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## Abstract

Inverse models are developed that use data and dynamics to estimate optimally the breaking-wave-driven setup and alongshore current, as well as the cross-shore forcing, alongshore forcing, and drag coefficient. The inverse models accurately reproduce these quantities in a synthetic barred-beach example. The method is applied to one case example each from the Duck94 and SandyDuck field experiments. Both inverse solutions pass consistency tests developed for the inverse method and have forcing corrections similar to a roller model and significant cross-shore variation of the drag coefficient. The inverse drag coefficient is related to the wave dissipation, a bulk measure of the turbulence source, but not to the bed roughness, consistent with the hypothesis that breaking-wave-generated turbulence increases the drag coefficient. Inverse solutions from a wider range of conditions are required to establish the generality of these results.

## Abstract

Inverse models are developed that use data and dynamics to estimate optimally the breaking-wave-driven setup and alongshore current, as well as the cross-shore forcing, alongshore forcing, and drag coefficient. The inverse models accurately reproduce these quantities in a synthetic barred-beach example. The method is applied to one case example each from the Duck94 and SandyDuck field experiments. Both inverse solutions pass consistency tests developed for the inverse method and have forcing corrections similar to a roller model and significant cross-shore variation of the drag coefficient. The inverse drag coefficient is related to the wave dissipation, a bulk measure of the turbulence source, but not to the bed roughness, consistent with the hypothesis that breaking-wave-generated turbulence increases the drag coefficient. Inverse solutions from a wider range of conditions are required to establish the generality of these results.

## Abstract

This is Part 1 of a two-part study of infragravity-frequency (nominally 0.005–0.05 Hz) motions on the continental shelf. Data from a large aperture (250 m × 250 m) array of 24 bottom-mounted pressure transducers deployed in 13 m depth is used to investigate the local forcing of infragravity motions by nonlinear difference-frequency interactions of surface gravity waves. Second-order nonlinear theory (Hasselmann) and observed swell-sea frequency-directional spectra are used to predict the energy levels of forced infragravity waves. For a wide range of wave conditions, the predicted forced wave levels are lower than the observed energy levels, suggesting that the infragravity band contains a mix of free and forced waves. Bispectral analysis is used to estimate the relative amounts of free and forced infragravity energy. Good agreement between bispectrum-based estimates and theoretical predictions of forced wave energy confirms that second-order nonlinear theory accurately predicts locally forced infragravity motions. The contribution of forced waves to the total infragravity energy, ranging from less than 0.1% to about 30%, is largest when the infragravity energy is maximum, consistent with previously noted trends in similar water depths. The bispectral technique developed here to estimate the energy of forced and free infragravity waves is used in Part 2 to investigate, with data from single-point pressure gauges, the shelfwide variability of free infragravity energy.

## Abstract

This is Part 1 of a two-part study of infragravity-frequency (nominally 0.005–0.05 Hz) motions on the continental shelf. Data from a large aperture (250 m × 250 m) array of 24 bottom-mounted pressure transducers deployed in 13 m depth is used to investigate the local forcing of infragravity motions by nonlinear difference-frequency interactions of surface gravity waves. Second-order nonlinear theory (Hasselmann) and observed swell-sea frequency-directional spectra are used to predict the energy levels of forced infragravity waves. For a wide range of wave conditions, the predicted forced wave levels are lower than the observed energy levels, suggesting that the infragravity band contains a mix of free and forced waves. Bispectral analysis is used to estimate the relative amounts of free and forced infragravity energy. Good agreement between bispectrum-based estimates and theoretical predictions of forced wave energy confirms that second-order nonlinear theory accurately predicts locally forced infragravity motions. The contribution of forced waves to the total infragravity energy, ranging from less than 0.1% to about 30%, is largest when the infragravity energy is maximum, consistent with previously noted trends in similar water depths. The bispectral technique developed here to estimate the energy of forced and free infragravity waves is used in Part 2 to investigate, with data from single-point pressure gauges, the shelfwide variability of free infragravity energy.

## Abstract

The energy of seaward and shoreward propagating ocean surface gravity waves on a natural beach was estimated with data from an army of 24 bottom-mounted pressure sensors in 13-m water depth, 2 km from the North Carolina coast. Consistent with a parameterization of surface wave reflection from a plane sloping beach by Miche, the ratio of seaward to shoreward propagating energy in the swell-sea frequency band (0.044–0.20 Hz) decreased with increasing wave frequency and increasing wave height, and increased with increasing beach-face slope. Although most incident swell-sea energy dissipated in the surf zone, reflection was sometimes significant (up to 18% of the incident swell-sea energy) when the beach face was steep (at high tide) and the wave field was dominated by low-energy, low-frequency swell. Frequency-directional spectra show that reflection of swell and sea was approximately specular. The ratio of seaward to shoreward propagating energy in the infragravity frequency band (0.010–0.044 Hz) varied between about 0.5 and 3 and increased with increasing swell energy. This trend suggests that infragravity waves generated in very shallow water, and refractively trapped on the sloping seabed, are significantly dissipated over a 50-km wide shelf during storms.

## Abstract

The energy of seaward and shoreward propagating ocean surface gravity waves on a natural beach was estimated with data from an army of 24 bottom-mounted pressure sensors in 13-m water depth, 2 km from the North Carolina coast. Consistent with a parameterization of surface wave reflection from a plane sloping beach by Miche, the ratio of seaward to shoreward propagating energy in the swell-sea frequency band (0.044–0.20 Hz) decreased with increasing wave frequency and increasing wave height, and increased with increasing beach-face slope. Although most incident swell-sea energy dissipated in the surf zone, reflection was sometimes significant (up to 18% of the incident swell-sea energy) when the beach face was steep (at high tide) and the wave field was dominated by low-energy, low-frequency swell. Frequency-directional spectra show that reflection of swell and sea was approximately specular. The ratio of seaward to shoreward propagating energy in the infragravity frequency band (0.010–0.044 Hz) varied between about 0.5 and 3 and increased with increasing swell energy. This trend suggests that infragravity waves generated in very shallow water, and refractively trapped on the sloping seabed, are significantly dissipated over a 50-km wide shelf during storms.

## Abstract

Airborne light detecting and ranging (lidar) systems can survey hundreds of kilometers of shoreline with high spatial resolution (several elevation estimates per square meter). Sequential surveys yield spatial change maps of beach and dune sand levels. However, lidar data include elevations of the exposed, subaerial beach and, seaward of the waterline, the ocean surface. Here, a simple method is developed to find the waterline and eliminate returns from the ocean surface. A vertical elevation cutoff is used, with the waterline elevation (*W*) above the known tide level because of the superelevation from wave setup and runup. During each lidar pass, the elevation cutoff (*W*) is assumed proportional (*C*) to the offshore significant wave height *H _{s}.* Comparison of in situ and lidar surveys on a moderately sloped, dissipative California beach yields

*C*≈ 0.4, which is qualitatively consistent with existing observations of runup and setup. The calibrated method rejects ocean surface data, while retaining subaerial beach points more than 70 m seaward of the mean high waterline, which is often used as a conservative default waterline.

## Abstract

Airborne light detecting and ranging (lidar) systems can survey hundreds of kilometers of shoreline with high spatial resolution (several elevation estimates per square meter). Sequential surveys yield spatial change maps of beach and dune sand levels. However, lidar data include elevations of the exposed, subaerial beach and, seaward of the waterline, the ocean surface. Here, a simple method is developed to find the waterline and eliminate returns from the ocean surface. A vertical elevation cutoff is used, with the waterline elevation (*W*) above the known tide level because of the superelevation from wave setup and runup. During each lidar pass, the elevation cutoff (*W*) is assumed proportional (*C*) to the offshore significant wave height *H _{s}.* Comparison of in situ and lidar surveys on a moderately sloped, dissipative California beach yields

*C*≈ 0.4, which is qualitatively consistent with existing observations of runup and setup. The calibrated method rejects ocean surface data, while retaining subaerial beach points more than 70 m seaward of the mean high waterline, which is often used as a conservative default waterline.

## Abstract

Steady surf-zone longshore currents and directional properties of the incident wave field were measured on a beach with nearly straight and parallel depth contours. Selected data were processed into 64 segments, each of 68.2 min. duration, irregularly spaced throughout an 18-day period. A wide variety of incident wave and longshore current conditions were observed. The radiation stress spectrum [*S _{xy}
*(

*f*)] was estimated from a slope array and two current meters located seaward of the surf zone. In many cases the total radiation stress [

*S*= Σ

_{xy}^{T}*S*(

_{xy}*f*)Δ

*f*] contains important contributions from a wide range of frequencies. In a few instances, sea and swell approach the beach from different directions quadrants resulting in a new zero

*S*. The strong shears and direction reversals of the longshore current that could conceivably occur in this circumstance were not observed. An EOF decomposition of the mean longshore current pattern shows that most (<90%) of the current spatial variation in the 64 runs is contained in a classical parabolic shape. The temporal expansion coefficients of the first EOF are equally highly correlated with both

_{xy}^{T}*S*, and a scale velocity suggested by radiation stress-based longshore current theories.

_{xy}^{T}## Abstract

Steady surf-zone longshore currents and directional properties of the incident wave field were measured on a beach with nearly straight and parallel depth contours. Selected data were processed into 64 segments, each of 68.2 min. duration, irregularly spaced throughout an 18-day period. A wide variety of incident wave and longshore current conditions were observed. The radiation stress spectrum [*S _{xy}
*(

*f*)] was estimated from a slope array and two current meters located seaward of the surf zone. In many cases the total radiation stress [

*S*= Σ

_{xy}^{T}*S*(

_{xy}*f*)Δ

*f*] contains important contributions from a wide range of frequencies. In a few instances, sea and swell approach the beach from different directions quadrants resulting in a new zero

*S*. The strong shears and direction reversals of the longshore current that could conceivably occur in this circumstance were not observed. An EOF decomposition of the mean longshore current pattern shows that most (<90%) of the current spatial variation in the 64 runs is contained in a classical parabolic shape. The temporal expansion coefficients of the first EOF are equally highly correlated with both

_{xy}^{T}*S*, and a scale velocity suggested by radiation stress-based longshore current theories.

_{xy}^{T}## Abstract

The radiation stresses *S _{ij}
* associated with the propagation of wind-generated waves are principal driving forces for several important surf-zone processes. The accurate estimation of the onshore flux of longshore-directed mean momentum

*S*, using a linear array of pressure sensors, is considered here. Three analysis methods are examined: integration of two high-resolution directional-spectrum estimators [maximum likelihood (MLM) and a modified version (IMLM)], and a direct estimator of the

_{yx}*S*directional moment (DMM

_{yx}_{v}) which is developed here.

The *S _{yx}
* estimation methods are compared using numerical simulations and field data from two experiments at Torrey Pines Beach, California. In the first field experiment, IMLM and DMM, estimates of

*S*(from a 3-element, 99 m long linear array) showed excellent agreement with a slope array (Higgins

_{yx}*et al*., 1981) in the frequency range 0.05–0.15 Hz. In the second experiment, IMLM and DMM, estimates of

*S*(from a 5-element, 360 m long array) agreed with values of

_{yx}*S*obtained from a nearby orthogonal-axis current meter for the frequency range 0.06–0.11 Hz. The integration of the MLM directional spectrum estimates yields biased (low) values of

_{yx}*S*. Although the DMM method is used here for the estimation of

_{yx}*S*, it can easily be adapted for the calculation of any arbitrary directional moment. While conventional methods are shown to be deficient in

_{yx}*S*estimation, they provide accurate estimates of

_{yx}*S*, the onshore flux of onshore-directed momentum.

_{xx}## Abstract

The radiation stresses *S _{ij}
* associated with the propagation of wind-generated waves are principal driving forces for several important surf-zone processes. The accurate estimation of the onshore flux of longshore-directed mean momentum

*S*, using a linear array of pressure sensors, is considered here. Three analysis methods are examined: integration of two high-resolution directional-spectrum estimators [maximum likelihood (MLM) and a modified version (IMLM)], and a direct estimator of the

_{yx}*S*directional moment (DMM

_{yx}_{v}) which is developed here.

The *S _{yx}
* estimation methods are compared using numerical simulations and field data from two experiments at Torrey Pines Beach, California. In the first field experiment, IMLM and DMM, estimates of

*S*(from a 3-element, 99 m long linear array) showed excellent agreement with a slope array (Higgins

_{yx}*et al*., 1981) in the frequency range 0.05–0.15 Hz. In the second experiment, IMLM and DMM, estimates of

*S*(from a 5-element, 360 m long array) agreed with values of

_{yx}*S*obtained from a nearby orthogonal-axis current meter for the frequency range 0.06–0.11 Hz. The integration of the MLM directional spectrum estimates yields biased (low) values of

_{yx}*S*. Although the DMM method is used here for the estimation of

_{yx}*S*, it can easily be adapted for the calculation of any arbitrary directional moment. While conventional methods are shown to be deficient in

_{yx}*S*estimation, they provide accurate estimates of

_{yx}*S*, the onshore flux of onshore-directed momentum.

_{xx}## Abstract

Low-salinity water from Chesapeake Bay forms an intermittent buoyant gravity current that propagates more than 100 km southward along the coast. During five events when wind and surface gravity-wave forcing were weak, the buoyant coastal current 90 km south of Chesapeake Bay was less than 5 km wide, was 5–10 m thick, and propagated alongshore at ∼50 cm s^{−1}. The density decreased 2–3 kg m^{−3} over a few hundred meters at the nose of the buoyant coastal current, which was located about 1 km offshore in ∼8 m of water. Water up to 4 km ahead of the advancing nose was displaced southward and offshore (maximum velocities near the nose of 20 and 10 cm s^{−1}, respectively). The southward alongshore current increased abruptly to ∼50 cm s^{−1} at the nose and continued to increase to a supercritical maximum of ∼70 cm s^{−1} about 1 km behind the nose. An onshore flow of between 5 and 15 cm s^{−1}, which extended at least 5 km behind the nose, supplied buoyant water to the onshore region of weak, subcritical alongshore flow. The observed flow structure is qualitatively similar to theoretical predictions and laboratory measurements of buoyant gravity currents propagating along a sloping bottom.

## Abstract

Low-salinity water from Chesapeake Bay forms an intermittent buoyant gravity current that propagates more than 100 km southward along the coast. During five events when wind and surface gravity-wave forcing were weak, the buoyant coastal current 90 km south of Chesapeake Bay was less than 5 km wide, was 5–10 m thick, and propagated alongshore at ∼50 cm s^{−1}. The density decreased 2–3 kg m^{−3} over a few hundred meters at the nose of the buoyant coastal current, which was located about 1 km offshore in ∼8 m of water. Water up to 4 km ahead of the advancing nose was displaced southward and offshore (maximum velocities near the nose of 20 and 10 cm s^{−1}, respectively). The southward alongshore current increased abruptly to ∼50 cm s^{−1} at the nose and continued to increase to a supercritical maximum of ∼70 cm s^{−1} about 1 km behind the nose. An onshore flow of between 5 and 15 cm s^{−1}, which extended at least 5 km behind the nose, supplied buoyant water to the onshore region of weak, subcritical alongshore flow. The observed flow structure is qualitatively similar to theoretical predictions and laboratory measurements of buoyant gravity currents propagating along a sloping bottom.

## Abstract

Surf-zone dispersion is studied using drifter observations collected within about 200 m of the shoreline (at depths of less than about 5 m) on a beach with approximately alongshore uniform bathymetry and waves. There were about 70 individual drifter releases, each 10–20 min in duration, on two consecutive days. On the first day, the sea-swell significant wave height *H _{s}
* was equal to 0.5 m and mean alongshore currents |

*υ*

^{−1}). On the second day, the obliquely incident waves were larger, with

*H*equal to 1.4 m, and at some surf-zone locations |

_{s}*υ*

^{−1}. The one-particle diffusivity was larger, with larger waves and stronger currents. On both days, the one-particle diffusivity tensor is nonisotropic and time-dependent. The major axis is initially parallel to the cross-shore direction, but after a few wave periods it is aligned with the alongshore direction. In both the along- and cross-shore directions, the asymptotic diffusivity is reached sooner within, rather than seaward of, the surf zone. Two-particle statistics indicate that relative dispersion grows like

*D*

^{2}(

*t*) ∼

*t*

^{3/2}and that the relative diffusivity is scale-dependent as

*μ*∼

*l*

^{2/3}, with

*l*being the particle separation. The observed scalings differ from 2D inertial-subrange scalings [

*D*

^{2}(

*t*) ∼

*t*

^{3}and

*μ*∼

*l*

^{4/3}]. Separations have a non-Gaussian self-similar distribution that is independent of time. The two-particle statistics are consistent with a nonconstant-coefficient diffusion equation for the separation probability density functions. The dispersion is explained by neither irrotational surface gravity waves nor shear dispersion. The observations imply the existence of a 2D eddy field with 5–50-m length scales, the source of which is speculated to be alongshore gradients in breaking-wave height associated with finite crest lengths.

## Abstract

Surf-zone dispersion is studied using drifter observations collected within about 200 m of the shoreline (at depths of less than about 5 m) on a beach with approximately alongshore uniform bathymetry and waves. There were about 70 individual drifter releases, each 10–20 min in duration, on two consecutive days. On the first day, the sea-swell significant wave height *H _{s}
* was equal to 0.5 m and mean alongshore currents |

*υ*

^{−1}). On the second day, the obliquely incident waves were larger, with

*H*equal to 1.4 m, and at some surf-zone locations |

_{s}*υ*

^{−1}. The one-particle diffusivity was larger, with larger waves and stronger currents. On both days, the one-particle diffusivity tensor is nonisotropic and time-dependent. The major axis is initially parallel to the cross-shore direction, but after a few wave periods it is aligned with the alongshore direction. In both the along- and cross-shore directions, the asymptotic diffusivity is reached sooner within, rather than seaward of, the surf zone. Two-particle statistics indicate that relative dispersion grows like

*D*

^{2}(

*t*) ∼

*t*

^{3/2}and that the relative diffusivity is scale-dependent as

*μ*∼

*l*

^{2/3}, with

*l*being the particle separation. The observed scalings differ from 2D inertial-subrange scalings [

*D*

^{2}(

*t*) ∼

*t*

^{3}and

*μ*∼

*l*

^{4/3}]. Separations have a non-Gaussian self-similar distribution that is independent of time. The two-particle statistics are consistent with a nonconstant-coefficient diffusion equation for the separation probability density functions. The dispersion is explained by neither irrotational surface gravity waves nor shear dispersion. The observations imply the existence of a 2D eddy field with 5–50-m length scales, the source of which is speculated to be alongshore gradients in breaking-wave height associated with finite crest lengths.

## Abstract

The performance of the Datawell Directional Waverider and the National Data Buoy Center (NDBC) 3-m discus buoy, widely used to measure the directional properties of surface gravity waves, are evaluated through comparisons to an array of six pressure transducers mounted 14 m below the sea surface on a platform in 200-m depth. Each buoy was deployed for several months within a few kilometers of the platform. The accuracy of the platform ground-truth array was verified by close agreement of wavenumber estimates with the theoretical linear dispersion relation for surface gravity waves. Buoy and array estimates of wave energy and directional parameters, based on integration of the directional moments across the frequency band of energetic swell (0.06–0.14 Hz), are compared for a wide range of wave conditions. Wave energy and mean propagation direction estimates from both buoys agree well with the platform results. However, the Datawell buoy provides significantly better estimates of directional spread and skewness than the NDBC buoy.

## Abstract

The performance of the Datawell Directional Waverider and the National Data Buoy Center (NDBC) 3-m discus buoy, widely used to measure the directional properties of surface gravity waves, are evaluated through comparisons to an array of six pressure transducers mounted 14 m below the sea surface on a platform in 200-m depth. Each buoy was deployed for several months within a few kilometers of the platform. The accuracy of the platform ground-truth array was verified by close agreement of wavenumber estimates with the theoretical linear dispersion relation for surface gravity waves. Buoy and array estimates of wave energy and directional parameters, based on integration of the directional moments across the frequency band of energetic swell (0.06–0.14 Hz), are compared for a wide range of wave conditions. Wave energy and mean propagation direction estimates from both buoys agree well with the platform results. However, the Datawell buoy provides significantly better estimates of directional spread and skewness than the NDBC buoy.

## Abstract

Shear waves (instabilities of the breaking wave–driven mean alongshore current) and gravity waves both contribute substantial velocity fluctuations to nearshore infragravity motions (periods of a few minutes). Three existing methods of estimating the shear wave contribution to the infragravity velocity variance are compared using extensive field observations. The iterative maximum likelihood estimator (IMLE) and the direct estimator (DE) methods use an alongshore array of current meters, and ascribe all the velocity variance at non–gravity wavenumbers to shear waves. The ratio (*R*) method uses a collocated pressure gauge and current meter, and assumes that shear wave pressure fluctuations are small, and that the kinetic and potential energies of gravity waves are equal. The shear wave velocity variance 〈*q*
^{2}
_{sw}
*q*
^{2}
_{sw}〉*q*
^{2}
_{sw}〉*q*
^{2}
_{sw}〉*R* estimates of *q*
^{2}
_{sw}〉*R* method attributes 15% more of the total horizontal velocity variance to shear waves than is attributed by the IMLE method. When mean currents and shear waves are weak, all three estimators are noisy and biased high.

## Abstract

Shear waves (instabilities of the breaking wave–driven mean alongshore current) and gravity waves both contribute substantial velocity fluctuations to nearshore infragravity motions (periods of a few minutes). Three existing methods of estimating the shear wave contribution to the infragravity velocity variance are compared using extensive field observations. The iterative maximum likelihood estimator (IMLE) and the direct estimator (DE) methods use an alongshore array of current meters, and ascribe all the velocity variance at non–gravity wavenumbers to shear waves. The ratio (*R*) method uses a collocated pressure gauge and current meter, and assumes that shear wave pressure fluctuations are small, and that the kinetic and potential energies of gravity waves are equal. The shear wave velocity variance 〈*q*
^{2}
_{sw}
*q*
^{2}
_{sw}〉*q*
^{2}
_{sw}〉*q*
^{2}
_{sw}〉*R* estimates of *q*
^{2}
_{sw}〉*R* method attributes 15% more of the total horizontal velocity variance to shear waves than is attributed by the IMLE method. When mean currents and shear waves are weak, all three estimators are noisy and biased high.