Energy dissipation in the inner surf zone: new insights from 1 LiDAR-based roller geometry measurements

The spatial and temporal variation of energy dissipation rates in breaking waves controls 16 the mean circulation of the surf zone. As this circulation plays an important role in the 17 morphodynamics of beaches, it is vital to develop better understanding of the energy dis- 18 sipation processes in breaking and broken waves. In this paper we present the ﬁrst direct 19 ﬁeld measurements of roller geometry extracted from a LiDAR dataset of broken waves to 20 obtain new insights into wave energy dissipation in the inner surf zone. We use a roller 21 model to show that most existing roller area formulations in the literature lead to consid- 22 erable overestimation of the wave energy dissipation, which is found to be close to, but 23 smaller than, the energy dissipation in a hydraulic jump of the same height. The role of 24 the roller density is also investigated, and we propose that it should be incorporated into 25 modiﬁed roller area formulations until better knowledge of the roller area and its link with 26 the mean roller density is acquired. Finally, using previously published results from deep- 27 water wave breaking studies, we propose a scaling law for energy dissipation in the inner 28 surf zone, which achieves satisfactory results at both the time-averaged and wave-by-wave 29 scales. 30


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The surf zone is the part of the nearshore characterized by breaking and broken waves,  Over the last few decades, numerical models based on the full Navier-Stokes equations 50 have been increasingly used to study wave breaking processes [e.g., see Lin and Liu, 1998; mean density ρ r . The quantity A represents the area of the surface roller located in front 120 of the breaker above the oscillatory wave motion and characterized by turbulent and aer-121 ated flows [Basco, 1985]. Although the value of A will by definition influence the value 122 of ρ r , no threshold for the void fraction which represents the underside of the roller area 123 has been proposed. In practice, A and ρ r are very difficult to consistently and accurately 124 measure due to complex hydrodynamics of the aerated region of the breaker (e.g., see 125 Duncan [1981], Govender et al. [2002], Kimmoun and Branger [2007], and the recent re-126 view of Lubin and Chanson [2017]). The tangent to the smooth water surface below the 127 hydrofoil-generated steady breaker was used by D81 to define A. However, this bound-128 ary is much harder to define for developed breakers, for instance forcing Govender et al. 129 [2002] to define A as the 'aerated region' only. The difficulty in measuring and defining 130 the roller area has led to the existence of numerous formulations in the literature as shown 131 in Table 2. A simple analysis assuming H = 1 m, L r = 1 m, tan θ = 0.1, and the beach 132 slope tan β = 0.01 demonstrates that it is possible to have an order of magnitude dif-133 ference between the formulations of D81 and Tajima [1996]. This suggests that energy  In this paper, we present a novel field dataset of surface roller properties (θ and L r ) ex-143 tracted from a 2D LiDAR dataset of inner surf zone waves collected by Martins et al. 144 [2017a]. The methodology to obtain this dataset is first described and it is then com-145 pared to the empirical relations obtained by D81 for steady spilling breakers generated by 146 a hydrofoil. Thanks to these direct measurements of roller properties, the number of un-147 knowns in the parameterization of Duncan [1981] (Eq. 1) is reduced to ρ r and A. We use 148 the classic model of Svendsen [1984] and the dissipation term given by Duncan [1981] to 149 investigate the capacity of various formulations of A for predicting the energy dissipation 150 rates observed in our inner surf zone data. The role of ρ r in particular in the definition 151 of A is also discussed in this analysis. Finally, we present an attempt to scale the energy 152 dissipation in the inner surf using local wave properties, which is less reliant on wave ge-153 ometric properties and could easily be implemented in a phase-averaged model or used by 154 remote-sensing techniques to estimate energy dissipation in broken waves. 2017a,b]. The field experiments and the raw data processing are described in these two 163 references, but some basic information is repeated here. Three eye-safe 2D LiDAR scan-

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We hence make the assumption that, in the inner surf zone, the internal structure of the 209 roller has a slope similar to that of the surface of the breaking region, which is consistent 210 with observations [e.g., Duncan, 1981;Kimmoun and Branger, 2007] and the compar-211 isons of A presented in Section 3.2.

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In the surf zone, good estimates of the wave celerity are required to accurately describe 213 the incident wave energy flux [Svendsen et al., 2003]. The traditional approach for esti-214 mating the wave celerity c relies in the following estimate: c ≈ ∆x/∆t where ∆x is the 215 distance travelled by the wave in the time ∆t [e.g., see Suhayda and Pettigrew, 1977]. The

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Radon Transform [Radon, 1917] has also been used to estimate individual wave celerities  As is commonly observed in the inner surf zone [e.g., Thornton and Guza, 1982], every 254 wave from the present dataset is found to be depth-limited with a correlation r 2 = 0.87 255 between the individual wave height H and the period-averaged water depth h w ( Figure   256 4a). In a first attempt to parameterize the roller angle, θ is compared with the wave height 257 (Figure 4b), and the product L r tan θ is shown against the surf zone similarity parameter 258 ( Figure 4c). There appears to be a linear trend between tan θ and H, however, more data where k is the wave number. In shallow water, the hyperbolic tangent can be approxi-

Modelling energy dissipation rates in broken waves with a roller model 301
The novel surface roller dataset presented in Section 2 allows the number of unknowns 302 in the parameterization of Duncan [1981] (Eq. 1) to be reduced to A and ρ r only. In this 303 Section, we use this dataset and the roller concept initially developed by Svendsen [1984] 304 with the dissipation term from Duncan [1981] to investigate the influence of different for-305 mulations of A and the role of ρ r on the modelling of the incident wave energy flux. We 306 first describe the model and the assumptions upon which it is based.
where α is the mean wave angle relative to shore normal. For waves propagating in the moves at the same speed c as the carrier wave [Svendsen, 1984]. To account for the ex-319 tra kinetic energy present in the roller, S84 separated the incident wave energy flux into a 320 wave and a roller contribution as follows: with 322 where ρ is the water density, g is the gravity constant, T is the wave period, η is the time-323 varying surface elevation, and ρ r and A the surface roller mean density and area (see also 324 Deigaard and Fredsøe [1989]). In practice, the surface roller constitutes the rotational part 325 of the broken wave and accounts for the extra kinetic energy found in breaking and broken 326 waves, see Svendsen [1984], Battjes [1988], and also the description of the roller model contributions to the dissipation such as that from air entrainment which are known to be 340 significant in the outer surf zone but whose effect is diminished in the inner surf zone neglected as it was found to be negligible on sandy beaches compared to that by breaking 343 processes [e.g., see Boers, 2005].

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The growth of the surface roller is compensated by the energy dissipation D τ that occurs 345 through shear stresses at the wave/roller interface and the dissipation that originates from 346 mass exchanges between the wave and the roller [Nairn et al., 1990;Deigaard, 1993;Stive 347 and de Vriend, 1994; Reniers and Battjes, 1997]. Deigaard [1993] (see also the note in 348 Stive and de Vriend [1994]) showed that the contribution of the mass exchanges to the en-349 ergy dissipation is similar to the spatial variation of the roller kinetic energy so that with 350 the assumptions made above, we can write: The energy balance system from Eq. 5 hence simplifies to a single differential equation: 353 From his hydrofoil experiments, D81 related the energy dissipation in steady breakers to 354 the Reynolds stresses at the boundary between the roller and the underlying layers of fluid 355 (see Eq. 1). The dissipation term due to shear stresses corresponds to the work done by 356 the roller averaged over the wave period see also Eq. 1:

Energy dissipation terms
In the following, we will also use the original model of Svendsen [1984] as a reference: The approach of S84 follows the seminal work of Le Méhauté [1962] on non-saturated 359 breakers, and that of Svendsen et al. [1978] to approximate the energy dissipation in a bro-360 ken wave with that of a hydraulic jump of the same height such that: where h w is the period-averaged water depth, and h c and h t are the water depths below 362 crest and trough respectively [e.g., Svendsen, 2006, p. 286], see Figure 1. by modelling E f ,w and comparing it to our observations. Eq. 10 and 12 are solved numer-372 ically with a finite difference modelling approach to estimate the cross-shore variation of 373 E f ,w (Eq. 7) and E f ,r (Eq. 8). Starting at an initial position x 0 , the model uses measured 374 wave quantities (H, c, θ and L r ) and local quantities (h w , h t ) to compute the roller con-377 9 reads: 378 where the subscripts i and i − 1 refer to the evaluation of the quantity at the successive 379 grid points x i and x i−1 respectively. δx = x i − x i−1 is the spatial discretization step, taken  Eq. 14. More information on this group is given in the Appendix. The wave and roller 387 properties of this group were extracted using the methodology presented in Section 2.2 388 and ensemble-averaged.

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The basic analysis on the order of magnitude of A presented in Section 1 showed poten-390 tial for large discrepancies between the different formulations presented in Table 2. The energy dissipation rates about 10 times greater than given by Engelund [1981] (Table 2).

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Considering the number of studies that have estimated the energy dissipation rates to be 409 close to that of a bore, and that the formulations from Tajima [1996] and Okayasu et al.

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Starting with the formulation by Engelund [1981], the best fit with observations is ob-413 tained with a density ratio of ρ r /ρ = 0.87 (Figure 6b), corresponding to a RMSE of where γ is the wave height to water depth ratio. For the present dataset, Eq. 18 cor- parameterization (e.g. Figure 4). In the following, we investigate the performance of the where, k is the wave number and has been calculated using the measured surf zone quan-521 tity cT.

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The dissipation coefficient b computed with Eq. 19 and 20 for the ensemble averaged 523 wave group (Appendix A1) demonstrates contrasting cross-shore evolution (Figure 8). In this paper, we present a high-resolution LiDAR dataset from which the geometrical 563 properties of surface rollers (θ and L r ) are extracted. This dataset constitutes the first di-564 rect measurements of these properties from field experiments. We report roller angle val-565 ues up to 6 times greater than the value of 5.7°typically used in energy balance-based 566 numerical models that use the parameterization of Duncan [1981] to model the energy 567 dissipation in broken waves ( Eq. 1 and 11). Future deployment of LiDAR scanners at dif-568 ferent field sites will enable this dataset to be extended for a range of wave conditions and 569 beach types, and will potentially allow the parameterization of L r and θ as a function of 570 wave and beach parameters. to answer questions such as: is there a void fraction that clearly defines the wave/roller in-592 terface or is it only related to the roller hydrodynamics (e.g. the most turbulent region).

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Further work could also investigate wave setup and undertow, probably in a more con-594 trolled environment, as it could lead to a better understanding of A and ρ r and a better 595 knowledge of the contribution of surface rollers in surf zone mean flow.

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The incorporation of ρ r /ρ into the formulations for A and the uncertainties regarding 597 these two parameters do not alone explain the modification of the original roller area for-598 mulation obtained by Duncan [1981], and that later derived by Svendsen [1984]. Another

636
In shallow water, B 0 is generally found to vary in the cross-shore direction: it is close to 637 0.125 in the shoaling region [Basco and Yamashita, 1986], but rapidly decreases towards 638 the break point and then slowly varies in the inner surf zone to a value close to a typi-639 cal value of 0.075 due to a more skewed wave profile [Svendsen, 1983[Svendsen, , 1984 to the typical value of 0.075 [Svendsen, 1983]. It is worth noting that to retrieve the local 647 wave height from the modelled wave energy flux in the present study (e.g. Section 3.2), 648 Eq. A.1 has to be used with the value B 0 = 0.0625. At the wave-by-wave scale, we note 649 more variability; this can be observed in the greater standard deviations obtained with the 650 integral form (Eq. 7). There are two potential reasons for this:

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• There can be a great variability in shape from one wave to another (e.g. Figure   652 A.2), and the formulation of Eq  Figure 1. Definition sketch of the broken wave geometry. The mean water depth h is defined as the vertical distance between the bed and Mean Water Level (MW L). The bore propagates at speed c in water depth h t and has a height H, corresponding to the distance between the crest (white dot) and the preceding trough (white square). The instantaneous water depth below the bore crest is expressed as h c = H + h t . The surface roller is defined from the wave crest (white dot) to the bore toe (red dot), defined as the point where ∂η ∂ x = 0.2 tan θ max , where θ max is the maximum angle found over the roller region. The surface roller has an angle with the horizontal of θ and a length L r . Finally, the surface roller area is noted A but is only represented schematically here, due to the lack of definition and knowledge on this quantity and on ρ r . To facilitate the calculation of the roller area, the interface between the roller and the wave was assumed to have an ellipsoidal shape close to the roller toe.