogous to control points. The location of transition points are identified and used to generalize the notion of hydraulic control in continuously stratified flows.

Description: An extended Korteweg-de Vries (KdV) equation is derived that describes the evolution and propagation of long interfacial gravity waves in the presence of a strong, space-time varying background. Provision is made in the derivation for a spatially varying lower depth so that some topographic effects can also be included. The extended KdV model is applied to some simple scenarios in basins of constant and varying depths, using approximate expressions for the variable coefficients derived for the case when the background field is composed of a moderate-amplitude ultra-long wave. The model shows that energy can be transferred either to or from the evolving wave packet depending on the relative phases of the evolving waves and the background variation. Comparison of the model with laboratory experiments confirms its applicability and usefulness in examining the evolution of weakly nonlinear waves in natural systems where the background state is rarely uniform or steady.

Description: A simple model is developed, based on an approximation of the Boussinesq equation, that considers the weakly nonlinear evolution of an initial interface disturbance in a closed basin. The solution consists of the sum of the solutions of two independent Korteweg-de Vries (KdV) equations (one along each characteristic) and a second-order wave-wave interaction term. It is demonstrated that the solutions of the two independent KdV equations over the basin length [O, L] can be obtained by the integration of a single KdV equation over the extended reflected domain [O, 2L]. The main effect of the second-order correction is to introduce a phase shift to the sum of the KdV solutions where they overlap. The results of model simulations are shown to compare qualitatively well with laboratory experiments. It is shown that, provided the damping timescale is slower than the steepening timescale, any initial displacement of the interface in a closed basin will generate three types of internal waves: a packet of solitary waves, a dispersive long wave and a train of dispersive oscillatory waves.

Description: Mechanisms for the degeneration of large-scale interfacial gravity waves are identified for lakes in which the effects of the Earth's 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.