Mixing Coefficient in Stably Stratified Flows

Lakshmi Kantha,Hubert Luce

doi:10.1175/jpo-d-18-0139.1

Abstract

AbstractTurbulent mixing in the interior of the oceans is not as well understood as mixing in the oceanic boundary layers. Mixing in the generally stably stratified interior is primarily, although not exclusively, due to intermittent shear instabilities. Part of the energy extracted by the Reynolds stresses acting on the mean shear is expended in increasing the potential energy of the fluid column through a buoyancy flux, while most of it is dissipated. The mixing coefficient χm, the ratio of the buoyancy flux to the dissipation rate of turbulence kinetic energy ε, is an important parameter, since knowledge of χm enables turbulent diffusivities to be inferred. Theory indicates that χm must be a function of the gradient Richardson number. Yet, oceanic studies suggest that a value of around 0.2 for χm gives turbulent diffusivities that are in good agreement with those inferred from tracer studies. Studies by scientists working with atmospheric radars tend to reinforce these findings but are seldom referenced in oceanographic literature. The goal of this paper is to bring together oceanographic, atmospheric, and laboratory observations related to χm and to report on the values deduced from in situ data collected in the lower troposphere by unmanned aerial vehicles, equipped with turbulence sensors and flown in the vicinity of the Middle and Upper Atmosphere (MU) radar in Japan. These observations are consistent with past studies in the oceans, in that a value of around 0.16 for χm yields good agreement between ε derived from turbulent temperature fluctuations using this value and ε obtained directly from turbulence velocity fluctuations.

Highlights

Apart from the well-mixed turbulent layers adjacent to the air–sea interface and the ground and above the ocean bottom, for the most part, the fluid columns in both the oceans and the atmosphere are stably stratified
From measurements made over the past few decades, oceanographers have discovered that tracer experiments in the upper ocean show that an excellent agreement is obtained for tracer diffusion if the mixing coefficient xm is assumed to be 0.20. Theory requires that it be a function of the strength of stratification as indicated, for example, by the gradient Richardson number Ri
Shigaraki UAV-Radar Experiment (ShUREX) 2016 observations reported above are consistent with past studies in the oceans, the atmosphere, and the laboratory

Summary

Introduction

When the fluid is stably stratified, part of the energy input to creating turbulence goes to increase the potential energy of the fluid column through a vertical buoyancy flux. There have been some studies of the turbulent mixing coefficient in stably stratified flows by atmospheric scientists, starting with Lilly et al (1974), who suggested a value of 0.33 corresponding to the upper limit on the flux Richardson number of 0.25. More often than not, the gradient Richardson number Ri associated with the observed values of the mixing coefficient is not measured and remains unknown [see the review by Gregg et al (2018)]. Most of these studies cannot shed light on the issue of Ri dependence of xm or why a value of around 0.2 yields acceptable diffusivities.

Governing equations

ShUREX 2016 observations

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Findings

Concluding remarks