Headland-bay Beach Research Articles

Iterative Mathematical Models Based on Curves and Applications to Coastal Profiles

The objective of this study is iterative systems based on general types of curves, not only on circumference arcs. We begin by presenting some implementations and generalizations of constructions based on arcs of circumference. Then we consider constructions based on general curves and give a “universal property” relating to the primary construction that exploits arcs of circumference. With the prospect of applying these theoretical models also to coastal geomorphology in the future, and inspired by one of the best-known models on the subject, the logarithmic spiral one for the so-called headland-bay beaches (HBBs), we study geometrically some cases in which the constructions are based on arcs of the golden spiral. Simultaneously we concretely illustrate and explain the universal property above. Finally we dedicate a section to discuss the possibility of how to numerically evaluate and compare the (infinite) lengths originating from our theoretical geometric constructions. Some explicit examples, calculations and comparisons will be provided by the use of infinity computing which is one of the various possible assets that contemporary non-standard mathematics makes available.

Headland-Bay Beaches: Static Equilibrium Concept for Shoreline Management

Headland-Bay Beaches: Static Equilibrium Concept for Shoreline Management

Classifying Headland-Bay Beaches and Dynamic Coastal Stabilization

In this paper, a framework is developed for classifying bay types using stability and a sediment supply source. The framework is used to classify a total of 212 headland-bay beaches in Southeast Asia. The results show that static bays, bays with no sediment supply, and dynamic bays (with a sediment supply), account for 36% and 64%, respectively, while stable bays, bays that can maintain their long-term shoreline stability, and unstable bays (changing their shape over time), account for 69% and 31%, respectively. The results reveal the importance of dynamic bays. The dynamic parabolic bay shape and bay characteristic equations have been verified to bridge the knowledge gap of coastal stabilization and management in dynamic bays. The verification of bay characteristic equations shows an efficiency index of more than 78%. The bay characteristic equation shows that dynamic bays are highly sensitive to low sediment supply and become less sensitive when the sediment supply increases. Knowledge of the coastal stabilization concept successfully implemented for static unstable bays has been extended in this study and applied to stabilize dynamic unstable bays using the verified equations. Sediment control and the combined method are developed in this study, and a case study is presented on the stabilization of a dynamic unstable bay.

An analytical model for beach erosion downdrift of groins: case study of Jeongdongjin Beach, Korea

Abstract. Beach erosion at the unprotected downdrift end of a groin is common with waves approaching the structure obliquely. This phenomenon has often occurred on the downdrift side of natural groins on the east coast of South Korea during high waves in winter months. The resulting planform assumes a distinctive crenulate shape with a maximum indentation point landward of the erosion. An analytical model is employed to study the beach erosion at the downdrift end of a natural rock groin at Jeongdongjin Beach in Korea, using mathematical equations derived from the parabolic model for headland-bay beaches in static equilibrium, to predict the downdrift control point and maximum indentation of the eroded shoreline. These equations are solved using the prevailing wave height, wave angle at breaking and wave direction derived from analyzing NOAA's wave data over 40 years and the longshore sediment transport rate calculated from the wave data. The location of the calculated maximum indentation is also verified using shoreline video monitoring data and compared with the result of a one-line numerical model for shoreline change. The limitation of the proposed analytical model is discussed as is the effect of sediment bypassing the groin.

MeePaSoL: A MATLAB-based GUI software tool for shoreline management

Headland-bay beaches (HBBs) are ubiquitous on Earth and many coastal scientists and engineers have attempted to define their planform since the 1940s, by employing laboratory tests, analytical tools, numerical models, and empirical equations. One of the latest attempts in numerical modeling is the development of a MATLAB-based GUI software called “MeePaSoL” to facilitate the application of the empirical parabolic model for HBBs. This visualization package can be applied to determine the predominant wave direction to the shoreline planform of a HBB in static equilibrium, and to minimize the uncertainty in locating downdrift control point while using other software. The background concepts (e.g., parabolic model, equilibrium beach and wave orthogonality hypothesis) are introduced, and the operation procedure with stepwise guides for applying MeePaSoL are given to assist the users. The reliability of MeePaSoL is validated by comparing the wave direction given by this package with that by the wave energy flux approach derived from field data. Examples of engineering applications are then demonstrated, covering local to regional wave direction, the effect of jetty/breakwater length and direction, harbor siltation in a macro-tidal environment, and devising engineering works for shoreline management in Korea. Sensitivity and uncertainty are also discussed.

Flow dynamics over a high, steep, erosional coastal dune slope

Flow dynamics over a high, unvegetated, and steep scarp slope that fronts a severely eroded relict transgressive dunefield were investigated at Salmon Hole (also known as Post Office Rock), a small headland-bay beach located near Beachport, south east South Australia. The ~15-metre-high steep dune at Salmon Hole provided the opportunity to conduct a wind flow experiment on a larger, higher and longer stoss slope than previously studied. The scarp slope is comprised of segments of varying slope that have a significant impact on flow dynamics over the dune. Percentage speed up and a decrease in turbulence were recorded up the stoss slope due to streamline convergence and flow compression. However, flow expansion at a change in gradient on the upper stoss slope caused a significant drop in wind speed and an increase in turbulence, contrary to what has previously been found in the literature where maximum percentage speed up is primarily recorded at the crest. Topographic steering typically seen in wind flow over scarps and foredunes was observed at Salmon Hole along with flow separation and the formation of a reversing vortex on the lee slope. This study also demonstrates how a lack of sediment delivery back to the beach and thence to the dune between storm events results in the inability for dune recovery or translation. However, the Salmon Hole study shows that blowouts can still develop and grow through dune cannibalization regardless of the lack of sediment supply from the beach and the recession of the shoreline.

Coastal erosion in sandy beaches along a tectonically active coast: The Chile study case

Coastal erosion in 45 sandy beaches covering nearly 2000 km along the tectonically active Chilean coast is assessed during the last four decades. The historical analysis is based on the assessment of decadal changes of the shoreline position extracted from topographic surveys, aerial photographs, satellite images and survey maps using the DSAS software. Results show that 80% of the sites presented erosion rates (>−0.2 m/y), 7% beaches accreted (>0.2 m/y) while 13% remained stable. Eroded beaches include headland bay beaches, embayed and pocket beaches. A discussion on the possible causes explaining these results is conducted. While changes in offshore wave climate are spatially smooth within the region, relative mean sea level changes are highly variable and modulated by tectonic activity; the reduction of the sediment supply explains erosion rates in few cases.

Application of the Static Headland-Bay Beach Concept to a Sandy Beach: A New Elliptical Model

The headland-bay beach is one of the most common coastal types in the world. Its morphology reflects the changes that occurred during long-term evolution of the sandy coast. Several headland-bay beach models have been proposed to simulate the coastline’s configuration in equilibrium. In this paper, a new elliptical model is proposed, described, and applied. On the east coast of Laizhou Bay in Shandong Province from Longkou Port to Diaolongzui, four typical headland-bay beaches have developed, and four headland-bay beach models are used in this paper to simulate the morphology of these beaches to assess the applicability of each model. The simulation results of the elliptical model verify that it is applicable to the study area. In addition, the elliptical model is easy to use. Through simulation and field investigations, we concluded that most of the coastal segments in this area will remain in an erosion state, and the human activity has a significant impact on the shoreline’s evolution.

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.

Longitudinal distribution of slope and sediment characteristics in headland-bay beaches in Cádiz, Spain

Rodriguez-Polo, S.; Del Rio, L., and Benavente, J., 2018. Longitudinal Distribution of Slope and Sediment Characteristics in Headland-bay Beaches in Cadiz, Spain. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 306–310. Coconut Creek (Florida), ISSN 0749-0208.Headland-bay beaches are systems with an asymmetric curved planform that develop downdrift of a natural or artificial coastal headland. These beaches have some common features found along the coasts of the world, presenting a well differentiated zonation. The main objective of this work is to elucidate how parameters such as grain size, sorting and beachface slope in the different zones (shadow, central and distal sectors) determine the final morphology of the beach. For this purpose, twelve headland-bay systems in the coast of Cadiz (South Spain), with contrasting wave and tide regimes, were monitored in order to analyze the longitudinal gradient in slope and sediment distribution. Topographic surveys were conducted and sediment samples were collected and analyzed for grain size. Shore-normal transects were surveyed with a RTK-DGPS (Real Time Kinematics - Differential Global Positioning System) and beach slope was computed on each sector of the beaches. Three sediment samples were taken in each transect, corresponding to the dry beach, the high intertidal zone and the lower intertidal zone. A total of 152 topographic profiles and about 456 sediment samples were analyzed. All this information has allowed to gain insight into how the different parameters vary along the twelve headland-bays, as well as to establish common trends among them and to see how these parameters affect the final morphology of the beach. In a further step, future work will be focused on the detailed analysis of wave processes on each system, in order to more accurately determine the relations between the above parameters and the hydrodynamic regime.

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.

Using Wind Direction and Shoreline Morphology to Model Sand Dune Mobilization

Carvalho, A.M.C.; Ellis, J.T.; Lamothe, M., and Maia, L.P., 2016. Using wind direction and shoreline morphology to model sand dune mobilization.This paper aims to establish a relationship between wind direction, shoreline position, and aeolian transport directions. Ceará State in NE Brazil, which mainly comprises barchans, barchanoids, parabolic dunes, and large flattened composed dunes was used to formulate and test a dune transport model that can be applied to north and east facing beaches. The headland-bay beaches of Ceará progressively increase in angle from E to W. This is accompanied by a corresponding change in the near-beach dune migration direction. A significant change in the wind pattern for the northern portion of Ceará was found, which resulted in a shift in dune migration from E-W to ENE-WSW as the shoreline orientation changed from NW-SE to W-E. The different aeolian transport conditions along the stretches of the headland-bay beach gradually shifted from transport parallel to the shoreline to unimpeded aeolian transport that moves sediments inland. Beach feeding occurs from headland bypass as dunes occupy the area landward of the headland. A zone of composite wind transport directions, winds carrying sand directly from the beach and those transporting sediment landward of the headland, was identified. This variation of wind transport conditions is confirmed by the dune morphology and influenced by the relationship between shoreline position and wind angle approach. The proposed model uses the sine of the angle between wind direction and shoreline orientation to predict aeolian transport and determines the volume of sediment transported by the wind. This model identifies stretches of shoreline under different aeolian transport directions and exhibits compatibility with measured dune migration.

Typical equilibrium beach profile models and their significances from different segments of a headland-bay beach

This study introduces three typical models on equilibrium beach profile, and discusses the application limitations of these models. Then this study examines the selections for applying these models on different coastal segments of a headland-bay beach in west Guangdong, South China, and explores the physical significances of those parameters in the models. The results indicate that: (1) Bodge’s model is more in line with the equilibrium beach profile of the tangential or transitional segment, whereas Lee’s model is more consistent with the shadow profile; (2) most of the parameters in three models have clear physical significances in accordance with the actual characteristics of this headland-bay beach; and (3) both the selections for the equilibrium beach profile from different segments and significances of most of the parameters in three models are in essence correlated with the morphodynamic states at various coastal locations.

Assessing Embayed Equilibrium State, Beach Rotation and Environmental Forcing Influences; Tenby Southwest Wales, UK

The morphological change of a headland bay beach—Tenby, West Wales, UK—was analysed over a 73-year period (1941–2014). Geo-referenced aerial photographs were used to extract shoreline positions which were subsequently compared with wave models based on storm event data. From the 1941 baseline, results showed shoreline change rates reduced over time with regression models enabling a prediction of shoreline equilibrium circa 2061. Further temporal analyses showed southern and central sector erosion and northern accretion, while models identified long-term plan-form rotation, i.e., a negative phase relationship between beach extremities and a change from negative to positive correlation within the more stable central sector. Models were then used in conjunction with an empirical 2nd order polynomial equation to predict the 2061 longshore equilibrium shoreline position under current environmental conditions. Results agreed with previous regional research which showed that dominant south and southwesterly wave regimes influence south to north longshore drift with counter drift generated by less dominant easterly regimes. The equilibrium shoreline was also used to underpin flood and inundation assessments, identifying areas at risk and strategies to increase resilience. UK shoreline management plans evaluate coastal vulnerability based upon temporal epochs of 20, 50 and 100 years. Therefore, this research evaluating datasets spanning 73 years has demonstrated the effectiveness of linear regression in integrating temporal and spatial consequences of sea level rise and storms. The developed models can be used to predict future shoreline positions aligned with shoreline management plan epochs and inform embayed beach shoreline assessments at local, regional and international scales, by identifying locations of vulnerability and enabling the development of management strategies to improve resilience under scenarios of sea level rise and climate change.

Impacts of enhanced central Pacific ENSO on wave climate and headland-bay beach morphology

Wave climate and Pacific basin coastal behaviour associated with El Niño Southern Oscillation (ENSO) is understood at a reconnaissance level, but the coastal response to different central Pacific (CP) versus eastern Pacific (EP) flavours of ENSO is unknown. We show that CP ENSO events produce different patterns of directional wave power to EP ENSO along the southeast Australian shelf and southwest Pacific region, because of significant variability in trade-wind wave generation. The modulation of the trade wind wave climate during CP ENSO has thus far been neglected in existing coastal process studies. We also show that coastal change between CP and EP ENSO cannot be inferred from shifts in the deepwater wave climate. This is because variability in trade wind wave generation is masked in deepwater by the persistence of high power extra-tropical waves that have reduced impact on nearshore processes due to high wave refraction. Morphodynamic modelling in a headland-bay beach indicates that CP ENSO leads to higher coastal erosion potential and slower post-storm recovery than EP ENSO during an El Niño/La Niña cycle. We show that the alongshore variability in beach morphological type can be used to model the static equilibrium planform response for each ENSO phase. Results indicate that shoreline response to ENSO in most headland-bay beach coasts is not as simple as the existing paradigm that (anti-) clockwise rotation occurs during El Niño (La Niña). Our methods provide a second-order approach to project coastal response and predict the discrete shoreline rotations for ENSO flavours.

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.

Mesoscale Morphological Change, Beach Rotation and Storm Climate Influences along a Macrotidal Embayed Beach

Cross-shore profiles and environmental forcing were used to analyse morphological change of a headland bay beach: Tenby, West Wales (51.66 N; −4.71 W) over a mesoscale timeframe (1996–2013). Beach profile variations were attuned with longer term shoreline change identified by previous research showing southern erosion and northern accretion within the subaerial zone and were statistically significant in both sectors although centrally there was little or no significance. Conversely a statistically significant volume loss was shown at all profile locations within the intertidal zone. There were negative phase relationships between volume changes at the beach extremities, indicative of beach rotation and results were statistically significant (p < 0.01) within both subaerial (R2 = 0.59) and intertidal (R2 = 0.70) zones. This was confirmed qualitatively by time-series analysis and further cross correlation analysis showed trend reversal time-lagged associations between sediment exchanges at either end of the beach. Wave height and storm events displayed summer/winter trends which explained longer term one directional rotation at this location. In line with previous regional research, environmental forcing suggests that imposed changes are influenced by variations in southwesterly wind regimes. Winter storms are generated by Atlantic southwesterly winds and cause a south toward north sediment exchange, while southeasterly conditions that cause a trend reversal are generally limited to the summer period when waves are less energetic. Natural and man-made embayed beaches are a common coastal feature and many experience shoreline changes, jeopardising protective and recreational beach functions. In order to facilitate effective and sustainable coastal zone management strategies, an understanding of the morphological variability of these systems is needed. Therefore, this macrotidal research dealing with rotational processes across the entire intertidal has significance for other macrotidal coastlines, especially with predicted climate change and sea level rise scenarios, to inform local, regional and national shoreline risk management strategies.

Condiciones geodinámicas derivadas del terremoto y tsunami de 2010 en la costa de Chile central: El caso de Pichilemu

The earthquake and tsunami of 2010, triggered changes in the geomorphological and sedimentary dynamics along the coast of Pichilemu, in central Chile. These changes were evidenced through the retreat and erosion of sandy beaches and coastal sand dunes. Modifi cation patterns were identifi ed through analysis of satellite images prior to and after the earthquake/tsunami, and compared with records from geomorphological fi eldwork carried out in 2004 that measured essential features of representative beach profi les in the headland bay beaches. The variability of beaches dominated by waves was analyzed. The results indicate that there is coastal erosion and a change in the sedimentary balance associated with the subsidence of the coast, and that four years after the earthquake/tsunami, after the erosive phase, a change in tendency towards progradation of sandy coastlines in the headland bay beaches can be observed

The sedimentary source, planform stability and shore normal morphological change of the Xichong beach on the southern coast of the Dapeng Peninsula of Shenzhen, Guangdong Province, China

The coast of the Dapeng Peninsula has been honored as one of “the eight most beautiful coasts in China”. The most precious tourism resource for the peninsula is headland bay beaches, among which the beach at Xichong on the southern coast of the peninsula is the longest and the most important one. The information of the stability, sedimentary source and shape change of the beach is very important for maintaining the beach in terms of sustainable development of the peninsula. Heavy minerals in the sand of the beach and the inland stream at Xichong are compared with those of a nearby beach on the same coast to determine the beach sand source; with help of a computer software, MEPBAY, the equilibrium planforms of the beaches on the peninsula are compared with those of an island without rivers to evaluate the stream’s effects on the beach stability; cross shore profiles along the Xichong beach are also surveyed in different seasons of a year to assess the annual shore normal beach changes affected by the stream input, and the relation between the equilibrium planform state and cross shore changes of the beach. It is shown that (1) stream is the main sedimentary source of the beach and the weathering materials of the rocky headlands on both sides of the bay transported by waves are the second source for the beach but it is limited, sand from an inner shelf is not the sedimentary source for the beach at present and was not even during the Holocene transgression; (2) the Xichong beach cannot reach static equilibrium around the entire bay shoreline, the segment of the shoreline where a stream outlet is located is in dynamic equilibrium, and the unstable section occurs in the wave shadow region in the lee of an offshore island; (3) no matter whether the section of the beach shoreline at Xichong is in an equilibrium state or not, it is eroded in the typhoon season and recovered after the season, the maximum change in erosion and accretion occurs in the unstable segment; (4) the Xichong beach can only have small sand body since it is supplied with sand mainly form inland streams, resulting in a possible danger in which sand loss induced by human activities or huge storms cannot be replenished naturally.