A headland is defined in common language as: (1) a point of usually high land jutting out into a body of water: promontory; (2) high point of land or rock projecting into a body of water. Therefore, a headland-bay beach is a beach whose shape is mainly conformed by the fact that it is located between such headlands, or at least adjacent to one. Some of the synonymous terms that can be found in literature to describe a headland-bay beach are: bay-shaped beach, pocket beach, zeta bay, bow-shaped bay, and half-heart bay. This type of feature shows a gradually changing curvature which Krumbein (1944) noted resembled that of a logarithmic spiral curve.
Johnson (1919) gave an incisive description of wave refraction caused by headlands along an embayed coastline, and Krumbein (1944) showed a simplified diagram of wave refraction into a bay lying to the lee of a headland. According to Yasso (1965), a headland was considered to be any natural or artificial obstruction that extended seaward from the coastline and caused a change in some element of the coastal wave pattern because of its presence.
Observations of headland beach morphology in nature Krumbein (1944) was the first author to describe a beach planshape— Halfmoon Bay, California (USA)—as being similar to the increasing radius of curvature found in the logarithmic spiral, although the restricted classification was removed some years later and the paper was reprinted in 1947. Yasso (1965) selected four US beaches for testing goodness of fit to the logarithmic spiral approximation. Poles for three of the best- fitting logarithmic spirals were located in close proximity to the seaward end of each headland. The spiral angle α was found to range between 41.26° and 85.64°. Berenguer and Enríquez (1988) reviewed data from 24 beaches around Spain and derived empirical correlations between geometrical characteristics of the layout of the offshore breakwaters and beach planshape. Moreno and Kraus (1999) performed fittings of analytical planshapes to a data set of 46 beaches in Spain and the United States and derived preliminary engineering design guidance including the proposal of a new functional shape.
Observations of headland beach morphology in the laboratory Silvester (1960) tracked the time evolution of a beach in a physical model and observed that the beach between headlands tended to reach an equilibrium shape—logarithmic spiral shape—in response to persistent swell directions under a certain wave angle of attack. The model coastline was allowed to erode without replenishment of sand at the updrift end. Yasso (1965) pointed out that the conditions of Silvester’s physical-model test (lack of continuous sediment supply to the updrift end of the model and close spacing of the headlands) suggested that the equilibrium form achieved in the model may not be identical to that achieved under natural conditions of sediment supply and wider separation of headlands.
Silvester (1970) performed additional model tests in which three different wave conditions were generated at three different angles of incidence to the alignment of a headland on an initially straight sandy beach. It was observed that the coastline developed three distinct curvature zones: first, a near circular section in the lee of the upcoast headland; second, a logarithmic spiral; and finally, a segment tangential to the downcoast headland. The time evolution of the spiral constant angle was plotted for the three incident wave angles tested, but only one wave condition was run long-enough as to see an asymptotic trend. A graphical linear relationship between the logarithmic spiral constant angle α and the wave angle β was provided based on the three angles tested.
Spătaru (1990) studied Romanian Black Sea beaches subjected to normal wave incidence by means of physical models. Equilibrium beach planshapes were considered to be described by arcs of circumferences, and provided design guidance based on geometry.
Numerical Model Approaches
Mashima (1961) constructed wave-energy diagrams from wind-rose diagrams and studied the configuration of stable coastlines based on energy considerations. When the wave energy is a semi-ellipse, the configuration of the stable coastline is approximately parabolic on which the tangent direction at the apex of the parabola coincides with the direction of the major axis of the wave-energy diagram.
LeBlond (1972) attempted to study how wave-induced longshore currents in the presence of a headland could erode a linear beach by developing a numerical model. LeBlond (1972) stated that if there existed a planimetric shape which the headland beach asymptotically approached, it must have the following properties: (1) it should be concave outwards, near the headland, and then convex outwards. (2) the sand transport should increase monotonically along it. (3) erosion, by causing the beach to be displaced normal to itself, should not qualitatively change the shape of the beach. LeBlond also pointed out that the logarithmic spiral did not satisfy the first condition because it is always concave outwards, and that there may be other curves satisfying all of the above three conditions, and one could not decide “a priori” which one will be the equilibrium one.
The main modification implemented by Rea and Komar (1975) with respect to previous numerical modeling efforts, was the combination of two orthogonal one-dimensional grids to simulate beach configurations, so that beach erosion could proceed in two directions without the necessity of a full two-dimensional array. Testing of the model in a hooked beach coastline configuration indicated that the coastline would always attempt to achieve an equilibrium configuration governed by the pattern of offshore wave refraction and diffraction and the distribution of wave-energy flux.
Walton (1977) presented an analytical model to describe the equilibrium shape of a coastline sheltered by a headland using a continuous wave-energy diagram consisting of representative offshore ship wave height and direction observations. It was found to produce coastline shapes similar to the logarithmic spiral shape for sheltered beaches in Florida. The model worked by establishing that the coastline orientation at a certain point was normal to the average direction of the so-called energy of normalized wave attack—that is, the energy which is allowed by the headland to reach the shore.
Yamashita and Tsuchiya (1992) constructed a numerical model for three-dimensional beach change prediction to simulate a pocket beach formation. The model consisted of three modules to calculate waves, currents, and sediment transport and beach change. The wave transformation module was based on the mild-slope equation of hyperbolic type; the current module was horizontally two-dimensional with direct interaction with sea-bottom change, which was evaluated by the sediment transport model formulated by Bailard (1982) in the third module.
Equilibrium Planshapes of Headland-Bay Beaches
Three functional shapes have been proposed to describe the equilibrium planshape of headland-bay beaches, namely the logarithmic spiral shape, the parabolic shape, and the hyperbolic tangent shape.
Values of α for headland-bay beaches reported in the literature range from about 45° to 75°. As α becomes smaller, the log spiral becomes wider or more open. There are two singular values for a: if α = 90°, the log spiral becomes a circle, and if α = 0°, the log spiral becomes a straight line.
Various authors have noted that fitting of the log-spiral shape is difficult in the downdrift section of the beach. It is a particular concern in attempting to fit to long beaches or to beaches with one headland. However, even in these situations, a good fit could be achieved for the stretch near the headland (Moreno and Kraus 1999).
For this parabolic shape, the focus of the parabola is taken to be the diffraction point. The three coefficients needed to define the shape (see Silvester and Hsu 1991, 1993) are functions of the predominant wave angle with respect to a control line. The control line is defined similarly to the case of the log-spiral shape as the line that extends from the diffraction point to a reference point, at an angle β between the control line and the predominant wave crest orientation. Downdrift of the reference point, the coastline is assumed to be aligned parallel to the incident wave crests. This shape pertains to that of a long straight beach with shape controlled by one headland. Values of β ranged in prototype beaches from 22.5° to 72.0°, whereas the variation in model beaches was from 30° to 72°.
Sensitivity tests were performed (Moreno and Kraus 1999), where the response of the parabolic shape to a change in the value of the characteristic angle β and of R0 was analyzed. The results proved that R0 is a scaling parameter—the length of the control line, the alongshore extent of the shape decreases as β increases because the parabolic shape is defined only for θ ≥ β. In summary, the angle β controls the shape of the parabola, and R0 controls its size. Because the control line intersects the beach at the point where the curved section meets the straight section of the beach, the sensitivity of the parabolic shape to errors in the estimation of the control point was examined. This was done by jointly changing R0 and β while keeping the distance from the headland to the straight coastline constant. This observation means that the control point is not well-defined, that is uncertainty in selection of the control point and hence the corresponding joint combination the radius R0 and β has little influence on the final result.
According to Moreno and Kraus (1999), the parabolic shape provides good fits for beaches with a single headland, because they consist of a curved section (well described by the portion of the beach protected by the headland) and a straight section (well describes the down-drift section).
For practical use of the parabolic shape, five unknowns need to be solved for: location of focus (two coordinates), characteristic angle β, scaling parameter R0, and the orientation of the entire parabolic shape in plan view.
This shape has three useful engineering properties. First, the curve is symmetric with respect to the x-axis. Second, the values y = ± a define two asymptotes; in particular of interest here is the value y = a giving the position of the downdrift coastline beyond the influence of the headland. Third, the slope dy/dx at x = 0 is determined by the parameter m, and the slope is infinite if m < 1. This restriction on slope indicates m to be in the range m < 1.
According to these three properties, the relative coordinate system should be established such that the x-axis is parallel to the general trend of the coastline with the y-axis pointing onshore. Also, the relative origin of coordinates should be placed at a point where the local tangent to the beach is perpendicular to the general trend of the coastline. These intuitive properties make fitting of the hyperbolic-tangent shape relatively straightforward as compared to the log-spiral and parabolic shapes, making it convenient in design applications.
Sensitivity testing of the hyperbolic tangent shape was performed (Moreno and Kraus 1999) to characterize its functional behavior and assign physical significance to its three empirical coefficients: a controls the magnitude of the asymptote (distance between the relative origin of coordinates and the location of the straight coastline); b is a scaling factor controlling the approach to the asymptotic limit; and m controls the curvature of the shape, which can vary between a square and an S-curve. Larger values of m (m ≥ 1) produce a more rectangular and somewhat unrealistic shape, whereas smaller values produce more rounded, natural shapes.
To fit the hyperbolic tangent shape to a given coastline, we must solve for six unknowns: the location of the relative origin of coordinates (two coordinates), the coefficients a, b, and m, and the rotation of the relative coordinate system with respect to the absolute coordinate system. Because of the clear physical meaning of the parameters, fitting of this shape can be readily done through trial and error.
Headland-Control Concept of Shore Protection
The headland-control concept of shore protection was first proposed by Silvester (1976) and further discussed by Silvester and Ho (1972), and was described as a combination of groins and offshore breakwaters at alongshore and seaward spacings such to create long lengths of equilibrium-bay beaches (Silvester and Hsu 1993). The structural dimensions in proportion to beach length are much smaller, and headlands are spaced much farther apart than offshore breakwaters. Therefore, it is intended to be a “regional” means of shore protection. Because headlands form pocket beaches, they might best be applied in a sediment-deficient area or for stabilizing an entire littoral reach of coast.
Headland beaches compartmentalize the coastline and reorient it in the local compartments to be parallel to the wavecrests of the predominant wave direction. If a coast has a substantial change in wave direction annually, the headland-bay beach might not be as stable as beaches behind traditional detached breakwaters, or shorter headland-bay beach compartments would be required.
A headland-bay beach design requires that a tombolo forms or be created (as by beach fill) behind the anchoring headland. If this connection is lost, the pocket beach is destroyed, and sediment can move alongshore, between adjacent compartments. Headland-bay beaches function and have their main attribute in creating independent pocket beaches, for which there is little or no communication of sand alongshore. Therefore, a headland beach presents a total barrier to littoral drift and can only be considered as a shore-protection alternative if such a barrier would not pose a problem to adjacent beaches.
The assumption that there is a single predominant wave direction which controls the final coastline shape is questionable where long distances are involved. If the design goal is to stabilize a regional extent by multiple pocket beaches, the headland-control concept might be appropriate.
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