Journal of Coastal Research 00 0 000–000 Coconut Creek, Florida Month 0000 Analysis of Surface Foam Holes Associated with Depth- Limited Breaking Charlotte A. Benbow, Jamie H. MacMahan*, and Edward B. Thornton

Oceanography Department Naval Postgraduate School Monterey, CA 93943, U.S.A.

ABSTRACT

Benbow, C.A.; MacMahan, J.H., and Thornton, E.B., 2017. Analysis of surface foam holes associated with depth-limited breaking. Journal of Coastal Research, 00(0), 000–000. Coconut Creek (Florida), ISSN 0749-0208.

The behavior of surf zone foam holes, as observed at the surface and associated with depth-limited breaking, was investigated. Aerial imagery of the surf zone was obtained with a small, unmanned quadcopter that supported an integrated, high-resolution camera. The quadcopter is an ideal platform for acquiring images directly above the surf zone, a requirement to obtain the requisite resolution. The images were georectified so that size, shape, orientation, and evolution of the wave-generated foam patterns could be quantified. Three hypotheses are proposed for foam-hole generation: obliquely descending eddies (ODEs), self-organization because of bubble rise, and bottom-generated turbulent boils. The fringe region was the most seaward foam region and was marked with circular foam rings that increased in area and were more distinct with time. The fringe region data were consistent with both the self- organization because of bubble rise and turbulent boil mechanisms. The gap region, located between the plunge point and the splash-up created by the bore collapse, was marked by horizontal foam tubes oriented in the cross-shore direction. The foam tubes were likely created in the convergent region between two counter-rotating vortices. The largest region encompassed nearly the entire surf zone and was described as a mat of foam that developed obvious foam holes. The foam holes located in the outer surf zone, near the break point, initially decreased in size, consistent with ODEs before increasing in size and elongation. The foam holes located in the inner surf zone, increased in both size and elongation during a wave period. Because of increasing size with time, the foam-hole generation was attributed to turbulent boils. The rate of increase in foam-hole growth significantly decreased as the waves shoaled from the fringe region in the outer to the inner surf zone, suggesting that growth rate and size decreased with depth.

ADDITIONAL INDEX WORDS: Foam holes, surf zone turbulence, breaking waves, drone remote sensing, video imaging.

INTRODUCTION spilling breaker because of gravity. More air is entrained into In the process of waves breaking at the shore, air is trapped the foam patch along the front edge as the patch rides down the below the ocean surface creating bubbles (Thorpe et al., 1999). wave, whereas more water is entrained into the foam patch The size of the bubble determines the speed at which they rise from below, along the foam–water interface. The foam damps to the surface, where larger bubbles rise the fastest. Some of the growth of the crest. Turner and Turner (2011) also the bubbles pop as soon as they reach the surface; others discussed how foam is generated behind a spilling breaker. remain on or just below the surface, creating patches of foam. Their theory originated with the ‘‘entraining plume’’ model, Directly in front of, above, and behind the wave crest, the but they added that foam generated in front of the crest can be bubbles exist as a dense foam mat with few gaps or holes. entrained downward and will pass under the crest. The foam is Thorpe et al. (1999) noted that because of the nature of the less dense and more buoyant than the surrounding water, so it turbulent processes in the surf zone, the bubbly foam mat will float to the surface and will join with the foam that passed quickly morphs into coherent patterns, recognizable to the over the crest. This foam patch remains behind as the bore continues to progress toward the shore. naked eye as foam holes. Quantifying foam holes is important As opposed to the gentle breaking process of a spilling to the understanding and modeling of turbulent structures breaker, a plunging breaker kinematically breaks as the water within the surf zone from wave breaking. particle velocities at the crest exceed the phase speed of the Longuet-Higgins and Turner (1974) described how a spilling wave causing the wave face to steepen and then overturn, breaker generates foam, using the ‘‘entraining plume’’ model in throwing a jet of water forward of the wave (Peregrine, 1983). which the waves break gently at the crest, trapping air. The Derakhti and Kirby (2014) outlined three air-entrainment air–water mixture stays at the surface because the foamy, air– mechanisms for plunging breakers. First, as the wave crest jet water mixture is less dense than the water surrounding it. The hits the surface, the tube closes and entrains air, creating foam patch begins to grow and ride down the front of the bubbles and foam. Then, as the jet crashes into the surface, splashes are sent forward of the bore. As this secondary splash DOI: 10.2112/JCOASTRES-D-16-00010.1 received 21 January 2016; falls back to the surface, even more air is entrained in the accepted in revision 24 January 2017; corrected proofs received 31 March 2017; published pre-print online XX Month XXXX. water, and more bubbles and foam are produced. As the bore *Corresponding author: [email protected] collapses entraining air, a cloud of bubbles and water are ÓCoastal Education and Research Foundation, Inc. 2017 thrown skyward created by an explosive splash on the backside

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of the bore. This process of jet formation and splash down can crest in front of the wave followed by ODEs with a near-vertical be repeated several times as the wave moves toward the shore. axis that descend toward the bottom and remain behind as the Once the wave becomes a self-similar bore, more air is wave propagates shoreward. Following the work by Watanabe, entrained in a process similar to that described above for a Saeki, and Hosking (2005), Farahani and Dalrymple (2014) spilling breaker. numerically showed that, where the curvature of the span-wise Turbulent processes in the wave-breaking region are not well horizontal vortex structure generated by the breaking waves is understood, partly because of a lack of measurements in the relatively high, coherent, reversed horseshoe, or hairpin vortex field. In the past, hot-film and laser-Doppler anemometers structures, are initiated. Two legs of the reversed horseshoe were used to measure turbulence in the surf zone (George, structures have counter-rotating vortices previously described Flick, and Guza, 1994). However, those point measurements as ODEs. The vortices are most intense at the surface where were not sufficient to describe the three-dimensional (3D) they are generated and decrease in intensity toward the aspects of turbulence. Ruessink (2010) obtained 3D turbulent bottom. Similar coherent, turbulent structures were found in data deploying three acoustic–Doppler velocimeters over the laboratory measurements for incipient, regular, plunging vertical at the macrotidal Truc Vert Beach, France, during breakers by Ting and Reimnitz (2015). high-energy waves. The large changes in depth over the tide The second possible mechanism for foam-hole generation is allowed measurements representative of different relative the process of self-organization by rising bubbles. Derakhti and locations with the surf zone. He found that the cross-shore Kirby (2014) showed by numerical modeling that, in both Reynolds stress hu0w0i increased with the ratio of wave height spilling and plunging breakers, the drag force between the to water depth to a maximum for depth-limited wave breaking bubbles and the liquid surrounding the bubbles was the and for the most part decreased in magnitude toward the bed. primary mechanism for momentum exchange in the vertical The decrease in magnitude of the Reynolds stress with depth direction. The larger bubbles rise faster than the smaller suggests the origin of the turbulence is at the surface during bubbles, imparting upward momentum on the water surround- wave breaking. He also found that, for 15–20% of the time, the ing them (Thorpe et al., 1999). When the larger bubbles reach opposite occurred with bottom-generated turbulence dominat- the surface, the upward-induced motion continues briefly, ing. The alongshore Reynolds stress hu0w0i was consistently causing divergence at the surface and a flow of smaller bubbles positive and decreased in height from the bed, suggesting radially outward, creating a region of no foam or bubbles. They bottom-generated turbulence. The total turbulent kinetic also noted that there is a downward motion (because of energy was found to be of a uniform depth. However, the continuity), which occurs at a greater radial distance. cross-shore turbulent intensity was at a maximum at the A third possible mechanism for foam-hole generation is by bottom and decreased toward the surface. bottom-generated coherent structures resulting in surface Most of our knowledge of turbulence from wave breaking is boils. Thorpe (2005, 2007) suggest boils are generated by the based on laboratory experiments with long period or solitary surging, wave-induced flows over the seabed, similar to waves (Ruessink, 2010). In recent years, particle image coherent turbulent structures resulting in boils at the surface velocimetry (PIV) and volumetric three-component velocimetry of rivers. Omidyeganeh and Piomelli (2013) numerically (V3V) have been used to measure breaking wave–generated showed that the boils in channel flow are due to horseshoe turbulence in laboratory wave tanks (see for example Cox and vortices originating in shear near the bed. Ejection of Kobayashi, 2000; Kimmoun and Branger, 2007; Ting, 2013; turbulence occurs behind the horseshoe head that causes a Ting and Reimnitz, 2015). However, those techniques have yet near-circular jet impinging on the surface with divergence of to be adapted to field measurements. Nadaoka and Kondoh flow vectors. The scale of the boils is comparable to the depth. (1982) observed that the intensity of turbulence at the bottom Boil surface expression was measured in a tidal river using of the water column was correlated with the turbulence at the particle-image velocimetry of surface-infrared images and top of the water column. Therefore, the foam on the surface associated turbulence over the vertical by Talke et al. (2013). may be used as a tracer to describe total water-column They found boils are areas of upwelling and horizontal turbulence. Those turbulent properties are important to divergence, which are caused by the interaction of coherent categorize because they drive processes such as sediment turbulent structures with the kinematic surface boundary transport, water clarity, and the transport of pollutants in the condition. Measurements showed turbulent intensity de- surf zone. The basic hypothesis of this study was that the scale creased toward the surface, implying the boils are generated of turbulence at the surface from breaking waves was related to near the bottom and that they are an important mechanism for the observed scale of the foam holes within the surf zone. vertical transport of turbulent kinetic energy away from the Thorpe (2005, 2007) and Thorpe et al. (1999) suggested three bottom. The authors have observed boils generated behind mechanisms for generating foam holes in the surf zone: (1) plunging breakers viewed overhead from piers. The boils were obliquely descending eddies (ODEs)/hairpin vortices, (2) self- identified as circular, turbulent, upwelled water from the organization by rising bubbles, and (3) bottom-generated bottom, as evidenced by sediments at the surface of the boil. coherent structures resulting in surface boils. Several wave- Herein, the patches of foam and the nearly circular foam tank studies on ODEs and hairpin vortices have been holes that emerge behind plunging breakers and bores inside conducted using entrained bubbles as a flow visualization tool the surf zone at the surface were quantified from imagery to describe and track the evolution of an ODE. Nadaoka, Hino, obtained with a small, aerial vehicle. The imagery of the foam and Koyano (1989) observed that when waves break horizontal patches was used as a passive tracer to infer the turbulent vortices are generated with an axis aligned parallel to the wave processes that occur because of wave breaking.

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Figure 2. Example of ‘‘raw’’ aerial image of the surf zone before Figure 1. Resolution (top) and FOV (bottom) versus elevation. FAA georeferencing, in which black boxes indicate search areas, and neon-green maximum allowed flight altitude (dashed line). pixels are pixels with color matches to the GCP.

METHODS calculations of pixel resolution determined the size of the The DJI (2015) Phantom 2 Visionþ (hereafter, referred to as GCPs required to obtain subpixel accuracy. the Phantom 2) quadcopter, an unmanned aerial vehicle GCPs are needed to georectify and georeference images. The (UAV), was used to acquire aerial imagery of the surf zone. GCPs were 0.61-m diameter plywood circles painted neon pink. The lightweight, quad rotor, battery-powered aircraft was The flight elevation during data acquisition was approximately chosen for this project because of its ease of flight, image 84 m. Approximately 8 pixels fell across the width of the GCP, capabilities because of its ability to hover in a station-keeping which is called subpixel accuracy. The pink color was chosen so mode, and relatively low cost ($1000). The Phantom 2 has the GCPs would stand out against the wet sand, water, foam, approximately 20 minutes of flight time. The Phantom 2 and breaking waves. Five GCPs mounted atop pipes jetted into supports a gimbaled, fish-eye lens camera located on the the sand were located in the surf zone (n¼1), swash (n¼3), and underside of the vehicle. The camera can record high-definition high-water line (n ¼ 1). Four additional GCPs were placed on video at 1080p30, defined as 1080 horizontal lines of vertical the sand just above the high-water line and within the field of resolution progressively scanned with every horizontal line view (Figure 2). The one GCP located within the surf zone drawn in sequence, and an image capture rate of 30 frames/s. broke during active wave breaking and had to be removed. The The camera can also continuously record still images at a locations of the GCPs were surveyed using GPS with horizontal accuracy better than 3 cm. resolution of 4384 3 3288 pixels at 1 frame every 3 seconds. The imagery of the surf zone foam patterns was obtained at Video mode, rather than still image mode, was chosen to Sand City, Monterey Bay, California, on 12 February 2015. maximize the frequency at which images were acquired. The The beach slope of the shore-connected shoals within the user can choose between either 858 or 1108 recording field of active surf zone was 1:100, which steepens to a slope of 1:20 view angles (http://www.dji.com/product/phantom-2-vision- seaward of the surf zone (MacMahan et al., 2005). Because of plus/spec). Based on camera specifications, the field of view the two slopes, waves initially break as plunging waves at the and resolution in centimeters per pixel were calculated for change in slope and then progress to shore as spilling breakers various flight altitudes assuming the UAV was directly above or self-similar bores. Because of the presence of the headlands the surf zone (Figure 1). The Federal Aviation Administration to the N and S narrowing the wave aperture and refraction (FAA) maximum flight altitude for unmanned aerial systems is over the submarine canyon in Monterey Bay, the waves at 152 m (shown as a horizontal black line in Figures 1c and d) Sand City approach near-normal incident, resulting in weak (FAA, 2015). The field of view was decreased by 20% to account alongshore currents (MacMahan et al., 2005). Directional for the portion of the image that is lost during image wave spectra measured by the National Data Buoy Center processing. Based on these calculated values, a flight altitude wave buoy 46042, located just outside Monterey Bay, were between 76 and 91 m was chosen to observe the estimated refracted to the 15-m water depth just offshore of the study site width of the surf zone O (100 m), to conform to FAA regulations as part of the California Data Information Program (CDIP) on UAVs, and to resolve the features of interest. It is desirable and Monitoring and Prediction (MOP) system, archived, to acquire aerial imagery from directly above the surf zone, hindcast data sets (O’Reilly et al., 2016). The day of collection with no tilt to the camera, to minimize image distortion. was chosen for the narrow-banded, long-period, 14-second However, to capture the entire width of the surf zone swell because it allowed time to capture the foam evolution (approximately 100 m) at a high enough resolution and to between waves. The calculated wave height at 15 m was capture all of the ground control points (GCPs), some of which transformed to the shore at the experimental site using the were located on the beach, it was necessary to apply a slight tilt wave transformation model of Thornton and Guza (1983). The to the camera and use the 1108 field-of-view angle. The modeled, significant wave-breaking height was 1 m, with a

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mean breaker depth of 1.6 m. Wave height varied in time because of the groupiness of the narrow-banded swell. Eleven videos were acquired, with video length limited to less than 2 minutes to make the data size more manageable. Continuous, in-flight, video downlink was enabled with the DJI Phantom cell phone application, and the video feed was constantly monitored on a cell phone to ensure that the GCPs were in the center of the field of view. The video was downloaded, and each frame was converted to a still image. Every 10th image was selected, providing three frames per second to keep the data size manageable while keeping the image-collection frequency high enough to capture the features of interest. To remove the fish-eye lens distortion, the camera’s intrinsic parameters were calculated using the methods described in Heikkila¨ and Silven´ (1997) and Zhang (2000). Those camera parameters are then applied to each image to remove the fish-eye distortion. Each undistorted photo was georectified and georeferenced similar to a process described by Brouwer et al. (2015). Georeferencing is essential to compare georectified foam hole images spatially between frames. Using a projective transfor- Figure 3. Aerial image of the surf zone, in which rectangle A depicts the mation matrix, the image pixels containing the GCPs were fringe region, rectangle B depicts the gap region, rectangle C is the plunging matched to the latitude and longitude of the GCP measured in breaker, circle D represents a foam hole, rectangle E depicts an OSB, the field. Because of possible movement of the UAV from wind, rectangle F depicts an ISB, rectangle G depicts a self-similar bore, and W an automated process was developed to find the GCPs, where a depicts the width of the surf zone. The yellow circles indicate the GCPs, and the pink circle shows the UAV operator. set of search boxes in the image, each containing a GCP, were created (outlined in black in Figure 2). Each pixel within each search box was then evaluated to determine whether the pixel’s corresponded to the foam holes, and the white regions red–green–blue color value fell within a range of values corresponded to the foam. However, that mechanized method matching that of the pink GCP (all matching pixels displayed of analyzing the foam patterns in the fringe region was in green in Figure 2). The center of the mass of each set of unsuccessful because the primary foam rings were filled with matching pixels was calculated, and that pixel was then stored less-dense foam that did not allow a determination of a as the image pixel location corresponding to a GCP. Each pixel threshold value to classify foam from no-foam regions. in an image was converted to an x and y reference value. The Therefore, to determine the shape and size of those foam images were then rotated to make the bore parallel to the x-axis patterns, the cross-shore and along-shore diameter of the foam so that the analysis is described in cross-shore and along-shore rings were calculated manually, and from those diameters, the directions relative to the bore (Figure 3). The undistorted, areas and elongations of the foam rings were estimated. rectified, referenced, and rotated images were then individu- The next region, henceforth called the gap region, occurred ally analyzed, and only a small portion of each image, the part only during the largest plunging waves and was located filled with foam and holes, was saved for further analyses. between the plunge point and the location of the splash-up There appears to be three distinct foam-pattern regions as because of the bore collapse behind the wave front (Figure 3, the plunging wave broke and progressed toward the shore as a box B). In that gap region, there were several horizontal foam self-similar bore (Figure 3). The first of those regions, hereafter streaks oriented in the cross-shore direction, which appeared to called the fringe region, existed seaward of the wave immedi- be foam tubes connecting the primary bore to the splash-up ately after the wave broke (Figure 3, box A). To determine the (Figure 3, box C). To characterize the size and separation of shape, size, and evolution of the foam patterns in the fringe these foam tubes, the gap region was extracted and converted region, the part of the image containing the fringe region was to a binary image. A line was selected in the along-shore extracted and converted to grayscale images, in which the direction, which bisected these foam tubes. That bisecting line grayscale pixels represented values between 0 and 1, and then was then used to determine the width and spacing of the foam into binary black and white images, in which the pixel values tubes as a function of time. represented either 0 or 1. The goal of conducting that image The third region of interest was the primary foam region analysis was to determine the borders between the foam and described as a dense mat of foam that appeared directly behind the no-foam regions and to use those borders to calculate the the self-similar bore after the wave breaks (Figure 3, boxes E statistics of the foam patterns. To convert an image from and F). That mat of foam grew in extent as the bore progressed grayscale to binary, a threshold value had to be determined. toward the shore; 1–2 seconds after the wave broke, the dense Any pixel that had an intensity value greater than the mat of foam began to develop a visible pattern of nearly circular threshold value was converted to a 1 (white), and any pixel foam holes (Figure 3, circle D) starting first at the most that had an intensity value less than the threshold value was seaward extent of the foam mat and extending toward the converted to a 0 (black). The black regions in the binary image shore as the wave propagated. Those foam holes appeared to

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Figure 4. Georectified, georeferenced, and rotated images depicting the OSB region (a), and then, 3 s later, separating into OSB and ISB regions (b).

grow and become less circular (elongated) with time elapsed regions in the binary image corresponded to the foam holes, because of wave breaking and distance from wave crest. and the white regions corresponded to the foam. Initially, a 0.5 Finally, the self-similar bore ran up the beach face (Figure 3 threshold value was applied to all the images, but after close box G). inspection, it was found that using only one threshold value for As a wave broke, the area just behind the wave was selected all the images did not appropriately classify the holes and foam, as the outer surf zone box (OSB) (Figure 4a). The left and right and aspects of the image were misrepresented. Carini et al. bounds of the OSB were selected so that any wave reflections (2015) and Podobna et al. (2010) created histograms of image from the shoreline in the image would not be included in the intensity to account for changes in image intensity over a set of box. As the bore progressed toward the shore in each images to determine the appropriate threshold value to use for subsequent image, the selected OSB became larger in the each image. In a manner similar to that approach, the mean cross-shore direction, extending from either the edge of the intensity value of each image was calculated and used as the foam or from the front of the next bore on the seaward side to threshold value. Again, this mean value method was not the back edge of the bore on the shoreward side. sufficient to appropriately represent foam and hole. Therefore, As the bore progressed toward the shore, there was a point a manual approach was adopted. where the foam holes in the OSB stopped growing. To avoid An interactive selection technique (Grande, 2000) was biasing the statistics of the mean hole size, a second inner surf performed to manually determine the threshold value. For zone box (ISB) was initiated that included ‘‘new’’ holes. Thus, each second (every third image analyzed), a subset grayscale two regimes were defined (Figure 3, boxes E and F; Figure 4b, image (Figure 5a) from the longest data set of OSBs was boxes A and B) depending on the size of the holes. The OSB was displayed on the screen, along with five binary images for located between either the front edge of the bore and the back of comparison, each with a different threshold value based on the the ISB (Figure 3, box E) or between the seaward side of the mean intensity (Figure 5). In this example, the image displayed foam and the back of the ISB (Figure 4b, box A). The ISB was is 13 seconds after wave breaking, and the threshold values located further shoreward, between the bottom of the OSB and displayed on the image are 0.525, 0.5, 0.475, 0.45, and 0.425. If the backside of the bore (Figure 3, box F; Figure 4b, box B). The the threshold value was set too high, gray pixels that should be OSB and ISB were allowed to grow and change as the bore turned white were turned black, which caused several holes to progressed toward the shore and represent different depths. As become connected and appear much larger than they actually a new wave broke, new OSB and ISB were selected. In total, 26 were. If the threshold value was set too low, the gray pixels that individual bores and their associated OSBs and ISBs were should connect to other gray pixel and form a hole turned analyzed. white, making the holes appear much smaller than they In a manner similar to that used to analyze the fringe region, actually were. The threshold value that best represented the the OSB and ISB images were converted to binary black and foam and holes was recorded. In this case, the threshold value white images. A threshold value had to be determined a priori selected was 0.475 (Figure 5d). Note, there is some subjectivity to convert the images into binary images. Again, the black to the analysis. However, the human eye is a very effective

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Figure 5. (a) Subset grayscale image of the surf zone foam and correspond- ing binary images estimated with different threshold values (0.525, 0.5, Figure 6. Threshold values for converting grayscale imagery into binary 0.475, 0.45, and 0.425), corresponding to (b–f), respectively. In this case, a imagery (see Figure 5) as a function of bore time estimated by the interactive- threshold value of 0.475 was selected. selection technique and a third-degree polynomial best-fit line. The solid line is for the OSB boxes, and the dashed line is for the ISB.

filter, and it is much more obvious to differentiate between the different threshold values and their impact on the image when where l is a measure of the largest length of the shape, and w is viewed on an expanded high-resolution computer screen. A the shortest width of the shape. An elongation value of one third-degree polynomial was fit to the threshold values for all represents a circle, and an elongation value greater than one times using the longest bore record (Figure 6, grey line) and represents a stretched ellipse. Principle component analysis was then applied to all the bores. The threshold determination (PCA) was used to calculate the principle axes of individual was similarly estimated on the longest data set of the ISBs and foam holes, which were then averaged for each image within another third-degree polynomial equation of threshold values the surf zone, where l and w are the magnitudes of the major was determined (Figure 6, black dashed line). Note that the and minor axes. Those axes were then used to calculate the threshold values were different between the OSBs and ISBs. orientation of the foam holes within the surf zone. The boundaries between the black regions and white regions were determined and used to calculate the area of the foam RESULTS holes. During a quality check of the areas, it became apparent There were eight breaking waves that produced pro- that some of the regions that were identified as foam holes nounced fringe regions (Figure 3, box A). A consistent line were, in fact, small, unimportant features within the image, of boil-like foam rings, aligned in the along-shore direction, which were artifacts of the conversion to a binary image. A mark the fringe region. Those circular boils appear to grow minimum area was determined that removed the unimportant slightly and become much more pronounced with elapsed features but maintained the features of interest. To determine time after wave breaking. The boils within the fringe region that minimum area, four of the same binary images were appear more elongated if the wave breaks closer to the shore displayed, each with a different minimum allowable area. The (shallower water) or in a region still covered by the previous foam holes with an area larger than the minimum allowable waves’ foam mat. The mean of the areas of the five foam rings area were displayed in black, and foam holes with an area along the fringe region was calculated at each time step for smaller than the minimum allowable area were displayed in each of the eight breaking waves (Figure 8). The 95% red (Figure 7). When consistency between images was reached confidence interval and linear regression were calculated in rejecting small areas close together, a cutoff area of 0.35 m2 and plotted, along with the mean ensemble areas. The mean 2 was chosen, and any black region with an area less than that area of the foam rings was initially 5 m and increased at a 2 cutoff was disregarded and removed from analysis. Although rate of 1.1 m /s. Assuming a circular foam ring, the calculated this technique is somewhat subjective, three researchers diameter of the foam rings was initially 2.5 m. After about 6 independently came to the same cutoff area, which provided seconds, the foam rings stop growing, maintaining a constant 2 confidence in the method. area of about 12 m , with a corresponding circular diameter Various shape factors were calculated to describe the of 3.9 m. Initially, the foam rings within the fringe region were filled with tiny bubbles. As time elapsed, the center of evolution of the foam holes. Shape factors are nondimensional the foam rings appears dark, indicating a lack of bubbles quantities calculated from the dimensions of the shape, which present. The elongation of the foam rings was calculated but describe the shape of an object relative to another standard did not exhibit appreciable change. object (Wojnar, 2000). The only shape factor that showed Only one wave produced sufficiently intense breaking to appreciable change was elongation, provide conditions necessary to visualize the foam tubes in the l gap region (Figure 4). The foam tubes were only visible from the Elongation ¼ ð1Þ w surface for approximately 4 second, with an average initial

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Figure 7. Example of binary images, in which surf zone foam holes are depicted as black, and foam holes smaller than a particular cutoff value are depicted in red for cutoff areas: (a) 0.4 m2, (b) 0.35 m2, (c) 0.3 m2, and (d) 0.25 m2. In this case, a cutoff area of 0.35 m2 was chosen.

width of approximately 0.3 m and decreasing to a width of 0.1 m diameter of about 1.2 m, assuming a circular foam hole. For the (not shown here). first 4 seconds, the area of the foam holes decreased linearly at Twenty-six bores were evaluated to quantitatively describe arateof0.08m2/s, with a minimum mean area of approx- the evolution of the foam holes within the foam mat. The areas imately 0.9 m2, which equates to a diameter of 1.1 m assuming of the foam holes were calculated for each bore at each time step a circular foam hole. After t ¼ 4 seconds, the area of the foam for the OSBs. A mean area was then calculated for each of the holes increased linearly at a rate of 0.11 m2/s, reaching a 26 bores at each time step (Figure 9a). The 95% confidence maximum mean area of 2.3 m2, corresponding to a circular intervals and the linear regressions were also plotted. Each diameter of 1.7 m at t ¼ 15.6 seconds. For the ISBs, the mean bore existed for a different length of time. Near the end of the area of the foam holes at t ¼ 0 seconds was about 0.7 m2, with a time series, there were fewer data points to average, causing corresponding circular diameter of 0.9 m. The mean area the confidence interval to spread. Therefore, linear regression increased linearly at a rate of 0.08 m2/s and reached a was only plotted for the first 10 seconds, and the data were maximum area of 1.9 m2, corresponding to a circular diameter truncated past 16 seconds. That same process was repeated for of 1.6 m at t ¼ 9.3 seconds. The initial decrease in area in the the ISBs (Figure 9b). For the OSBs, the mean area of the foam holes was initially between 1 and 1.2 m2, which equates to a

Figure 9. Mean area of foam holes for all bores as a function time. (a) OSB. Slopes of the best-fit lines:0.081 m2/s with an R2 value of 0.78, and 0.11 m2/s Figure 8. Mean area of the foam rings as estimated in the fringe region as a with an R2 value of 0.94. (b) ISB. Slope of the best-fit line: 0.084 m2/s with an function of time. Slope of the best-fit line is 1.05 m2/s, with an R2 value of R2 value of 0.81. Gray line is the 95% confidence interval, black dashed line is 0.96. The gray line is the 95% confidence interval, and black, dashed line is the best-fit line, and the black vertical line is the peak wave period. Note that the best-fit line. the y-axis does not extend to zero.

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Figure 10. Mean elongation of the foam holes as a function of time. (a) OSB. Figure 11. The mean absolute orientation of the foam holes for all bores as a 2 Exponential decay of the best-fit line: 0.018/s with an R2 value of 0.83. (b) function of time. (a) OSB. Slopes of the best-fit line: 1.528/s with an R value 2 ISB. Exponential decay of the best-fit line: 0.021/s with an R2 value 0.72. of 0.88. (b) ISB. Slope of the best-fit line: 1.818/s with an R value of 0.74. Gray Gray line is the 95% confidence interval, black dashed line is the best-fit line, line is the 95% confidence interval, and the black dashed line is the best-fit and the black vertical line is the peak wave period. Note that the y-axis does line. not extend to zero. over 3.63 seconds of the OSB provided details on how the foam OSBs was associated with a splash-down cycle of the plunging holes evolved (Figure 12). The areas of some of the foam holes breaker, creating larger foam holes initially. The ISBs do not grew by merging with other foam holes (Figure 12, red circles show that initial decrease because the foam holes in that region and pink ovals), whereas the areas of other foam holes grew were generated by self-similar bores. independently. Foam circles also appeared to split apart A process similar to that used to calculate mean areas above (Figure 12, blue squares), thus decreasing in size. Examination (Equation 1) was used to calculate the mean elongation of the of sequences from the ISB provided similar results. The foam holes as a function of time (Figure 10). The mean sequence of images show that the foam holes translated elongation of the foam holes for the OSBs (Figure 10a) offshore because of orbital wave motion in the trough of the increased from an initial near-circular value of 1.0 at a rate wave behind the bore front. of 0.02 per second for the first 10 seconds and then flattened at a rate of 0.005 per second. The mean elongation of the foam holes DISCUSSION for the ISBs increased from an initial value of 1.1 at a rate of Regions of foam holes were identified, from offshore to 0.02 per second. The elongations of all the foam holes for one onshore as the (1) fringe region, (2) gap region, and (3) foam bore were plotted against the areas of all the foam holes for the mat divided into outer and inner surf zones. Thorpe (2005, same bore. The elongations and areas were not correlated 2007) and Thorpe et al. (1999) hypothesized three physical statistically. processes that generate foam holes in the surf zone. The first of In a manner similar to that used to calculate the mean area those processes are the ODEs/hairpin vortices that develop as a and the elongation of the foam holes, the orientation was by-product of breaking waves. Observing that phenomenon in calculated from the angle of the principal component relative to the natural surf zone is difficult, and to our knowledge, no the onshore direction for each foam hole with PCA analysis. observations or measurements of ODEs/hairpin vortices have Given that the measured angles for individual foam holes were both positive and negative, absolute angles were used; been made in the field. The notion of using surface foam as a otherwise, the angles tended to cancel each other out in the passive tracer to identify those features, and quantify their size mean. An orientation angle of 08 indicates that the foam hole and evolution is dependent on the ODE/hairpin vortex was oriented in the along-shore direction, and an orientation penetrating the ocean surface. Watanabe, Saeki, and Hosking angle of 908 indicates an orientation in the onshore direction. (2005) described the process of air entrainment in an ODE, The foam holes within the OSBs and the ISBs had initial based on laboratory observations. They observed that bubbles absolute orientations between 508 and 608, with a narrow were captured by the ODE, and those trapped bubbles spread. Over time, the foam holes elongated in the cross-shore concentrated along the axis of rotation of the ODE, where the direction, with a final absolute orientation for the foam holes in pressure is smallest. If the ODE does penetrate the surface, both the OSBs and the ISBs of between 708 and 808 (Figure 11). then the circular pattern on the surface, depicting the ODE, Although the time series of mean areas, elongations, and would be filled with bubbles and not free of bubbles, as is orientations provided a general view of the foam hole evolution, evident in the collected imagery. Watanabe, Saeki, and they do not describe the details of the evolution of individual Hosking (2015) further observed through numerical modeling foam holes or how the foam holes moved relative to a fixed that, because of the conservation of angular momentum, the reference frame. A sequence of 12 images every 0.33 seconds angular velocity increases and the radius of the ODE

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Figure 12. Sequence of 12 zoomed-in images (10 by 20 m) in the OSB region every 0.33 s over a total 3.63 s. The red, circled region depicts the growth of a foam hole by merging with other foam holes. The blue, squared regions show foam holes splitting apart. The pink, oval regions depict the merging of foam holes and the elongation in the cross-shore direction

decreases. In this study, foam holes consistently grew with The second hypothesis for foam hole generation is the theory time (Figure 9). Therefore, the foam holes within the dense of self-organization by rising bubbles. There are two pieces of foam mat were not consistent with an ODE/hairpin vortex in evidence in this study that do not support the theory of self- the water column below, except possibly during the beginning organization by rising bubbles of various sizes entrained into of the OSB, when the foam holes decreased in size. the water column during wave breaking. The first is the

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upward momentum imparted by the large bubbles as they rise. circular diameter for the mean maximum areas were Shortly after the large bubbles reach the surface, the upward approximately 3.9, 1.6, and 1.4 m. Hypothesis t tests that momentum and resulting velocity divergence ceased. However, compared the slopes of the foam hole growth and the mean the observed foam holes continued to grow, and not just for a maximum areas showed they were significantly different few seconds, as would be expected from the self-organization with 95% confidence, with the growth rates and mean argument. The second piece of evidence that is incompatible maximum areas decreasing shoreward. with the self-organization by rising bubbles theory was the Natural-scaling parameters for the foam holes are the observed darkness of the foam holes immediately after breaking wave height and depth of water. However, the formation. According to the self-organization by rising bubbles breaker wave height and self-similar bores are a constant theory, there would be a stream of small bubbles rising after function of depth within the surf zone. Therefore, depth is the the large bubbles that would make the center of the foam hole relevant parameter. Although the water depth was not appear gray or white for some time. measured directly, an approximate measure of the depth was The theory of self-organization by rising bubbles could, obtained using the modeled depth at breaking of 1.6 m. however, describe the foam rings that developed in the fringe Assuming a plane slope and dividing the surf zone into an region (Figure 3, box A). Those foam rings initially appeared as outer fringe region and two equal surf zone regions for OSB and bright rings of white foam with lighter, grayer foam in the ISB, the calculated mean depths were 1.6, 1.2, and 0.4 m, center. That is consistent with the slow rate at which the small respectively. The decreasing diameters of the foam holes and bubbles rose into the center of the void that was produced by depths between the fringe OSBs and ISBs suggest that the the bursting of the large bubbles. The center of the foam rings foam holes may be a function of depth. also became darker and less gray with time elapsed from wave These observations were not a definitive test for the breaking because of the outward movement of the small hypothesis that the foam holes were due to bottom-generated bubbles. The fringe-region foam holes might also be surface turbulent boils. A more-comprehensive and elaborate experi- boils filled with sediment from the bottom. The rings grew for ment was required. The waves, depth, and morphology should approximately 6 seconds and then remained a constant size be measured in a cross-shore array. For image processing, in (Figure 8), which corresponds with the outward momentum hindsight, the high-resolution, still images acquired at three ceasing shortly after the large bubbles reached the surface. per second by the Phantom 2 would have been preferable for The foam tubes that developed in the gap region only obtaining coverage of the entire surf zone. In addition, images occurred for the largest plunging breakers (Figure 3, box B) close enough to the surface from a tower within the surf zone and were similar to the aerated vortex filaments and surface should have been obtained to perform a PIV to obtain the scars predicted in numerical simulations. Aerated vortex turbulent velocity structure on the surface within a foam hole, filaments, modeled by Lubin and Glockner (2015), form along with the velocity structure beneath the surface. The most between the plunge point and the splash-up location and are obvious approach would be a prototype-scale laboratory oriented in a cross-shore direction. Those cross-shore, horizon- experiment where 3D particle-image techniques could be tal vortices are formed from the stretching of fluid during wave employed. However, the generation of foam is very different breaking and develop in counter-rotating pairs. In the in fresh and salt water. Laboratory experiments use fresh upwelling region between a pair of vortices, the surface water only because of the problem of salt water disposal. experiences divergent flow, and in the down-welling region of Therefore, the experiment would need to be performed in the a pair of vortices, the surface experiences convergent flow field where particle-imaging techniques over the vertical have (Saruwatari, Watanabe, and Ingram, 2009). The surface foam yet to be adapted. The alternative is to measure the turbulent formed by the breaking wave is concentrated along the region velocity field over the vertical (Ruessink, 2010; Talke et al., of convergence and downwelling. Because the foam bubbles are 2013) to infer whether the processes creating the foam holes are buoyant, they remain at the surface and form distinct foam from the surface or from the bottom. tubes in the cross-shore direction. In the upwelling region, bubble-free water is brought to the surface and forms the CONCLUSIONS bubble-free gaps between the foam tubes. Unique surf zone imagery, acquired from a UAV at Sand The third hypothesis for foam hole generation is by bottom- City, Monterey Bay, California, was analyzed to describe the generated, coherent structures that result in surface boils. foam patterns that evolved within the surf zone. The ability to Thorpe (2005, 2007) suggests boils are generated by the obtain images almost directly overhead at close range allowed surging, wave-induced flows over the seabed, similar to high-resolution imagery to be obtained to resolve foam holes in coherent, turbulent structures resulting in boils at the the surf zone. On the day of collection, the narrow-band waves surface observed in rivers. Field measurements in rivers with a period of 14 seconds were categorized as plunging find boils are areas of horizontal divergence at the surface breakers approaching near shore-normal. Three distinct foam and that the scale of the boils is comparable to the water pattern regions were observed within the imagery (Figure 3). depth (Nimmo Smith, Thorpe, and Graham, 1999; Talke et The fringe region existed on the seaward edge of the foam mat al., 2013). and is marked by rings of foam aligned in the alongshore Qualitatively, it appears the foam holes decrease in size direction. Those foam rings became more defined and larger when both the slopes of the growth of foam holes and the with time. There was also a gap region that existed only during mean maximum areas between the fringe region are the largest plunging waves. The gap region forms between the compared in the OSB and the ISB. The corresponding plunge point and the secondary splash and is marked by foam

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tubes, aligned in the cross-shore direction, which appear to bottom-generated turbulent boils was found to be consistent connect the plunge point to the secondary splash. The final with the observations and was identified as a possible region was a dense mat of foam that developed into coherent mechanism. patterns of foam holes that grew and elongated with time. There are three hypotheses that describe the generation of ACKNOWLEDGMENTS the foam holes within the surf zone. The first hypothesis is that C.A.B. was supported by the U.S. Navy for her Master of the ODEs/hairpin vortices created by the breaking waves and Science degree. J.H.M. was supported by ONR and technol- that cause vortices to penetrate the surface (Thorpe, 2007; ogy development by GOMRI. The UAV was obtained from an Thorpe et al., 1999). The second hypothesis is that self- ONR DURIP. Appreciation is extended to Mike Cook, Keith organization by rising bubbles causes foam holes within the Wyckoff, Mathias Roth, Darin Keeter, and Tucker Freismuth. surf zone (Thorpe, 2007; Thorpe et al., 1999). The third We wish to thank the five anonymous reviewers and Steve hypothesis is that bottom-generated, coherent structures from Thorpe for their constructive critical reviews that substan- surging, wave-induced flows over the seabed result in surface tially improved the manuscript. boils (Thorpe, 2005, 2007). Those theories were tested by analyzing the UAV imagery of the surf zone. LITERATURE CITED The foam rings in the fringe region were nearly circular and Brouwer, R.L.; de Schipper, M.A.; Rynne, P.F.; Graham, F.J.; Reniers, did not elongate with time. The corresponding circular A.J.H.M., and MacMahan, J.H., 2015. Surfzone monitoring using diameter of the foam rings was initially 2.5 m and increased rotary wing unmanned aerial vehicles. 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