## 1. Introduction

[2] Shoreline undulations are episodically or persistently found along many sandy coasts. A well known example are megacusps, associated with surfzone rhythmic sandbars and rip channels, with a typical alongshore length scale of *O*(10^{2} m) and a timescale of *O*(10^{1} day) [*Short*, 1999]. The formation and dynamics of these rhythmic patterns have been explained as a self-organized behavior of the morphodynamic system [*Coco and Murray*, 2007]. The patterns are not simply dictated by a template in the hydrodynamic forcing but they emerge from the feedback between morphology and hydrodynamics via sediment transport and they posses their own characteristic length and timescales [e.g., *Garnier et al.*, 2008; *Castelle et al.*, 2010].

[3] In the present study we will focus on shoreline undulations with a larger alongshore length scale *O*(10^{3} m) and a longer timescale *O*(10^{3} day). These large scale undulations have been observed on various coasts around the world [see, e.g., *Bruun*, 1954; *Stewart and Davidson-Arnott*, 1988; *Verhagen*, 1989; *Thevenot and Kraus*, 1995; *Gravens*, 1999; *Guillen et al.*, 1999; *Ruessink and Jeuken*, 2002; *Davidson-Arnott and van Heyningen*, 2003; *Alves*, 2009; *Ryabchuk et al.*, 2011; *Kaergaard et al.*, 2012]. They are in general unrelated to surfzone sandbars and they will be referred to as large scale shoreline sand waves or simply sand waves. Sand waves cause a spatial and temporal variability of the shoreline position that can be greater than the uniform trend and their dynamics are therefore of great importance for coastal management [*Stive et al.*, 2002].

[4] A potential mechanism for the formation and dynamics of sand waves was provided by *Ashton et al.* [2001]. They showed that a rectilinear sandy coast exposed to very oblique wave incidence (angle of wavefronts in deep water with respect to coastline orientation larger than a critical angle, *θ* ≃ 42°) may be unstable leading to the formation of sand waves, cuspate features and spits (hereinafter referred to as high angle wave instability or HAWI). Although there are no direct observations of this instability working in nature (an important difficulty is the large length and time scales), coastlines with a wave climate dominated by very oblique incidence commonly feature large scale undulations, suggesting that this instability could be responsible for the formation of sand waves [*Ashton and Murray*, 2006b; *Falqués*, 2006; *Medellín et al.*, 2008; *Ryabchuk et al.*, 2011; *Kaergaard et al.*, 2012].

[5] The approach of *Ashton et al.* [2001]is based on the reasonable assumption that, for the description of shoreline changes on a large temporal and spatial scale (at least one order of magnitude bigger than that of rip channels and the rhythmicity of surfzone bars), the one-line shoreline modeling concept can be applied and that the details of surfzone morphodynamics can be ignored. The changes in shoreline position are simply governed by the gradients in the total alongshore transport rate*Q* (m^{3}/s) driven by obliquely breaking waves [*Komar*, 1998]. Negative gradients in *Q*lead to shoreline advance (deposition) and positive gradients lead to shoreline retreat (erosion). It is assumed that on a long timescale the cross-shore profile attains an equilibrium shape and that it shifts together with the shoreline position. Therefore, when an undulation is present in the shoreline, the bathymetric lines follow this undulation.*Q* is commonly described as a function of the wave height, *H*_{b}, and the angle between the wavefronts at breaking and the shoreline orientation, *α*_{b} = *θ*_{b} − *ϕ* (e.g., CERC formula) [*Komar*, 1998]. However, *H*_{b} and *θ*_{b} can not be considered external parameters for shoreline evolution. When an undulation is present in the shoreline, refraction over the associated bathymetry leads to alongshore gradients in *θ*_{b} and refractive wave energy spreading leads to alongshore gradients in *H*_{b}. This feedback between the shoreline changes, the associated bathymetry and the wavefield is the essential physical mechanism behind HAWI. For low wave incidence angles (*θ* ≲ 42°) the gradients in *α*_{b} are dominant for *Q* and they cause a positive transport gradient along a shoreline perturbation, which leads to diffusion of the perturbation and a stable shoreline. However, for high wave incidence angles (*θ* ⪆ 42°) the gradients in *H*_{b} become dominant for *Q* and they cause a negative transport gradient along a shoreline perturbation, which leads the growth and migration of the perturbation and therefore an unstable shoreline [*Ashton and Murray*, 2006a, 2006b; *Falqués and Calvete*, 2005; *Falqués et al.*, 2011a].

[6] *Ashton et al.* [2001] used a cellular shoreline model and in order to include the feedback mechanism associated to HAWI they defined *Q* in terms of the wave height and angle at the base of the shoreface, before nearshore wave transformation takes place. This depth is equivalent to the wave base and here the wave height and angle are independent of the shoreline and we refer to them as deep water waves (*H*_{∞} and *θ*_{∞}). A crucial step in this approach is the computation of *H*_{b} and *θ*_{b} as a function of *H*_{∞}, *θ*_{∞}, the wave period (*T*) and the nearshore bathymetry. To this end, they assumed: i) wave transformation over rectilinear depth contours that are parallel to the evolving shoreline and ii) that any gradient in the alongshore transport leads to an instantaneous shift of the whole cross-shore profile. Assumption i is inconsistent with the undulating shape of the bathymetry and, most importantly, it assumes indirectly that shoreline undulations extend offshore in the bathymetry down to the wave base. Assumption ii restricts the applicability of the model to large timescales (much larger than the reaction time of the cross-shore beach profile) and both assumptions are only suitable in the limit of very large scale features. A validation of the main results of this study requires a model that can describe bathymetric perturbations with a finite offshore extent and curvilinear depth contours.

[7] A linear stability analysis with these two characteristics was presented by *Falqués and Calvete* [2005]. They found that instability could still develop, provided that the offshore extent of the perturbations in the bathymetry, which was a free parameter in their analysis, was large enough. Furthermore, the range of unstable angles was significantly reduced for long period waves. Thus, the critical angle proposed by *Ashton et al.* [2001], *θ*_{∞} ≃ 42*°*, is actually a lower bound and instability in general requires larger angles and short wave periods. A very important output of the linear stability analysis was a wavelength selection for the initial development of the shoreline sand waves, *λ* ∼ 3–15 km [see also *Uguccioni et al.*, 2006]. This is one order of magnitude larger than that of surfzone instabilities and, together with the characteristic growth time predicted by the model, *O*(1 yr), it confirms that the one-line approach is appropriate for dealing with HAWI. Although the study of*Falqués and Calvete* [2005] confirmed the existence of HAWI and provided some new insight, it had several limitations. First, it relied on the assumption of an instantaneous reaction of the bathymetry to shoreline changes (assumption ii above). Furthermore, the offshore extent of the bathymetric perturbations was fixed beforehand. This is a crude approximation as in nature this distance is dynamic. Finally, a basic assumption of the linear stability analysis is that the amplitude of the sand waves was considered to be small and the analysis does not describe the actual evolution of the sand waves and possible nonlinear effects.

[8] *Van den Berg et al.* [2011] used a nonlinear quasi 2D morphodynamic model to study the evolution of nourished beaches under high angle wave incidence. This model is an extension of the linear model of *Falqués and Calvete* [2005]. It can describe shoreline undulations with a large amplitude and a parametrization of cross-shore dynamics was introduced. Because of the latter, the coupling between the shoreline and the bathymetry is no longer instantaneous and the offshore extent of the bathymetric perturbations is dynamic. In the present study this model is used to investigate the formation and dynamics of shoreline sand waves. In particular, we investigate to what extent the predictions of*Ashton et al.* [2001] and *Falqués and Calvete* [2005]depend on their idealizations and we look at new aspects of sand wave dynamics that were not caught by the previous models. The influence of wave height and period, the important role of cross-shore transport and the effect of variable wave incidence angles are investigated. Furthermore, new insight is provided into the physical mechanism behind HAWI and the wavelength selection and we compare the generic results with observations, from a qualitative point of view.