Introduction

Sandy beaches are extremely valuable natural resources, providing the first line of defence against coastal storm impacts, as well as other ecosystem services (Barbier et al., 2011) such as ecological habitats and recreation areas. These beaches often are an essential part of a nation’s heritage. However, many of the world’s coastlines suffer erosion, due to interruption of sand flows from upstream (Syvitski et al., 2005) and alongshore, sand mining and sea-level rise effects, which is especially the case in the vicinity of tidal inlets (Ranasinghe et al., 2013). Science-based strategies for managing complex sandy coasts are, therefore, of the utmost importance, requiring stable, robust and rapid evaluation methods.

Coastline changes along sandy beaches at timescales beyond events and seasons often are dominated by gradients in wave-driven longshore transport. The first practical concept for predicting coastline change due to interruption of this wave-driven longshore transport was developed by Pelnard-Considère (1956), who derived a diffusion-type equation based on the assumptions of a small angle of incidence and a constant cross-shore profile shape. The first very limiting assumption of a small wave angle was relaxed to some extent by numerical, one-dimensional (1D) model approaches, such as GENESIS (Hanson, 1989), LITPACK (Kristensen et al., 2016), and UNIBEST (Tonnon et al., 2018). These models, which we will refer to as ‘standard 1D models’ also applied increasingly powerful and more advanced physics-based approaches using newer transport formulations, (e.g., Bijker, 1967; Kamphuis, 1991; van Rijn, 2014), to estimate the transport rate as a function of incident wave conditions, sediment parameters and profile shape. However, the main characteristic of the transport curve as a function of the wave angle remained a sine curve, with a maximum at roughly 45^o. For relative angles beyond this critical angle, longshore transport decreases for increasing angles and the morphological behavior of the coastline becomes fundamentally unstable.

While the existing coastline models had no real solution for this instability, the Coastal Evolution Model (CEM) proposed by Ashton et al. (2001) addressed this point using a grid-based, upwind approach, which they showed to be able to explain a variety of coastal forms found in nature. The high angle wave instability mechanism (HAWI) has been studied extensively through linear and non-linear stability analysis,(Falqués & Calvete, 2005; e.g., Ashton & Murray, 2006; van den Berg et al., 2011; Falqués et al., 2017) the latter included both this and the low angle instability mechanism (LAWI) proposed by Idier et al. (2011). These analyses pointed out the importance of the refraction on the foreshore of shoreline undulations, which generally stabilize the coastline relative to the original HAWI mechanism proposed by Ashton et al. (2001). Recently, Robinet et al. (2018, 2020), starting from the same grid-based one-line approach, extended the concept by coupling it with a 2D wave refraction model and including cross-shore transport. Such models are quite powerful in describing complex coastal shapes but at the cost of requiring complex and relatively time-consuming codes.

The standard 1D models approaches so far address either a single, possibly curving, coastline or disparate sections of coastline represented by separate models. However, there are many cases where islands, shoals and spits shield other parts of the coast from waves. In other situations, spits weld onto the coast to form lagoons that in turn may break up into different parts, or islands migrate towards the coast and weld onto it. Even though some gradual reshaping from a straight beach to a bay shape was achieved with numerical models (e.g., Hanson, 1989), still the grid definition in existing models only allows for a small re-curvature of the coast, generally much less than 90^o. As a result, research often focuses on a final stage of the morphological development (e.g., a static bay shape; Hsu et al., 2010). The flexible generation of the grid at each time step is a requisite to deal with complex shoreline shapes which may change substantially over time, such as spits, salients and tombolos.

A vector-based approach to represent the coastline was first used by Kaergaard and Fredsoe (2013a), as part of a system that coupled a one-line coastline approach with a two-dimensional description of the wave and sediment transport, on an unstructured mesh. They applied a quite complicated approach to ensure the volume balance, using triangular and trapezoidal elements. The original coastline representation proposed by Kaergaard and Fredsoe (2013a) was adopted by Hurst et al. (2015) which followed a similar approach to the one presented in this manuscript but for a single coastline (i.e. no splitting or merging was included). Kaergaard & Fredsoe (2013a) and Hurst et al.(2015) implicitly assumed sediment rich environments. Payo et al. (2017) extended the use of vector-based coastline models to sediment-starved environments and applied it to a field study case to simulate the coastal change after defence removal (Payo et al., 2018). In all these vector-based models the coastline followed is at the top of the active profile. This choice implies a strong need to correct the volume balance for strongly curved coasts. As we will explain in detail, our approach follows a more representative coastline, situated approximately at Mean Sea Level (MSL). This recognizes the fact that typically, over longer periods aimed at by this model, the active profile extends from a closure depth to the crest of foredunes; these are at approximately equal vertical distances from the MSL contour. The approach has two advantages: the contour usually extracted from satellite imagery is the MSL contour, and as we will show, complex curvature corrections to the sediment balance are not needed, which makes our method much simpler.

It is noted that computations can be made with complex two-dimensional horizontal (2DH) process-based models, as was shown for the recent case of the ‘Sand Engine’ in Holland (Luijendijk et al., 2017) as well as for other complex coastal forms. However, this comes at great computational expense and requires considerable expertise. Even though an effort was made to increase the robustness and efficiency of the process-based morphological models (Kaergaard & Fredsoe, 2013a), still these detailed field models are appropriate for the investigation of specific processes (i.e. science), but are often less suited to apply in engineering (design phase) for data poor environments where quick scenario evaluations are needed. So, engineers are rather stuck with one approach which does not capture the complexity of coastal evolution and another which is too expensive.

Consequently, a new approach is needed to robustly follow coastal features through complete lifecycles at reasonable computational cost. We demonstrate here the capabilities that arise when a model with a flexible, coastline-following grid is used: namely, straightforward definition of a complex planform, freedom to allow coastal evolution in any direction, and the possibility to merge and split coastline sections where needed. The characteristic features of the model are presented for a selection of analytical and principle test cases. In addition, the practical simulation of coastline evolution is validated for a large-scale, manmade land reclamation and a case of a groyne field filling in. We do not yet consider event-scale and seasonal variations, which are mostly due to cross-shore transport. Methods for including this as recently described by e.g. Vitousek (2017), Robinet (2018), Antolínez (2019), Palalane and Larson (2020), and Tran and Barthélemy (2020) are currently under consideration.