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## Abstract

An approximate dispersion relation for near-inertial internal waves propagating in geostrophic shear is formulated that includes straining by the mean flow shear. Near-inertial and geostrophic motions have similar horizontal scales in the ocean. This implies that interaction terms involving mean flow shear of the form (v·Δ)V as well as the mean flow itself [(V·Δ)v] must be retained in the equations of motion. The vorticity ζ shifts the lower bound of the internal waveband from the planetary value of the Coriolis frequency *f* to an effective Coriolis frequency *f*
_{
e
}π = *f* + ζ/2. A ray tracing approach is adopted to examine the propagation behavior of near-inertial waves in a model geostrophic jet. Trapping *and* amplification occur in regions of negative vorticity where near-inertial waves' intrinsic frequency &omega_{0} can be less than the effective Coriolis frequency of the surrounding ocean. Intense downward-propagating near-inertial waves have been observed at the base of upper ocean negative vorticity in the North Pacific Subtropical Front, warm-core rings, a Gulf Stream cold-core ring and an anticyclonic eddy in the Sargasso Sea. Waves that are not trapped are focussed into tight beams as they leave the jet.

## Abstract

An approximate dispersion relation for near-inertial internal waves propagating in geostrophic shear is formulated that includes straining by the mean flow shear. Near-inertial and geostrophic motions have similar horizontal scales in the ocean. This implies that interaction terms involving mean flow shear of the form (v·Δ)V as well as the mean flow itself [(V·Δ)v] must be retained in the equations of motion. The vorticity ζ shifts the lower bound of the internal waveband from the planetary value of the Coriolis frequency *f* to an effective Coriolis frequency *f*
_{
e
}π = *f* + ζ/2. A ray tracing approach is adopted to examine the propagation behavior of near-inertial waves in a model geostrophic jet. Trapping *and* amplification occur in regions of negative vorticity where near-inertial waves' intrinsic frequency &omega_{0} can be less than the effective Coriolis frequency of the surrounding ocean. Intense downward-propagating near-inertial waves have been observed at the base of upper ocean negative vorticity in the North Pacific Subtropical Front, warm-core rings, a Gulf Stream cold-core ring and an anticyclonic eddy in the Sargasso Sea. Waves that are not trapped are focussed into tight beams as they leave the jet.

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## Abstract

Using velocity and temperature profile surveys collected beside a seamount, the dependence of the energy ratio PE/KE on the aspect ratio (*Nk _{H}
*/

*fk*)

_{z}^{2}is compared with theoretical relations for linear internal waves and geostrophy. The kinetic and potential energies are partitioned between these two types of motion. The vertical wavelengths of 50–400 m and horizontal wavelengths of 1–20 km spanned by the measurements have traditionally been associated with internal waves. Most of the energy is in the largest vertical and horizontal scales. This larger-scale energy is split between internal waves and geostrophy in one survey and is all geostrophic in the other. Both internal waves and geostrophy contribute to smaller scales.

## Abstract

Using velocity and temperature profile surveys collected beside a seamount, the dependence of the energy ratio PE/KE on the aspect ratio (*Nk _{H}
*/

*fk*)

_{z}^{2}is compared with theoretical relations for linear internal waves and geostrophy. The kinetic and potential energies are partitioned between these two types of motion. The vertical wavelengths of 50–400 m and horizontal wavelengths of 1–20 km spanned by the measurements have traditionally been associated with internal waves. Most of the energy is in the largest vertical and horizontal scales. This larger-scale energy is split between internal waves and geostrophy in one survey and is all geostrophic in the other. Both internal waves and geostrophy contribute to smaller scales.

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## Abstract

Velocity and temperature profiles collected in warm-core ring 82-I in January 1983 are used to describe the mean and fluctuating fields of the ring. The ring was relatively small, containing no Sargasso See water. Its signature vanished below 600 m and its velocity maximum was 50 cm s^{−1} at 35 km radius. The mean flow was in cyclogeostrophic balance and much of the core in solid body rotation with vorticity −0.5*f*. Potential vorticity was uniform along isopycnals below the depth of atmospheric influence. Horizontal Reynolds-stress forces did not contribute significantly to the mean momentum balance.

Downward-propagating near-inertial waves account for the most energetic of the fluctuating velocities. Energy levels four times greater than typically observed were found inside the ring, consistent with the expectation of trapping and critical-layer amplification of near-inertial waves in regions of negative vorticity as a consequence of a depressed lower bound of the internal waveband, *f*
_{eff} = *f* + ζ/2. A single wavepacket at the base of the core was responsible for the excess. The waves vertical and horizontal wavelengths were 250 and 60 km. respectively. Enhanced wave energy was not found in the verlocity maximum.

## Abstract

Velocity and temperature profiles collected in warm-core ring 82-I in January 1983 are used to describe the mean and fluctuating fields of the ring. The ring was relatively small, containing no Sargasso See water. Its signature vanished below 600 m and its velocity maximum was 50 cm s^{−1} at 35 km radius. The mean flow was in cyclogeostrophic balance and much of the core in solid body rotation with vorticity −0.5*f*. Potential vorticity was uniform along isopycnals below the depth of atmospheric influence. Horizontal Reynolds-stress forces did not contribute significantly to the mean momentum balance.

Downward-propagating near-inertial waves account for the most energetic of the fluctuating velocities. Energy levels four times greater than typically observed were found inside the ring, consistent with the expectation of trapping and critical-layer amplification of near-inertial waves in regions of negative vorticity as a consequence of a depressed lower bound of the internal waveband, *f*
_{eff} = *f* + ζ/2. A single wavepacket at the base of the core was responsible for the excess. The waves vertical and horizontal wavelengths were 250 and 60 km. respectively. Enhanced wave energy was not found in the verlocity maximum.

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## Abstract

Internal-wave-driven dissipation rates ε and diapycnal diffusivities *K* are inferred globally using a finescale parameterization based on vertical strain applied to ~30 000 hydrographic casts. Global dissipations are 2.0 ± 0.6 TW, consistent with internal wave power sources of 2.1 ± 0.7 TW from tides and wind. Vertically integrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt topography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong. Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameterizations of tidal mixing. Average diffusivities *K* = (0.3–0.4) × 10^{−4} m^{2} s^{−1} depend only weakly on depth, indicating that *ε* = *KN*
^{2}/*γ* scales as *N*
^{2} such that the bulk of the dissipation is in the pycnocline and less than 0.08-TW dissipation below 2000-m depth. Average diffusivities *K* approach 10^{−4} m^{2} s^{−1} in the bottom 500 meters above bottom (mab) in height above bottom coordinates with a 2000-m *e-*folding scale. Average dissipation rates *ε* are 10^{−9} W kg^{−1} within 500 mab then diminish to background deep values of 0.15 × 10^{−9} W kg^{−1} by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Circumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation rates for semidiurnal or diurnal internal tides equatorward of 28° and 14° latitudes, respectively, although elevated *K* is found about 30° latitude in the North and South Pacific.

## Abstract

Internal-wave-driven dissipation rates ε and diapycnal diffusivities *K* are inferred globally using a finescale parameterization based on vertical strain applied to ~30 000 hydrographic casts. Global dissipations are 2.0 ± 0.6 TW, consistent with internal wave power sources of 2.1 ± 0.7 TW from tides and wind. Vertically integrated dissipation rates vary by three to four orders of magnitude with elevated values over abrupt topography in the western Indian and Pacific as well as midocean slow spreading ridges, consistent with internal tide sources. But dependence on bottom forcing is much weaker than linear wave generation theory, pointing to horizontal dispersion by internal waves and relatively little local dissipation when forcing is strong. Stratified turbulent bottom boundary layer thickness variability is not consistent with OGCM parameterizations of tidal mixing. Average diffusivities *K* = (0.3–0.4) × 10^{−4} m^{2} s^{−1} depend only weakly on depth, indicating that *ε* = *KN*
^{2}/*γ* scales as *N*
^{2} such that the bulk of the dissipation is in the pycnocline and less than 0.08-TW dissipation below 2000-m depth. Average diffusivities *K* approach 10^{−4} m^{2} s^{−1} in the bottom 500 meters above bottom (mab) in height above bottom coordinates with a 2000-m *e-*folding scale. Average dissipation rates *ε* are 10^{−9} W kg^{−1} within 500 mab then diminish to background deep values of 0.15 × 10^{−9} W kg^{−1} by 1000 mab. No incontrovertible support is found for high dissipation rates in Antarctic Circumpolar Currents or parametric subharmonic instability being a significant pathway to elevated dissipation rates for semidiurnal or diurnal internal tides equatorward of 28° and 14° latitudes, respectively, although elevated *K* is found about 30° latitude in the North and South Pacific.

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## Abstract

The upwelling diapycnal limb of the ocean’s meridional overturning circulation is driven by divergence of diabatic turbulent buoyancy fluxes 〈*w*′*b*′〉 across density surfaces. A global assessment of zonally averaged internal-wave-driven turbulent diapycnal buoyancy fluxes from a strain-based finescale parameterization is used to infer mean diapycnal transports in the interior and near the bottom boundary. Bulk interior diabatic transports dominate above 2500-m depth (buoyancies |*B*| = *gγ*
_{
n
}/*ρ*
_{0} < 0.267 m s^{−2}, neutral densities *γ*
_{
n
} < 27.9 kg m^{−3}), upwelling at 10–11 Sv (1 Sv = 10^{6} m^{3} s^{−1}); 2, 5, and 3–4 Sv in the Indian, Pacific, and Atlantic, respectively, but are weak in the abyss. Boundary water-mass transformations peak at 18–25 Sv (4–6, 10–14, and 4–5 Sv in the Indian, Pacific, and Atlantic) near buoyancy |*B*| ~ 0.268 m s^{−2} (*γ*
_{
n
} ~ 28.1 kg m^{−3}, 4500-m depth) between bottom and lower deep waters, consistent with published 20–30-Sv global Antarctic Bottom Water (AABW) transport estimates. Interior transports above 2500-m depth fall below inverse estimates, consistent with a more adiabatic ocean interior where diapycnal mixing occurs at Southern Hemisphere high-latitude surface density outcrops.

## Abstract

The upwelling diapycnal limb of the ocean’s meridional overturning circulation is driven by divergence of diabatic turbulent buoyancy fluxes 〈*w*′*b*′〉 across density surfaces. A global assessment of zonally averaged internal-wave-driven turbulent diapycnal buoyancy fluxes from a strain-based finescale parameterization is used to infer mean diapycnal transports in the interior and near the bottom boundary. Bulk interior diabatic transports dominate above 2500-m depth (buoyancies |*B*| = *gγ*
_{
n
}/*ρ*
_{0} < 0.267 m s^{−2}, neutral densities *γ*
_{
n
} < 27.9 kg m^{−3}), upwelling at 10–11 Sv (1 Sv = 10^{6} m^{3} s^{−1}); 2, 5, and 3–4 Sv in the Indian, Pacific, and Atlantic, respectively, but are weak in the abyss. Boundary water-mass transformations peak at 18–25 Sv (4–6, 10–14, and 4–5 Sv in the Indian, Pacific, and Atlantic) near buoyancy |*B*| ~ 0.268 m s^{−2} (*γ*
_{
n
} ~ 28.1 kg m^{−3}, 4500-m depth) between bottom and lower deep waters, consistent with published 20–30-Sv global Antarctic Bottom Water (AABW) transport estimates. Interior transports above 2500-m depth fall below inverse estimates, consistent with a more adiabatic ocean interior where diapycnal mixing occurs at Southern Hemisphere high-latitude surface density outcrops.

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## Abstract

In the decade or so below the Ozmidov wavenumber (*N*
^{3}/*ε*)^{1/2}, that is, on scales between those attributed to internal gravity waves and isotropic turbulence, ocean and atmosphere measurements consistently find *k*
^{1/3} horizontal wavenumber spectra for horizontal shear *u*
_{
h
} and horizontal temperature gradient *T*
_{
h
} and *m*
^{−1} vertical wavenumber spectra for vertical shear *u*
_{
z
} and strain *ξ*
_{
z
}. Dimensional scaling is used to construct model spectra below as well as above the Ozmidov wavenumber that reproduces observed spectral slopes and levels in these two bands in both vertical and horizontal wavenumber. Aspect ratios become increasingly anisotropic below the Ozmidov wavenumber until reaching ~*O*(*f*/*N*), where horizontal shear *u*
_{
h
} ~ *f*. The forward energy cascade below the Ozmidov wavenumber found in observations and numerical simulations suggests that anisotropic and isotropic turbulence are manifestations of the same nonlinear downscale energy cascade to dissipation, and that this turbulent cascade originates from anisotropic instability of finescale internal waves at horizontal wavenumbers far below the Ozmidov wavenumber. Isotropic turbulence emerges as the cascade proceeds through the Ozmidov wavenumber where shears become strong enough to overcome stratification. This contrasts with the present paradigm that geophysical isotropic turbulence arises directly from breaking internal waves. This new interpretation of the observations calls for new approaches to understand anisotropic generation of geophysical turbulence patches.

## Abstract

In the decade or so below the Ozmidov wavenumber (*N*
^{3}/*ε*)^{1/2}, that is, on scales between those attributed to internal gravity waves and isotropic turbulence, ocean and atmosphere measurements consistently find *k*
^{1/3} horizontal wavenumber spectra for horizontal shear *u*
_{
h
} and horizontal temperature gradient *T*
_{
h
} and *m*
^{−1} vertical wavenumber spectra for vertical shear *u*
_{
z
} and strain *ξ*
_{
z
}. Dimensional scaling is used to construct model spectra below as well as above the Ozmidov wavenumber that reproduces observed spectral slopes and levels in these two bands in both vertical and horizontal wavenumber. Aspect ratios become increasingly anisotropic below the Ozmidov wavenumber until reaching ~*O*(*f*/*N*), where horizontal shear *u*
_{
h
} ~ *f*. The forward energy cascade below the Ozmidov wavenumber found in observations and numerical simulations suggests that anisotropic and isotropic turbulence are manifestations of the same nonlinear downscale energy cascade to dissipation, and that this turbulent cascade originates from anisotropic instability of finescale internal waves at horizontal wavenumbers far below the Ozmidov wavenumber. Isotropic turbulence emerges as the cascade proceeds through the Ozmidov wavenumber where shears become strong enough to overcome stratification. This contrasts with the present paradigm that geophysical isotropic turbulence arises directly from breaking internal waves. This new interpretation of the observations calls for new approaches to understand anisotropic generation of geophysical turbulence patches.

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## Abstract

Regions of negative vorticity are observed to trap and amplify near-inertial internal waves, which are sources of turbulent mixing 10–100 times higher than typically found in the stratified ocean interior. Because these regions are of finite lateral extent, trapped waves will not form a continuum but be quantized in modes. A model for the radial structure of near-inertial azimuthal modes in an axisymmetric vortex is described in order to explain intense near-inertial motions observed in the cores of a Gulf Stream warm-core ring and a vortex cap above Fieberling Seamount. Observed signals exhibit little variability of the rectilinear phase *ϕ* = arctan(*υ*/*u*) in the core and evanesce rapidly outside the swirl velocity maximum, where *u* is the zonal velocity and *υ* the meridional velocity. The authors focus on azimuthal mode *n* = −1 (propagating clockwise around the vortex) and the gravest radial mode (no zero crossing) that appears to dominate observations. Model solutions resemble Bessel functions inside the velocity maximum and modified Bessel function decay outside, consistent with observations and solutions previously found by Kunze et al. using a less complete model. The improved model supports their conclusions concerning radial wavelengths, vertical group velocities, and energy fluxes for trapped waves.

## Abstract

Regions of negative vorticity are observed to trap and amplify near-inertial internal waves, which are sources of turbulent mixing 10–100 times higher than typically found in the stratified ocean interior. Because these regions are of finite lateral extent, trapped waves will not form a continuum but be quantized in modes. A model for the radial structure of near-inertial azimuthal modes in an axisymmetric vortex is described in order to explain intense near-inertial motions observed in the cores of a Gulf Stream warm-core ring and a vortex cap above Fieberling Seamount. Observed signals exhibit little variability of the rectilinear phase *ϕ* = arctan(*υ*/*u*) in the core and evanesce rapidly outside the swirl velocity maximum, where *u* is the zonal velocity and *υ* the meridional velocity. The authors focus on azimuthal mode *n* = −1 (propagating clockwise around the vortex) and the gravest radial mode (no zero crossing) that appears to dominate observations. Model solutions resemble Bessel functions inside the velocity maximum and modified Bessel function decay outside, consistent with observations and solutions previously found by Kunze et al. using a less complete model. The improved model supports their conclusions concerning radial wavelengths, vertical group velocities, and energy fluxes for trapped waves.

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## Abstract

Small-scale internal waves are numerically ray-traced through random 3D Garrett and Munk internal wave backgrounds. Past ray-tracing investigations of the transfer of internal wave energy toward small scales have considered only interactions with background vertical shear **V**
_{
z
} = (*U*
_{
z
}, *V*
_{
z
}). Here, scale analysis and numerical simulations indicate a nonnegligible role for vertical divergence *W*
_{
z
}. Background low-frequency vertical shear **V**
_{
z
} transfers high-frequency test waves toward small scales while background vertical divergence *W*
_{
z
} transfers low-frequency test waves toward small scales. These ray-tracing results resemble those of weak resonant-triad interaction theory, although the interactions considered here are not limited to be weak or resonant.

## Abstract

Small-scale internal waves are numerically ray-traced through random 3D Garrett and Munk internal wave backgrounds. Past ray-tracing investigations of the transfer of internal wave energy toward small scales have considered only interactions with background vertical shear **V**
_{
z
} = (*U*
_{
z
}, *V*
_{
z
}). Here, scale analysis and numerical simulations indicate a nonnegligible role for vertical divergence *W*
_{
z
}. Background low-frequency vertical shear **V**
_{
z
} transfers high-frequency test waves toward small scales while background vertical divergence *W*
_{
z
} transfers low-frequency test waves toward small scales. These ray-tracing results resemble those of weak resonant-triad interaction theory, although the interactions considered here are not limited to be weak or resonant.

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## Abstract

The spectral transfer of internal wave energy toward high vertical wavenumber *k*
_{
z
} and turbulence production ε is examined by ray tracing small-scale test waves in a canonical Garrett and Munk background wave field. Unlike previous ray-tracing studies, interactions with internal-wave vertical divergence *W*
_{
z
} as well as vertical shear **V**
_{
z
} are allowed. The resulting spectral energy transfer rates are 2 to 4 times shear-only predictions depending on the degree of vertical and horizontal scale separation imposed. An upper limit is obtained by imposing vertical scale separation *K*
_{
z
} < *k*
_{
z
} and no restriction on horizontal wavenumber or frequency. This violates the WKB approximation in the horizontal and time. A lower limit is obtained by imposing stricter vertical scale separation *K*
_{
z
} < 0.5*k*
_{
z
} and horizontal scale separation *K*
_{
H
} < *k*
_{
h
} for background frequencies Ω > 11*f.* Imposing WKB horizontal scale separation at all background frequencies causes some test waves to get stuck at low horizontal wavenumber and low frequency. This is not realistic. The shear and strain lower limit produces turbulence production rates close to the upper limit of shear-only calculations and therefore consistent with observations.

## Abstract

The spectral transfer of internal wave energy toward high vertical wavenumber *k*
_{
z
} and turbulence production ε is examined by ray tracing small-scale test waves in a canonical Garrett and Munk background wave field. Unlike previous ray-tracing studies, interactions with internal-wave vertical divergence *W*
_{
z
} as well as vertical shear **V**
_{
z
} are allowed. The resulting spectral energy transfer rates are 2 to 4 times shear-only predictions depending on the degree of vertical and horizontal scale separation imposed. An upper limit is obtained by imposing vertical scale separation *K*
_{
z
} < *k*
_{
z
} and no restriction on horizontal wavenumber or frequency. This violates the WKB approximation in the horizontal and time. A lower limit is obtained by imposing stricter vertical scale separation *K*
_{
z
} < 0.5*k*
_{
z
} and horizontal scale separation *K*
_{
H
} < *k*
_{
h
} for background frequencies Ω > 11*f.* Imposing WKB horizontal scale separation at all background frequencies causes some test waves to get stuck at low horizontal wavenumber and low frequency. This is not realistic. The shear and strain lower limit produces turbulence production rates close to the upper limit of shear-only calculations and therefore consistent with observations.

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## Abstract

A transect of velocity profiles across Gulf Stream warm-core ring 82-B shows that the thermostad contained virtually no vertical shear. Enhanced downward-propagating near-inertial wave shear was present at the base of the core, consistent with these waves encountering critical layers as they try to leave the region of negative vorticity. Wave amplification and a shrinking vertical wavelength at the critical layer should lead to instability and shear production of turbulence. High dissipation rates have been observed at the base of ring 81-D's core.

## Abstract

A transect of velocity profiles across Gulf Stream warm-core ring 82-B shows that the thermostad contained virtually no vertical shear. Enhanced downward-propagating near-inertial wave shear was present at the base of the core, consistent with these waves encountering critical layers as they try to leave the region of negative vorticity. Wave amplification and a shrinking vertical wavelength at the critical layer should lead to instability and shear production of turbulence. High dissipation rates have been observed at the base of ring 81-D's core.