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- Author or Editor: C. P. Caulfield x

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## Abstract

The authors explore the theoretical and empirical relationship between the nonlocal quantities of the entrainment ratio *E*, the appropriately depth- and time-averaged flux coefficient * _{o}* in density currents. The main theoretical result is that

*E*= 0.125

_{o}

^{2}(

*C*

_{U}^{3}/

*C*)/cos

_{L}*θ*, where

*θ*is the angle of the slope over which the density current flows,

*C*is the ratio the turbulent length scale to the depth of the density current, and

_{L}*C*is the ratio of the turbulent velocity scale to the mean velocity of the density current. In the case of high bulk Froude numbers

_{U}_{o}

^{−2}and (

*C*

_{U}^{3}/

*C*) =

_{L}*C*∼ 1, so

_{ϵ}*E*∼ 0.1, consistent with observations of a constant entrainment ratio in unstratified jets and weakly stratified plumes. For bulk Froude numbers close to one,

*E*∼ Fr

_{o}^{2}, again in agreement with observations and previous experiments. For bulk Froude numbers less than one,

## Abstract

The authors explore the theoretical and empirical relationship between the nonlocal quantities of the entrainment ratio *E*, the appropriately depth- and time-averaged flux coefficient * _{o}* in density currents. The main theoretical result is that

*E*= 0.125

_{o}

^{2}(

*C*

_{U}^{3}/

*C*)/cos

_{L}*θ*, where

*θ*is the angle of the slope over which the density current flows,

*C*is the ratio the turbulent length scale to the depth of the density current, and

_{L}*C*is the ratio of the turbulent velocity scale to the mean velocity of the density current. In the case of high bulk Froude numbers

_{U}_{o}

^{−2}and (

*C*

_{U}^{3}/

*C*) =

_{L}*C*∼ 1, so

_{ϵ}*E*∼ 0.1, consistent with observations of a constant entrainment ratio in unstratified jets and weakly stratified plumes. For bulk Froude numbers close to one,

*E*∼ Fr

_{o}^{2}, again in agreement with observations and previous experiments. For bulk Froude numbers less than one,

## Abstract

Broadband measurements of the internal wavefield will help to unlock an understanding of the energy cascade within the oceanic realm. However, there are challenges in acquiring observations with sufficient spatial resolution, especially in horizontal dimensions. Seismic reflection profiling can achieve a horizontal and vertical resolution of order meters. It is suitable for imaging thermohaline fine structure on scales that range from tens of meters to hundreds of kilometers. This range straddles the transition from internal wave to turbulent regimes. Here, the authors analyze an 80-km-long seismic image from the Falkland Plateau and calculate vertical displacement spectra of tracked reflections. First, they show that these spectra are consistent with the Garrett–Munk model at small horizontal wavenumbers (i.e., *k*
_{x} ≲ 3 × 10^{−3} cpm). There is a transition to stratified turbulence at larger wavenumbers (i.e., *k*
_{x} ≳ 2 × 10^{−1} cpm). This transition occurs at length scales that are significantly larger than the Ozmidov length scale above which stratification is expected to modify isotropic Kolmogorov turbulence. Second, the authors observe a rapid onset of this stratified turbulence over a narrow range of length scales. This onset is consistent with a characteristic energy injection scale of stratified turbulence with a forward cascade toward smaller scales through isotropic turbulence below the Ozmidov length scale culminating in microscale dissipation. Finally, they estimate the spatial pattern of diapycnal diffusivity and show that the existence of an injection scale can increase these estimates by a factor of 2.

## Abstract

Broadband measurements of the internal wavefield will help to unlock an understanding of the energy cascade within the oceanic realm. However, there are challenges in acquiring observations with sufficient spatial resolution, especially in horizontal dimensions. Seismic reflection profiling can achieve a horizontal and vertical resolution of order meters. It is suitable for imaging thermohaline fine structure on scales that range from tens of meters to hundreds of kilometers. This range straddles the transition from internal wave to turbulent regimes. Here, the authors analyze an 80-km-long seismic image from the Falkland Plateau and calculate vertical displacement spectra of tracked reflections. First, they show that these spectra are consistent with the Garrett–Munk model at small horizontal wavenumbers (i.e., *k*
_{x} ≲ 3 × 10^{−3} cpm). There is a transition to stratified turbulence at larger wavenumbers (i.e., *k*
_{x} ≳ 2 × 10^{−1} cpm). This transition occurs at length scales that are significantly larger than the Ozmidov length scale above which stratification is expected to modify isotropic Kolmogorov turbulence. Second, the authors observe a rapid onset of this stratified turbulence over a narrow range of length scales. This onset is consistent with a characteristic energy injection scale of stratified turbulence with a forward cascade toward smaller scales through isotropic turbulence below the Ozmidov length scale culminating in microscale dissipation. Finally, they estimate the spatial pattern of diapycnal diffusivity and show that the existence of an injection scale can increase these estimates by a factor of 2.

## Abstract

Two mechanisms are proposed whereby internal gravity waves (IGW) may radiate from a linearly unstable region of Boussinesq parallel flow that is characterized in the far field by constant horizontal velocity and Brunt-Väisälä frequency. Through what is herein referred to as “primary generation,” IGW may be directly excited by linear instability of the initial-state parallel shear flow. Characteristically, these waves propagate with horizontal phase speed and wavenumber equal to that of the most unstable mode of linear stability theory. Through the second mechanism, referred to as “secondary generation,” IGW may be excited via nonlinear modification of the initial instability into a form that couples strongly to a large amplitude outgoing internal wave field. The authors propose that the primary generation of IGW may occur provided a penetration condition, which is derived on the basis of linear theory, is satisfied. The penetration condition provides a limit on the growth rate of a disturbance of any particular frequency that is capable of propagating into the far field. This hypothesis is supported by a sequence of representative nonlinear numerical simulations in two spatial dimensions for both free mixing layer and jet flows with horizontal velocity profiles *U*(*z*) = tanh (*z*) and *U*(*z*) = sech^{2}(*z*), respectively. For the purpose of these analyses, the fluid density is taken to be such that the square of the Brunt–Väisälä frequency is given by *N*
^{2}(*z*) = *J* tanh^{2}(*z/R*). Such stratification allows both for the development of large-scale eddies in the region of low static stability and, in the far field where *N*
^{2} ≈ *J* is positive and approximately constant, for the radiation of a broad frequency spectrum of IGW.

## Abstract

Two mechanisms are proposed whereby internal gravity waves (IGW) may radiate from a linearly unstable region of Boussinesq parallel flow that is characterized in the far field by constant horizontal velocity and Brunt-Väisälä frequency. Through what is herein referred to as “primary generation,” IGW may be directly excited by linear instability of the initial-state parallel shear flow. Characteristically, these waves propagate with horizontal phase speed and wavenumber equal to that of the most unstable mode of linear stability theory. Through the second mechanism, referred to as “secondary generation,” IGW may be excited via nonlinear modification of the initial instability into a form that couples strongly to a large amplitude outgoing internal wave field. The authors propose that the primary generation of IGW may occur provided a penetration condition, which is derived on the basis of linear theory, is satisfied. The penetration condition provides a limit on the growth rate of a disturbance of any particular frequency that is capable of propagating into the far field. This hypothesis is supported by a sequence of representative nonlinear numerical simulations in two spatial dimensions for both free mixing layer and jet flows with horizontal velocity profiles *U*(*z*) = tanh (*z*) and *U*(*z*) = sech^{2}(*z*), respectively. For the purpose of these analyses, the fluid density is taken to be such that the square of the Brunt–Väisälä frequency is given by *N*
^{2}(*z*) = *J* tanh^{2}(*z/R*). Such stratification allows both for the development of large-scale eddies in the region of low static stability and, in the far field where *N*
^{2} ≈ *J* is positive and approximately constant, for the radiation of a broad frequency spectrum of IGW.

## Abstract

Direct numerical simulations of stratified turbulence are used to test several fundamental assumptions involved in the Osborn, Osborn–Cox, and Thorpe methods commonly used to estimate the turbulent diffusivity from field measurements. The forced simulations in an idealized triply periodic computational domain exhibit characteristic features of stratified turbulence including intermittency and layer formation. When calculated using the volume-averaged dissipation rates from the simulations, the vertical diffusivities inferred from the Osborn and Osborn–Cox methods are within 40% of the value diagnosed using the volume-averaged buoyancy flux for all cases, while the Thorpe-scale method performs similarly well in the simulation with a relatively large buoyancy Reynolds number (Re_{b} ≃ 240) but significantly overestimates the vertical diffusivity in simulations with Re_{b} < 60. The methods are also tested using a limited number of vertical profiles randomly selected from the computational volume. The Osborn, Osborn–Cox, and Thorpe-scale methods converge to their respective estimates based on volume-averaged statistics faster than the vertical diffusivity calculated directly from the buoyancy flux, which is contaminated with reversible contributions from internal waves. When applied to a small number of vertical profiles, several assumptions underlying the Osborn and Osborn–Cox methods are not well supported by the simulation data. However, the vertical diffusivity inferred from these methods compares reasonably well to the exact value from the simulations and outperforms the assumptions underlying these methods in terms of the relative error. Motivated by a recent theoretical development, it is speculated that the Osborn method might provide a reasonable approximation to the diffusivity associated with the *irreversible* buoyancy flux.

## Abstract

Direct numerical simulations of stratified turbulence are used to test several fundamental assumptions involved in the Osborn, Osborn–Cox, and Thorpe methods commonly used to estimate the turbulent diffusivity from field measurements. The forced simulations in an idealized triply periodic computational domain exhibit characteristic features of stratified turbulence including intermittency and layer formation. When calculated using the volume-averaged dissipation rates from the simulations, the vertical diffusivities inferred from the Osborn and Osborn–Cox methods are within 40% of the value diagnosed using the volume-averaged buoyancy flux for all cases, while the Thorpe-scale method performs similarly well in the simulation with a relatively large buoyancy Reynolds number (Re_{b} ≃ 240) but significantly overestimates the vertical diffusivity in simulations with Re_{b} < 60. The methods are also tested using a limited number of vertical profiles randomly selected from the computational volume. The Osborn, Osborn–Cox, and Thorpe-scale methods converge to their respective estimates based on volume-averaged statistics faster than the vertical diffusivity calculated directly from the buoyancy flux, which is contaminated with reversible contributions from internal waves. When applied to a small number of vertical profiles, several assumptions underlying the Osborn and Osborn–Cox methods are not well supported by the simulation data. However, the vertical diffusivity inferred from these methods compares reasonably well to the exact value from the simulations and outperforms the assumptions underlying these methods in terms of the relative error. Motivated by a recent theoretical development, it is speculated that the Osborn method might provide a reasonable approximation to the diffusivity associated with the *irreversible* buoyancy flux.

## Abstract

It is shown that geostrophic vertical shear estimates can be recovered from seismic (i.e., acoustic) images of thermohaline structure. In the Southern Ocean, the Antarctic Circumpolar Current forms a loop within the Falkland Trough before it flows northward into the Argentine Basin. Seismic profiles that cross this loop show the detailed structure of different water masses with a horizontal resolution of *O*(10 m). Coherent seismic reflections are tilted in response to current flow around the Falkland Trough. Average slopes were measured on length scales that are large enough to ensure that the geostrophic approximation is valid (i.e., with a Rossby number <0.1). By combining shear estimates with satellite altimetric measurements and acoustic Doppler current profiles, geostrophic velocities can be calculated throughout the data volume. This technique for estimating geostrophic vertical shear from legacy seismic images yields useful information about the spatial and temporal variation of mesoscale circulation.

## Abstract

It is shown that geostrophic vertical shear estimates can be recovered from seismic (i.e., acoustic) images of thermohaline structure. In the Southern Ocean, the Antarctic Circumpolar Current forms a loop within the Falkland Trough before it flows northward into the Argentine Basin. Seismic profiles that cross this loop show the detailed structure of different water masses with a horizontal resolution of *O*(10 m). Coherent seismic reflections are tilted in response to current flow around the Falkland Trough. Average slopes were measured on length scales that are large enough to ensure that the geostrophic approximation is valid (i.e., with a Rossby number <0.1). By combining shear estimates with satellite altimetric measurements and acoustic Doppler current profiles, geostrophic velocities can be calculated throughout the data volume. This technique for estimating geostrophic vertical shear from legacy seismic images yields useful information about the spatial and temporal variation of mesoscale circulation.