Kraig B. Winters

Available potential energy and mixing in density-stratified fluids

A conceptual framework for analysing the energetics of density-stratified Boussinesq fluid flows is discussed. The concept of gravitational available potential energy is used to formulate an energy budget in which the evolution of the background potential energy, i.e. the minimum potential energy attainable through adiabatic motions, can be explicitly examined. For closed systems, the background potential energy can change only due to diabatic processes. The rate of change of background potential energy is proportional to the molecular diffusivity. Changes in the background potential energy provide a direct measure of the potential energy changes due to irreversible diapycnal mixing. For open systems, background potential energy can also change due to boundary fluxes, which can be explicitly measured. The analysis is particularly appropriate for evaluation of diabatic mixing rates in numerical simulations of turbulent flows. The energetics of a shear driven mixing layer is used to illustrate the analysis.

The stochastic nature of larval connectivity among nearshore marine populations

Many nearshore fish and invertebrate populations are overexploited even when apparently coherent management structures are in place. One potential cause of mismanagement may be a poor understanding and accounting of stochasticity, particularly for stock recruitment. Many of the fishes and invertebrates that comprise nearshore fisheries are relatively sedentary as adults but have an obligate larval pelagic stage that is dispersed by ocean currents. Here, we demonstrate that larval connectivity is inherently an intermittent and heterogeneous process on annual time scales. This stochasticity arises from the advection of pelagic larvae by chaotic coastal circulations. This result departs from typical assumptions where larvae simply diffuse from one site to another or where complex connectivity patterns are created by transport within spatially complicated environments. We derive a statistical model for the expected variability in larval settlement patterns and demonstrate how larval connectivity varies as a function of different biological and physical processes. The stochastic nature of larval connectivity creates an unavoidable uncertainty in the assessment of fish recruitment and the resulting forecasts of sustainable yields.

Diascalar flux and the rate of fluid mixing

We define the rate at which a scalar θ mixes in a fluid flow in terms of the flux of θ across isoscalar surfaces. This flux θ d is purely diffusive and is, in principle, exactly known at all times given the scalar field and the coefficient of molecular diffusivity. In general, the complex geometry of isoscalar surfaces would appear to make the calculation of this flux very difficult. In this paper, we derive an exact expression relating the instantaneous diascalar flux to the average squared scalar gradient on an isoscalar surface which does not require knowledge of the spatial structure of the surface itself. To obtain this result, a time-dependent reference state θ( t,z *.) is defined. When the scalar is taken to be density, this reference state is that of minimum potential energy. The rate of change of the reference state due to diffusion is shown to equal the divergence of the diffusive flux, i.e. (∂/∂ z *)θ d . This result provides a mathematical framework that exactly separates diffusive and advective scalar transport in incompressible fluid flows. The relationship between diffusive and advective transport is discussed in relation to the scalar variance equation and the Osborn–Cox model. Estimation of water mass transformation from oceanic microstructure profiles and determination of the time-dependent mixing rate in numerically simulated flows are discussed.

Measurements of diapycnal diffusivities in stratified fluids

Linearly stratified salt solutions of different Prandtl number were subjected to turbulent stirring by a horizontally oscillating vertical grid in a closed laboratory system. The experimental set-up allowed the independent direct measurement of a root mean square turbulent lengthscale L,, turbulent diffusivity for mass K ρ , rate of dissipation of turbulent kinetic energy e, buoyancy frequency N and viscosity ν, as time and volume averaged quantities. The behaviour of both L t and K ρ was characterized over a wide range of the turbulence intensity measure, ∈/νN 2 , and two regimes were identified. In the more energetic of these regimes (Regime E, where 300 1000, our measured diffusivities diverge from the model prediction. For example, at ∈/νN 2 ≃ 10 4 there is at least an order of magnitude difference between the measured and predicted diffusivities.

Three-dimensional wave instability near a critical level

The behaviour of internal gravity wave packets approaching a critical level is investigated through numerical simulation. Initial-value problems are formulated for both small- and large-amplitude wave packets. Wave propagation and the early stages of interaction with the mean shear are two-dimensional and result in the trapping of wave energy near a critical level. The subsequent dynamics of wave instability, however, are fundamentally different for two- and three-dimensional calculations. Three-dimensionality develops by transverse convective instability of the two-dimensional wave. The initialy two-dimensional flow eventually collapses into quasi-horizontal vortical structures. A detailed energy balance is presented. Of the initial wave energy, roughly one third reflects, one third results in mean flow acceleration and the remainder cascades to small scales where it is dissipated. The detailed budget depends on the wave amplitude, the amount of wave reflection being particularly sensitive.

A Spectral Model for Process Studies of Rotating, Density-Stratified Flows

A numerical model designed for three-dimensional process studies of rotating, stratified flows is described. The model is freely available, parallel, and portable across a range of computer architectures. The underlying numerics are high quality, based on spectral expansions, and third-order time stepping. Optional submodels include accurate calculation of Lagrangian trajectories. Special consideration has been taken to ensure ease of use by geophysical, as distinguished from computational, scientists. The mathematical and computational methods underlying the model are presented here as are several illustrative applications highlighting the model capabilities and the types of flows amenable to simulation using the model. Sample applications include forced inertial gravity waves, parametric subharmonic instability, shear-driven mixing layers, instability of a low Froude number vortex street, and geostrophic adjustment of intermittent, isolated mixing patches.

Turbulence and Mixing in Holmboe Waves

Abstract Motivated by the tendency of high-Prandtl-number fluids to form sharp density interfaces, the authors investigate the evolution of Holmboe waves in a stratified shear flow through direct numerical simulation. Like their better-known cousins, Kelvin–Helmholtz waves, Holmboe waves lead the flow to a turbulent state in which rapid irreversible mixing takes place. In both cases, significant mixing also takes place prior to the transition to turbulence. Although Holmboe waves grow more slowly than Kelvin–Helmholtz waves, the net amount of mixing is comparable. It is concluded that Holmboe instability represents a potentially important mechanism for mixing in the ocean.

Turbulent mixing in a sloping benthic boundary layer energized by internal waves

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 e tr ≈ 15 vN 2 , 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.

Instability of internal waves near a critical level

Abstract The three-dimensional stability problem is investigated for a family of velocity and density profiles similar in form to those expected for large-amplitude internal gravity waves near a critical level. These profiles exhibit regions of high shear and stable stratification alternating with regions of weak shear and unstable stratification. Analytical solutions are given for inviscid, neutral modes that are similar to those found under neutral conditions with stable stratification. Neutral modes form closed streamline patterns centered at locations of maximal shear, and are not strongly influenced by nearby regions of unstable stratification. Unstable modes are computed numerically. It is shown that the instability mechanism for these wave-like flows fundamentally three-dimensional in character and exhibits both shear and convective dynamics. For flows with parameter values below the neutral curves, unstable modes oriented in the streamwise direction undergo shear instability, while modes oriented orthogonally are convectively unstable. In addition to their intrinsic physical relevance, the results of this study have important implications for the physics and the numerical modeling of breaking internal gravity waves. Two-dimensional models will bias the breaking dynamics by eliminating the possibility for convection oriented in the transverse plane.

Direct Simulation of Internal Wave Energy Transfer

Abstract A three-dimensional nonhydrostatic numerical model is used to calculate nonlinear energy transfers within decaying Garrett–Munk internal wavefields. Inviscid wave interactions are calculated over horizontal scales from about 1 to 80 km and for vertical mode numbers less than about 40 in an exponentially stratified model ocean 2000 m deep. The rate of energy transfer from these scales to smaller, numerically damped scales is used to make predictions of the dissipation rate e in the open ocean midlatitude thermocline. In agreement with the theoretical analyses based on resonant interaction and eikonal theories, the simulation results predict e ∝ Ē2 N2, where Ē and N are the internal wave energy density and the ambient buoyancy frequency respectively. The magnitudes of the simulated dissipation rates are shown to be in good agreement with the dissipation measurements taken from six diverse sites in the midlatitude thermocline. The results suggest that the rates of dissipation and mixing in the ocean...