1. Introduction
Apart from the usual WKB assumptions for a slowly varying medium, the use of the RTE requires that wave components are statistically independent (Komen et al. 1994), which is valid for quasi-homogeneous (and Gaussian) wave fields. In deep water, where the evolution is mostly dominated by the source term balance, this is generally true. However, in shallow water, where the waves interact with slowly varying topography (or currents), the coherent scattering of narrowband wave fields can result in inhomogeneities and fast-scale variations in wave energy (Janssen et al. 2008), which can have an O(1) effect on the mean statistics (Smit and Janssen 2013, hereinafter SJ13) and—as a consequence—affect wave-driven circulation and transport processes. These effects are not accounted for in models based on the RTE.
The quasi-coherent (QC) theory presented in SJ13 is a fundamental generalization of the RTE to account for coherence (which for constant depth was also considered by Pedersen and Lokberg 1992) and incorporates inhomogeneities and scattering effects by resolving (and transporting) cross-phase information in the wave field statistics. However, the form of the model equations as derived in SJ13 [see Eqs. (15) and (16) in SJ13] is quite different from the conventional RTE-type transport models, which hamper the physical interpretation of the inhomogeneity contributions and make it more difficult to combine the two different modeling approaches in a single numerical model. As part of this work we revisit and expand on the theoretical results from SJ13 to show that the QC approximation can be written in a similar form as Eq. (1), but with an additional scattering term SQC on the RHS, that describes spatial energy variations resulting from wave interference and depends on local cross correlations between spectral wave components.
We further develop a consistent numerical implementation for this model, validate the model against laboratory and field observations, and explore the interpretation and use of the cross correlators implicit in the coupled-mode (CM) spectrum (as opposed to the variance density spectrum). In that regard, we briefly summarize the principal results from SJ13 (section 2) and derive and discuss a consistent approximation for medium variations based on the wave field decorrelation length scale (section 2). We present simulations with the new model of laboratory flume experiments (section 3) and field observations of ocean waves interacting with a submarine canyon (section 4), discuss the effects of wave inhomogeneity in the observational data and model results, and sum up our principal findings in section 5.
2. Evolution of inhomogeneous wave fields
Medium variations in a coherent wave field
Combined, Eqs. (4) and (6) summarize the principal theoretical result from SJ13 and represent the starting point of this work. In the following, we will derive a consistent form of this model using the decorrelation length scale of the wave field, making the physical interpretation more intuitive, relating it explicitly to the RTE, and making it suitable for numerical evaluation. In particular, the Fourier transform on the right side of Eq. (4) makes the evolution of the coupled-mode spectrum dependent on medium variations throughout the entire spatial domain. Since random ocean waves have a finite decorrelation length scale, this is not only impractical but also unnecessary from a physical point of view.
To make this explicit, we consider that the slow medium variations are characterized by a small parameter 
Here, we replace 
For waves with a finite coherent radius 
In what follows, we will refer to Eq. (15) as the quasi-coherent model (QCM). In the simulations presented in this work, we consider steady-state solutions to Eq. (15) (such that 
3. Wave deformation by an elliptical shoal
A monochromatic, initially unidirectional wave field that interacts with topography can be considered as the archetype of a coherent scattering problem. Moreover, it represents an excellent test on the limits of the stochastic model since the QCM explicitly assumes a finite coherent radius (or finite 
To illustrate the behavior of the QCM and the statistical information that is inherently available in the model, and test its performance under such conditions, we consider the wave basin experiment by Berkhoff et al. (1982), where monochromatic (period 1 s), unidirectional waves (wave height H = 0.0464 m) were generated at the wavemaker (at x = −10, depth 0.45 m) and propagated over a shoal (crest located at x = 0 m, depth of 0.135 m) situated on a 1:50 slope (see Fig. 1). Wave heights were measured along eight transects at regular intervals, of which we consider three (indicated in Fig. 1).
Plan view of the experimental setup by Berkhoff et al. (1982), including ray trajectories for unidirectional monochromatic waves (thin gray lines) that start at x = −10 m and propagate toward the shoal. Depth contours are indicated by the black solid lines; instrumented transects used in this study (identified by boxed numerals) are indicated by dashed lines, whereas the solid dot (identified by the boxed A) indicates the point A discussed in the main text.
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
The RTE and QCM are numerically evaluated on a rectangular spatial (20 × 20 m2) and spectral domain [10 × 10 rad2 m−2, starting at k = (−0.05, −5) rad m−1], uniformly discretized with mesh sizes Δx = Δy = 5 cm and Δkx = Δky = 0.1 rad m−1. The finite bandwidth Δq = 0.2 rad m−1 implies that the maximum resolvable coherent scale in the model is 
To augment the observations with data where no observations are available, we include model simulations with the deterministic model Surface Waves till Shore (SWASH; Zijlema et al. 2011), which solves the 3D Euler equations for a free-surface fluid of constant density. This highly detailed model reproduces the laboratory observations in great detail (see, e.g., Stelling and Zijlema 2003), and we use it here to provide a ground truth for the QCM to validate its ability to capture wave interferences and its representation of the complete second-order statistics. Since the CQ model is linear, the SWASH model is linearized also by reducing the incident wave height to H = 0.001 m. In this way we have a direct comparison with the QCM and can identify nonlinear effects in the observations.
Results
The refractive focusing of the waves produces a lateral interference pattern in the wake of the shoal (e.g., Fig. 1). The finescale pattern is reproduced by the QCM, and normalized wave heights correspond well with observations (Fig. 2). In contrast, the quasi-homogeneous model (RTE) underestimates wave heights along central transect 7 and does not reproduce the wave heights in transect 4. These shortcomings have been noted earlier by O’Reilly and Guza (1991) in comparisons of a ray-based spectral refraction model (equivalent to RTE) and a phase-resolved refraction–diffraction model. The RTE (Figs. 3a–c) cannot resolve the finescale pattern due to a lack of cross-phase information. Also, note that the differences seen between the QCM and observations along transect 8 are mostly because of nonlinearity, as confirmed by the comparison to the nonlinear and linearized deterministic model results (the latter is in close agreement with the QCM).
The normalized wave height H′ (normalized with incident wave height) along the indicated transects. Comparison between observations (open circles), and relative wave heights obtained with the QCM (solid line), RTE (dashed line), and (linearized) SWASH (crosses). For reference, results from the fully nonlinear SWASH model forced with the actual wave height are also included (gray dotted lines).
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
Plan view of (a)–(c) normalized wave heights H′ and (d)–(f) normalized covariance function 
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
The shortcomings of the RTE are fundamental and a consequence of the fact that it strictly transports variance contributions and omits cross-covariance contributions entirely. In other words, the RTE approximation transports the covariance function 
Upwave from the shoal, the spatial covariance function takes on the form of a long-crested wave field, with lines of equal phase (or more precisely, equal phase difference) that are alternately in phase at the maxima (or “wave crests”) and out of phase at the minima (or “wave troughs”) with the wave field at the point 
4. Swell over submarine canyons
Just offshore and to the north of San Diego, stretching from Black’s Beach down to La Jolla point (see Fig. 4), the seafloor bathymetry is characterized by two steep submarine canyons: Scripps Canyon (approximately 150 m deep and 250 m wide) and La Jolla Canyon (approximately 120 m deep and 350 m wide). Along these canyons (see Fig. 4), which extend to 200 m from shore, strong wave refraction occurs because of the steep slopes along the canyon walls (locally exceeding 45°). Especially for long-period swell waves, refraction causes extreme spatial gradients in wave height, and locally, coherent interference effects associated with waves arriving along different ray paths are expected to be important (e.g., Magne et al. 2007).
Bathymetry near (right) Scripps and La Jolla Canyon and the (left) surrounding area. The numbered black dots indicate the instrument locations during the NCEX experiment. Red contour lines correspond to depths of 10, 15 and 50 m. The origin is located at 32°49.7′N and 117°21.9′W, with the x and y axes pointing positive east and north, respectively.
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
The Nearshore Canyon Experiment (NCEX) was conducted in the fall of 2003 to study wave transformation over the canyons (Thomson et al. 2005, 2007; Magne et al. 2007), with a particular focus on Scripps Canyon. Pressure sensors (locations 13–17 and 20–31), Waverider directional buoys (locations 21 and 32–37), and NORTEK vector current meters (PUVs, location 1–12) were deployed around the canyons (see Fig. 4b for locations). The offshore wave conditions were recorded by the permanently deployed Torrey Pines Outer directional Waverider buoy (TPB hereafter) that is located approximately 12.5 km offshore at 549-m depth (see Fig. 4). The La Jolla Outer Buoy (LJB) is located directly to the west of the NCEX area, but it is situated in relatively shallow water (200 m) and near a steep slope so that the wave field recorded at this location is generally not suited as an offshore boundary condition.
At all pressure sensor and PUV sites, surface height variance density spectra are obtained from the detrended 3-h pressure records. Each record is subdivided in windowed segments with 50% overlap and ensemble averaging of the resulting periodograms yields estimates of the bottom pressure spectrum with 120 degrees of freedom and frequency resolution 
From the 3-month field campaign, we selected three cases to compare the QCM and RTE with observed wave conditions. Because coherent effects are most dominant for directionally narrow fields (cf. Fig. 5 in SJ13), we consider cases where clearly distinguishable swell waves were observed at the TPB, incident from either the south (cases I and II) or from the west (case III). For each case, the bulk parameters are summarized in Table 1 with spectra shown in Fig. 5. Because we focus on swell, the mixed sea swell system, incident from the west, as present in case II (separated from the distinct southern swell peak; see Fig. 5), was discarded in the present analysis by a low-pass filter (
Significant wave height 
(top) Directional wave spectra and (bottom) integrated frequency spectra as observed at the TPB in the fall of 2003 for (a),(d) case I, (b),(e) case II, and (c),(f) case III. The red dashed line corresponds to the observed spectra at the LJB.
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
Near Scripps Canyon (location 3), the particle velocities 
Absolute phase differences between surface elevation and the velocity components in the dominant wave direction (
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
a. Model setup
The spectral models are numerically evaluated on a set of nested rectangular spatial grids (see Table 2). The coarsest grid (A) is forced by directional spectra derived from the buoy data on the western boundary (Fig. 4), whereas on the southern and northern boundaries 
Model parameters used for the different nested grids A to C. The spectral resolution was set to 
For southerly waves (cases I and II), the boundary derived from the TPB is not optimal. For these cases, waves that arrive at the NCEX site are refracted over the continental shelf well south of the TPB, whereas southerly waves that arrive at the TPB do not reach the NCEX site. These effects are seen in the large differences between the swell peaks observed at the TPB and LJB for southerly swells (see case I and II in Fig. 5). Nevertheless, the ratio R between the predicted to measured significant wave heights at the LJB was near unity for cases II (R = 1.01) and III (R = 0.94), and only for case I did the ratio differ significantly (R = 0.74). For the latter case the spectra at the boundary were rescaled with 1/R2 to obtain more realistic conditions at the LJB and presumably the NCEX area.
b. Results
For the southerly swell cases (I and II), a significant part of the energy is refracted toward the coast before it arrives at the NCEX site because of the relatively shallow region south of the canyons (see Fig. 4). Hence, wave energy is already much reduced when it arrives at the NCEX site (Figs. 7a,b). The waves subsequently refract strongly over the steep canyon walls, which is visible in the bands of enhanced wave height along the canyon walls. In between the canyons, the convex shape of the topography focuses wave energy so that a mild focal zone emerges. For westerly swells, the pattern is again dominated by the local geometry of the canyons, with a band of enhanced wave energy along the canyon walls and a mild focal region in between the two canyons (Figs. 7c,d).
Wave heights (normalized with the incident wave height) predicted over the canyons for (a),(d) case I, (b),(e) case II, and (c),(f) case III by the (top) QCM and (bottom) RTE. Superimposed arrows correspond to the mean wave direction.
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
Although the QCM and RTE predictions appear to be fairly similar, there are some important differences. To intercompare the models in more detail, and compare simulation results from both models with observations, we consider transects of wave height estimates along different depth contours (Fig. 8). Overall correspondence between the observations and both models is reasonable, specifically for the 50- and 15-m contour lines. However, along the 10-m contour line (around s = 2.5 km, where s represents the along-contour distance measured from the starting point on the north edge of the NCEX area), significant differences between the models are seen. It is in this region that interference occurs between waves that travel in a western direction and waves that are refracted out of the canyon. This results in rapid oscillations of the wave heights, which is particularly visible in the observations for case II. The QCM reproduces this oscillatory behavior almost perfectly (at least near s = 3 km), whereas the variations in the RTE model are much less extreme and do not capture the rapid changes in the mean wave heights.
Normalized wave height 
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
As highlighted in the analysis of the laboratory data, the QCM captures the complete second-order statistics and thus inherently contains spatial information of coherent patterns and standing wave fields. To analyze the wave pattern surrounding the canyon heads, we consider the covariance function as calculated by the QCM at site 31 on the northern edge of Scripps Canyon and at site 24 in the focal area between the two canyons (Fig. 9). For illustrative purposes, we include a few ray trajectories that are initiated at x = 0 using the predicted mean wavenumber 
Normalized covariance functions 
Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0219.1
In the mean wave direction (approximately aligned with the wave rays), the covariance function attains its typical oscillatory pattern, indicative of predominantly propagating wave motion. However, near site 31 (Figs. 9a–c), the covariance function shows a nodal structure in the lateral direction, which implies fast variations of the statistics associated with crossing waves (as also indicated by the crossing rays in this region). In case III (Fig. 9c), a clear nodal pattern emerges where in the lateral (or along crest) direction the covariance function alternates between positive and negative values. The covariance function centered at site 24 retains a structure more in line with the assumptions of quasi-homogeneous theory, with a modulated oscillatory structure in the wave direction (the limited extend of the correlation function due to the finite width of the spectrum) and a slowly decaying correlation with constant phase difference in the lateral direction (indicative of the finite directional width of the wave field). This is consistent with the observation that quasi-homogeneous theory (RTE) and quasi-coherent theory (QCM) predict similar wave heights in this region. In fact, the covariance functions predicted by the RTE are similar to those of the QCM near site 24, although they decay more rapidly (not shown).
From the covariance functions, we see that because of the strong refraction by the canyons, the waves just north and south of Scripps Canyon are statistically nearly independent in cases I and III. That is, the dominant swells, traveling almost parallel to the canyon axis, are trapped on the north side of the canyon, and a weak, uncorrelated component that crosses the canyon from more oblique angles dominates the wave motion south of the canyon. For case II, the wave field is so narrow that some correlation between waves on opposite sides of the Scripps Canyon remains. The directional narrowness of the incident waves during case II is also apparent from the large spatial extent of the correlation bands just north of the canyon.
5. Conclusions
In this study, we developed a stochastic modeling framework for describing the effects of coherent wave interference on spectral wave evolution. By considering a finite coherent footprint of the wave field, the quasi-coherent theory of SJ13 was reformulated in a form similar to a radiative transfer equation, commonly used in operational wave prediction models, but with an additional source term to account for the coherent effects in the wave evolution. The transported variable in this equation is a coupled-mode spectrum, essentially a generalization of the variance density spectrum, which allows for the evolution of cross-phase information and thus the complete spatial covariance function. We verified that the model captures the complete second-order statistics, including mean wave heights and the wave covariance function, for coherently interfering waves by comparison of model results to laboratory observations of a wave focal zone behind a submerged shoal and with Monte Carlo simulations with a deterministic model. Comparison of model predictions to field observations obtained during the Nearshore Canyon Experiment (NCEX) at Scripps Canyon, a submarine canyon on the southern California coast, demonstrates the improved predictive capability of the new stochastic model. In particular, the QCM accurately predicts the observed interference patterns of crossing waves (near the canyon head) that are neglected in traditional models based on the radiative transfer equation. These results, and in particular the comparison to field observations, show that the QCM can resolve finescale structures in nearshore wave statistics associated with crossing wave fields, which contributes to our understanding of these dynamics near the coast and improves the ability to model nearshore wave statistics, wave-driven circulation, and transport processes near the coast.
Acknowledgments
This research is supported by the U.S. Office of Naval Research (Littoral Geosciences and Optics Program and Physical Oceanography Program) and by the National Oceanographic Partnership Program. We thank M. Zijlema, who supplied the initial version of the BiCGSTAB solver used in the present work; A. Reniers, who provided financial support for PBS; and L. Holthuijsen, for continued support of PBS.
APPENDIX A
Fourier Transform Operators
APPENDIX B
Discrete Model
To exclude interactions between waves and topographical variations on the infrawave scale, which are excluded at the order 
a. Coefficients
b. Iterative solution technique
The resulting set of equations involves 
Hereto, 
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Also known as the Wigner or Wigner–Ville spectrum (see, e.g., Wigner 1932; Ville 1948).
