Field Observations of the Evolution of Plunging‐Wave Shapes

There are few high‐resolution field observations of the water surface during breaking owing to the difficulty of collecting spatially dense measurements in the surf zone, and thus the factors influencing breaking‐wave shape in field conditions remain poorly understood. Here, the shape and evolution of plunging breakers is analyzed quantitatively using three‐dimensional scans of the water surface collected at high spatial and temporal resolution with a multi‐beam terrestrial lidar scanner. The observed internal void shapes in plunging breakers agree well with previously developed theoretical shapes at the onset of breaking, and become more elongated and less steep as breaking progresses. The normalized void area increases as the local bottom slope steepens and as the breaking depth decreases. The void shape becomes more circular as the local bottom slope and the ratio of breaking water depth to wavelength increase, as well as in conditions with opposing winds.


Introduction
Depth-induced wave breaking transfers momentum into the surfzone water column, driving cross-and alongshore currents, increasing shoreline water levels, and generating turbulence and vorticity (Peregrine, 1998). Breaking waves often are categorized based on their shapes and properties, with plunging and spilling breakers the primary types on most beaches (Galvin, 1968;Peregrine, 1983;Battjes, 1988). Plunging breakers are characterized by the formation of an internal air cavity (or void) as the crest of the wave curls forward and connects with the wave face. This void eventually collapses, dissipating energy, generating turbulence, and entraining air bubbles (Kiger & Duncan, 2012;Peregrine, 1983). Relative to spilling breakers, plunging breakers generate high levels of turbulence that extend deeper in the water column (Flick et al., 1981;Ting & Kirby, 1994, 1995Aagaard et al., 2018) and increase suspended sediment concentrations (Aagaard et al., 2018;Beach & Sternberg, 1996;Inman et al., 1980;Nielsen, 1984;Voulgaris & Collins, 2000).
Observing the rapid transformation of the shapes of waves as they shoal and begin to break requires spatially dense measurements in and near the surf zone, which are difficult to obtain in the field, particularly in high-energy, plunging conditions. Most studies focusing on the shape of depth-limited plunging breakers have utilized laboratory experiments (Blenkinsopp & Chaplin, 2007; Chanson & Jaw-Fang, 1997; Abstract There are few high-resolution field observations of the water surface during breaking owing to the difficulty of collecting spatially dense measurements in the surf zone, and thus the factors influencing breaking-wave shape in field conditions remain poorly understood. Here, the shape and evolution of plunging breakers is analyzed quantitatively using three-dimensional scans of the water surface collected at high spatial and temporal resolution with a multi-beam terrestrial lidar scanner. The observed internal void shapes in plunging breakers agree well with previously developed theoretical shapes at the onset of breaking, and become more elongated and less steep as breaking progresses. The normalized void area increases as the local bottom slope steepens and as the breaking depth decreases. The void shape becomes more circular as the local bottom slope and the ratio of breaking water depth to wavelength increase, as well as in conditions with opposing winds.

Plain Language Summary
Breaking waves inject momentum into the surf zone that drives nearshore currents, raises water levels, and generates turbulence. However, there are few highresolution field observations of the water surface during breaking because of the difficulty of collecting spatially dense measurements of the water-surface elevation at the location of wave breaking, leaving open questions related to the factors controlling the shape and properties of breaking waves. In this study, three-dimensional scans of the water surface in the surf zone collected using a multi-beam terrestrial lidar scanner are used to analyze quantitatively the shape and evolution of plunging breakers. The shape of the internal barrel (or void) created as the plunging lip intersects with the front face of the wave agrees well with theoretical void shapes at the start of the breaking process, and then becomes more elongated and less steep through the breaking process. The normalized area of the void is larger when waves break on steeper bottom slopes and in shallower water. The void is more circular when waves break on steeper bottom slopes and with larger ratios of breaking water depth to wavelength, and also when the wind is blowing in a direction opposing the direction of wave propagation.
O'DEA ET AL. © 2021. The Authors. This article has been contributed to by US Government employees and their work is in the public domain in the USA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Ting & Kirby, 1994, 1995Yasuda et al., 1997) or numerical simulations (Grilli & Svendsen, 1990;Grilli et al., 1997Grilli et al., , 2001Lin & Liu, 1998;Lubin et al., 2006;Vinje & Brevig, 1981;Yasuda et al., 1997), many of which are extensions of earlier numerical and parametric studies focused on deep-water plunging wave shapes (Longuet-Higgins, 1982;Longuet-Higgins & Cokelet, 1976Peregrine et al., 1980). A depth-independent parametric form of the equations of motion (Longuet-Higgins, 1982) has been used to assess depth-limited plunging wave shapes in the laboratory (Blenkinsopp & Chaplin, 2008) and the field (Mead & Black, 2001). In these studies, a modified curve was fit to optical images of breaking waves, and the resulting curve parameters were used to characterize the plunging wave shapes, and to analyze the factors contributing to the differences between observed and theoretical shapes. However, the field study (Mead & Black, 2001) provided only limited quantitative analysis owing to the use of unrectified optical snapshots during breaking and the lack of spatially and temporally overlapping wave and environmental data. The factors controlling plunging-wave shapes in field conditions, and thus their contribution to nearshore circulation, air-sea gas exchange, and sediment transport, remain poorly understood.
Here, the shape and evolution of plunging waves in the field are analyzed quantitatively using a novel multi-beam lidar collection approach that produces three-dimensional scans of the water surface at high spatial and temporal resolution. The shape of the internal void in finite-crested plunging waves is reconstructed by combining information from multiple beams surrounding the leading edge of the breaking crest. This multi-beam system allows the dual free surface formed by the plunging lip to be measured, which is not possible with two-dimensional lidar scanners used previously (Blenkinsopp et al., 2010;Brodie et al., 2015;Martins et al., 2017Martins et al., , 2018 owing to the shadowing that occurs from the plunging lip when measured along a single transect. For each wave, the shape of the internal void created by the plunging lip is compared with theory (Longuet-Higgins, 1982), and the spatio-temporal evolution of the plunging wave and the factors influencing the void shape, angle from horizontal, and area are assessed at the onset of breaking.

Observations
Breaking-wave shapes were determined from three-dimensional lidar scans of the water surface collected at the U.S. Army Engineer Research and Development Center's Field Research Facility (FRF) on the Atlantic coast of North Carolina. A Velodyne HDL-32E lidar scanner was mounted on a horizontal arm attached to a platform on an amphibious vehicle (CRAB) (Birkemeier & Mason, 1984), enabling unobstructed views of the water surface ( Figure 1a). The lidar simultaneously emits 32 beams (903 nanometer wavelength) that each scan along a single line in space. The lidar returns provide the location (distance and angles, converted to Cartesian coordinates) where each beam intersects the water, creating a three-dimensional snapshot of the surface with a typical accuracy of up to 0.02  m. The beams are spaced at 1.33 increments resulting in a 41.34 field of view. The alongshore extents of the scans thus increase with distance from the scanner from 7 m (below the scanner) to 20 m (∼±25 m cross-shore from the scanner). The azimuthal (cross-shore) resolution is a function of the adjustable speed of rotation (10 Hz here) and the fixed firing time (46.08 s), resulting in a resolution of 0.166. The cross-shore extent of quality returns depends on the water surface roughness, foam, and sediment concentrations, and often was less than the effective terrestrial range (100 m). During post-processing, the scans were limited to a 60-× 25-m cross-and alongshore range, although most scans extended 30-50 m in the cross-shore and up to 15 m in the alongshore direction ( Figure 1c).
Scans were collected for eight 30-min periods during conditions with plunging breakers on September 7, September 8, and October 25, 2017. The CRAB was positioned near the surfzone break point (Figure 1a) for each collection period, and relocated after each 30-min period to account for changes in wave-breaking location owing to changing wave conditions and water levels (circles in Figure 1b). The cross-shore position of the CRAB varied up to 75 m over the collection periods. The alongshore location of the CRAB was approximately fixed on each of the three days, but varied up to 632 m between days. On each day of data collection the bathymetry was surveyed along a cross-shore transect from the dune to 5-m water depth (mean cross-shore resolution 0.7 m) at the alongshore location of the CRAB (

Geophysical Research Letters
All scans within each 30-min collection period were transformed into a local, globally aligned, lidar-centric coordinate system. The roll and pitch components of the transformation were determined by leveling the returns from the water surface, neglecting any small tilts due to setup or setdown. The location and yaw orientation of the lidar were determined using the known orientation of the arm on which the lidar was mounted relative to the body of the CRAB, combined with the GPS coordinates of the four corners of the CRAB, which were visible in the lidar scans. To account for the changes in the lidar orientation and location resulting from the occasional settling of the CRAB into the sediment (up to 0.3 m during the 30-min collections), the water surface was leveled using approximately 1.5 min of lidar data centered on each wave of interest. The mean water-surface elevation over that period was set to the elevation measured every 6 min at the end of the FRF pier in approximately 8-m water depth (blue curves in Figures 2g and 2h). Individual waves were identified using a zero-up-crossing technique within the region directly below the scanner.
Nearshore hydrodynamic conditions were determined from a 16-element array of bottom-mounted pressure gauges located in 8-m water depth, which provided hourly estimates of frequency-directional wave spectra (Long & Oltman-Shay, 1991), significant wave height s H (4 times the standard deviation of sea-surface elevation fluctuations), peak wave period p T , and the mean wave direction at the peak frequency p   (d) shows the angle of the wave crest relative to shore parallel used to generate the cross-and along-crest coordinate system. into a wave-oriented coordinate system (described in Section 2.2.1). Wind stress w  was estimated following Large and Pond (1981).

Plunging Wave Identification
Lidar observations of finite-crested plunging waves enable reconstruction of the shape of the internal void by combining information from multiple beams surrounding the leading edge of the breaking crest (Figures 1f and 1g). A total of 30 plunging waves with at least three consecutive frames with clearly defined internal voids were identified manually. For each plunging wave, the angle of wave propagation within the lidar field of view was determined (Figure 1d, red line), and all scans within that wave were rotated into a wave-oriented coordinate system where the y-axis is parallel to the wave crest and the x-axis is parallel to the direction of wave propagation. The angle was estimated by finding the mean rotation angle about the z-axis in several frames that minimized the vertical spread of the water surface in x z  space. This rotation collapses lidar returns on top of each other in the along-crest direction, creating a cross-crest transect of water-surface elevation when viewed in x z  space (Figures 1f and 1g). and (e), (f) p  from the 8-m array (black curves) and (g), (h) tidal elevation at the end of the Field Research Facility pier (blue curves) for the three days of data collection, with September 7-8 shown in the left panels and October 25 shown in the right panels. Blue circles in panels (a-d) show the same parameters determined from a 30-min time series of mean water-surface elevation from the lidar scans. Red circles in panels (a-f) show the wave-by-wave parameters extracted from the lidar scans and red circles in (g), (h) show the water depth at the location of wave breaking for each wave analyzed. variability on wave shape, an along-crest window centered on the leading edge of the breaking crest (3-4 m along the wave) was determined manually for each wave (Figure 1d, black parallel lines), and only points within this window were used in the analysis of the evolution of void shapes (Figure 1g).

Void Fitting
To characterize the plunging-wave shapes, a curve adapted from a parametric solution to the equations of motion (Longuet-Higgins, 1982) was fit to the lidar data: where L and W are the length and width of the internal void in the plunging breaker, respectively, and the x-axis aligns with the major axis of the function (  Black, 2001), L and W were allowed to vary independently, allowing comparisons of field observations with the theoretical aspect ratio. In addition, Equation 1 provides a framework to describe the void shape quantitatively, and to compare void shapes through the breaking process and for a range of waves. Equation 1 was fit to the points on the inside edge of the internal void in each frame with a clearly defined void (3-5 frames per wave) by solving a nonlinear minimization problem using a variant of the simplex method. To extract the points on the inside edge of the void, the function was fit to all points within manually determined x and z bounds. Points greater than 0.1 m above or below the function were discarded, and the process was repeated (Figures 3b and 3c). For each fit, the resulting void length L, width W , aspect ratio L W / , angle from horizontal (i.e., the vertical angle between the major axis of the curve and the cross-crest x-axis) v  , and area A were calculated, with A determined from (Blenkinsopp & Chaplin, 2008)

Wave-By-Wave Parameters
To assess sources of variability in observed void shapes, void parameters at o t were compared with wave and environmental parameters. Local bulk parameters (significant wave height s H and peak period p T , blue circles in Figures 2a-2d) over the 30-min collection periods were determined from a time series of the mean water-surface elevation over a 0.2 m (cross-shore) × 1.0 m (alongshore) rectangle directly below the lidar. Wave-by-wave parameters including breaking wave height b H , period b T , direction wb  (red circles in Figures 2a-2f) and speed b C were estimated from the lidar scans. The breaking location for each wave was determined manually and defined as the location of the wave crest in the first frame that showed the crest starting to curl forward. The wave height at breaking b H was estimated from the frame immediately prior to the initiation of breaking by finding the vertical difference between the maximum elevation of the crest (excluding spray) and the preceding minimum elevation. The period b T was estimated as the average time between three successive wave crests centered on the breaking wave. The angle wb  was estimated during the data rectification and processing (Section 2.2.1). The wave speed b C was estimated from the mean displacement of the wave crest from the time the wave initially enters the field of view until the internal void begins to collapse and a wave crest cannot be defined (at least 18 frames). The breaking wavelength b  was defined individually for each wave as the mean seafloor slope from the location of breaking to one half of a wavelength offshore, with the depth at breaking b h defined as the mean water depth over the same area (red circles in Figures 2g and 2h). The estimated cross-crest wind stress w  was averaged over 20 s centered on the time of breaking.

Void parameters were compared with individual wave and environmental parameters (
h , and w  ), as well as parameters combining wave and bathymetric characteristics (

Results and Discussion
The breaking heights b H (red circles, Figures 2a and 2b) of each wave, as well as bulk, 30-min local (blue circles in Figures 2a and 2b) and offshore (black curves in Figures 2a and 2b) Figures 2a and 2b), suggesting that the plunging waves were among the largest waves observed during each collection period.
At the time step closest to the moment the plunging lip intersects with the front face of the wave o t , the void aspect ratio / A H were compared with a range of wave-by-wave parameters estimated from the lidar scans and environmental data. Breaking-wave characteristics are influenced by wave properties (primarily wave height and steepness), bathymetry, and wind (Battjes, 1974;Douglass, 1990;Galvin, 1968;Goda, 2010). On planar beaches, breaking type and location often are estimated using the surf-similarity parameter   / ( / ) / H b dw 1 2 (Battjes, 1974). However, the relevant bottom slope and other breaking parameters in field conditions remain poorly defined (Apotsos et al., 2008;Robertson et al., 2013;Weishar & Byrne, 1978). Additionally, wave breaking over a bar or reef can differ from breaking on a planar slope (Blenkinsopp & Chaplin, 2008;Smith & Kraus, 1991), further complicating analyses of the role of bathymetry in controlling breaking-wave characteristics. Here, the bathymetry differed between the three days of data collection, with wave breaking occurring on a relatively planar slope on September 7 and 8 and over a pronounced bar on October 25 (Figure 1b). The bottom slope between the location of wave breaking and one half of a wavelength offshore had a stronger relationship with void parameters than the slope over one quarter or one full wavelength offshore.
The local bathymetry influenced void shapes, which become rounder (i.e., A H ( 2 R  0.1), highlighting the importance of the local bathymetry to wave breaking. Some of the scatter in the relationships between wave shape and environmental parameters may be owing to neglect of alongshore variability in bathymetry that may influence the location and type of wave breaking.
Wave shape ( / o o L W ) is weakly correlated with wind stress w  , with offshore-directed (negative) winds decreasing the aspect ratio ( 2 R = 0.2, Figure 4h). Increased spray off the crest of the waves indicative of strong, opposing winds was visible in many of the lidar scans with small / o o L W (Figure 4a). Cross-shore winds influence the breaking type and location in both the laboratory (Douglass, 1990;King & Baker, 1996) and the field (Galloway et al., 1989), with offshore (onshore) winds resulting in an increase in plunging (spilling) breakers and an onshore (offshore) displacement of the breaking location. Onshore-directed winds also alter the shape of waves during the shoaling process (Feddersen & Veron, 2005;Zdyrski & Feddersen, 2021   weak relationship between wind stress and wave shape was observed, the present analysis may be limited by the distance (429-544 m) between the lidar data collection in the surf zone and the wind measurements at the end of the pier. Further work is necessary to understand and quantify the role of wind stress in controlling plunging-wave shapes.
The high temporal resolution of the lidar scans allows for analyses of the spatio-temporal evolution of the void shape beyond o t . In general, the voids were rounder (i.e., smaller aspect ratio) and steeper (i.e., larger void angle from horizontal) as the plunging lip first intersected the front face of the wave, becoming more elongated and less steep through the breaking process (Figures 3c, 3d, 4a and 4b). These changes in void shape and steepness occurred rapidly over 1-3 m and within half a second of o t . The aspect ratios ranged from 1.7 to 5.0, and the void angles from horizontal ranged from 32 to 72   Figure 3d).
The geometric, kinematic, and dynamic properties of breaking waves influence the rate and location of wave-energy dissipation in the surf zone, which has implications for nearshore circulation, air-sea gas exchange, and sediment transport. The three-dimensional scans of the water surface presented here allowed for analyses of breaking-wave shapes at unprecedented spatial and temporal resolution in an effort to quantify the variability in void shapes and the spatio-temporal evolution of void shape during the early stages of breaking.