A New Probabilistic Wave Breaking Model for Dominant Wind-sea Waves Based on the Gaussian Field Theory
Caio Eadi Stringari, Marc Prevosto, Jean François Filipot, Fabien Leckler, Pedro Veras Guimarães
mmanuscript submitted to
JGR: Oceans
A New Probabilistic Wave Breaking Model forDominant Wind-sea Waves Based on the GaussianField Theory
C. E. Stringari , M. Prevosto , J.-F. Filipot , F. Leckler , P. V. Guimar˜aes , France Energies Marines, Plouzan´e, France Institut Fran¸cais de Recherche pour l’Exploitation de la Mer, Plouzan´e, France PPGOceano, Federal University of Santa Catarina, Florian´opolis, 88040-900, Brazil
Key Points: ● A new probabilistic wave breaking model based on Gaussian field theory is pre-sented for dominant, wind-sea waves. ● Wave breaking probabilities are modeled from the joint probability density betweenwave phase speed and particle orbital velocity. ● The proposed model performs well when compared to six other historical mod-els using three field datasets.
Corresponding author: C.E. Stringari,
Corresponding author: J.F. Filipot,
[email protected] –1– a r X i v : . [ phy s i c s . a o - ph ] J a n anuscript submitted to JGR: Oceans
Abstract
This paper presents a novel method for obtaining the probability wave of break-ing ( P b ) of deep water, dominant wind-sea waves (that is, waves made of the energy within ±
30% of the peak wave frequency) derived from Gaussian wave field theory. For a giveninput wave spectrum we demonstrate how it is possible to derive a joint probability den-sity function between wave phase speed ( c ) and horizontal orbital velocity at wave crest( u ) from which a model for P b can be obtained. A non-linear kinematic wave breakingcriterion consistent with the Gaussian framework is further proposed. Our model wouldallow, therefore, for application of the classical wave breaking criterion (that is, wave break-ing occurs if u / c >
1) in spectral wave models which, to the authors’ knowledge, has notbeen done to date. Our results show that the proposed theoretical model has errors inthe same order of magnitude as six other historical models when assessed using three fielddatasets. With optimization of the proposed model’s single free parameter, it can be-come the best performing model for specific datasets. Although our results are promis-ing, additional, more complete wave breaking datasets collected in the field are neededto comprehensively assess the present model, especially in regards to the dependence onphenomena such as direct wind forcing, long wave modulation and wave directionality.
Plain Language Summary
Waves will break if the speed of the water particles on the wave crest is greater thanthe speed of the wave itself, causing the wave crest to overtake the front part of the wave,leading to wave breaking. Precisely simulating real ocean waves requires, therefore, a particle-by-particle description of the water motion, which is too expensive for the current com-puters to handle in real-world applications. Instead, wave models describe waves by meansof their statistical properties, that is, averaged over a large number of waves. In this pa-per, we present a mathematical formulation that allows to calculate the combined prob-ability between the speed of particles on the wave crest and the wave speed based onlyon statistical properties. From these combined probabilities, we model the probabilityof wave breaking. Our results indicate that our model performed relatively well whencompared to six other models using three historical datasets. Because of a lack of ob-served data to assess our model, we recommend that future research should focus on col-lecting more wave breaking data measured in the field. Future advances on this line of –2–anuscript submitted to
JGR: Oceans research could lead, for example, to improvements on operational weather forecast mod-els.
A robust description of wave breaking is a crucial aspect of wave modelling. It isvia wave breaking that most of the wave energy is dissipated and a precise formulationof this phenomenon is required to obtain reliable models. Despite of its importance, en-ergy dissipation due to wave breaking is still modelled as a semi-empirical process dueto the difficulty to represent physically-derived wave breaking criteria on phase-averagedwave models (Battjes & Janssen, 1978; Thornton & Guza, 1983; Banner et al., 2000; Fil-ipot et al., 2010; Filipot & Ardhuin, 2012; Banner et al., 2002; Ardhuin et al., 2010; Ban-ner et al., 2014; Zieger et al., 2015; Ardag & Resio, 2020). The available probabilistic(that is, parametric, or empirical) formulations included in these models have been de-rived from limited datasets and without rigorous theoretical frameworks and, therefore,they currently lack a solid physical background. While the current operational (spectral)models are capable of reproducing field observations of integrated spectral parameters(for example, significant wave height, peak wave period and peak wave direction) withgood accuracy, it remains unclear if their wave breaking parameterizations are entirelyreliable. This knowledge gap partly occurs because limited research has focused on wavebreaking statistics derived from field data, especially when it comes to wave breaking ob-servations distributed as a function of wave scales (for example, wave frequency or wavephase speed). The research developed here has, therefore, important implications for air-sea flux parameterizations (Kudryavtsev et al., 2014), safety at sea (Kjeldsen et al., 1980)and design of offshore structures (Filipot et al., 2019), all of which directly rely on theproperties of breaking waves.Historically, parametric wave breaking models have been constructed from two dif-ferent approaches: the first approach considers wave statistics (wave steepness, most fre-quently) derived from a wave-by-wave analysis of the surface elevation timeseries collectedat a single point location where wave breaking occurrences are synchronously identified(using video data, most frequently). The wave breaking probability (that is, the ratiobetween the total number of breaking waves over the total number of waves during a givenperiod of time) can then be expressed as a bulk quantity (Thornton & Guza, 1983; Chawla& Kirby, 2002; Alsina & Baldock, 2007; Janssen & Battjes, 2007) or can be distributed –3–anuscript submitted to
JGR: Oceans over wave frequency ( f ), wavenumber ( k ), or wave speed ( c ) ranges, referred as to “wavescales” by the wave modelling community (Eldeberky & Battjes, 1996; Banner et al., 2002;Filipot et al., 2010).The second approach follows from Phillips (1985) who defined the distribution Λ ( c ) dc as the “average total length per unit surface area of breaking fronts that have velocitiesin the range c to c + dc ”. This approach therefore relates to the analysis of sea surfaceimages in which individual wave breaking patches are tracked in space and time. Themain motivation for introducing this new concept was clearly stated in Phillips (1985):“There is clearly some association of the breaking events with waves of different scales,but it is difficult to make the association in an unambiguous way if we consider only thesurface configuration at one given instant. A breaking crest may indeed be a local max-imum in the instantaneous surface configuration but there is no guarantee that a localwavelength of the breaking wave can be defined clearly. It seems more satisfactory touse the velocity c of the breaking front as a measure of the scale of the breaking”. Thisquotation clearly identify the limitations of directly relying on the analysis of single pointelevation timeseries. Different parameterizations have been proposed to quantity Λ ( c ) dc from theoretical (Phillips, 1985) or empirical considerations (Melville & Matusov, 2002;Sutherland & Melville, 2013; Romero, 2019). However, Phillips’ (1985) framework re-mains controversial, particularly regarding its practical application, given that differentinterpretations of his concepts can generate differences of several orders of magnitudein the calculations of Λ ( c ) dc and its moments (Banner et al., 2014). For a detailed re-view of commonly used parametric wave breaking models please refer to Appendix A.Interestingly, while the ratio between the horizontal orbital velocity at the crest ( u )to wave phase speed ( c ) appears the most reliable parameter to determine wave break-ing occurrence (Saket et al., 2017; Barthelemy et al., 2018; Derakhti et al., 2020; Var-ing et al., 2020), it was not used by any of the approaches mentioned above. This pa-per provides a new promising wave breaking model by revisiting Rice (1944) and Longuet-Higgins (1957) statistical descriptions of Gaussian processes (that is, for linear waves)to obtain the theoretical joint probability density between c and u ( p ( c, u ) ). We thenmodel P b assuming a kinematic wave breaking criterion consistent with non-linear waves,that is, a wave breaks if the fluid velocity at the wave crest is greater than the wave phasespeed ( u > c ). This study focuses on analysing dominant waves, defined as waves thathave frequencies within ±
30% of the spectral peak frequency of the wind-sea (Banner –4–anuscript submitted to
JGR: Oceans et al., 2000). Future research will be dedicated to extend our efforts to broader wave scales.This paper is organized as follows: Section 2 describes the proposed model, Section 3 presentsthree historical datasets used to evaluate the model, Section 4 presents the results, Sec-tion 5 discusses and Section 6 concludes.
The kinematic wave breaking criterion u / c = u / c exceeds0.85 in deep and shallow water. Further numerical simulations showed that wave break-ing occurs when the maximum orbital velocity ( u max ) equals c somewhere along the waveprofile and not necessarily at the wave crest (Varing et al., 2020). Although the relation-ship u / c provides a solid physical background to establish the onset of wave breaking,this approach has never been applied to spectral wave models because it requires phase-resolving the wave field. In the sections below, we circumvent this difficulty by defininga wave breaking probability model using the joint probability density between c and u corresponding to a given wave energy spectrum ( E ( f ) ). The efforts in this paper are con-sistent with part of the recent work from Ardag and Resio (2020) in the sense that bothworks aim to solidify the use of the kinematic wave breaking criterion as the standardapproach for modelling wave breaking. Longuet-Higgins (1957) published a very complete work on the statistics of Gaus-sian wave fields. In particular, Longuet-Higgins (1957) studied the probability densityof the speed of zero-crossings along a given line that is of interest for us in this work. Inhis paper, the speed of zero-crossings were applied in particular to the zero-crossings ofthe space derivative of a Gaussian process, that is, the velocities of the local maxima inspace (Longuet-Higgins (1957), pp. 356-357). The present work describes how the samemethodology can be extended to derive the joint density of the speed of space local max-ima (or local crests) and simultaneous wave horizontal orbital velocity for a one-dimensional –5–anuscript submitted to
JGR: Oceans
Gaussian sea state. For simplicity, this paper follows the same notations as those of Longuet-Higgins (1957) and the reader is directed to Section 2.5 in Longuet-Higgins (1957) forfurther details.As explained in Longuet-Higgins (1957), if ξ ( x, t ) is a stationary-homogeneous pro-cess and we are interested in the points (for example, in space) were this process crossesa level x , the joint distribution of the space derivative of ξ noted ξ , with other relatedprocesses ξ , ξ , . . . at ξ = x is given by: p ( ξ , ξ , ξ , ... ) x = ∣ ξ ∣ p ( ξ , ξ , ξ , ξ , ... )∣ ξ = x N ( x ) (1)where N ( x ) is the number of crossings of the level x by ξ (see Equation 2.2.5 in Longuet-Higgins (1957)). In this paper we are interested in joint distributions at the local max-ima in space of the wave elevation process ξ . Therefore, ξ is the space derivative of thewave process and local maxima correspond to down-crossings of the zero level by ξ = ∂ξ / ∂x . ξ = ∂ξ ∂x , ξ = ∂ ξ ∂x = ∂ξ ∂x . (2)In the case of Gaussian processes, N − ( x ) is: N − ( x ) = π √ m m exp (− x m ) , N − = N − ( ) = π √ m m (3)where m , m , . . . , m i are the i -th wavenumber spectral moments and the minus signindicates that we consider only down-crossings. Following Longuet-Higgins (1957), if we are interested in the speed c of the localmaxima in space, that is, the speed of the down-crossings of ξ , we have: c = − ∂ξ / ∂t∂ξ / ∂x = − ξ ξ with ξ = ∂ ξ ∂x and ξ = ∂ξ / ∂t. (4)Using Equation 1, –6–anuscript submitted to JGR: Oceans p ( ξ , ξ ) = ∣ ξ ∣ p ( ξ , ξ , ξ )∣ ξ = N − (5)with p ( ξ , ξ , ξ ) the point joint distribution of the three Gaussian processes ∂ξ ∂x , ∂ ξ ∂x , ∂ ξ ∂x∂t is: p ( ξ , ξ , ξ ) = p ( ξ ) p ( ξ , ξ ) = e − ξ m π √ m e − ⎛⎜⎜⎜⎜⎜⎝[ ξ ξ ] Q − c ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ ξ ξ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎞⎟⎟⎟⎟⎟⎠ √( π ) det ( Q c ) (6)and covariance matrix: Q = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ m m m ′ m ′ m ′′ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎣ m Q c ⎤⎥⎥⎥⎥⎥⎦ . (7)Note that following Longuet-Higgins (1957) notations, m ′′ i indicates the mixed wavenumber-frequency i -th spectral moment, where the number of quotes indicates the order of thefrequency spectral moment, for example, m ′ = ∫ ∞ πf k E ( k ) dk, (8)where E ( k ) is a given wavenumber spectra.Classically, to introduce c in the joint density and obtain p ( c, ξ ) , we apply a changeof variables ξ = − ξ c , ξ = ξ (9)and after the integration of p ( c, ξ ) over all the domain of definition of ξ , we obtainthe distribution of c (Longuet-Higgins (1957), Eq. 2.5.19): p ( c ) = m m ′′ − m ′ √ m ( c m + cm ′ + m ′′ ) / (10) –7–anuscript submitted to JGR: Oceans
Note that the sign on c (or on m ′ ) depends on the convention on the wave propagationdirection. We have kept the convention used by Longuet-Higgins (1957) here. As indicated in Equation 1, we can introduce in the formula a variable which rep-resents the horizontal orbital velocity. For Gaussian waves the horizontal orbital veloc-ity u is defined as u = H t ( ∂ξ ∂t ) (11)with H t the Hilbert transform in time domain. Which means that ξ = ∑ i a i cos ( k i x − ω i t ) (12)is transformed in u = ∑ i a i ω i cos ( k i x − ω i t ) , (13)with a i the wave amplitude, k i the wavenumber and ω i the angular wave frequency ofthe wave component i . As the Hilbert transform is a linear operator, u is also Gaussian.As previously, at the local maxima we have: p ( ξ , ξ , u ) = ∣ ξ ∣ p ( ξ , ξ , ξ , u )∣ ξ = N − (14)with a new covariance matrix for ξ , ξ , ξ and u : Q = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ m m m ′ m ′ m ′ m ′′ m ′′ m ′ m ′′ m ′′ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ⎡⎢⎢⎢⎢⎢⎣ m Q c ⎤⎥⎥⎥⎥⎥⎦ . (15)As previously, we can apply a similar change of variables –8–anuscript submitted to JGR: Oceans ξ = − ξ c , ξ = ξ , u = u, (16)or the easiest to deal with, ξ = − cξ , ξ = ξ , u = u (17)and integrate p ( c, ξ , u ) over all the domain of definition of ξ . The result is a more com-plicated but again semi-analytical. The body of the integral has the forme − [ ξ ( c ) ξ + β ( c,u ) ξ + α ( u )] ξ (18)and its integration in ξ on the down-crossings space ] − ∞ , ] gives I ( c, u ) = (( φ + ) √ π ( erf ( φ ) + ) e φ + φ )√ ξ / e − α / (19)with φ = φ ( c, u ) = √ β ( c, u )√ ξ ( c ) , α = α ( u ) , (20)∆ = det ( Q c ) = m ′ ( m ′ m ′′ − m ′ m ′′ ) + m ( m ′′ m ′′ − m ′′ ) + m ′ ( m ′ m ′′ − m ′ m ′′ ) , (21) α ( u ) = m m ′′ − m ′ ∆ u , (22) β ( c, u ) = m ′ m ′′ − m ′ m ′′ ∆ u + m m ′′ − m ′ m ′ ∆ uc (23)and ξ ( c ) = m ′′ m ′′ − m ′′ ∆ + m ′ m ′′ − m ′ m ′′ ∆ c + m m ′′ − m ′ ∆ c . (24)The joint probability density of ( c, u ) is then: –9–anuscript submitted to JGR: Oceans p ( c, u ) = N − ( π ) √ m ∆ I ( c, u ) = I ( c, u ) π √ m ∆ . (25)Note again that the sign on c and u (or on m ′ and m ′ ) depends on the convention onthe wave propagation direction and Longuet-Higgins (1957)’s convention is still used here.The coefficients ( α, β, ξ ) can be calculated directly numerically and ∆ is the determi-nant of Q c , the sub-matrix of Q , and after the inverse of Q c is calculated: Q − c = ⎡⎢⎢⎢⎢⎢⎣ R ss t r ⎤⎥⎥⎥⎥⎥⎦ (26)we find α ( u ) = ru , (27) β ( c, u ) = [ c ] s u, (28) ξ ( c ) = [ c ] R ⎡⎢⎢⎢⎢⎢⎣ c ⎤⎥⎥⎥⎥⎥⎦ . (29)An example of the joint density of the couple (phase speed, horizontal particle velocity)at local maxima in space is shown in Figures 1-a and b for a JONSWAP spectrum. P b from p ( c, u ) By using Equation 25 applied to the dominant spectral wave band (that is, thatcontained in the interval [0 . f p , 1 . f p ], where f p is the peak wave frequency), the prob-ability of dominant wave breaking can be computed by integrating Equation 25 over allphase speeds and for orbital velocities over a threshold Ac , with A a constant that willbe in the next section: P b = ∫ u > Ac ∫ ∞ p ( c, u ) dcdu. (30) P b will be modelled following Equation 30 hereafter. Note that from the definitions inEquation 3, the proposed P b is defined as number of breaking local maxima over the to- –10–anuscript submitted to JGR: Oceans tal number of local maxima. From the analysis of p ( c, u ) we observed that spurious, non-moving local maxima may exist around c = 0 and u = 0; therefore, to avoid artificiallyincreasing P b , we adopted a practical integration range of c, u ∈ [ . , +∞] here. Notethat this range may, however, only be valid for very narrow spectra. Further, we drawattention that, following from Equation 1, our P b model is defined in space domain, whereasall the previous P b models and data are (at least partially) defined in time domain (seeAppendix A for details). For the very narrow spectral band used here, the differencesbetween temporal and spatial definitions of P b are negligible. This is discussed furtherin Section 5.Finally, the proposed model can be extended to accommodate two-dimensional spec-tra without changes on how p ( c, u ) is calculated. This is done by applying an appropri-ated spreading function to any given one-dimensional spectra (or directly inputting a di-rectional spectra) and by recalculating the moments in Equations 8 to take direction-ality into account or, more explicitly, m i = ∫ π ∫ ∞ ( f cos θ cos α + f sin θ sin α ) i E ( f, θ ) df dθ. (31)An example considering the simplified cosine spreading law ( D ( θ ) = cos ( θ − ¯ θ ) s ) with s = 20, ¯ θ = 0 and α =0 applied to same JONSWAP spectrum shown in Figure 1-a is shownin Figure 1-c. Note that the differences in p ( c, u ) between the one-dimensional (Figure1-b) and the two-dimensional (Figure 1-d) spectra are negligible for the present assump-tions. This relatively simple extension allows for the consideration of two-dimensionalwave spectral but we caution the reader that it may not be fully complete. A follow-uppublication will be dedicated to include and assess the effects of wave directionality inour method more rigorously. The previously introduced joint probability density distribution p ( c, u ) is based onGaussian theory and therefore assumes that waves are linear. Breaking waves are, how-ever, highly non-linear. For real non-linear waves, as detailed in the introduction, it iswidely accepted that wave breaking starts when the water particle horizontal velocityat its crest ( u nl ) reaches the wave phase speed ( c nl ). A non-linear wave breaking crite- –11–anuscript submitted to JGR: Oceans
Figure 1.
Example of the application of the method. a) JONSWAP spectrum for H m =15m, T p =10s and shape parameter γ js =10. b) Obtained joint probability density between the wavephase speed ( c ) and the horizontal particle velocity at wave crest ( u ) calculated using Equation25. Note that the joint probability density was computed using only the spectral energy between0 . f p and 1 . f p , that is, corresponding to the dominant wave band only. c) Directional spectrumfor the same parameters as in a) and directional spreading D ( θ ) = cos ( θ − ¯ θ ) s with s = 20 and ¯ θ = 0. d) Obtained p ( c, u ) considering only the spectral energy in the direction α = 0.–12–anuscript submitted to JGR: Oceans rion can be thus be defined as A nl = u nl / c nl = 1. Therefore, we assume that it is pos-sible to obtain an equivalent kinematic criterion, A lin = constant that relates Gaussianwaves to non-linear waves.Based on numerical experiments, Cokelet (1977) provided the potential and kineticenergy of a fully non-linear regular wave in deep-water at the onset of wave breaking (seethe last row of his Table A.0). Based on his results, we define the kinematic criterion asthe linear wave that has total energy equals to the nearly breaking non-linear regularwave computed by Cokelet (1977). Following Cokelet (1977), where k , g and ρ are ex-pressed as non-dimensional variables, a deep-water wave at the breaking onset (see lastrow of his table A.0) has kinetic energy T = 3 . × − and potential energy V = 3 . × − . The energy-equivalent linear wave (denote with subscript eq ) has, therefore, am-plitude: a eq = √ × E = √ × ( V + T ) = . . (32)For this particular case, the linear dispersion relation reads: ω = gk = , (33)the fluid velocity at crest of the energy-equivalent linear wave is: u eq = ωa eq = . , (34)and the phase speed of the linear wave is: c eq = √ gk = . (35)Given these constants, we obtain: A lin = u eq c eq = . = . . (36)Following this approach, we define the correction coefficient A = A lin = 0 .
382 thatwill be used as reference value hereafter for our tests. This result is consistent with re- –13–anuscript submitted to
JGR: Oceans cent findings from Ardag and Resio (2020) who reported from the re-analysis of Duncan’s(1981) experimental results, a wave breaking threshold between 0.75 and 1.02 (see theirFigure 1). Note, however, that these authors defined their wave breaking threshold as u / c g , where c g is the group velocity and u was obtained from linear wave theory. Replac-ing wave group velocity ( c g ) by the wave phase speed ( c ) yields a range of possible val-ues between 0.35 and 0.50, which is consistent with A lin .Figure 2 illustrates the sensitivity in wave breaking probability with changes in thewave breaking threshold A . For the given p ( c, u ) in Figure 2-a, letting A to vary from0 to 1 resulted in a exponential increase in P b at A ≤ . A = A lin =0.382 and letting the significant wave height ( H m )and wave peak period ( T p ) vary in the definition of the JONSWAP spectrum, the resultsindicate that steeper waves are more probable to break, which is expected (Figure 2-c).Finally, note that the wave breaking threshold A might be sensitive to other wave andatmospheric parameters such as wave directionality or direct wind forcing (or, equiva-lently, wave age). In the next sections, the accuracy of our model is assessed using fieldobservations and our results are compared with other parametric wave breaking formu-lations. Figure 2. a) Example of joint probability density between u and c obtained from Equation30. The colored lines indicate different values of A and the red dashed line shows A = A lin =0.382.b) Possible values of P b for varying A calculated using the joint PDF from a). The verticaldashed line shows A = A lin =0.382. c) Obtained P b for varying H m and T p and fixed A (0.382)and γ js (10). The dashed blue lines and marker indicate the H m and T p values used in a) andb). Note that as in Figure 1, these results only consider dominant waves, that is, they werecalculated from the spectrum between 0 . f p and 1 . f p .–14–anuscript submitted to JGR: Oceans
Three historical datasets were used to evaluate the present model. Further, six his-torical models (detailed in Appendix A) were chosen to contextualize our model in re-lation to the state-of-the-art. These historical models range from baseline models in whichthe only inputs are known environmental parameters (wind speed in Melville and Ma-tusov (2002) or wave steepness in Banner et al. (2000), for example) to fairly complexmodels that account for combinations of several phenomena (Romero (2019), for exam-ple).
The first data are from Thomson (2012) and Schwendeman et al. (2014), hereafterTSG14, and were collected in the Strait of Juan de Fuca, Washington. These data werecollected by a gray scale video camera with a resolution of 640 ×
480 pixels installed abovethe wheelhouse of Research Vessel R/V
Robertson which recorded at an acquisition rateof 30 Hz (Schwendeman et al., 2014). These data were then projected into a metric co-ordinate grid with resolution of 0.25m (cross wave) and 0.075m (along wave) using themethod proposed by Holland et al. (1997) and were then used to obtain Λ ( c ) using thespectral approach of Thomson and Jessup (2009). The data were collected in a (usually)fetch-limited region and for a young sea state; note, however, that the particular sea-statesanalyzed here may not be fetch-limited. Figure 3-a shows the measured wave spectra,Figure 3-b shows Λ ( c ) distributions, and Table 1 shows a summary of these data. Forthese data, P b was calculated using the measured Λ ( c ) distributions combined with themethod described below in Equation 37. Additional information regarding the data col-lection is available from Thomson (2012) and Schwendeman et al. (2014). The second dataset is from Sutherland and Melville (2013), hereafter SM13, andwas collected using the Research Platform R/P
FLIP during a two-day field campaignin the Southern California Bight under the scope of the SoCal 2010 experiment (Sutherland& Melville, 2013). Here, we focus only on the visible imagery collected by these authorsto keep consistency with the previously presented data. Stereo video data were collectedby a pair of video cameras mounted on the R/P FLIP for 10 minutes at the start of each –15–anuscript submitted to
JGR: Oceans hour and Λ ( c ) was obtained using a variation of the method of Kleiss and Melville (2011),that is, tracking the temporal evolution of breakers obtained via pixel intensity thresh-old. Figure 3-c shows the measured wave spectra, Figure 3-d shows Λ ( c ) distributions,and Table 1 shows a summary of these data. Note that because wave breaking was notobserved for frequencies below 0.2 Hz and from numerical simulations (not shown) thesewaves corresponded to a cross-swell not forced by the wind, our analyses only considerwaves in the frequency range 0 . < f < . Hz . Additional information regarding thedata collection is available from Sutherland and Melville (2013). For these and TSG14data, P b was calculated using the measured Λ ( c ) distributions combined with the for-mulas from Banner and Morison (2010): P b = ∫ c c c Λ ( c ) dc ∫ c c c Π ( c ) dc (37)where c = g π . f p , c = g π . f p , Π ( c ) = χg /( πc ) and χ = .
6. The implication ofthis choice is discussed in further detail in Section 5.
The third dataset is from Banner et al. (2000), hereafter B00, and was collectedin the Black Sea (BS), Lake Washington (LW) and the Southern Ocean (SO). These au-thors directly provide values for significant wave height H m , peak period ( T p ) and thewave breaking probability in their Tables 1 (Black Sea, denoted as BS here) and 2 (South-ern Ocean, denoted as SO here). The majority of the data were collected in the BlackSea (13 data runs) and two data runs are from the Southern Ocean. Given that the orig-inal spectral data were not published alongside their paper, we approximate the observedspectra using the provided pairs H m , T p assuming a JONSWAP shape with γ js = . –16–anuscript submitted to JGR: Oceans
Table 1.
Data summary for the two experiments described in Sections 3.1 and 3.2. Note thatthe parameters obtained from wave spectra were computed specifically for the bands shown inFigure 3 for TSG14 and SM13 cases. The wave height ( H p ) and wave steepness ( (cid:15) ) parametersfor dominant waves were calculated as per Banner et al. (2002) (see Section A1 for details). Thewave age parameter was calculated as c p / u ∗ . Dataset Date Length H m T p H p (cid:15) U u ∗ c p Wave age P b [−] [ min ] [ m ] [ s ] [ m ] [−] [ ms − ] [ ms − ] [ ms − ] [−] [−] TSG14 14/02/2011 20:33 6.5 0.75 2.88 0.66 0.160 11.50 0.373 4.50 12.07 3.54E-03TSG14 14/02/2011 20:58 5.1 0.75 2.96 0.66 0.152 12.55 0.417 4.62 11.08 9.57E-03TSG14 14/02/2011 21:30 6.5 0.91 2.99 0.82 0.184 15.07 0.561 4.67 8.33 6.29E-02TSG14 14/02/2011 21:44 8.5 1.09 3.17 1.00 0.200 15.73 0.599 4.94 8.25 1.01E-01TSG14 14/02/2011 22:29 6 1.21 3.44 1.09 0.186 17.24 0.636 5.36 8.44 1.51E-01TSG14 14/02/2011 22:37 4.8 1.37 3.53 1.24 0.199 18.01 0.660 5.52 8.36 7.61E-02TSG14 15/02/2011 19:04 10 0.87 3.29 0.79 0.146 14.45 0.360 5.13 14.28 3.75E-03TSG14 15/02/2011 19:19 6 0.90 3.31 0.81 0.149 13.11 0.477 5.17 10.85 4.05E-02SM13 06/12/2010 21:59 10 0.61 3.51 0.52 0.085 6.46 0.205 5.48 26.68 7.96E-03SM13 06/12/2010 23:00 10 0.61 3.33 0.54 0.097 7.55 0.342 5.20 15.22 1.95E-03SM13 07/12/2010 00:00 10 0.73 3.45 0.66 0.112 8.62 0.319 5.38 16.85 3.24E-03SM13 08/12/2010 00:00 10 0.34 2.04 0.23 0.110 5.24 0.160 3.19 19.96 1.65E-02B00 (SO) 10/6/1992 5 9.20 13.46 8.02 0.089 19.80 0.835 21.01 25.17 2.70E-02B00 (SO) 11/6/1992 9 4.20 12.04 3.66 0.051 16.00 0.626 18.78 30.02 0.00E+00B00 (BS) 1993 34-68 0.39 2.78 0.34 0.089 11.70 0.414 4.34 10.49 3.80E-02B00 (BS) 1993 34-68 0.49 2.94 0.43 0.100 12.70 0.461 4.59 9.96 6.50E-02B00(BS) 1993 34-68 0.53 3.33 0.47 0.084 14.00 0.524 5.20 9.93 6.00E-02B00 (BS) 1993 34-68 0.54 3.23 0.47 0.092 14.40 0.544 5.04 9.26 5.20E-02B00 (BS) 1993 34-68 0.38 2.27 0.34 0.131 15.00 0.574 3.55 6.18 6.30E-02B00 (BS) 1993 34-68 0.45 2.56 0.40 0.121 14.60 0.554 4.00 7.23 6.70E-02B00 (BS) 1993 34-68 0.45 2.44 0.40 0.134 13.70 0.509 3.81 7.49 8.40E-02B00 (BS) 1993 34-68 1.19 5.88 1.04 0.061 8.70 0.295 9.18 31.10 0.00E+00B00 (BS) 1993 34-68 1.32 6.24 1.15 0.060 11.20 0.391 9.74 24.91 0.00E+00B00 (BS) 1993 34-68 0.83 6.24 0.73 0.038 9.50 0.322 9.74 30.22 0.00E+00B00 (BS) 1993 34-68 0.89 5.88 0.78 0.045 10.70 0.368 9.18 24.91 0.00E+00B00 (BS) 1993 34-68 0.99 3.71 0.87 0.127 10.00 0.339 5.79 17.06 3.40E-02B00 (BS) 1993 34-68 0.88 4.00 0.77 0.097 8.70 0.295 6.24 21.14 5.80E-02 –17–anuscript submitted to
JGR: Oceans
Figure 3.
Field data. a) Spectral data from TSG14. b) Λ ( c ) data from STG14. c) Spectraldata from SM13. c) Λ ( c ) data from SM13. The coloured circular markers show in a) and c) showthe peak frequency ( f p ) and the coloured circular markers show in b) and d) show the peak wavespeed ( c p ). The red dashed line in b) and d) shows the theoretical c − decay predicted by Phillips(1985). In all plots, the color scale shows the wave age ( c p / u ∗ ).–18–anuscript submitted to JGR: Oceans
Figure 4 shows the comparison between estimated (or observed) (x-axis) and mod-elled (y-axis) values of P b for each model. In general, no model was able to closely re-produce the trends seen in the combined observed data, regardless of the underlying math-ematical or physical formalism. Furthermore, orders of magnitude of difference betweenthe models and, more worryingly, between the models and the measured data were ob-served. In general, models based on a wave steepness-derived wave breaking criterion (Banneret al. (2000), Banner et al. (2002), for example) overestimated data derived from Λ ( c ) while models based on Λ ( c ) (Melville and Matusov (2002) and Sutherland and Melville(2013), for example) underestimated P b data that was not derived from Λ (that is, B00data). The model from Filipot et al. (2010) was found to be the most consistent model.From Figure 4-g, the formulation presented in Section 2 with A = A lin = 0.382 under-estimated the observed P b for B00 and SM13 data (note that P b was too low to be dis-played on the plot) but performed relatively well for the majority of TSG14 data. Us-ing the mean absolute error (MAE) as a convenient metric to assess the models, it wasfound that the present model has errors in the same order of magnitude as the previ-ous models. Given the spread in the results seen in Figure 4, no model could be consid-ered a clear winner. For the discussion of these results, see Section 5. From the analysis of Figure 2, minor changes in A can lead to major variations in P b . Further, from the analysis of Figure 4, the proposed model underestimated P b for A = A lin = 0.382 particularly for S13 and B00 data. Given that it is a common prac-tice to optimize wave breaking models for particular datasets, we present two methodsto do so using TSG14 data as an example. The same could be done for B00 and SM13data but, for brevity, this is not done here. Given that the present model is not compu-tationally expensive, the first approach consisted of varying A from 0.1 to 0.5 in 0.001intervals and finding the value of A that resulted in the lowest squared error ( √( p db i − p mb i ) ,where the superscripts d and m indicate observed and modelled data, respectively) foreach data run. Figure 5-a shows the results of this procedure. The value A = A opt = 0.24was, on average, the optimal values of for this particular dataset. The second approach –19–anuscript submitted to JGR: Oceans
Figure 4.
Compassion between measured and computed P b for different models and data.a) Banner et al. (2000), b) Banner et al. (2002), c) Melville and Matusov (2002), d) Filipotet al. (2010), e) Sutherland and Melville (2013), f) Romero (2019), and g) present model with A = A lin =0.382. The thick black line shows the linear regression between measured and modelled P b and the blue dashed line indicates the one-to-one correspondence in all panels. Data pointswith modelled P b < − or observed P b = 0 are not shown in this plot. In all plots, r xy is Pear-son’s correlation coefficient and MAE indicates the mean absolute error. Note the logarithmicscale. –20–anuscript submitted to
JGR: Oceans consisted in parameterizing the optimal value of A for each data run as a function of aknown environmental variable, in this example, the waveage c p / u ∗ (Figure 5-b). The re-sults of these two approaches are show in Figures 5-c and d, respectively. Both approachesconsiderably improved the model results from the baseline model presented in Figure 4,with the parametric model (Figure 5-d) performing slightly better when considering Pear-son’s correlation coefficient ( r xy ) as a comparison metric. Figure 5.
Results of the optimization procedures. a) Optimization curves for each data record(coloured lines) and the global averaged (black line). The vertical dashed line show s A = A opt = 0.24. b) Parametrization of A as a function of c p / u ∗ . The blue swath indicates the 95% confi-dence interval. For this particular case, A = 0 . c p / u ∗ + .
16. Note the logarithmic scale in a),c) and d). In all plots, the color scale shows the wave age ( c p / u ∗ ). In b) to d) r xy is Pearson’scorrelation coefficient. We have introduced a new model for obtaining the probability of wave breaking( P b ) for dominant waves based on the theoretical joint probability density distributionbetween wave phase speed ( c ) and horizontal orbital velocity at the wave crest ( u ) for –21–anuscript submitted to JGR: Oceans unidirectional Gaussian wave fields. The present model has only one parameter for defin-ing the wave breaking threshold ( A ), which makes it relatively easy to optimize for a givendataset (as shown in Section 4.2). While the proposed model performed relatively wellfor one of the investigated datasets (TSG14), it greatly underestimated P b for the twoother datasets (SM13 and B00). For the data investigated here, such underestimationdid not result in a high mean absolute error (MAE) and, in fact, our model had one ofthe lowest MAE. Recent results of Barthelemy et al. (2018), Derakhti et al. (2020) andVaring et al. (2020) showed that waves with horizontal fluid velocity that exceeds 0.85times the phase velocity will inevitably break. These results suggest that the breakingthreshold derived from Cokelet (1977) in Section 2.3 could be reduced by ≈ A = . × . = .
324 which would helpto reduce the underestimation of P b , but not significantly. It is more probable that otherenvironmental phenomena such as direct wind forcing, directional spreading and longwave modulation, which are not accounted in our model, are the reason for such differ-ences.One of the most challenging aspects when assessing our model is, nevertheless, re-garding the field data. The attribution of wave breaking occurrences to wave scales us-ing timeseries analysis, as done in Banner et al. (2000) or Filipot et al. (2010), is diffi-cult because several wave scales can be present at the same time and space. This leadus to use Λ ( c ) observations as well as data from Banner et al. (2000) to investigate ourmodel. Different interpretations of how Λ ( c ) dc is computed from field data can, how-ever, generate orders of magnitude of difference in its moments (Gemmrich et al., 2013;Banner et al., 2014) and, consequently, in P b . Next, it is difficult to relate the speed ofthe wave breaking front to the phase speed of the carrying wave because small, slowerbreaking waves could merely be traveling on top of longer, much faster waves. In par-ticular, we believe that these wave breaking events can significantly contribute to the ob-served Λ ( c ) dc distribution as they would have c close to the peak wave phase speed. Thiswave breaking “sub-population” has not receive much research interest because of its ap-parent small contribution to energy dissipation but, for our particular case, they directlyimpact model validation.Further, relating Λ ( c ) to P b is also challenging. Here, we adopted the convenientformula from Banner and Morison (2010). While this formula has some support fromthe literature (Ardhuin et al., 2010), the actual functional form of Π ( c ) and the value –22–anuscript submitted to JGR: Oceans for the constant χ (see Equation 37) are unknown and changes in these will lead to changesin P b . The Gaussian framework developed in Section 2.1 provides an alternative methodto obtain Π ( c ) (from Equation 3, for example) but this is beyond the scope of this in-troductory paper and will be the focus of a future publication.Finally, we would like to re-emphasize that our model is derived in the space do-main whereas P b data is (at least partially) obtained in the time domain. For the nar-row spectral band investigated here, Monte-Carlo simulations of linear waves indicatethat the difference between P b modelled in space is less than five percent from P b mod-elled in time (not shown). Given all these complications and the fact that some histor-ical models are being compared to data that was used to create them (Banner et al. (2000)and Sutherland and Melville (2013), for example), we are unable to provide an accurateranking of the existing models. Future research should focus, therefore, on obtaining P b data that is unambiguous and widely available. In this regard, and despite its own lim-itations, wave tank experiments could bring further insight on the statistics of dominant(or not) breaking waves. Such a dataset would ultimately allow researchers to focus onmodels derived from physical and mathematical concepts (such as ours) rather than onempirical concepts. We have presented a new statistical wave breaking model derived from Gaussianfield theory that we have applied to obtain the probability of wave breaking for domi-nant, wind-sea waves. Although more mathematically complex than previous formula-tions, the present model relies on the ratio between the crest orbital velocity and the phasespeed and uses only on a single free parameter, the wave breaking threshold A . Usingtheoretical results obtained by Cokelet (1977) for regular nearly breaking waves, we de-rived a wave breaking threshold to adapt our linear model to non-linear waves. The presentmodel has errors in the same order of magnitude as six other historical models when as-sessed using three field datasets. For a particular dataset (TSG14), our model performedwell, especially if the free-parameter A is fine tuned. Additional observations are how-ever required, to further understanding and quantifying the dependence of A on envi-ronmental parameters that are not accounted for in our model (for example, wind forc-ing, wave directionality or modulation by long waves). Future research should be ded-icated to collect more wave breaking observations in different and repeatable environ- –23–anuscript submitted to JGR: Oceans mental conditions to provide reliable constraints for the optimization of the present andother wave breaking models. Still and although the research presented here is in earlystages, the present model should be extendable to waves of any scale and, therefore, hasthe potential to be implemented in current state-of-the-art spectral wave models as a newwave breaking dissipation source term with relatively little effort.
Appendix A Historic Parametric Wave Breaking Models
A1 Banner et al. (2000)
Banner et al.’s (2000) is a popular model for calculating wave breaking probabil-ities for deep water, dominant waves. This model follows from observations and resultsfrom Donelan et al. (1972), Holthuijsen and Herbers (1986) and Banner and Tian (1998)who demonstrated the importance of the wave group modulation on the wave breakingonset. These authors conveniently obtained a parameterization for the probability of wavebreaking ( P b ) based solely on the spectral steepness of the dominant wave scale ( (cid:15) p ), as-suming that their formulas would capture the influence of the wave group modulationon the wave breaking onset. Their formulation was derived using a dataset of measure-ments collected in various environments ranging from lakes to open ocean conditions (Banneret al., 2000). From these observations, these authors were then able to obtain a wave break-ing threshold behaviour for the dominant waves as a function of the dominant spectralwave steepness given by: (cid:15) p = H p k p k p is the wavenumber at peak frequency ( f p ) and H p is the significant wave heightof the dominant waves calculated as: H p = ¿``(cid:192)(∫ . f p . f p E ( f ) df ) (A2)where E ( f ) is the spectra of wave heights as a function of frequency. For their data, P b was then parameterized as a single equation with three free parameters ( p , p , p ): P b = p + ( (cid:15) p − p ) p , (A3) –24–anuscript submitted to JGR: Oceans
For the available field data, Banner et al. (2000) found optimal values of p = p = . p = .
01. Note that hereafter free parameters for the different models willbe denoted as p n where n is a sequential number. A2 Banner et al. (2002)
This work extended Banner et al. (2000) model to shorter wave scales (up to 2.48times the peak wave frequency). From field data Banner et al. (2002) reported that thewaves were breaking if the saturation spectrum σ ( f ) = π f E ( f )/ g = σ ( k ) = k E ( k ) exceeded a threshold that was frequency dependent. These author’s related this depen-dence to the directional spreading θ ( k ) which later led Banner and Morison (2010) toexplicitly define the following empirical formulation: P b ( k c ) = H h ( ˜ σ ( k c ) − p ) × p × ( ˜ σ ( k c ) − ˜ σ t ) , (A4)in which H h is the Heaviside step function, k c is the central wavenumber for a given wavenum-ber range, ˜ σ ( k c ) = σ ( k c )/ θ ( k c ) is the saturation spectrum normalized by the averageddirectional spreading, p = . p = 33 are constants obtained from their ob-servations. Following Banner et al. (2002), the directional spreading angle is calculatedaccording to Hwang et al. (2000) (their equation 19a): θ ( kk p ) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ . + . ( − kk p ) if kk p < . . + . ( kk p − ) if 1 . ≤ kk p < θ is the directional spreading angle as a function of the wavenumber. A3 Filipot et al. (2010)
This method follows from the original works of Le M´ehaut´e (1962), Battjes and Janssen(1978) and Thornton and Guza (1983) and assumes that the probability distribution func-tion (PDF) of breaking wave heights in the dominant wave scale is parameterized by itscentral frequency f c or, equivalently, by its representative phase speed c ( f c ) and the prod-uct between a Rayleigh PDF for the wave heights –25–anuscript submitted to JGR: Oceans P ( H, f c ) = HH rms ( f c ) exp ⎡⎢⎢⎢⎢⎣− ( HH rms ( f c ) ) ⎤⎥⎥⎥⎥⎦ (A6)in which H r ( f c ) = √ √∫ ∞ U fc ( f ) E ( f ) df (A7)and U f c = . − . ( πδ [ ff c − − δ ]) (A8)where δ is the bandwidth of a Hann window (in this study, δ = . W ( H, f c ) = p [ β r β ] { − exp [− ( β ˜ β ) p ]} (A9)in which β = kH / tanh ( kh ) , and p and p are free parameters. In order to extend theformulation outside the shallow water domain, these authors replaced Thornton and Guza’s(1983) breaking criterion based on the wave height ( H ) to water depth ( h ) ratio ( γ = H / h = .
42) with an adaptation of Miche’s (1944) wave breaking parameter: β r = k r ( f c ) H r ( f c ) tanh ( k r ( f c ) h ) (A10)in which k r ( f c ) = ∫ ∞ U f c ( f ) k ( f ) E ( f ) df ∫ ∞ U f c ( f ) E ( f ) df (A11)and ˜ β = b ( b tanh ( kh ) − b tanh ( kh ) + b tanh ( kh ) − b ) (A12)in which b = . b = . b = . b = . b = . β was obtained via numerical calculations of regular nearly break- –26–anuscript submitted to JGR: Oceans ing waves using the stream wave theory of Dean (1965). Finally, the wave breaking prob-ability is obtained as: P b ( f c ) = ∫ ∞ P ( H, f c ) W ( H, f c ) dH ≤ . (A13)To keep consistency with Section A2, P b will be only considered at the spectral peak;other definitions are, however, also possible. A4 Models based on Phillips’ (1985) Λ ( c ) The major issue with the previous models is the difficulty to obtain reliable obser-vations of the wave breaking probabilities as a spectral distribution solely from point mea-surements. Due to the presence of different wave scales at the time and location, it isindeed difficult to assign the breaking occurrence to a given wave frequency of wave num-ber. To avoid this problem, Phillips (1985) proposed to use the speed of the breakingfront as a proxy for the phase speed of the carrying wave. Phillips (1985) defined the pa-rameter Λ ( c ) dc as the “average total length per unit surface area of breaking fronts thathave velocities in the range c to c + dc ” and then defined the following quantities: L = ∫ Λ ( c ) dc (A14)and R = ∫ c Λ ( c ) dc (A15)which represent the “total length of breaking fronts per unit area” (Equation A14) and“the total number of breaking waves of all scales passing a given point per unit time”(Equation A15). Assuming that Phillips (1985) assumptions hold, it is possible to ob-tain parametric models for Λ from known variables (e.g., wind speed) and, consequently,for P b (see Equation 37). A41 Melville and Matusov (2002)
Melville and Matusov’s (2002) model for Λ ( c ) relies only on the wind speed mea-sured at 10m ( U ) to obtain Λ ( c ) . Following Melville and Matusov (2002) and using –27–anuscript submitted to JGR: Oceans the explicit formula given by Reul and Chapron (2003), this parameterization is writ-ten as: Λ ( c ) = p [ U ] − exp [−( p c )] (A16)in which p and p are constants. For their data, Melville and Matusov (2002) found p = . p = .
64. As discussed by Reul and Chapron (2003), this formulation approachesPhillips’s (1985) theoretical c − but may overly estimates the amount of small breakers. A42 Sutherland and Melville (2013)
Sutherland and Melville (2013) used dimensional analysis to scale Λ ( c ) and obtaina parameterization that is a function of the wind drag ( u ∗ ), peak wave phase speed ( c p ),significant wave height ( H s ) and three constants. From Sutherland and Melville’s (2013)Equation 9 and their Figure 4, Λ ( c ) is calculated as:Λ ( c ) = p gc p ( u ∗ c p ) p ( c √ gH s ( gH s c p ) p ) − (A17)where p = . p = .
5, and p = . c − frequency dependency but does nothave the typical roll-off at low c as these authors chose to use infrared (other than vis-ible) imagery to obtain and model their Λ ( c ) . This choice included the contribution ofmicro-scale breakers that do generate visible bubbles in their model, hence the difference. A43 Romero (2019)
Recently, Romero (2019) developed and implemented a new wave breaking param-eterization in WaveWatchIII which relies exclusively on Λ ( c ) . Differently from previousparameterizations, Romero’s (2019) takes into account both the modulations due to windsand long waves on Λ ( c ) . His model is fairly general but depends on six free parametersthat needed to be laboriously obtained by comparing WaveWatchIII’s significant waveheight outputs with available measured significant wave heights from buoy data. In Romero’s(2019) model, Λ was modelled assuming that it is proportional to the crest lengths ex-ceeding a slope threshold: –28–anuscript submitted to JGR: Oceans Λ ( f, θ ) = ( ( π ) p g ) f exp [− ( p B ( f, θ ) )] M LW M W (A18)where p = . x − and p = x − are constants to be obtained from the data, M LW is the modulation due to long waves, M W is the modulation due to winds and B ( f, θ ) is the directional wave breaking saturation spectra: B ( f ) = ∫ π B ( f, θ ) dθ = E ( f ) ( πf g ) . (A19)The modulation due to long waves is calculated according to Guimar˜aes (2018): M LW = [ + p √ cmss ( E ( f )) cos ( θ − ˆ θ )] p (A20)where p =
400 and p = / = ∫ ∞ E ( f ) ( ( π ) f g ) df. (A21)and ˆ θ = tan ( ∫ E ( f, θ ) sin ( θ ) df dθ ∫ E ( f, θ ) cos ( θ ) df dθ ) (A22)The modulation due to the wind is computed as: M W = ( + p max ( , ff ))( + p ) (A23)with f = p u ∗ g π (A24)where p = . p = /
28 is yet anotherconstant. Finally, the conversion from Λ ( f ) to Λ ( c ) is done using the relation Λ ( c ) dc = Λ ( f ) df and the linear dispersion relation (see Romero’s (2019) Eqs. 17-23 for details). –29–anuscript submitted to JGR: Oceans
Acknowledgments
This work benefited from France Energies Marines and State financing managedby the National Research Agency under the Investments for the Future program bear-ing the reference numbers ANR-10-IED-0006-14 and ANR-10-IEED-0006-26 for the projectsDiME and CARAVELE. The authors’ thank Peter Sutherland and Jim Thompson forkindly sharing their data.
Data Availability
All data used in this publication has been previously published by Banner et al.(2000), Sutherland and Melville (2013), Schwendeman et al. (2014).
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