Study on the Set-up and Set-down Induced by Breaking Waves Over A Reef

This paper proposes an equation to calculate breaking wave induced wave set-up and set-down along reef flat. The mathematical equation was derived based on the theory of radiation stress and the conservation of wave energy. The equation is primarily determined by several physical variables including the breaking wave index, the stable wave index, the attenuation coefficient of wave energy flux, and the flow velocity in the re-stabilization zone. A series of laboratory experiments were carried out to calibrate the theoretical equations. Specifically, the breaking wave index, the stable wave index, and the velocity over the reef flat were measured in the laboratory. The attenuation coefficient of wave energy flux in our theoretical equation was determined by calibration by comparing with the laboratory measured wave height. Furthermore, it has been put forward that the velocity based on cnoidal wave theory could be used to determine the velocity over the reef flat if there is no velocity measurement available. Overall, the proposed equation can provide satisfactory prediction of wave set-up and set-down along the reef flat.


Introduction
The increase and decrease of the mean water level during wave propagation are commonly known as wave set-up and wave set-down, respectively. The wave set-up and setdown are important factors that affect the change of water level above reef flat, and thus it is crucial to determine the design water level of buildings on the reef flat. Research on wave set-up and set-down above the reef terrain could be traced back to the 1940s. The earlier research on wave variation above the reef terrain mainly relied on in-situ measurement and made little progress. In 1972, with the development of the radiation stress theory, Tait (1972) studied the wave set-up above reef terrain based on the momentum flux conservation principle and proposed an equation to calculate the maximum wave set-up, which is η max where is the maximum wave set-up; h f is the water depth above the reef flat; h b is the water depth where wave breaks; and γ b is the breaker index (H b /h b ). η max Gerritsen (1980) and Seelig et al. (1983) were the early researchers who carried out laboratory experiments to study wave set-up above the reef flat. Later on, there were more studies of wave set-up above reefs by laboratory experiments, theoretical analysis or field observation, producing fruitful results. For example, Gourlay (1996aGourlay ( , 1996b studied the variation of maximum wave set-up above the reef flat with respect to wave height, wave period, and water depth above the reef flat through a series of laboratory experiments, and the author proposed an equation to calculate the maximum wave set-up based on the conservation of wave energy as follows: K p k r where is the reef shape parameter; is the reflection China Ocean Eng., 2020, Vol. 34, No. 6, P. 853-862 DOI: https://doi.org/10.1007/s13344-020-0077-6, ISSN 0890-5487 http://www.chinaoceanengin.cn/ E-mail: coe@nhri.cn η max K p coefficient; H 0 is the off-reef wave height (equivalent deep water value); T is the wave period; and g is the gravity acceleration. This equation has been considered as one of the most comprehensive formulas for calculating the maximum wave set-up above island-reef terrain, and it has been widely used by scholars in China and abroad. However, there are two difficulties in using this equation: (1) It is an implicit function that has the unknown maximum wave set-up on both sides of the equation, as a result, it requires iterative calculation; and (2) it is difficult to determine the reef shape parameter . By using the wave refraction-diffraction equation, Massel and Gourlay (2000) proposed an equation to calculate wave propagation and wave set-up above 2D reef terrain. However, their equation requires simultaneous iterative calculation of wave height and wave set-up, thus the calculation process is complicated. Zhu et al. (2004) measured the wave on the reef flat of Yongshu Reef in the South China Sea and found that the attenuation rate of wave height per 100 m distance was 31.35% on average, and the friction coefficient at the bottom was about 10 times larger than that of the normal sea bottoms. Jago et al. (2007) measured the mean water level and waves above the reef flat of Lady Elliot Island in Australia. It was found that wave set-up above the reef flat is closely related to the change of tide level. At the middle tide level, they observed two wave set-up phenomena above the reef flat. That is to say, there was wave set-up at both the reef edge and the shoal, while there was wave set-down at the middle of the reef edge and the shoal. Vetter et al. (2010) observed wave set-up above the reef flat of Guam, established a good linear correlation between the wave setup above the reef flat and the incident wave height. Buckley et al. (2016) compared the variation of wave set-up on rough and smooth reef surfaces by inlaying 1.8cm-long concrete cube blocks on a generalized reef terrain. It was found that there is no significant difference between wave set-up on rough and smooth surfaces. However, according to the numerical simulation of wave set-up above smooth reef surfaces and rough reef surfaces by Apotsos et al. (2007) and Franklin et al. (2013), wave set-up would significantly increase up to 22% or more as the surface roughness increases. η max Based on the mass conservation theory, Yao et al. (2016) proposed an equation to calculate the maximum wave set-up above the shallow reef flat as follows: where β is the non-uniform coefficient. All of the above studies only considered the influence of waves and ignored the influence of tidal currents. Yao et al. (2017) studied wave set-up using wave flume experiments by considering the coexistence of waves and tidal currents.
They found that when waves and tidal currents move in the same direction, tidal currents reduce wave set-up above the reef flat, while they move in reverse directions, tidal currents increase wave set-up. Furthermore, there is a significant linear relationship between the maximum wave set-up and the tidal flowrate.
Recently, Zheng et al. (2020) conducted a series of laboratory experiments to investigate the characteristics of the wave set-up in the 2DH Reef-Lagoon-Channel system. The test results show that Cross-shore wave set-up decreases from seaward edge of the reef flat to the lagoon and Alongshore wave set-up decreases from the central reef flat to the channel through the reef.
It is seen that the aforementioned studies have mainly addressed the wave set-up above reef terrain involving the influences of wave condition, reef shape, tidal current, and bed roughness on wave set-up. However, most of them focused on the characteristics and calculation of the maximum wave set-up, with less attention paid to the distribution of wave set-up and set-down along the reef flat. From the perspective of engineering practice, it is of great significance to understand the detailed forecast of water level for the purpose of building layout design, thus a complete calculation of the wave set-up and set-down along reef flat is necessary. This paper aims to fill this research gap and concentrates on breaking waves induced wave set-up and setdown along reef flat by theoretical analysis and laboratory experiments.
This paper is organized as follows. Section 2 carries out theoretical analysis of wave set-up and set-down along reef flat based on the theory of radiation stress and the conservation of wave energy, and we propose an explicit equation to calculate wave set-up and set-down. Section 3 presents the details of a set of laboratory experiments which were used to determine the parameters in our proposed equation. Section 4 compares the result of laboratory experiments and theoretical analysis. Section 5 summarizes the findings of this work.

Theoretical analysis of wave set-up and set-down along reef flat
It is a common practice to generalize the terrain to a streamwise domain of a slope connected to a flat when studying the characteristics of reef terrain (see Fig. 1), as done by Gourlay (1996a), Quiroga and Cheung (2013), Ren et al. (2018), and Zhang et al. (2019). The generalized streamwise domain was also adopted in this research. The wave set-up and set-down along the reef flat is related to the breaking position of waves. This study focuses on wave setup and set-down caused by breaking waves at the reef edge and above the reef flat. Furthermore, it was assumed that section A−A is the initial breaking position and section B−B is the position where stable waves form after breaking, as shown in Fig. 1.

Wave set-down at the breaking point
According to the radiation stress theory, we have the following relation: η where is the wave set-up/set-down; ρ is the water density; h is the local water depth; S xx is the radiation stress, which could be expressed as: where E is the wave energy density, and and k is the wave number. Eq. (5) could be further expressed as: σ where c g is the wave group velocity; c is the wave phase velocity; and is the wave angular frequency. The dispersion relation of waves is given as: /g It was assumed that and for simplicity in the following.η Outside the surf zone, the wave set-up and set-down is usually considered small relative to the depth h, so . By ignoring the energy loss due to bottom friction, the wave energy flux remains unchanged during wave propagation and the frequency also remains unchanged. By combining Eqs. (4), (6) and (7), we can obtain that Integration of Eq. (8) gives where B is an undetermined constant. By assuming that the boundary condition of the wave set-up in deep water is zero, B is set as 0.

By substituting
ds dp and into Eq, (9), we obtain that . (10) Since kh<1 and the breaking wave height at the breaking point (i.e., the reef edge), Eq. (10) is further simplified as: 2.2 Wave set-up behind the breaking point According to the principle of wave energy conservation, where K is the attenuation coefficient of wave energy flux; is the wave energy flux at Section A−A; and is the wave energy flux at Section B−B. The wave can be regarded as shallow water wave after breaking, therefore, The wave starts to break at Section A−A, and its height . The wave becomes stable at Section B−B, and the stable wave height , where is the stable wave index.
Eq. (12) can be expressed as: The change of wave height across the surf zone above the reef flat was obtained by integral calculation, which is, The radiation stress S xx on the reef flat is still expressed as: The momentum equation in the surf zone was expressed as (Gourlay and Colleter, 2005): where q is the unit-width discharge above the reef flat and ; v is the flow velocity; and f r is the bed friction coefficient. In Eq. (16), the third term is the convection acceleration, and the fourth term is the friction resistance. Since convection acceleration is generally small, its influence was ignored. Then Eq. (16) is simplified to (15) and into Eq. (17), we can obtain that g ∂η ∂x The water depth on the reef flat is relatively shallow, and the flow velocity is generally evenly distributed along the vertical direction. To simplify the calculation, it is suggested that the flow velocity v is related to the shallow water wave velocity , that is, the flow velocity was determined as , where k f is the velocity coefficient. As a result, Eq. (18) Eq. (19) can be further arranged as: (20) For the horizontal reef flat, the water depth h f above the reef flat does not change with respect to the position x, so Eq. (20) can be further expressed as: (21) Substituting Eq. (14) into Eq. (21) gives By assuming and , Eq.
(22) can be obtained as follows: Integrating Eq. (23) gives is a function of the still water depth h f above the reef flat. It can be determined with Eq. (11), which is used to calculate the wave set-down at the breaking point.
Eq. (24) is further simplified into (25) Rearrangement of Eq. (25) by taking square root givesη (26) Eq. (26) is established to calculate wave set-up/set-down along reef flat. However, the attenuation coefficient of wave energy flux K, the breaking index γ b , the post-breaking stability index γ r , the bed friction coefficient f r , and the velocity coefficient k f are still undetermined.
The wave friction coefficient f w can be taken as the bed friction coefficient f r . The friction coefficient f w of different flow regimes (laminar flow, smooth turbulent flow and rough turbulent flow) can be expressed by Zhang et al. (2002) as follows: (28) where is the friction coefficient under the condition of laminar flow: (30) where A m is the orbital amplitude of water particle at the bottom of the wave. The following formula was used to cal- The velocity coefficient k f was determined by the following equation, It should be noted that although the formula of Gourlay (2005) for calculating the flow rate v in Eqs. (18) and (37) can be used, it will make our formula very complicated and deviate from the original intention of formula establishment in this paper. Therefore, to make the equation convenient for practical application, the maximum velocity u max is adopted in the following analysis in this paper, instead of the calculation formula of Gourlay and Colleter (2005) for wave generated flow, namely (38) In addition, the attenuation coefficient of wave energy flux K, the breaking index γ b and the post-breaking stability index γ r were still to be determined in Eq. (26). To determine the three parameters and verify the calculation equation concerning the wave set-up and set-down, the following laboratory experiments were conducted.

Laboratory experiments
A series of laboratory experiments were conducted in the wave flume of Nanjing Hydraulic Research Institute. The wave flume is 40 m long, 0.8 m wide and 1.0 m deep, as shown in Fig. 2. One end of the flume was equipped with a push-plate wave maker capable of active reflection absorption, and the other end was with a gentle slope for wave dissipation. In this paper, a 1:1 slope was piled up 25 m away from the pushing plate of the wave maker, and the slope was connected with a horizontal flat. The flat was 50 cm high and 8 m long, as shown in Fig. 3. The slope and surface of the flat had been plastered with cement mortar. Thirteen wave gauges were arranged (see Fig. 3) to measure the change of water surface. Among them, wave gauges #1−#3 were arranged before the slope to measure the wave reflection coefficient, and their positions varied slightly with the incident wave condition; #4−#7 were respectively arranged in front of the slope, at the toe, the middle and top of the slope; and #8−#13 were arranged on the reef flat. Details on the respective distance between wave gauges #4−#13 and the reef edge are shown in Table 1. A velocity measuring point was placed 3.55 m away from the reef edge (between wave probes #11−#12) to measure the vertical change of velocity. The velocity was measured by an ADV current meter.
Regular wave was used in this test. Parameters such as incident wave height, wave period, and water depth before reef are shown in Table 2. The wave breaking points were all located at the reef edge or slightly behind the reef edge. Velocity measurement was conducted for tests #1 and #2.

Data analysis method of wave set-up and set-down
The value of wave set-up and set-down was obtained by subtracting the still water level from the mean water level measured in the laboratory experiment. A positive value indicated wave set-up, and a negative value indicated wave set-down. The mean water level was obtained by averaging the surface level measured at a wave gauge for a certain period of time.
where M is the number of waves in the selected time period; N is the frequency of water surface data collection in each wave period; and is the measured real-time water level; is the static water level at the beginning of the test. At the beginning of each test in this paper, the still water level is set to be zero, that is .

Breaker type and position
The breaker type and position are the factors affecting the wave set-up and set-down, but the surf-similarity parameter is not applicable to the reef topography (Yao et al., 2013;Zhu et al., 2018). A large number of experimental studies showed that the water depth of the reef flat is an important factor affecting the breaker type and position. Figs. 4 and 5 show the changes of the breaker type and position obtained by the author through previous experiments with the relative water depth of the reef flat. It can be seen from Fig. 4 and Fig. 5 that when the relative depth of the reef flat h f /H 0 < 1.8, the breaker type is plunging or surging. When the relative depth of the reef flat h f /H 0 =0.9−1.5, the breaker position changes around the reef edge.
According to the observation of the test process, the detailed breaker type and position of each test are shown in Table 3. It can be seen from Table 3 that the relative depth of the reef flat h f /H 0 in the six tests was between 0.91 and 1.45, and the breaker type was all of plunging, and the breaker points were all near or slightly behind the reef edge.   Fig. 6 shows the vertical distribution of the measured velocity in the re-stabilization zone of the reef flat and the comparison with the theoretical value of different finite amplitude wave theory. The vertical coordinate is z/h along the vertical relative position of the measuring point. z/h=−0.6 means 0.6 times the depth below the still water level, and z/h=−1 means the bed surface. The x-coordinate is the dimensionless relative velocity , and u max is the maximum velocity measured in each water depth. As can be seen from Fig. 6, the measured flow velocity in the re-stabilization zone is evenly distributed along the vertical direction without obvious attenuation. From the comparison between the theoretical value and the experimental value, the theoretical value of the cnoidal wave theory is closest to the measured value. Therefore, if the theoretical velocity is adopted, the cnoidal wave theory is recommended in this paper.

Selection of characteristic velocity
4.4 Wave set-up and set-down along the reef flat Fig. 7 shows the comparison between the calculated values based on theoretical equations and laboratory measured values of wave height and wave set-up and set-down along the reef flat when the incident wave height H=0.07 m, wave period T=1.0 s, and the still water depth above the flat h f =0.1 m (Test #1 in Table 2). The breaking index γ b and the stable wave height index γ r were determined based on the laboratory measurements. Specifically, the measured breaking index γ b = 0.75, and the measured stable wave height index γ r = 0.37. Fig. 7a shows that the calculated value of wave height using Eq. (14) with a calibrated attenuation coefficient of wave energy flux K = 0.13 is in good agreement with the measured wave height profile in the laboratory. Fig. 7b shows the comparison between the calculated value using Eq. (26) and the laboratory measured value of wave set-up and set-down along the reef flat. The bed friction coefficient f r is calculated by Eqs. (27)−(36) with the measured bottom velocity u m . According to the experience that the average vertical velocity of one-way flow is about 0.6 times the depth velocity, the velocity coefficient is calculated by Eq. (38) with the measured velocity u max at 0.6 times local water depth. For this test run, the bed friction coefficient of f r =0.018 and the velocity coefficient (black line in Fig. 7b). It can be seen that the calculated value of wave set-up and set-down from our theoret- 1.10 Plunging 0.10 0.30 3.1 Note: the position in the table is the distance relative to the reef edge, which is negative on the seaside and positive on the landside.  Table #2). For this test run, the measured breaking index γ b = 0.90, and the measured stable wave height index above the reef flat γ r = 0.38. According to Fig. 8a, when the attenuation coefficient of wave energy flux K = 0.12, the calculated wave height variation along the reef flat is in good agreement with the measured value in the laboratory. According to Fig. 8b, when the bed friction coefficient f r =0.017 and the velocity coefficient (black line in Fig. 8b), the calculated value of wave set-up and set-down is basically consistent with the laboratory measured value, and the calculated value is slightly larger than the measured value. The discrepancy might be caused by the fact that the convective acceleration term in Eq. (16) was ignored in Eq. (26).
The above analysis shows that the wave height and wave set-up and set-down along the reef flat could be calcu-lated by using Eqs. (14) and (26) with the measured wave profile and flow velocity. However, the flow velocity above the reef flat is often unknown in engineering practice. Our study above has shown that the theoretical velocity based on cnoidal wave theory is most consistent with the laboratory measured velocity in the re-stabilization zone above the reef flat (see Fig. 6). Therefore, it is suggested that in the absence of the measured velocity data, the velocity u m in Eq. (36) takes the theoretical value of the near-bottom velocity of the cnoidal wave; the velocity u max in Eq. (38) takes the theoretical value of the cnoidal wave at 0.6 times local water depth. Figs. 7b and 8b show the comparison between the calculated wave set-up and set-down (red line) based on the above theoretical velocity and the other one based on the laboratory measured velocity (black line). It can be seen that there is a slight difference between the calculated wave variation based on the above theoretical velocity and that based on the measured velocity. The result based on the theoretical velocity is also consistent with the laboratory measured value, especially in the area between the reef edge and the maximum wave set-up point. The difference between two methods increases with the maximum difference up to 8% at the end of the reef flat.
In addition to the above two test runs, the comparison between the calculated and laboratory measured wave height and wave set-up and set-down in the other four test runs is shown in Figs. 9−12. The only calculated wave setup and set-down according to the theoretical velocity was compared with the laboratory measured value, since the velocity was not measured in the four test runs. The comparison shows that wave set-down happens at the reef edge due to wave breaking. The mean water level changes most rapidly from set-down to set-up after the reef edge. The wave set-up diminishes gradually as the wave continues to travel shoreward after it passes through the maximum wave set-up point. Eq. (26) can better predict wave variation along the reef flat.
Since Dally et al. (1985) did not consider the influence of wave generated flow after breaking, if the energy con-sumed by the wave generated flow is taken into account, the stable wave height H r in the re-stabilization zone will be slightly reduced, and the calculated value of Dally et al. (1985) will be slightly larger, which may be one of the reasons why the experimental value of the stable wave height H r is in general slightly smaller than the calculated value.
Figs. 7−12 show that the attenuation coefficient of wave energy flux K varies from 0.10 to 0.20, and its value is related to the intensity of wave breaking. According to our experiment observation, a larger K value should be used for a severe breaking process while a smaller one for a mild breaking process. The laboratory measurement showed that the breaking index γ b varies from 0.75 to 1.05, generally greater than that of solitary wave on the flat bottom (0.78). The stable wave height index γ r above the reef flat varies from 0.35 to 0.42. Friction coefficient f r and velocity coeffi- cient k f are closely related to the velocity of re-formed stable wave after breaking. Measured velocity data are directly used when measured velocity data are available. If there are no measured velocity data, the theoretical velocity based on the cnoidal wave theory can be used in the re-stabilization zone.

Conclusions
The work has studied the breaking wave induced wave set-up and set-down along reef flat with both theoretical analysis and laboratory experiments. The laboratory experiment was mainly used to calibrate the parameters in our the-oretical analysis. Major findings are summarized as follows: η/h f (1) Based on the radiation stress theory and conservation theory of wave energy, this paper proposes Eq. (26) to calculate the breaking wave induced wave set-up and setdown along the reef flat. As the wave energy propagation velocity c g and the breaking wave index γ b are simplified in the formula derivation process in this paper, when the relative wave set-up is large, such simplified treatment will lead to certain errors. The comparison with the experimental values shows that the formula can give satisfactory calculation results of wave set-up and set-down along the reef flat   (2) Six laboratory wave flume test runs have been conducted to calibrate the proposed equations. Among all runs, the breaking index γ b varies from 0.75 to 1.05 and the stable wave height index γ r above the reef flat from 0.35 to 0.42. The attenuation coefficient of wave energy flux K varies from 0.10 to 0.20. A larger K value should be used for a severe breaking process while a smaller one for a mild breaking process.
(3) Friction coefficient f r on the bed surface and velocity coefficient k f are closely related to the velocity of reformed stable wave over the reef flat; and they can be calculated according to Eqs. (27)−(38). Measured velocity data are directly used to calculate friction coefficient f r and velocity coefficient k f if measurements are available. However, if there are no measured velocity data, the theoretical velocity based on cnoidal wave theory can be used.