| Citation: | Yuan Li, Chi Zhang, Hongshuai Qi, Jiacheng Song, Weiqi Dai, Shanhang Chi, Jian Shi, Dake Chen. Analysis of wave shoaling and shore-breakers on a low tide terrace beach based on in-situ measurements at Xisha Bay on South China coast[J]. Acta Oceanologica Sinica, 2023, 42(7): 175-184. doi: 10.1007/s13131-023-2188-4
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Sandy coasts provide biological habitats and sites for human recreational activities. Under different driving forces of waves and tides, natural beaches can be classified into various types, i.e., dissipative, intermediate and reflective beach, judged by the dimensionless fall velocity (Wright and Short, 1984). Dissipative beaches are usually found on the high-energy open coastlines. The configuration of dissipation beaches includes a mild sloping intertidal zone and extremely mild subtidal zone (Moulton et al., 2021; Wright et al., 1979). Reflective beaches always develop on low energy headland coastlines, and equips a steep beachface and shoreface. Sandbars are absent on both reflective and dissipative beaches (Chi et al., 2023; Li et al., 2021). Masselink and Short (1993) considered the effect of tidal range on beach states and further dividing the intermediate beach states using the dimensionless relative tidal range. Low tide terrace beach is one typical intermediate beach that often occurs on low-to-medium-energy macrotidal coastlines such as South China coasts (Cai et al., 2022). The configuration of the low tide terrace beach includes a relative steep upper beachface and a low-gradient intertidal zone. A better understanding of tidal effects on hydrodynamic processes on the low tide terrace beach is important for beach nourishments and restorations on South China coasts.
Beach states are highly controlled by surf states, especially on wave-dominated beaches. Although low tide terrace beaches are macrotidal, the role of waves in shaping the morphology cannot be neglected. Surf states on low tide terrace beaches are usually unsaturated. Here, the unsaturated surf means that the increase in offshore wave energy cannot be dissipated totally by wave breaking and thus the inshore wave heights change with offshore wave heights (Baldock et al., 1998). Surf states are judged using the surf similarity parameter proposed by Battjes (1974). Unsaturated surf is usually found on steeper beaches or with low steepness incident waves. The wave breaking zone is narrow or sometimes absent of unsaturated surf and wave breaking can occur near the shoreline very close to or even at the shoreline (Baldock et al., 1998), thus it was defined as “shore-breakers”.
The shore-breaker serve as a direct motivation for shoreline changes. It drives swash flows upward, longshore currents downstream and rip currents seaward. Besides, the shore-breaker is also a main factor that threatening the safety of beachgoers. Muller (2018) found the shore-breaker was a contributing factor in over 82% of reported surf-zone injuries in Ocean City, Maryland from 2006 to 2015. However, there is still a scarcity of quantitative research on shore-breakers. For example, how much energy remains at the initialization of shore-breakers? How much is the energy dissipation rate of shore-breakers on the low tide terrace beaches?
The beaches at Xisha Bay on the South China coasts are equipped with a typical “low tide terrace type” morphology, i.e., with a mild dissipative terrace and a steep reflective beachface. Besides, shore-breakers are frequently observed near the shoreline. Therefore, in-situ measurements were conducted in this area to provide quantitative description of wave transformations and shore-breakers under the tidal effects.
Field observations were carried out on the sandy beach of Xisha Bay located on the South China coast. The sandy beach is located on the north side of the Taiwan Strait. The width of the dry beach is approximately 200 m (Fig. 1a). The tidal regime is macrotidal and semi-diurnal with the annually mean tidal range of 4.2 m. Medium-to-coarse sediment with d50 = 0.33 mm on the low tide terrace and d50 = 0.60 mm on the beachface (Cai et al., 2012). During the beach nourishment project in Xisha Bay, the sediments were filled on the berm to increase the width of the dry beachface. The grain size of filling sediments is approximately two times coarser than native sediments. This is because using coarse sediments can effectively decrease the intensity of sediment motions and to enhance the stability of the upper beachface. As a preliminary estimation, the dimensionless sediment fall velocity and relative tidal range are 1.4 and 10.5, respectively. The dimensionless sediment fall velocity defined as H/wsT (in which, H and T are the representative wave height and period, and ws is the sediment settling velocity) describes relative time needed for sediment suspension and settling. The relative tidal range is defined as the magnitude of the breaking wave height and tidal range. The beach state is therefore classified as the low tide terrace according to classic conceptual models of beach types (Wright and Short, 1984; Masselink and Short, 1993). The beach profile in the intertidal zone consists of two parts, i.e., a dissipative low tide terrace with a slope of 1/30 and a reflective high tidal beachface with a slope of 1/10. As can be seen in Fig. 1b, above the high-water level, rhythmic beach cusps can be also found on the beach, corresponding well with the classification of Masselink and Short (1993). As can be seen in Fig. 1c, the x-axis (z-axis) origins at the location (elevation) of dune toe and the positive offshore (upward). The surveyed beach profile ends at the point where the mean low waterline intersects the shoreface. The total length of the survey beach is 142 m.
Field measurements were carried out from 10:00 December 6, 2020 to 10:00 December 8, 2020. Beach profile was measured using the Real-time-kinematic (RTK) at low tides. RTK can obtain real-time positioning data in the field, and the accuracy of the instrument is at the centimeter level, which can greatly improve the efficiency of field work. By comparing the beach profiles measured before and after the two-day field measurements, the morphological changes are found negligible. Time series of wave pressure was recorded using a wave-tidal gauge (RBRsolo³ D wave16), which is fixed on the bottom and works with sampling frequency of 4 Hz. Six gauges were deployed on the low tide terrace from the seaward boundary to shoreward boundary of the terrace, as can be seen in Fig. 1c.
Offshore wave climates are extracted from a high-resolution wave hindcast database for the Chinese coast. The wave hindcast database (CWAVE) was established using the wave model TOMAWAC forced with the wind field from CFSR database over the whole domain along the coast of the Chinese mainland (Shi et al., 2019). The database can provide wave parameters with high spatial and temporal resolutions. The accuracy of wave hindcasts have been verified using buoy data and satellite altimeter data. The offshore location exhibited in Fig. 1a is chosen by extending the current cross-shore domain to the 20 m water depth. Offshore Root-mean-square (RMS) wave height and mean wave period are provided in Fig. 2, in which t = 0 h corresponds to 10:00 December 6, 2020. It can be found that the offshore wave characteristics are not significantly modulated by tides. Offshore wave heights and periods generally decrease in the first 20 h then increase in the next 20 h, and finally achieve a peak at t = 42 h. Offshore wave heights and periods range from 0.73 m to 1.11 m and from 4.68 s to 7.05 s, respectively. Wave climate in the study area at the time period is modal and storm events are not found.
Bottom pressure power density spectrum is computed using the Fast Fourier Transforms, with 10 Hanning windowed, 50% overlapping segments (20 degrees of freedom). The spectrum of bottom pressure is converted into the spectrum of water surface elevation using the classic transferring function method based on the linear wave theory (Bishop and Donelan, 1987). Then, the time series of water surface elevation is obtained using an Inverse Fast Fourier Transform. Raw data was separated into 15 min time series bin to ensure that the still wave level is stationary compared with the changing tidal level, following Power et al. (2010). The bin will be discarded from analysis if the gauge could not be submerged for the full period of 15 min. Wave parameters are calculated after band-pass filter at 0.04 Hz and 1 Hz to remove low frequency motions and instrument noises. The RMS wave height Hrms and mean wave period Tm are calculated as
| $$ {H_{{\rm{rms}}}} = \sqrt {8{m_0}} , $$ | (1) |
| $$ {T_{\rm{m}}} = {m_0}/{m_1} , $$ | (2) |
where m0 and m1 are the 0th and 1st order moment variance of the band-passed filtered water surface. The wave breaking types or surf states can be described using a dimensionless surf similarity parameter
| $$ {\zeta _0} = \frac{m}{{\sqrt {{H_{{\rm{rms}}0}}/{L_0}} }} , $$ | (3) |
where m is the bottom slope of the low tide terrace, Hrms0 and L0 are offshore RMS wave height and wave length.
Wave shape asymmetry relative to horizontal and vertical axes are described by wave skewness Sk and asymmetry As, which are defined as
| $$ Sk = \frac{{\left\langle {{{\left( {\eta - \overline \eta } \right)}^3}} \right\rangle }}{{{{\left\langle {{{\left( {\eta - \overline \eta } \right)}^2}} \right\rangle }^{3/2}}}} , $$ | (4) |
| $$ As = \frac{{\left\langle {{{{ {\bf{H}}}}}{{\left( {\eta - \overline \eta } \right)}^3}} \right\rangle }}{{{{\left\langle {{{\left( {\eta - \overline \eta } \right)}^2}} \right\rangle }^{3/2}}}} , $$ | (5) |
in which,
| $$ Ur = \frac{{{H_{{\rm{rms}}}}L_{\rm{m}}^2}}{{{d^3}}} , $$ | (6) |
where the mean wave length calculated with the mean wave period and nonlinear wave celerity in Booij (1981) .
Figure 3 provides the sketch of the shore-breaker near the shoreline. The event of a shore-breaker is identified as the last local maximum wave height near the shoreline in Fig. 3, in which the He is the representative wave height of the most shoreward gauge, i.e., wave height at the end of the shore-breaker. To ensure the shore-breaker dissipates sufficiently in the distance
The mean energy dissipation Dbs of shore-breakers is calculated as
| $$ {D_{{\rm{bs}}}} = \frac{{F{w_i} - F{w_e}}}{{\Delta x}} , $$ | (7) |
where Fw =
| Symbol | Definition | Unit |
| $\eta$ | wave surface elevation | m |
| Hrms | root-mean-square wave height | m |
| Tm | wave mean period | s |
| mn | nth-moment variance of the band-passed filtered water surface elevation | m2/sn |
| H0 | offshore significant wave height | m |
| Sk | wave skewness | – |
| As | wave asymmetry | – |
| Ur | ursell number | – |
| d | mean water depth | m |
| $\rho$ | water density | kg/m3 |
| g | gravity acceleration | m/s2 |
| cg | wave group velocity | m/s |
| c | wave celerity | m/s |
| n | ratio of group velocity to wave celerity | – |
| Dbs | mean energy dissipation rate of shore-breaker | N/ms |
| Fw | wave energy flux | N/s |
| fp | peak wave frequency | Hz |
| k | wave number | – |
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As can be seen in Fig. 4a, there are approximately four tidal cycles in the study period. The wave height generally increases to the shoreline (i.e., instantaneous shorelines) and then decreases. Remarkable shore-breakers can be found in front of the shoreline. The wave height of shore-breakers can be up to 0.4 m, which can be almost four times larger than the wave height at the offshore boundary at high tidal levels. At the low tidal levels, wave shoals and breaks in a very short distance on the low tide terrace. At the high tidal levels, a large part of the terrace is dominated by wave shoaling and shore-breakers occur only near the shoreline, as can be seen in Fig. 4b.
To better clarify the wave transformation on the low tide terrace, the cross-shore profiles of RMS wave height and setup at a certain moment t = 28.75 h is provided in Fig. 5a. At this moment, wave shoals on the first 60 m of the low tide terrace. The shore-breaker occurs at x = 82 m with the maximum wave height of 0.20 m and breaking depth of 1.29 m. The wave setup decreases as wave shoals and achieves a maximum wave set down at the shore-breaker. Shoreward of the shore-breaker, the magnitudes of RMS wave height and wave setup are very small.
The wave height transformation on the low tide terrace beach is different from that on the dissipative beach, on which the wave height decreases from the offshore boundary of the intertidal zone to the shoreline without obvious wave shoaling processes (Li et al., 2022a). On the low tide terrace, the cross-shore profile of wave height is a typical characteristic of an unsaturated surf with much wave energy remained and finally released near the shoreline, i.e., shore-breakers.
The time series of Sk, As and water depth measured with the most offshore gauge are provided in Fig. 6a, since the offshore gauge is submerged most of the time. The magnitudes of Sk and As are highly modulated by tidal levels, corresponding well with the recent observations on a dissipative beach by Li et al. (2022c). The magnitudes of Sk and As range from 0.07 to 1.24 and from −1.65 to 0.04, respectively. Their magnitudes generally decrease as water depths increase. Similar observations can also be found on the low-crest breakwater, fringing reef or shoreface nourishment in surf zones (Chen et al., 2019; Li et al., 2022a; Peng et al., 2009). Cross-shore variations of Sk and As at the certain moment t = 28.75 h (same moment as Fig. 5a) are provided in Fig. 6b. It can be found that Sk increases as wave shoals and achieves the maximum value of 0.87 at the shore-breaker, and then decreases rapidly. The magnitude of As also increases from 0 to the maximum value of 2.85 at the shore-breaker and then decreases to 0 rapidly. Therefore, the position of the shore-breaker is the critical position where the wave nonlinearity reaches its maximum.
Time series of mean energy dissipation rate of shore-breakers (Dbs) are provided in Fig. 7, with 147 events of shore-breakers identified in total. Note that the event of shore-breaker in this section is identified based on the bulk wave analysis mentioned in Section 2.3 rather than the actual occurrences of shore-breakers in wave-by-wave analysis. Dbs generally increases with the offshore water depths, i.e., tidal levels. It can be found that Dbs ranges from 0 to 50 N/ms and the surf similarity parameter
Positive relations are found between Dbs and surf similarity parameter
The net near-bed sediment transport rate is well affected by the magnitudes of wave nonlinearities of wave profile, since the near-bed wave orbital velocity is well controlled by the water surface elevation (Rocha et al., 2017; Ruessink et al., 2013). Since estimations of Sk and As from period-resolving water surface elevations would lead to a huge computational effort when these two parameters are embedded in a beach morphodynamic model, Sk and As are usually parameterized using Ur in the period-averaged framework (Doering and Bowen, 1995; Elfrink et al., 2006; Isobe and Horikawa, 1982; Ruessink et al., 2012). Li et al. (2022c) recently found that the differences between nonlinearities of water surface elevation and wave orbital velocity is not significant when water depths are not very small. In this study, dataset collected on the low tide terrace is also used to verify most recently developed parameterizations of Li et al. (2022a).
Compared with previous formulas that parameterized on the limited bottom slopes (e.g., on the low-crested breakwater by Peng et al. (2009) and on the fringing reef by Chen et al. (2019)), Li et al. (2022a) proposed new parameterizations of Sk and As taking both Ur and bottom slope into consideration recently. The new parameterizations were found to improve the prediction accuracy especially at shallow water depth in the inner surf zone. These two parameterizations are written as
| $$ Sk = ( - 23.08{m^2} + 4.99m + 0.1) \ln (Ur) + 95.64{m^2} - 20.03m + 0.62 , $$ | (8) |
| $$ As = 1.19\tanh [(92.93m + 12.96)/Ur] - 1.2 , $$ | (9) |
where m is the bottom slope of the low tide terrace. The Sk and As measured in this study are used to evaluate these two new parameterizations. Comparisons of measured and predicted Sk and As are provided in Fig. 9. Note that the most shoreward gauge is excluded in this comparison, since the magnitudes of measured wave nonlinear parameters are very small and the gauge is above the water most of time. It can be found that new parameterizations of Li et al. (2022a) can provide good predictions when the water depth is large, i.e., at a more offshore position or at high tidal levels. However, the Sk is overestimated as the water depth decreases. The phenomenon is obvious when the gauge is placed near the shoreline, where the overestimation is more significant. The predictions of As are generally better than Sk, though underestimations of the magnitudes of As are found at the low tidal levels. The consistent between predicted and measured As is good even near the shoreline. This is because Eqs (8) and (9) were also parameterized with intense wave breaking on the shoreface nourishment, where the waves are highly pitched-forward and asymmetric. Thus, the new parameterizations are naturally suitable for predicting As of intense shore-breakers.
As mentioned above, the occurrence of shore-breakers indicates the surf processes are unsaturated. Wave height variations in the unsaturated surf can be expressed empirically with the relationship between the relative wave height (Hrms/d) and local water depths, beach slopes or offshore wave characteristics (Power et al., 2010; Yao et al., 2020). Based on these empirical relationships, local wave heights can be efficiently calculated when these easy-obtained parameters such as water depths, beach slopes and offshore wave heights are known. In this study, no direct relationship is found between Hrms/d and beach slopes (or other combinations of slopes such as m/kd used in Raubenheimer et al. (1996). Therefore, dimensionless parameters combined with local water depth and offshore significant wave height, i.e., kd and d/H0 are used following Power et al. (2010) and Yao et al. (2019). As can be seen in Fig. 10a, Hrms/d generally decreases with kd, and tends to be stable when kd>1. The curve of Hrms/d to kd is fitted using the method of nonlinear least squares. The expression of the curve is read as
| $$ \frac{{{H_{{\rm{rms}}}}}}{d} = 1.10\;\tanh ( - 2.69kd) + 1.14 , $$ | (10) |
for which, the squared correlation coefficient is 0.54, implying the overall fitness is not very good. Especially when the kd is small and the data points are more scattered. This is in general agreement with previous studies and the same phenomenon can be found in Power et al. (2010), who also conducted observations of wave height variations in the unsaturated surf.
As can be seen in Fig. 10b, Hrms/d is found to well relate to d/H0. Hrms/d decreases with increasing of d/H0 and then tend to be stable. This relationship implies the RMS wave heights in the intertidal zone are affected by offshore wave heights significantly, and they increase with the increasing of the offshore wave heights to a certain level. The phenomenon can provide a direct field evidence for the definition of the unsaturated surf from Baldock et al. (1998). The expression for the fitted curve is written in Eq. (11).
| $$ \frac{{{H_{{\rm{rms}}}}}}{d} = - 0.78\tanh \left(1.33\frac{d}{{{H_0}}}\right) + 0.84 . $$ | (11) |
The squared correlation coefficient is 0.85, implying the overall fitness is excellent. However, the fitted curve seems to underestimate the Hrms/d at small water depth and tends to a specific upper limit at the shoreline. This is not realistic since the upper limit should not be reached since Hrms/d is infinite at the shoreline. Therefore, Eq. (11) is recommended to be used when d/H0 > 0.5 in the unsaturated surfs.
Previous studies provided qualitative descriptions of an unsaturated surf such as the positive relations between inshore wave heights and offshore wave heights, and the occurrence of shore-breakers carrying remaining part of wave energy breaking near the shoreline (Alexander and Holman, 2004; Baldock et al., 1997, 1998; Carini et al., 2021; Power et al., 2010, 2013; Wright et al., 1979). In this section, the ratio of the remaining wave energy flux at the initialization of the shore-breaker to offshore wave energy flux is used as an indicator for quantifying the degree of the unsaturated surf. The remaining wave energy flux at the initialization of the shore-breaker is denoted as Fwi in Eq. (7) and the offshore wave energy flux is calculated based on the linear wave theory. The remaining wave energy flux at the initialization of the shore-breaker is only a small part of offshore wave energy flux with Fwi/Fw0 ranging from 1% to 12%, though in the unsaturated surf. Fwi/Fw0 in the first two tidal cycles in the studying period is generally larger than the last two cycles, as can be seen in Fig. 11a. Especially at the high tide of the second tidal cycle, the ratio of energy flux can be up to 12%. Since the offshore wave height achieves its minimum at the second high tide, as can be seen in Fig. 2a, indicating the wave breaking is more unsaturated at this moment. Fwi/Fw0 in the last two tidal cycles are more stable and is not modulated by tidal level changes significantly. Hence, Fwi/Fw0 is more affected by offshore wave conditions than tidal levels.
A quantitative relationship between Fwi/Fw0 and offshore wave heights is illustrated in Fig. 11b. The relationship is read as
| $$ \frac{{F{w_i}}}{{F{w_0}}} = - 0.51\tanh ({H_0}) + 0.49 , $$ | (12) |
for which, the squared correlation coefficient is 0.70, implying the fitness is overall good. Although the dimensions at both ends of the equation are not uniform, this formula focuses on expressing a quantitative relationship rather than detailed physical mechanisms. Fwi/Fw0 generally decreases with increasing of offshore wave heights. This is expected since a larger Fwi/Fw0 represents a higher degree of the unsaturated surf, which is likely to occur at the milder wave condition (i.e., with smaller offshore wave heights and larger surf similarity parameters). Conversely, a lower degree of the unsaturated surf with a smaller Fwi/Fw0 is likely to be found at the storm wave condition with larger offshore wave heights and smaller surf similarity parameters. This can be further explained with the breaking intensification mechanism and breaking resistance mechanism proposed by Zhang et al. (2021a). The breaking intensification mechanism is applied to storm wave conditions. The larger offshore waves evolve to asymmetric shapes and break as water depth decreases on the low tide terrace with the wave breaker index decreases with decreasing of water depth, i.e., waves are easier to break at smaller water depth. In such cases, the remaining wave energy flux near the shoreline is smaller, indicating the degree of the unsaturated surf is lower. The breaking resistance mechanism is applied to mild wave conditions. Under this circumstance, the wave breaker index increases with decreasing of water depth. Smaller waves adapted more sufficiently to the local bathymetry due to shoaling, and they will struggle not to break until a larger breaker index is achieved, i.e., near the shoreline. Thus, the degree of the unsaturated surf is higher.
The most important implication of this study relies on its potential guideness on the design of beach nourishment project on low tide terrace beach. The most intense sediment transport and morphological changes occur on the steep beachface at high tides. Shore-breakers in the plunging type release wave energy rapidly thus pick sediments up and mix them in the water column (Ting and Kirby, 1995; Wang et al., 2003; Li et al., 2022b). High concentration of sediments in the water column are either washed upwards by swash flows or carried downstream by alongshore currents, resulting in imbalances of net sediment transport rate and local sediment budget (Chi et al., 2021; Fang et al., 2013; Shi and Kirby, 2008; Zheng et al., 2014; Zhu and Dodd, 2020). Beach nourishments on low tide terrace beaches are mainly conducted by filling sediments on the beachface to enlarge the width of the dry beach (Qi et al., 2010; Shi et al., 2013; Zhao et al., 2021). It is suggested that using coarse sediments is more appropriate for beach nourishments on the low tide terrace beach. On the one hand, the beachface filled with coarse sediments can form a steep slope under the natural adjustments, promoting swash-based wave reflections (Zhang et al., 2021b). On the other hand, relatively low mobility of coarse-grained sediments can enhance the stability of the beachface.
Field measurements were conducted on a low tide terrace beach at Xisha Bay on South China coasts. Offshore wave characteristics are obtained by a regional wave hindcast model at water depth of 20 m. Wave pressures were recorded along the low tide terrace and then converted to water surface elevations. RMS wave height, wave skewness and asymmetry are calculated to provide wave shoaling processes. Then, mean energy dissipation rate of shore-breakers as well as the remaining energy flux at the initialization of shore-breakers are analyzed and parameterized. Main conclusions are provided below.
Wave breaking is unsaturated on the low tide terrace beach at the Xisha Bay. The wave height generally increases to the shoreline and then decreases rapidly. Cross-shore profiles of wave height indicate typical characteristics of shore-breakers that considerable wave energy remained and finally released near the shoreline. Inshore wave heights are found to be well parameterized by local water depths and offshore wave heights.
The magnitudes of wave skewness and asymmetry are highly modulated by tidal levels. Magnitudes of wave skewness and asymmetry increases as wave shoals and achieves the maximum value at the shore-breaker, and then decreases rapidly. Overall trends of the wave skewness and asymmetry are well predicted by recent parameterizations.
Mean energy dissipation rates of shore-breakers are found to be modulated by tidal levels. Mean energy dissipation rates are larger at high tides since waves break on the steep beachface in the plunging type. Conversely, mean energy dissipation rates are small at low tides because waves break on the mild terrace in the spilling type. Besides, the remaining wave energy flux at the initialization of the shore-breaker is only a small part of offshore wave energy flux, ranging from 1% to 12%. The energy flux ratio is found to decrease with the increasing offshore wave heights.
This study highlights the most intense wave breaking and sediment motions modulated by changing tidal levels would occur near the shoreline. Wave attenuation at shore-breakers can be estimated directly from offshore wave conditions based on findings in this study, favoring designs of seawalls or beach nourishment projects. However, only incident short-waves are analyzed in this study. Complete wave energy budget that takes infragravity waves and reflected waves into consideration will be analyzed in future works.
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| 1. | Hang Yin, Feng Cai, Hongshuai Qi, et al. Analysis of Tidal Cycle Wave Breaking Distribution Characteristics on a Low-Tide Terrace Beach Using Video Imagery Segmentation. Remote Sensing, 2024, 16(24): 4616. doi:10.3390/rs16244616 | |
| 2. | Xiangming Cao, Jian Shi, Chi Zhang, et al. The combined effects of tide and storm waves on beach profile evolution. Ocean Engineering, 2024, 299: 117416. doi:10.1016/j.oceaneng.2024.117416 | |
| 3. | Yuan Li, Jialin Tang, Chi Zhang, et al. Analysis and machine-learning-based prediction of beach accidents on a recreational beach in China. Anthropocene Coasts, 2024, 7(1) doi:10.1007/s44218-024-00067-5 |
Figures(11) / Tables(1)
Supported by:
Beijing Renhe Information Technology Co. Ltd
Yuan Li, Chi Zhang, Hongshuai Qi, Jiacheng Song, Weiqi Dai, Shanhang Chi, Jian Shi, Dake Chen. Analysis of wave shoaling and shore-breakers on a low tide terrace beach based on in-situ measurements at Xisha Bay on South China coast[J]. Acta Oceanologica Sinica, 2023, 42(7): 175-184. doi: 10.1007/s13131-023-2188-4
| Symbol | Definition | Unit |
| $\eta$ | wave surface elevation | m |
| Hrms | root-mean-square wave height | m |
| Tm | wave mean period | s |
| mn | nth-moment variance of the band-passed filtered water surface elevation | m2/sn |
| H0 | offshore significant wave height | m |
| Sk | wave skewness | – |
| As | wave asymmetry | – |
| Ur | ursell number | – |
| d | mean water depth | m |
| $\rho$ | water density | kg/m3 |
| g | gravity acceleration | m/s2 |
| cg | wave group velocity | m/s |
| c | wave celerity | m/s |
| n | ratio of group velocity to wave celerity | – |
| Dbs | mean energy dissipation rate of shore-breaker | N/ms |
| Fw | wave energy flux | N/s |
| fp | peak wave frequency | Hz |
| k | wave number | – |
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