Bay Shape Equation Research Articles

Evaluation of beach response due to construction of submerged detached breakwater

Submerged detached breakwaters (SDBWs) have increasingly been used in recent times as an alternative against their emergent counterpart (EDBWs) to mitigate erosion because the former do not spoil the seascape. Both of these structures are (usually) constructed using precast concrete blocks or natural granite rocks, hence becoming permeable structures. For an EDBW, a parabolic bay shape equation can be readily used to estimate the planar shape of the shoreline behind the structure, but there is still no approach to estimate how the shoreline behind the SDBW is formed. In this study, we estimated how the shoreline is balanced by examining how the dominant wave direction changes due to the diffraction of the transmitted wave generated after the installation of the SDBW from the long-term wave directional spectrum. The change in dominant wave direction was determined under the shoreline gradient condition where littoral drift does not occur, considering the diffraction phenomenon due to the difference in transmitted waves. This means that the shape of the equilibrium shoreline changes to face perpendicular to the dominant wave direction. As a meaningful result, when the transmittance is 0, it converges to the well-known empirical equation of EDBW. The present methodology is validated by comparing the observed data (wave and shoreline change) from two beaches (Anmok and Bongpo-Cheonjin Beaches) on the eastern coast of Korea. This rational approach to shoreline changes behind permeable SDBWs will help in proactive review work for coastal management as well as beach erosion mitigation.

NEW PROCESS BASED EQUATION FOR A STATIC EQUILIBRIUM BEACH PLANFORM

Equilibrium planform formulations of beaches are engineering tools for providing solutions for beach erosion problems. They aim at defining the final orientation of the beach on a scale of years and thus they are used for evaluating the shoreline response due to human interference (ports, breakwaters, protection works…etc). These formulations create engineering solution to numerous coastal stability problems. The Parabolic Bay Shape Equation (PBSE) proposed by Hsu and Evans (1989) is the most used equation and allow to solve problems. However, these shape equations are valid under certain simplifications. Therefore, they are unable to determine the beach planform when the bathymetry is complex, with rocky areas or/and islands. The aim of this study is to present a new equilibrium beach planform equation (Gainza et al. 2018) that overcomes the problem previously mentioned. What is more, an extension of the model is presented in order to modeled the shape of a shoreline on a given day.

Equilibrium planform of pocket beaches behind breakwater gaps: On the shape of the equilibrium shoreline.

Bay beaches are common coastal landforms along the world's coastlines and have frequently been used as equilibrium coastal systems to mitigate erosion problems and stabilize coasts. Throughout the literature, several formulae can be found to obtain the static equilibrium planform (SEP) of such beaches on the leeward sides of single protruding headland structures. However, equations used to define SEP behind breakwater gaps are rare and based on a limited number of studies, especially when the obliquity angle ( β ) is large. This paper proposes a new formula for modelling the SEP of bay beaches that includes cases with planform shapes characterized by large obliquity angles ( β > 75°) for which the SEP is almost quasi-semicircular. The formula represents a general form of the parabolic bay shape equation (PBSE) with modified coefficients to alter the planform's curvature to be quasi-semicircular, mimicking the behavior of such bays in nature. The coefficients are expressed as functions of both the obliquity angle ( β ) and the curvature-adjustment angle ( φ ), which was determined based on field observations of the best-fitting SEP of 26 bay beaches with ( β > 75°) along the coasts of Spain, Portugal and Brazil. Additionally, 32 beach cases characterized by smaller obliquity angles ( β < 70°) were included in the derivation of the curvature adjustment angle, which was expressed as a function of ( β ). The model showed good results in modelling the SEP, with an RMSE of 0.90° obtained when estimating the planform's curvature-adjustment angle ( φ ) to obtain quasi-semicircular planform shapes for the prototype cases, confirming its utility as a valuable tool for engineering applications. • Static equilibrium planforms (SEPs) characterized by large obliquity angles are modelled. • A new formula was derived to plot the (SEP) of bay beaches behind gaps when (β > 75°). • A curvature-adjustment angle is used to alter the SEP's curvature based on field cases. • The SEP is quasi-semicircular for pocket beaches behind breakwater gaps when (β > 75°). • The curvature-adjustment angle is directly proportional to the obliquity angle (β).

Simulation of Bay-Shaped Shorelines after the Construction of Large-Scale Structures by Using a Parabolic Bay Shape Equation

Among the various causes of coastal erosion, the installation of offshore breakwaters is considered the main cause that influences the most serious changes in shorelines. However, without a proper means for predicting such terrain changes, countries and regions continue to suffer from the aftermath of development projects on coastal land. It has been confirmed that the parabolic bay shape equation (PBSE) can accurately predict shoreline changes under the wave climate diffracted as a result of such development projects. This study developed a shoreline change model that has enhanced the previous shoreline change models by applying PBSE to shoreline changes into bay-shaped features. As an analytical comparison with the second term of the GENESIS model, which is an existing and well-known shoreline change model, a similar beach erosion width was obtained for a small beach slope. However, as the beach slope became larger, the result became smaller than that of the GENESIS model. The validity of the model was verified by applying it to satellite images that demonstrated the occurrence of shoreline changes caused by breakwaters for seaports on the eastern coast of Korea; Wonpyeong beach, Yeongrang beach, and Wolcheon beach. As a result, each studied site converged on the static equilibrium planform within several years. Simultaneously, the model enabled the coastal management of the arrangement of seaports to evaluate how the construction of structures causes serious shoreline changes by creating changes to wavefields.

Impact Assessment of Beach Erosion from Construction of Artificial Coastal Structures Using Parabolic Bay Shape Equation

Impact Assessment of Beach Erosion from Construction of Artificial Coastal Structures Using Parabolic Bay Shape Equation

A MODIFIED HYPERBOLIC TANGENT EQUATION TO DETERMINE EQUILIBRIUM SHAPE OF HEADLAND BAY BEACHES

When designing any artificial beach, it’s desirable to avoid (or minimise) future maintenance commitments by arranging the initial beach planshape so that it remains in equilibrium given the incident wave climate. Headlands bays, or embayments, where a sandy beach is held between two erosion resistant headlands, tend to evolve to a stable beach planshape with little movement of the beach contours over time. Several empirical bay shape equations have been derived to fit curves to the shoreline of headland bay beaches. One of the most widely adopted empirical equations is the parabolic bay shape equation, as it is the only equation that directly links the shoreline positions to the predominant wave direction and the point of diffraction. However, the main limitation with the application of the parabolic bay shape equation is locating the downcoast control point. As a result of research presented in this paper a new equation, based on the hyperbolic tangent shape equation was developed, which eliminates the requirement of placing the down coast control point and relies on defining a minimum beach width instead. This modified equation was incorporated into a new ArcGIS tool.

Dynamic equilibrium planform of embayed beaches: Part 1. A new model and its verification

Equilibrium beach formulations are useful tools for diagnosing and managing coastal engineering problems, providing solutions for beach erosion problems. Headland Bay Beaches (HBBs) can be used as equilibrium coastal systems for stabilizing coastlines and mitigating erosion problems. These embayed beaches may exist in a state of static or dynamic equilibrium. Throughout the literature, several equations can be found for obtaining the Static Equilibrium Planform (SEP) of Headland Bay Beaches (HBBs) with almost negligible net littoral drift rates. However, the formulations used to define the Dynamic Equilibrium Planform (DEP) of embayed beaches with specific net sediment transport rates are scarce, and based on a limited number of studies. This paper proposes a new derived formula for obtaining the planform shape of HBBs in dynamic equilibrium conditions. The formula represents a general form of the Parabolic Bay Shape Equation (PBSE) with modified coefficients as a function of both the wave obliquity (β) and the net littoral drift rate passing through the bay (Q). The angular difference (γd) between the direction of the mean wave energy flux at the diffraction point of the headland and the beach orientation down-coast is also incorporated in the proposed formula. The model was verified against natural HBBs in dynamic equilibrium with different net littoral drift rates along the Brazilian coast, producing good results.

On the influence of wave directional spreading on the equilibrium planform of embayed beaches

Equilibrium beach formulations are engineering tools for estimating a response of beaches in the long term. They aim at defining the final orientation of a beach on a scale of years and thus they are used for evaluating the shoreline response due to human interference (ports, breakwaters, protection works, etc). Several equations can be found in the literature for obtaining the Static Equilibrium Planform (SEP) of Headland Bay Beaches (HBBs), one such being the Parabolic Bay Shape Equation (PBSE). The SEP is strongly dependent on the location of the down-coast control point (Po) which is the downdrift limit from which the PBSE is applicable. The literature recommends the determination of the (Po) point by means of the direction of the mean wave energy flux at the diffraction point. However, when this is applied to equilibrium embayed beaches in zones with a wide variability of wave climate directionality and/or in cases where the diffraction point is located far from the equilibrium shoreline, this approach is no longer valid and thus the resulting planform shape does not properly fit the SEP for such conditions. This paper investigates the methodology for locating the (Po) point of the SEP for such cases, exploring the role of wave climate directional spreading, employing 44 HBBs in Spain and Latin America. It correlates the planform shape in the long term with the directional variability of the wave climate at the diffraction point. Additionally, an extensive series of numerical simulations using a spectral wave model was carried out to model the combined effects of refraction-diffraction in the lee of a breakwater, defining the part affected by the coastal structure under different wave conditions. The results clarify the importance of wave directional spreading in locating the (Po), indicating that the wider the directional spreading the farther the (Po) point on the shoreline. Moreover, the farther the diffraction point from the coast, the smaller the part of the beach affected by the coastal barrier. A new formula has been derived to locate the down-coast control point (Po) of the parabolic part of the shoreline corresponding to the PBSE as a function of the directional variance of the wave climate and the location of the diffraction point. The model produced good results with (R2 = 0.9117 and a RMSE = 1.8297°) in estimating the location of the (Po) point in various natural and man-made beaches with different degrees of wave directionality.

Experimental study on dynamic equilibrium of headland-bay beaches

A dynamic equilibrium bay (DEB) is an embayment with continuous sediment supply and its shoreline planform can remain stable over a long period of time without erosion or accretion. For coastal conservation of sandy headland-bay beaches (HBB), the concept of using a static equilibrium bay (SEB) is well known, but that for DEB has received little attention. Moreover, an empirical equation for the stability of a DEB is not yet available. Experiments on DEB shape that aim to derive new coefficients in the parabolic bay shape equation (PBSE) for DEB are now being conducted in the laboratory. The work commences from an initial artificial HBB in static equilibrium with sediment supply source from the lee of an upcoast headland. A final equilibrium planform is obtained for the condition with a specific wave obliquity and sediment supply rate until no further shoreline change is found. In order to fit the PBSE for a DEB, a new parameter called SSR (sediment supply ratio) that represents the ratio of sediment supply rate from the source and the potential longshore sediment transport rate is introduced to quantify the balance of sediment to the bay. Alternative C coefficients in the PBSE for DEB, which include wave obliquity and the SSR, are then calculated. These new coefficients for DEB can now be used to evaluate the influence of sediment supply from a riverine source on a DEB and to classify its equilibrium status for planning sediment management strategies in coastal conservation.

Equilibrium of Crenulated Bays in Thailand

The C coefficients in the now well-known parabolic bay shape equation (PBSE) are the most important parameters for verifying the bay beach stability or designing a headland bay beach in static equilibrium. These coefficients are revised specifically for bays in static and dynamic conditions in Thailand. Three techniques, namely Least Square (LS), Genetic Algorithms (GA), and Differential Evolution (DE) are used to determine the alternative coefficients. The newly derived C coefficients for static equilibrium bays (SEB) differ slightly from the original coefficients. Dynamic equilibrium bays (DEBs) with sediment within the embayment are analyzed to obtain a new set of C coefficients, which are a function not only of wave obliquity but also sediment supply ratio (SSR), which is a new parameter representing the ratio between the rate of sediment supplied from a river source to the bay and the longshore sediment transport rate. The resulting new C coefficients for DEBs vary uniformly and systematically, similar to those for the SEBs, and are expected to be applicable to DEBs elsewhere for coastal management.

Wave energy distribution and morphological development in and around the shadow zone of an embayed beach

BackgroundThe curved shoreline shape of embayed beaches is one of its most notable characteristics and can be described using the parabolic bay shape equation (PBSE). Wave diffraction in and around the shadow zone is often regarded as the primary forcing mechanism leading to the prominent curvature of the shoreline. However, wave climate variables (wave direction, directional spreading and wave height) are shown to be influential in redistributing wave energy throughout the bay and in the shadow zone. MethodsIn this study, a process-based morphological model (Delft3D) is used for hydrodynamic and morphodynamic simulations of a schematic embayed beach. Wave forcing conditions are systematically varied between a mixture of time-invariant and time-varying cases. ResultsThe role of diffraction is shown to be dominant only when the wave conditions are both narrow-banded (<20°) and when the PBSE angle β is high (>30°). Otherwise, as little as 6% variation in wave direction within a 90° range can account for the shoreline curvature in and around the shadow zone. ConclusionThe degree to which wave direction and directional spreading vary through time therefore has a large effect on the equilibrium orientation and shoreline planform of the bay.

A free software for sand and gravel embayed beach modelling: PhoEbuS- parabolic equation shape

An in-depth knowledge of the littoral system is essential for its conservation and for an accurate planning of due interventions. The parabolic bay shape equation, used to study the embayed beach planform, is a qualitatively assessed instrument to obtain information about beach equilibrium conditions. Nowadays the equation has been extended in order to be applicable not only to sand beaches but also to gravel beaches. In this paper Phoebus user-friendly software for an automatic use of the extended parabolic bay shape equation is presented. The software enables both technical experts and non-professionals to conduct expeditious analysis in order to evaluate the evolution and equilibrium conditions of embayed littoral beaches lying between natural headlands or man-made structures, as well as to artificially recreate stable embayed beaches by building new artificial structures on open beaches. The software can be also used for coastal planning, to anticipate beach modifications in response to new building or to changes in pre-existing structures.

Application of the parabolic bay shape equation to sand and gravel beaches on Mediterranean coasts: Reply to the comments of D.E. Reeve

The authors wish to focus on certain aspects which had not been discussed previously in Schiaffino et al. (2012). These aspects cover the methodology adopted to obtain the coefficients proposed for the parabolic bay shape equation and the approach adopted for the evaluation of their applicability and reliability for sand and gravel beaches. In particular, results concerning shoreline variability as regards control point and wave direction are submitted.

Comment on ‘Application of the parabolic bay shape equation to sand and gravel beaches on Mediterranean coasts’, by Schiaffino et al. (2011)

The authors have provided an interesting proposal for modifying existing methods for predicting the shape of equilibrium bays. They have proposed separate formulae for sand and gravel beaches, based on an analysis of beaches across the Mediterranean and an optimised fitting procedure. This discussion explores the applicability of the proposed model in the context of the uncertainties in specifying the key parameters.

Application of the parabolic bay shape equation to sand and gravel beaches on Mediterranean coasts

The equation most commonly used to describe the bay planform was proposed by Hsu and Evans (1989) and it was obtained through empirical analysis of sand beach planforms. In the last decade interest in gravel sediments increased owing to their greater stability on beaches, compared with sand sediments. Due to the differences between the morphodynamics of sand and gravel beaches, which is strictly influenced by their different hydraulic characteristics, it was necessary to create a predictive instrument for this beach type as well. Therefore, in this study the standard Hsu and Evans equation (1989) was modified in order to make it applicable not only to sand beaches but also to gravel beaches. The shoreline was computed according to a parabolic model in polar coordinates whose coefficients are considered linearly dependent on the wave direction and related to the beach type. The estimation of the free parameters of the model was performed according to a statistical analysis of a shorelines data set of Mediterranean sand and gravel embayed beaches.

Study on headland-bay sandy coast stability in South China coasts

Headland-bay beach equilibrium planform has been a crucial problem abroad to long-term sandy beach evolution and stabilization, extensively applied to forecast long-term coastal erosion evolvement and the influences of coastal engineering as well as long-term coastal management and protection. However, little concern focuses on this in China. The parabolic relationship is the most widely used empirical relationship for determining the static equilibrium shape of headland-bay beaches. This paper utilizes the relation to predict and classify 31 headland-bay beaches and concludes that these bays cannot achieve the ultimate static equilibrium planform in South China. The empirical bay equation can morphologically estimate beach stabilization state, but it is just a referential predictable means and is difficult to evaluate headland-bay shoreline movements in years and decades. By using Digital Shoreline Analysis System suggested by USGS, the rates of shoreline recession and accretion of these different headland-bay beaches are quantitatively calculated from 1990 to 2000. The conclusions of this paper include that (a) most of these 31 bays maintain relatively stable and the rates of erosion and accretion are relatively large with the impact of man-made constructions on estuarine within these bays from 1990 to 2000; (b) two bays, Haimen Bay and Hailingshan Bay, originally in the quasi-static equilibrium planform determined by the parabolic bay shape equation, have been unstable by the influence of coastal engineering; and (c) these 31 bays have different recession and accretion characters occurring in some bays and some segments. On the one hand, some bays totally exhibit accretion, but some bays show erosion on the whole. Shanwei Bay, Houmen Bay, Pinghai Bay and Yazhou Bay have the similar planforms, characterized by less accretion on the sheltering segment and bigger accretion on the transitional and tangential segments. On the other hand, different segments of some bays have two dissimilar evolvement characters. Dacheng Bay, Shenquan Bay, Hudong Bay, Wukan Bay, Fengjia Bay, Wuchang Bay, Lingshui Bay and Tufu Bay produce accretion on the tangential segment, erosion on the transitional segment and accretion on the sheltering segment. However, Guang’ao Bay, Haimen Bay, Jinghai Bay, Sanya Bay(a), Dajiao Bay, Hailingshan Bay, Hebei Bay, Fuhu Bay, Shuidong Bay, Wangcun Bay and Bomao Bay generate erosion on the tangential part, accretion on the transitional part and accretion on the sheltering part. It seems to imply some relations between headland-bay beach evolvement and controls on headland-bay beaches, which may possibly to classify headland-bay beach types and should be further studied.

Uncertainty in the application of the Parabolic Bay Shape Equation: Part 1

This paper aims to quantify the uncertainty in the application of the Parabolic Bay Shape Equation (PBSE) for a bay in static equilibrium. By means of expert elicitation, a database was generated consisting of the position of the control points that determine the Static Equilibrium Planform (SEP). The elicitation consisted of two parts. In Part 1, 22 expert volunteers were asked to place the three control points that determine the SEP on a vertical aerial photograph of Taquaras–Taquarinhas Bay, a morphologically stable bay in the south of Brazil, approximately 1800 m in width and with an indentation of 750 m. The distribution of the location of the SEP along four evenly spaced (200 m) profiles in the southern part of the bay was determined. The overall bias of the location of the SEP calculated over the four profiles was in the order of 41 m (landward) and the average standard deviation was 116 m. These parameters increased when moving alongshore toward the curved section of the bay. This means that the uncertainty in the application of the PBSE is dependent on the particular point of interest along the bay. In Part 2 of the elicitation, 30 volunteers participated. This time the consequence of the placement of the control points (the corresponding SEP) was visible. Comparing the results from Parts 1 and 2, it was observed that when volunteers were directly confronted with the result of the placement of the control points (a plotted SEP) a much smaller variation in the position of the SEP occurred. This implies that when feedback on the result is provided the PBSE is a more robust method.

Uncertainty in the application of the parabolic bay shape equation: Part 2

This paper attempts to highlight issues that are relevant in the application of the Parabolic Bay Shape Equation (PBSE) to a non-equilibrium bay. For this case, the bay of Imbituba in southern Brazil was chosen. The construction of a breakwater to shelter the port of Imbituba in the south of the bay was accompanied by an increase in sedimentation to the port area from an eroding downdrift beach. Superimposed plots of the coastline of the Bay of Imbituba from different years confirm a general trend of accretion of the southern part of the bay accompanied with a retreat of the coastline in the northern part. After the application of the PBSE it became clear that the breakwater caused a change in the equilibrium state of the bay. Between 1947 and 2001 the Bay of Imbituba has changed from a dynamic equilibrium to a natural beach reshaping or self-reshaping. The tendency of the sedimentation of the southern part of the bay can be explained by the SEP associated with the new updrift diffraction point (tip of the breakwater): The seaward position of the SEP predicts a need for additional sediment in order to achieve a stable plan form.

Static bay beach concept for scientists and engineers: A review

Headland-bay beach (HBB) is one of the most prominent physiographic features on the oceanic margin of many countries in the world. Under the influence of a predominant swell, its curved periphery in natural environment may reach static equilibrium and remains stable without sediment supply from updrift and/or a riverain source within its own embayment. Coastal scientists and engineers have attempted to develop mathematical expressions to quantify this ideal bay shape since the 1940s. As the scenario with depleting sediment supply has become a common reality on many parts of the world coastline in more recent time, some coastal engineers have advocated a rational approach to mimicking the static bay shape found in nature in order to mitigate beach erosion as well as for coastal management. Nowadays, many useful applications have emerged since the publication of the parabolic bay shape equation (PBSE) developed for static equilibrium planform (SEP) in late 1980s. The advance in modern computer technologies and international collaboration has further facilitated the exchange of knowledge and applications of this static bay beach concept (SBBC). This paper reviews the progress on the research in HBB made by geographers and geologists since the 1940s, together with the involvement of coastal engineers working on shore protection and management from the 1960s. Aided by software MEPBAY developed in 2003 for the PBSE, several practical examples are demonstrated in this paper for the stability verification of natural and man-made bay beaches, shore protection, mitigating downdrift erosion of a harbour, design of recreational bay beach, as well as for environmental impact assessment. Limitations on and clarifications for engineering applications based on the SBBC are also discussed in this paper.

Appreciation of Static Bay Beach Concept for Coastal Management and Protection

Sandy bay beach in static equilibrium is the only physiographic feature that could remain stable without sediment supply in a persistent swell environment anywhere around the world. With a continuing trend of erosion to sandy beaches worldwide, a conservative and practical hypothesis is not to consider the mobile dynamic wave parameters but to safeguard against a sandy beach under vanishing supply in the future. To achieve this, an empirical bay shape equation is strongly recommended as one of the best coastal management options. Combining with MEPBAY, software derived for implementing the static bay beach concept on a display unit, this empirical bay shape equation is readily applicable to verify the stability of numerous headland bay beaches in natural and man-made condition. It is also adequate for coastal managers to apply this empirical equation as a valuable tool to aid in the creation of headland bay beaches for shore protection, recreation, and coastal management before embarking on sophisticated numerical modeling.