Dynamically induced uncertainty in stratified turbulent mixing models
The interaction of turbulence with buoyancy effects due to gravity is of central importance in a wide range of environmental and industrial flows. On both global and regional scales in the ocean, it is necessary to model small-scale turbulent processes and the way in which they enhance mixing in order to form an accurate picture of the vertical transport of heat, nutrients and anthropogenic pollutants. This applies to parameterisation schemes in coarsely resolved large-scale numerical simulations, as well as the processing and interpretation of spatio-temporally sparse observational field measurements. Most studies of stratified turbulence modelling support the conclusion that models of mixing should depend on parameters of the flow derived from diagnosable turbulent statistics in addition to (or in some cases, instead of) those associated with the ambient background state and the physical properties of the fluid. However, there is no consensus on precisely what these parameters should be, how many are necessary, and what functional form these dependencies should take. Using data from a variety of idealised direct numerical simulations (DNS), and with careful consideration of the underlying fluid dynamics, this thesis serves to highlight the compound importance of the many varieties of uncertainty that can arise during the different stages of model application, from selecting and pre-processing suitable inputs to then transforming these inputs into estimates for relevant mixing properties. For the purposes of illustration, we frame our discussion in terms of two classical and commonly used models for calculating local values of turbulence-enhanced vertical diffusivity in the ocean, described in chapter 1.
We first investigate mixing processes driven by Kelvin-Helmholtz instability, which acts to overturn density interfaces in a stratified, vertically sheared flow. Chapters 2 and 3 illustrate two distinct ways in which geophysically realistic departures from an idealised model background flow contribute towards generating uncertainty in various important mixing diagnostics. In particular, chapter 2 tackles issues surrounding the `suitable' definition of a local mixing region and associated background density profile in the presence of inverted local density gradients, as well as considering the relevance of different approximate overturning length scales for estimating the mixing efficiency of the flow. Chapter 3 utilises a time-dependent forcing scheme to highlight the modification of mixing behaviour by a competition between shear and convective processes driving turbulence. Chapters 4 and 5 are concerned with turbulence in a strongly stratified environment that exhibits the formation of a characteristic vertically ‘layered’ structure in the velocity and density fields. The results are motivated by considering the inferences that can be made from typical observational datasets consisting of a limited number of measured velocity and density gradients in the flow. Chapter 4 looks at the transient pathways to strongly stratified turbulence in a freely evolving purely horizontal shear flow, discussing the potential links to parameterisations based on individual overturning events. Looking to the future, chapter 5 considers the emerging relevance of machine learning tools for problems in stratified turbulence. We introduce a novel data-driven method to investigate and improve upon some of the existing uncertainties due to small-scale anisotropy that arise when making estimates of energy dissipation rates in flows with low levels of turbulence intensity, as measured by the buoyancy Reynolds number Reb.