Slope stability formulae for rubble mound structures are usually developed for head-on conditions. Often, the effects of oblique waves are neglected, mainly because it is assumed that for oblique wave attack, the reduction in damage compared to perpendicular wave attack is insignificant. When the incident waves are oblique, the required armour size can be reduced compared to the perpendicular wave attack case. Therefore, it is important to consider the wave obliquity influence on slope stability formulae as a reduction factor. One of the most recent formulae for estimating the stability of rock-armoured slopes, referred to as Etemad-Shahidi et al. (2020), was proposed for perpendicular wave attack. The aim of this study is to develop a suitable wave obliquity reduction factor for the above-mentioned stability formula. To achieve this, first, laboratory experiment datasets from existing reliable studies were selected and analysed. Then, previously suggested reduction factors were evaluated and a suitable reduction factor for the mentioned stability formula were suggested. The suggested reduction factor includes the effect of wave obliquity and directional spreading explicitly. It is shown that the stability prediction is improved by using the wave obliquity reduction factor.

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Conventional rubble mound structures such as breakwaters, seawalls, and revetments are the most common type of coastal structures around the world used to protect harbour basins and embankments from wave action. To have a safe and economic design, two aspects need to be considered. The first one is the structural stability where the required armor size (weight) must be determined. The second aspect is the safety, where the crest freeboard of the structure is usually determined based on the allowable mean wave overtopping rate. Several semi-empirical formulas have been developed for these purposes. These formulas, which have evolved over time, are generally semi-empirical and based on the small-scale laboratory experiments where both incident wave characteristics and the structure configuration are considered. This paper aims to provide a comprehensive overview of the performance of existing formulas developed for the assessing the stability and mean overtopping rate of conventional rubble mound structures, while also introducing the recent ones. The Rock Manual formulas for the slope stability and EurOtop formula for estimating the mean overtopping rate will be discussed, and their performances will be compared with those of more recent and comprehensive ones using both lab and field data. It will be shown that the recent formulas that utilize the spectral energy mean period for stability analysis and run-up for the mean overtopping rate are more robust and physically sound. Finally, design formulas and uncertainty estimates are presented, along with guidance for practitioners.

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During storms, ensuring the protection of people, vehicles and infrastructure on the crest of coastal structures from wave overtopping hazards is crucial. The thickness of the wave overtopping layer is a key variable used for assessing safety and maintaining a secure design. Traditionally, this parameter is associated with the height difference between the fictitious wave run-up level exceeded by 2% of waves and the crest freeboard of coastal structures. This study aims to investigate the wave overtopping layer thickness on the crest of rubble mound seawalls. To achieve this, a series of 125 small-scale 2D physical model tests were conducted on a two-layer rubble mound seawall with an impermeable core and slopes of 1:1.5 and 1:2. The obtained results indicated that the existing empirical formulas, originally developed for dikes, underestimate the overtopping layer thickness on the studied seawall. Therefore, modifications were made to the formulas found in the literature specifically tailored for rubble mound seawalls. The newly proposed formulas for estimating overtopping layer thickness at both the seaward edge and the middle of the crest showed improvements compared to the existing formulas.

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In this study four experiments were conducted to investigate uncertainty in future longshore sediment transport (LST) projections due to: working with continuous time series of CSIRO CMIP6-driven waves (experiment #1) or sliced time series of waves from CSIRO-CMIP6-Ws and CSIRO-CMIP5-Ws (experiment #2); different wave-model-parametrization pairs to generate wave projections (experiment #3); and the inclusion/exclusion of sea level rise (SLR) for wave transformation (experiment #4). For each experiment, a weighted ensemble consisting of offshore wave forcing conditions, a surrogate model for nearshore wave transformation and eight LST models was used. The results of experiment # 1 indicated that the annual LST rates obtained from a continuous time series of waves were influenced by climate variability acting on timescales of 20-30 years. Uncertainty decomposition clearly reveals that for near-future coastal planning, a large part of the uncertainty arises from model selection and natural variability of the system (e.g., on average, 4% scenario, 57% model, and 39% internal variability). For the far future, the total uncertainty consists of 25% scenario, 54% model and 21% internal variability. Experiment #2 indicates that CMIP6 driven wave climatology yield similar outcomes to CMIP5 driven wave climatology in that LST rates decrease along the study area’s coast by less than 10%. The results of experiment #3 indicate that intra- and inter-annual variability of LST rates are influenced by the parameterization schemes of the wave simulations. This can increase the range of uncertainty in the LST projections and at the same time can limit the robustness of the projections. The inclusion of SLR (experiment #4) in wave transformation, under SSP1-2.6 and SSP5-8.5 scenarios, yields only meagre changes in the LST projections, compared to the case no SLR. However, it is noted that future research on SLR influence should include potential changes in nearshore profile shapes.@en

Although there have been many efforts in the literature to hindcast the patterns of longshore sediment transport (LST), they mainly disregarded uncertainty issues. Forcing datasets, wave transformation methods, and LST models are among the main sources of uncertainty in LST estimations. The combination of the aforementioned sources of uncertainty makes the estimation of LST patterns challenging for non-straight coastlines as the uncertainty ranges might vary from site to site. In this paper, a simple ensemble modeling framework was employed to investigate LST rate uncertainty at seven sites along a non-straight coastline (Gold Coast, Australia). The ensemble was formed by two different forcing datasets (i.e., integral parameters of total wave energy from two hindcast datasets of ERA5 and CAWCR), two different wave transformation methods (i.e., inclusion and exclusion of local wind effects), and eight LST models (i.e., bulk formulae and process-based models). Moreover, the relative importance of each source of uncertainty was ranked using the ANOVA-variance-based model. Finally, the weighted ensemble mean was used to investigate intra- and inter-annual variability of LST rates. The results showed that the range of uncertainty of LST rates for open coasts of Gold Coast is much higher than that of semi-sheltered coasts. On annual scale, for open coasts, 40% to 50% of total uncertainty was due to the choice of wave transformation methods, while for semi-sheltered coasts, it was 20%–30%. Moreover, almost for all sites, 30% to 50% of total uncertainty was controlled by the choice of LST models and the interaction of wave transformation methods and LST models. Although the weighted ensemble mean could provide an estimate of LST patterns along the coast, addressing the residual uncertainties (arising from other sources, discussed at the end of this paper), in future works, would help increase the certainty of the estimations.

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Seawalls play a significant role in protecting coastal areas against wave attack and flooding. The accurate estimation of wave overtopping at seawalls is therefore crucial to adequately protect people and infrastructure in these regions. In this study, the mean wave overtopping rate at rubble mound seawalls was investigated through 140 small-scale physical model tests which adds to the limited existing data for this structure type in the extended CLASH database called EurOtop (2018). The combined dataset is used to evaluate the prediction skill of existing empirical formulae and to identify their limitations. The role of wave steepness on the mean overtopping rate is closely examined as it has not yet been considered properly in the EurOtop (2018) formulation. A new formula was derived using dimensional analysis and physical justifications of the overtopping phenomenon. The formula was found to provide a 40% decrease in RMSE in comparison to that of the EurOtop (2018). In addition, the new formula yields a BIAS ≈0, a significant improvement compared to −0.38 (non-dimensional discharge) of the EurOtop (2018) formula. The proposed formula has a simple form where non-dimensional overtopping discharge depends only on the relative crest freeboard and wave steepness, which were found to be the most important variables based on a sensitivity analysis.

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Estimation of the mean overtopping discharge is a major task in the design and assessment of the crest level of rubble mound structures such as breakwaters and seawalls. The tolerable mean overtopping rates are given based on the associated risk and wave characteristics. Several empirical formulas have been developed for the prediction of mean overtopping discharge at coastal structures. These formulas can be applied to a wide variety of coastal structures, but have limited accuracy and/or do not reflect the physics of the phenomena. The main aim of this study is to overcome these issues for rubble mound structures by considering the physics of the process in the formula development. To achieve this, first, the references used in the extended CLASH database (also called the EurOtop-2018 database), were scrutinized, the reported wave characteristics were corrected (if required) and the rubble mound structure subset was extended using a recent study. Then noting that overtopping occurs when the wave runup exceeds the freeboard, the difference between the maximum wave runup and crest freeboard was considered as the governing parameter in the mean overtopping discharge formula. In the developed formula, a semi-empirical relationship between the mean overtopping and wave runup has been established. The performances of the developed formulas and existing ones were evaluated both qualitatively and quantitively. Accuracy metrics such as RMSE and BIAS indicated the superiority of the developed simple formula. Finally, a design formula to consider uncertainty and some guidelines are provided for practitioners.

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For a safe design of a rubble mound seawall, overtopping characteristics such as the mean overtopping discharge (q) and the maximum individual overtopping volume (V_max) should be limited. Unlike q, the estimation of V_max is more complex and requires a wave-by-wave analysis of overtopping as well as a statistical analysis. The present study contributes to the knowledge of the distribution of individual overtopping volumes and the estimation of V_max at rubble mound seawalls. A total of 135, small-scale 2D physical model tests were conducted across a practical range of crest freeboards and considered the slopes of 1:1.5 and 1:2. The well-known 2-parameter Weibull and Exponential distributions were first fitted to the experimental data to estimate the V_max. Different approaches to sample the observed distribution of wave-by-wave overtopping volumes were evaluated including a threshold method using the top 10%, 30%, and 50% of individual overtopping volumes, and a method that applies a greater weighting to the larger events. For both Weibull and Exponential distributions, the weighted method was found to be the best one providing a 23% and 17% decrease in scatter index (SI) values compared to the best of existing methods. To facilitate the estimation of V_max for design purposes, a simple empirical formula was developed as a function of the dimensionless mean overtopping discharge (q*) and the number of overtopping waves (N_ow). This formula with SI = 37% outperformed the distribution-based methods as well as the best of existing formulae for V_max. In the case of the normalised bias (NBIAS), the distribution methods underestimated V_max by −21% (Weibull) and −31% (Exponential) whereas the new formula yielded NBIAS = −6%.

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This study quantifies the uncertainties in the projected changes in potential longshore sediment transport (LST) rates along a non-straight coastline. Four main sources of uncertainty, including the choice of emission scenarios, Global Circulation Model-driven offshore wave datasets (GCM-Ws), LST models, and their non-linear interactions were addressed through two ensemble modelling frameworks. The first ensemble consisted of the offshore wave forcing conditions without any bias correction (i.e., wave parameters extracted from eight datasets of GCM-Ws for baseline period 1979–2005, and future period 2081–2100 under two emission scenarios), a hybrid wave transformation method, and eight LST models (i.e., four bulk formulae, four process-based models). The differentiating factor of the second ensemble was the application of bias correction to the GCM-Ws, using a hindcast dataset as the reference. All ensemble members were weighted according to their performance to reproduce the reference LST patterns for the baseline period. Additionally, the total uncertainty of the LST projections was decomposed into the main sources and their interactions using the ANOVA method. Finally, the robustness of the LST projections was checked. Comparison of the projected changes in LST rates obtained from two ensembles indicated that the bias correction could relatively reduce the ranges of the uncertainty in the LST projections. On the annual scale, the contribution of emission scenarios, GCM-Ws, LST models and non-linear interactions to the total uncertainty was about 10–20, 35–50, 5–15, and 30–35%, respectively. Overall, the weighted means of the ensembles reported a decrease in net annual mean LST rates (less than 10% under RCP 4.5, a 10–20% under RCP 8.5). However, no robust projected changes in LST rates on annual and seasonal scales were found, questioning any ultimate decision being made using the means of the projected changes.@en

Toe design is an important task for coastal engineers as it ensures the stability of the main armor layer and prevents scour in front of the armor slope. Several laboratory experiments have been conducted to investigate the toe stability using different testing approaches, i.e. damage due to a single test condition and cumulative damage due to a number of conditions. In addition, the methods of measuring and reporting damage to the toe are not the same as some researchers have counted only stones that were washed away from the toe; while others counted all the stones that have been displaced. Several formulas with different levels of success have been developed based on these studies. However, the scatter in the results is large and effects of some parameters are disregarded. The aims of this study are (a) to briefly review the abovementioned differences and exiting formulas, and (b) to develop a and physically sound formula for common design conditions, which considers the effect of all governing parameters. To achieve this, first a comprehensive data base from existing reliable studies was collected. Then nondimensional parameters which capture effects of governing parameters such as wave height, wave period, water depth, toe depth, toe width and foreshore slopes were deployed to develop a stability formula using physical reasoning and regression analysis. The new formula outperforms existing formulae for toe stability. The coefficient of variation of the formula was also determined to be used for probabilistic design applications. Finally, some design hints are provided for practitioners.

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Conventionally, allowable mean overtopping discharge is used as a design criterion for coastal structures. The mean overtopping discharge needs to be limited to ensure structural stability as well as the safety of people, vehicles, and properties behind the structure. Nowadays, limits for the maximum individual overtopping volumes are also specified in the design manuals, which requires the study of wave overtopping in wave-by-wave form. In some cases, in order to achieve more reliable safety for pedestrians and vehicles or to assess the stability of the inner slope, the maximum velocity and thickness of the overtopping flow need to be considered as well. The present paper aims to review the basic concepts of the individual wave overtopping such as the estimation of the probability of overtopping and the statistical methods to study the distribution of individual overtopping volumes. The temporal evolutions of the overtopping flow thickness / velocity along with the calculation of the overtopping discharge are discussed. Moreover, since the continuous recording of parameters is required for the study of individual wave overtopping, the most common experimental measurement and data analysis techniques with an emphasis on their advantages and limitations are discussed.

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