Dynamics of large-scale shoreline perturbations

Universitat Polit`ecnica de Catalunya Departament de F´ısica Dynamics of large-scale shoreline perturbations. Mem`oria presentada per Jaime Alonso Arriaga Garc´ıa per optar al grau de Doctor en Ci`encies. Directors: Francesca Ribas Prat Albert Falqu´es Serra Barcelona, Febrer del 2018 Contents 1 Introduction 1 1.1 Motivation ..................................... 1 1.1.1 Shorenourishments ............................ 1 1.1.2 Shoreline self-organized patterns . . . . . . . . . . . . . . . . . . . . . 3 1.2 Shorelinemodelling................................. 6 1.3 Shorelineinstabilities................................ 8 1.4 Research objectives and outline . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Description of the models 13 2.1 Introduction..................................... 13 2.2 Q2Dmorfomodel.................................. 15 2.2.1 Grid, equilibrium profile and bathymetry . . . . . . . . . . . . . . . . 15 2.2.2 Wave transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Bed evolution and sediment transport . . . . . . . . . . . . . . . . . . 17 2.2.4 Numerical implementation and boundary conditions . . . . . . . . . . 21 2.2.5 Application 1: comparison with the previous model version . . . . . . 21 2.2.6 Application 2: sea-level rise as a cause of beach erosion . . . . . . . . 24 2.3 1Dmorfomodel................................... 26 2.3.1 Governingequations............................ 26 2.3.2 Basic reference state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.3 Perturbed dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.4 Bathymetric perturbation function . . . . . . . . . . . . . . . . . . . . 31 2.3.5 Application 1: role of the bathymetric perturbation . . . . . . . . . . 32 2.3.6 Application 2: 1Dmorfo comparison with Q2Dmorfo in Denmark . . . 34 2.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Mega-nourishment modelling and model validation 41 3.1 Introduction..................................... 41 3.2 Site description and model bathymetry . . . . . . . . . . . . . . . . . . . . . . 42 ii CONTENTS 3.2.1 Wavesandtides .............................. 42 3.2.2 Morphology................................. 42 3.3 Calibration and validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Modelcalibration.............................. 44 3.3.2 Modelvalidation .............................. 46 3.3.3 Computation of shoreline diffusivity . . . . . . . . . . . . . . . . . . . 49 3.4 Long-term evolution and feeding capability . . . . . . . . . . . . . . . . . . . 51 3.4.1 Wave climate scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.2 Diffusion and feeding properties . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Discussion...................................... 56 3.5.1 Calibrated parameter values . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5.2 The role of HAWI in the ZM evolution . . . . . . . . . . . . . . . . . . 58 3.5.3 Feedingasymmetry............................. 59 3.6 Conclusions..................................... 60 4 Diffusion of mega-nourishments 63 4.1 Introduction..................................... 63 4.2 Methodology .................................... 64 4.2.1 Design of an analytic mega-nourishment . . . . . . . . . . . . . . . . . 66 4.2.2 Design of a synthetic wave climate . . . . . . . . . . . . . . . . . . . . 69 4.3 Effect of varying the mega-nourishment shape . . . . . . . . . . . . . . . . . . 74 4.3.1 Sensitivity to the initial asymmetry . . . . . . . . . . . . . . . . . . . 74 4.3.2 Sensitivity to shape ratio . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.3 Sensitivity to volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Effect of varying the wave forcing . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.1 Design of SWC with different obliquity occurrence . . . . . . . . . . . 78 4.4.2 Effect of obliquity occurrence in mega-nourishment dynamics . . . . . 79 4.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Formation of shoreline sand waves 87 5.1 Introduction..................................... 87 5.2 Observations .................................... 88 5.2.1 Site ..................................... 88 5.2.2 Qualitative description of the shoreline sand waves . . . . . . . . . . . 90 5.2.3 Shorelineanalysis.............................. 92 5.3 Modelling ...................................... 93 5.3.1 Wave transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.2 Correlation between shoreline sand wave presence and high-angle wave incidence .................................. 94 5.3.3 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 96 CONTENTS iii 5.4 Discussion......................................101 5.4.1 Role of HAWI on Dungeness shoreline undulations . . . . . . . . . . . 101 5.4.2 Role of tides on Dungeness shoreline undulations . . . . . . . . . . . . 102 5.4.3 Justification of the setup chosen in the 1Dmorfo model . . . . . . . . . 103 5.5 Conclusions.....................................104 6 Synthesis 105 6.1 Mainfindings....................................105 6.2 FutureResearch ..................................108 Bibliography 111 iv CONTENTS Chapter 1 Introduction 1.1 Motivation The beaches are complex environments where many processes occur at the same time. The boundary between the wet beach and the dry beach, i.e. the shoreline, is a visible feature and its behaviour is an indication of the dominating processes and constraints of the system. Shorelines around the world are rarely smooth and they can present undulations and cuspate shapes. In this thesis, the term coastline perturbations is used to refer to deviations of the shoreline from an otherwise straight shoreline. On the one hand, human actions can cause shoreline perturbations via structures or beach nourishments. This human interference introduces a perturbation into the system, either in the wave field, which drives morphological changes, or in the beach itself. On the other hand, nature alone can also introduce perturbations to the coastal system via geological constraints or by positive feedbacks between the driving forces and the morphological changes. This thesis combines field observations and morphological models to improve the understanding of large-scale coastal perturbations. Two sites are mainly studied: the Delfland Dutch coast, where a very large nourishment called Sand Engine was constructed in 2011, and the Dungeness Cuspate Foreland, which is a natural perturbation that has been evolving for thousands of years. 1.1.1 Shore nourishments Shore nourishments consist of dumping sand on the coastal system and are used to mitigate beach erosion. This excess of sand is typically transported in time out of the nourishment area in the cross-shore and in the alongshore directions. The most well-known nourishment is the beach nourishment, perhaps due to its immediate visual impact. Sand is deposited on the sub-aerial beach, advancing the shoreline seaward (Dean, 2002). As a consequence, 2CHAPTER 1. INTRODUCTION a) b) c) Figure 1.1.1: Conceptual diagram of the different nourishment strategies: a) beach nourishment, b) shoreface nourishment, and c) localized mega-nourishment. The yellow lines indicate the expected direction of sediment transport. Image modified from Stive et al., 2013. the initial cross-shore profile is steeper than the original beach profile. The modification of the beach profile favours an initially rapid diffusion in the cross-shore direction while the alongshore diffusion is directly related to gradients in the alongshore sediment transport. The latter process is easily understood through a parametrisation of the alongshore sediment transport like the CERC equation (Komar, 1998) Q(y) = µH5/2 bsin(2αb),(1.1.1) where Hbis the wave height at breaking and αb=θb−φsis the angle between the wave angle at breaking, θb, and the local shoreline, φs. Therefore, even with a uniform wave field, the nourishment is expected to experience alongshore gradients due to the φsdependence (i.e. the perturbation in the shoreline angle) ant thereby diffuse. Other wave alongshore sediment transport parametrisations give similar qualitative results. Another type of nourishment is the shoreface nourishment, for which the sand is deposited as a submerged berm within the active zone of the cross-shore profile, i.e. the zone were wave stirring affects the sandy bed, (Dean, 2002). Two mechanisms are present in this type of nourishment. The first one is the cross-shore feed due to the modification of the beach profile. Instead of the sand being transported offshore, as occurs for beach nourishments, the sand is transported onshore because the modified beach profile is less steep than the original one. The second one is the lee effect related to the perturbation of the wave transformation. The submerged sand causes wave focusing leading to gradients in alongshore sediment transport. In this case, the terms initially affected in Eq. (1.1.1) are the ones comprising the wave 1.1 Motivation 3 transformation Hband θb. More specifically, there will be accretion in the lee side of the submerged sand. However, this can also cause a down drift erosion zone (van Duin et al., 2004). Mega-nourishments: The Sand Engine Nourishment strategies, that is their size and time periodicity, varies in different countries. Spain, Italy and France have an interest in coastal development (e.g. harbours) and apply a reactive strategy when negative impacts induced by these projects require coastal stabilization (Hamm et al., 2002). In The Netherlands, coastal protection follows a preventive strategy and is a high-level priority as reflected by its coastal policy of maintaining the coastal position at its 1990 position (de Ruig & Hillen, 1997; Mulder et al., 2011). In 2000 it was decided to extend the policy and also maintain the volume in the area between 20- m depth and the landward boundary of the dune area. The annual average nourishment volume since 1990 of about 6 Mm3was raised to 12 Mm3(see the trend in Fig. 2 of Stive et al., 2013). More recently, a Dutch State Committee delivered recommendations on how to keep the Netherlands flood proof over the next century in the light of human-induced climate change and accelerated sea-level rise (Stive et al., 2013). In line with their recommendations, The Netherlands adopted an innovative intervention approach named “The Sand Engine” or Zand Motor in Dutch. From now on it will be here referred also as ZM. The ZM is a hook-shaped mega-nourishment of 17 Mm3which contains 1.5 times more sand than the quantity of sand used in the whole Dutch coast since the policy extended to also maintaining sand volume. The ZM had an initial alongshore length of 2.5 km and offshore extension of 1 km. It was constructed from March 2011 to July 2011 (Fig. 1.1.2) within the 17-km long beach section (Delfland coast) bounded by the harbours of Scheveningen and Hoek van Holland. Furthermore, the design contained a small lake to prevent the freshwater lens in the dunes to migrate seaward. This mega-nourishment presents characteristics of both the beach nourishments and the shoreface nourishments. In fact, this perturbation affects the wave transformation and the local shoreline angle, i.e. every term in Eq. (2.2.9). Therefore, the mega-nourishments have more complex dynamics than the traditional nourishments and the study of their dynamics is very recent. 1.1.2 Shoreline self-organized patterns The early studies of alongshore rhythmic patterns in the coast assumed the pre-existence of templates-spatially organized structures in either the hydrodynamics or the underlying geology. More specifically, it was assumed that its shape would be imprinted on unconsolidated sand (Coco & Murray, 2007). Indeed, this occurs for example in the Carchuna system (Southern Spain) where the observed shoreline undulations are related to the wave refraction influenced by submerged canyons (Ortega-S´anchez et al., 2014). As a consequence, these features are stable and quasi static. In contrast, there are shoreline undulations that are highly dynamic, exhibiting growth and migration. The self-organization theory is now the most accepted theory to explain such rhythmic undulations in the shoreline (see Coco & Murray, 2007). The main difference between the template-forcing theory and the self- 4CHAPTER 1. INTRODUCTION Figure 1.1.2: Aerial photograph of the Sand Engine in July 2011. organization theory is that the morphological changes driven by a template forcing do not affect the template forcing itself, while in the framework of self-organization a positive feedback occurs between the hydrodynamics and the morphological changes driven by it. This positive feedback implies an instability of a featureless situation and is precisely the cause of the pattern formation. Potentially self-organized shoreline undulations can be observed at different spatial and temporal scales and are related to different processes. Beach cusps are the smallest features with metres/hours characteristic scales and are related to swash zone processes (Werner & Fink, 1993; Coco et al., 1999, e.g. Fig. 1.1.3a). Mega-cusps are larger features with hundreds- of-metres/weeks characteristic scales and are related to rip channels connected to rhythmic sand bars (Deigaard et al., 1999; Calvete et al., 2005, e.g. Fig. 1.1.3b). Shoreline sand waves are even larger features with kilometres/years characteristic scales and have been related to the energy dispersion produced by wave refraction of highly-oblique incoming waves (Zenkovitch, 1959; Ashton et al., 2001; van den Berg et al., 2012; Kaergaard & Fredsoe, 2013a, e.g. Fig. 1.1.3c). The present work deals with these larger features and due to its characteristic scales will be referred to as Kilometric-scale Shoreline Sand Waves (KSSW). The driving mechanisms are discussed in detail in Section 1.3. Shoreline sand wave observations From the observational point of view, the dynamics of the formation and evolution of KSSW has been difficult to study due to their large length and time scales. On one hand, these scales complicate the systematic measurement of the beach. Although the advances in satellite imagery have increased the observations of shorelines at large scales, bathymetric measurements are scarce. These bathymetric observations are necessary to understand the role played by wave-related processes. Nevertheless, there are many sites where km-scale 1.4 Research objectives and outline 11 is it linked to the wave climate? 7) How morphodynamic models results compare with observed shoreline sand wave formation? This thesis is divided in 6 chapters. In chapter 2, the Q2Dmorfo model is described and modified to further expand the model capabilities to study large amplitude perturbations. In particular, a new algorithm to deal with the sediment transport in the shoreline is implemented. The new model version is compared with the previous one. Also, sea-level rise is implemented and compared against the Bruun rule prediction. Also, the 1Dmorfo model is briefly described and the cross-shore links between the shoreline and the bathymetry are tested to further understand and exemplify the HAWI and LAWI instabilities. In chapter 3, the data of the first three years of evolution of the ZandMotor is used to calibrate and validate the Q2Dmorfo model. In particular, the parameters controlling the alongshore sediment transport strength and the cross-shore sediment transport strength and offshore reach are calibrated. Afterwards, realistic wave climates constructed with historical wave data are used to make long-term predictions of the Zandmotor behaviour. In chapter 4, the general form of the 2D-Gaussian is used to imitate the perturbation of the ZandMotor. Once this is validated, the parameters of the 2D-Gaussian are varied to investigate the role played by its asymmetry, the shape ratio, and the volume of the meganourishment on its own long-term evolution. Also, the procedure to synthesize a real wave climate is optimized and then the probability of occurrence of the oblique waves is varied to better understand its effect on the long-term behaviour of a mega-nourishment. In chapter 5, a 2-km stretch of beach located in the north-eastern flank of the Dungeness Cuspate Foreland is studied in detail. There, two formation events of shoreline sand waves are observed. The correlation of the wave climate characteristics with the formation events is investigated. Finally, the observed data is used to validate the 1Dmorfo model. Finally, in chapter 6 a summary of the most relevant conclusions of each chapter is done by answering the proposed research questions. Also, future research lines are discussed. 12 CHAPTER 1. INTRODUCTION Chapter 2 Description of the models 2.1 Introduction In this thesis two morphodynamic models are used: the non-linear model Q2Dmorfo and the linear stability model 1Dmorfo. Both models provide complementary information about the shoreline dynamics, the 1Dmorfo model is very useful to study the initial formation of shoreline sand waves while the Q2Dmorfo model is useful to study the non-linear dynamics of beaches in the long-term. An important part of this thesis consists on modifying the Q2Dmorfo model. The Q2Dmorfo applications of this Chapter have the objective of showing the effect of the improvements made while the two research studies performed with this model are found in Chapters 3 and 4. Also in this Chapter, the 1Dmorfo is applied to simple scenarios to explain the model strengths, limitations and outputs while the research study performed with this model is found in Chapter 5. The Q2Dmorfo model (or quasi-two-dimensional morphodynamic model) is a non-linear model for large-scale shoreline dynamics (1 km-100 km) on a medium to long-term time scale (months to years). In order to handle such scales in a computationally efficient way, the model is based on the assumption that the shoreline dynamics is controlled by the gradients in the alongshore transport and that the cross-shore beach profiles tend to a previously defined equilibrium profile. The alongshore sediment transport is parametrised directly from the wave transformation and follows the typical shape of the alongshore current. The resulting accumulated/eroded sediment causes profile perturbations. The perturbed profile Subsection 2.2.5 is largely based on Arriaga et al. (2014): Arriaga, J., Ribas, F., Marino-Tapia, I. J. & Falqu´ es, A. 2014 Km-scale shoreline sand waves: numerical modelling and observations. In Coastal Eng. 2014. Doi: 10.9753/icce.v34.sediment.68 Subsections 2.3.5 and 2.3.6 are partly based on Idier et al. (2017): Idier, D., Falqu´ es, A., Rohmer, J. & Arriaga, J. 2017 Self-organized kilometre-scale shoreline sandwave generation: sensitivity to model and physical parameters. J. Geophys. Res. 122, doi:10.1002/2017JF004197 and on Falqu´es et al. (2017): Falqu´ es, A., Ribas, F., Idier, D. & Arriaga, J. 2017 Formation mechanisms for self-organized km-scale shoreline sand waves. J. Geophys. Res. Earth Surf. 122, 10.1002/2016JF003964 14 CHAPTER 2. DESCRIPTION OF THE MODELS is modelled to tend to the equilibrium profile by means of a diffusive cross-shore transport. Such transport is assumed to be stronger near the shoreline and to decay to a residual value beyond the active depth. The wave transformation computed over the evolving bathymetry takes into account refraction and shoaling over the curvilinear contours up to the breaking point. These simplifications limit the application to large-scale features. Consequently, surf-zone morphodynamic patterns such as near-shore sand bars and rip currents can not be modelled with the Q2Dmorfo model. Moreover, the tides and aeolian transport are neither taken into account. A first version of the model was presented in Falqu´es et al. (2008). The next version, presented by van den Berg et al. (2012), included improvements in the discretization and the boundary conditions. This last version had two important shortcomings. First, the evolving shoreline was treated as a sharp boundary between the dry and wet beach, which was difficult to implement numerically. In particular, the model could not handle correctly the shoreline evolution when the shoreline deviated from the mean shoreline (y-axis) more than ≈13◦. Second, the direction of the sediment transport followed the global x-axis and y-axis directions regardless of the shoreline orientation. The first model improvement in the present thesis is the implementation of the fuzzy shoreline algorithm: the dynamic equations are now solved throughout the whole domain and the shoreline is treated as a transition zone. This allows the description of larger shoreline deviations. The second improvement is that the cross-shore direction is computed locally as the direction of maximum bed level gradient (i.e. the normal direction to the local contours) of a smoothed bathymetry. Finally, the model has also been recently adapted to handle sea level variations. The 1Dmorfo linear stability model, introduced by Falqu´es & Calvete (2005), aims to describe the dynamics of the small departures of the coastline with respect to a rectilinear shoreline. Therefore it is ideal to investigate the initial formation of shoreline sand waves and has previously been used to investigate such conditions (Falqu´es, 2006; Falqu´es et al., 2011; Idier et al., 2011; Falqu´es et al., 2017; Idier et al., 2017). The alongshore sediment transport is computed in a similar fashion as the Q2Dmorfo model but the cross-shore sediment transport assumes instantaneous changes of the beach profile. Unlike the Q2Dmorfo, the 1Dmorfo can describe diverse cross-shore transport dynamics, i.e. the shape of the bathymetric perturbations associated to the shoreline. This flexibility allowed to study diverse shoreline instability mechanisms (Falqu´es & Calvete, 2005; Idier et al., 2011). In Section 2.2 the improved Q2Dmorfo model is described with an emphasis on the improvements done in the sediment transport computation with respect to the model version presented by (van den Berg et al., 2012). Also tests are performed to (i) compare the improved version of the model with the one of van den Berg et al. (2012) for the case of an evolving train of shoreline sand waves and to (ii) evaluate the sea level rise implementation by reproducing the Bruun rule (Bruun, 1962). In this thesis the 1Dmorfo model is not modified but a brief description is done in Section 2.3 with an emphasis on the shape of the assumed bathymetric perturbation. The model details can be found in Falqu´es & Calvete (2005). Also, tests are performed to (i) understand/explain the role played by two types of bathymetric perturbations, the first one related to the high-angle wave instability and the second one related to the low-angle wave instability, and to (ii) compare the 1Dmorfo model and the Q2Dmorfo model using the site of Srd. Holmsland Tange. 2.2 Q2Dmorfo model 15 x y i=0 i=ni=1 ic=1 ic=n x=xs(y) j=0 j=m j=1 jc=1 jc=m i=2 j=2 jc=2 ic=2 ic=ishore(jc) x=Lx y=Ly zb> 0 zb ≤ 0 Figure 2.2.1: Geometry of the modelling domain and the staggered grid. 2.2 Q2Dmorfo model 2.2.1 Grid, equilibrium profile and bathymetry The model uses a Cartesian frame of reference, where the y-axis is parallel to the mean shoreline and the x-axis is pointing offshore, and a rectangular domain (0 < x < Lx,0< y < Ly), Lxand Lybeing the cross-shore and the alongshore domain lengths. A staggered grid is applied with xcell grid size, ∆x, and ycell grid size, ∆y(Fig. 2.2.1). The bed level is defined by zb(x, y). The shoreline position xs(y) is determined by interpolating in the cross-shore direction between the last dry cell (zb≥0) and the first wet cell (zb<0), indicated by ishore(y) in Fig. 2.2.1. The model assumes that the beach has an equilibrium profile, which can be an analytic profile or can be read from an external file. For the former case, the model can be used with the modified Dean profile zbe(x) = −b(x+x0)2/3−x2/3 0,(2.2.1) 16 CHAPTER 2. DESCRIPTION OF THE MODELS where x0is a small shift to avoid an infinite slope at the shoreline and b= (3/2)βx01/3, βbeing the slope of the swash zone (used by Falqu´es & Calvete, 2005). Alternatively, the model can be used with the double slope profile without bars zbe(x) = −a11−β2 β1tanh β1x a1−β2x, (2.2.2) were β1is the slope near the shoreline and β2is the slope defined at the depth a1(used by Yu & Slinn, 2003). The initial bathymetry, zb, can be read from an external file or it can be constructed as an alongshore repetition of a cross-shore profile following an initial defined shoreline. Various initial perturbations can be added to the bathymetry (e.g. random small scale perturbations, alongshore rhythmic perturbations, and a localized large scale perturbation). In general, it is recommended to create the bathymetry externally, taking care that the profiles are similar to the equilibrium profile. 2.2.2 Wave transformation The wave module takes into account refraction and shoaling over the curvilinear contours by assuming monochromatic waves with T=Tp(peak period), H=Hrms (root-mean-square wave height) and a wave angle θ. The waves are propagated from the offshore boundary (H0, T0, θ0) by solving in cascade a set of three decoupled equations: the dispersion relation, the equation for wave number irrotationality and the wave energy conservation equation: w2=gk tanh(kD),(2.2.3) ∂ky ∂x =∂kx ∂y ,(2.2.4) ∂ ∂x cgH2kx k+∂ ∂y cgH2ky k= 0.(2.2.5) Here, ωis the radian frequency, gis the gravity acceleration, ~ k= (kx, ky) = k(−cos θ, sin θ) is the wave number vector (where θis the angle between wave crests and the y-axis), cgis the group celerity, and D=−zb+ηthe local depth. Here, zbis the bed level and ηrepresents the sea surface level. As default, ηis set to 0. These equations assume steady conditions and ignore wave diffraction, and wave energy dissipation by bottom shear stresses and wave breaking. From the computed wave field, we extract the breaker wave height, Hb, and the corresponding wave angle, θb, to feed the sediment transport equations. The breaking point is defined as the most onshore position where H≤γbD, where γbis the saturation ratio of H/D in the surf zone and the value γb= 0.5 is used. The details about how Eqs. (2.2.3)- (2.2.4)-(2.2.5) are solved numerically can be found in van den Berg (2012). 2.2 Q2Dmorfo model 17 2.2.3 Bed evolution and sediment transport The changes in the bed level are computed with the sediment mass conservation equation ∂zb ∂t +∂qx ∂x +∂qy ∂y = 0 ,(2.2.6) where tis the time and ~q = (qx, qy) is the depth-integrated sediment flux, which includes the bed porosity factor. This is the main governing equation and it is solved throughout the whole domain. The shoreline position, xs(y, t) is computed from the modelled depth D=−zb+ηinterpolating between the last wet cell and the first dry cell and is assumed to be a uni-valued function of y. This uni-value limitation implies that spit dynamics can not be modelled, which will have implications in Chapter 3. Fuzzy shoreline The model treats the shoreline as a transition zone; i.e. a fuzzy shoreline, which can be physically interpreted as the swash zone. Due to its global implications, the fuzzy shoreline is not a single algorithm but rather a general treatment for every sediment transport equation. This means that all the variables and functions change smoothly from certain values corresponding to the wet cells to other values corresponding to the dry cells. For example, the wave-driven alongshore transport is assumed to have a standard cross-shore distribution in the surf zone and decays to zero across the swash zone; i.e. taking into account the width of the swash zone. In addition, the factor in front of the cross-shore transport is assumed to have a certain distribution in the surf and shoaling zones and it is imposed to decay exponentially to zero across the swash, in this case an extra distribution is required for the dry cells (the mathematical details are described later in this section). This rather simple concept facilitates the numerical implementation of the sediment transport equations. Evolving cross-shore direction The cross-shore direction is defined for every grid point by taking into account the curvature of the shoreline and its associated bathymetric contours (Fig. 2.2.2). This helps to better describe the dynamics of largely perturbed beaches or with curvilinear shapes. The local normal direction to the bathymetric contours is represented by an averaged orientation, φ, evaluated as sin φ= ∂¯zb ∂y q(∂¯zb ∂y )2+ (∂¯zb ∂x )2 ,(2.2.7) where the spatially averaged bed level ¯zbis computed within a rectangular box Ll×Lc. Here, Ll= 100 m and Lc= 50 m are used. For the coastline angle, φs, the boxes do 18 CHAPTER 2. DESCRIPTION OF THE MODELS not take into account the dry cells in order to avoid the influence from the dry beach. Following the model convention, the normal and tangential vectors to the bathymetric lines are ˆn= (cos φ, −sin φ) and ˆ t= (sin φ, cos φ), respectively. Sediment flux decomposition The depth integrated sediment flux ~q is decomposed as ~q =~qL+~qN+~qD,(2.2.8) where the first term, ~qL, represents the littoral drift driven by breaking waves and is evaluated by first computing the total sediment transport rate Q. As default, the CERC formula (Komar, 1998) is chosen, Q(y0) = µH5/2 bsin(2αb).(2.2.9) Here, Hbis the (rms) wave height at breaking and αb=θb−φsis the angle between wave fronts at breaking and the coastline. Here, y0(instead of y) indicates that the variables Hb, θband φsassociated to each point correspond to the position found following the direction normal to the local coastline (instead of the global xdirection, see Fig. 2.2.2). The µconstant is related to the non-dimensional Kconstant of the original CERC formula by µ=K 16(s−1)(1 −p)rg γb ,(2.2.10) where sand pare the relative density and porosity of sediment, respectively. By setting s= 2.65, p= 0.4 and γb= 0.5, the range K∼0.2−1.6 suggested by Komar (1998) gives a range µ∼0.06 −0.45 m1/2s−1. The total Qis then redistributed across the profile with a normalized shape function, which is assumed to be similar to an alongshore current profile: f(x0) = 4 √πL3x02e−(x0/L)2,(2.2.11) where x0is the distance to the shoreline and L= 0.7X0 b+X0 sz, with X0 bbeing the width of the surf zone and X0 sz being the width of the swash zone. The cross-shore coordinate x0, and the distances X0 band X0 sz are calculated in the direction normal to the local coastline by using the corresponding φ(Fig. 2.2.2). The cross-shore distribution f(x0) (Eq. 2.2.11) is based on alongshore current measurements reported by Komar (1998) for a wide range of beach profiles. Finally, we impose that the transport ~qLis directed tangent to the local bathymetric lines, 2.2 Q2Dmorfo model 19 Figure 2.2.2: Sketch representation of the local cross-shore direction and the alongshore transport computation. The first asterisk indicates the position of φsand the second asterisk indicates the position of Hband θbfor the signaled grid cell. ~qL=Q(y0)f(x0)ˆ t. (2.2.12) The second term in Eq. (2.2.8), ~qN, stands for the transport that drives the bathymetry to a certain cross-shore equilibrium profile, i.e. it parametrises the cross-shore transport processes, and reads ~qN=−γN(∇zb·ˆn+βe)ˆn. (2.2.13) As can be seen, it is proportional to the difference between the equilibrium slope βe, at the local depth D=−zb+η, and the actual slope in the local shore-normal direction. An implicit assumption of this approach is that the equilibrium profile must be monotonic (without bars). The cross-shore diffusivity factor γNis related to the influence of orbital velocities and turbulence produced by incoming waves on the sea bed. Its order of magnitude has been estimated from the expression of momentum mixing (Battjes, 1975) and it is scaled with a power of wave height at breaking, γN(x) = νNγ−1/6 bH11/6 bX−1/3 bψ(x),(2.2.14) 20 CHAPTER 2. DESCRIPTION OF THE MODELS / Figure 2.2.3: Sketch of the shape function ψ(Eqs. 2.2.15 and 2.2.16), which controls the crossshore transport magnitude and is a proxy of the cross-shore behavior of wave stirring. A large residual value value fhas been used in this sketch to allow a clear visualization. where νNis a non-dimensional parameter that requires calibration. The factor γNvaries throughout the bathymetry with a shape function ψ, which has a maximum at the shoreline and then decays offshore and onshore (Fig. 2.2.3). In the wet cells the expression ψ(D) = 1 + b+ tanh((αDc+zb)/Ld) 1 + b+ tanh(αDc/Ld)(2.2.15) is adopted, which becomes 1 at the shoreline and decays to a given value fat D=Dc(here, f= 0.02 and is controlled by the parameter b). The model instantaneous depth of closure, Dc, is computed as a fraction of the depth at which the sediment particles are first mobilized by the waves, Dm(Dc=fcDm, where the parameter fchas to be calibrated). The residual value of ψat deep water is controlled by the parameter α. The default αvalue is set to 0.46 so that ψ(∞)∼f/2 = 0.01. The decay rate of ψis controlled by Ldand it is set to Ld= 0.5αDc. In the dry cells, ψ(x) decays to 0 in the onshore direction as ψ(x) = exp −x−xs Xsz 4!,(2.2.16) where x−xsis the distance to the shoreline and the width of the swash zone Xsz controls the decay distance. The third term in Eq. (2.2.8), ~qD, is an alongshore diffusive transport that represents the tendency of small bumps to be flattened by breaking waves and it tends to stabilize the numerical solution by diffusing the small-scale morphodynamic noise, ~qD=−γD(∇zb·ˆ t)ˆ t. (2.2.17) 2.3 1Dmorfo model 27 0 200 400 600 800 time (d) 0 20 40 60 80 100 120 140 xc (m) 0 500 1000 1500 x (m) -14 -12 -10 -8 -6 -4 -2 0 2 zb (m) Dc=7m, Xzs=30m Dc=7m, Xzs=400m Dc=20m, Xzs=400m Figure 2.2.7: Initial beach profile (dashed line) and beach profiles after 2 years of simulation (left) and the shoreline retreat xc troughout time (right). where ¯ Dis the averaged active depth of the profile, and alongshore sediment transport rate, Q, is computed with the CERC equation (Eq. 2.2.9) and requires the computation of Hb(y, t) and αb(y, t). The wave transformation considers that the bathymetric perturbations related to the coastline have a finite offshore extension and takes into account curvilinear depth contours. This is achieved by using the dispersion equation, the conservation of the wave front equation, and the wave energy equation (Eqs. 2.2.3-2.2.5). The model is forced uniformly offshore (x=x∞) with H(x∞, y, t) = H∞(t) and θ(x∞, y, t) = θ∞(t). 2.3.2 Basic reference state The linear stability analysis requires a basic reference state, i.e. a state where the shoreline undulations are absent. Given an alongshore uniform topography D=D0(x) with D0(0) = 0 and given an uniform offshore wave input, a steady and alongshore uniform solution for the wave field is found. The wave number k0(x) is directly computed from Eq. (2.2.3). The uniform topography allows for Eq. (2.2.4) to be cast into the Snell law d dx(k0sin(θ0)) = 0 (2.3.21) 28 CHAPTER 2. DESCRIPTION OF THE MODELS and Eq. (2.2.5) gives d dx(cg0cos(θ0)H02)=0,(2.3.22) from where the wave incidence angle, θ0(x), and wave height, H0(x), are found. The uniform wave field means that Hband θbare alongshore uniform, therefore also the sediment transport rate Qis uniform. This immediately gives the next solution hereinafter considered the basic reference state xs= 0, k =k0(x), θ =θ0(x), H =H0(x).(2.3.23) with the position of the breaker line given by H0(x0 b) = γbD0(x0 b). 2.3.3 Perturbed dynamics Now a perturbed state is considered by imposing an infinitesimal-amplitude undulation on an initially rectilinear shoreline defined as: xs(y, t) = A 2eσt+iKy +c.c. (2.3.24) where Ais the amplitude, Kis the shoreline perturbation wave number (λ= 2π/K) and σ=σr+iσiis the complex growth rate. The expression of the shoreline perturbation is expressed in Fourier modes (Eq. 2.3.24) for convenience. The model aims at providing σrfor a given K, from which the characteristic growth time σr−1and the migration celerity V= σr/K can be computed. A positive growth rate σrmeans that the shoreline perturbation of the corresponding wavelength λgrows. Regarding the unperturbed state, the main inputs of the model are the cross-shore equilibrium beach profile, zb(x) = −D0(x), and the given wave parameters offshore, i.e. significant wave height, peak period and angle. Regarding the topographic perturbation h(x, y, t), the main inputs are its alongshore wavelength, λ, its cross-shore shape function, g(x) (which is link between the shoreline and the bathymetry is discussed in depth in the next section), and the depth of its offshore reach, Dc. Thus, the perturbed bathymetry associated with the sand wave defined in Eq. (2.3.24) is given by D(x, y, t) = D0(x)−h(x, y, t) = D0(x)−βsg(x)xs(y, t) (2.3.25) To compute the growth rate (σ), Eq. (2.3.24) is inserted into the one-line sediment conservation equation (Eq. 2.3.20). The effect of the constant µused in Qhas an effect only 2.3 1Dmorfo model 29 on the time scale, but the sign of the growth rate is insensitive to the magnitude of this parameter. Computing the left hand side of Eq. (2.3.20) is straightforward from Eq. (2.3.24), but estimating the right hand side requires first the computation of the perturbed wave field defined by k=k0(x) + ˆ k(x)0eσt+iKy +c.c. θ=θ0(x) + ˆ θ(x)0eσt+iKy +c.c. H=H0(x) + ˆ H(x)0eσt+iKy +c.c. (2.3.26) The perturbed wave number can be computed from the linearised dispersion relation (Eq. 2.2.3) ˆ k0 k0 =2p 2p+ sinh(2p) ˆ h D0 (2.3.27) where p=k0D0. The linearised eikonal equation (Eq. 2.2.4) is dS dx −iK(tan θ0)S=−iK cos θ0 ˆ k0(2.3.28) where S(x) = koˆ θ0cos θ0+ˆ k0sin θ0is proportional to the phase Φ = −(i/K)S. This can be solved numerically as an initial value problem with S(x∞) = 0 from offshore up to the coastline. Once θ(x)0is known, the linearized wave energy equation (Eq. 2.2.5) dΨ dx −iK(tan θ0)Ψ = iKcgH2 0 cos θ0 ˆ θ0(2.3.29) can also be solved as an initial value problem like Eq. (2.3.28), where Ψ(x)=2cgH0ˆ H0cos θ0+ ˆcg 0H2 0cos θ0−cgH2 0ˆ θ0sin θ0. Once the perturbed wave transformation is known, the wave variables at the perturbed breaking line can be evaluated. The perturbed breaking condition H(x0 b+x0 b) = γbD(x0 b+x0 b) allows to determine the perturbed breaking line x0 b=H0(x0 b, y, t) + γbh(x0 b, y, t) γbdD0 dx (x0 b)−dH0 dx (x0 b)(2.3.30) The perturbed wave breaking variables 30 CHAPTER 2. DESCRIPTION OF THE MODELS θ0 b(y, t) = θ(x0 b+x0 b, y, t)−θ0(x0 b)) H0 b(y, t) = H(x0 b+x0 b, y, t)−H0(x0 b)) (2.3.31) are computed as θ0 b=θ0(x0 b, y, t) + dθ0 dx (x0 b)x0 b H0 b=H0(x0 b, y, t) + dH0 dx (x0 b)x0 b(2.3.32) Using the linearised relationship αb=θb−φ≃θb−∂xs ∂y (2.3.33) together with the linearisation of Eqs. (2.2.9)-(2.3.20), the morphodynamic governing equation becomes ∂xs ∂t =2µH0 b5/2 ¯ D∂2xs ∂y2−∂θ0 b ∂y cos 2θ0 b−5 4H0 b ∂H0 b ∂y sin 2θ0 b(2.3.34) Finally, by inserting the perturbed expressions in the perturbed governing equation (Eq. 2.3.34, the complex growth rate σcan be computed as σ=−2Kµ ¯ DH0 b5/2 K+iˆ 2θ0 b A!cos 2θ0 b+5iˆ H0 b 2AH0 b sin 2θ0 b!.(2.3.35) The real part of σindicates the rate at which the perturbation will either grow or decay (K) = −σr(K) K2,(2.3.36) and a wavelength-dependent celerity can be defined from the imaginary part of σ V(K) = −σi(K) K.(2.3.37) 2.3 1Dmorfo model 31 2.3.4 Bathymetric perturbation function The feedback between the morphology and the wave field dictates the beach response. In the morphodynamic models where the coastline evolves as a result of the changes in bathymetry driven by wave-induced sediment transport, such as the Q2Dmorfo model, the dynamics of the shoreline and the bathymetry are linked in a natural way (van den Berg et al., 2012). However, in models based on the one-line concept such as the 1Dmorfo model, a link must be explicitly defined between shoreline and bathymetric perturbations. This link could be, in principle, defined from observations. However, given the lack of detailed observations of shoreline sand waves, such information is not available nowadays. Hence, the link must be inferred from physical principles. Considering that sand waves evolve at large time scales O(1 – 10 yr) in comparison with the short-term event scale of storms, the assumption of a bathymetric perturbation corresponding to a cross-shore shift of the equilibrium profile (following the shoreline displacement) has been used in the past by Ashton et al. (2001). Some studies assumed this profile shift but imposing a zero perturbation beyond the closure depth Dc(Falqu´es & Calvete, 2005; Kaergaard & Fredsoe, 2013a). Falqu´es & Calvete (2005) also considered a perturbation in the bed level which could decrease exponentially or linearly from a maximum value at the shoreline up to 0 at Dc. So, the shape perturbations used in the existing studies can be divided in two classes: type P1, based on profile shift assumptions, and type P2, based on a prescribed decay of the bed level perturbation. Here, the representative perturbations P1 : g(x) = 1 βs ∂D0 ∂x (2.3.38) and P2 : g(x)=1−x xc (2.3.39) are examined, where xcis the cross-shore distance at which Dcis found, after which the perturbation is forced to 0, g(x>xc) = 0. By inserting the shape function P1 in Eq. (2.3.25) and considering Eq (2.3.24), it is readily seen that it corresponds to horizontally shifting the profile by the same amount as the shoreline displacement (blue and red lines in Fig. 2.3.8b). The shape function P2 is based on a linear decay of the bed level perturbation (yellow line in Fig.2.3.8b) which is a limit of the exponential perturbation (when the decay distance is very large) used by Falqu´es & Calvete (2005). The perturbation shapes can have serious implications in the perturbed bathymetric contours. For a high gradient profile (D1, see Fig. 2.3.8a), the bathymetric contours perturbed with P1 and with P2 are very similar until 10-m depth (Fig. 2.3.8c). In contrast, for a low gradient profile (D2, see Fig. 2.3.8a) the perturbed contours differ substantially. Notoriously, the contours perturbed with P2 reach a maximum at a certain distance from the coast at D≈2 m, and decrease to 0 at D≈6 m (Fig. 2.3.8d). 32 CHAPTER 2. DESCRIPTION OF THE MODELS Figure 2.3.8: a) Modified Dean profiles using b= 0.19 m1/3and b= 0.047 m1/3, with βs= 0.02 for both profiles. b) Shape g(x) of the perturbations P1 (blue line for the high bathymetric gradient profile D1 and red line for the low bathymetric gradient profile D2) and P2 (yellow line for both profiles). The lower panels show the depth contours with a perturbation λ= 1 km using P1 (solid lines) and P2 (dashed lines), corresponding to the profile D1 (c) and to the profile D2 (d). 2.3.5 Application 1: role of the bathymetric perturbation The relative cross-shore amplitude of the depth contours with respect to the amplitude of the shoreline, constant for all contours for P1 and variable for every contour when using P2, strongly influences the wave transformation. Thereby, the alongshore sediment transport patterns and the resulting growth rate (positive in case of shoreline instability, negative in case of shoreline stability) also depends a lot on the bathymetric perturbation. The main purpose of this first model application is to illustrate this dependency by showing the growth rate curves (perturbation wavelength vs growth rate) for each of the perturbed bathymetries seen in Fig. 2.3.8. Since this model is applied to real conditions in Chapter 5, it is important to first understand the role of the bathymetric perturbations. In particular, its association to the high-angle wave instability mechanism or to the low-angle wave instability mechanism must be investigated. The objective is not to perform a systematic study but to 2.3 1Dmorfo model 33 do a qualitative exemplification of the role played by the bathymetric perturbation on the instability mechanisms. To this end, only a low-gradient profile and a high-gradient profile are used to create four general perturbed states (following the profile shift perturbation P1 and the linear bed-level decay of the perturbation P2, shown in Fig. 2.3.8). Here, the wavelength of the perturbation range from λ= [100 m −30 km]. Finally, the growth rate of these perturbed states is computed for three wave conditions: almost normal waves (H= 1 m, T= 1 s, θ= 5◦), oblique waves (H= 1 m, T= 1 s, θ= 45◦), and very oblique waves (H= 1 m, T= 1 s, θ= 65◦). Given that the aim is to explain the qualitative behaviour of the growth rate curves rather than its quantitative behaviour, σris normalized as σref = max[ |min(σr)|∗(−sign (max (σr))) ,max(σr) ].(2.3.40) When an instability occurs, the curve σr/σref has positive values and the preferred wavelength is where σr(λ)/σref = 1. For the oblique wave condition (θ= 45◦), the predicted curves show the expected behaviour of HAWI theory (i.e. a growth rate curve with a single maximum value at a certain wavelength, see Fig. 2.3.9b). The damping of short wavelengths in HAWI is controlled by the focusing/defocusing of wave energy induced by the undulating contours (van den Berg et al., 2014). For relatively long wavelengths, in comparison with the offshore reach of the bathymetric perturbation, the focusing is always near the sandwave crests (i.e. the prograding section) and the defocusing near the embayments. In this case, only one maximum is found at wavelengths λ= 3,12,16,24 km. After that, the curve decays to 0 with a λ−2factor. For the very oblique wave condition (θ= 65◦, Fig. 2.3.9c), the behaviour of the curves is similar except for the combination D2-P1 where three local maxima can be seen. This behaviour was also found by Uguccioni et al. (2006). The reason is that for very oblique waves, the rays can cross the shoals corresponding to several crests instead of only one. So, the focusing/defocusing becomes more complex with the result that the energy focal point is not necessarily located at the crests. For this reason, the instability curve is sensitive to small changes in the wavelength of the perturbation. Low-gradient profiles are the most prone to this phenomenon. Nevertheless, for the bathymetry D2-P2 only one maxima occurs. The reason may be related to the fact that the more perturbed offshore contours produce larger refraction and it becomes harder for the rays to cross more than one shoal. Indeed, by forcing the model with θ= 85◦several maxima can be found for both perturbations and for both profiles (not shown here). When diffusion is predicted for every wavelength, the curve σr/σref has only negative values. In this case, the normalization of the curve is done with the lowest σrvalue (following Eq. 2.3.40). The diffusion of the shorter-perturbation wavelengths is several orders larger than that of the larger-perturbation wavelengths. For this reason, the curves D1-P1, D1-P2, and D2-P1 seem to be a straight line touching 0 while in fact they approach to 0 (Fig. 2.3.9a). This occurs only for shore-normal waves (θ= 5◦) except for the special case D2, P2. In fact, this instability mechanism is different from the HAWI one. Idier et al. (2011) found that under certain conditions a critical high-wave-angle threshold is not necessary for destabilization of the shoreline. This instability is now known as low-angle wave instability (LAWI, see Idier et al., 2011). Idier et al. (2017) described two conditions for LAWI to occur. The first is that the cross-shore amplitude of the undulations in the bathymetry should be 34 CHAPTER 2. DESCRIPTION OF THE MODELS 0 10 20 30 λ (km) -1 -0.5 0 0.5 1 σr/σref θ=5 ° D1,P1 D1,P2 D2,P1 D2,P2 0 10 20 30 λ (km) θ=45 ° 0 10 20 30 λ (km) θ=65 ° Figure 2.3.9: Growth rate curves normalized for the four bathymetries described in Fig. 2.3.8c-d. For wave incidence θ= [5◦,45◦,65◦]. larger than the cross-shore amplitude of the undulation in the shoreline. The second is that the region where this occurs has to be located offshore the surf zone. So, it is dependent on the wave conditions and the bathymetric perturbation. It can be seen that these conditions are met by D2-P2. 2.3.6 Application 2: 1Dmorfo comparison with Q2Dmorfo in Denmark Srd. Holmsland Tange, located in the west coast of Denmark, has been used to test the theory for the formation of KSSW (Kaergaard & Fredsoe, 2013a; Falqu´es et al., 2017; Idier et al., 2017) after Kaergaard et al. (2012) described in detail the observed shoreline undulations and the associated bathymetric contours. The present application has two purposes: i) to investigate the capacity of the Q2Dmorfo model to describe these undulations by comparing the results in the linear regime (using the 1Dmorfo model) with those in the non-linear regime (using the Q2Dmorfo model) and ii) to substantiate with this site the existence in nature of the counter-intuitive P2 bathymetric perturbation. Here, the KSSW observed in Denmark are described and the discrepancies between two previous linear model findings are explained (Kaergaard & Fredsoe, 2013b; Falqu´es et al., 2017). After that, a comparison between the 1Dmorfo model and the Q2Dmorfo model is done. For this the 1Dmorfo model is applied to this site with perturbations P1 or P2 (i.e. two different cross-shore links are used), but the cross-shore link in the Q2Dmorfo model is a result of its cross-shore transport formulation. Since the bathymetric shape perturbation cannot be imposed in the sediment transport formula of the Q2Dmorfo model, and in order to imitate the configuration of the 1Dmorfo model, the initial state of the former has been 2.3 1Dmorfo model 35 constructed with the equilibrium profile plus a shoreline sand wave of small amplitude and a cross-shore shape following the perturbations P1 or P2. Then, the initial temporal evolution of the amplitude is used to compute the growth rate and compare it with the linear model. Finally, the consistence of the bathymetric perturbations in nature with P2 is discussed. Site: Srd. Holmsland Tange Kaergaard et al. (2012) described field measurements in Srd. Holmsland Tange (see Fig. 2.3.10a). Particularly, km-scale shoreline sand waves were reported with a wavelength λ≃5- 6 km, an amplitude (cross-shore distance between the crest of a shoreline undulation and the adjacent embayment) of ≃100 m, and a migration rate of 0.37 km/yr. Kaergaard et al. (2012) hypothesized that the undulations were correlated with the HAWI mechanism since an important amount of wave energy arrives with a very oblique wave incidence. This hypothesis was reinforced by the weak correlation between the offshore position of alongshore bars and the shoreline undulations. Later on, Kaergaard & Fredsoe (2013b) used the average wave conditions (θ= 42◦,Hs= 1.8 m, Tp= 6.1 s) measured 14-km offshore to model the shoreline undulations and assuming a Dcof 5-7 m. They found that these wave conditions would diffuse the undulations (negative growth rates). In order to find an unstable regime (i.e. positive growth rates), they forced the model with a wave angle of 55◦(13◦more than the measured average) and found a shoreline wavelength of 5-9 km. More recently, Falqu´es et al. (2017) modelled this same site and argued that a reasonable range for Dcis 9-13 m. Using both the averaged wave conditions and the real wave series, they examined the sensitivity to Dc. For the wave series, a Dc= 11 −15 m was required to obtain the observed wavelength while for the constant wave conditions, a Dc= 7.5−15 m was required. The disagreement between the studies of Kaergaard & Fredsoe (2013b) and Falqu´es et al. (2017) is partly due to the different models and assumptions, e.g. the former used smaller depth of closures and took into account the directional spreading. Nevertheless, the key difference is related to the cross-shore link between the shoreline and the bathymetric contours. Kaergaard & Fredsoe (2013b) used a cross-shore shift (P1) and Falqu´es et al. (2017) used a linear decay of the bed-level perturbation (P2). Comparison between 1Dmorfo and Q2Dmorfo Here, the Srd. Homsland Tange site is modelled with the 1Dmorfo and the Q2Dmorfo models. Given that the main difference between previous works is related to the link between the shoreline and the bathymetry, two assumptions are examined: the cross-shore shift of the profile (perturbation P1) and the vertical linear decay of the perturbation from the shoreline to the depth of closure (perturbation P2). This is straightforward in the 1Dmorfo model. However, the cross-shore transport is fixed in the Q2Dmorfo model, which assumes that the beach profile tends to the equilibrium one, i.e. a cross-shore shift in the longterm. Since this can not be changed, only the initial bathymetries follow the perturbation P1 or the perturbation P2 in the Q2Dmorfo simulations. After a while, the Q2Dmorfo bathymetry will look like that corresponding to a P1 perturbation. Three wave conditions are used: θ= [30◦,42◦,60◦], Hs= 1.8 m, Tp= 6.1 s. As unperturbed basic profile, the observed one is used (Fig. 2.3.10b) and a Dcof 12 m is assumed following the findings of 36 CHAPTER 2. DESCRIPTION OF THE MODELS 0 2000 4000 6000 8000 10000 12000 alongshore coordinate (m) -100 0 100 200 300 400 500 600 cross-shore coordinate (m) -2 m -1 m 0 m 1 m 2 m 3 m 4 m 5 m a) b) c) Figure 2.3.10: a) Satellite image of Srd. Holmsland Tange downloaded from Google Earth, b) Cross-shore profile digitized (solid line) from Kaergaard & Fredsoe (2013b) and its Dean-profile adjustment (dashed line), c) Depth contours digitized from Kaergaard et al. (2012). Falqu´es et al. (2017). The 1Dmorfo model allows a quick computation of the growth rate for several wavelengths but the Q2Dmorfo requires larger computational effort. This is why the Q2Dmorfo simulations are done only for perturbations with wavelengths 2,4,6,8,10 km and until the first 100 d. The amplitude of the initial perturbations is of 6 m. Thus, the comparison is the closest possible by focusing the Q2Dmorfo model analysis to the initial evolution time and to small perturbation amplitudes, i.e. the assumptions of the 1Dmorfo model. The growth rate σrof the shoreline undulations in the non-linear model is obtained following A2=A1eσr∗(t2−t1)(2.3.41) where A1is the initial amplitude and A2is the final amplitude. Two growth rates are computed: σr(t1= 0 d, t2= 50 d) (asterisks in Fig. 2.3.11) and σr(t1= 50 d, t2= 100 d) (circles in Fig. 2.3.11). For P1, the growth rates computed with both models have similar behaviour and are within the same order of magnitude. Although no positive σris observed (see left panels in 3.2 Site description and model bathymetry 43 NL BE UK x y a) 3.6% 7.2% 10.8% 14.4% 18% 0% EW N S Hs≥4 3≤Hs< 4 2≤Hs< 3 1≤Hs< 2 0≤Hs< 1 500 1000 1500 2000 2500  12  10  8  6  4  2 0 x (m) zb(m) N waves W waves b) d) c) Figure 3.2.1: (a) ZandMotor location, with the model coordinate system, (b) ZM picture provided by Jantien Rutten, (c) directional distribution of Hsat the Europlatform buoy (32 m depth), and (d) time-and-space-averaged bed elevation, zb, versus distance xin the ZM area (blue line) and the adjusted profile (red line) of Yu & Slinn (2003). from the long-term JarKus data set by averaging the profiles spatially and temporally. The JarKus annual profiles usually start in the dune area and end at about 800 m seaward with 250 m alongshore spacing. Every 5 yr, coastal profiles are surveyed up to about 2500 m seaward with 1 km alongshore spacing. The alongshore spatial distance for the derivation of the averaged profile is of 10 km around the ZM, and the temporal period chosen is from 1990 to 2009 (de Ruig & Hillen, 1997). The equilibrium profile for the model is obtained from the averaged profile (Fig. 3.2.1d) by adjusting the Yu & Slinn (2003) profile equation (Eq. 2.2.2). Also, the sensitivity of the model to using different equilibrium beach profiles was analysed, (varying from an area 1 to 10 km around the ZM and from 5 to 40 yr before the ZM construction), and no appreciable changes were observed. In the framework of the ZM project, bathymetric surveys were performed every month in the first year after the installation, and every two months in the subsequent years. The bathymetries extend 1.5 km offshore and 4.5 km alongshore. The grid resolution is 2 m and 25 m in the cross-shore and alongshore coordinates, respectively. The initial bathymetry for the model simulations corresponds to the survey of 17 January 2012 (Fig. 3.2.2a), once the initial hook-shape had connected to the adjacent beach (Fig. 3.2.2a), creating a second 44 CHAPTER 3. MEGA-NOURISHMENT MODELLING AND MODEL VALIDATION Figure 3.2.2: (a) Bathymetric survey from 17 January 2012 with volume control boxes and (b) input bathymetry of the model with the bars filtered out and the lagoons adjusted. enclosed water body. The initial model bathymetry is made by combining bathymetric data from the intensively surveyed area of the ZM with the equilibrium profile extracted from the Jarkus data set for the remainder of the modelled domain. In the bathymetries of the ZM area, we filter out the bars using the volume approach (Kaergaard et al., 2012) to meet the model assumption of a monotonic equilibrium profile. First, for each depth the bed level is integrated over a vertical range, the resulting volume is converted to distance from a fixed location on the beach, obtaining a clean profile with the volume conserved and without bars (see thick dashed line in Fig. 3.3.3). Second, the surveyed dry beach area is added, with the inner water bodies treated as 0.1 m high dry beach. Third, the contours in the model domain outside of the ZM area are constructed following the equilibrium profile assuming a straight shoreline (i.e. the overall position of the shoreline previous to the ZM construction). Finally, the bathymetry is interpolated from the overlapped contours (Fig. 3.2.2b). 3.3 Calibration and validation 3.3.1 Model calibration The Q2Dmorfo model described in chapter 2.2 is calibrated with the ZM data (see chapter 3.2.2), starting with the measured bathymetry from 17 January 2012 (Fig. 3.2.2). The three most influential parameters are calibrated by comparing the modelled and the measured bathymetries after 400 d (to take seasonality into account), and by forcing the model with the wave data from the Europlatform buoy. The first parameter, µ, controls the magnitude of the alongshore sediment transport (Eq. 2.2.9), the second parameter, fc, controls the depth where the cross-shore and diffusive transports (Eqs. 2.2.13 and 2.2.17) drop to ∼0 (i.e. it controls the active depth for sediment transport), and the third parameter, νcontrols the magnitude of the cross-shore and diffusive transports (i.e. it controls the relaxation time to equilibrium). In this contribution, the calibration process is simplified by using the same values for νin both transport terms, diffusive and cross-shore, that is ν=νN=νD, as in van den Berg et al. (2012). Similarly, the same value of fchas been used for both transport terms. Thereby, γD=γN. The range of values used for the calibration are µ= [0.01; 0.04; 0.07; 0.10] m1/2s−1,ν= [0.01; 0.03; 0.05] and fc= [0.05; 0.15; 0.25; 0.35; 0.45]. 3.3 Calibration and validation 45 mean sea level depth range depth of interest new profile Figure 3.3.3: Sketch of the volume approach. Thick solid line corresponds to the original profile. Thick dashed line corresponds to the profile after filtering out the bar with the volume approach. Larger µvalues are not included because preliminary model simulations showed that they largely over-predicted the ZM diffusion. Also, the values for fcand νare chosen because they are physically meaningful and still prevent numerical instabilities. Too low values lead to the growth of small scale morphological noise and numerical instabilities. From observations, it is clear that a factor fc>0.50 is not plausible (e.g. using fc= 0.50, the active depth would be 9.31 m for Hrms = 1 m and Tp= 6 s). On the other hand, if there are areas of significant alongshore sediment transport convergence and there is not enough capacity to redistribute it cross-shore, ”unphysical” islands tend to grow and the simulations blow up. Finally, a very high νvalue is equivalent to an unrealistic instantaneous shift of the profile as in the one-line models. This gives a constraint on the ratio µ/ν. The model performance is evaluated with the root-mean-square skill score, RMSSS = 1−RMSE(Y, X)/RMSE(B, X), of the modelled contours (until 10 m depth). In the definition of the RMSSS,RMSE stands for the root-mean-square error, Xis a set of n measurements, x1, x2, ..., xn,Yis a set of corresponding predictions, y1, y2, ..., yn, and Bis the prediction of no change (i.e. the initial survey), also called baseline prediction (Sutherland et al., 2004). The contours of the bathymetric survey are extracted using the volume approach (Fig. 3.3.3). The root-mean-square errors are weighted over the depth contours with a coefficient of 0.9D(Dbeing the water depth) so that the coastline and shallow contours have more weight than deeper contours. Perfect agreement (i.e. RMSE(Y, X) = 0) gives a RMSSS of 1. If the model prediction is further away from the measured condition than the baseline prediction, the RMSSS becomes negative. In general, after 400 d the RMSSS improved with decreasing fc(Fig. 3.3.4a). For fc= 0.05 the simulations became unstable for µ≥0.04 m1/2s−1and low νvalues, which 46 CHAPTER 3. MEGA-NOURISHMENT MODELLING AND MODEL VALIDATION 0.1 0.2 0.3 0.4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 fc RMSSS a) ν= 0.05 ν= 0.03 ν= 0.01 0.1 0.2 0.3 0.4 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 fc b) µ= 0.10 m1/2s−1 µ= 0.07 m1/2s−1 µ= 0.04 m1/2s−1 µ= 0.01 m1/2s−1 Figure 3.3.4: Root-mean-square skill score of the bathymetric lines (a) after 400 d and (b) after 1150 d, as a function of fc,µ, and ν. can be explained by a lack of capacity to redistribute the accumulated sediment in the crossshore direction. The best RMSSS was obtained for fc= 0.15 and µ= 0.04 m1/2s−1. When using these values, the RMSSS was similar for ν= 0.03 and ν= 0.05. We have chosen the latter to ensure the simulations stability in energetic situations. 3.3.2 Model validation To validate the model, the RMSSS is also computed after 1150 d for the same range of parameter values of the previous section, confirming that the calibrated parameter values have the best performance (Fig. 3.3.4b). In particular, after 400 d, µ= 0.01 m1/2s−1 and µ= 0.04 m1/2s−1have similar performance but after 1150 d their performance gap increases and µ= 0.04 m1/2clearly reproduces the observations more accurately. This can be explained by the initially fast cross-shore dynamics in the model (see Section 3.3.3), adapting rapidly (i.e. faster than in reality) the profile (hence, the contours) to a quasiequilibrium state. This adaptation initially disguises the role of µ. The RMSSS of the calibrated model after 1150 d is about a factor 3 larger than after 400 d (Fig. 3.3.4). The skill score of the calibrated model increases continuously in time because RMSE(B, Y ) experiences a continuous increase (Fig. 3.3.5) due to the ZM diffusive 3.3 Calibration and validation 47 Jan12 May12 Sep12 Jan13 May13 Sep13 Jan14 May14 Sep14 Jan15 0 30 60 90 120 150 RMSE (m) RMSEsho(B,Y) RMSE (B,Y) RMSEsho(X,Y) RMSE (X,Y) Figure 3.3.5: Comparison of RMSE of the no-change prediction (blue lines) and RMSE of the calibrated model (red lines). The solid lines correspond to the shoreline and the dashed lines correspond to the bathymetric lines until 10 m depth. nature whereas RMSE(X, Y ) hardly grows (Fig. 3.3.5). In fact, the RMSSS increases for every set of parameter values, so that a sub-optimal set of tuning calibration parameters (µ,fc,ν) may eventually reach high RMSSS values. Therefore, we have to interpret the RMSSS values carefully. The Q2Dmorfo is based on the one-line approach and, as such, is expected to represent better the shoreline than the bathymetric lines. Indeed, the root-mean-square error of the shoreline, RMSEsho(X, Y ), shows an initial increment then a decay and a subsequent stabilization while oscillating around 30 m (Fig.3.3.5). The modelled shoreline differs from the observed one in the north-east side (Fig. 3.3.6a) because the model does not take the interaction between the lagoon and the sea into account. Also, small scale undulations in the bathymetric lines (related to processes such as surfzone dynamics) are not captured in the simulations and are a persistent error source in the quantification (Fig. 3.3.6b). To further validate the model results, we also compared how the volumes of sand changed over 1150 d in three control boxes (CB) representative of the ZM tip (B) and the adjacent beaches (A, to the SW, and C, to the NE, see Fig. 3.2.2a). Here, CB-B is expected to loose sand while the CB-A and the CB-C are expected to gain sand. A quantification of the performance is given with the following averaged volume error E∗=s1 N N P i=1 (V∗(i)sim −V∗(i)mes)2 max(V∗mes)−min(V∗mes)(3.3.1) where V∗stands for the volume in box *, ifor the survey number, Nfor the number of 48 CHAPTER 3. MEGA-NOURISHMENT MODELLING AND MODEL VALIDATION 4000450050005500600065007000750080008500 600 800 1000 1200 1400 1600 y (m) x (m) a) 4000450050005500600065007000750080008500 1000 1500 2000 y (m) x (m) b) 0 m 2 m 4 m 6 m 8 m 10 m Figure 3.3.6: (a) Shoreline position on January 2012 (blue), after 400 d, on March 2013 (red), and after 1150 d, on March 2015 (green). (b) Bathymetric lines every 2 m until 10 m depth after 1150 d. Measured (solid line) and modelled (dashed line). surveys, sim for simulations and mes for measurements. Overall, the diffusion of the ZM over the adjacent beaches is well represented by the model (Fig. 3.3.7). The modelled loss of sand in the tip (CB-B) resembles the measured one (EB= 0.09). The initial offset in volume is a result of the linear interpolation used in the construction of the modelled bathymetry (the modelled wet area had 0.5 % less sand than the survey). To reveal more detail, the tip area, CB-B, is decomposed into its south-west (Fig. 3.3.7BA) and north-east (Fig. 3.3.7BC) sides. The CB-BC has a lower error (EBC = 0.07) than the CB-BA (EBA = 0.17), and their behaviour is consistent with their respective tip sides (EC= 0.06 and EA= 0.19). The model generally underestimates the volume in CB-A except for the last survey, while for CB-C the differences are small throughout time except for the under-feeding observed in the last survey. In general, the modelled volume change (Fig. 3.3.7, right axis) of CB-B, CB-BA and CB-BC follow the measured trend with small differences of magnitude while the modelled and measured volume changes of CB-A and CB-C (the ones being fed) show more significant differences. The long-term trend in the total volume behaviour is well captured. 3.3 Calibration and validation 49 6.9 7.1 7.3 7.5 Volume (M m3) −50 0 50 100 Volume change (k m3) 11 11.6 12.2 12.8 13.4 Volume (M m3) −300 −200 −100 0 100 Volume change (k m3) 7.3 7.5 7.7 7.9 8.1 Volume (M m3) −50 0 50 100 150 Volume change (k m3) 5 5.5 6 6.5 Volume (M m3) −200 −100 0 100 Volume change (k m3) 5.2 5.7 6.2 6.7 Volume (M m3) −200 −100 0 100 Volume change (k m3) Jan12 Jul12 Jan13 Jul13 Jan14 Jul14 Jan15 0 1 2 3 4 5 Hs(m) Time (mo−yr) A B C BC BA Figure 3.3.7: Modelled (red) and measured (green) total volume (solid lines) and volume change (dashed lines) in the control boxes defined in Fig. 3.2.2a. The asterisks indicate the surveyed data points. The significant wave height is plotted in the lower panel. 3.3.3 Computation of shoreline diffusivity The diffusivity of the ZM, both in reality (obs) and in the simulations (Q2D), is here examined using the concept of shoreline diffusivity which is easily formulated within the framework of the one-line approximation for shoreline dynamics. By assuming a certain alongshore sediment transport simplification/parametrisation and neglecting the feedback of 50 CHAPTER 3. MEGA-NOURISHMENT MODELLING AND MODEL VALIDATION bathymetric changes into the wave propagation, the Pelnard-Consid`ere equation is obtained (Pelnard-Consid`ere, 1956) ∂xs ∂t =∂2xs ∂y2(3.3.2) where is the diffusivity coefficient, which depends on the wave condition and the depth of closure (see Eq.3.3.6). An analytic solution of Eq. (3.3.2) is derived by approximating the initial shoreline, xs(y, 0), to a Gaussian shape (e.g. after 1150 d the surveyed shoreline has a 18.6 m mean square error with respect to a Gaussian shape) and then expanding it as a Fourier integral, leading to xs(y, 0) = A0e−((y−ya)/L)2=A0 L √πZ∞ 0 e−k2L2/4cos(k(y−ya))dk (3.3.3) where A0is the initial amplitude, Lis the initial Gaussian width and yais the alongshore location of the crest. Using the boundary conditions: xs(−∞, t) = xs(∞, t) = 0, and performing some computations, the analytic solution of Eq. (3.3.2) can be cast into: xs(y, t) = A(t) exp −(y−ya)2 L2+ 4t (3.3.4) where the amplitude is A(t) = A0 p1+4t/L2(3.3.5) The classical diffusivity coefficient, cla, obtained using the CERC formula for the alongshore transport in Eq. (3.3.2), is =cla = 2µH5/2 b Dc cos(2θb).(3.3.6) To evaluate cla a constant Dcof 8 m is assumed. This depth-of-closure value is inferred from the measured contours and is consistent with the analysis of de Schipper et al. (2016). To compute the instantaneous Hband θbin Eq. (3.3.6), waves are propagated from the buoy until the breaking point with the Snell law and the energy conservation, assuming parallel contours to a straight shoreline. Then, we average the resulting instantaneous diffusivity coefficient over the three years of evolution, giving cla = 0.0052 m2/s. By using Eq. (3.3.5), time dependent values of the modelled and observed diffusivity, Q2Dand obs, can be inferred from the corresponding A(t). The initial amplitude, A0, and width, L, are obtained by fitting the Gaussian function to the initial shoreline, xs(y, 0), and the subsequent amplitudes, A(t), to the instantaneous shoreline, xs(y, t). Notice that Q2Dand obs represent the effective diffusivity between the initial state and time t.obs decreases in time and stabilizes after ∼200 d (April 2013) to 0.0022 m2/s (Fig. 3.3.8a). Until this moment the diffusion may not only be driven by alongshore transport (the assumption behind Eq. (3.3.2)) but also by cross-shore transport, since the perturbed profile is far from the characteristic local equilibrium profile. Similarly, Q2Dstabilizes to 0.0021 m2/s but the model over-predicts the initial cross-shore transport contribution (Fig. 3.3.8a). During the 3.4 Long-term evolution and feeding capability 51 Figure 3.3.8: (a) The diffusivity coefficients and (b) the amplitudes, based on the measurements (blue line) and on the Q2Dmorfo simulations (red line), versus time. first days the modelled amplitude decays by 40 m (Fig. 3.3.8b), suggesting a misrepresentation of the cross-shore transport when the initial profiles are far from the defined equilibrium profile. However, the time evolution of the modelled cumulative diffusivity presents a change in slope around 200 d (Fig. 3.3.8a), which agrees with the stabilization time of the measurements, and after 500 d the model catches up with the measurements. A lower νvalue could reduce the cross-shore transport over-prediction but numerical instabilities may then arise during energetic events. 3.4 Long-term evolution and feeding capability 3.4.1 Wave climate scenarios For the long-term analysis, a total simulation time of 30 yr has been chosen which is safely longer than the envisaged time of 15-20 yr (de Schipper et al., 2014; Stive et al., 2013). Considering the validation time of 3 yr, the long-term modelling is performed over 27 yr. To account for variability in the future wave climate (hereafter referred to as WC), five different WC scenarios have been designed based on the available wave data prior to the last validated simulation (01 March 2015). First, a time interval of myr is defined and then is repeated until reaching the 27 yr duration. The chosen intervals are m= 1,3,5,10,20 yr, so that when m= 1 the interval is from March 2014 to March 2015 and repeats itself 27 times, for m= 3 the interval is from March 2012 to March 2015 and repeats itself 9 times, etc. 52 CHAPTER 3. MEGA-NOURISHMENT MODELLING AND MODEL VALIDATION Table 3.4.1: Statistics of the modelled wave climate scenarios, where His computed in Hs terms and Tin Tpterms. WC m ¯ HW(m)¯ HN(m)¯ TW(s)¯ TN(s)¯ θW¯ θNPT108 (W/m) PW PN%θoblique 1 1 1.48 1.00 5.87 5.90 60.2 45.2 1.93 2.71 61.6 2 3 1.42 1.06 5.77 5.94 61.7 45.9 1.92 2.02 61.9 3 5 1.41 1.10 5.76 6.00 61.4 45.3 2.01 1.80 60.9 4 10 1.40 1.13 5.78 6.00 60.1 44.9 2.02 1.66 59.4 5 20 1.39 1.13 5.82 6.01 59.8 44.2 1.90 1.62 58.0 The WC characteristics, evaluated at the buoy depth of 32 m, are analysed by first separating the waves coming from the west i.e. θ0<0◦with respect to the global shoreline orientation (hereafter referred to as W waves) from the waves coming from the north i.e. θ0>0◦with respect to the global shoreline orientation (hereafter referred to as N waves). Then, the averaged H,Tand θare computed for W and N waves. Also, the alongshore component of the wave energy flux is calculated as P=1 8ρgH2Cgsin θ(3.4.7) where ρis the water density. The accumulated module of Pis computed as PT=P(|P|), and the wave power asymmetry is evaluated as the ratio PW/PN. Finally, the percentage of oblique waves (angle larger than |45◦|), %θoblique, is computed. The average conditions of the five WC are similar (Table 3.4.1). In general, the W waves are more oblique and more energetic than the N waves. The dominant W wave energy flux is consistent with the known net alongshore sediment transport direction from SW to NE (van Rijn, 1997). The maximum ¯ Hdifference among the five different WC is 0.134 m for ¯ H(for the W waves), while for ¯ Tis 0.11 s (for the N waves), and for θis 1.72◦(for the N waves). The resemblance in wave statistics gives small differences in PTbut the wave power asymmetry shows a decay with larger mvalues (e.g. PW(m= 1) = 2.71PNand PW(m= 20) = 1.62PN). A similar tendency is observed in %θoblique, which decreases with increasing m, indicating an increment of wave obliquity in recent years. 3.4.2 Diffusion and feeding properties The long-term simulations performed show that the ZM is expected to exhibit continuous diffusion during more than 30 yr (Figs. 3.4.9 and 3.4.10). Therefore, the wave obliqueness of the WC scenarios is not large enough to trigger the formation of self-organized large-scale shoreline-sand waves. The long-term effective diffusivity is inferred by using the modelled ZM amplitude, A(t), and Eq. (3.3.5) with A0and Lcorresponding to the first long-term simulation (01 March 2015). After some initial variability (during about 5 yr), the effective diffusivity stabilizes and the averaged value of the last 5 yr is shown in Table 3.4.2. The least diffusive wave climate is WC2, and the most diffusive is WC5, with a 40 m difference in amplitude after 27 yr. 3.5 Discussion 59 HAWI is induced by a positive feedback between the undulations in the depth contours and the associated perturbations in wave refraction and shoaling while damped by the undulations in the coastline. For relatively low wave incidence angles, the instability source is negligible, the stabilizing effect dominates and the shoreline perturbation diffuses with a diffusivity that is nearly independent on the angle. This is quite well reproduced by the classical one-line approach. In contrast, for relatively high wave angles the diffusivity depends on wave angle and eventually becomes negative above some threshold. Therefore, the significant influence of the wave angle on the diffusivity found with Q2Dmorfo model suggests that the ZM is far from the purely diffusive situation described by the classical one-line approach and ‘near’ the HAWI threshold. For example, by jumping from 58% (WC5) to 62% (WC2) of high-angle waves the diffusivity drops by 16%. Furthermore, the over-prediction of the diffusivity by a 2.5 factor by the classical one-line approach is clearly a result of neglecting wave obliquity. Indeed, by forcing the real wave climate to have normal incidence (the period and wave height still vary) we find a diffusivity of = 0.0053 m2/s, which is near the classical theory with cla = 0.0052 m2/s, confirming the important role of wave incidence on the diffusivity. According to Falqu´es (2003) (see Fig. 5 of that paper), this 2.5 factor means that HAWI would be reached by increasing wave obliquity roughly by 18%. Thus, as already suggested by Falqu´es (2006), the Dutch coast is near the HAWI threshold. Finally, a perfect diffusive behaviour would show a constant , while the diffusivity of the modelled shorelines drops from 0.0021 m2/s, for the three-year validation period, to 0.0013- 0.0018 m2/s, for the 27-yr long-term period. Initially, the ZM perturbation is pronounced at the shoreline but relatively weak at the depth contours, which results in a relatively strong diffusive behaviour. However, through time, the mismatch between depth contours and shoreline tends to decrease, resulting in stronger de-stabilizing effects. If the diffusivity continues declining, a relict of the ZM may eventually survive. Therefore, feedback processes underlying HAWI are clearly active at the ZM even if stabilizing effects slightly dominate under the present wave climate. 3.5.3 Feeding asymmetry The idea behind the ZM project is a mega-nourishment that feeds sand to adjacent beaches on a decadal time scale (de Schipper et al., 2016). Both the measurements of the first 3 yr (Fig. 3.3.7) and the long-term simulations of 30 yr (Fig. 3.4.11) indicate that there is an asymmetry between the feeding to the NE beaches and the SW beaches, the latter feeding being smaller. de Schipper et al. (2016) already detected an asymmetry in the feeding by analysing the first year of the ZM evolution, and suggested that this is a consequence of the dominant NE alongshore transport direction although in our analysis there is no indication of this dominance. Most important, the changes in shoreline position are governed by gradients in transport, not by the transport itself. Such transport gradients can be interpreted with the following three time-varying characteristics of the perturbation: diffusivity, , migration, V, and the shoreline asymmetry, SA. The magnitude of the feeding is primarily controlled by , which decreases the ZM amplitude and increases its width, whilst the feeding asymmetry, FA, may be related to Vand SA. Large northward V should produce larger feeding to the NE beaches, and hence a larger FA, but this effect is weakened if the perturbation has negative SA. Note also that the larger the wave power asymmetry, PW/PN, the larger SA 60 CHAPTER 3. MEGA-NOURISHMENT MODELLING AND MODEL VALIDATION and V(Tabs. 3.4.1 and 3.4.2). Table 3.4.2 suggests that Vand SA compensate each other, resulting in very similar FA (for WC1, WC3, WC4, and WC5). Note that the arguments in this paragraph rely on an idealization, but in reality the transport gradients are more complex. For instance, a wave climate with a larger percentage of oblique waves, where HAWI processes become more important, adds complexity to the sediment transport. This may explain why FA for WC2 is 20% larger than the FA from the other climate scenarios. Therefore, a model, such as the Q2Dmorfo, is required to predict such details in the longterm. 3.6 Conclusions A morphodynamic model called Q2Dmorfo has been successfully calibrated and validated with bathymetric measurements of a mega-nourishment constructed in July 2011 on the Dutch coast (Zandmotor, ZM), which is characterized by a bimodal wave climate with a significant percentage of high-angle waves. After being calibrated with the bathymetries measured during 1 yr, the model can properly reproduce the observed ZM evolution during the next 2 yr, not only the shoreline but also the depth contours so that sand volumes are well represented. The calibration of the model provides a value of the non-dimensional K parameter of the CERC formula of K= 0.14, which is at the lowest limit of the values reported in the literature. Long-term model simulations have been performed using five different wave climate scenarios, WC. Results show that the shoreline will behave diffusively, so that the amplitude of the perturbation will have decayed from the initial 960 m (immediately after construction) to about 350 m, 30 yr after the ZM installation. At the same time, the shoreline of the adjacent beaches, 2.5 km at each side, will have shifted seaward (on average) by about 100 m at the NE defined section and about 80 m at the SW defined section. These results are very robust since they are reproduced with the five applied WC. The model predicts small alongshore migration rates (due to the bidirectional WC) and a maintenance of the shape asymmetry, SA, and both parameters correlate with the wave power asymmetry. The diffusivity is smallest for the WC showing the largest percentage of high-angle waves. An effective diffusivity of the shoreline, due to the alongshore sediment transport, has been evaluated by analysing the shoreline evolution during the first 3 yr, obtaining similar results for the measured and the modelled shorelines, obs = 0.0022 m2/s and Q2D= 0.0021 m2/s, respectively. In contrast, the classical one-line approach over-predicts the diffusion by a factor of 2.5, cla = 0.0052 m2/s. Therefore, the ZM lifetime, here defined as the time needed to reduce the amplitude after construction by a factor 5, predicted by cla is of only ∼35 yr instead of the ∼90 yr computed with the Q2D. It is found that the alongshore-driven effective diffusivity must be evaluated at least 1 yr after the meganourishment construction to avoid the strong influence of cross-shore transport at the initial states when the perturbed profiles are far from equilibrium. Although the measurements over the first three years and the model predictions for 30 yr show a diffusive behaviour of the ZM, the significant reduction in coastline diffusivity compared with the classical oneline approach, attributable to wave obliquity, confirms that the Dutch coast is not far from high-angle wave instability. 3.6 Conclusions 61 A morphodynamic model like the Q2Dmorfo, which includes more physical processes than the one-line approach but still allows performing long-term simulations, is especially suited to predict the shoreline evolution of mega-nourishments. In particular, the model can be a useful tool for the design of mega-nourishments since it can accurately reproduce the diffusion, the alongshore migration, and the feeding asymmetry to adjacent beaches. 62 CHAPTER 3. MEGA-NOURISHMENT MODELLING AND MODEL VALIDATION Chapter 4 Diffusion of mega-nourishments 4.1 Introduction The ZandMotor (ZM) is the largest beach nourishment in the world and one of the few human actions that have the purpose of mitigating the effects of climate change and the associated sea-level rise (Stive et al., 2013). In Chapter 3 of this thesis, the Q2Dmorfo model was calibrated with 3-yr measured morphological data, starting from 17 January 2012, and was used to predict the ZandMotor evolution in the long term. The results shed light in the diffusion dynamics of the mega-nourishment (MN) and other propertires such as its migration and the feeding of sand to adjacent beaches. For example, the lifetime predicted by the model is at least three times larger than the expected one (i.e. in the initial design). This is attributed to the diffusion effect of the incoming oblique waves (Arriaga et al., 2017). However, the role of the shape and size of the mega-nourishment was not investigated. Also, due to the variability in wave characteristics of the real wave climate used in the simulations, it was not possible to pinpoint the diffusion contribution of the different types of wave conditions. The same occurred for more complex nourishment traits such as the sand feeding asymmetry to adjacent coasts and the shape asymmetry. Tonnon et al. (2018) designed analytic mega-nourishments (from now on referred to as AMN) assuming a trapezoidal shoreline, and the associated bathymetry followed an equilibrium profile. They principally used the UNIBEST model (based on the one-line concept) to study the evolution of the AMNs. The validation was done by comparing its results with the Delft3D but the actual modelled perturbations had different volume sizes in the two models. This was attributed to the fact that the UNIBEST assumes a uniform crossshore shape and the Delft3D applies more volume in deeper water for larger nourishments. Therefore, it was difficult to control the sand volume in the AMN design. This chapter is the basis of the article in preparation Arriaga et al. (2018b): Arriaga, J., Ribas, F., Falqu´ es, A., Rutten, J. & Ruessink, B. 2018bLong-term performance of mega-nourishments: role of the initial shape and the wave climate. Coast. Eng. In preparation 64 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS The synthesizing of the wave climate has been used in previous studies to reduce the complexity of the forcing and understand more easily the associated large-scale morphological responses (Ashton & Murray, 2006a; van den Berg et al., 2011; Kaergaard & Fredsoe, 2013b). The most extreme case is to consider only one wave condition as representative of the real wave climate (see the application of the West coast of Denmark in Kaergaard & Fredsoe, 2013b). However, this means that there is only one wave direction, which is very unrealistic when the wave climate is bimodal and can result in large artificial migration celerity of nourishments, as noted by van den Berg et al. (2011). In the context of high-angle wave instability modelling, four wave conditions with the same wave height and period have often been used to analyse the percentage of oblique waves needed to generate shoreline sand waves: two assumed to favour shoreline stability, θ= 30◦and θ=−30◦, and two assumed to favour shoreline instability θ= 60◦and θ=−60◦(van den Berg et al., 2012; Kaergaard & Fredsoe, 2013b). Nevertheless, the focus of these studies was theoretical and did not take into account a real wave climate (from now on referred to as RWC) as a starting point for building the synthetic wave climate (from now on referred to as SWC). In this Chapter, a new way to design an AMN is explored that allows to take into account orientation, shape, and size (Section 4.2.1). This is validated by comparing its evolution with the one of the real ZM by forcing both with a 50-yr real wave climate based on 5-yr historical data (see Section 3.4). Once validated, the roles of the orientation, shape and volume, in the mega-nourishment evolution, are investigated (Section 4.3). Independently, the best procedure to create a SWC is investigated taking into account (i) the method to extract the representative wave condition of a wave series divided into sectors, (ii) the optimal number of sectors that represent a RWC, (iii) and the duration of each wave condition (Section 4.2.2). The real ZandMotor from January 2012 is used as the initial bathymetry (Fig. 3.2.2 in Section 3.2). Afterwards, the sensitivity of the ZM evolution to the probability of occurrence of the oblique wave sectors is investigated (Section 4.4). The simulations are performed with the Q2Dmorfo model. 4.2 Methodology The simulations of the analytic mega-nourishment are done with the 50-yr real wave climate constructed from repeating ten times the wave data from 2010 to 2015 of the Europlatform buoy (see Section 3.4 for more details). The synthetic wave climate is extracted from the same wave data and the initial bathymetry used in those simulations is the measured ZandMotor in January 2012. The morphological characteristics of the mega-nourishment that are computed in the long-term simulations are its diffusivity, its feeding capacity to adjacent coasts, its alongshore migration, and its shape asymmetry. The computation of the diffusivity assumes that the shoreline has a Gaussian shape (the details of the computation are explained on Section 3.3.3). The feeding performance of the mega-nourishment is measured by defining monitoring areas of equal size, located at both sides of the mega-nourishment. Then the beach linear metre is computed for every survey in each area (following Eq. 3.4.8). The area limits are such that they have a width of 2.5 km and begin where the perturbation initially affects the shoreline (see Fig. 4.2.1). The feeding asymmetry is defined by Eq. 3.4.9 (but instead of 4.2 Methodology 65 x-axis (km) y-axis (km) 20 15 10 5 0 1 1.5 0.5 area 1 area 2 (+) Figure 4.2.1: Sketch of the simulations. The wave angle θindicated has a positive sign and the shape asymmetry is negative. The areas defined to compute the feeding asymmetry are marked with a dashed rectangle. NE and SW, the equation refers to area 1 and area 2). Strictly speaking, the migration of the perturbation is complicated to compute because the shape of the shoreline changes continuously. Therefore, as an indication, the alongshore Gaussian displacement with respect to its initial location is computed by minimizing the error between a symmetric Gaussian and the shoreline. The shape asymmetry is computed with the same equation of the feeding asymmetry but the limits of the areas are defined by the lateral boundaries and the center of the adjusted Gaussian (i.e. from the dashed line in the middle of the mega-nourishment in Fig. 4.2.1 to the lateral boundaries of the domain). This implicitly assumes that the mega-nourishment influence is limited to the areas where there are shoreline changes. The SA is negative when the tip is oriented towards the negative y direction (as in Fig. 4.2.1). The used model setup is practically the one obtained after the calibration of Section 3.3. The three most important parameters are (i) the factor in the wave-driven alongshore transport (parameter µ=0.04 m1/2s−1), (ii) the diffusivity factors ~qNand ~qD(parameter ν=0.05), (iii) and the depth of closure Dc(parameter fc= 0.15). In some preliminary simulations the SWC had long periods of fair weather and then the fcvalue of 0.15 could cause large amounts of sediment accumulation near the shore and, as a consequence, numerical instabilities arose. However, simulations with a slightly larger fcvalue (of 0.20), not tested in Section 3.3.1 (only the values of 0.15, 0.25, 0.35 and 0.45 were tested), showed that using fc= 0.20 the Brier skill score is lower at first but the performance improves in time and is even slightly better after three years than that obtained using fc= 0.15. Therefore, a fcvalue of 0.20 is chosen in this application. The alongshore length of the simulations, Lx, is of 20 km and the cross-shore length is of 4 km, with the mean shoreline (before applying the perturbation) located at x=600 m. Finally, the fact that the SWCs largest waves are lower than the storm waves found in the RWC allows to use a smaller time step. So, ∆t=0.001 s is set in the SWC simulations and ∆t=0.006 s is set in the AMN simulations. The rest of the setup values are those used in Chapter 3. 66 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS 4.2.1 Design of an analytic mega-nourishment The general form of a two-dimensional elliptical Gaussian function is used to perturb a bathymetry in equilibrium. The parameters of the Gaussian function are optimized with the submerged bathymetry of the ZandMotor from January 17 of 2012. The resulting dry beach is set to a constant berm thus arriving to a pseudo analytic perturbation. This is validated by comparing its evolution with the evolution of the real ZandMotor, both are forced with the RWC. Construction of the analytic mega-nourishment The perturbation of the ZM is obtained by subtracting the measured bathymetry from the idealized equilibrium bathymetry previous to the construction. The latter consists of a straight shoreline with parallel depth contours following the profile of Yu & Slinn (2003) adjusted to the real profile. Afterwards, the wet area is used to optimize the 2D-Gaussian function f(x, y) = Aexp(−(a(x−x0)2−2b(x−x0)(y−y0) + c(y−y0)2)),(4.2.1) where a=cos2Θ 2σ2 x +sin2Θ 2σ2 y b=−sin 2Θ 4σ2 x +sin 2Θ 4σ2 y c=sin2Θ 2σ2 x +cos2Θ 2σ2 y . Here, x0and y0determine the center position, σy(σx) is the standard deviation of the alongshore (cross-shore) decay of the perturbation, Θ is the rotation angle, which controls the initial shape asymmetry (note that the initial ZM orientation Θ is negative), and A determines the height of the perturbation and therefore modules the perturbation in the alongshore and in the cross-shore. To reduce the number of variables in the optimization process, x0is fixed to the shoreline and y0explores a small distance near the center of the ZM. The rest of the Gaussian parameters are varied to optimize the depth RMSE of the measured ZM wet area. Once the best parameter values are obtained, the analytic perturbation is added to the idealized equilibrium bathymetry. However, the dry beach has 160% more sand than the dry beach of the ZM (all the volume differences are computed with respect to the ZM) and resembles a hill with a maximum height over 30 m. In order to have a consistent dry-sand volume, the berm (height of the dry beach) is set to 2 m. This berm height is obtained from averaging the dry beach height of the real ZM. The RMSE of this first approximation is 0.41 m: the dry volume difference is 5.7% and the wet volume difference is -7%. A last step is taken to diminish the dry and wet volume differences by slightly varying the Gaussian parameters obtained before. In particular, the amplitude (A) and both shape variables (σx, σy) are slightly varied but maintaining the aspect ratio (σx/σy=cte). This last step gives a slightly higher RMSE of 0.44 m: a dry volume difference of 0.1% and a wet volume difference of -0.01%. This is a reasonable compromise between the RMSE and 4.2 Methodology 67 a) b) c) d) simplified ZM m g t analytic depth (m) depth (m) perturbation (m) perturbation (m) Figure 4.2.2: Simplified ZandMotor with the dry beach set to 2 m (a) and the analytic meganourishment (b). The associated perturbations can be seen in the lower panels. the discrepancy in volume. The obtained values are A= 36 m, Θ = −53◦,σx= 660 m and σy= 520 m. To further investigate the implications of assuming a constant berm, another initial bathymetry is modelled, corresponding to the measured ZandMotor but changing the dry beach to a berm height uniform of 2 m (Fig. 4.2.2a). The resulting most offshore dry point is located at 1080 m in the real bathymetry and at 1070 m in the analytic bathymetry. The associated bathymetric contours present similar amplitudes and reach similar offshore distances. Validation of the analytic mega-nourishment The performance of the previously obtained analytic perturbation (defined as case 3) is evaluated by comparing its evolution with the evolution of the original ZM (defined as case 1) and with the simplified ZM where the berm is set to 2 m (defined as case 2). These three perturbations are forced with the 5-yr RWC. The amplitude decay of the three mega-nourishments present small differences (Fig. 4.2.3, left panel). The amplitude is almost identical during the first 5 years. Afterwards, case 1 slightly deviates from the other two cases. This long-term discrepancy appears to be caused 68 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS 0 10 20 30 40 50 time (yr) 200 300 400 500 600 700 800 900 Amplitude (m) case 1 case 2 case 3 0 10 20 30 40 50 time (yr) 1 1.5 2 2.5 3 ǫ (m2/s) ×10-3 Figure 4.2.3: Predicted amplitude of the mega-nourishments (left panel) and the corresponding mega-nourishment diffusivity (right panel) over 50 yr. Case 1 is the ZM, case 2 is the simplified ZM with a uniform berm height, and case 3 is the analytic mega-nourishment. by setting the dry beach to a uniform berm height. The diffusivity allows to identify with better detail the evolution of the perturbation (Fig. 4.2.3, right panel). At the beginning, the diffusivity of case 1 and case 2 are similar. After 5 years, the curve diffusivity of cases 2 and 3 tend to converge. Therefore, the medium-term discrepancy is attributed to the initial bathymetric differences and the long-term discrepancy is attributed to setting the dry beach to a constant berm. These discrepancies are small and the AMN (case 3) is considered to reproduce well the diffusivity of the ZandMotor since, at the end of the simulation, the AMN diffusivity is only 5% larger than the ZM diffusivity. The diffusion of the mega-nourishment produces a widening of the perturbation and feeds the adjacent beaches, increasing the dry beach area. Fig. 4.2.4 (left panel) shows that the three cases feed more sand to area 1 than to area 2, i.e. the dry beach gained in area 1 is larger than the one gained by area 2. The feeding gap between the AMN (case 3) and the real ZM slightly increases in time, reaching a 6% difference after 50 years. Half of this difference is attributed to setting a constant dry beach (by seeing the differences with respect to case 2). This is more easily observed in the feeding asymmetry (Fig. 4.2.4, right panel). In the short-term the feeding asymmetry (FA) of cases 1 and 2 are practically equal. The three cases show a strong variability during the first 5 years due to the initially fast morphodynamic evolution. This trend will be observed in every simulation of this Chapter. After 20 years the FA of case 2 tends to the FA of case 3. Therefore, as occurs with the diffusivity, the short-tem differences are related to the initial bathymetric discrepancies and the long-term differences are related to using a constant dry beach. The three perturbations migrate in the same direction (to the NE in the ZM coordinates, 4.3 Effect of varying the mega-nourishment shape 75 -100 0 100 orientation ° 1 1.5 2 2.5 3 ǫ (m2/s) ×10-3 0 20 40 time (yr) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 FA -60 ° -30 ° 0 ° 30 ° 60 ° Figure 4.3.12: Diffusivity inferred from 50-yr simulations (left panel) and the FA during 50 years (right panel) for perturbations with different orientations. fusive than a perturbation with rotation. However this difference is small, the amplitude difference between the scenario with a rotation of 0 degrees and the scenario with a rotation of 60 degrees is of only 10 m after 50 years. Given the scales of the perturbation and the differences between the simulations, it can be considered that the diffusivity is robust to the orientation of the perturbation. However, the initial orientation has a larger impact on the feeding asymmetry (Fig. 4.3.12, right panel). First, the Θ =0◦scenario indicates that part of the feeding asymmetry is due to the wave climate (slightly dominated by waves with negative incident angles). Second, the initial orientation can enhance this asymmetry (negative angles) or diminish it (positive angles). The significant differences of the absolute values of FA between positive and negative orientations is again due to the wave climate. The impact of the initial orientation can have long term effects since the FA for positive orientation angles still remains negative (although small) after 50 yr. Regarding the mega-nourishment migration, the long-term trend is independent of the initial orientation angle (Fig. 4.3.13, left panel). The mega-nourishment moves towards the negative y-axis and with similar rates for the five cases. However, during the first 5 years the migration is controlled by the initial orientation angle. This initial migration is due to the strong diffusion occurring in the side with the largest shoreline slope. The shape asymmetry values are very low for every scenario but the wave climate ends up causing negative SA values in the long term (Fig. 4.3.13 right panel). 76 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS 0 10 20 30 40 50 time (yr) -150 -100 -50 0 50 gaussian displacement (m) 0 10 20 30 40 50 time (yr) -0.04 -0.02 0 0.02 0.04 SA -60 ° -30 ° 0 ° 30 ° 60 ° Figure 4.3.13: Gaussian displacement (left) and shape asymmetry (right) for different orientations of the perturbation over 50 yr. 4.3.2 Sensitivity to shape ratio The original shape ratio of the perturbation, controlled by σx/σy, is here modified with a factor ranging from 0.25 up to 2 every 0.25 of the original one. Nevertheless, the shoreline slopes of the perturbed beach are highly sensitive to this change (i.e. the asymmetry) even though the perturbation orientation is the same. For this reason, the orientation is set to 0◦, which also helps to further isolate the role of the shape ratio. The volume is kept constant by modifying σxand A. The symmetric perturbation is wider for smaller shape ratios. For example, the mega-nourishment has an amplitude of about 500 m for the case with a shape ratio 75% smaller than the original one (factor of 0.25). The amplitude for high factors (above 1.25) reach similar values quite fast and the difference becomes imperceptible after 20 yr. However, the diffusivity is more sensitive to the shape since it also depends on the initial perturbation width. The diffusivity is smaller for larger shape ratios with an exponential decay (Fig. 4.3.14, left panel). The initial variability in the feeding asymmetry is larger for the more concentrated nourishments (higher shape ratio), which is probably due to the initially larger shoreline slopes (Fig. 4.3.14, right panel). The long-term feeding asymmetry tends to be slightly positive even for wide nourishments with a small amplitude. The Gaussian displacement is negative for every scenario and the displacement rate is larger for more localized nourishments (Fig. 4.3.15, left panel). As occurs with the FA, the variability of the shape asymmetry is larger during the first 5 years for higher shape ratios. This initial noise fades in time tending to similar small values for the five cases (Fig. 4.3.15, right panel). 4.3 Effect of varying the mega-nourishment shape 77 0 0.5 1 1.5 2 shape factor 1 1.5 2 2.5 3 ǫ (m2/s) ×10-3 0 10 20 30 40 50 time (yr) -0.5 0 0.5 1 FA factor 0.25 factor 0.75 factor 1.25 factor 1.75 Figure 4.3.14: Diffusivity inferred from 50-yr simulations (left panel) and feeding asymmetry for different shape ratios of the perturbation over 50 yr (right panel). 0 10 20 30 40 50 time (yr) -150 -100 -50 0 50 gaussian displacement (m) 0 10 20 30 40 50 time (yr) -0.04 -0.02 0 0.02 0.04 SA factor 0.25 factor 0.75 factor 1.25 factor 1.75 Figure 4.3.15: Gaussian displacement (left) and shape asymmetry (right) for different shape ratios over 50 yr. 4.3.3 Sensitivity to volume The amplitude of the perturbation Ais varied together with σxto change the meganourishment volume. In this scenarios, the orientation of the perturbation and the shape ratio (σx/σy) are maintained equal to the ones of the default case. The original volume of the ZM is modified with a factor ranging from 0.25 to 2 every 0.25. As expected, the initial amplitude of the perturbation is larger for larger volumes. Fig. 4.3.16 (left panel), shows that the diffusivity is larger for lower volumes than the original 78 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS 012 volume factor 1.2 1.4 1.6 1.8 2 2.2 2.4 ǫ (m2/s) ×10-3 0 20 40 time (yr) -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 FA factor 0.25 factor 0.75 factor 1.25 factor 1.75 Figure 4.3.16: Diffusivity inferred from 50-yr simulations (left panel) and feeding asymmetry for different volumes of the perturbation over 50 yr (right panel). volume and stabilizes for larger volumes. The long-term feeding asymmetry is higher for larger volumes (Fig. 4.3.16, right panel). The smallest nourishment migrates slower than the other three cases (Fig. 4.3.17, left panel) and no clear difference can be seen between the two larger mega-nourishments. The shape asymmetry does not seem to be sensitive to the sand volume since all scenarios present variations around -0.0075 (Fig. 4.3.17, right panel). 4.4 Effect of varying the wave forcing 4.4.1 Design of SWC with different obliquity occurrence The obtained parameters of the synthetic wave climate in Section 4.2.2 are used to create two different set of simulations, in which the probability of occurrence of the four different wave sectors is varied to study the effect of varying the occurrence of high-angle waves. First, sectors I and IV are considered to belong to the oblique waves category and their combined probability is pO=pI+pIV . In the real wave climate, pO≃0.6 and this value is varied from 0 to 1 in steps of 0.1 in the first set of simulations. This is done by increasing/decreasing pIand pIV (keeping their ratio) and decreasing/increasing pII and pIII (also keeping their ratio) (see Fig. 4.4.18, above panel). In the second set of simulations, only pIis increased while the other three are decreased (keeping their ratios), so that pOchanges from 0.6 to 1 every 0.05 (Fig. 4.4.18, lower panel). Therefore, the first set of simulations has a bimodal 4.4 Effect of varying the wave forcing 79 0 10 20 30 40 50 time (yr) -150 -100 -50 0 50 gaussian displacement (m) 0 10 20 30 40 50 time (yr) -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 SA factor 0.25 factor 0.75 factor 1.25 factor 1.75 Figure 4.3.17: Gaussian displacement (left) and shape asymmetry (right) for different volumes of the perturbation over 50 yr. wave climate with different degrees of obliquity whilst the bimodality in the second set of simulations decreases with increasing obliquity. The probability of occurrence of the fifth sector pV(i.e. the one that does not produce morphological changes), of about 20%, is kept unchanged. 4.4.2 Effect of obliquity occurrence in mega-nourishment dynamics For wave climates with a large percentage of high-angle waves (≥80%), erosion areas are formed at the sides of the mega-nourishment (Fig. 4.4.19), which is the initial stage in the HAWI instability (van den Berg et al., 2011). In the first set of simulations, after 50 years the 100% case of oblique waves shows a maximum erosion of 150 m at y=6.5 km. In the second set of simulations, the run with 100% of oblique waves crashes after 40 years because at this moment all the initial dry beach (of 600 m) is eroded at y=6.5 km. The diffusivity decays linearly with an increasing percentage of wave obliquity in the SWC (Fig. 4.4.20). The decay is slightly larger for the second set of simulations, where only the occurrence of sector I increases. This might be related with the fact that this sector contains more energetic and oblique waves than sector IV. However, for SWCs with a large percentage of high-angle waves (≥80%) the inferred diffusivity might be invalid because of the erosion areas formed in the sides of the mega-nourishment, which deviates the shoreline shape from a Gaussian function and thus renders invalid the supposition taken to infer the diffusivity. Indeed, the obtained diffusivity in these cases should be  < 0 (Falqu´es & Calvete, 2003; Ashton & Murray, 2006a). It can be seen that the feeding asymmetry is larger for the scenarios with larger wave obliquity (Fig. 4.4.21, right panel). This might be related to the fact that the Sector-I waves 80 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS 0.5 0.6 0.7 0.8 0.9 1 1.1 oblique-wave occurrence 0 0.5 1 occurrence 0 0.2 0.4 0.6 0.8 1 1.2 oblique-wave occurrence 0 0.5 1 occurrence Figure 4.4.18: Probability of occurrence of the four Sectors for defined oblique waves occurrence. The upper panel assumes that Sectors I and IV are modified proportionally and the lower panel assumes that only the probability of occurrence of Sector I increases. 00.511.52 alongshore direction (m) ×104 0 500 1000 1500 cross-shore direction (m) Figure 4.4.19: Shorelines after 40 yr of evolution. The blue shoreline is forced with a 60% oblique bimodal wave climate, the red shoreline is forced with a 100% oblique bimodal wave climate, and the orange shoreline is forced with a a 100% oblique uni-modal wave climate. are more energetic than the Sector-IV waves. At the same time, the feeding rate is smaller for larger obliquity because the diffusivity decreases (Fig. 4.4.21, left panel). The strange behaviour of FA in the 100% wave obliquity case is due to the erosion areas formed from the beginning at the sides of the mega-nourishment. The long-term migration of the meganourishment is controlled by the percentage of oblique waves (Fig. 4.4.22, left panel). For small pO, the sector III is dominant (Fig. 4.4.18, above panel) and the mega-nourishment strongly migrates to the positive y-axis. For large pO, sector I is slightly more frequent than 4.4 Effect of varying the wave forcing 81 0 0.2 0.4 0.6 0.8 1 percentage of oblique waves 0 1 2 3 4 5 ǫ (m2/s) ×10-3 first set second set Figure 4.4.20: Averaged diffusivity over the last 5 years of simulations in function of wave obliquity occurrence. The blue line corresponds to the first set of simulations and the red line to the second set of simulations. sector IV (and the former is more energetic and with a higher incidence angle) so migration to the negative y-axis is observed. Notice that the short-term migration is the same for every climate because it is controlled by the initial mega-nourishment shape asymmetry. The Gaussian displacement and the shape asymmetry present an important correlation in the long term (Fig. 4.4.22). The larger the negative displacement is, the larger the shape asymmetry is. In the second set of simulations, the behaviour of FA when increasing only the occurrence of sector-I waves is more complex (Fig. 4.4.23). Actually, after only increasing the wave obliquity (pO) to 70%, the FA remains constant at around 0.5 instead of decreasing as in all the previous situations. For larger oblique percentages (pO), the FA increases continuously in time. The behaviour of the FA in this set of simulations is more complex because it is not only influenced by the diffusivity and the erosion areas (as for the first set of simulations) but also by an increasingly strong migration of the mega-nourishment. Indeed, the meganourishment strongly migrates to the negative y-axis for cases with more than 80% of oblique waves (Fig. 4.4.24, left panel) because the wave climate is dominated by sector-I waves. The Gaussian displacement and the SA show again a strong correlation (Fig.4.4.24, right panel) so the SA is very large for the very oblique wave scenarios. Notice that both the migration and SA experience strong increments when increasing the percentage of obliqueness from 80% to 100%. 82 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS 0 10 20 30 40 50 time (yr) 0 50 100 150 ∆xs(m) 0 10 20 30 40 50 time (yr) 0 0.2 0.4 0.6 0.8 1 FA 0% 20% 40% 60% 80% 100% Figure 4.4.21: Beach linear metre gained in the defined sections along the coast area 1 (solid line) and the coast area 2 (dashed line) over 50 yr (left panel) and the associated feeding asymmetry over 50 yr (right panel). These curves correspond to the percentage of oblique waves of the first set of simulations. 0 20 40 time (yr) -300 -200 -100 0 100 gaussian displacement (m) 0 20 40 time (yr) -0.1 -0.05 0 0.05 0.1 0.15 SA 0% 20% 40% 60% 80% 100% Figure 4.4.22: Gaussian displacement (left panel) and shoreline shape asymmetry (right panel) for the first set of simulations over 50 yr. 4.5 Discussion and conclusions The 50-yr evolution of an analytic mega-nourishment, constructed with a 2D-Gaussian bed level perturbation, has been compared with the evolution of the ZandMotor. Both cases have 4.5 Discussion and conclusions 83 0 20 40 time (yr) 0 50 100 150 ∆xs(m) 0 20 40 time (yr) -1 -0.5 0 0.5 1 FA 60% 70% 80% 90% 100% Figure 4.4.23: Beach linear metre gained in the defined sections along the coast area 1 (solid line) and the coast area 2 (dashed line) over 50 yr (left panel) and the associated feeding asymmetry over 50 yr (right panel). These curves correspond to the percentage of oblique waves of the second set of simulations. 0 10 20 30 40 50 time (yr) -1500 -1000 -500 0 gaussian displacement (m) 0 10 20 30 40 50 time (yr) -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 SA 60% 70% 80% 90% 100% Figure 4.4.24: Gaussian displacement (left panel) and shoreline shape asymmetry (right panel) for the second set of simulations over 50 yr. been computed with the Q2D-morfo model and the same real wave climate. Since the AMN has the same berm height in the dry beach, a simplified ZM with the same berm height has also been modelled. In order to quantify the performance of the mega-nourishments four characteristics have been examined: the diffusivity, the feeding capacity, the alongshore Gaussian displacement, and the shape asymmetry. The diffusivity of the AMN and the 84 CHAPTER 4. DIFFUSION OF MEGA-NOURISHMENTS simplified ZM is 10% larger than the one of the real ZM after 20 years of simulation and this difference was maintained over the next 30 years. The feeding asymmetry of the AMN is 20% larger than the one of the ZM during the first 3 years. Afterwards, the three FA curves reach similar values with only a 5% difference. The shape asymmetry of the 3 perturbations do not show significant differences. However, the Gaussian displacement of the real ZM starts deviating after 20 years from the other two cases. In general, the short-term differences are attributed to the discrepancies in the wet beach while the long-term differences are attributed to the assumption of a constant berm height in the AMN. Therefore, these findings show that a 2D-Gaussian perturbation can be used to mimic real mega-nourishments such as the ZandMotor and that the dry beach design is relevant for the long-term behaviour of the mega-nourishment. The role of the AMN shape on its own evolution has also been analysed by varying its asymmetry, the shape ratio and the volume. The shape asymmetry is varied using the orientation of the 2D-Gaussian function. It turns out that it has little relevance on the diffusivity but it has a large influence on the feeding asymmetry, determining its sign in the short and in the long term (over 50 years). A symmetric perturbation with a 0◦orientation, still has a FA value of 0.1, which can be attributed to the wave climate. The rest of the FA value is due to the initial orientation design. The long-term alongshore displacement shows the same rate (typically small) and direction (to the negative y-axis) for all the orientations but during the first 5 years the displacement follows the orientation sign. In the same line, the SA sign follows the sign of the orientation in the short term but in the long-term the SA curves tend to similar values for all orientations. The second parameter studied, the shape ratio (i.e. the ratio of the standard deviations of the Gaussian), has been tested using an orientation of 0◦. The original shape ratio, obtained after the optimization of the AMN to mimic the ZM, is modified from a fourth of the original value until doubling it. Its principal effect is seen on the diffusivity: for wider perturbations the diffusivity is larger (the widest perturbation doubles the diffusivity of the narrowest perturbation) and vice versa. This suggests that the larger cross-shore amplitude of the depth contours linked to the narrower perturbations (more localized) produce larger perturbations in the wave field. This in turn diminishes the diffusivity and enhances the processes leading to HAWI. The FA is smaller for wider perturbations, practically 0 for the widest one. The Gaussian-displacement rate is smaller for wider perturbations and the SA values have a similar behaviour at first. However, the SA reaches similar values at the end of the simulations (for all shape factors). The initial volume size has a similar impact as the shape ratio: for smaller nourishments the diffusivity was larger but this trend is not observed for larger volumes than the original one. Both the FA and the Gaussian displacement are in general smaller for smaller nourishments. The volume size seems to have no impact in the SA. In order to study the role of wave obliqueness on mega-nourishments, the real wave climate has been reduced to a few representative wave conditions defined by different wave incidence sectors, creating a synthetic wave climate. It has been found that the best way to extract the representative wave condition of each sector is by averaging the waves with its significant wave height to the 5/2 power, i.e. following the power in the CERC equation. Also, the number of bins necessary to represent the real wave climate has been tested. It is found that four wave conditions, two representing nearly shore-normal waves and two representing oblique waves, are optimal for representing the wave climate and that adding more wave conditions do not improve the performance of the SWC. 5.2 Observations 91 change (m) elevation (m) 8 7 6 5 4 3 2 1 0 -1 -2 -3 200 400 600 200 400 600 200 400 600 200 400 600 200 400 600 200 400 600 cross-shore distance (m) 2000 1800 1600 1400 1200 1000 800 600 400 200 2000 1800 1600 1400 1200 1000 800 600 400 200 alongshore distance (m) 200 400 600 200 400 600 200 400 600 200 400 600 200 400 600 200 400 600 200 400 600 2010 2011 2012 2013 2014 2015 2016 6 5 4 3 2 1 0 -1 -2 -3 Figure 5.2.3: Surface elevation plots (up) for the years 2010-2016 around July and the corresponding elevation change plots (down) between years. The North direction points upwards. shore resolution of about 5 m (to some z=−2 m) and an alongshore spacing of about 60 m. This second data set is also obtained from the Channel Coastal Observatory. The shorelines have been derived from the two available data sets by interpolating the position corresponding to z= 0. In total, 36 shorelines of 2-km length are extracted from February 2003 until February 2016. Fig. 5.2.4 displays the evolution of the shoreline in time, in which undulations of various wavelengths can be observed. In general, the shorelines can be classified visually in three types: straight (e.g. September 2005, red line), with small-scale undulations (around 200 m wavelength, e.g. March 2006, purple line), and with larger-scale undulations (around 400 m wavelength, e.g. February 2007, orange line, and July 2014, green line). The shoreline data confirm the presence of undulations of ∼450 m wavelength from July 2014 until February 2016. Moreover, in the shoreline of February 2007, undulations with a slightly smaller wavelength are also visible (of ∼350 m). Finally, smaller-scale undulations with a wavelength of ∼200 m appear and disappear throughout the whole study period in many of the shorelines. 92 CHAPTER 5. FORMATION OF SHORELINE SAND WAVES Feb03 Jul04 Nov05 Apr07 Aug08 Dec09 May11 Sep12 Feb14 Jun15 Oct16 0 500 1000 1500 2000 2500 alongshore distance (m) Figure 5.2.4: Shorelines extracted from February 2003 until February 2016. The time is displayed in the x-axis (the lower end of each shoreline indicates its survey time) but the shorelines have the same spatial scale in both axes. The thicker lines correspond to the four dates analysed in Fig. 5.2.5 and discussed in the text. 5.2.3 Shoreline analysis The Discrete Fourier Transform (DFT) technique is used to quantify the shoreline undulations. The shorelines are first smoothed using a running average (with a window size of 500 m) to obtain average shorelines. Then, the average shorelines are subtracted from the original ones obtaining the rectified shorelines, which show the characteristics of the undulations around an approximately constant straight line. Finally, the DFT of the rectified shorelines is computed. Fig. 5.2.5 shows the DFT magnitude of the three shorelines shown with colours in Fig. 5.2.4. The first one (September 2005) represents a shoreline with no clear dominant wavelength, the second one (March 2006) has a clear peak around 200 m, the third one (February 2007) shows a dominant wavelength of about 350 m, and the fourth one (July 2014) shows a dominant wavelength of about 450 m. In order to have a condensed and robust representation of the time evolution of the undulations, the DFT magnitude of the two wavelengths that best characterize the observed undulations (350 and 450 m) are extracted for each shoreline (upper panel of Fig. 5.3.6). In particular, the average of the DFT magnitude is computed within a range of ±30 m around each of the two wavelengths. From November 2006 until May 2007, there is a peak corresponding to the undulations at wavelengths of 350 m. These latter undulations decay gradually and they are no longer observed during spring-summer 2009. From July 2014 until the end of the study period a second (and the largest) peak is detected, corresponding to the undulations at wavelengths of 450 m. The DFT magnitude of the 200-m wavelength has also been computed (not shown) and it displays a different behaviour from that of the larger wavelengths, with smaller values and a shorter term variability, apparently not correlated with any characteristic of the wave climate. Moreover, these smaller wavelengths are close to the alongshore resolution of the surveyed profiles (60 m) so that the alongshore spacing between cross-shore sections may be 5.3 Modelling 93 200 400 600 λ(m) 0 100 200 300 400 500 DFT magnitude (m) 200 400 600 λ(m) 0 100 200 300 400 500 200 400 600 λ(m) 0 100 200 300 400 500 200 400 600 λ(m) 0 100 200 300 400 500 a) b) c) d) Figure 5.2.5: Magnitude of the Discrete Fourier Transform of the rectified shorelines of September 2005 (a), March 2006 (b), February 2007 (c), and July 2014 (d). too large to properly resolve such scale. Given that the detection of such wavelengths is not completely reliable and that they seem to be related to different physical processes (because they show a different behaviour), the rest of this study focuses on the larger-scale features. 5.3 Modelling 5.3.1 Wave transformation In order to investigate the relationship between the observed shoreline sand waves and the high-angle wave instability, the wave parameters at the depth of closure in front of the sand wave area are needed. Thereby, the 10-yr hourly wave parameters at the offshore buoy are propagated to this onshore location using the SWAN model. This is a third-generation wave model that represents the sea state in two dimensions by solving the spectral action balance equation. The details of the model are reported by Booij et al. (1999) (version 41.10 is used here). Some physical processes that can be described with the SWAN model are ignored in the present simulations. In particular, the wind wave generation, and the quadruplet and triad interactions are ignored. Instead, diffraction and dissipation by depth-induced breaking and bottom friction are accounted for. The bathymetry is created by combining two surveys. The first one has a resolution of 0.5×1.0 km (in the north and east directions respectively) and reaches depths of 50 m. The second survey has a resolution of ∼2×2 m and reaches ∼1.0 km offshore. For optimization purposes, the SWAN model is first applied to two coarse grids, one of 46 ×45 km for SW 94 CHAPTER 5. FORMATION OF SHORELINE SAND WAVES waves (Fig. 5.3.7) and another of 37 ×65 km for NE waves (not shown). The cell size in these two coarse grids is 380 ×380 m. A nested grid of 2 ×5 km is also used for both types of waves (Fig. 5.3.8), which has a resolution of 8 ×50 m. A sensitivity analysis to the size of the grids and their resolution has shown that this configuration is optimal and provides an acceptable accuracy. The lateral and offshore boundaries are forced uniformly using a JONSWAP spectrum with the peak enhancement parameter default of 3.3 and a directional spreading of 10 degrees (for each wave condition). The directional spreading is chosen accordingly to the work done by Kaergaard & Fredsoe (2013b) in the context of HAWI. After applying the SWAN model, the parameters of the propagated 10-yr wave series are averaged along the 4 m contour depth in front of the area of interest (i.e. the 2 alongshore kilometres shown in Fig. 5.2.1). Fig. 5.3.9 shows the wave conditions at the buoy and at 4-m depth from October 2013 until June 2014, as an example. The SW waves experience strong refraction during their propagation towards the sand wave area, which causes a significant energy dispersion. This is translated into a significantly smaller Hsin shallow waters in comparison with the buoy (Fig. 5.3.7, 5.3.8 and 5.3.9). In contrast, the Hsof NE waves is kept relatively unchanged. Moreover, the waves with a very high incidence angle (with respect to the studied shoreline) do not necessarily have an effect on the study area as only waves with an angle smaller than 90◦(from now on, the incidence wave angles are referenced with respect to shore normal) contribute to alongshore sediment transport. For example, at 6 m depth only 23% of SW waves have a wave angle smaller than 90◦, while this is true for 63% of SW waves at 4 m depth. This percentage increases for decreasing depths due to the wave transformation over the fan-shaped bathymetric contours (i.e. the bathymetric contours gradually diverge northwards, as can be seen in the left panel of Fig. 5.3.8). 5.3.2 Correlation between shoreline sand wave presence and highangle wave incidence There is some debate on the definition of high angle waves, i.e. on the precise value of the wave angle at the depth of closure, θc, above which HAWI develops. Ashton et al. (2001) gave the value of θc= 42◦. However, this theoretical prediction can vary depending on the sediment transport formula (Ashton & Murray, 2006b), on the assumptions of the model (Falqu´es & Calvete, 2005; van den Berg et al., 2012; Kaergaard & Fredsoe, 2013a) and on the shape of the bathymetric undulations associated to the shoreline undulations (Idier et al., 2017). In this section we adopt the value θc= 45◦for being representative of most of the predicted values. To quantify the degree of dominance of destabilizing over stabilizing waves in the wave climate (i.e. high-angle waves, |θ|>45◦, versus low-angle waves, |θ|<45◦), the ”strength” of the high-angle waves for each time survey, tk, is defined as E(tk)>45◦=Z89◦ 45◦ dθ Ztk tk−1 H5/2dt +Z−45◦ −89◦ dθ Ztk tk−1 H5/2dt (5.3.1) where Hand θin these formulas are those computed at 4 m depth in front of the shoreline wave area. The 5/2 power is introduced because, according to the widely used CERC formula 5.3 Modelling 95 m 0 5 10 R Oct05 Dec06 Feb08 Apr09 J un10 Aug11 Oct12 Dec13 Feb15 -5 0 5 (s-1) ×10-7 300 m 400 m 500 m 0 500 magnitude (m) 350 m 450 m 50 DFT magnitude (m) 0 500 DFT magnitude (m) Figure 5.3.6: DFT magnitude of the two wavelengths characterising the observed shoreline sand waves (upper panel), coefficient Raveraged during the time period between consecutive shoreline surveys (middle panel) and growth rate of the LSA for three wavelengths (lower panel). (Komar, 1998), the total alongshore transport rate is proportional to H5/2 b. Similarly, the ”strength” of the low-angle waves is defined as E(tk)<45◦=Z45◦ −45◦ dθ Ztk tk−1 H5/2dt (5.3.2) Afterwards, the wave-dominance ratio, R, is defined as R(tk) = E(tk)>45◦/E(tk)<45◦. The time intervals with R > 1 correspond to a dominance of high-angle waves over low-angle waves. The larger R, the stronger the relative influence of high-angle waves. The middle panel of Fig. 5.3.6 shows the time series of R, obtained during the time period between consecutive shoreline measurements, from October 2005 to November 2015. The coefficient Ris almost always above 1 and there are two main peaks of about R= 11 and R= 6. The peak of R= 6 coincides with the formation event of sand waves around February 2007, as can be seen in the corresponding DFT-magnitude peak for λ= 350 m. Relative low values of Rduring April 2009 to September 2013 are roughly consistent with small DFT magnitudes of the two analysed wavelengths. The largest peak in Rtakes place around February 2014, clearly related with the formation event of 2014 that is visible as an increase in DFT magnitude for λ= 450 m. Notice that the Rpeaks and the DFT-magnitude peaks display a small time lag of 3-6 months. This makes sense because on the one hand the surveys are available every 3-4 months and on the other hand the shoreline undulations 96 CHAPTER 5. FORMATION OF SHORELINE SAND WAVES 5 15 25 35 45 x coordinate (km) 5 15 25 35 45 y coordinate (km) Bathymetry (m) -40 -30 -20 -10 0 5 15 25 35 45 x coordinate (km) Hs (m) 0 0.5 1 1.5 2 Figure 5.3.7: Coarse grid bathymetry (left panel) and significant wave height (right panel) for the SW wave averaged conditions The North direction points upwards. can only be detected when they have reached a significant amplitude. 5.3.3 Linear stability analysis 1Dmorfo setup Here, the linear stability model 1Dmorfo, described in Chapter 2.3, is used to investigate the formation events of the undulations observed in Dungeness. Regarding the model setup, the equilibrium cross-shore beach profile is extracted from combining the high-resolution intertidal topographic surveys (which extend to about 3 m depth) and the bathymetry extending 1 km offshore. The wave conditions at Dcare obtained from those propagated with SWAN and two different types of simulations are performed: one type uses the full 10-yr wave series and the other type applies constant wave conditions. The empirical constant µ(in the CERC equation, Eq. 2.2.9), which modulates the strength of the alongshore transport, is set to µ=0.15 m1/2s−1. Regarding the perturbation, several values of λin the range of the observed ones (λ=200-1000 m) are used. Regarding the cross-shore shape of the perturbation, a cross-shore shift of the profile is used as the shoreline moves onshore/offshore. Given that the available high-resolution topographic measurements show that the shoreline sand waves extend across the whole measured domain, from z= +4 m until z=−3 m (Fig. 5.2.3), a default depth of closure of Dc= 4 m is used. The latter two choices are motivated and discussed in Chapter 5.4.3. 5.3 Modelling 97 33 33.5 34 34.5 x coordinate (km) 8 8. 5 9 9. 5 10 y coordinate (km) Bathymetry (m) -10 0 0 0 -10 -30 -25 -20 -15 -10 -5 0 5 Wave direction -6 -6 -4 -4 -2 -2 32.5 33 33.5 34 34.5 x coordinate (m) 33 33.5 34 34.5 x coordinate (m) 0 0 0 -10 -10 Hs (m) 0 0.5 1 1.5 2 Figure 5.3.8: Nested grid bathymetry (left panel), significant wave height (central panel) and wave direction (right panel) for the SW averaged conditions. The North points upwards. Wave series simulations The 10-yr alongshore averaged wave series at 4 m depth (e.g. red lines in Fig. 5.3.9) is used as input for the 1Dmorfo model. For each (hourly) record of the time series, tj, and for a number of wavelengths, λ, the growth rate σ(tj) = σr(tj) + iσi(tj) for each wavelength is computed. This means that the amplification factor of a small amplitude sand wave with wavelength λfrom tjto tj+ ∆t, where ∆t= 1 h is exp(σr(tj)∆t). Thus, when σr(t) for a given λis positive, sand waves of this wavelength are expected to grow. On the contrary, when σr(t) for a λis negative, such sand waves are expected to decay. The wave time series has a strong variability at hourly level while the morphology reacts much more slowly. For this reason, instead of examining the raw σr(t) time series, a filtered time series with a running average (window of 90 days) is used. The wave series simulation has been done for λ= 200 m to λ= 1000 m with a 20 m spacing. Notice that the 1Dmorfo model can only resolve length scales significantly larger than the surf zone width (which is about 30 m for the energetic periods in this area). The lower panel of Fig. 5.3.6 shows the time series of the filtered σrfor λ= 300, 400 and 500 m, which are relevant wavelengths according to the observations. It is seen that the growth rate for λ= 300 m is always negative, except during December 2006, preceding the sand wave formation event that is observed during the winter 2006-2007. In December 2006, the σrcorresponding to 500 m also has a positive value although smaller. This σr corresponding to λ= 500 m is always negative except in December 2006 and in winter 2013-2014. During this second period it shows a remarkable peak, preceding the observed sand wave formation event of spring 2014. Thereby, the LSA model accurately predicts the observed shoreline sand wave formation moments, with the modelled wavelengths slightly under-predicting/over-predicting the observed ones in the first/second event. The characteristic growth time of the instability, σ−1 r, is of 23 days in the peak of winter 2013-2014 (and slightly larger). This is in the correct order of magnitude in account of the observed reaction time of this morphological system. 98 CHAPTER 5. FORMATION OF SHORELINE SAND WAVES Oct13 Nov13 Dec13 Feb14 Mar14 May14 J un14 0 1 2 3 4 5 Hs (m) at 43-m depth at 4-m depth Oct13 Nov13 Dec13 Feb14 Mar14 May14 J un14 -50 0 50 100 150 200 θ° Figure 5.3.9: Time series of the significant wave height (upper panel) and wave direction with respect to the shore normal of the studied shoreline stretch (lower panel) from October 2013 until June 2014. The blue line shows the offshore wave conditions recorded at the buoy (43-m depth) and the red line are the wave parameters at 4-m depth in front of the shoreline sand waves area. It is found that the positive σrperiods occur when the Rratio is large enough (R > 5), which is consistent with the HAWI theory. After the formation, although the modelled σr switches from positive to negative values, the observed sand waves can persist or even keep on growing for a while. We must recall that the linear stability analysis is valid only for small amplitude sand waves and it is possible that, after the initial formation, non-linear interactions drive their dynamics. Finally, it is important to notice that the growth rate for λ= 400 m is always negative while one would expect positive σrfor this wavelength during the two formation events (since this λis close to the observed ones). Finally, the model predicts an averaged migration rate of 600 m/yr, for λ=500 m, during the period 2014-2016 (the period when undulations of this wavelength are observed to migrate, see Fig. 5.2.3). The migration direction and its order of magnitude agree with the observed ones (about 200 m/yr, see Chapter 5.2). The over-prediction might be related to the fact that the model assumes perturbations of infinitesimal amplitude whereas the migration is observed for fully developed shoreline undulations. In other studies of coastal morphodynamic patterns, the migration rate in the non-linear regime is half of that obtained in the linear regime (Garnier et al., 2008). 5.3 Modelling 99 200 300 400 500 600 700 800 900 1000 -1 0 1 σO (s-1) ×10-5 θ =65 ° θ =70 ° θ =75 ° θ =80 ° θ =85 ° 200 300 400 500 600 700 800 900 1000 -1 0 1 σN (s-1) ×10-5 θ =10 ° θ =15 ° θ =20 ° θ =25 ° θ =30 ° 200 300 400 500 600 700 800 900 1000 λ (m) -1 0 1 σ (s-1) ×10-6 Figure 5.3.10: Growth rate curves for different incident angles representing high-angle waves (upper panel) and low-angle waves (middle panel) during winter 2013-2014. Average growth rate curve (lower panel) using a weighting associated to the frequency of high- and low-angle waves. Constant waves simulations In order to understand the results of the wave series simulations, an analysis for constant (in time) wave parameters is very convenient. First of all, the average wave statistics of high-angle (|θ|>45◦) and low-angle (|θ|<45◦) waves are computed for the period period prior to the second formation event (December 2013 - March 2014). This gives Hs= 0.85 m, θ= 75◦and Tp= 6.8 s for high angles and Hs= 0.58 m, θ= 20◦and Tp= 5.1 s for low angles. During this period, high-angle waves occur 93% of the time and only 7% of the waves are low-angle (this can be appreciated in Fig. 5.3.9). The 1Dmorfo model is then run for high-angle and for low-angle waves with the corresponding values of Hsand Tp. Five different angles are considered for each case: the average one and the latter ±5◦and ±10◦. In this way, five growth rate curves (σr-λ) for high-angle and five for low-angle waves are computed and shown in Fig. 5.3.10. As can be seen in that figure, the low-angle waves damp all the wavelengths, with a stronger damping for smaller wavelengths and smaller angles. In contrast, the high-angle waves produce positive σrbut, remarkably, each instability curve has three local maxima (in the range λ > 200 m). The larger the the angle, the larger the wavelengths of the maxima and the smaller the corresponding maximum σrvalues. Typically, the instability curves for HAWI present a single maximum (Falqu´es & Calvete, 2005). However, as was found by 100 CHAPTER 5. FORMATION OF SHORELINE SAND WAVES Nov05 Mar07 Aug08 Dec09 May11 Sep12 Jan14 Jun15 -5 0 5 σ (s-1) ×10-7 Dc = 4 m Dc = 5 m Figure 5.3.11: Growth rate sensitivity of λ= 500 m to the depth of closure. van den Berg et al. (2014), secondary maxima can sometimes occur. This happens for a marginal region in the parameter space, for very high wave angles, for some combinations of bathymetric profiles and wave periods, and for relatively short wavelengths (λ∼1 km or less), and it can be interpreted as follows. The damping of short wavelengths in HAWI is controlled by wave energy focusing/defocusing by the undulations (van den Berg et al., 2014). For relatively long wavelengths, in comparison with the offshore reach of the bathymetric perturbation, the focusing is always near the sand wave crests (i.e. the prograding section) and the defocusing near the embayments. This was also explained by Uguccioni et al. (2006) as a result of each wave ray crossing only one of the shoals associated to the shoreline undulation. However, for relatively short wavelengths, large angle and small periods, each wave ray can cross several shoals. In this situation, the wave focusing can take place away from the crests and is highly sensitive to the wavelength, wave angle and bathymetry. As a result, the instability curve becomes quite complex for short wavelengths featuring several local maxima that look somewhat erratic (very sensitive to small changes in the angle and the bathymetry). A way to filter out such strong sensitivity, in account of the inherent uncertainty sources coming from the parameters and the model, is to make an average of the different growth rate curves with a weighting associated to the frequency of high and low-angle waves. The corresponding averaged curve shows a local positive maximum at λ≈540 m and another at λ≈220 m (lower panel of Fig. 5.3.10). Since the 220 m wavelength peak is in the lower limit of the length scales that can be resolved by the 1Dmorfo model (given the surf zone width in this case) this averaging lends support to the robustness of λ≈500 m as an output of the LSA for the time series of wave parameters. This is in good agreement with the 450-m wavelength of the formation event in 2014. Rather, the wavelengths below 500 m must be taken with care and this explains why the linear stability analysis can predict growth for λ= 300 m and decay for λ= 400 m during the 2006-2007 event, where the observed wavelength is λ≈350 m. 6.1 Main findings 107 the diffusion by a factor of 2.5. The diffusivity can be used to compute the ZM lifetime, here defined as the time needed to reduce the amplitude after construction by a factor 5. The classical diffusivity predicts a lifetime of only ∼35 yr whereas the Q2Dmorfo diffusivity predicts a lifetime of ∼90 yr. 4) What is the effect of wave obliquity in the evolution of a mega-nourishment? Could a mega-nourishment trigger the formation of shoreline sand waves? The wave angle has an important influence on the dynamics of mega-nourishments and should be taken into account in their design. If the wave climate is dominated by lowangle waves (<45◦) the diffusivity of a mega-nourishment is larger than in case where both low- and high-angle waves are present. In particular, the diffusivity decreases linearly with increasing frequency of high-angle waves. If high-angle waves dominate (frequency above 80%) erosional hotspots are formed at the sides of the mega-nourishment. This is specially strong when the high-angle waves come mostly from one side (unimodal wave climate), in which case a dramatic alongshore migration of the mega-nourishment is also produced (e.g. a displacement of 800 metres occurs after 50 years for 90% of high-angle waves). A smaller but significant migration can also occur with bimodal wave climates that have a certain directional asymmetry. In this cases there is also an asymmetry in the feeding to adjacent beaches. According to van den Berg et al. (2011), the formation of erosional hotspots precedes the triggering of km-scale shoreline sand waves due to high-angle wave instability (HAWI). In the present simulations it is found that, as explained above, this occurs above a frequency of 80% of high-angle waves. However, the simulation time of 50-yr used in this thesis was not large enough to develop a full train of sand waves. Only in the case of a 100% of unidirectional high-angle waves, the formation of sand waves has been directly observed. In the case of the ZandMotor, the beach is safe from being destabilized by the mega-nourishment because the wave climate only has 62% of high-angle waves. A collateral result of the importance of wave obliquity is that, when building synthetic wave climates, four wave incidence sectors must be considered: two representing low-angle waves and two representing high-angle waves. Using one or two wave sectors gives large errors and using more than four wave sectors does not improve the performance. 5) What is the role played by the initial shape and size of a mega-nourishment on its own evolution? The role of the mega-nourishment shape on its own evolution has been analysed focusing on its initial shape asymmetry, the shape ratio and the volume. The initial shape asymmetry only affects the asymmetry in the feeding to adjacent beaches. The beach at the side of the mega-nourishment with the largest shoreline slope receives more sediment, not only in the short term but also in the long term. The shape factor and the volume control the diffusivity. Smaller and wider mega-nourishments are more diffusive than larger and narrower (more localized) ones. This might be due to the fact that the larger cross-shore perturbations of the depth contours linked to the narrower and larger mega-nourishments produce larger perturbations in the wave field. This, together with the obliquity of the wave climate, in turn enhances the processes leading to shoreline instability, thereby decreasing the diffusivity. 108 CHAPTER 6. SYNTHESIS 6) Can the formation of km-scale shoreline sand waves be observed in nature, and if so, is it linked to the wave climate? Detecting the formation of km-scale shoreline sand waves is in general very difficult due to the large time and length scales involved. In the present thesis, KSSW have been observed along the north-east flank of Dungeness foreland (U.K.). Two clear formation moments have been detected during the study period, the first one on February 2007 with a wavelength of 350 m and the second one on July 2014 with a wavelength of 450 m. A gradual decay of the 2007 shoreline undulations is observed and they are no longer visible on autumn 2009. The undulations formed at the second event persist at least until February 2016 (with some decay) and migrate northward at a mean rate of about 200 m/yr. The role of HAWI on the formation and dynamics of these shoreline undulations has been examined. A ratio quantifying the degree of dominance of high-angle waves over lowangle waves, R, has been computed at 4 m depth and a good correlation between high Rvalues and the formation of the shoreline sand waves has been found. In particular, the undulations occur when R > 5 implying that a climate with 80% of high-angle waves is needed to trigger their formation, in agreement with previous HAWI studies and with the results about mega-nourishments of this thesis (see the previous question). Therefore, it is concluded that HAWI is the primary cause of the shoreline sand wave formation in Dungeness. This is a unique data set that shows for the first time the formation moment of KSSW showing a clear correlation with the dominance of high-angle waves at that time. 7) How morphodynamic models results compare with observed shoreline sand wave formation? The 1Dmorfo linear stability model was used to model the formation of KSSW in Dungeness. The model predicts positive growth rates previous to the two observed formation moments for wavelengths similar to the observed ones. It is also found that a bathymetric perturbation corresponding to a cross-shore profile shift is required to reproduce the growth of sand waves with the observed characteristics. Moreover, the morphological response occurs at time scales of the order of the observed ones (characteristic growth time of about 1 month and migration rates of hundreds of metres per year). Despite the applied linear model has been able to represent the initial formation of the undulations it can not to reproduce the dynamics of the finite-amplitude features. 6.2 Future Research The design and prediction of beach nourishment evolution has been traditionally done with one-line numerical models, which assume an instantaneous profile shift as cross-shore link between the shoreline and the depth contours. This approach has been successful because the correct description of the depth contours (over which waves are transformed) can be neglected for small nourishments and moderate wave angles. However, larger nourishments, such as the ZandMotor, perturb significantly the depth contours. Under these conditions, the one-line modelling studies of mega-nourishments implicitly confirm that an accurate description of the bathymetric contours is key for the correct prediction of its evolution. For 6.2 Future Research 109 example, in the ZM context, Tonnon et al. (2018) tested two different depths of closure (6 m vs 19 m) and found a large discrepancy in the obtained one-year net alongshore sediment transport. A long-term comparative study between one-line models and models that better describe the depth-contours evolution, such as the Q2Dmorfo model, is necessary to understand the limitations of the one-line approximation and improve the design of meganourishments. The Q2Dmorfo model, once calibrated with the evolution of the ZandMotor, initially overpredicts the cross-shore diffusion, i.e. with a decay rate of the amplitude higher than the observed one. This is due to the initial bathymetric slope, which is much larger than the equilibrium one. Although the cross-shore diffusion can be modulated in the model with the parameter ν, using a low νvalue gives rise to artificial shoals. In order to solve this problem, it is necessary to use a cross-shore transport that does not need to prescribe an equilibrium profile but which, at the same time, reaches an equilibrium under constant wave forcing. Nevertheless, the previous issue is only problematic in the first months of simulation and the model as it is allows to further investigate the mega-nourishment dynamics in the mid and long term. Another limitation of the Q2Dmorfo model is that the shoreline curve has to be univalued and thereby can not describe the formation of spits, which occur when there is a high frequency of unidirectional high-angle waves (Ashton et al., 2001). The use of a fuzzy shoreline algorithm can certainly help to make this improvement. Finally, the present sealevel rise implementation is not yet fully operative because the lateral boundary conditions must be improved. This is an important issue given the long time scales involved in meganourishment dynamics, for which the sea level rise induced by climate change may play a role. Some of the results here obtained with the Q2Dmorfo model deserve further attention. In particular, the reason of the diffusivity decay with volume and shape ratio increments must be analysed in depth. Intuitively, these morphological conditions are characterised by large cross-shore perturbation in the depth contours, which, in case of oblique wave climates, would enhance the instability mechanisms and reduce the diffusivity. This hypothesis could be tested by modelling mega-nourishments with different shape ratios and volumes using different wave climates (i.e. with different frequencies of high-angle waves). Shoreline sand waves have been observed in a shoreline stretch of the Dungeness Cuspate Foreland. The wave climate is ideal for testing the HAWI and LAWI hypothesis. In agreement with the former, the shoreline sand waves are formed during events were the high-angle incidence wave energy is largely dominant. The 1Dmorfo model captures these formation events despite ignoring the observed 6-m tidal range. However, the model can not describe the subsequent behaviour, i.e. maintenance and even growth of the perturbations, due to the linearity assumption. The good-quality data and the relative small characteristic scales of the observed features allow to use more complex models to simulate its behaviour. In particular, simulations with a two-dimensional non-linear morphodynamic model are required to investigate the finite-amplitude regime and understand the role played by the tides. 110 CHAPTER 6. SYNTHESIS Bibliography Arriaga, J., Falqu´ es, A., Ribas, F. & Crews, E. 2018aFormation events of shoreline sand waves on a gravel beach. Ocean Dynamics 68 (6), 735–748. 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