Attempting to predict inertial oscillation amplitude: an approach following Nikurashin and Ferrari 2010 

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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 wave-mean flow interactions through critical layers for instance (Booker and Brether-ton 1967). More recently, Nikurashin and Ferrari (2010) suggested from numeri-cal 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.

Micro-structure: a local measure

A seemingly basic step towards understanding the energetics of the ocean’s interior is to measure in situ energetic variables. This requires deploying instru-ments such as Vertical Microstructure Profilers (VMP, Carter and Imberger 1986) in the ocean interior, and tracking the flow characteristics on small scales. The field of interest for this study is the turbulent kinetic energy dissipation rate, a primary proxy towards diapycnal mixing rates (Eq. 1.3b).
Turbulent kinetic energy dissipation occurs on scales of centimeters (micro-scales). This makes measuring TKE dissipation rates quite a technical challenge, for several reasons. Since differences measured on centimeter scales are bound to have a large uncertainty compared to the absolute signal, the accuracy of the measures is critical to obtain precise data. Moreover, measuring micro-scales requires probes arranged as dipoles, each pole being distant from the other by about a centimeter. These probes are delicate and onerous. Finally, as move-ments of the ship would transmit motions to the instrument and disrupt the local measure and local dissipation rate, the profiler is required to be detached from the ship, free falling towards the bottom. A free falling device must be very finely tuned. If it comes back up too early, all the bottom dissipation would not be measured. But if it sinks too deep, it might touch the ocean floor, which would lead to the complete loss of the instrument. All of this contributes to difficult technical issues, high instrument cost and the need of large specific knowledge for its operation.
As such, micro-structure measurements are very accurate and provide great and insightful knowledge, however they are very sparse and do not cover the very bottom of the ocean, which is the main limitation in their applicability.

The fine-scale parameterization: inferring mixing from shear and stress

Since directly measuring TKE dissipation rate is so tricky, progress was made to infer less reliable dissipation rates from more easily obtained data. Most ocean datasets contain information on velocity and density structure on scales of tens to hundreds of meters (hereafter called fine-scales, Polzin et al. 2014). The aim of fine-scale parameterizations was to estimate rates of turbulent kinetic energy dissipation from such datasets. They use density and/or velocity profiles obtained from standard CTD and ADCP measurements, far more extensive, easy to obtain and of resolution of tens to hundreds of meters. Since the leading order of the ocean’s interior energy field is attributed to internal waves (Ferrari and Wunsch 2009), the fine-scale parameterizations make use of the wave properties (shear and strain) to infer the turbulent kinetic energy dissipation rate. The use of such parameterizations has greatly increased over the recent years, and although the results are extremely sensitive (to the scheme used as well as to the way the data is handled) and approximative (up to an order of magnitude different from the micro-structure data), some great insight into the turbulent kinetic energy dissipation rate was achieved.
The fine-scale parameterization considers that the rate of turbulent produc-tion P in the internal wave field is determined by the spectral energy transport in the vertical wave-number domain F(m) (integrated over frequencies). Although a diversity of studies parameterise P depending on the fields available (Polzin 2004, Olbers and Eden 2013), we will focus on the study by Polzin (2004) for simplicity.
Through combining heuristic arguments and dimensional parameters, Polzin (2004) arrive at Z F(m, w)dw!m ! ¥ P , (1.4). the rate of turbulent production arises from the total energy transport.

Ocean recipes: from topography and stratifica-tion to mixing estimates

We have seen that fine-scale parameterizations can provide turbulent kinetic energy dissipation rates from shear and strain ratios, using ray tracing theory. In short, knowing the velocity and buoyancy profile, as well as the energy spec-trum, is enough to apply the fine-scale parameterization.
Current knowledge could allow one to predict the energy spectrum in the fluid column from information on the flow and the topography. Assembling both theories would enable one the predict the one-dimensional profile of the turbulent kinetic energy dissipation rate from precise knowledge of bottom to-pography and flow characteristics.
As discussed in the introduction, increasing interest has been taken in inter-nal lee waves, especially in the Southern Ocean. Unfortunately, ocean recipes have been derived for internal tide dissipation (Polzin 2009), but have yet to be published for internal lee wave dissipation (work in progress, also by K. Polzin). We will here focus on the published internal tide problem, and add comments on how the internal lee wave problem can be adapted to it. Such theories, named ocean recipes, have the advantage that they only re- quire external variables of the problem, such as a buoyancy profile, a topographic spectrum and either geostrophic velocity profile (in the case of internal lee wave) or barotropic tide (in the case of internal tides). These variables have the advan-tage that they are nearly all available outputs of ocean models (apart from the ill-known topographic spectrum), and are readily available.

Getting the bottom energy conversion

As just said, the major step towards obtaining the energy dissipation rate is to compute the energy spectrum, before the fine-scale parameterizations can take over. To compute the energy spectrum, the energy sources and their spectral distribution have to be considered. Several energy sources for internal waves are available, and can mainly be divided into wind generation at the surface and generation by interaction with topography (at the bottom). The ocean recipes described here focus on the bottom generation. Polzin (2009) further asserts that additional energy sources would easily be taken into account in the recipe.

Getting a global coverage of TKE dissipation

As explained above, the diapycnal mixing is most commonly inferred from the turbulent kinetic energy dissipation rate. Moreover, the global distribution of TKE dissipation rate is generally represented as depending solely on its horizon-tal location. That is, the amplitude of the TKE dissipation rate is set at a given longitude and latitude, whereas its vertical distribution obeys the same scaling whatever the horizontal location (St. Laurent et al. 2002, Melet et al. 2013, Saenko et al. 2012): E = qE(x, y)F(z) (1.13).
where E(x, y) is the energy flux into internal waves, q is the fraction of this en-ergy flux that is assumed to be dissipated in the water column and F(z) describes the vertical structure of TKE dissipation. Such a parameterization is poorly constrained and lacks most of the physics going on, but it has the advantage of being easily implemented into ocean mod-els.

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Description of the Turbulent Kinetic Energy dis-sipation rate

Let us first look at the general behavior of the TKE dissipation rate through a few simple cases. The basic settings we will take as references consist in a 2 km wide topography of amplitude ranging from 20 to 80 m with free-slip bottom boundary conditions: the L2 f s cases.
Figure 2.3 shows the evolution versus time of the TKE dissipation integrated over 2000 m for all the simulations. For near linear cases where hT = 20 m, integrated TKE dissipation increases slowly with time, and does not reach any steady state within the time of integration. The other two simulations reach a statistically steady state (hereafter named saturation) after a few inertial periods. These simulations then show a slow decay of TKE dissipation rate with time. We shall attempt to explain this behavior later in the manuscript. A difference between the lowest topographic amplitude case and the others can be expected for practically any observation, since the non-linearities greatly influence the flow as a whole.
This figure shows that, in these three reference cases, the TKE dissipation rate is clearly enhanced near the topography. The bottom 1000 m show an ele-vation of up to an order of magnitude in turbulent dissipation. This means that the internal lee waves generated at the topography dissipate strongly shortly after their emission, before continuing their propagation upwards with a rather constant amplitude, without any TKE dissipation. The lowest amplitude case (hT = 20 m) holds a low amount of TKE dissipation. This can be expected, since the waves are quasi-linear: the available energy carried by the waves is smallest, and little turbulence is observed.
Using Eq. (1.3b), N = 10 3 s 1 and assuming g = 0.2, the averaged rate of TKE dissipation over the lower 500 m in simulation H80 L2 f s gives a vertical diffusivity of about 4.10 3 m2.s 1. This value is strong but only slightly above the range of TKE dissipation rate in the deep Southern Ocean inferred from vertical micro-structure profilers (Waterman et al. 2014).
Figure 2.5 shows profiles of the TKE dissipation above a given height for all simulations. The same observations as in Fig. 2.4 can be observed. The profiles show that for all the non-linear cases (hT 40 m), dissipation is clearly enhanced near topography, systematically under 1000 m. The quasi-linear cases do not show much signal in the TKE dissipation field, since wave-wave interactions is very small.
Since there is no interaction with the atmosphere or any initial disturbance apart from the topography, the energy input results entirely from the conversion from the geostrophic flow to the internal lee wave field at the bottom. Subse-quently, the TKE dissipation rate can be directly compared with the bottom en-ergy conversion rate Pup = p0w0xjz=0. Figure 2.6 shows profiles of the integrated TKE dissipation above a given height, z0, and below H = 2000 m for different simulations, scaled by Pup. R H E¶z is the energy dissipated in the domain z0 comprised between the topography and height z0. If all the energy were to be dissipated at a given height z0, then we would have the equality R0z0 E¶z = Pup, or R H E¶z = 0. This way, what is plotted on figure 2.6 is a measure of the rela- z0 tive importance between the sink and source terms (given that any motion that radiates above 2000 m is dissipated in the sponge layer).
In figure 2.7 R H Edz is scaled by the total TKE dissipation integrated over the z0 bottom 2000 m, R0H Edz. The profile represented is identical to that of Fig. 2.6, except that the sink term is scaled by the total dissipation in the bottom 2000 m. Thus, it ranges from 0 to 1.
Figures 2.7 and 2.6 show that the vertical profile of kinetic energy dissipation is largely independent on the topographic features (hT, kT). From the two differ-ent normalizations, two types of conclusion can be made: either on the fraction of emitted energy that is dissipated, or on the proportion of the dissipation that occurs below (or above) a certain depth. For our range of parameters we observe on figure 2.6 that at most 20% of the internal lee wave energy produced at the topography is dissipated in the water column. Similar ratios of dissipated to emitted energy have been observed in the ocean by (Sheen et al. 2013, Brearley et al. 2013). In simulations that reach saturation, we observe from figure 2.7 that at least 80% of the dissipated energy occurs below 600 m. In a nutshell, these results confirm those observed in the simulations by Nikurashin and Ferrari (2010).

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 
A Annexes 
A.1 Resonant interaction theory
A.1.1 Deriving the resonant interaction theory
A.1.2 Calculation of the growth rate
A.2 Asymptotic theory, from Nikurashin and Ferrari (2010) and more .
A.2.1 Summary of Nikurashin and Ferrari (2010)
A.2.2 Moving on from Nikurashin and Ferrari (2010)
A.2.3 Vertical structure and propagation of inertial oscillations
Bibliography 

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