▪ Abstract Passive scalar behavior is important in turbulent mixing, combustion, and pollution and provides impetus for the study of turbulence itself. The conceptual framework of the subject, strongly influenced by the Kolmogorov cascade phenomenology, is undergoing a drastic reinterpretation as empirical evidence shows that local isotropy, both at the inertial and dissipation scales, is violated. New results of the complex morphology of the scalar field are reviewed, and they are related to the intermittency problem. Recent work on other aspects of passive scalar behavior—its spectrum, probability density function, flux, and variance—is also addressed.

Passive Scalars in Turbulent Flows

Atmospheric boundary layers with weak stratification are relatively well described by similarity theory and numerical models for stationary horizontally homogeneous conditions. With common strong stratification, similarity theory becomes unreliable. The turbulence structure and interactions with the mean flow and small-scale nonturbulent motions assume a variety of scenarios. The turbulence is intermittent and may no longer fully satisfy the usual conditions for the definition of turbulence. Nonturbulent motions include wave-like motions and solitary modes, two-dimensional vortical modes, microfronts, intermittent drainage flows, and a host of more complex structures. The main source of turbulence may not be at the surface, but rather may result from shear above the surface inversion. The turbulence is typically not in equilibrium with the nonturbulent motions, sometimes preventing the formation of an inertial subrange. New observational and analysis techniques are expected to advance our understanding of...

Stably Stratified Atmospheric Boundary Layers

The evolution of unsheared grid-generated turbulence in a stably stratified fluid was investigated in a closed-loop salt-stratified water channel. Simultaneous single-point measurements of the horizontal and vertical velocity and density fluctuations were obtained, including turbulent mass fluxes central in understanding the energetics of the fluctuating motion. When the buoyancy lengthscale was initially substantially larger than the largest turbulent scales, the initial behaviour of the velocity and density fields was similar to that in the non-stratified case. With further downstream development, the buoyancy lengthscale decreased while the turbulence scale grew. Deviations from neutral behaviour occurred when these two lengthscales became of the same order, after the initially larger inertial forces associated with the initial kinetic energy had become weaker and buoyancy forces became important.Buoyancy forces produced anisotropy in the largest scales first, preventing them from overturning, while smaller-scale isotropic turbulent motions remained embedded within the larger-scale wave motions. These small-scale motions exhibited classical turbulent behaviour and scaled universally with Kolmogorov length and velocity scales. Eventually even the smallest scales of the decaying turbulence were affected by buoyancy, all isotropic motions disappeared, and Kolmogorov scaling failed. The turbulent vertical mass flux decreased to zero for this condition, indicating that the original turbulent field had been completely converted to random internal wave motions.The transition from a fully turbulent state to one of internal waves occurred rapidly in a time less than the characteristic time of the turbulence based on the largest-scale eddies found in the flow at transition. The dissipation rate for complete transition to a wave field was found to be of the order of et = 24.5νN2, where ν is the kinematic viscosity and N the Brunt-Vaisala frequency. This is in fairly good agreement with the value 30νN2 predicted by Gibson (1980, 1981).The present experiments have determined quantitative limits on the range of active turbulent scales in homogeneous stratified turbulence, in terms of an upper limit near the buoyancy lengthscale and a lower limit determined by viscosity in the usual way. This description has been used here to help explain and assimilate the results from the earlier stratified grid-turbulence experiments of Lin & Veenhuizen (1975) and Dickey & Mellor (1980). While some of the features of the present observations may be qualitatively seen in the numerical simulations of the problem of Riley, Metcalfe & Weissman (1981), there are fundamental differences, probably due in part to large differences in initial lengthscale ratios and in the limited range of scales attainable in numerical simulations. The present experiments may serve as a useful test case for future modelling and interpretation of the behaviour of turbulence in stratified flows observed in the oceans and atmosphere.

Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid

The validity of the assumption of local isotropy is investigated using measurements of three orthogonal components of the turbulent velocity fields associated with initially high-Reynolds-number geophysical turbulence The turbulent fields, generated by various large-scale internal motions caused by tidal flows over an estuarine sill, decay under the influence of stable mean density gradients With measurements from sensors mounted on a submersible, we examine the evolution of spectral shapes and of ratios of cross-stream to streamwise components, as well as the degree of high-wavenumber universality, for the observational range of the parameter I≡ ks/kb = lb/ls This ratio is a measure of separation between the Kolmogoroff wavenumber ks≡ (e/ν3)¼ ≡ 2π/ls typical of scales by which turbulent kinetic energy has been dissipated (at rate e), and the buoyancy wavenumber kb ≡ (N3/e)½ ≡ 2π/lb typical of scales at which the ambient stratification parameter N ≡ (−gρz/ρ0)½ becomes important For values of I larger than ∼ 3000, inertial subranges are observed in all spectra, and the spectral ratio ϕ22/ϕ11 of cross-stream to streamwise spectral densities reaches the isotropic value of 4/3 for about a decade in wavenumber As ks/kb decreases, inertial subranges vanish, but spectra of the cross-stream and streamwise components continue to satisfy isotropic relationships at dissipation wavenumbers We provide a criterion for when e may safely be estimated from a single measured component of the dissipation tensor, and also explore questions of appropriate low-wavenumber normalization for buoyancy-modified turbulence

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112084001592

Local isotropy and the decay of turbulence in a stratified fluid

The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances.
The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri [approximate] 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy.
Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid.

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Direct numerical simulation of stratified homogeneous turbulent shear flows

To distinguish rigorously between wave motion and turbulence in a stratified fluid seems impossible, although useful approximations seem feasible. The task is made more difficult by the fact that properties of turbulence are very little like those described in most theories of turbulence, even when the Reynolds number is quite high.

R. W. Stewart

Papers

Turbulence and Waves in a Stratified Atmosphere