Remote Sensing Observations of Dominant Breaking Waves in Intermediate to Deep Water from a Lighthouse During Storm Conditions

Wave breaking is one of the most important yet poorly understood water wave phe1 nomena. It is via wave breaking that waves dissipate most of their energy and the occurrence 2 of wave breaking directly influences several environmental processes, from ocean-atmosphere 3 gas exchanges to beach morphodynamics. Large breaking waves also represent a major threat for 4 navigation and for the survivability of offshore structures. This paper provides a systematic search 5 for intermediate to deep water breaking waves with particular focus on the elusive occurrence 6 of plunging breakers. Using modern remote sensing and deep learning techniques, we identify 7 and track the evolution of over four thousand unique wave breaking events using video data 8 collected from La Jument lighthouse during ten North Atlantic winter storms. Out of all identified 9 breaking waves (Nb=4683), ≈22% were dominant breaking waves, that is, waves that have speeds 10 within [0.77cp, 1.43cp], where cp is the peak wave speed. Correlations between the occurrence 11 rate of dominant breaking waves (that is, waves per area and time per peak wave period) and 12 wave steepness and wave age were observed. As expected, the number of identified plunging 13 waves was small and six waves of all detected breaking waves, or 0.13%, could undoubtedly be 14 considered as plunging waves. Two waves were also identified as unusually large, or rogue waves. 15 Although afflicted by several technical issues, the data presented here provides a good indication 16 that the probability of occurrence of plunging waves should be better incorporated into the design 17 of offshore structures, particularly the ones that aim to harvest energy in offshore environments. 18


Introduction
Wave breaking is still one of the least understood water wave phenomena despite decades of continuous research, especially during extreme storm conditions. Longuet- 23 Higgins [1] summarizes the occurrence of large scale intermediate to deep water wave 24 breaking into two main categories: spilling and plunging breakers. In a spilling breaker, 25 a region of strong turbulence, visible as white foam, develops on the crest of the wave 26 as it propagates. Plunging breakers are far more dramatic as the wave profile becomes 27 so unstable that the wave crest curls and a forward moving (plunging) jet is formed. 28 Breaking waves in intermediate to deep water result from wave group interactions [2] or 29 may be generated by the hydrodynamic modulation of short waves by underlying long 30 waves, which increases the probability of wave breaking of the short waves [3]. While 31 the occurrence of spilling breakers is easily observed during moderate to strong wind 32 conditions, direct field observations of intermediate to deep water plunging breakers 33 remains rare and, to the authors' knowledge, only laboratory data [4][5][6] or reports from 34 mariners [7] are available. Further, contrarily to their shallow water counterparts, no 35 non-dimensional number to differentiate between the two types of breakers is currently 36 available in the literature, which makes their identification very challenging using 37 conventional instruments such as wave buoys or pressure transducers. 38 This paper will focus on the occurrence of dominant breaking waves, which are 39 defined as waves that have energy contained within ±30% of the spectral peak frequency 40 ( f p ) and are assumed to correspond to the peak enhancement region in the JONSWAP 41 spectrum [2]. Assuming that the linear dispersion relation is valid, dominant waves are, 42 consequently, waves that travel at speeds close to the peak spectral speed (c p ), within 43 the range [0.77c p , 1.43c p ] (not accounting for bottom interaction). Despite this definition 44 being rather arbitrary, it is practical and supported by the fact that the most intense 45 and dangerous breaking waves are the dominant waves, since they carry most of the 46 energy of the wave field. The research presented here will, therefore, heavily rely on 47 such definition. 48 The main candidates for intermediate to deep water dominant breakers in well 49 developed, storm seas may be the unusually large waves known was rogue waves. 50 For example, a laboratory investigation of the Draupner wave has recently shown that 51 plunging breaking happened for one of the possible simulated scenarios [5]. There 52 also is some debate regarding if another famous rogue wave, the Andrea wave, was 53 a breaking wave or not [8] (the evidence points towards no). In contrast, the rogue energies, it is important to make a clear distinction between the occurrence of plunging 66 and spilling breakers in the design of offshore structures given that the forces that 67 these waves generate on structures are very distinct. While spilling breakers generate 68 continuous forces, the forces from plunging breakers may be felt as sharp peaks [15], 69 known as "slamming forces". Considering the famous Goda formula [16], the slamming 70 force of a breaking wave is: where ρ w is the water density, D is the structure diameter, c b is the breaking wave speed, 72 η b is the maximum water elevation above the still water level, C s is a constant (equals 73 to π in Goda's work [16]) and λ is a wave curling factor which gives the proportion 74 of the crest that is active in the slamming load.  25 20.00 N, 5 0 8 2.28 W) using the same instruments described in 103 Filipot et al. [14]. The data used in this paper were collected by a pair of synchronized  Figure 2-a for the location of the buoy). A total of ninety 30-minute long video data 114 acquisition runs were recorded but only ten 20-minute records, representative of storm 115 conditions, will be analysed here, as it will be explained below. Note that the wave buoy 116 data collection started after 1 month from the start of the campaign; therefore, the data 117 reported here previous to this date was extracted from the HOMERE database [17,18].

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The wind data reported here is from the European Center for Medium Range Weather
From the analysis of Table 1, the significant wave height for these storms varied from 126 approximately 4m up to 10m, the peak period varied from 10s to 18s and the peak  field. These two problems make the three-dimensional stereo reconstruction for the vast 142 majority of the data either impossible or very unreliable. It was possible, nevertheless, to use the standard ARGUS [21] methodology to project pixel to metric coordinates to track the evolution of breaking waves. To do so, 250 ground control points (GCPs) 146 were extracted from a successful stereo reconstruction (from the day the cameras were   The wave breaking detection technique used in this paper is an extension from 156 the method developed in Stringari et al. [22]. These authors developed different neural 157 networks trained on a relatively diverse dataset to classify images that contained active 158 wave breaking, that is, breaking waves that are actively generating bubble entrainment.

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Differently from the original method, which is limited to a binary image classification  wave crests are travelling (c b ). Finally, wave breaking crest lengths were calculated using the α-shape algorithm [28] (red lines in Figure 3-f) and ellipses were fitted to each 189 detected event as per Stringari et al. [22]. While these data were not analysed in detail 190 here, it allows, for example, to obtain Phillips' Λ(c)dc distribution [29] from which wave 191 breaking probabilities can be derived.  g 2π T 2 p ) and, to a lesser extent, with wave age (c p /u * ) and not with wave height (H m0 ).

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Here, the occurrence rate of waves per unit of area and time per peak wave period is 224 used as a proxy to investigate such correlations. This parameter is calculated as: where N is the number of breaking waves indicated, respectively, by the subscripts 226 t, d and p for total, dominant and plunging waves, A is the total analysed area in 227 square meters, T is the total analysis duration in seconds and T p is the wave peak 228 period in seconds. Table 2 shows the values of R d,t,p for all analysed storms. Figure   229 4 shows the correlations between R d and H m0 , S p and c p /u * for the present data. As Date  Correlation between the wave steepness (S p ) and R d . (f) Correlation between wave age (c p /u * ) and (R d ). In (d), (e) and (f) the blue swash indicates the 95% confidence interval for the regression, r xy is Pearson's correlation coefficient and each marker corresponds to a storm from Table 2.

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From the results presented above, six waves could be considered intermediate   . In all plots, data were from the HOMERE and were linearly interpolated from a non-structured grid to a regular grid with 25m resolution. Vectors were re-sampled to a coarser resolution to improve visualization.

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We have presented a systematic search for intermediate to deep water, dominant 260 breaking waves with particular focus on plunging waves using remote sensing tech-261 niques from a lighthouse that is highly exposed to severe sea-states. that the occurrence of these waves are far more likely than previously thought [38,39]. Goda's formula will predict a much more intense slamming force for these waves than 333 for spilling or non-breaking waves, which should be considered with increased attention 334 by the engineering community in the design of future offshore structures.

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Finally, while it is impossible to precisely know the interconnections between waves, 336 winds and currents that led to the occurrence of waves #1 and #2, it is possible to search 337 for the most likely causes. From the analysis of Figure 1-d, the directional spreading was 338 low for both waves, which indicates that crossing-seas were not a direct cause. Although 339 the synoptic wind field seems to be aligned with the peak wave direction for both waves, in the surf zone [13] and may have been a decisive factor creating this wave. While no 343 opposing current was present for wave #1, a strong opposing current jet is clearly seen in Figure 6-e. It is well established that strong opposing currents lead to enhanced wave 345 steepness and rogue waves [12], hence this is the most likely cause of wave #2. Finally, 346 for both waves, the classic modulational instability cannot be ruled out as cause. The   [40]. The data necessary to reproduce the results seen in this paper are also available from Zenodo 388 [41].