Mixing in stably stratified turbulent flows: improved parameterizations of diapycnal mixing in oceanic flows
Date
2018
Authors
Garanaik, Amrapalli, author
Venayagamoorthy, Subhas Karan, advisor
Bienkiewicz, Bogusz, committee member
Barnes, Elizabeth, committee member
Julien, Pierre Y., committee member
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Abstract
Mixing of fluid with different properties across a gravitationally stable density interface, due to background turbulence is an ubiquitous phenomenon in both natural and engineered flows. Fundamental understanding and quantitative prediction of turbulent mixing in stratified flows is a challenging problem, with a broad range of applications including (but not limited to) prediction of climate, ocean thermohaline circulation, global heat and mass budget, pollutant and nutrients transport, etc. Large scale geophysical flows such as in the ocean and atmosphere are usually stably stratified i.e. the density increases in the direction of gravitational force. The stabilizing nature of the density layers has a tendency to inhibit the vertical motion. In such flows, diapycnal mixing, i.e. mixing of fluid across the isopycnal surfaces of constant density, plays a crucial role in the flow dynamics. In numerical models of large scale flows, turbulent mixing is inherently a small scale phenomenon that is difficult to resolve and is therefore generally parameterized using known bulk parameters of the flow. In oceans, the mixing of water masses is typically represented through a turbulent (eddy) diffusivity of mass Kρ. A widely used formulation for Kρ in oceanic flows is given as Kρ = Γϵ/N2, ϵ is the rate of dissipation of turbulent kinetic energy, N = √(-g/ρ)(∂ρ/∂z) is the buoyancy frequency of the background stratification, ρ is the density, Γ = Rƒ/(1 - Rƒ) is a mixing coefficient and Rƒ is the mixing efficiency, that is widely (but questionably) assumed to be constant or sometimes parameterized. However, a robust and universal parameterization for the mixing efficiency remains elusive to date despite numerous studies on this topic. This research focuses on improved parameterizations of diapycnal mixing through an integration of theoretical knowledge with observational and high resolution numerical simulation data. The main objectives are: (1) to provide a better assessment of field microstructure data and methodology for data analysis in order to develop/test appropriate parameterization of the mixing efficiency, (2) to determine the relevant length and velocity scales for diapycnal mixing, (3) to provide improved parameterization(s) of diapycnal mixing grounded on physical reasoning and scaling analysis, (4) to provide a practical field method to identify the dynamic state of turbulence in stably stratified flows from measurable length scales in the ocean. First, an analysis of field microstructure data collected from different locations in the ocean was performed to verify existing parameterizations. A key finding is that the mixing efficiency, Rƒ does not scale with buoyancy Reynolds number, Rℓb, as been proposed previously by others. Rather, Rƒ depends on the strength of background stratification. In a strongly stratified thermocline, a constant value for the mixing efficiency is found to be reasonable while for weakly stratified conditions (e.g. near boundaries) a parameterization is required. A discussion on different methods to estimate the background shear and stratification from field data is provided. Furthermore, the present state-of-the-art microstructure instruments measure the small scale dissipation rate of turbulent kinetic energy ϵ from one dimensional components by invoking the small scale isotropy assumption that is strictly valid for high Reynolds number flows. A quantitative assessment of the departure from isotropy in stably stratified flows is performed and a pragmatic method is proposed to estimate the true three dimensional dissipation (ϵ3D) from one dimensional dissipation (ϵ1D) obtained from microstructure profilers in the ocean. Next, a scaling analysis for strongly stratified flow is presented to show that, the true diapycnal length scale Ld and diapycnal velocity scale wd can be estimated from the measurable Ellison length scale, LE and a measurable root mean square vertical velocity, w´, using a turbulent Froude number defined as Fr = ϵ/Nk, where k is the turbulent kinetic energy. It is shown that the eddy diffusivity Kρ can be then directly inferred from LE and w´. For weakly stratified flow regimes, Fr > O(1), Kρ ~ w´LE and for strongly stratified flow regimes, Fr < O(1), Kρ ~ w´LE x Fr. This finding is confirmed with direct numerical simulation (DNS) data for decaying as well as sheared stratified turbulence. This result indicates that Fr is a relevant non-dimensional parameter to identify strength of stratification in stably stratified turbulent flows. DNS with particle tracking is performed to separate isopycnal and diapycnal displacements of fluid particles, an analysis that is not possible from an Eulerian approach or from standard field measurements. The Lagrangian analysis show that LE is indeed an isopycnal length scale. Furthermore, having established that Fr is the signature parameter which can describe the state of stratified turbulence, a parameterization of mixing coefficient, Γ (or Rƒ) as a function of turbulent Froude number Fr is developed using scaling arguments of energetics of the flow. Proposed parameterization is then verified using DNS data of decaying, sheared and forced stratified turbulence. It is shown that for Fr << O(1), Γ ~ Fr0, for Fr ~ O(1), Γ ~ Fr-1 and for Fr >> O(1), Γ ~ Fr-2. Finally, a practically useful method to identify the dynamic state of turbulence in stably stratified flows is developed. Two commonly measurable length scales in the ocean are the Thorpe overturning length scale, LT and the dimensionally constructed Ozmidov length scale, LO. From scaling analysis and DNS data of decaying, sheared and forced stratified turbulence a new relation between Fr and the ratio of the length scales, LT/LO is derived. The new scaling is, for LT/LO > O(1), Fr ~ (LT/LO)-2 and for LT/LO < O(1), Fr ~ (LT/LO)-2/3.
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Subject
direct numerical simulations
parameterizations of mixing
turbulent mixing
geophysical flows
diapycnal
stratified turbulence