Progression of Wave Breaker Types on a Plane Impermeable Slope, Depending on Experimental Design

Plane slopes are the most common type of coastal protection because of their ability to transform incident wave energy by reflection, transmission, and dissipation. They can also be designed and built with different angles, materials, and sizes. Their hydrodynamic performance depends on the kinematic and dynamic regimes that develop while the wave train interacts with the slope. Those regimes are directly related to the type of wave breaker (Battjes, 1974; Díaz-Carrasco et al., 2020).

, where X is a dimensionless variable (output) for many overall properties of the breaking waves and  is a surf similarity parameter, defined by: Recently, Derakhti et al. (2020) investigated the framework for predicting the wave breaking onset for surface gravity waves in an arbitrary water depth. They numerically calculated the progression of regular waves over a plane impermeable beach with different slopes (1:5, 1:10, 1:20, 1:40, 1:100, and 1:200). Except for the slope, 1:200, the relative water depth (h = 0.5 m) at the toe of the slope and the wave period (T w = 4.0 s) were constant;  / 0.0557 h L was also constant. On the 1:200 beach slope, the water depth was h = 0.3 m, and  / 0.044 h L . By increasing  0 , Derakhti et al. (2020) (their Figures 3a, 3c, 3e, 3g, and 3i), found that there was a progression/transition from spilling to collapsing and surging. In their study, given that  0 was defined by using the wave steepness 0 0 / H L in deep water, it thus followed that    0 .
It should be highlighted that there are different locations where the incident wave characteristics (experimental input) are specified. In Galvin (1968), these were the "hypothetical" deep water wave steepness (offshore parameter) and breaker steepness parameter 2 / b H gT , known as the "inshore parameter", based on quantities measured along the breaker line. Furthermore, in Figures 5 and 6, the classifying parameters used were the offshore parameter divided by the square of the beach slope, and the inshore parameter divided by the beach slope, respectively. Battjes (1974) used the wave height at the toe of the slope and the deep water wave length, whereas Derakhti et al. (2020) used the deep water wave steepness. In this study, following Baldock and Torres-Freyermuth (2020), Díaz-Carrasco et al. (2020), Hughes (2004), and Moragues et al. (2020), both the incident wave height, H, and the wave length, L, of the regular train is specified at the toe of the slope, where the water depth, h, is constant.
Based on those distinctions, Moragues et al. (2020) extended Galvin's classification to six wave breaker types: spilling (1-Sp), weak plunging (2-WPl), strong plunging (3-SPl), strong bore (4-SB), weak bore (5-WB), and surging (6-S). Importantly, the collapsing breaker is split into the strong and weak bore. Galvin´s well developed plunging and plunging are changed to strong and weak plunging, respectively, and his ID numbers are also renamed. Furthermore, they provide a brief description of the transition between consecutive breaker types. This paper applies the method of dimensional analysis to reduce the number of experimental variables affecting the wave-slope interaction and to improve the scalability of the results. The main objective of this study was to analyze the progression of the extended breaker types on plane impermeable slopes, depending on the experimental design defined by the slope angle and the experimental space defined by the incident wave characteristics, such as relative water depth and wave steepness,   / , / I h L H L , as derived from the dimensional analysis (see Appendix A). This research is a natural continuation and logical extension of Battjes (1974), Díaz-Carrasco et al. (2020), Galvin (1968), and Moragues et al. (2020), inter alia. A breaker type classification may eventually be useful in coastal protection design against coastal risks such as flooding (Del-Rosal-Salido et al., 2019), the coastal hazard from extreme storms , human interaction with large-scale coastal morphological evolution (Baquerizo & Losada, 2008) and uncertainty in the assessment of coastline changes, Kroon et al. (2020), among others.
The rest of this paper is organized as follows. Section 2 verifies Battjes' assumption on the importance of the relative water depth at the structure. The physical experiments of Galvin (1968) and recent numerical results of Derakhti et al. (2020) were used for this purpose. Section 3 presents the experimental design along with the physical (CIAO) and numerical experiments (IH-2VOF) conducted on an impermeable slope 1:10. As explained in Section 4, the results of these experiments were used to delimit the approximate regions of the experimental space defined by   (log / ,log( / )) I H L h L , where each of the six breaker types most frequently occurs. The extended breaker type classification in Moragues et al. (2020) is used. Section 5 shows the link between the experimental technique and the expected breaker types. Section 6 examines the implications of the transitional intervals of the breaker type classification for its applicability, based on the continuous hypothesis of the breaker progression. Next, in Section 7, the need to define the characteristics of the incident train at the toe of the slope and the approximation of the non-dimensional energy dissipation on the slope, considering the wave-reflected energy flux and the wave breaker type is discussed. Finally, the conclusions derived from this research are presented in Section 8. Appendix A revises the dimensional analysis for the experimental characterization of a regular wave train impinging on a plane impermeable slope. Appendix B describes the tests carried out by (Galvin, 1968) and (Derakhti et al., 2020) and specifies numerical data pertaining to the IH-2VOF model (Lara et al., 2008). It also includes information regarding the physical tests carried out in the CIAO wave flume (Addona et al., 2018;Andersen et al., 2016;Lira-Loarca et al., 2019;Moragues et al., 2020).

Verifying Battjes's Assumptions
By applying the method of dimensional analysis (see Appendix A) for a given slope angle, it was possible to obtain a functional relationship between the flow characteristics on a plane impermeable slope and the non-dimensional variables, / h L and / I H L. The π−theorem does not provide the form of the functional relationship. This form can only be obtained by physical or numerical experimentation or by theoretically solving the problem (Sonin, 2001). Based on experimental evidence that flow characteristics and breaker types are two facets of the same process, in contrast to Battjes (1974), it was expected that for each slope, , there would be a functional relationship between the wave breaker types and the non-dimensional variables, / h L and / I H L, determined at the toe of the structure.
Galvin's experimental data pertain to three slopes: (a) m = 1:5; (b) m = 1:10; and (c) m = 1:20. For each one, the stroke, w S , of the generator and the wave period, T varied. Following linear theory (Dean & Dalrymple, 1991), the pair of values   , I H L at the toe of the structure were computed. Figures 1a-1c show Galvin's experimental data for the three slopes and the reported wave breaker type. The experimental space for each slope is approximately defined by: where A and B are the parameters of a straight line fitted to experimental points / h L and / I H L. It should be highlighted that the approximated linear relationship between the value pairs of the log-transform / h L and / H L determines a very narrow experimental input. The use of the log-transform should help to capture the variability of the experimental output (observed dominant breaker types) under very small changes of the experimental input.

 
log m ] under regular waves, (see their Table 1 "Input parameters for the simulated cases" and the output breaker types in their Figure 3). The value of the incident wave height I H and of the wave period at the toe of the structure is assumed to be approximately equal to w H , the wave height, and wave period, w T ≈ 4 s, at the wavemaker. Regardless of the value of  0 , (equal to the inverse of the square root of Galvin`s offshore parameter), the values of / , the breaker type evolves with the values of m and the chosen value of h/L. Therefore, this dataset for regular waves only confirms that once a pair of characteristic wave values at the toe of the slope is selected and as m progressively decreases, the six breaker types of the classification can be observed.
In summary, the dominant breaker types and their progression/gradation depend on three quantities: (i) the characteristics of the incident waves at the foot of the slope, / h L and / I H L; and (ii) the slope, m. Obviously, these three can be used to construct a 3D graph, where the prevalence regions of the different breaker types can be observed. However, the interplay of the three quantities is most accurately and usefully represented by graphic representations in a 2D system in terms of value pairs (taken two by two) while the other quantity remains constant.   (Galvin, 1968), for three impermeable smooth slopes: (a) 1:5; (b) 1:10; and (c) 1:20. Breaker types as classified in Table 3 by Galvin (1968). WD plunging and plunging ARW identify well developed plunging and plunging altered by a reflected wave, respectively. For each slope, breaker types evolve with the pair of values, H I /L, and h/L; The purple line represents the linear fit to the experimental data (Equation 3).

Classification of the Wave Breaker Types and Their Expected Variability
Any classification of the breaker types is somewhat subjective and clearly dependent on the observation technique and the experience of the observer. In general, as in Galvin (1968), the experimental information is an ID and a breaker type, based on an overall description of the profile transformation on the slope with emphasis on the steepening and overturning of the wave front and its spatial evolution on the slope. This research used Galvin's classification as extended by Moragues et al. (2020). Table 1 contains both classifications of the breaker types.
Furthermore, seawards of the slope in the water of constant depth, in the transition from a non-breaking wave to a visible breaking process at the crest (Derakhti et al., 2020) might be adopted once the maximum wave steepness at the toe of the slope fulfills Miche's criterion.
As can be observed, this criterion is a crude simplification of the concept of wave breaking, as analyzed in depth by Derakhti et al. (2020). On the other hand, the transition from surging breaker to standing wave (full reflection) might be adopted once overturning at the toe (leading edge) of the wave has ceased to occur (Derakhti et al., 2020) and/or the wave tip on the slope is no longer turbulent ( Spilling (Sp): The wave volute begins, but disappears in turbulence before it impacts the slope or the wave. The jet evolves at the crest of the wave, above the mean level.
Well-developed plunging: Crest curls over a large air pocket forming a volute.
Smooth splash-up usually follows. (ID 2) Weak plunging (WPl): The wave volute appears and impacts on the wave itself, around the mean level, generating a roller that propagates with the wave.
Plunging: The volute is smaller than in 2. (ID 3) Strong plunging (SPl): The wave volute appears impacting the slope, hitting it and bouncing back. There is a lot of splashed water and the wave loses a lot of energy. The development of a bump on the leeside of the wave causes the strong jet.
Plunging altered by reflected wave: Small waves reflected from the preceding wave peak up the breaking crest. Breaking otherwise unaffected. (ID 6) Strong bore (SB): There is an attempt to plunge, but, before it can finish the plunge, the front collapses, generating an inclined plane, mixing water and air bubbles generating a lot of turbulence. The water surface behind the crest is almost plane.
Collapsing: Breaking occurs over lower half of wave. There is no air pocket and no splash-up. There are bubbles and foam. (ID 4) Weak bore (WB): The inclined plane becomes more vertical, becomes unbalanced, and collapses in the middle or bottom of the water column without volute.
Surging: Wave slides up the slope with little or no bubble production. The water surface remains almost plane except on the beach face during runback. (ID 5) Surging (S): The wave trains oscillate (like a standing wave), generating no turbulence in the profile. The period of the water rising and falling along the slope is considerably larger than the wave period.

Physical (CIAO) and Numerical (IH-2VOF) Tests and Experimental Design
To determine the dependence of the potential breaker types observed in the wave generation devices, and to delimit the prevalence region of each of the six types, a new set of physical and numerical tests was carried out on a smooth impermeable slope, 1:10.
Numerical tests were performed in the IH-2VOF model (Lara et al., 2008). In this research, a new numerical experimental set-up (ID code IH-2VOF) was implemented. See Moragues et al. (2020) for the set-up and the details of the physical experiments in the Atmosphere-Ocean Interaction Flume (CIAO) wave flume (ID code CIAO). Photographs and video cameras were used to record the breaker types.
Regular wave trains were simulated with a combination of H and T to cover as much area as possible inside the experimental space. To compare physical and numerical results, some tests were run with the same input as in the CIAO flume. For more information about the numerical and physical tests, see Appendix B and supporting information.
The conventional experimental technique (Galvin, 1968;Van Der Meer, 1988) involves building a ramp with a fixed slope, and running sets of tests with a fixed wave period, while progressively increasing the wave height. Since this study involved a ramp with a constant slope (e.g., 1 : 10 m ), the breaker types depended on the combination of / h L and / I H L values, Figure 3. The pairs of values were thus selected so as to be able to observe the largest possible number of breaker types and their progression, and also to plot them, according to the wave characteristics and the alternate similarity parameter,  (Equation 4). Díaz-Carrasco et al., (2020) showed that this provides an accurate description of the interplay between the characteristic wave pairs ( / h L, / I H L) in the wave energy transformation processes on a given slope.

Observed Predicted Breaker Types and Their Expected Variability
The extended classification of six breaker types  is mainly based on the work by Kirby (1995, 1996) for spilling and weak and strong plunging, Lakehal and Liovic (2011)   Galvin (1968) and Derakhti et al. (2020) for spilling, plunging, collapsing and surging. The numerical and physical model by the authors completes the existing database. It is obvious that the adscription of observation to a breaker type has a subjective component, in particular in the case of the transitions of breaker types To minimize the subjectivity, in addition to the classical descriptions reflected in Figure 11 of Galvin (1968) and Figure 3 of Derakhti et al. (2020), some distinctive features of each breaker types are used: • The jet of a weak plunging hits the front slope of the wave around the mean water level, while in the spilling breaker the jet evolves locally at the crest and above the mean water level. • The jet of a strong plunging breaker hits the water surface or the return flow landwards of the front slope. The existence of a strong jet is related to the development of a hump on the leeside of the crest. This increases the pressure and strengthens the water circulation from the bottom toward the jet, see Figure 1 of Galvin (1968) and Figure 9 of Lakehal and Liovic (2011). Straight "furrows" moving parallel to the crest, and behind it, identifying the sequence of developed vortices with horizontal axes. • This hump is not observed in a strong bore. The water surface behind the crest is "almost" plane and collapses at different parts of the front above the bottom, see Figure 5 of Zhang and Liu (2008). • A weak bore develops a turbulent front face and at the toe, a slip thin up-rush layer with a turbulent front, see  Figure 4a shows photographs of the six breaker types included in the extended classification. They are similar but not exactly the same as those in Moragues et al. (2020). It is not unusual for one observer to associate the same breaker type in two wave trains, whose progression shows certain differences, and for another observer to consider them to be two different breaker types. Figure 4b shows the photographs of two breaking waves both classified as strong plunging (SPl). The first one is more similar to (or shares characteristics with) a strong bore (SB), whereas the second one is closer to a weak plunging (WPl) breaker. Indeed, the volume of each wave impacts at different points, such as the front slope or the toe of the wave, and both waves show a "hump" on the sea side of the wave crest (a distinct feature of plunging breakers), although of different dimensions. Nevertheless, both breaker types are classified as SPl. Figure 5 shows the location of the photographs in the experimental space.

Prevalence and Progression of Breaker Types
One cause of these discrepancies, as discussed further on, is that the classification is based on "transitional intervals" of the breaker types, whereas observed reality progresses more gradually. Moreover, thanks to the improvement in visual techniques and advanced numerical codes, it is now possible to obtain very detailed information regarding the progression of the kinematics and dynamics of the breaking process, small-scale behaviors, flow features, and the fluid-air interaction (Derakhti et al., 2020;Zhang & Liu, 2008). For this reason, discrepancies regarding interval classification, visual wave channel observation, and the quasi-continuous description of the numerical models will doubtlessly increase.

Regions of Prevalent Breaker Type
Based on previous work (Díaz-Carrasco et al., 2020;Moragues et al., 2020) and after the visualization of a large number of photographs, videos, and numerical results, green strips were plotted in the experimental space of the test performed in the CIAO wave flume and with the numerical model IH-2VOF on an impermeable 1:10 slope, Figure 6. Each strip signals the border between contiguous breaker types. They mark the boundaries of the regions in the experimental space where each of the six breaker types is prevalent. Their width merely indicates that at the moment, with the current data set and the intrinsic variability of breaker types, it is difficult to be more precise. It should be highlighted that the strips follow straight lines of the   Figure 7 shows one of the common experimental techniques used in many laboratories for testing hydrodynamic performance on sloping structures. It involves maintaining the Iribarren number or wave steepness constant (in this case, for the same 1:10 slope) in the coordinate sys- Figure 7a shows three vertical lines fitted to three experimental datasets, each with a different wave steepness value. For the same three datasets, Figure 7b shows the corresponding breaker types observed versus the log-transform of the alternate similarity parameter,  (Equation 4). For each set impinging on a 1:10 slope, only two or three breaker types can be observed.

The Link Between Experimental Technique, Data Analysis, and Progression of Breaker Types
The white diamonds represent the breaker types observed on some of the sides that delimit the parallelogram of the sample space. The three lines fitted to the three sets of points in Figure 7b correspond to the three vertical lines in Figure 7a, which link the observations to a constant / I H L. It should be highlighted that in Figure  . Table 2 shows how the Iribarren number alone is not able to predict the type of breaking as it does not take into account the influence of relative depth. However, the  parameter takes into account both variables for a given slope angle.  Figure 8b shows that for each set impinging on a 1:10 slope at a given relative water depth, three or four breaker types can be observed. As shown in Figure Figure 9a shows three lines fitted to three experimental datasets within the parallelogram. For the three datasets, Figure 9b shows the corresponding breaker types versus the log-transform of the alternate similarity parameter, ). For oblique lines with a positive slope (purple line) and a shallow or intermediate water depth, all of the breaker types were observed. However, as the relative water depth increased (blue line), some of the breaker types did not occur. Moreover, for the oblique lines with a negative slope (green line) following a green strip, only one breaker type (strong bore) was observed. As shown in Figure 9a, points with the same breaker type (e.g., SB) are located within their region, whereas in Figure  For purposes of comparison, Figure 10 shows, for the set of experiments conducted by Galvin (1968) over a 1:10 slope (Figure 1b), the corresponding breaker types versus the log-transform of the alternate similarity parameter  (Equation 4). The breaker types observed by Galvin (1968) Figure 7, From top to bottom given slope depends on the experimental design and technique, and depends on the interplay of the wave characteristics as described by the alternate similarity parameter,  .

Table 2 Values of the Iribarren Number, Iribarren Number for Deep Water Conditions, Wave Height, Wavelength, Breaker Types (BT), the Alternate Similarity Parameter, Wave Steepness and Relative Water Depth, for the Middle Fitted Line in
The link between experimental design and technique, data analysis, and progression of breaker types, is also present when the values of m and / I H L are combined and / h L remain constant. Thus, for a given relative water depth, the progression of breaker types depends on the interplay of the slope and wave steepness, see Figure 2 with the data by Derakhti et al. (2020). Battjes (1974)   Regardless of the fact that the observed dataset is obtained by varying the slope two orders of magnitude, the fitted straight line connecting the data with   0.001, describes the progression of breaker types as a continuous process, depending on log(m). Because those data all have the same relative water depth value, the progression of breaker types depends on the interplay of the slope and wave steepness.

The Transitional Intervals and the Continuous Hypothesis of Breaker Type Progression
A careful analysis of the photographs (Figure 4b)  , approaching a weak plunging breaker. However, once the observation is ascribed to a breaker type, its vertical position is determined, and all the observations ascribed to the same breaker type, are located on the corresponding horizontal line. The value of    log determines their ubication on the x-axes and furthermore the separation between them. The resulting plot is the consequence that the classification is based on "transitional intervals" of the breaker types, whereas observed reality progresses more gradually.
The straight lines fitted to the observed breaker types (i.e., Figures 7b, 8b, and 9b) are assumed to be the geometric locus of the continuous process triggered by the corresponding set of experimental input (i.e., Figures 7a, 8a, and 9a). Therefore, to fulfill the continuous hypothesis, the observed breaker types of each MORAGUES AND LOSADA 10.1029/2021JC017211 11 of 21    Figure 1b). For comparison, the fitted straight line (see Figure 9b) to the CIAO data is included.

Discussion
This paper presents a physical and numerical data set of observed wave breaker types over a plane impermeable slope under a regular wave. These data are used to determine the progression of breaker types under different relationships of / h L and / I H L at the toe of the slope. This study derives the non-dimensional energy dissipation on the slope, considering the wave reflected energy flux on the slope and the breaker type. This energy dissipation is parametrized following Duncan (1981) and Martins et al. (2018), but expressing C, g C , I H and T at the toe of the slope.

Specification of the Input Wave Characteristic at the Toe of the Slope
The characteristics of the incident wave train of experiments by Galvin (1968) and Derakhti et al. (2020) were not given at the toe of the slope. To uniquely specify any of the tests, Galvin chose the following four experimental variables: beach slope, depth at the toe of the beach, generator stroke, and period. For this research, the generated incident wave height was estimated by applying the linear theory of wave generation (Dean & Dalrymple, 1991). In addition, it was assumed that the estimated wave height near the paddle was representative of the incident wave height at the toe of the depth. This implies that the generation of the incident wave was not affected by the reflected wave train, and also that the energy dissipation down at the bed and the side walls of the flume was negligible. Table 1 of Derakhti et al. (2020) provides the input parameters, namely, the slope and wave height and wave period of the regular wave train at the wavemaker, the distance of the wavemaker to the toe of the slope L 1 , and the surf similarity parameter in deep water. For all the simulated cases, P1-r-LV to P6-r-LV, it was considered that L 1 = 0. The LES/VOF model specifies the total instantaneous free surface and the liquid velocity at the model upstream boundary. It was then assumed that the incident wave height and wave period at the toe of the structure were approximately equal to their respective values at the wavemaker. Moreover, the output breaker type was taken from Figure 3. Note that this model neglects the interfacial surface tension and viscous stress. Moreover, the subgrid-scale (SGS) stress is estimated using an eddy viscosity assumption and the Dynamic Smagorinsky model, which includes water/bubble interaction effects.
MORAGUES AND LOSADA 10.1029/2021JC017211 13 of 21  The experimental space, depicted in Figures 1 and 2 (Galvin, 1968), and constant / h L (Derakhti et al., 2020). Independently of the rough method applied to obtain the wave characteristics at the toe of the slope, the breaker types observed by Galvin (1968) and by Derakhti et al. (2020) follow the derived functional relationship.

Estimated Energy Dissipation for the Observed Wave Breaker Types
The wave energy conservation equation is formulated in a control volume, (CV), extending landwards from the toe of the slope to the shoreline: the spatial variation in the CV of the time-averaged wave energy fluxes is equal to the amount of wave-averaged energy dissipated per unit area, where, , (J/(ms)), is the time-averaged wave energy flux per unit surface at the toe of the slope, toe x , and the subindexes I and R denote the incident and the reflective wave train, respectively. The fluxes in and out at the shoreline, top x , are assumed to be zero;   cot l h is the horizontal length of the CV: and h is the water depth at the toe of the slope. Note that all the tests were carried out with the slope  1 : 10 m . Only the energy of the most progressive incident and reflective wave trains were considered. The equation can be expressed in non-dimensional quantities, where 2 R K is the module of the reflection coefficient for the most progressive mode in the wave train, defined as the quotient of the reflected and incident wave energy fluxes per unit surface area, and  d is the non-dimensional energy dissipation rate, Similarly to the wave breaker type analysis, it is hypothesized that    CV D can be described as a function of the characteristics of the incident wave train , I h H L L at the toe of the slope. Thus,following Duncan (1981) and Martins et al. (2018), but expressing C, g C , I H and T at the toe of the slope, where S B is a bulk dissipation coefficient. Figure 13 shows the non-dimensional bulk coefficient, S B versus    log . Depending on the wave breaker types, two regions can be identified. For small values of l   og 6, the reflected wave energy flux modulates the wave evolution over the slope. B s increase slightly as the breaker type progresses from surging to weak and strong bore. For large values of , the dissipation by breaking prevails over the reflected energy flux. S B increases its value as the breaker type progresses from strong bore to strong and weak plunging. Unfortunately, there are not enough numerical data for spilling Two lines fitted to the data points show the relation between the coefficient and  .
breakers but despite this, the slope of the straight line is not expected to change. The equation for  d can be modified to make explicit the dependence of the wave period, wave steepness, and relative water depth at the toe of the slope.

Conclusions
The main objective of this research was to analyze the progression of breaker types on a smooth impermeable slope. This study used dimensional analysis to demonstrate that relative water depth is a key explanatory quantity and that its omission very likely results in an incorrect dimensional analysis model. New physical and numerical experiments conducted on an impermeable 1:10 slope were conducted to approximately delimit the regions of the experimental space where each of the six breaker types, as described by Moragues et al. (2020), occurred most frequently. Based on the analysis of four datasets, Galvin (1968), Derakhti et al. (2020), and the present physical and numerical experiments, this research showed that the breaker types on a given slope can be well approximated by the log-transform of the alternate similarity parameter The progression of breaker types observed inside the experimental space defined by the parallelogram of blue dashed sides was determined by the curve connecting the experimental wave input. In this paper, straight lines, Equation 3, drawn in the experimental space, were used. For each input straight line, there is a corresponding output straight line, which relates the progression of breaker types and    log . As the wave steepness and relative water depth increase (Figure 7a), there is a continuous gradation in the type of breaking from surging to spilling. The actual observation of breaker types depends on the initial value pairs of h/L and H I /L selected, the experimental space, and the slope of the straight line (Equation 3). In Table 2, it can be observed that the quality of the breaker type prediction improves significantly when adding the relative water depth. Furthermore, when wave steepness decreases and relative water depth increases but  remains constant, the breaker type does not change (green line in Figure 7a). In other words, there is no progression of breaker type.
Since the classification of breaker types is discontinuous, the data assigned to each type were placed on horizontal lines, depending on the value of . Because the breaking of a wave train on a smooth slope is assumed to progress continuously, the location of the displaced data was corrected to satisfy that assumption. The line thus obtained establishes, for a given slope (m = 1:10), a linear relationship between the continuous progression of breaker types and the value of . There is a functional relationship between the sets of the experimental space and those of the breaker types. Thus, the breaker types included in the extended classification should be considered as milestones in the continuous wave breaking process on a plane impermeable slope.
The non-dimensional wave energy dissipation on the slope is derived, considering the wave reflected energy flux on the slope. The results show how it is proportional to a dimensionless bulk dissipation coefficient which depends on the breaker type and, therefore, on the value of the alternate similarity parameter  at the toe of the slope.
Finally, for future research, of wave evolution over a beach or ramp of constant slope m, the experimental design for dimensional analysis should define a complete set of value pairs for / h L and / I H L, which makes it possible to identify the progression of the observed breaker types. To elucidate the mechanisms which could trigger variations of the wave front it would be convenient to use transversal arrays of gauges that do not perturb the flow (acoustic or similar).
If the motion on the slope is fully turbulent, "the actual value (of  3 Π Re) does not significantly affect the resultant motion on the slope" (Battjes, 1974).  s 2 gL is a dimensionless quantity, which indicates the relative importance of surface tension and gravity. It shows up in the surface boundary condition at z = 0 (Crapper, 1984). For all the runs carried out in the CIAO flume (see supporting information), this quantity is O(10 −6 ). The actual value of 4 Π does not affect the propagation, wave length, and wave celerity of the incident wave trains in the flume.
Since omitting the relative water depth is contrary to the logic that led to Buckingham's  -theorem, then The π−theorem does not provide the form of the functional relationship expressed by equation (5). Indeed, the form has to be obtained by physical or numerical experimentation, or by solving the problem theoretically. Díaz-Carrasco et al. (2020), Moragues et al. (2020), and this study, based on numerical and experimental data, show that for each slope angle, the interplay of / h L and / , H L as defined by      / / , h L H L is a convenient form of the relationship that can be used to explore the type of wave breaker, the flow characteristics, and the transformation of the wave energy on a specific plane slope. Finally, the slope angle of the structure should be regarded as a characteristic of the structure. Consequently, the tests for each slope should be analyzed independently as Galvin (1968)

The IH-2VOF Numerical Model
Numerical tests were performed in the IH-2VOF model (Lara et al., 2008). It has a uniform grid on the y-axis with a cell size of 5 mm and three regions on the x-axis: (a) the generation region with cell size from 8 to 5 mm; (b) the central region (where the slope is located) with a constant cell size of 5 mm; and (c) a third region with cell size from 5 to 10 mm. The total number of cells in the numerical domain is 2993 × 201. All the setups were run with the reflection absorption activated in the generation paddle. Moreover, active wave absorption was used at the generation boundary and at the end of the flume to simulate the dissipation ramp.
The free surface output recorded with 31 wave gauges mostly located along the slope ( Figure B1) were used to determine the breaker type. Wave conditions are summarized in Table B3. Using the three gauge method of Baquerizo (1995), the incident wave train and reflected wave train were separated. The zero-upcrossing mean wave height was estimated from the incident ware train.

The Ciao Wave Flume, University of Granada
The Atmosphere-Ocean Interaction Flume (CIAO) is part of the Environmental Fluid Dynamics Laboratory and focuses on the study of the coupling processes between the sea and the atmosphere. The wave generation system (wave flume), has a width of 1 m, a water depth of 0.70 m, a length of 15 m, and it can generate waves with a period of 1-5 s and a height of up to 25 cm. One of its main features is the presence of two paddles on opposite sides, which are controlled by a software program, for the complete absorption or partial reflection of waves, including a possible phase shift of the reflected wave (Addona et

Table B2
Test Characteristics of Derakhti et al., (2020). Parameters H and T are Input Values, Namely, the Values Given to the Generation System and L the Wave Length Figure B1. Scheme of the simulated impermeable and non-overtoppable slope in the IH-2VOF and the locations of the 31 wave gauges. Note. Parameters H and T are input values, namely, the values given to the generation system. Tz, L, and H I are the zeroupcrossing mean wave period, wavelength, incident wave height, respectively, obtained from the separation method of incident and reflected wave trains and the statistical analysis of the surface elevation data. For this study, an impermeable wooden ramp, with a slope angle of 1:10 ( Figure B2) was used. Test conditions are summarized in Table B4. The three-gauge method of A. sunción. Baquerizo (1995) was applied to separate the incident wave train and reflected wave train. The zero-upcrossing mean wave height was estimated from the incident wave train.

A Comparison Between Data Sets Used in This Paper
MORAGUES AND LOSADA   Note. Parameters H and T are input values, namely, the values given to the generation system. T z , L, H I are, the zeroupcrossing mean wave period, wavelength, and incident wave height, respectively.

Table B4
Test Characteristics Carried on in the CIAO wave Flume research group TEP-209 (Junta de Andalucía) and by the following projects: "Protection of coastal urban fronts against global warming-PROTOCOL" (917PTE0538), "Integrated verification of the hydrodynamic and structural behavior of a breakwater and its implications on the investment project-VIVAL-DI" (BIA2015-65598-P). This work was funded by the projects PCI2019-103565-SUSME and PID2019-107509GB-I00-ROMPEOLAS (SRA (State Research Agency)/10.13039/501100011033). M. A. Losada was partially funded by the emeritus professorship mentoring program of the University of Granada. We would like to thank the three reviewers for providing helpful comments on earlier drafts of the manuscript.