Evolution of shoreline position moments

Abstract

This paper considers the problem of forecasting coastline position in response to a fluctuating wave climate. Statistical moments are introduced as a method for describing shoreline evolution and variability. Equations are derived and solved for the statistical moments of shoreline position. These equations describe analytically the time-dependent ensemble averaged solution, and its dependence on wave climate without the need for computationally intensive Monte-Carlo simulations. The average is understood to be taken over the ensemble of possible wave sequences. An example application, based on a time series of nearshore synthetic wave conditions, is used to illustrate how the technique might be used in practice to account for both short- and long-term variability in wave climate.

Introduction

Changes in shoreline morphology are the result of physical processes with a range of temporal and spatial scales. From a practical perspective, there is interest in beach response to distinct storm events (short-term) and in coastal evolution over some years and decades (long-term). Long-term changes trace the general trend in morphological variations over a time span approximately coinciding with the period of design life for coastal structures (De Vriend et al., 1993). Storm events can lead to large but often transient deformations of the beach, with the effects being smoothed over a period of months. From a long-term perspective, the morphological response to storms is analogous to ‘noise’ about the long-term trend, caused by the fluctuating morphological forcing of the waves, tides and surges. Estimation of long-term shoreline movement and its variability remains a difficult open problem.

The ability to predict changes in coastal morphology is hampered by a lack of observational data, and by the prohibitive computational complexity of applying deterministic dynamical equations for fluid flow and sediment transport over even relatively short periods of a single storm (De Vriend et al., 1993). Therefore, much research has focused on simplified models for longshore or cross-shore transport. An equation governing the longshore transport of sand on a beach was derived by Pelnard-Considere (1956). This has been subsequently extended to account for variations in wave direction and sediment transport along the shore (Larson et al., 1997). At its simplest, the equation for the position of a single chosen depth contour, y(x,t), from a fixed datum line takes the form of a linear diffusion equation with constant coefficient, and has gained the epithet of the ‘one-line equation’. This model has proved remarkably robust, and over the past decade has been used widely, typically to examine shoreline changes over periods of months or years (see Hanson and Kraus, 1989, Kamphuis, 1991, Larson et al., 1997).

In practical applications, the one-line model has been used with sequences of forcing conditions that are considered representative of the conditions likely to occur in the future. Wave conditions are required to drive a one-line model and these are often available only as summary statistics. These conditions might be monthly average conditions determined from several years of data or, synthetic time sequences composed by combining fragments of records together. LeMehaute and Soldate (1979) addressed the problem of using wave statistics to drive the one-line model and proposed a procedure for constructing representative wave conditions. A more rigorous approach is to use Monte Carlo simulations. Vrijling and Meyer (1992) applied the one-line model to perform Monte Carlo simulations of the shoreline position near a port, while Dong and Chen (1999) included random temporal variability in a Monte Carlo study based on a one-line model modified to account for some cross-shore sediment exchanges. In both cases, assumptions were made about the statistics of the forcing conditions that restrict the application of the techniques to more general situations. An alternative is to use a formal averaging procedure and solve the resulting equation for the mean shoreline position. Reeve and Fleming (1997) used a time-averaged form of the one-line model and historical shoreline positions to infer the distribution of time-averaged sediment sources and thence to estimate likely future mean position of the shoreline. While providing an indication of the typical position of a beach and its sensitivity to variations in the boundary conditions, none of these approaches provides a direct method for determining the mean and variance of the shoreline in the presence of fluctuating wave conditions.

In this paper, we derive solutions to the equations for the first and second moments of the shoreline position (i.e., the coastal plan shape), as governed by the one-line equation. Specifically, we formulate equations for the ensemble average shoreline position (or coastal plan shape) and its variance. General solutions are obtained for forcing conditions with arbitrary probability density functions and temporal autocorrelation function. The solution expresses the spatial variability of the coastal plan shape explicitly with time in terms of the statistics of the underlying wave climate. An illustration of how the methods can be applied in practice is given, together with some results for specific cases.

The derivation of the equations is given in Section 2, together with their general solutions for the case of the evolution of an arbitrary initial coastline. Solutions for a specific case of beach nourishment are presented in Section 2.4. In Section 3, the steps needed to apply the method to practical situations are illustrated using hindcast wave data covering a period of almost 30 years. A discussion of the results and conclusions are contained in Section 4.