Diapycnal mixing in the ocean: from ocean sampling to ocean modelling

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Role of internal lee waves

As we have seen in the previous section, the horizontal distribution and vertical structure of the diapycnal mixing, and through Eq. 1.3b turbulent kinetic energy dissipation rate, is more important than their integrated value. Indeed, mixing processes are key in the transformation of the AABW. The two main sources of mixing occur through the interaction of different ingredients, and occur in different places. Through the interaction of the barotropic tide and bottom topography, internal tides are mostly generated along the great ridges North of 50-60°S. Internal lee waves, on the other hand, require deep reaching jets above rough topography, which are mostly found along the Antarctic Circumpolar Current (ACC) South of 50-60°S (Ferrari and Wunsch 2009).
The AABW, during their journey northward, will therefore first encounter internal lee wave generation and dissipation before that of internal tides, giving the former waves a very special role. To date, little is known of internal lee wave (ILW) propagation and dissipation, all the more so concerning diapycnal mixing in the interior.

Energy pathways of internal lee waves

Although it is still lacking, knowledge of the energy pathways of ILW has grown in recent years. Polzin et al. (1997) observed large amplitude of diapycnal mixing in mountainous areas with strong deep reaching currents and suggested it was due to the presence of an intense internal lee wave field. The internal lee wave generation mechanisms at play and estimations of the energy they carry lead to further studies (Scott et al. 2011, Wright et al. 2014, Nikurashin and Ferrari 2011). Other studies were interested in the propagation and final dissipation of the waves through diverse interaction mechanisms, such as wavemean flow interactions through critical layers for instance (Booker and Bretherton 1967). More recently, Nikurashin and Ferrari (2010) suggested from numerical simulations that a degenerate case of internal wave (the inertial oscillations, or IO) could be key factors in the energy pathways and dissipation of internal lee waves.
Despite these numerous studies, the energy pathways of internal lee waves remain very intricate, and an accurate representation of the energy emission, transfers and dissipation is still missing. For instance, the energy conversion rate into internal lee waves is subject to much debate (Waterman et al. 2014, Wright et al. 2014), and several mechanisms that appear to be important are often lacking in interaction representations (such as the IOs suggested by Nikurashin and Ferrari (2010) or momentum deposition from breaking internal waves to the mean flow). As such, understanding and representing the dissipated energy responsible for diapycnal mixing seems complex and might require considering a number of energy pathways at once. This can only be achieved when the nature of these energy pathways is known, before tackling their implications on diapycnal mixing.

Outline of this manuscript

In this manuscript, we propose to investigate the energy pathways taken by internal lee waves in the deep ocean and describe the implications of this evolution on turbulent kinetic energy dissipation. As seen above, doing so raises quite a few questions, from the observations available from the real ocean, to the sources of the energy pathways, and the role of inertial oscillations and turbulent kinetic energy dissipation in these pathways. These questions can be organized into four different topics:
To investigate TKE dissipation, the foremost step is to gather field measures of the energy dissipation rates in the ocean. Several techniques can be used, mainly depending on the spatial scale of interest. These techniques are differently done, rely on various hypotheses and different ranges of validity.
This raises the questions of how estimates of TKE dissipation are gathered, and under which conditions can these estimates be inferred.
The energy pathways in the ocean are intricate and complex, start from various sources and take different routes. There exist several major energy reservoirs available, which are not equivalent regarding to their impact on the ocean currents. Regarding the case of AABW consumption, there may exist an energy source and process particularly relevant for investigating the energy pathways. One may wonder what processes and energy reservoirs should be taken into account to shed light on AABW consumption.
It has been suggested in previous studies that inertial oscillations might be key to understanding the energy pathways of internal lee waves. The kinetics of this interaction are of great interest, and could inform on the conditions when these two wave motions can interact. The specific role of the inertial oscillations in the energetic route of the lee waves deserves due consideration. This points to the need of investigating and quantifying the role of inertial oscillations in the energy pathways of the internal lee waves.
Of the processes thought to impact on internal waves, some are fundamentally dependent on energy dissipation, some are not. Non-linear effects could be sufficient for creating strong interactions in the flow and for conditioning internal waves to ulterior energy dissipation. The respective roles of dissipative and non-dissipative interactions in the energy pathways must be clarified in order to fully understand the nature of the mechanisms at play.
We will attempt to answer these questions by allying literature reviews, numerical simulations and theoretical computations. Through these three aspects we aim at joining together as much as possible the three main ways of studying the ocean: field campaigns, computing and black-boarding.
In chapter 1 we will investigate the existing approaches and methods of inferring diapycnal mixing. These evaluations are called parameterizations since they make use of diverse assumptions, depending on the available data. We will review parameterizations of different scales, from the millimeter level to global estimates.
In chapter 2 the simulations used to ground and evaluate the theory will be described. Two dimensional simulations were used for the rapidity of their calculation and for comparison with previous works. All the simulations are non-hydrostatic, meaning they permit the existence of full vertical pressure gradients, and the full resolution of internal waves.
Chapter 3 describes an extension of the dissipative asymptotic theory of Nikurashin and Ferrari (2010) in the vertical direction. From so doing we will attempt to describe the temporal numerical evolution of inertial oscillations and TKE dissipation, as well as their vertical structure.
In chapter 4 we will make use of a different non-linear theory that does not fundamentally rely on energy dissipation: the Resonant Interaction Theory (RIT). The RIT will be used to investigate the rate of growth of the inertial oscillations in the light of the numerical simulations. The comparison of two separate theories for the same phenomenon will shed light on the mechanisms at play in the energy pathway.
Chapter 5 is a discussion on the implications of the two dimensional hypothesis and how to free the calculations from any unphysical dynamics.
We will then attempt to build a three dimensional setting that would permit to fully explore the full panel of mechanisms present in the ocean. We will show that the Transformed-Eulerian Mean (TEM) framework may be an appropriate tool for finely investigating wave-mean flow interactions.

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Table of contents :

1 Diapycnal mixing in the ocean: from ocean sampling to ocean modelling
1.1 From Turbulent Kinetic Energy dissipation rates to diapycnal mixing: energy conservation
1.2 Micro-structure: a local measure
1.3 The fine-scale parameterization: inferring mixing from shear and stress
1.4 Ocean recipes: from topography and stratification to mixing estimates
1.4.1 Getting the bottom energy conversion
1.4.2 The nonlinear propagation model
1.4.3 Caveats
1.5 Towards global estimates
1.5.1 Getting a global coverage of TKE dissipation
1.6 Summary of the diapycnal mixing parameterizations
2 A first look at the phenomenology: numerical simulations 
2.1 Introduction of the numerical case study
2.1.1 Physical configuration
2.1.2 Numerical set-up
2.1.3 Off the hat behavior
2.2 Description of the Turbulent Kinetic Energy dissipation rate
2.3 Description of inertial oscillations
2.4 On the link between IO amplitude and TKE dissipation rate
3 Attempting to predict inertial oscillation amplitude: an approach following Nikurashin and Ferrari 2010 
3.1 The asymptotic theory
3.1.1 Assumptions of the theory
3.1.2 Keeping the vertical coordinate
3.2 Comparison with the simulations
3.2.1 Growth rate of the inertial oscillations
3.2.2 Vertical extent of the inertial oscillations
4 Attempting to predict inertial oscillation amplitude: on the importance of resonant triad interactions  
4.1 The resonant interaction theory
4.1.1 Pros and cons of the underlying hypotheses
4.1.2 Derivation
4.2 Analyzing the numerical experiments in light of the resonant interaction theory
4.2.1 Applying the RIT to the interaction involving inertial oscillations and internal lee waves
4.2.2 Comparison with the simulations
4.3 The computation of ¶zu0w0 in the RIT
5 Towards a three dimensional description 
5.1 Possible implications of three-dimensional dynamics
5.1.1 Specificities of two and three-dimensional studies
5.1.2 Implications of momentum deposition on large scale circulation
5.2 Design of the numerical experiment
5.2.1 A new numerical code
5.2.2 Defining the experimental setup
5.2.3 Practical implementation
5.3 Analysis of the simulations
5.3.1 Overview of the analysis
5.3.2 Energy reservoirs and fluxes
5.3.3 The Transformed-Eulerian Mean (TEM) framework
5.3.4 Preliminary results
5.4 Conclusion
Conclusions and perspectives


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