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## Abstract

Nonlinear effects in Lagrangian sea surface motions are important to understanding variability in wave-induced mass transport, wave-driven diffusion processes, and the interpretation of measurements obtained with moored or free-drifting buoys. This study evaluates the Lagrangian vertical and horizontal motions of a particle at the surface in a natural, random sea state using second-order, finite-depth wave theory. In deep water, the predicted low-frequency (infragravity) surface height fluctuations are much larger than Eulerian bound wave motions and of the opposite sign. Comparison to surface elevation bispectra observed with a moored buoy in steady, high-wind conditions shows good agreement and confirms that—in contrast to the Eulerian sea surface motion with predominant phase coupling between the spectral peak and double-frequency harmonic components—nonlinearity in Lagrangian wave observations is dominated by phase-coupled infragravity motions. Sea surface skewness estimates obtained from moored buoys in deep and shallow sites, over a wide range of wind–sea and swell conditions, are in good agreement with second-order theory predictions. Theory and field data analysis of surface drift motions in deep water reveal energetic [*O*(10) cm s^{−1}] infragravity velocity fluctuations that are several orders of magnitude larger and 180° out of phase with Eulerian infragravity motions. These large fluctuations in Stokes drift may be important in upper-ocean diffusion processes.

## Abstract

Nonlinear effects in Lagrangian sea surface motions are important to understanding variability in wave-induced mass transport, wave-driven diffusion processes, and the interpretation of measurements obtained with moored or free-drifting buoys. This study evaluates the Lagrangian vertical and horizontal motions of a particle at the surface in a natural, random sea state using second-order, finite-depth wave theory. In deep water, the predicted low-frequency (infragravity) surface height fluctuations are much larger than Eulerian bound wave motions and of the opposite sign. Comparison to surface elevation bispectra observed with a moored buoy in steady, high-wind conditions shows good agreement and confirms that—in contrast to the Eulerian sea surface motion with predominant phase coupling between the spectral peak and double-frequency harmonic components—nonlinearity in Lagrangian wave observations is dominated by phase-coupled infragravity motions. Sea surface skewness estimates obtained from moored buoys in deep and shallow sites, over a wide range of wind–sea and swell conditions, are in good agreement with second-order theory predictions. Theory and field data analysis of surface drift motions in deep water reveal energetic [*O*(10) cm s^{−1}] infragravity velocity fluctuations that are several orders of magnitude larger and 180° out of phase with Eulerian infragravity motions. These large fluctuations in Stokes drift may be important in upper-ocean diffusion processes.

## Abstract

In this paper, the combined effects of refraction and nonlinearity on the evolution of ocean surface wave statistics are considered and possible implications for the likelihood of extreme waves, also known as freak or rogue waves, are examined. A frequency-angular spectrum model is derived that accounts for cubic nonlinear dynamics and weak lateral homogeneity of the medium. Through Monte Carlo simulations, the evolution of wave statistics in freely developing waves, waves over an opposing shearing current, and waves refracted over an isolated topographical feature is modeled. The simulations show that freely developing, directionally spread wave fields generally maintain near-Gaussian statistics, which was also found in earlier model studies. However, the enhanced nonlinearity caused by the refractive focusing of narrowband wave fields can result locally in strongly non-Gaussian statistics and an associated increased likelihood of extreme wave events.

## Abstract

In this paper, the combined effects of refraction and nonlinearity on the evolution of ocean surface wave statistics are considered and possible implications for the likelihood of extreme waves, also known as freak or rogue waves, are examined. A frequency-angular spectrum model is derived that accounts for cubic nonlinear dynamics and weak lateral homogeneity of the medium. Through Monte Carlo simulations, the evolution of wave statistics in freely developing waves, waves over an opposing shearing current, and waves refracted over an isolated topographical feature is modeled. The simulations show that freely developing, directionally spread wave fields generally maintain near-Gaussian statistics, which was also found in earlier model studies. However, the enhanced nonlinearity caused by the refractive focusing of narrowband wave fields can result locally in strongly non-Gaussian statistics and an associated increased likelihood of extreme wave events.

## Abstract

A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number *N* of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large *N*, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for *N* = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δ*t* = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δ*t*.

## Abstract

A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number *N* of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large *N*, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for *N* = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δ*t* = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δ*t*.

## Abstract

This is Part 1 of a study of nonlinear effects on natural wind waves. Array measurements of pressure at the sea floor and middepth, collected 30 km offshore in 13-m depth, are compared to an existing theory for weakly nonlinear surface gravity waves. In this depth, free surface waves (obeying the linear dispersion relation) an weakly attenuated at the sea bed at sea and swell frequencies (0.05–0.3 Hz) but very strongly attenuated at frequencies higher than about 0.35 Hz. Only nonlinearly driven motions can reach the sea floor at these high frequencies. Nonlinear interactions between free (primary) waves of about the same frequency, travelling in nearly opposing directions, theoretically excite long-wavelength, double-frequency forced (secondary) waves that are only weakly attenuated at the sea door and form a mechanism for the generation of microseisms at great depth. In 13-m depth, wind-generated free waves and corresponding long-wavelength, high-frequency forced waves can be simultaneously observed on the sea floor, and the coupling, between the two examined in some detail.

Bottom-pressure spectra observed over a 4-day period show large [O(10^{2})] fluctuations in high-frequency (0.35–0.6 Hz) forced-wave energy levels at the sea floor occurring in only a few hours. Correspondingly rapid changes in estimates of the free-wave frequency-directional spectrum show that forced-wave energy levels are weak in unidirectional seas and increase dramatically in response to nearly opposing seas, consistent with the theoretical generation mechanism. On one occasion, directionally opposing seas, and a corresponding double-frequency forced-wave peak, followed a rapidly veering wind. However, comparable increases in forced-wave energy levels were observed in response to the arrival of nonlocally generated seas with directions much different than local winds and seas.

Although the accuracy of theoretical forced-wave predictions is limited by the directional resolution of the small aperture (20 m × 20 m) middepth array, predicted and observed forced-wave energy levels agree within about a factor of 2. The observed weak decay between middepth and sea-floor wave pressure at double sea frequencies is also consistent with theoretically expected long wavelengths. Wavelengths, propagation directions, and phase coupling between free and forced waves are examined using the bottom-pressure array data in Part 2.

## Abstract

This is Part 1 of a study of nonlinear effects on natural wind waves. Array measurements of pressure at the sea floor and middepth, collected 30 km offshore in 13-m depth, are compared to an existing theory for weakly nonlinear surface gravity waves. In this depth, free surface waves (obeying the linear dispersion relation) an weakly attenuated at the sea bed at sea and swell frequencies (0.05–0.3 Hz) but very strongly attenuated at frequencies higher than about 0.35 Hz. Only nonlinearly driven motions can reach the sea floor at these high frequencies. Nonlinear interactions between free (primary) waves of about the same frequency, travelling in nearly opposing directions, theoretically excite long-wavelength, double-frequency forced (secondary) waves that are only weakly attenuated at the sea door and form a mechanism for the generation of microseisms at great depth. In 13-m depth, wind-generated free waves and corresponding long-wavelength, high-frequency forced waves can be simultaneously observed on the sea floor, and the coupling, between the two examined in some detail.

Bottom-pressure spectra observed over a 4-day period show large [O(10^{2})] fluctuations in high-frequency (0.35–0.6 Hz) forced-wave energy levels at the sea floor occurring in only a few hours. Correspondingly rapid changes in estimates of the free-wave frequency-directional spectrum show that forced-wave energy levels are weak in unidirectional seas and increase dramatically in response to nearly opposing seas, consistent with the theoretical generation mechanism. On one occasion, directionally opposing seas, and a corresponding double-frequency forced-wave peak, followed a rapidly veering wind. However, comparable increases in forced-wave energy levels were observed in response to the arrival of nonlocally generated seas with directions much different than local winds and seas.

Although the accuracy of theoretical forced-wave predictions is limited by the directional resolution of the small aperture (20 m × 20 m) middepth array, predicted and observed forced-wave energy levels agree within about a factor of 2. The observed weak decay between middepth and sea-floor wave pressure at double sea frequencies is also consistent with theoretically expected long wavelengths. Wavelengths, propagation directions, and phase coupling between free and forced waves are examined using the bottom-pressure array data in Part 2.

## Abstract

An improved method for estimating the directional spectrum of linear surface gravity waves from in Situ observations is presented. The technique, a refinement and extension of the inverse method of Long and Hasselmann, is applicable to multicomponent wave measurements at fixed locations in constant or slowly varying depth water. On a frequency band by frequency band basis, an estimate of the directional distribution of wave energy *S*(θ) is obtained by minimizing a roughness measure of the form ∫*d*θ[*d*
^{2}
*S*(θ)/*d*θ^{2}]^{2} subject to the constraints: (i) *S*(θ) is nonnegative with unit integral, (ii) *S*(θ) fits the data within a chosen statistical confidence level, and (iii) *S*(θ) is zero on any directional sectors where energy levels are always relatively low because of the influence of geographic surroundings. The solution to this inverse problem is derived through a variational formulation with Lagrange multipliers.

A series of simulations using the new estimator show the fundamental limitations of sparse array data and the importance of using all available data-independent information [i.e., constraints (i) and (iii)] for achieving optimal estimates. The advantages of smoothness optimization are illustrated in a comparison of the present and Long and Hasselmann methods. The present method yields smooth estimates where Long and Hasselmann obtained rough estimates with multiple spurious peaks. A smooth solution to the inverse problem that has only truly resolved features is both easier to interpret and more readily evaluated numerically than wildly spurious solutions. The examples also demonstrate the subjectivity of intercomparing estimation techniques.

A few illustrative examples are presented of *S*(θ) estimates obtained from a two-dimensional array (aperture 120 m × 96 m) of 14 pressure transducer in 6 m water depth. Estimates using the full array and no geographic constraints are smooth and exhibit the expected refractive columnation of shoreward propagating energy towards normal incidence. Additionally, reflection from the mildly sloping beach 310 m shoreward of the center of this array is very weak at wind wave and swell frequencies. Estimates of *S*(θ) made using only the sensors on a longshore line, and a constraint of no reflected energy, are very similar to *S*(θ) obtained with the full array and no constraint.

## Abstract

An improved method for estimating the directional spectrum of linear surface gravity waves from in Situ observations is presented. The technique, a refinement and extension of the inverse method of Long and Hasselmann, is applicable to multicomponent wave measurements at fixed locations in constant or slowly varying depth water. On a frequency band by frequency band basis, an estimate of the directional distribution of wave energy *S*(θ) is obtained by minimizing a roughness measure of the form ∫*d*θ[*d*
^{2}
*S*(θ)/*d*θ^{2}]^{2} subject to the constraints: (i) *S*(θ) is nonnegative with unit integral, (ii) *S*(θ) fits the data within a chosen statistical confidence level, and (iii) *S*(θ) is zero on any directional sectors where energy levels are always relatively low because of the influence of geographic surroundings. The solution to this inverse problem is derived through a variational formulation with Lagrange multipliers.

A series of simulations using the new estimator show the fundamental limitations of sparse array data and the importance of using all available data-independent information [i.e., constraints (i) and (iii)] for achieving optimal estimates. The advantages of smoothness optimization are illustrated in a comparison of the present and Long and Hasselmann methods. The present method yields smooth estimates where Long and Hasselmann obtained rough estimates with multiple spurious peaks. A smooth solution to the inverse problem that has only truly resolved features is both easier to interpret and more readily evaluated numerically than wildly spurious solutions. The examples also demonstrate the subjectivity of intercomparing estimation techniques.

A few illustrative examples are presented of *S*(θ) estimates obtained from a two-dimensional array (aperture 120 m × 96 m) of 14 pressure transducer in 6 m water depth. Estimates using the full array and no geographic constraints are smooth and exhibit the expected refractive columnation of shoreward propagating energy towards normal incidence. Additionally, reflection from the mildly sloping beach 310 m shoreward of the center of this array is very weak at wind wave and swell frequencies. Estimates of *S*(θ) made using only the sensors on a longshore line, and a constraint of no reflected energy, are very similar to *S*(θ) obtained with the full array and no constraint.

## Abstract

This is Part 2 of a study of nonlinear effects on natural wind-generated surface gravity waves in 13-m depth, 30 km offshore of Virginia. At the sea floor in this depth, free surface gravity waves are only weakly attenuated at sea and swell frequencies (0.05–0.30 Hz) but are very strongly attenuated at frequencies higher than about 0.35 Hz. Hence, above 0.35 Hz, relatively long wavelength forced waves, excited by nonlinear interactions between directionally opposing free wind waves, are exposed at the sea floor. An array of pressure transducers at middepth was used to estimate the frequency-directional spectrum of (free) primary sea and swell waves, and the associated (forced) secondary pressure fluctuations were measured with an array on the sea floor. In Part 1, it was shown that forced-wave energy levels at the sea floor increase sharply in response to directionally opposing wind waves, in agreement with weakly nonlinear theory. In Part 2, wavelengths, propagation directions, and non-Gaussian phase coupling between free and forced waves are examined on three occasions with relatively high forced-wave energy levels.

A root-mean-square wavenumber magnitude and a vector-averaged mean wave propagation direction (both functions of frequency) can be expressed accurately in terms of the pressure array cross-spectra. The wavenumber estimates at the sea floor show the theoretically expected sharp transition between a 0.05–0.30 HZ frequency range dominated by free sea and swell waves and a 0.35–0.60 Hz range dominated by forced waves with wavelengths that are long relative to free waves of the same frequency. In the “free-wave frequency range,” wavenumber estimates are usually well within 10% of the linear dispersion relation and wave direction estimates are in excellent agreement with the directional spectra extracted from the middepth array. In the “forced-wave frequency range,” wavenumber and direction estimates agree with nonlinear theory predictions, confirming that the observed forced waves have the sum vector wavenumber of the interacting directionally opposing wind waves.

Phase coupling between free and forced waves is examined through third-order statistics of the sea floor pressure data. Consistent with theory, the normalized bispectrum has small imaginary parts scattered approximately randomly about zero and relatively large negative real parts at frequencies that correspond to directionally opposing seas and swell. Estimates of the bispectrum integrated for constant sum frequency confirm that nearly all the energy at double sea frequencies is nonlinearly coupled to directionally opposing wind waves. In qualitative agreement with nonlinear theory predictions, bispectral levels are sometimes significantly reduced by directional spreading of the interacting free waves.

## Abstract

This is Part 2 of a study of nonlinear effects on natural wind-generated surface gravity waves in 13-m depth, 30 km offshore of Virginia. At the sea floor in this depth, free surface gravity waves are only weakly attenuated at sea and swell frequencies (0.05–0.30 Hz) but are very strongly attenuated at frequencies higher than about 0.35 Hz. Hence, above 0.35 Hz, relatively long wavelength forced waves, excited by nonlinear interactions between directionally opposing free wind waves, are exposed at the sea floor. An array of pressure transducers at middepth was used to estimate the frequency-directional spectrum of (free) primary sea and swell waves, and the associated (forced) secondary pressure fluctuations were measured with an array on the sea floor. In Part 1, it was shown that forced-wave energy levels at the sea floor increase sharply in response to directionally opposing wind waves, in agreement with weakly nonlinear theory. In Part 2, wavelengths, propagation directions, and non-Gaussian phase coupling between free and forced waves are examined on three occasions with relatively high forced-wave energy levels.

A root-mean-square wavenumber magnitude and a vector-averaged mean wave propagation direction (both functions of frequency) can be expressed accurately in terms of the pressure array cross-spectra. The wavenumber estimates at the sea floor show the theoretically expected sharp transition between a 0.05–0.30 HZ frequency range dominated by free sea and swell waves and a 0.35–0.60 Hz range dominated by forced waves with wavelengths that are long relative to free waves of the same frequency. In the “free-wave frequency range,” wavenumber estimates are usually well within 10% of the linear dispersion relation and wave direction estimates are in excellent agreement with the directional spectra extracted from the middepth array. In the “forced-wave frequency range,” wavenumber and direction estimates agree with nonlinear theory predictions, confirming that the observed forced waves have the sum vector wavenumber of the interacting directionally opposing wind waves.

Phase coupling between free and forced waves is examined through third-order statistics of the sea floor pressure data. Consistent with theory, the normalized bispectrum has small imaginary parts scattered approximately randomly about zero and relatively large negative real parts at frequencies that correspond to directionally opposing seas and swell. Estimates of the bispectrum integrated for constant sum frequency confirm that nearly all the energy at double sea frequencies is nonlinearly coupled to directionally opposing wind waves. In qualitative agreement with nonlinear theory predictions, bispectral levels are sometimes significantly reduced by directional spreading of the interacting free waves.

## Abstract

Conventional observations of waves carried out with a buoy in open sea conditions were supplemented with simultaneous visual observations of whitecaps to identify breaking events in the buoy records. A statistical wave-by-wave analysis of these records indicates that such seemingly obvious parameters as wave steepness or wave asymmetry cannot be used to separate breakers from nonbreakers and the breaking occurs at wave steepness values much less than the theoretically expected steepness of a limiting wave. The observed fraction of breaking waves varied from about 0.10 to about 0.16, depending on wind speed. Two-thirds of the breaking waves were breaking in one-third of the wave groups for which a *H*
_{rms}-threshold definition was used.

## Abstract

Conventional observations of waves carried out with a buoy in open sea conditions were supplemented with simultaneous visual observations of whitecaps to identify breaking events in the buoy records. A statistical wave-by-wave analysis of these records indicates that such seemingly obvious parameters as wave steepness or wave asymmetry cannot be used to separate breakers from nonbreakers and the breaking occurs at wave steepness values much less than the theoretically expected steepness of a limiting wave. The observed fraction of breaking waves varied from about 0.10 to about 0.16, depending on wind speed. Two-thirds of the breaking waves were breaking in one-third of the wave groups for which a *H*
_{rms}-threshold definition was used.

## Abstract

Acoustic Doppler current profilers (ADCPs) are widely used for routine measurements of ocean currents and waves in coastal environments. These instruments have the basic capability to measure surface wave frequency–directional spectra, but the quality of the estimates is not well understood because of the relatively high noise levels in the velocity measurements. In this study, wave data are evaluated from two 600-kHz ADCP instruments deployed at 20- and 45-m depths on the Southern California continental shelf. A simple parametric estimation technique is presented that provides robust estimates of the gross directional wave properties, even when the data quality is marginal, as was often the case in this benign wave environment. Good agreement of mean direction and (to a lesser degree) directional spreading estimates with measurements from a nearby surface-following buoy confirms that reliable wave information can generally be extracted from ADCP measurements on the continental shelf, supporting the instrument’s suitability for routine wave-monitoring applications.

## Abstract

Acoustic Doppler current profilers (ADCPs) are widely used for routine measurements of ocean currents and waves in coastal environments. These instruments have the basic capability to measure surface wave frequency–directional spectra, but the quality of the estimates is not well understood because of the relatively high noise levels in the velocity measurements. In this study, wave data are evaluated from two 600-kHz ADCP instruments deployed at 20- and 45-m depths on the Southern California continental shelf. A simple parametric estimation technique is presented that provides robust estimates of the gross directional wave properties, even when the data quality is marginal, as was often the case in this benign wave environment. Good agreement of mean direction and (to a lesser degree) directional spreading estimates with measurements from a nearby surface-following buoy confirms that reliable wave information can generally be extracted from ADCP measurements on the continental shelf, supporting the instrument’s suitability for routine wave-monitoring applications.

## Abstract

Data from a cross-shore array of nine collocated pressure sensors and bidirectional current meters, extending from the shoreline to approximately 4.5-m depth, are used to estimate the relative contributions of gravity waves (e.g., edge and leaky waves) and instabilities of the alongshore current (shear waves) to motions in the infragravity (frequencies nominally 0.004–0.05 Hz) band. The ratio between frequency-integrated velocity and pressure variances is shown to be approximately equal to *g*/*h* for a broad spectrum of gravity waves independent of the mode mix of edge and leaky waves. Since shear waves have velocity to pressure variance ratios ≫ *g*/*h,* this ratio can be used to estimate the relative contributions of gravity and shear waves to the infragravity band. Outside the surf zone where the shear in the alongshore current is relatively weak, the observed velocity to pressure variance ratios are approximately equal to *g*/*h,* consistent with a gravity-dominated wave field. Inside the surf zone where alongshore currents are strongly sheared, these ratios are up to a factor of 4 larger, indicating that shear waves contribute as much as 75% of the velocity variance in the infragravity band. Observed shear-wave-dominated infragravity band motions are confined to a narrow region of strong shear on the seaward side of the alongshore current maximum, and their cross-shore structure appears to be insensitive to changes in the beach profile, qualitatively consistent with theoretical predictions by linear stability analysis.

## Abstract

Data from a cross-shore array of nine collocated pressure sensors and bidirectional current meters, extending from the shoreline to approximately 4.5-m depth, are used to estimate the relative contributions of gravity waves (e.g., edge and leaky waves) and instabilities of the alongshore current (shear waves) to motions in the infragravity (frequencies nominally 0.004–0.05 Hz) band. The ratio between frequency-integrated velocity and pressure variances is shown to be approximately equal to *g*/*h* for a broad spectrum of gravity waves independent of the mode mix of edge and leaky waves. Since shear waves have velocity to pressure variance ratios ≫ *g*/*h,* this ratio can be used to estimate the relative contributions of gravity and shear waves to the infragravity band. Outside the surf zone where the shear in the alongshore current is relatively weak, the observed velocity to pressure variance ratios are approximately equal to *g*/*h,* consistent with a gravity-dominated wave field. Inside the surf zone where alongshore currents are strongly sheared, these ratios are up to a factor of 4 larger, indicating that shear waves contribute as much as 75% of the velocity variance in the infragravity band. Observed shear-wave-dominated infragravity band motions are confined to a narrow region of strong shear on the seaward side of the alongshore current maximum, and their cross-shore structure appears to be insensitive to changes in the beach profile, qualitatively consistent with theoretical predictions by linear stability analysis.

## Abstract

Refractive focusing of swell waves can result in fast-scale variations in the wave statistics because of wave interference, which cannot be resolved by stochastic wave models based on the radiative transport equation. Quasi-coherent statistical theory does account for such statistical interferences and the associated wave inhomogeneities, but the theory has thus far been presented in a form that appears incompatible with models based on the radiative transfer equation (RTE). Moreover, the quasi-coherent theory has never been tested against field data, and it is not clear how the coherent information inherent to such models can be used for better understanding coastal wave and circulation dynamics. This study therefore revisits the derivation of quasi-coherent theory to formulate it into a radiative transport equation with a forcing term that accounts for the inhomogeneous part of the wave field. This paper shows how the model can be nested within (or otherwise used in conjunction with) quasi-homogeneous wave models based on the RTE. Through comparison to laboratory data, numerical simulations of a deterministic model, and field observations of waves propagating over a nearshore canyon head, the predictive capability of the model is validated. The authors discuss the interference patterns predicted by the model through evaluation of a complex cross-correlation function and highlight the differences with quasi-homogeneous predictions. These results show that quasi-coherent theory can extend models based on the RTE to resolve coherent interference patterns and standing wave features in coastal areas, which are believed to be important in nearshore circulation and sediment transport.

## Abstract

Refractive focusing of swell waves can result in fast-scale variations in the wave statistics because of wave interference, which cannot be resolved by stochastic wave models based on the radiative transport equation. Quasi-coherent statistical theory does account for such statistical interferences and the associated wave inhomogeneities, but the theory has thus far been presented in a form that appears incompatible with models based on the radiative transfer equation (RTE). Moreover, the quasi-coherent theory has never been tested against field data, and it is not clear how the coherent information inherent to such models can be used for better understanding coastal wave and circulation dynamics. This study therefore revisits the derivation of quasi-coherent theory to formulate it into a radiative transport equation with a forcing term that accounts for the inhomogeneous part of the wave field. This paper shows how the model can be nested within (or otherwise used in conjunction with) quasi-homogeneous wave models based on the RTE. Through comparison to laboratory data, numerical simulations of a deterministic model, and field observations of waves propagating over a nearshore canyon head, the predictive capability of the model is validated. The authors discuss the interference patterns predicted by the model through evaluation of a complex cross-correlation function and highlight the differences with quasi-homogeneous predictions. These results show that quasi-coherent theory can extend models based on the RTE to resolve coherent interference patterns and standing wave features in coastal areas, which are believed to be important in nearshore circulation and sediment transport.