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## Abstract

The finite angular width of Doppler sonar beams introduces errors into the measurement of ocean velocity from moving ships. These errors are exacerbated as ship speed increases and as the acoustic scatter field becomes more inhomogeneous in space. Zooplanktonic scatters often reside in distinct quasi-isopycnal scattering layers. When measured from a moving ship, such layers appear to have a distinct velocity signature, even if the ocean is quiescent. Here, this effect is explored in a simple analytic model and a least squares algorithm is presented for remediating its signature. The algorithm corrects the measured velocity at any given depth, with the correction being a weighted sum of the depth gradient of log acoustic intensity at surrounding depths. The correction weights are a property of the specific sonar beam pattern. Once determined, they can be considered constant. The algorithm, when applied to a 50-kHz sonar on the Research Vessel (R/V) *Roger Revelle*, is found to remove roughly 90% of the error in vertical wavenumber spectra of shear. More sophisticated algorithms can be developed as experience with the present approach is gained.

## Abstract

The finite angular width of Doppler sonar beams introduces errors into the measurement of ocean velocity from moving ships. These errors are exacerbated as ship speed increases and as the acoustic scatter field becomes more inhomogeneous in space. Zooplanktonic scatters often reside in distinct quasi-isopycnal scattering layers. When measured from a moving ship, such layers appear to have a distinct velocity signature, even if the ocean is quiescent. Here, this effect is explored in a simple analytic model and a least squares algorithm is presented for remediating its signature. The algorithm corrects the measured velocity at any given depth, with the correction being a weighted sum of the depth gradient of log acoustic intensity at surrounding depths. The correction weights are a property of the specific sonar beam pattern. Once determined, they can be considered constant. The algorithm, when applied to a 50-kHz sonar on the Research Vessel (R/V) *Roger Revelle*, is found to remove roughly 90% of the error in vertical wavenumber spectra of shear. More sophisticated algorithms can be developed as experience with the present approach is gained.

## Abstract

In May 1980 an 18-day sequence of oceanic velocity profiles was obtained off the coast of Southern California. The measurements were made using a pair of Doppler sonars mounted on the research platform FLIP and angled downward 45°. The profiles extend to a depth of 600 m. Depth resolution is approximately 30 m. From these profiles the vertical wavenumber-frequency spectrum of the oceanic shear field, Φ(κ, ω)≡〈(∂*u*/∂*z*)^{2}〉/*d*κ*d*ω is estimated.

The shear spectrum is resolved between vertical wavenumbers 1/530 and 1/28 cpm. It is band-limited in wavenumber in the frequency region encompassing near-inertial waves and semidiurnal tides. Motions of vertical wavelength between 100 and 300 m have the greatest shear spectral density. As frequency increases, the band of most energetic motion shifts to ever higher wavenumbers. At frequencies above 8 cpd only the low-wavenumber side of the energetic band can be resolved by the sonars. The wavenumber dependence here appears blue.

It is unlikely that the high-frequency, high-wavenumber shear is a result of linear internal wave activity. The spectrum Φ(κ, ω) is not consistent with previous estimates of the spectrum of isotherm vertical displacement if linear internal wave scaling is used. The vertical displacement spectrum becomes progressively more red (low-mode dominated) with increasing frequency while the shear spectrum becomes progressively more blue. In ignorance of the dynamics of these motions, it is unwise to use internal wave (WKB) scaling to describe the vertical variation of the shear field.

## Abstract

In May 1980 an 18-day sequence of oceanic velocity profiles was obtained off the coast of Southern California. The measurements were made using a pair of Doppler sonars mounted on the research platform FLIP and angled downward 45°. The profiles extend to a depth of 600 m. Depth resolution is approximately 30 m. From these profiles the vertical wavenumber-frequency spectrum of the oceanic shear field, Φ(κ, ω)≡〈(∂*u*/∂*z*)^{2}〉/*d*κ*d*ω is estimated.

The shear spectrum is resolved between vertical wavenumbers 1/530 and 1/28 cpm. It is band-limited in wavenumber in the frequency region encompassing near-inertial waves and semidiurnal tides. Motions of vertical wavelength between 100 and 300 m have the greatest shear spectral density. As frequency increases, the band of most energetic motion shifts to ever higher wavenumbers. At frequencies above 8 cpd only the low-wavenumber side of the energetic band can be resolved by the sonars. The wavenumber dependence here appears blue.

It is unlikely that the high-frequency, high-wavenumber shear is a result of linear internal wave activity. The spectrum Φ(κ, ω) is not consistent with previous estimates of the spectrum of isotherm vertical displacement if linear internal wave scaling is used. The vertical displacement spectrum becomes progressively more red (low-mode dominated) with increasing frequency while the shear spectrum becomes progressively more blue. In ignorance of the dynamics of these motions, it is unwise to use internal wave (WKB) scaling to describe the vertical variation of the shear field.

## Abstract

During January 1977 a cruise was conducted off the California coast on the Research Platform FLIP. Repeated temperature profiling devices were used to sense the internal wavefield in the top 400 m of the sea. From a sequence of 8192 profiles, vertical-velocity spectra and vertical coherence were calculated. Near-surface coherence was found to increase with increasing frequency between local inertial and Väisälä frequencies. Below 200 m the coherence was approximately constant with frequency. The near-surface change in the vertical coherence patterns results from the selective attenuation of the longer vertical wavelengths as the surface is approached. From frequency-depth changes in the near-surface coherence, variations of the internal-wave spectral form can be inferred, in spite of the fact that the deep vertical coherence remains constant. This near-surface effect is not so apparent in data from horizontal-velocity sensors, as only the vertical component of motion is constrained to vanish at the sea surface.

## Abstract

During January 1977 a cruise was conducted off the California coast on the Research Platform FLIP. Repeated temperature profiling devices were used to sense the internal wavefield in the top 400 m of the sea. From a sequence of 8192 profiles, vertical-velocity spectra and vertical coherence were calculated. Near-surface coherence was found to increase with increasing frequency between local inertial and Väisälä frequencies. Below 200 m the coherence was approximately constant with frequency. The near-surface change in the vertical coherence patterns results from the selective attenuation of the longer vertical wavelengths as the surface is approached. From frequency-depth changes in the near-surface coherence, variations of the internal-wave spectral form can be inferred, in spite of the fact that the deep vertical coherence remains constant. This near-surface effect is not so apparent in data from horizontal-velocity sensors, as only the vertical component of motion is constrained to vanish at the sea surface.

## Abstract

A Doppler sonar has been developed at the Marine Physical Laboratory for the remote measurement of the upper ocean velocity field. The sonar transmits a 1° beam at 87.5 kHz, with 3 KW peak power. The sound scatters off drifting plankton, as well as other organisms in the upper ocean. From the Doppler shift of the backscattered sound, the component of water velocity parallel to the beam can be determined at many ranges. During an early test of this system, from FLIP in May 1975, two large propagating events were observed. The events had peak velocity >20 cm s^{−1}, scale wavelength <20 m, and a phase velocity of 40 cm s^{−1}. The horizontal strain rate associated with their passage exceeded 0.01 s^{−1}. Estimates of the Doppler spectral width of the backscattered sound were significantly higher within the core of the events. These events could be internal solitary waves propagating on a high-density gradient layer in the thermocline. Efforts to compare the observations to existing theoretical predictions of weakly nonlinear, two-dimensional solitary waves are not conclusive, perhaps because either the observed events are too nonlinear for the theoretical models to apply or the complex background velocity field of the thermocline is inadequately modeled in the simplified theory.

## Abstract

A Doppler sonar has been developed at the Marine Physical Laboratory for the remote measurement of the upper ocean velocity field. The sonar transmits a 1° beam at 87.5 kHz, with 3 KW peak power. The sound scatters off drifting plankton, as well as other organisms in the upper ocean. From the Doppler shift of the backscattered sound, the component of water velocity parallel to the beam can be determined at many ranges. During an early test of this system, from FLIP in May 1975, two large propagating events were observed. The events had peak velocity >20 cm s^{−1}, scale wavelength <20 m, and a phase velocity of 40 cm s^{−1}. The horizontal strain rate associated with their passage exceeded 0.01 s^{−1}. Estimates of the Doppler spectral width of the backscattered sound were significantly higher within the core of the events. These events could be internal solitary waves propagating on a high-density gradient layer in the thermocline. Efforts to compare the observations to existing theoretical predictions of weakly nonlinear, two-dimensional solitary waves are not conclusive, perhaps because either the observed events are too nonlinear for the theoretical models to apply or the complex background velocity field of the thermocline is inadequately modeled in the simplified theory.

## Abstract

The irregular nature of vertical density profiles is a ubiquitous characteristic of the ocean thermocline. This distortion can be quantified by tracking a set of constant-density (isopycnal) surfaces over time. Examination of 30 000 km of vertical density profile data from seven Pacific Ocean sites indicates that the statistics of isopycnal vertical separation follow the gamma probability distribution, the continuous representation of a Poisson process. All aspects of this process are specified by a single parameter *κ*
_{0}, of order 0.5–2 m^{−1} across the Pacific. When vertical wavenumber spectra of vertical strain are nondimensionalized by *κ*
_{0}, the variability in these pan-Pacific spectra reduce from a factor of 20 to a factor of 2. Given that numerous dimensionless metrics such as the Richardson number, Froude number, Burger number, etc., are required to specify dynamical balances in the sea, it is intriguing that a single-parameter model describes all aspects of the statistics of vertical strain over the range of scales ~2–200 m. While both internal wave and vortical motions are present in the data, the waves dominate the strain signal at these sites. The high-wavenumber cutoff in the strain spectrum is set by the nonsinusoidal waveform of short-vertical-scale internal waves. As large-scale numerical models improve in resolution, they should replicate this Poisson structure in order to properly model plankton variability, vertical diffusion, horizontal dispersion, sound propagation, and other fine-scale phenomena.

## Abstract

The irregular nature of vertical density profiles is a ubiquitous characteristic of the ocean thermocline. This distortion can be quantified by tracking a set of constant-density (isopycnal) surfaces over time. Examination of 30 000 km of vertical density profile data from seven Pacific Ocean sites indicates that the statistics of isopycnal vertical separation follow the gamma probability distribution, the continuous representation of a Poisson process. All aspects of this process are specified by a single parameter *κ*
_{0}, of order 0.5–2 m^{−1} across the Pacific. When vertical wavenumber spectra of vertical strain are nondimensionalized by *κ*
_{0}, the variability in these pan-Pacific spectra reduce from a factor of 20 to a factor of 2. Given that numerous dimensionless metrics such as the Richardson number, Froude number, Burger number, etc., are required to specify dynamical balances in the sea, it is intriguing that a single-parameter model describes all aspects of the statistics of vertical strain over the range of scales ~2–200 m. While both internal wave and vortical motions are present in the data, the waves dominate the strain signal at these sites. The high-wavenumber cutoff in the strain spectrum is set by the nonsinusoidal waveform of short-vertical-scale internal waves. As large-scale numerical models improve in resolution, they should replicate this Poisson structure in order to properly model plankton variability, vertical diffusion, horizontal dispersion, sound propagation, and other fine-scale phenomena.

## Abstract

The irregular nature of vertical profiles of density in the thermocline appears well described by a Poisson process over vertical scales 2–200 m. To what extent does this view of the thermocline conflict with established models of the internal wavefield? Can a one-parameter Poisson subrange be inserted between the larger-scale wavefield and the microscale field of intermittent turbulent dissipation, both of which require many parameters for their specification? It is seen that a small modification to the Poisson vertical correlation function converts it to the corresponding correlation function of the Garrett–Munk (GM) internal wave spectral model. The linear scaling relations and vertical wavenumber dependencies of the GM model are maintained provided the Poisson constant *κ*
_{0} is equated with the ratio of twice the displacement variance to the vertical correlation scale of the wavefield. Awareness of this Poisson wavefield relation enables higher-order strain statistics to be determined directly from the strain spectrum. Using observations from across the Pacific Ocean, the average Thorpe scale of individual overturning events is found to be nearly equal to the inverse of *κ*
_{0}, the metric of background thermocline distortion. If the fractional occurrence of overturning *ϕ* is introduced as an additional parameter, a Poisson version of the Gregg–Henyey relationship can be derived. The Poisson constant, buoyancy frequency, and *ϕ* combine to create a complete parameterization of energy transfer from internal wave scales through the Poisson subrange to dissipation. An awareness of the underlying Poisson structure of the thermocline will hopefully facilitate further improvement in both internal wave spectral models and ocean mixing parameterizations.

## Abstract

The irregular nature of vertical profiles of density in the thermocline appears well described by a Poisson process over vertical scales 2–200 m. To what extent does this view of the thermocline conflict with established models of the internal wavefield? Can a one-parameter Poisson subrange be inserted between the larger-scale wavefield and the microscale field of intermittent turbulent dissipation, both of which require many parameters for their specification? It is seen that a small modification to the Poisson vertical correlation function converts it to the corresponding correlation function of the Garrett–Munk (GM) internal wave spectral model. The linear scaling relations and vertical wavenumber dependencies of the GM model are maintained provided the Poisson constant *κ*
_{0} is equated with the ratio of twice the displacement variance to the vertical correlation scale of the wavefield. Awareness of this Poisson wavefield relation enables higher-order strain statistics to be determined directly from the strain spectrum. Using observations from across the Pacific Ocean, the average Thorpe scale of individual overturning events is found to be nearly equal to the inverse of *κ*
_{0}, the metric of background thermocline distortion. If the fractional occurrence of overturning *ϕ* is introduced as an additional parameter, a Poisson version of the Gregg–Henyey relationship can be derived. The Poisson constant, buoyancy frequency, and *ϕ* combine to create a complete parameterization of energy transfer from internal wave scales through the Poisson subrange to dissipation. An awareness of the underlying Poisson structure of the thermocline will hopefully facilitate further improvement in both internal wave spectral models and ocean mixing parameterizations.

## Abstract

From October 1997 through October 1998, the Surface Heat Budget of the Arctic (SHEBA) ice camp drifted across the western Arctic Ocean, from the central Canada Basin over the Northwind Ridge and across the Chukchi Cap. During much of this period, the velocity and shear fields in the upper ocean were monitored by Doppler sonar. Near-inertial internal waves are found to be the dominant contributors to the superinertial motion field. Typical rms velocities are 1–2 cm s^{−1}. In this work, the velocity and shear variances associated with upward- and downward-propagating wave groups are quantified. Patterns are detected in these variances that correlate with underlying seafloor depth. These are explored with the objective of assessing the role that these extremely low-energy near-inertial waves play in the larger-scale evolution of the Canada Basin. The specific focus is the energy flux delivered to the slopes and shelves of the basin, available for driving mixing at the ocean boundaries. The energy and shear variances associated with downward-propagating waves are relatively uniform over the entire SHEBA drift, independent of the season and depth of the underlying topography. Variances associated with upward-propagating waves follow a (depth)^{−1/2} dependence. Over the deep slopes, vertical wavenumber spectra of upward-propagating waves are blue-shifted relative to their downward counterparts, perhaps a result of reflection from a sloping seafloor. To aid in interpretation of the observations, a simple, linear model is used to compare the effects of viscous (volume) versus underice (surface) dissipation for near-inertial waves. The latter is found to be the dominant mechanism. A parallel examination of the topography of the western Arctic shows that much of the continental slope is close to critical for near-inertial wave reflection. The picture that emerges is consistent with “one bounce” rather than trans-Arctic propagation. The dominant surface-generated waves are substantially absorbed in the underice boundary layer following a single roundtrip to the seafloor. However, surface-generated waves can interact strongly with nearby (<300 km) slopes, potentially contributing to dissipation rates of order 10^{−6}–10^{−7} W m^{−3} in a zone several hundred meters above the bottom. The waves that survive the reflection process (and are not back-reflected) display a measurable blue shift over the slopes and contribute to the observed dependence of energy on seafloor depth that is seen in these upper-ocean observations.

## Abstract

From October 1997 through October 1998, the Surface Heat Budget of the Arctic (SHEBA) ice camp drifted across the western Arctic Ocean, from the central Canada Basin over the Northwind Ridge and across the Chukchi Cap. During much of this period, the velocity and shear fields in the upper ocean were monitored by Doppler sonar. Near-inertial internal waves are found to be the dominant contributors to the superinertial motion field. Typical rms velocities are 1–2 cm s^{−1}. In this work, the velocity and shear variances associated with upward- and downward-propagating wave groups are quantified. Patterns are detected in these variances that correlate with underlying seafloor depth. These are explored with the objective of assessing the role that these extremely low-energy near-inertial waves play in the larger-scale evolution of the Canada Basin. The specific focus is the energy flux delivered to the slopes and shelves of the basin, available for driving mixing at the ocean boundaries. The energy and shear variances associated with downward-propagating waves are relatively uniform over the entire SHEBA drift, independent of the season and depth of the underlying topography. Variances associated with upward-propagating waves follow a (depth)^{−1/2} dependence. Over the deep slopes, vertical wavenumber spectra of upward-propagating waves are blue-shifted relative to their downward counterparts, perhaps a result of reflection from a sloping seafloor. To aid in interpretation of the observations, a simple, linear model is used to compare the effects of viscous (volume) versus underice (surface) dissipation for near-inertial waves. The latter is found to be the dominant mechanism. A parallel examination of the topography of the western Arctic shows that much of the continental slope is close to critical for near-inertial wave reflection. The picture that emerges is consistent with “one bounce” rather than trans-Arctic propagation. The dominant surface-generated waves are substantially absorbed in the underice boundary layer following a single roundtrip to the seafloor. However, surface-generated waves can interact strongly with nearby (<300 km) slopes, potentially contributing to dissipation rates of order 10^{−6}–10^{−7} W m^{−3} in a zone several hundred meters above the bottom. The waves that survive the reflection process (and are not back-reflected) display a measurable blue shift over the slopes and contribute to the observed dependence of energy on seafloor depth that is seen in these upper-ocean observations.

## Abstract

During the spring tides of late November 1992, early January and early February 1993, solitary internal wave packets were observed at 2°S, 156°15′E in the western equatorial Pacific. Apparently generated in the Nuguria island group (3°S, 153°E), the waves propagate northeastward at 2.4–2.8 m s^{−1}, appearing in fixed phase with the underlying semidiurnal baroclinic tide. The initial solitary wave crests have downward displacements in excess of 60 m and peak velocities greater than 80 cm s^{−1}. Groups of 1–3 crests are observed, with vertical structure resembling a first-mode internal wave and horizontal variability consistent with third-order comb KdV solitons. Many of the packets exhibit an overall organization reminiscent of undular bores. This borelike behavior is confined primarily to the upper 70 m, not sharing the mode-one dependency of the crests. The solitons displace the ambient equatorial currents, including the Equatorial Undercurrent, both vertically (80 m), and laterally (1–2 km), with little apparent interaction.

A 161-kHz Doppler sonar mounted on the R.V. *John Vickers* provided ocean velocity measurements with 3-m vertical resolution and 2-min time resolution in the upper 250 m of the sea. Merged with GPS-derived ship’s navigation, the resultant depth–time records of absolute velocity enable estimation of flow streamlines, given the two-dimensional nature of the wave trains. During passage of the soliton crests, the vertical displacement of streamlines is in good agreement with the observed vertical displacement of biological scattering layers.

Noticeable increases in acoustic scattering strength are associated with the passage of all soliton groups, suggesting the (turbulent) production of small-scale (0.46 cm) structure in the sound speed field. However, the shear in these mode-one solitons is small compared to ambient equatorial background shears. The minimum Richardson number intrinsic to the soliton packet is of order 15. The crests apparently trigger small-scale instabilities on the background shear. Several of the soliton packets display pronounced internal wave “tails.” These too are apparently triggered disturbances on the preexisting flow. High-frequency shears are oriented nearly orthogonal to the low-frequency background, independent of the propagation direction of the soliton.

The energy density of the larger soliton groups approaches 0.2 gigajoules per meter of crest, a value comparable to the underlying baroclinic tide, and also comparable to the energy lost from the *M*
_{2} barotropic tide over a ∼1500 km propagation path through the western tropical Pacific (given a mean barotropic dissipation rate of 3 × 10^{−3} W m^{−2} for the region).

## Abstract

During the spring tides of late November 1992, early January and early February 1993, solitary internal wave packets were observed at 2°S, 156°15′E in the western equatorial Pacific. Apparently generated in the Nuguria island group (3°S, 153°E), the waves propagate northeastward at 2.4–2.8 m s^{−1}, appearing in fixed phase with the underlying semidiurnal baroclinic tide. The initial solitary wave crests have downward displacements in excess of 60 m and peak velocities greater than 80 cm s^{−1}. Groups of 1–3 crests are observed, with vertical structure resembling a first-mode internal wave and horizontal variability consistent with third-order comb KdV solitons. Many of the packets exhibit an overall organization reminiscent of undular bores. This borelike behavior is confined primarily to the upper 70 m, not sharing the mode-one dependency of the crests. The solitons displace the ambient equatorial currents, including the Equatorial Undercurrent, both vertically (80 m), and laterally (1–2 km), with little apparent interaction.

A 161-kHz Doppler sonar mounted on the R.V. *John Vickers* provided ocean velocity measurements with 3-m vertical resolution and 2-min time resolution in the upper 250 m of the sea. Merged with GPS-derived ship’s navigation, the resultant depth–time records of absolute velocity enable estimation of flow streamlines, given the two-dimensional nature of the wave trains. During passage of the soliton crests, the vertical displacement of streamlines is in good agreement with the observed vertical displacement of biological scattering layers.

Noticeable increases in acoustic scattering strength are associated with the passage of all soliton groups, suggesting the (turbulent) production of small-scale (0.46 cm) structure in the sound speed field. However, the shear in these mode-one solitons is small compared to ambient equatorial background shears. The minimum Richardson number intrinsic to the soliton packet is of order 15. The crests apparently trigger small-scale instabilities on the background shear. Several of the soliton packets display pronounced internal wave “tails.” These too are apparently triggered disturbances on the preexisting flow. High-frequency shears are oriented nearly orthogonal to the low-frequency background, independent of the propagation direction of the soliton.

The energy density of the larger soliton groups approaches 0.2 gigajoules per meter of crest, a value comparable to the underlying baroclinic tide, and also comparable to the energy lost from the *M*
_{2} barotropic tide over a ∼1500 km propagation path through the western tropical Pacific (given a mean barotropic dissipation rate of 3 × 10^{−3} W m^{−2} for the region).

## Abstract

Upper-ocean velocity and shear data were obtained from Doppler sonars operated at the Surface Heat Budget of the Arctic Ocean (SHEBA) ice camp during the camp’s year-long drift across the western Arctic Ocean. These are used to estimate wavenumber–frequency spectra of shear *E*(*κ _{z}
*,

*σ*) during three selected time intervals. The Arctic shear spectra are similar in form to typical oceanic spectra, except that they have roughly an order-of-magnitude less variance. The slope of the frequency dependence is also steeper (

*σ*

^{−3}–

*σ*

^{−4}for

*σ*< −

*f*, where

*f*is the Coriolis frequency) in the internal-wave band, and the vertical wavenumber dependence is centered at higher wavenumber. Given the small vertical scales and low velocities of Arctic signals, a careful assessment of sonar precision is performed. Fluctuations at vertical scales > 10 m and time scales > 1 h are deemed significant. At subinertial frequencies, a vortical (quasigeostrophic) contribution to the shear spectrum is seen. The vertical wavenumber dependence of the shear spectrum in this frequency range is distinctly red, in contrast to the band-limited form of the superinertial spectrum; that is,

*E*(

*κ*,

_{z}*σ*) ∼

*κ*

^{−1}

_{ z }for |

*σ*| <

*f*. A fundamental characteristic of both the internal-wave and vortical spectral contributions is that the observed frequency bandwidth increases linearly with increasing vertical wavenumber magnitude. This is interpreted as the signature of the Doppler shifting of the observations by random “background” currents and by ice motion and is responsible for the distinctly “nonseparable” nature of the shear spectrum. As a consequence, the vertical wavenumber spectrum of the subinertial motion field is white:

*E*

_{vort}(

*κ*) = ∫

_{z}*E*

_{vort}(

*κ*,

_{z}*σ*)

*dσ*∼

*κ*

^{0}

_{ z }.

## Abstract

Upper-ocean velocity and shear data were obtained from Doppler sonars operated at the Surface Heat Budget of the Arctic Ocean (SHEBA) ice camp during the camp’s year-long drift across the western Arctic Ocean. These are used to estimate wavenumber–frequency spectra of shear *E*(*κ _{z}
*,

*σ*) during three selected time intervals. The Arctic shear spectra are similar in form to typical oceanic spectra, except that they have roughly an order-of-magnitude less variance. The slope of the frequency dependence is also steeper (

*σ*

^{−3}–

*σ*

^{−4}for

*σ*< −

*f*, where

*f*is the Coriolis frequency) in the internal-wave band, and the vertical wavenumber dependence is centered at higher wavenumber. Given the small vertical scales and low velocities of Arctic signals, a careful assessment of sonar precision is performed. Fluctuations at vertical scales > 10 m and time scales > 1 h are deemed significant. At subinertial frequencies, a vortical (quasigeostrophic) contribution to the shear spectrum is seen. The vertical wavenumber dependence of the shear spectrum in this frequency range is distinctly red, in contrast to the band-limited form of the superinertial spectrum; that is,

*E*(

*κ*,

_{z}*σ*) ∼

*κ*

^{−1}

_{ z }for |

*σ*| <

*f*. A fundamental characteristic of both the internal-wave and vortical spectral contributions is that the observed frequency bandwidth increases linearly with increasing vertical wavenumber magnitude. This is interpreted as the signature of the Doppler shifting of the observations by random “background” currents and by ice motion and is responsible for the distinctly “nonseparable” nature of the shear spectrum. As a consequence, the vertical wavenumber spectrum of the subinertial motion field is white:

*E*

_{vort}(

*κ*) = ∫

_{z}*E*

_{vort}(

*κ*,

_{z}*σ*)

*dσ*∼

*κ*

^{0}

_{ z }.

## Abstract

Continuous depth–time measurements of upper-ocean velocity are used to estimate the wavenumber–frequency spectrum of shear. A fundamental characteristic of these spectra is that the frequency bandwidth increases linearly with increasing wavenumber magnitude. This can be interpreted as the signature of Doppler shifting of the observations by time-changing “background” currents as well as by instrument motion. Here, the hypothesis is posed that the apparently continuous wavenumber–frequency spectrum of oceanic shear results from the advective “smearing” of discrete spectral lines. In the Arctic Ocean, lines at the inertial (*ω* = −*f* ) and vortical (*ω* = 0) frequencies (where *f* is the Coriolis frequency) account for most of the variance in the shear spectrum. In the tropical ocean, two classes of inertial waves are considered, accounting for 70% of the observed shear variance. A simple model is introduced to quantify the effects of lateral advection, random vertical advection (“fine-structure contamination”), and deterministic (tidal) vertical advection on these “otherwise monochromatic” records. Model frequency spectra are developed in terms of the probability density and/or spectrum of the advecting fields for general but idealized situations. The model successfully mimics the increasing frequency bandwidth of the shear spectrum with increasing vertical wavenumber. Excellent fits to the observed frequency spectrum of shear are obtained for the Arctic (weak advection and short-spatial-scale inertial waves) and low-latitude (strong advection and long and short inertial waves) observations. While successfully replicating the wavenumber–frequency spectrum of shear, the model does not even consider motion at scales greater than ∼250 m, the “energy containing” scales of the internal wave field. To a first approximation, the waves with the majority of the kinetic and potential energy constitute a population apart from those with the momentum, shear, and strain.

## Abstract

Continuous depth–time measurements of upper-ocean velocity are used to estimate the wavenumber–frequency spectrum of shear. A fundamental characteristic of these spectra is that the frequency bandwidth increases linearly with increasing wavenumber magnitude. This can be interpreted as the signature of Doppler shifting of the observations by time-changing “background” currents as well as by instrument motion. Here, the hypothesis is posed that the apparently continuous wavenumber–frequency spectrum of oceanic shear results from the advective “smearing” of discrete spectral lines. In the Arctic Ocean, lines at the inertial (*ω* = −*f* ) and vortical (*ω* = 0) frequencies (where *f* is the Coriolis frequency) account for most of the variance in the shear spectrum. In the tropical ocean, two classes of inertial waves are considered, accounting for 70% of the observed shear variance. A simple model is introduced to quantify the effects of lateral advection, random vertical advection (“fine-structure contamination”), and deterministic (tidal) vertical advection on these “otherwise monochromatic” records. Model frequency spectra are developed in terms of the probability density and/or spectrum of the advecting fields for general but idealized situations. The model successfully mimics the increasing frequency bandwidth of the shear spectrum with increasing vertical wavenumber. Excellent fits to the observed frequency spectrum of shear are obtained for the Arctic (weak advection and short-spatial-scale inertial waves) and low-latitude (strong advection and long and short inertial waves) observations. While successfully replicating the wavenumber–frequency spectrum of shear, the model does not even consider motion at scales greater than ∼250 m, the “energy containing” scales of the internal wave field. To a first approximation, the waves with the majority of the kinetic and potential energy constitute a population apart from those with the momentum, shear, and strain.