Internal gravity waves, the subsurface analogue of the familiar surface gravity waves that break on beaches, are ubiquitous in the ocean. Because of their strong Internal gravity waves are propagating disturbances of the ocean's density stratification. Their physics resembles that of surface gravity waves but with buoyancy rather than gravity providing their restoring force -making them much larger (10's to 100's of meters instead of 1 to 10 meters) and slower (hours instead of seconds). Generated primarily by tidal flow past seafloor topography and winds blowing on the sea surface, and typically having multi-kilometer-scale horizontal wavelengths, their estimated 1 TW of deep-sea dissipation is understood to play a crucial role in the ocean's global redistribution of heat and momentum 12 . A major challenge is to improve understanding of internal wave generation, propagation, steepening and dissipation, so that the role of internal waves can be more accurately incorporated in climate models.The internal waves that originate from the Luzon Strait on the eastern margin of the South China Sea (SCS) are the largest documented in the global oceans ( Figure 1).As the waves propagate west from the Luzon Strait they steepen dramatically ( Figure 1a), producing distinctive solitary wave fronts evident in sun glint and synthetic aperture radar (SAR) images from satellites ( Figure 1b). When they shoal onto the continental slope to the west, the downward displacement of the ocean's layers associated with these solitary waves can exceed 250 m in 5 minutes 8 . On such a scale, these waves pose hazards for underwater navigation and offshore drilling 4 , and supply nutrients from the deep ocean that nourish coral reefs 1 and pilot whale populations that forage in their wakes 13 .Over the past decade a number of field studies have been conducted in the region; this work has been comprehensively reviewed 10,11 . All of these studies, however, focused on the propagation of the internal waves across the SCS and their interactions with the continental shelf of China. Until the present study there had been no substantial in situ data gathered at the generation site of the Luzon Strait, in large part because of the extremely challenging operating conditions. A consequence has been persistent 5 confusion regarding the nature of the generation mechanism 11 ; an underlying cause being the sensitivity of the models employed to the system parameters, such as the chosen transect for a two-dimensional model, the linear internal wave speed or the assumed location of the waves' origin within the Luzon Strait. Furthermore, the lack of in situ data from the Luzon Strait has meant an inability to test numerical predictions of energy budgets 9 and no knowledge of the impact of the Kuroshio on the emergence of internal solitary waves 11 .The goal of IWISE is to obtain the first comprehensive in situ data set from the Luzon Strait, which in combination with high-resolution three-dimensional numerical modeling supports a cradle-to-grave picture ...

High-resolution two- and three-dimensional numerical simulations are performed of first-mode internal gravity waves interacting with a shelf break in a linearly stratified fluid. The interaction of nonlinear incident waves with the shelf break results in the formation of upslope-surging vortex cores of dense fluid (referred to here as internal boluses) that propagate onto the shelf. This paper primarily focuses on understanding the dynamics of the interaction process with particular emphasis on the formation, structure and propagation of internal boluses onshelf. A possible mechanism is identified for the excitation of vortex cores that are lifted over the shelf break, from where (from the simplest viewpoint) they essentially propagate as gravity currents into a linearly stratified ambient fluid.

Scaling arguments are presented to quantify the widely used diapycnal (irreversible) mixing coefficient $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D716}_{PE}/\unicode[STIX]{x1D716}$ in stratified flows as a function of the turbulent Froude number $Fr=\unicode[STIX]{x1D716}/Nk$. Here, $N$ is the buoyancy frequency, $k$ is the turbulent kinetic energy, $\unicode[STIX]{x1D716}$ is the rate of dissipation of turbulent kinetic energy and $\unicode[STIX]{x1D716}_{PE}$ is the rate of dissipation of turbulent potential energy. We show that for $Fr\gg 1$, $\unicode[STIX]{x1D6E4}\propto Fr^{-2}$, for $Fr\sim \mathit{O}(1)$, $\unicode[STIX]{x1D6E4}\propto Fr^{-1}$ and for $Fr\ll 1$, $\unicode[STIX]{x1D6E4}\propto Fr^{0}$. These scaling results are tested using high-resolution direct numerical simulation (DNS) data from three different studies and are found to hold reasonably well across a wide range of $Fr$ that encompasses weakly stratified to strongly stratified flow conditions. Given that the $Fr$ cannot be readily computed from direct field measurements, we propose a practical approach that can be used to infer the $Fr$ from readily measurable quantities in the field. Scaling analyses show that $Fr\propto (L_{T}/L_{O})^{-2}$ for $L_{T}/L_{O}>O(1)$, $Fr\propto (L_{T}/L_{O})^{-1}$ for $L_{T}/L_{O}\sim O(1)$, and $Fr\propto (L_{T}/L_{O})^{-2/3}$ for $L_{T}/L_{O}<O(1)$, where $L_{T}$ is the Thorpe length scale and $L_{O}$ is the Ozmidov length scale. These formulations are also tested with DNS data to highlight their validity. These novel findings could prove to be a significant breakthrough not only in providing a unifying (and practically useful) parameterization for the mixing efficiency in stably stratified turbulence but also for inferring the dynamic state of turbulence in geophysical flows.

Oceanic density overturns are commonly used to parameterize the dissipation rate of turbulent kinetic energy. This method assumes a linear scaling between the Thorpe length scale L T and the Ozmidov length scale L O . Historic evidence supporting L T ; L O has been shown for relatively weak shear-driven turbulence of the thermocline; however, little support for the method exists in regions of turbulence driven by the convective collapse of topographically influenced overturns that are large by open-ocean standards. This study presents a direct comparison of L T and L O , using vertical profiles of temperature and microstructure shear collected in the Luzon Strait-a site characterized by topographically influenced overturns up to O(100) m in scale. The comparison is also done for open-ocean sites in the Brazil basin and North Atlantic where overturns are generally smaller and due to different processes. A key result is that L T /L O increases with overturn size in a fashion similar to that observed in numerical studies of Kelvin-Helmholtz (K-H) instabilities for all sites but is most clear in data from the Luzon Strait. Resultant bias in parameterized dissipation is mitigated by ensemble averaging; however, a positive bias appears when instantaneous observations are depth and time integrated. For a series of profiles taken during a spring tidal period in the Luzon Strait, the integrated value is nearly an order of magnitude larger than that based on the microstructure observations. Physical arguments supporting L T ; L O are revisited, and conceptual regimes explaining the relationship between L T /L O and a nondimensional overturn size c L T are proposed. In a companion paper, Scotti obtains similar conclusions from energetics arguments and simulations.

An alternative framework for parameterizing stably stratified shear-flow turbulence is presented. Using dimensional analysis, four non-dimensional parameters of interest are identified that consider the independent effects of stratification, shear, viscosity, and scalar diffusivity. In the interest of geophysical applications, the problem is further simplified by considering only high Reynolds number flow. This leads to a two-dimensional parameter space based on a buoyancy strength parameter (i.e., an inverse Froude number) and a shear strength parameter. Consideration for the gradient Richardson number allows the space to be divided into an unforced regime, a shear-dominated regime, and a buoyancy-dominated regime. On this basis, a large database of direct numerical simulation and laboratory data from various sources is evaluated. Of particular interest is the observed length scale of overturning. Overturns are found to scale with k1/2/N in the buoyancy-dominated regime, k1/2/S in the shear-dominated regime, and k3/2/ε in the unforced regime, where k, N, S, and ε are the turbulent kinetic energy, buoyancy frequency, mean shear rate, and turbulent kinetic energy dissipation rate, respectively. Implications for estimates of diapycnal mixing in the ocean are discussed and a new parameterization for eddy diffusivity is presented.

In this paper, we derive a general relationship for the turbulent Prandtl number Pr t for homogeneous stably stratified turbulence from the turbulent kinetic energy and scalar variance equations. A formulation for the turbulent Prandtl number, Pr t , is developed in terms of a mixing length scale L M and an overturning length scale L E , the ratio of the mechanical (turbulent kinetic energy) decay time scale T L to scalar decay time scale T ρ and the gradient Richardson number Ri . We show that our formulation for Pr t is appropriate even for non-stationary (developing) stratified flows, since it does not include the reversible contributions in both the turbulent kinetic energy production and buoyancy fluxes that drive the time variations in the flow. Our analysis of direct numerical simulation (DNS) data of homogeneous sheared turbulence shows that the ratio L M /L E ≈ 1 for weakly stratified flows. We show that in the limit of zero stratification, the turbulent Prandtl number is equal to the inverse of the ratio of the mechanical time scale to the scalar time scale, T L /T ρ . We use the stably stratified DNS data of Shih et al. (J. Fluid Mech., vol. 412, 2000, pp. 1-20; J. Fluid Mech., vol. 525, 2005, pp. 193-214) to propose a new parameterization for Pr t in terms of the gradient Richardson number Ri . The formulation presented here provides a general framework for calculating Pr t that will be useful for turbulence closure schemes in numerical models.

The flux Richardson number $R_{f}$ (often referred to as the mixing efficiency) is a widely used parameter in stably stratified turbulence which is intended to provide a measure of the amount of turbulent kinetic energy $k$ that is irreversibly converted to background potential energy (which is by definition the minimum potential energy that a stratified fluid can attain that is not available for conversion back to kinetic energy) due to turbulent mixing. The flux Richardson number is traditionally defined as the ratio of the buoyancy flux $B$ to the production rate of turbulent kinetic energy $P$. An alternative generalized definition for $R_{f}$ was proposed by Ivey & Imberger (J. Phys. Oceanogr., vol. 21, 1991, pp. 650–658), where the non-local transport terms as well as unsteady contributions were included as additional sources to the production rate of $k$. While this definition precludes the need to assume that turbulence is statistically stationary, it does not properly account for countergradient fluxes that are typically present in more strongly stratified flows. Hence, a third definition that more rigorously accounts for only the irreversible conversions of energy has been defined, where only the irreversible fluxes of buoyancy and production (i.e. the dissipation rates of $k$ and turbulent potential energy ($E_{PE}^{\prime }$)) are used. For stationary homogeneous shear flows, all of the three definitions produce identical results. However, because stationary and/or homogeneous flows are typically not found in realistic geophysical situations, clarification of the differences/similarities between these various definitions of $R_{f}$ is imperative. This is especially true given the critical role $R_{f}$ plays in inferring turbulent momentum and heat fluxes using indirect methods, as is commonly done in oceanography, and for turbulence closure parameterizations. To this end, a careful analysis of two existing direct numerical simulation (DNS) datasets of stably stratified homogeneous shear and channel flows was undertaken in the present study to compare and contrast these various definitions. We find that all three definitions are approximately equivalent when the gradient Richardson number $Ri_{g}\leqslant 1/4$. Here, $Ri_{g}=N^{2}/S^{2}$, where $N$ is the buoyancy frequency and $S$ is the mean shear rate, provides a measure of the stability of the flow. However, when $Ri_{g}>1/4$, significant differences are noticeable between the various definitions. In addition, the irreversible formulation of $R_{f}$ based on the dissipation rates of $k$ and $E_{PE}^{\prime }$ is the only definition that is free from oscillations at higher gradient Richardson numbers. Both the traditional definition and the generalized definition of $R_{f}$ exhibit significant oscillations due to the persistence of linear internal wave motions and countergradient fluxes that result in reversible exchanges between $k$ and $E_{PE}^{\prime }$. Finally, we present a simple parameterization for the irreversible flux Richardson number $R_{f}^{\ast }$ based on $Ri_{g}$ that produces excellent agreement with the DNS results for $R_{f}^{\ast }$.

Direct numerical simulations of stably stratified turbulence are used to compare the Thorpe overturn length scale, L T , with other length scales of the flow that can be constructed from large-scale quantities fundamental to shear-free, stratified turbulence. Quantities considered are the turbulent kinetic energy, k, its dissipation rate, , and the buoyancy frequency, N. Fundamental length scales are then the Ozmidov length scale, L O , the isotropic large scale, L k , and a kinetic energy length scale, L kN . Behavior of all three fundamental scales, relative to L T , is shown to be a function of the buoyancy strength parameter NT L , where T L = k/ is the turbulence time scale. When buoyancy effects are dominant (i.e., for NT L > 1), L T is shown to be linearly correlated with L kN and not with L O as is commonly assumed for oceanic flows. Agreement between L O and L T is only observed when the buoyancy and turbulence time scales are approximately equal (i.e., for the critical case when NT L ≈ 1). The relative lack of agreement between L T and L O in strongly stratified flows is likely due to anisotropy at the outer scales of the flow where the energy transfer rate differs from . The key finding of this work is that observable overturns in strongly stratified flows are more reflective of k than . In the context of oceanic observations, this implies that inference of k, rather than , from measurements of L T is fundamentally correct when NT L ≈ 1 and most appropriate when NT L > 1. Furthermore, we show that for NT L < 1, L T is linearly correlated with L k when mean shear is absent. C 2013 AIP Publishing LLC. [http://dx.