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- Author or Editor: Jerome Smith x

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## Abstract

The energy, momentum, and mass-flux exchanges between surface waves and underlying Eulerian mean flows are considered, and terms in addition to the classical wave “radiation stress” are identified. The formulation is made in terms of the vertically integrated flow. The various terms are identified with other analyses and interpreted in terms of physical mechanisms, permitting reasonable estimates of the associated depth dependencies. One term is identified with the integrated “CL vortex force” implemented, for example, in simulations of Langmuir circulation. However, as illustrated with a simple example of steady refraction across a shear zone, other terms of the same order can significantly alter the results. The classic example of long waves forced by short-wave groups is also revisited. In this case, an apparent singularity arising in shallow water is countered by finite-amplitude dispersion corrections, these being formally of the same order as the forced long-wave response, and becoming significant or dominant as shallow water is approached.

## Abstract

The energy, momentum, and mass-flux exchanges between surface waves and underlying Eulerian mean flows are considered, and terms in addition to the classical wave “radiation stress” are identified. The formulation is made in terms of the vertically integrated flow. The various terms are identified with other analyses and interpreted in terms of physical mechanisms, permitting reasonable estimates of the associated depth dependencies. One term is identified with the integrated “CL vortex force” implemented, for example, in simulations of Langmuir circulation. However, as illustrated with a simple example of steady refraction across a shear zone, other terms of the same order can significantly alter the results. The classic example of long waves forced by short-wave groups is also revisited. In this case, an apparent singularity arising in shallow water is countered by finite-amplitude dispersion corrections, these being formally of the same order as the forced long-wave response, and becoming significant or dominant as shallow water is approached.

## Abstract

Phased-array Doppler sonars (PADS) have been used to probe an area several hundred meters on a side with 8-m spatial resolution, sampling every second or less with under 2 cm s^{−1} rms velocity error per sample. Estimates from two systems were combined to produce horizontal velocity vectors. Here, concerns specific to use of PADS in shallow water are addressed. In particular, the shallower the water is, the larger the fraction of bottom backscatter, so the stronger the bias is toward zero Doppler shift in the estimates. First, direct comparisons are made with other current measurements made during the multi-investigator field experiment “SandyDuck,” sponsored by the Office of Naval Research, which took place in the autumn of 1997 off the coast of Duck, North Carolina. The coherences between PADS and in situ current measurements are high, but the amplitude of the sonar response is generally low. To explore this further, a simplified model of wave shoaling is developed, permitting estimates of wave-frequency velocity variances from point measurements to be extrapolated over the whole field of view of PADS for comparison. The resulting time–space movies of sonar response are consistent with quasi-steady acoustic backscatter intensity from the bottom competing with a variable backscatter level from the water volume. The latter may arise, for example, from intermittent injection of bubbles by breaking waves, producing patches of high or low acoustic response that advect with the mean flow. Once this competition is calibrated via the surface wave variance comparison, instantaneous measured total backscatter intensities can be compared with an estimated bottom backscatter level (which is updated on a longer timescale, appropriate to evolution of the water depth or bottom roughness) to provide corrected sonar estimates over the region.

## Abstract

Phased-array Doppler sonars (PADS) have been used to probe an area several hundred meters on a side with 8-m spatial resolution, sampling every second or less with under 2 cm s^{−1} rms velocity error per sample. Estimates from two systems were combined to produce horizontal velocity vectors. Here, concerns specific to use of PADS in shallow water are addressed. In particular, the shallower the water is, the larger the fraction of bottom backscatter, so the stronger the bias is toward zero Doppler shift in the estimates. First, direct comparisons are made with other current measurements made during the multi-investigator field experiment “SandyDuck,” sponsored by the Office of Naval Research, which took place in the autumn of 1997 off the coast of Duck, North Carolina. The coherences between PADS and in situ current measurements are high, but the amplitude of the sonar response is generally low. To explore this further, a simplified model of wave shoaling is developed, permitting estimates of wave-frequency velocity variances from point measurements to be extrapolated over the whole field of view of PADS for comparison. The resulting time–space movies of sonar response are consistent with quasi-steady acoustic backscatter intensity from the bottom competing with a variable backscatter level from the water volume. The latter may arise, for example, from intermittent injection of bubbles by breaking waves, producing patches of high or low acoustic response that advect with the mean flow. Once this competition is calibrated via the surface wave variance comparison, instantaneous measured total backscatter intensities can be compared with an estimated bottom backscatter level (which is updated on a longer timescale, appropriate to evolution of the water depth or bottom roughness) to provide corrected sonar estimates over the region.

## Abstract

The reintroduction of compressibility into the equations for surface gravity waves can permit mixed acoustic–gravity modes that are periodic in the vertical as well as horizontal directions. These modes interact with the bottom even in deep water, so bottom motion can excite them. Because they propagate rapidly, it has been suggested they may be useful as precursors of tsunamis. Here the equations are revisited, and, using some robust approximations, some physical understanding and interpretation of the phenomena are presented. It is posed that these new modes can alternatively be thought of as acoustic modes slightly modified by a gravity wave boundary condition at the surface, rather than as surface waves dramatically modified by compressibility. Their potential use is not diminished; indeed, this alternative perspective should help make implementation more practical.

## Abstract

The reintroduction of compressibility into the equations for surface gravity waves can permit mixed acoustic–gravity modes that are periodic in the vertical as well as horizontal directions. These modes interact with the bottom even in deep water, so bottom motion can excite them. Because they propagate rapidly, it has been suggested they may be useful as precursors of tsunamis. Here the equations are revisited, and, using some robust approximations, some physical understanding and interpretation of the phenomena are presented. It is posed that these new modes can alternatively be thought of as acoustic modes slightly modified by a gravity wave boundary condition at the surface, rather than as surface waves dramatically modified by compressibility. Their potential use is not diminished; indeed, this alternative perspective should help make implementation more practical.

## Abstract

Short, dissipative, surface waves superposed on longer waves cause a growth of the long wave momentum *M _{l}* at a ratewhere

*k*,

_{l}*a*are the amplitude and wavenumber of the long waves, so that

_{l}*k*

_{l}*a*is their steepness;

_{l}*S*is the radiation stress of the short waves and τ

_{a}*, the rate of transfer of momentum to the short waves by the wind; and the angle braces denote an average over the long-wave phase θ =*

_{s}*k*

_{l}*x*−ω

*.*

_{l}tThe first term in the above equation is the radiation stress interaction (Phillips, 1963; Hasselmann, 1971) and is generally negligible compared with the second term, neglected by Hasselmann (1971), which shows that long waves can grow if short wave generation (rather than dissipation) is correlated with the long wave orbital velocity.

Even if the modulation of τ* _{s}* is only O(

*k*

_{l}*a*) times 〈τ

_{l}*〉, this mechanism can contribute a significant fraction of long wave momentum. However, even a substantially greater modulation of τ*

_{s}*, perhaps due to varying exposure of short waves to the wind, is unlikely to account for all the alleged momentum input to long waves, due to the upper bound*

_{s}*k*

_{l}*a*on the efficiency of the process.

_{l}## Abstract

Short, dissipative, surface waves superposed on longer waves cause a growth of the long wave momentum *M _{l}* at a ratewhere

*k*,

_{l}*a*are the amplitude and wavenumber of the long waves, so that

_{l}*k*

_{l}*a*is their steepness;

_{l}*S*is the radiation stress of the short waves and τ

_{a}*, the rate of transfer of momentum to the short waves by the wind; and the angle braces denote an average over the long-wave phase θ =*

_{s}*k*

_{l}*x*−ω

*.*

_{l}tThe first term in the above equation is the radiation stress interaction (Phillips, 1963; Hasselmann, 1971) and is generally negligible compared with the second term, neglected by Hasselmann (1971), which shows that long waves can grow if short wave generation (rather than dissipation) is correlated with the long wave orbital velocity.

Even if the modulation of τ* _{s}* is only O(

*k*

_{l}*a*) times 〈τ

_{l}*〉, this mechanism can contribute a significant fraction of long wave momentum. However, even a substantially greater modulation of τ*

_{s}*, perhaps due to varying exposure of short waves to the wind, is unlikely to account for all the alleged momentum input to long waves, due to the upper bound*

_{s}*k*

_{l}*a*on the efficiency of the process.

_{l}## Abstract

Wave breaking and wave-forced flows are important to air–sea interactions and to the transport and dispersal of materials at sea. But recent measurements have shown a discrepancy in the Eulerian response to wave groups compared to scientists’ current theoretical understanding of wave–current interactions. Flow structures on scales of centimeters to meters occur underneath breaking waves, and larger-scale flows are driven by wave–current interactions (e.g., Langmuir circulation, alongshore flows). Such details of the vertically resolved flow are just beginning to be modeled, and observational guidance is needed. Here a new instrument is described that is intended to measure waves and currents over a 2D vertical plane underwater, resolving two components of velocity on this plane. Initial observations were made near the Scripps Pier (La Jolla, California), where steep waves and strong currents can be reliably found, yet logistics are not too burdensome. To get the spatial resolution desired using 200-kHz sound, ping-to-ping “coherent processing” would have be used for Doppler estimation; however, near shore the reverberations remain too strong for far too long to get any coherence, unlike the previous experience in deep water. In view of this, using much higher frequencies (>1 MHz) with “incoherent processing” is suggested; the increased attenuation at higher frequencies then would subdue the reverberation problem, but with comparable space–time resolution.

## Abstract

Wave breaking and wave-forced flows are important to air–sea interactions and to the transport and dispersal of materials at sea. But recent measurements have shown a discrepancy in the Eulerian response to wave groups compared to scientists’ current theoretical understanding of wave–current interactions. Flow structures on scales of centimeters to meters occur underneath breaking waves, and larger-scale flows are driven by wave–current interactions (e.g., Langmuir circulation, alongshore flows). Such details of the vertically resolved flow are just beginning to be modeled, and observational guidance is needed. Here a new instrument is described that is intended to measure waves and currents over a 2D vertical plane underwater, resolving two components of velocity on this plane. Initial observations were made near the Scripps Pier (La Jolla, California), where steep waves and strong currents can be reliably found, yet logistics are not too burdensome. To get the spatial resolution desired using 200-kHz sound, ping-to-ping “coherent processing” would have be used for Doppler estimation; however, near shore the reverberations remain too strong for far too long to get any coherence, unlike the previous experience in deep water. In view of this, using much higher frequencies (>1 MHz) with “incoherent processing” is suggested; the increased attenuation at higher frequencies then would subdue the reverberation problem, but with comparable space–time resolution.

## Abstract

The performance limitations of an acoustic Doppler sonar system are explored and compared with anticipated requirements for the measurement of surface wave directional/frequency spectra. To obtain measurements to a range *D* requires a delay Δ*t* between pings long enough for sound to propagate out to *D* and back: Δ*t*(*c*/2) ≥ *D*. This defines a Nyquist frequency, ω_{N} (radiances s^{−1}). Linear dispersion relates this to a “matched wavenumber,” *k _{N}* = ω

_{N}

^{2}/

*g*. Waves travelling obliquely and harmonics of longer waves appearing at ω

_{N}all have smaller wavenumbers,

*k*≤

*k*; thus,

_{N}*k*; defines a maximum wavenumber requirement, or (equivalently) a matched range resolution, Δ

_{N}*R*. From idealized surface wave spectra, the velocity resolution Δ

*V*required to measure spectra out to (ω

_{N},

*k*) can be estimated. For a given sonar “tone,” the error-product

_{N}*E*= Δ

*R*Δ

*V*is a constant, so velocity resolution and range resolution must be traded off. The error product decreases with increasing acoustic frequency

*f*

_{0}and number of tones. Higher frequency sound is also attenuated more rapidly, limiting the maximum range attainable. A practical approach is to define a desired range

*D*, find the highest frequency which can be detected to that range, and then determine the number of tones required to achieve the target velocity and range resolutions. If too many tones are needed, a slight retreat in range resolution yields a relaxation in the velocity requirement as well (because of the steep spectral slope of surface wave spectra). Electronic design and performance is neglected here, on the presumption that the physical limits discussed will eventually be the important ones.

## Abstract

The performance limitations of an acoustic Doppler sonar system are explored and compared with anticipated requirements for the measurement of surface wave directional/frequency spectra. To obtain measurements to a range *D* requires a delay Δ*t* between pings long enough for sound to propagate out to *D* and back: Δ*t*(*c*/2) ≥ *D*. This defines a Nyquist frequency, ω_{N} (radiances s^{−1}). Linear dispersion relates this to a “matched wavenumber,” *k _{N}* = ω

_{N}

^{2}/

*g*. Waves travelling obliquely and harmonics of longer waves appearing at ω

_{N}all have smaller wavenumbers,

*k*≤

*k*; thus,

_{N}*k*; defines a maximum wavenumber requirement, or (equivalently) a matched range resolution, Δ

_{N}*R*. From idealized surface wave spectra, the velocity resolution Δ

*V*required to measure spectra out to (ω

_{N},

*k*) can be estimated. For a given sonar “tone,” the error-product

_{N}*E*= Δ

*R*Δ

*V*is a constant, so velocity resolution and range resolution must be traded off. The error product decreases with increasing acoustic frequency

*f*

_{0}and number of tones. Higher frequency sound is also attenuated more rapidly, limiting the maximum range attainable. A practical approach is to define a desired range

*D*, find the highest frequency which can be detected to that range, and then determine the number of tones required to achieve the target velocity and range resolutions. If too many tones are needed, a slight retreat in range resolution yields a relaxation in the velocity requirement as well (because of the steep spectral slope of surface wave spectra). Electronic design and performance is neglected here, on the presumption that the physical limits discussed will eventually be the important ones.

## Abstract

Waves and currents interact via exchanges of mass and momentum. The mass and momentum fluxes associated with surface waves are closely linked to their Stokes drift. Both the variability of the Stokes drift and the corresponding response of the underlying flow are important in a wide range of contexts. Three methods are developed and implemented to evaluate Stokes drift from a recently gathered oceanic dataset, involving surface velocities measured continually over an area 1.5 km in radius by 45°. The estimated Stokes drift varies significantly, in line with the occurrence of compact wave groups, resulting in highly intermittent maxima. One method also provides currents at a fixed level (Eulerian velocities). It is found that Eulerian counterflows occur that completely cancel the Stokes drift variations at the surface. Thus, the estimated Lagrangian surface flow has no discernable mean response to wave group passage. This response is larger than anticipated and is hard to reconcile with current theory.

## Abstract

Waves and currents interact via exchanges of mass and momentum. The mass and momentum fluxes associated with surface waves are closely linked to their Stokes drift. Both the variability of the Stokes drift and the corresponding response of the underlying flow are important in a wide range of contexts. Three methods are developed and implemented to evaluate Stokes drift from a recently gathered oceanic dataset, involving surface velocities measured continually over an area 1.5 km in radius by 45°. The estimated Stokes drift varies significantly, in line with the occurrence of compact wave groups, resulting in highly intermittent maxima. One method also provides currents at a fixed level (Eulerian velocities). It is found that Eulerian counterflows occur that completely cancel the Stokes drift variations at the surface. Thus, the estimated Lagrangian surface flow has no discernable mean response to wave group passage. This response is larger than anticipated and is hard to reconcile with current theory.

## Abstract

In August 1990, tests were performed to investigate the usefulness of a horizontally scanning Doppler acoustic technique in shallow water. Comparisons of radial velocity estimates from a vertical fan beam versus a horizontally aimed pencil beam indicate no degradation attributable to multiple reflections from the surface and bottom. Further tests, in which ping-to-ping phase-coherent means are examined, indicate negligible stationary backscatter from the bottom. Tests in which the acoustic beams were directed shoreward indicate that an extremely dense bubble cloud formed by plunging breakers produces an impenetrable “wall” at the breakpoint, at acoustic frequencies near 195 kHz. Useful velocity estimates (one component) are obtainable everywhere seaward of the breakpoint of the incoming surf. The spatially extensive velocity estimates offered by this technique provide enormous potential for the study of horizontal currents and wave-current interactions in shallow water.

## Abstract

In August 1990, tests were performed to investigate the usefulness of a horizontally scanning Doppler acoustic technique in shallow water. Comparisons of radial velocity estimates from a vertical fan beam versus a horizontally aimed pencil beam indicate no degradation attributable to multiple reflections from the surface and bottom. Further tests, in which ping-to-ping phase-coherent means are examined, indicate negligible stationary backscatter from the bottom. Tests in which the acoustic beams were directed shoreward indicate that an extremely dense bubble cloud formed by plunging breakers produces an impenetrable “wall” at the breakpoint, at acoustic frequencies near 195 kHz. Useful velocity estimates (one component) are obtainable everywhere seaward of the breakpoint of the incoming surf. The spatially extensive velocity estimates offered by this technique provide enormous potential for the study of horizontal currents and wave-current interactions in shallow water.

## Abstract

A phased-array Doppler sonar (PADS) system is described that uses sound at frequencies near 200 kHz to probe an area several hundred meters on a side with 7–20-m spatial resolution. The area can be sampled every second or less with under 2 cm s^{–1} rms velocity error per sample. Radial velocity estimates from two or more systems can be combined to produce time series of horizontal velocity vector maps over the irregularly shaped overlapping region. Such extensive and continuous sampling in time and space permits analysis via direct 3D Fourier transformation, for example, producing complete wavenumber–frequency spectra. Free waves, bound harmonics, finite-amplitude effects, Doppler shifting by currents, etc., can be studied. Extended temporal sampling permits investigations into lower-frequency vortical and internal wave modes as well as surface waves, and of the modulation of these by tides. A pair of PADS was deployed as part of SandyDuck, a large collaborative field experiment held in 1997 near Duck, North Carolina. An example drawn from SandyDuck data illustrates use of the technique, demonstrating that both mean flow and oscillatory (wave) motions can be detected.

## Abstract

A phased-array Doppler sonar (PADS) system is described that uses sound at frequencies near 200 kHz to probe an area several hundred meters on a side with 7–20-m spatial resolution. The area can be sampled every second or less with under 2 cm s^{–1} rms velocity error per sample. Radial velocity estimates from two or more systems can be combined to produce time series of horizontal velocity vector maps over the irregularly shaped overlapping region. Such extensive and continuous sampling in time and space permits analysis via direct 3D Fourier transformation, for example, producing complete wavenumber–frequency spectra. Free waves, bound harmonics, finite-amplitude effects, Doppler shifting by currents, etc., can be studied. Extended temporal sampling permits investigations into lower-frequency vortical and internal wave modes as well as surface waves, and of the modulation of these by tides. A pair of PADS was deployed as part of SandyDuck, a large collaborative field experiment held in 1997 near Duck, North Carolina. An example drawn from SandyDuck data illustrates use of the technique, demonstrating that both mean flow and oscillatory (wave) motions can be detected.