Modelling Sediment Transport in the Swash Zone

Transcription

1 Modelling Sediment Transport in the Swash Zone Arie Arnold van Rooijen

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3 Title Client Delft University of Technology Pages 142 Keywords numerical modelling, XBeach, high frequency swash, low frequency swash, Le Truc Vert Abstract The swash zone is the part of the beach that reaches from the limit of wave run-up until the limit of wave run-down. It is recognized as being a dynamic area of the nearshore region, characterized by strong and unsteady flows, high turbulence levels, large sediment transport rates and morphological changes on a small timescale. Due to the complexity of the processes taking place in the swash zone, there are still great uncertainties about the driving forces for sediment transport. Morphodynamic process-based numerical models tend to overestimate the seaward directed sediment transport in the swash zone, especially for mild conditions. The main objective of this thesis is to obtain insight in the hydrodynamic processes responsible for sediment transport in the swash zone, and to use this knowledge to optimize a morphodynamic numerical model (XBeach) for simulating swash zone physics. First, an extensive literature review is carried out to provide the physical base. Second, a number of (theoretical) linear profile simulations are conducted to provide insight into the simulated swash characteristics for different beach states, and to assess the effect of including a number of swash processes (e.g. turbulence or groundwater flow) in the simulations. Third, measurements obtained during a field experiment in Le Truc Vert (France) are used to verify three hydrodynamic modelling approaches and two sediment transport models. Version Date Author Initials Review Initials Approval Initials aug. 211 A. A. van Rooijen J.S.M. van Thiel de Vries T. Schilperoort State final

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5 August 211 MODELLING SEDIMENT TRANSPORT IN THE SWASH ZONE by Arie Arnold van Rooijen A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the field of Civil Engineering at Delft University of Technology Delft, The Netherlands Submitted for approval on August 19, 211 i

6 August 211 Graduation committee Prof.dr.ir. M.J.F. Stive Prof.dr.ir. A.J.H.M. Reniers Dr.ir. J.S.M. van Thiel de Vries Ir. R.T. McCall Ir. M. Henriquez Ir. P.B. Smit Chairman, Delft University of Technology University of Miami (U.S.A.) Deltares / Delft University of Technology Deltares / University of Plymouth (U.K.) Delft University of Technology Delft University of Technology / A. A. van Rooijen / August 19, 211 This research was carried out at: Deltares Rotterdamseweg MH Delft The Netherlands and Rosenstiel School of Marine and Atmospheric Science University of Miami 46 Rickenbacker Causeway Miami, FL U.S.A. Arie Arnold van Rooijen 211. All rights reserved. Reproduction or translation of any part of this work in any form by print, photocopy or any other means, without the prior permission of either the author, members of the graduation committee and/or Deltares is prohibited. ii

7 August 211 Zeedag Schuimend zonlicht golft tot aan de horizon. Witte stapelwolken drijven door het blauw. Een snoer van meeuwen en strandlopertjes kringelt langs de branding, waar de hemel zich spiegelt in het natte zand. Wij stappen voort over het strand, over stroompjes en knisperende schelpjes, een snelvervagend spoor achterlatend. M. T. van Zweeden, 29 iii

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9 August 211 by Arnold van Rooijen Abstract The swash zone is the part of the beach that reaches from the limit of wave run-up until the limit of wave run-down. It is recognized as being a dynamic area of the nearshore region, characterized by strong and unsteady flows, high turbulence levels, large sediment transport rates and morphological changes on a small timescale. Due to the complexity of the processes taking place in the swash zone, there are still great uncertainties about the driving forces for sediment transport. Morphodynamic process-based numerical models tend to overestimate the seaward directed sediment transport in the swash zone, especially for mild wave conditions. The main objective of this thesis was to obtain insight in the hydrodynamic processes responsible for sediment transport in the swash zone, and to use this knowledge to optimize a morphodynamic numerical model (XBeach) for simulating swash zone physics. Numerous research experiments have been conducted over the past fifteen years, both in laboratory and in the field. The results of the literature review carried out in this thesis, show that wave asymmetry, wave skewness, turbulence and boundary layer effects are important processes considering sediment transport in the swash zone. Infiltration, exfiltration and groundwater flow are found to be dominant on steeper beaches with larger grain sizes. Swash-swash interactions, acceleration and horizontal pressure gradients are generally found to be important in the swash zone, but are, however, not well understood yet. To obtain insight in the simulated swash characteristics for different beach state levels (quantified by the Iribarren number), linear profile simulations were conducted with a typical wave steepness and beach slope combination. The results of these simulations show that the beach state level has a great effect on the predicted hydrodynamics and morphodynamics. Even though no comparison with measured data was carried out, it is most likely that the model overestimates the hydrodynamics and morphodynamics in offshore direction for reflective beaches, due to the less accurate method for solving of the high frequency waves within the model. It is questionable whether the Stokes drift / undertow concept used in the model is still applicable in the swash zone, especially for reflective beaches. The results of including wave asymmetry, wave skewness, groundwater flow, short wave turbulence and long wave turbulence in the model, show that all processes have a net onshore transport effect, except for the short wave turbulence. Since in literature turbulence is found to be onshore transport promoting, it is concluded that short wave turbulence is not implemented in the model correctly for sediment transport in the swash zone. However, this is also related to the Stokes drift / undertow concept. The last step in this thesis was to study three hydrodynamic modelling approaches (surf beat approach, hydrostatic approach and non-hydrostatic approach), in which the swash hydrodynamics are simulated in more or less detail, and two different sediment transport models (Van Rijn [27] transport model and Nielsen [1992] / Roelvink & Stive [1989] transport model). All approaches are verified with a dataset obtained from a field experiment in Le Truc Vert, France. The results show that the hydrostatic approach in combination with the Nielsen / Roelvink & Stive transport model provides a good prediction of the measured morphodynamics. There is, however, an underestimation in the predicted run-up, mainly for the accretive case. It is considered most likely, that the underestimation is due to two dimensional effects (e.g. wind), that are not accounted for in the one dimensional simulations. v

10 August 211 It was concluded that the surf beat approach is able to accurately predict erosion, but not accretion, while the non-hydrostatic approach overpredicts the velocity and acceleration magnitudes, leading to a substantial overprediction of the sediment transport rates. Finally, it was found that the Nielsen / Roelvink & Stive transport model performs better than the Van Rijn model, mainly due to the inclusion of the acceleration term in the Nielsen formulation for bed load transport. In general, it is concluded that the XBeach surf beat approach most likely overpredicts (offshore) sediment transport rates in the swash zone for more reflective beaches. The Van Rijn transport model is well able to predict erosive swash conditions, but accretion is not represented by the simulations. The hydrostatic approach in combination with the Nielsen/Roelvink & Stive transport model is able to predict accretional swash, mainly due to the inclusion of an acceleration term. There is, however, an underestimation in the predicted run-up, and it is therefore suggested more (field) experiments, similar to the Truc Vert swash experiment, should be carried out to increase the knowledge of swash zone processes in general, and to further improve morphodynamic models. vi

11 August 211 Acknowledgements Deltares is greatly thanked for supporting my thesis research, and for giving me the opportunity to spend three months in Florida, USA. Also, I would like to acknowledge the following people for their valuable support during my studies in general, and this thesis in particular: The members of my graduation committee are greatly thanked for their feedback, critics and discussions during my thesis work. Special thanks to Jaap van Thiel de Vries for his daily supervision, numerous open discussions and enthusiasm throughout the thesis period, and to Ad Reniers for his daily support and valuable input during my stay at the University of Miami. I would like to thank Ad and Stella, as well as Mandy and Claudio, for making my stay in Miami an unforgettable experience. Even though it was only for a short period, I would like to thank the staff at the USGS (St. Petersburg, Florida) for giving me the opportunity to work together with them, and for being so welcome and including me in all the parties and trips, making it a wonderful two weeks. I would like to thank Chris Blenkinsopp for sharing his Le Truc Vert swash zone dataset with us, making it possible to make an actual comparison between model predictions and natural behavior, thereby adding great value to this thesis. I want to thank all my friends in Katwijk and in Delft that I made over the past years, for all the great times, necessary distractions and support during my studies, and my Deltares graduation colleagues for the interesting discussions and fun talks during the coffee and lunch breaks. Special thanks to my family for always supporting and encouraging me throughout my education, and to Cláudia for her unconditional support, love and care, for the numerous valuable discussions and feedback that added great value to this thesis, and of course the great times we spend together. vii

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13 August 211 Contents Abstract Acknowledgements v vii 1 Introduction Background Problem description Research question Objectives Thesis outline 3 2 Swash zone physics Introduction Coastal terminology Waves and wave-induced swash Beach states and swash motions Sediment transport Hydrodynamics and morphodynamics in the swash zone Morphological response of the swash zone Numerical modelling and the swash zone Conclusions 26 3 Methodology Introduction Modelling approach Surf beat approach Hydrostatic approach Non-hydrostatic approach Overview 44 4 Modelling basic swash processes Introduction Model setup Simulation results Discussion Conclusions 7 5 Modelling additional swash processes Introduction Swash processes in the surf beat approach The non-hydrostatic approach Discussion Conclusions 88 6 Field case: Le Truc Vert Introduction Data description 9 ix

14 August Model setup Hydrodynamic results Morphodynamic results Discussion and model sensitivity Conclusions Conclusions and recommendations Conclusions Recommendations 131 Bibliography 133 List of Abbreviations and Symbols 139 x

15 August Introduction 1.1 Background Beaches are present on a great part of the coastlines worldwide, and are amongst the most dynamic physical systems on Earth. They are composed of sediments such as gravel, sand or mud, and are constantly affected by tides, waves, currents and wind. Sediment transport is the process responsible for the morphology of a sandy coast. The amount of transport is often a function of sediment characteristics (e.g. size and weight), and the local wave climate. Waves are responsible for stirring up the sediment and bringing it into suspension, while currents or wave-induced flows are responsible for transporting the sediment. In some cases the tide can also be a relevant contributor to sediment transport. A difference in transported sediment in and out of a specific area causes either erosion (retreat of the coastline) or accretion (advance of the coastline). The swash zone is defined as the region on the beach that reaches from the wave run-up level until the wave run-down level, and can be seen as the transition between sea and land. The water motion in the swash zone is the main driver for cross-shore sediment exchange between the dry and wet parts of the beach. Good knowledge of the swash zone is important for several practical purposes, e.g. storm impact on dunes and barrier islands and beach nourishments. The swash zone is the most dynamic area of the nearshore, characterized by strong and unsteady flows, high turbulence levels, large sediment transport rates and morphological changes on a small timescale. 1.2 Problem description Researchers have been able to conduct accurate measurements in the swash zone only over the past 15 years. The processes that occur in this region are not yet fully understood. It is rather complex to conduct accurate swash measurements due to the small water depths and the highly dynamic character of the swash zone. In addition most measurement equipment is either designed for wet or for dry conditions, while the swash zone contains both. In the last decades researchers have developed numerous process-based numerical models, which are used as a tool to predict sediment transport rates and morphological changes and to obtain insight into the morphodynamic processes taking place at a sandy coast. Examples of these models are Delft3D, MIKE21, UNIBEST and XBEACH. However, comparisons of the model predictions with field or lab data show that the offshore directed sediment transport and morphological changes in the swash zone are generally overestimated, especially for mild wave conditions. Comprehension of the processes in the swash zone is still limited, and a lot of scientific research in this field is currently being carried out all over the world, see for instance Figure 1-1. Not all the processes presently known, or even expected to be relevant for the swash zone, are included in the morphodynamic models (yet). 1 of 142

16 August 211 Figure 1-1 An example of field measurements in the swash zone at Santa Rosa Island, USA [Houser & Barrett, 29] 1.3 Research question Due to the complexity of the processes taking place in the swash zone, there are still great uncertainties about the driving forces for sediment transport. Morphodynamic process-based numerical models tend to overestimate the seaward directed sediment transport in the swash zone, especially for mild conditions. Also, some processes are known or expected to be relevant for sediment transport in the swash zone, but are not (yet) implemented in morphodynamic models. The aim of this study is, therefore, to gain insight in the physical processes playing a role in the swash zone, and explore whether these processes are implemented and, if so, how they are simulated within an existing morphodynamic numerical model, XBeach. The obtained insight is used to optimize the model for simulating swash zone physics, and to be able to give better predictions for sediment transport in the swash zone. Leading to the primary research question in this study: What are the main driving hydrodynamic processes for sediment transport in the swash zone and how can these processes be modelled (better) in a morphodynamic model such as XBeach? 1.4 Objectives The main objective of this thesis is to obtain insight in the hydrodynamic processes responsible for sediment transport in the swash zone, and to use this knowledge to optimize a morphodynamic numerical model for simulating swash zone physics. 2 of 142

17 August 211 To answer the research question and fulfill the main objective a number of secondary objectives are formulated: Identify the dominant (hydrodynamic) processes responsible for sediment transport in the swash zone. In literature a number of hydrodynamic processes are indicated as relevant for sediment transport in the swash zone. Identify which processes are or are not included in the model. Not all processes found to be relevant in literature are included in the model. Some processes are too complicated, computationally too expensive, or simply considered insignificant. Assess the model sensitivity to each of the included processes. Each process has a certain effect on the simulated morphodynamics which could vary from the effect described in literature. Assess the sensitivity of the model to changing wave conditions and beach geometry. Depending on the beach state, different characteristic swash motions are observed in nature and other physical processes may become dominant. Assess the performance of different hydrodynamic model approaches. Different model approaches, in which swash hydrodynamics are simulated in more or less detail, can be used. Verify and validate the model with a field dataset. A comparison of the model results with field data helps to assess the model skill. Study the effect of applying different sediment transport formulations. A large number of sediment transport formulations exist. By applying a different sediment transport formulation more insight can be obtained into the simulated swash hydrodynamics and morphodynamics. 1.5 Thesis outline The initial step in the present study is to identify the governing hydrodynamic processes in the swash zone. The insight in swash hydrodynamics is obtained by means of an extensive literature review (Chapter 2). Numerous research experiments have been conducted over the past fifteen years, both in laboratory and in the field, and have been described in literature. An inventarisation of the predominant hydrodynamic processes is made to serve as a physical base for the numerical simulations. In this study the morphodynamic model XBeach is used as a tool, because it takes into account water level variations under short wave group forcing. As a result there is an actual swash zone present in the model, which gets wet and dry due to wave motions. Using default settings XBeach tends to compute only erosion over a single swash cycle, while in reality accretion may also occur, especially during mild wave conditions. Tide-induced water level variations generally have a smoothening effect on simulated beach profile changes, thereby reducing the effect of the overpredicted erosion in the swash zone area. The focus in this thesis, however, is on the actual (small scale) swash zone processes and not on the (larger scale) tidal effect. Not all processes indicated in literature are presently implemented in XBeach. The next step is, therefore, to analyze how the swash characteristics are simulated, and which processes are and which processes are not included in the model (Chapter 3). The XBeach manual and program code serves as an information source for this step. The sensitivity of the model to changing wave conditions and the beach geometry is assessed (Chapter 4), as well as the effect of including the presently implemented swash processes (Chapter 5). A number of characteristic linear profile simulations are conducted and analyzed to provide this insight. 3 of 142

18 August 211 The last step in this thesis is to study the difference in model results for three hydrodynamic modelling approaches, in which the swash hydrodynamics are simulated in more or less detail, and two different sediment transport models. All approaches are verified and validated with a dataset obtained from a field experiment (Chapter 6). The goal of the verifcation and validation is to obtain insight in the capability of the model to predict sediment transport in the swash zone accurately. 4 of 142

19 August Swash zone physics 2.1 Introduction The swash zone is a highly dynamic and complex region where many different hydrodynamic and morphodynamic processes occur. Knowledge of the dominant swash processes is essential for the development and use of accurate process-based morphodynamic models. The aim of this chapter is to describe the physical processes occurring in the swash zone and the role of the swash zone in the nearshore system. Some extra background theory is provided here, for instance on waves. However, it is assumed the reader is familiar with the basic concepts of coastal engineering. Therefore, only concepts specifically related to sediment transport in the swash zone are discussed here. For further background reference is made to the various literature concerning coastal processes and engineering [e.g. Short, 1999; Dean & Dalrymple, 22; Bosboom & Stive, 21]. In section 2.2, some coastal terminology and definitions will be given, while section 2.3 discusses some background theory about waves and their relevance for the swash zone. Section 2.4 describes the effect of the beach state on different characteristic swash motions, and section 2.5 discusses sediment transport in general and specifically in the swash zone. Section 2.6 focuses on the several hydrodynamic processes found to be dominant in the swash zone. In section 2.7 the morphological response of the swash zone to hydrodynamic and morphodynamic processes is discussed, while section 2.8 gives a brief description of numerical model methods for the swash zone used in earlier research. Finally, section 2.9 contains the conclusions of the literature review. 2.2 Coastal terminology There are many different natural phenomena of interest in the field of coastal engineering e.g. coastal lagoons, river deltas and sandy coasts. The present study focuses on sandy coasts that are subject to water level variations induced by tides and waves that propagate, transform and eventually reach the coast. The domain of interest is known as the nearshore region and can be divided into a number of zones based on wave transformation along the domain and the local beach geometry. Additionally, the nearshore region can be seen as the region in which sediment is brought into motion by waves and the tide [Dean & Dalrymple, 22]. The zones within the nearshore region most relevant for the present study will be discussed next The nearshore region As long as wind or swell waves are offshore and the water depth is large enough waves will not interact with the bottom. However, approaching the coast, there will be a point at which the water depth has reduced to such an extent that the wave propagation velocity (c g ) decreases. Before breaking, the wave energy flux is conserved within propagating waves [e.g. Holthuijsen, 27]: P Ec g constant [2.1] where E is the wave energy and c g is the wave (group) velocity. From the energy flux conservation balance it follows that the wave energy will increase for decreasing propagation 5 of 142

20 August 211 velocity. The increase in energy results in an increase of the wave height. This phenomenon is called shoaling and the zone in which this occurs is known as the shoaling zone. In even shallower water waves willl become unstable and start breaking. The area in which this occurs is referred to as the breaker zone. After breaking a wave propagates as a bore (foam shaped broken wave) through the surf zone. When the bore is near the waterline, it will collapse and a thin layer of water, known as the swash lens, will travel up (run-up) on the beach and back down (run-down) in an area known as the swash zone. The sequence of wave-run up and run-down is known as the swash cycle. In Figure 2-1 a schematization of the nearshore region is given. Figure 2-1 Classification of the nearshore region. At the start of the nearshore region an incoming (short) wave will first shoal (in the shoaling zone) and later break due to the limited depth (in the breaker zone) and propagate further onshore as a bore (foam shaped broken wave) in the surf zone. When the bore collapses a thin layer of water (swash lens) will run up and down the beach in the swash zone The swash zone The swash zone is the particular part of the beach that is consecutively wet and dry due to the motions of the sea. This definition, however, is not as straight forward as it seems. There is discussion among scientists on how to accurately define the swash zone, since both landward and seaward limits are constantly changing [Puleo & Butt, 26], especially on beaches with a large tidal range [e.g. Short, 1999]. Therefore, some scientists rather define the point of bore collapse as the seaward boundary of the swash zone [Puleo & Butt, 26]. Herein, the definition according to Short [1999] is used, which states that the swash zone is the part of the beach located between the lower limit of wave run-down and the upper limit of wave run-up on the beach. The part of the beach between the low tide water line and the high tide water line, including the upper limit of swash action, is then referred to as the beachface. These definitions indicate that the swash zone is a smaller, but far more dynamic region than the beachface, and that it changes significantly, every time a wave runs up the beach. There is a practical reason for using Shorts definition here; it is rather difficult to define the point of bore collapse in a morphodynamic model such as the one (XBeach) used later on in this thesis. Besides the constantly moving land-water boundary, Short [1999] describes two additional characteristics that make the swash zone morphodynamically unique compared to the rest of the beach. First there is the fact that water depths in the swash can be very small, especially 6 of 142

21 August 211 during wave run-down, which results in a complicated flow pattern. Secondly, part of the bed in the swash zone is unsaturated which makes infiltration of water in the bottom an important aspect concerning sediment transport. Although the exact definition is disputable, it is globally agreed on that the swash zone is the most dynamic region of the nearshore, and that it is characterized by strong and unsteady flows, high turbulence levels, large sediment transport rates, infiltration in the beach and morphological changes on a small timescale [Butt & Russell, 1999; Masselink & Puleo, 26; Bakhtyar et al., 29]. Since the existence of the swash zone is a direct result of the presence of waves near the shore, some background theory considering waves will be discussed next. 2.3 Waves and wave-induced swash Surface waves can be characterized by their type and period (or length). They can originate from wind (wind waves, swell waves, capillary waves), gravitational forces between astronomical bodies (tide), seaquakes (tsunamis) or can be induced by other waves (low frequency waves). The wave period can vary from less than.1 seconds for capillary waves to more than 24 hours for trans-tidal waves. Three types of waves are particularly important for the swash zone: tide, high frequency (or short) waves and low frequency (or long) waves. Since the time and spatial scale of the tide is much larger than the swash zone scale, the effect of the tide in the swash zone can best be represented as a local water level variation, rather than a wave. Therefore, only high and low frequency waves will be discussed here. Both wave types induce a characteristic swash motion which will be discussed in section High frequency waves Wind waves are waves generated by wind and occur in the area of generation [e.g. Short, 1999]. The size of these waves is dependent of the wind velocity, wind duration, fetch (length over which the wind interacts with the sea surface) and the water depth. Their period is usually between.25 and 3 seconds, and they are referred to as surface gravity waves, short waves or high frequency waves. Wind waves are relatively short, and consist of rather random and irregular motions. Waves can propagate for a long distance, but due to the process of frequency dispersion (where waves are sorted on their wave frequency due to the difference in wave celerity), the wave sequence will become more regular (known as swell). Another effect of frequency dispersion, is that the waves tend to travel in so-called wave groups, see Figure 2-2. In deep water short wave groups travel at a group velocity, which is half of the individual wave celerity. In shallower water, where waves are radically altered by the breaking process, the waves ungroup. The propagation of wave groups can be observed by a gradual modulation of the short wave height in space and time. 7 of 142

22 August z s [m] High frequency wave motions Short wave group envelope x [m] Figure 2-2 Representation of the short wave groups. The solid line represents the high frequency wave motions, while the dashed line represents the short wave group envelope, which indicates the overall shape of the wave groups that propagates in space. The term high frequency waves or short waves will be used throughout this thesis when referring to either wind or swell waves Low frequency waves Besides high frequency waves longer wave motions exist which can reach a wave period up to five minutes (with a frequency of.3 to.3 Hz). These waves have a larger wave length, but generally much smaller amplitudes compared to high frequency waves, and are associated with the short wave groups. In literature they are referred to as long waves, low frequency waves, surf beat or infra-gravity waves. Throughout this thesis the term low frequency waves or long waves will be used when referring to these short wave group induced long waves. Here, two types of low frequency waves will be discussed; bound long waves and free waves. Bound long waves Munk [1949] was the first to report about long wave motions, and suggested they were caused by the variation in the short wave mass transport. Later Tucker [195] found a strong negative relationship between the high frequency wave envelope and the low frequency water level variations. Eventually, Longuet-Higgins & Stewart [1962] (LH-S hereafter) combined this low frequency wave with the short wave radiation stress and mass flux to explain the negative correlation. LH-S [1962] found that the low frequency wave travels with the short wave group velocity, and is therefore considered bound to the wave group. They further concluded that these bound long waves become free in the surf zone, when the high frequency waves start breaking, and can be reflected at the shoreline. The LH-S theory states that the variations in radiation stress have an effect on the water level elevations. Radiation stress is the momentum flux as a consequence of the presence of 8 of 142

23 August 211 waves. Horizontal gradients in the radiation stress induce a net wave-induced force on the water, which can be explained by the law of conservation of momentum, stating that the rate of change of momentum of a fluid element should be equal to the forces acting on that element. These forces can induce water level variations and currents. Radiation stress consists of an advection component (advection of momentum by the horizontal particle velocity) and a wave-induced pressure component. For simplicity, only the transport of x- momentum in x-direction (where x is the direction of wave propagation) is considered here (representing normal incidence waves propagating towards an alongshore uniform coast). The radiation stress is then given by: xx x x wave h S u u p dz [2.2] where is the water surface elevation, h is the water depth, is the water density, u x is the velocity in x-direction, p wave is the wave-induced pressure and the overbar indicates time averaging over the wave motion. According to linear wave theory radiation stress can be simplified to: S 1 xx n 2 E ne [2.3] where E is the wave energy and the ratio between wave group velocity and individual wave celerity (n) is given by: 1 2kh n 1 [2.4] 2 sinh2kh and where k is the wave number and h is the local water depth. The wave-induced force on the water level in cross-shore direction, for an alongshore uniform coast and normal incidence waves, is given by the derivative of the radiation stress: dsxx Fx [2.5] dx Finally, from the wave-induced force the resulting water level gradient can be computed : dsxx d d Fx gh gh [2.6] dx dx dx where g is the gravitational acceleration, is the water level variation, h is the still water depth and the overbar indicates time averaging over the wave motion. When combining equation [2.3] and [2.6], a simple model can be derived for the calculation of the variation in water level as a function of the variation in wave energy (for an alongshore uniform coast and normally incident waves): d 3 de [2.7] dx 2gh dx According to this model a positive spatial gradient in the wave energy is associated with a negative spatial gradient in the water level and vice versa. Since high frequency waves tend to travel in wave groups, and the high frequency wave energy varies spatially, areas with lower and higher average water levels are found, forming a bound long wave. In Figure 2-3 a bound long wave is schematically represented for a bichromatic (two wave components) short wave group. In case of a situation with a completely bound long wave, the short wave envelope will be (as shown in Figure 2-3) 18 degrees out of phase with the bound long wave. Maximum shortwave values correspond with minimum bound long wave values and vice versa. In practice, 9 of 142

24 August 211 however, short wave groups and the accompanying bound long waves are rather irregular (compared to the example shown in Figure 2-3). 4 2 z s [m] High frequency wave motions Mean water surface elevation x [m] Figure 2-3 Representation of a bound long wave forced by a bichromatic short wave group. The solid line represents the high frequency waves, while the dashed line represents the bound long wave. An additional generation mechanism of low frequency waves, known as the time-varying breakpoint model, was found by Symonds et al. [1982]. They divided the surf zone into the outer surf zone, a transition zone, and the inner surf zone and stated that outside of the surf zone the variation in radiation stress is negligible. Therefore, no generation of bound long waves. In the transition zone the point of breaking varies over the cross-shore due to the wave height variations over time and induces both onshore and offshore directed waves. The shoreward directed wave is reflected at the shoreline, resulting in a standing wave in the surf zone. The region outside the surf zone then contains the offshore directed wave directly from the forcing zone (breaker zone) as well as the initially onshore directed wave reflected at the shoreline. Experiments in both field and lab show that for mild slopes the LH-S generation mechanism is dominant, while for steeper beaches the time-varying breakpoint model dominates [e.g. Ruessink, 1998; Dong et al., 29]. Free long waves When short wave groups enter the breaker zone, higher waves will break and bound long waves are released. The long waves propagate farther into the surf zone as free waves that are reflected or dissipated (on very dissipative beaches [e.g. van Dongeren et al., 27]). Reflected waves induce a standing wave pattern in the swash zone due to the combination of incoming and outgoing wave motions and will either propagate out of the surf zone or become trapped in the surf zone. The free (wave group generated) low frequency waves which travel into the surf zone, and get reflected at the beach, propagating outside of the surf zone are referred to as leaky waves [Herbers et al., 1995]. Edge (or trapped) waves are similar to leaky waves with one difference: they don t travel out of the surf zone, but are trapped inside and travel along the 1 of 142

25 August 211 beach [e.g. Short, 1999]. Whether a wave will become trapped or not depends on the angle of incidence of the wave. Waves that enter the surf zone obliquely experience refraction before they get reflected at the coast. While travelling back in offshore direction they experience the refraction again, and when this effect is strong enough, the refraction will bend the wave towards the shore again and trap the wave (see Figure 2-4). Figure 2-4 Schematization of an edge wave The effect of refraction being stronger for reflected low frequency waves is due to the difference in bound long wave and free long wave characteristics. The incoming (bound) long wave initially travels with the short wave group velocity, while the angle of incidence typically differs from the mean wave direction of the high frequency wave components. The low frequency wave direction is a function of the cross-shore wave number. For the bound long wave the cross-shore wave number is calculated as a function of the high frequency wave numbers, while for the (reflected) free long waves, the wave number is obtained from the long wave dispersion relation. Because of its dependence on the dispersion relation, the refraction process is stronger for the reflected (free) wave, and can be so strong that the wave is directed onshore again. The criterion for whether free waves will become edge waves is given by [Short, 1999]: 2 gk y [2.8] where is the radian frequency, g is the gravitational acceleration and k y is the alongshore wave number. The process of an edge wave getting reflected and refracted back again can go on for a long distance along the coast, which is indicated in Figure The swash cycle Waves arriving at the shore induce a cyclic pattern of wave run-up and run-down. The run-up and run-down of flow due to a single wave is referred to as the swash cycle. A swash cycle consists of two separate phases, each having their own characteristics [Bakhtyar et al., 29]. The run-up of water on the beach is referred to as uprush. During uprush the flow velocity will decrease (due to bottom friction and gravity force) until it will reach zero. At that moment the water has reached its maximum run-up height and the water will start moving back. After this point the velocity increases again, but now directed offshore, until the next swash cycle is met. The rundown of the water from the beach towards the sea is referred to as backwash. A representation of a swash cycle is given in Figure of 142

26 August 211 High and low frequency swash There are two characteristic types of swash cycles due to the existence of dissipated high frequency waves (and in some cases dissipated low frequency waves [e.g. van Dongeren et al., 27]) and reflected low frequency waves in the swash zone [Short, 1999; Masselink & Puleo, 26]. The first is a result of the collapsing of wave bores driven by dissipating (high frequency) waves. This swash motion is here referred to as high frequency swash (although it can also be induced by dissipating low frequency waves). The second motion is a result of non-breaking (low frequency) waves that reflect at the beach and create standing waves, here referred to as low frequency swash. In literature often the terms incident or infragravity swash are used for respectively high- and low frequency swash. The division between both characteristic motions is not completely strict, and swash cycles can generally be classified into the three frequency ranges shown in Table 2-1, where subharmonic swash is associated with the alongshore propagating standing edge waves or the swash-swash interactions at the seaward boundary of the swash zone. However, to keep the classification of swash motions uniform with the classification of waves used in this thesis, subharmonic swash is considered as high frequency swash here. The classification used in this thesis is further clarified in Figure 2-6. Regular waves are subject to shoaling and breaking, resulting in a (regular) swash motion. For an irregular wave field, the high frequency waves are subject to shoaling and breaking, resulting in a high frequency swash component, while swash-swash interactions induce a subharmonic swash component. The low frequency waves generally reflect at the shoreline, resulting in a standing wave swash component. The (standing) edge waves propagating alongshore induce an additional subharmonic component. Table 2-1 Frequency ranges in the swash zone [after Short, 1999] Frequency Period [s] [Hz] High freq. swash Subharm. swash Low freq. swash of 142

27 August 211 Figure 2-5 Swash cycle schematically represented, simulated with the morphodynamic model XBeach. In panel (A) a wave bore is propagating towards the beach. In panel (B) the bore height decreases (collapses) and changes into a thin layer of water (swash lens) still traveling up the beach (uprush). In panel (C) the velocities are decreasing due to bottom friction and (mainly) gravity. In panel (D) and (E) the backwash is shown; the water travels back from the beach towards the sea. In panel (F) the swash meets the subsequent bore which will induce a new swash cycle. 13 of 142

28 August 211 Figure 2-6 Schematization of energy transfer from offshore waves to swash and the classification for high and low frequency swash used in this thesis (modified from Mase [1995]). Swash cycle asymmetry The difference in uprush and backwash during one swash cycle is referred to as swash asymmetry. In Figure 2-7 a short time series of the water level and velocity measured in the swash zone is shown, where swash asymmetry can clearly be observed. In contrast to the backwash, the uprush acceleration is short and strong and the velocity will reach a higher magnitude. Additionally, the uprush water levels are clearly higher than the backwash water levels. A combination of higher velocities and higher water levels would suggest that the discharge during uprush is larger than during backwash and that the net effect would be water transported towards the shore. This is, however, clearly not the case on a beach; two additional aspects have to be taken into account. The first aspect is the difference in uprush and backwash duration; in Figure 2-7 the uprush duration is shorter than the backwash duration. The second aspect is groundwater flow. Water infiltrates the (dry) beach during uprush and will exfiltrate during backwash, therefore part of the water brought upslope by the uprush is still in the bed during backwash. Both during uprush and backwash numerous hydrodynamic processes take place that have an effect on the sediment transport in the swash zone. These processes will be discussed next. 14 of 142

29 August 211 Figure 2-7 Field measurements of the horizontal velocity (solid line) and the water depth (dashed line) for a single swash cycle, measured at a location half way between the limit of run-up and run-down [Hughes et al., 1997]. 2.4 Beach states and swash motions Beach states According to Wright & Short [1984] beaches can be classified into three so-called beach states. Reflective beaches: relatively steep beaches that present a narrow surf and swash zone. The waves present at reflective beaches are of plunging to collapsing breaker type, or do not break at all, and get reflected (surging). The sediment present at the beach is relatively coarse and there are no breaker bars [Short, 1999]. Due to the low wave energy dissipation, these beaches are often referred to as low-energy beaches. Dissipative beaches: relatively flat beaches with a wide surf and swash zone and multiple breaker bars present in the cross-shore profile [Short, 1999]. The waves present at dissipative beaches are of the spilling breaker type and the sediment is relatively fine [Short, 1999]. The main swash motion consists of collapsed wave bores running up and down the beach. Because of the dissipation of a large part of the wave energy these beaches are often referred to as high-energy beaches. Intermediate beaches: beaches with a combination of the characteristics of the two other beach states (spilling to plunging/collapsing breaker type) and can be seen as semi-dissipative (or semi-reflective) beaches Dean number According to e.g. Gourlay [1968] and Dean [1973] the beach state can be determined with the following dimensionless parameter, often referred to as Dean number: H B [2.9] wt S 15 of 142

30 August 211 where H B is the breaker wave height, w S is the sediment fall velocity and T the wave period. In the Dean number expression a higher wave steepness (represented by the wave height over wave period ratio) or a smaller grain size (represented by the sediment fall velocity) leads to more dissipative conditions. Reflective beaches typically have a -value smaller than 1, while dissipative beaches have a -value of 6 or larger. Intermediate beaches have a value between 1 and Iribarren number Another indication for the beach state of a certain beach is the Iribarren number, also known as surf similarity parameter [Battjes, 1974; Guza & Inman, 1975]. The Iribarren number is a relation between the beach slope and the wave steepness and is given by: tan [2.1] H / L where is the beach slope, H is the deep water wave height and L is the deep water wave length. For reflective beaches a high Iribarren number (>5) and for dissipative beaches a low Iribarren number (in the order of.5 or smaller) can be expected High and low frequency wave dominance Several experiments have been conducted to study the difference in swash zone processes for a dissipative and reflective beach [e.g. Masselink & Russell, 26; Miles et al., 26]. On reflective beaches the lower frequency waves get reflected, while the higher frequency waves break rather abruptly (plunging or collapsing), making the higher frequency waves more dominant in the swash zone. On dissipative beaches the relatively gentler beach slope enhances lower frequency waves to develop more, shoal and eventually break. Due to the dissipation of lower frequency waves and the more gentle breaker type for higher frequency waves (spilling), dissipative beaches have dominant low frequency wave motion [Wright & Short, 1984; Short, 1999]. 2.5 Sediment transport Sediment transport in general, and specifically in the swash zone, has been analyzed in several studies, but is still not fully understood. It was found that the hydrodynamic processes described earlier all have an effect on the sediment transport in the swash zone, where large quantities of sand are transported during every swash cycle. However, the exact effect is still unknown and a lot of research still has to be conducted. Due to the large amount of interpretations and formulations that can be found in literature, only some basic concepts of sediment transport will be discussed next Cross-shore and longshore transport Sediment transport can be divided into cross-shore and longshore sediment transport. Crossshore sediment transport is the transport of sediment in onshore or offshore direction, while longshore sediment transport is the sediment transport along a coastline. The latter is the result of obliquely incoming waves, or a gradient in the wave height along the shore, that induce a longshore current. Sediment that is stirred up by the waves is then transported by 16 of 142

31 August 211 the longshore current, or, in some cases, by a tidal current propagating along a coast. In most cases, however, the cross-shore sediment transport is predominant in the swash zone Bed load, suspended load and sheet flow transport Another distinction in sediment transport can be made according to the characteristics of the transport [Nielsen, 1992; Fredsøe & Deigaard, 1992]. The three main transport types relevant for this study are bed load transport, suspended load transport and sheet-flow transport. Bed load transport is the transport of sediment grains in a (thin) layer close to the bottom. The moving sediment is in more or less continuous contact with the bottom. Suspended load is the transport of sediment suspended in the water column, during which the sediment is not in contact with the bottom. Sheet-flow transport is similar to bed load transport but due to high shear stresses (during strong flows) a highly concentrated layer (with a maximum thickness of a few centimeters) of moving sediment is created [Bosboom & Stive, 21]. Due to the unsteady character of swash flow and the small water depths, it is expected that bed load transport (or sheet flow) is the dominant type of transport in the swash zone. However, the sediment suspended at bore collapse and transported by the swash motion, might also be an important contributor, or even the dominant transport mode. Horn & Mason [1994] analyzed the ratio between bed and suspended load transport in the swash zone for a number of field experiments, and found that bed load generally dominates in the swash zone. In the uprush suspended load transport was found to be dominant only occasionally, while bed load transport generally dominates the backwash Sediment transport formulations As mentioned before, numerous empirical formulas to predict the amount of sediment transport have been proposed in the past, based on either laboratory or field experiments, or a combination of both. The main challenge in the prediction of sediment transport in the swash zone is the relatively small (net) difference between the relatively large gross transports during every uprush and backwash. Several authors proposed formulations to predict sediment transport rates in the swash zone based on the formula by Meyer-Peter & Müller [1948] that depends on the Shield parameter, while the energetics approach [e.g. Roelvink & Stive, 1989] is also widely used for sediment transport predictions near the shore [Bakhtyar et al., 29]. Due to the large amount of formulations, only three formulations will be described in Chapter Hydrodynamics and morphodynamics in the swash zone There is still a lot unknown about the hydrodynamic and morphodynamic processes taking place during a swash cycle. Conducting accurate experiments in the swash zone is rather complex because of the relatively small difference of the two large (onshore and offshore) gross transport rates [Masselink & Hughes, 1998] and due to the periodically wetting and drying of the zone [e.g. Short, 1999; Puleo et al., 2]. Most measurement equipment is designed either for the wet or for the dry part of the beach. Researchers have been conducting field experiments in the swash zone only over the last fifteen years [Elfrink & Baldock, 22; Masselink & Puleo, 26]. 17 of 142

32 August 211 The following processes have been found in literature to be the most relevant for swash zone cross-shore sediment transport: high and low frequency wave motions [Bakhtyar et al., 29], wave skewness and asymmetry [Grasso et al., 211], turbulence due to wave breaking [Puleo et al., 2; Longo et al., 22], swash-swash interaction [Erikson et al., 25; Blenkinsopp et al., 211], boundary layer flow and shear stress [Barnes & Baldock, 21] and infiltration/exfiltration [Li & Barry, 2; Butt et al., 21; Horn, 26; Steenhauer, 21]. In addition, there can be a longshore sediment transport component present in the swash zone [Elfrink & Baldock, 22]. In the present study only the cross-shore processes, which are usually predominant, are considered and will be discussed Wave skewness and asymmetry In theory waves are often characterized as sinusoidal water level functions, but in reality both high and low frequency waves are never regular. There are different forms of wave nonlinearities and research shows that these nonlinearities are relevant in the occurrence of sediment transport, especially near the shore [e.g. Austin et al., 29]. Even though the swash motion cannot be strictly seen as waves anymore, two wave nonlinearities, wave skewness and wave asymmetry, have been found to be relevant for sediment transport in the swash zone and will be discussed here. The phenomenon of sharp wave crests and flat wave troughs (referred to as Stokes wave) is referred to as wave skewness. A characteristic of this wave shape is the higher velocities under the crest in comparison with the velocities under the trough of the wave [Holthuijsen, 27]. Since the velocity differs, more sediment is mobilized under the crest, which generates more sediment transport under the crest, and thus, a net onshore transport. Wave skewness could also cause net offshore transport due to a phase lag between the mobilization and the transport of sediment. In that case sediment is mobilized by the higher crest velocities and transported by the trough velocities [Grasso et al., 211]. Whether a phase-lag between mobilization and transport exists depends on the sheet-flow layer, the wave period and the sediment settling velocity [Dohmen-Janssen et al., 22]. Another wave nonlinearity that plays a role in sediment transport is wave asymmetry [e.g. Grasso et al., 211]. Wave asymmetry is the occurrence of waves in saw-tooth shapes with a steep wave front and a gentler wave back [Bosboom & Stive, 21]. At the steep front strong fluid accelerations occur which enhances the mobilization of sediment. As explained for wave skewness this will result in a net onshore sediment transport. Grasso et al. [211] conducted experiments in a wave flume to study the relation between wave nonlinearities and sediment transport and found that a small skewness results in a net onshore sediment transport. In that case, the mobilization of sediment is rather weak, but the crest velocities are larger than the trough velocities, which results in an onshore sediment flux. A large skewness can also lead to onshore transport, when the wave asymmetry is large enough. However, when there is a weak wave asymmetry (due to phase-lag effects) a large skewness enhances offshore transport Turbulence Another phenomenon often clearly visible near the shoreline is the presence of wave bores. The unsteady character of a wave bore approaching the shore is a result of the turbulence 18 of 142

33 August 211 that is present within it [Masselink & Puleo, 26]. Turbulence is the highest frequency motion in the swash zone, and can originate from generally three different sources [Aagaard & Hughes, 26]: bed frictional processes, swash-swash interactions or the water movement in a wave bore. Research showed that turbulence generally plays a relevant role for sediment transport in the swash zone by stirring up the sediment and bringing it into suspension [e.g. Butt et al., 24]. Puleo & Butt [26] and Masselink & Puleo [26] concluded that the turbulence present during uprush is dominated by the wave bore, while turbulence during backwash is dominated by bed turbulence and the growing boundary layer (see also section 2.4.6). Puleo et al. [2] found that the influence of the wave bore induced turbulence is greater than the effect of the bed turbulence and boundary layer growth, concluding that turbulence plays a greater role in uprush than in backwash. However, this is only valid for very small water depths (where the bore is near the bed). The exact effect of turbulence on the sediment transport on (slightly) deeper water is not known due to a lack of field studies [Longo et al., 22]. It can, however, be stated that turbulence is responsible for lifting large volumes of sediment especially during uprush [Bakhtyar et al., 29]. This results in higher suspended sediment concentrations during uprush than during backwash, and would mean an increasing potential for onshore directed sediment transport (disregarding the velocity magnitude during uprush and backwash) Swash-swash interactions When a wave reaches the coast and travels up a beach, it is not always able to complete a full swash cycle before the next wave arrives. This generally occurs when the swash duration is larger than the incident wave period. The second wave will catch up and absorb the first wave (when the first wave is in uprush phase) or both waves will collide (when the first wave is in backwash phase) [Erikson et al., 25]. This phenomenon is known as swash-swash interaction and is schematically shown in Figure 2-8. There is only little described in literature about the effect of swash-swash interactions on sediment transport in the swash zone. Erikson et al. [25] concluded that it enhances the turbulence in the swash motion and that it has a large influence on the maximum run-up length and the swash duration. Blenkinsopp et al. [211] concluded that swash-swash interaction induces larger transport rates, either onshore or offshore. 19 of 142

34 August 211 Figure 2-8 Schematic representation of the two swash-swash interaction types, where the numbers (1,2,3 and 4) indicate the incoming waves in chronological order. The top left figure shows catch-up and absorption. The bottom left figure shows uprush (2nd wave) and backwash (1st wave) collision. The figures on the right show the evolution of the waterline in time for both cases [Erikson et al., 25] Acceleration and horizontal pressure gradient During uprush water initially accelerates due to the bore collapse and the high horizontal pressure gradient present at the front of a wave (as can be seen in Figure 2-7). This acceleration is of a rather short duration, and soon after acceleration the water flow will decelerate until the maximum run-up distance is reached (with a zero velocity), from that moment the backwash phase starts. The effect of acceleration in the swash zone is highly disputable and more research should still be conducted in order to obtain a better understanding of this subject. Puleo et al. [23] found for instance high suspended sediment concentrations and onshore pressure gradients during uprush accelerations, which suggests that acceleration can be seen as an additional onshore transport mechanism. They found more resemblance between their model and observations when they included an extra term for acceleration. Baldock & Hughes [26] measured instantaneous water surface gradients, which are a close approximation to the horizontal pressure gradient (see Figure 2-9), and found mainly offshore effects. At the leading edge the water surface slope is expected to be landward dipping during uprush, however, Baldock & Hughes only found a constant landward dipping water surface close to the location of bore collapse. Further on in the uprush phase they frequently found horizontal or even seaward dipping water surface slopes. In the area behind the leading edge they exclusively found seaward dipping surface slopes, both during the uprush and the entire backwash phase. Their observations suggest the uprush flow is reduced by an adverse pressure gradient while the pressure gradient is favorable for the flow during backwash. This indicates that fluid acceleration predominantly enhances offshore sediment transport. 2 of 142

35 August 211 Figure 2-9 Schematization of landward and seaward dipping water surface slopes. Landward (seaward) dipping water surface slopes are associated with onshore (offshore) horizontal pressure gradients Infiltration, exfiltration and the boundary layer thickness A typical characteristic of the swash zone that the bed can be partly unsaturated, which allows for infiltration of water into the bed. The resulting groundwater flow can be an important aspect regarding sediment transport. In general, water infiltrates the beach surface during uprush and exfiltrates during backwash. The amount of in-/exfiltration strongly depends on the groundwater level, beach slope and sediment characteristics [Elfrink & Baldock, 22]. Groundwater flow below the beach surface has a significant influence on the swash zone morphology [e.g. Bakhtyar et al., 29], and according to Li & Barry [2] infiltration is a dominant process. Infiltration has a stabilizing effect on a beach and can be promoted by artificially lowering the beach groundwater level (beach dewatering). Coastal engineers apply this method on eroding beaches to promote onshore sediment transport [e.g. Dean & Dalrymple, 22]. In literature three different effects of infiltration and exfiltration on sediment transport are mentioned and will be discussed here. Effective sediment weight Different authors [e.g. Butt et al., 21] found that infiltration has a stabilizing effect on the bed, while exfiltration destabilized the bed (see Figure 2-1 top panels). During infiltration downward directed flow gradients increase the effective weight of the sediment and therefore less sediment will be in suspension. During exfiltration the opposite occurs and the sediment mobility increases. Since the velocities during uprush and backwash are respectively onshore and offshore directed, the process of stabilization and destabilization increases the potential for offshore directed sediment transport (disregarding the velocity magnitudes for uprush and backwash). Boundary layer thickness Besides the influence on the effective sediment weight, in-/exfiltration affects the thickness of the boundary layer [Butt et al., 21]. The boundary layer is a small layer near the bed where the wave-induced flow is affected by the bed [Longuet-Higgins, 1953]. It is also present under waves in deeper water and usually has a thickness of 1 to 1 cm for short waves, depending on the roughness of the bed, and the local Reynolds number, which is a measure for the intensity of the turbulence (Re=Ud/, where U is a characteristic velocity difference, d is the 21 of 142

36 August 211 distance over which the velocity difference is found and is the kinematic viscosity). For longer waves (e.g. tide) the boundary layer thickness can be much larger. The flow in the boundary layer consists of an oscillatory flow induced by the wave motions and a non-zero wave-averaged horizontal flow. The latter flow is called streaming and plays a dominant role in onshore sediment transport [Longuet-Higgins, 1953]. The water that flows over the bed induces a shear stress which can set sediments into motion. An important characteristic of the boundary layer for sediment transport is its thickness. In a thinner boundary layer the near-bed velocity is higher, thereby making the potential for sediment transport higher. In the swash zone the boundary layer thickness is reduced by the process of infiltration, while exfiltration thickens the layer [Conley & Inman, 1994]. This process is schematized in the bottom panels of Figure 2-1. Due to the reduced boundary layer thickness the near-bed velocities are larger. Consequently, the sediment transport is potentially larger during infiltration, promoting onshore sediment transport. Swash flow asymmetry Masselink & Li [21] showed that infiltration enhances the swash cycle asymmetry by reducing the backwash velocity and increasing the backwash duration. The increased swash asymmetry enhances onshore sediment transport and this results in berm formation, and relatively steep beach gradients. However, they also found this effect only occurs when the infiltration volume (V i ) is more than two percent of the swash uprush volume (V u ). The infiltration volume can be related to the grain size (larger grains result in larger pores, therefore more infiltration). The threshold condition for increased swash asymmetry (V i >.2V u ) can therefore be translated into a critical grain size of D 5 =1.5 mm [Masselink & Li, 21]. This threshold value indicates that the swash asymmetry effect of infiltration only takes place on gravel beaches with a D 5 >1.5mm and not on sandy beaches where grain sizes are usually smaller than 1mm. 22 of 142

37 August 211 Figure 2-1 Schematic representation of sediment stabilization and boundary layer thinning during uprush (top left panel en bottom left panel respectively) and destabilization and boundary layer thickening during backwash (right top panel and right bottom panel respectively) [Butt et al., 21]. In summary it can be stated that the effect of infiltration and exfiltration on the effective sediment weight promotes offshore transport, while the modification in thickness of the boundary layer and the swash flow asymmetry enhance a net onshore transport. From literature it is not directly clear which process is dominant, although some suggestions have been made. Butt et al. [21] concluded, based on the research of Nielsen [1997] and Turner & Masselink [1998], that there must be a critical grain size below which effective weight effects dominate, and above which the boundary layer thickness effects dominate. This value should lie somewhere between.45 and.58 mm. However, more research is suggested in order to give a valid estimate. Nevertheless, it can be stated that effective weight effects dominate for fine sediments (<.45 mm), boundary layer effects dominate for greater grain sizes (>.58 mm) and swash flow asymmetry will be relevant for grain sizes greater than 1.5 mm. This indicates that for fine sediments infiltration and exfiltration will have the effect of a net offshore sediment transport, while for larger grain sizes infiltration and exfiltration will have a net onshore transport effect. Table 2-2 Overview of the infiltration and exfiltration effects in the swash zone and their dominance depending on the grain size, based on Nielsen [1997] and Turner & Masselink [1998] Dominant infiltration/exfiltration effect Grain size [mm] Transport effect Effective weight effect <.45 Offshore Boundary layer thickness effect >.58 Onshore Swash flow asymmetry >1.5 Onshore 23 of 142

38 August Morphological response of the swash zone The swash zone is a highly dynamic region. During a swash cycle large quantities of sediment are moved, nevertheless, the net result is often small. Erosion or sedimentation in the swash zone is the result of a small difference in (large) uprush and backwash sediment transport. Due to the constant movement of the bed in the swash zone it is difficult to determine whether erosion, sedimentation or a net zero effect occurs. A very common morphological feature that owes its existence to the swash motion is the swash berm. The swash berm is a result of the accumulation of sediment at the most landward part of the swash zone, usually during mild conditions. The height of the berm depends on how far the sediment is transported on the beach during uprush and can be predicted by [Takeda & Sunamura, 1982]: 38 Z.125H gt [2.11] berm b where H b is the breaker wave height and T is the wave period. The berm is an important feature on the beach profile because the wind can transport sediment from the berm to the dunes. In addition, it acts as a barrier against wave action. During storm conditions the berm is likely to be eroded. Another morphological feature that has drawn a lot of attention in science is the existence of beach cusps in the swash zone. Beach cusps are shoreline formations made up of various grades of sediment in an arc pattern [Dean & Dalrymple, 22], see Figure 2-11 for an example. Inman & Guza [1982] did research to the origin of beach cusps and they concluded there are two main types, namely surf zone cusps and swash cusps. Surf zone cusps are formed by the nearshore circulation system while swash cusps are formed by the swash motion (uprush and backwash). The authors also concluded that the dimensions of the swash cusps depend solely on the run-up height of the wave during uprush. Swash cusps morphology is mainly associated with reflective beach systems and the features are composed of medium to coarse sand [Short, 1999]. In literature a large amount of possible explanations for swash cusp formation have been proposed [Short, 1999]. A few examples are foreshore irregularities, instability of breaking waves or the presence of longshore currents along a coast. A more widely accepted theory is that swash cusps are formed due to the presence of edge waves [e.g. Sallenger Jr, 1979; Inman & Guza, 1982]. However, Werner & Fink [1993] found that swash cusps are a result of the feedback between swash flow and beachface morphology, which is in contradiction to the edge wave theory. Currently there is still much discussion concerning this subject in the coastal engineering community [Coco et al., 1999]. 24 of 142

39 August 211 Figure 2-11 Beach cusp on Praia do Moçambique, Florianópolis, Santa Catarina, Brazil (photo by A.A. van Rooijen, 21) 2.8 Numerical modelling and the swash zone Numerical models can be used to obtain better understanding of the hydrodynamic and sediment transport processes in the swash zone and to predict its morphological behaviour. In process-based numerical models the main hydrodynamic processes occurring in the swash zone are simulated with a hydrodynamic model, while bed level changes in the beach profile can be predicted by coupling it with a sediment transport model. This section discusses some commonly used numerical modelling methods for swash zone hydrodynamics Navier-Stokes equations The Navier-Stokes equations (momentum) in combination with the continuity equation form a closed set of equations able to accurately describe the flow of water. It is not possible to solve these equations analytically and therefore they are discretized and solved numerically. Although it is possible to solve the Navier-Stokes equations numerically, this can be very time consuming. Therefore, some simplifications are usually applied leading to the nonlinear shallow water equations or Boussinesq equations. There are, however, some examples of numerical models using the complete Navier-Stokes equations like RIPPLE [Kothe et al., 1991] and FLOW3D [Chopakatla et al., 28] Nonlinear shallow water equations The nonlinear shallow water equations (NSWE) describe the propagation of water surface elevation and depth-averaged velocity induced by waves with a relatively large wave length compared to the water depth. They can be derived from the Navier-Stokes equations when applying a few simplifications. It is assumed that water is incompressible and that the pressure is hydrostatic. In addition, decomposition and averaging of the velocity- and pressure terms is applied (referred to as Reynolds decomposition and averaging), which 25 of 142

40 August 211 results in more smoothly varying equations. Finally, it is assumed that the wave length is much larger than the water depth, allowing to neglect the vertical directed velocities and the effect of the vertical shear stress on the horizontal velocities [Randall, 26]. The NSWE are able to accurately simulate the propagation of long waves and are less time consuming compared to the full Navier-Stokes equations. However, the equations are only applicable under the shallow water assumption (L>>h). The NSWE are, for instance, included in RBREAK [Kobayashi et al., 1989] and in XBEACH [Roelvink et al., 29]. In Chapter 3 the nonlinear shallow water equations will be discussed further. Recently, a non-hydrostatic model based on the NSWE was developed [Zijlema et al., 211]. A compensation for the hydrostatic pressure assumption is added, resulting in a method that can be applied in intermediate water and shallow water. A depth-averaged version of the nonhydrostatic model has recently been added to the XBeach model [Smit et al., 21] Boussinesq equations Boussinesq-type equations are, like the NSWE, derived from the Navier-Stokes equations. The vertical coordinate is eliminated, hereby reducing the computational time compared to the full Navier Stokes equations. However, in comparison to the shallow water equations, Boussinesq equations contain some extra terms for the curvature of the water surface with which the non-hydrostatic pressure part is still present. Boussinesq equations are only accurate in shallow water although a number of researchers have suggested implementations to better approximate the dispersion relation for deeper water. However, these implementations generally also increase the computational time. Boussinesq-type equations are for instance used in the numerical models by Fuhrman & Madsen [28] and Karambas [26]. 2.9 Conclusions The swash zone is the most landward part of the (wet) beach where waves and tides have a large effect. It can be exposed to different types and magnitude of wave forces and the local morphology changes on a very small time scale. The processes occurring in the (inner) surf zone provide the seaward boundary conditions for the swash zone. The swash zone is a highly dynamic and complex region and many physical processes play a role. The dominance of either high frequency or low frequency waves on a certain coast determines the dominant type of swash motion (either characterized by dissipated or reflected waves) and has for instance a large correlation with the beach slope and the sediment characteristics. In literature both wave skewness and asymmetry are indicated as an important effect for the sediment transport in the swash zone. A small skewness results in a net onshore transport in the swash zone, while a large skewness can either lead to offshore transport (when there is a weak asymmetry due to phase-lag effects) or to onshore transport (when the wave asymmetry is large enough). Turbulence is also considered an important process, mainly for stirring up the sediment, especially during uprush. It can therefore generally be seen as a process promoting onshore sediment transport. It is found in literature that the thickness of the boundary layer greatly determines the transport rate. Since the boundary layer generally 26 of 142

41 August 211 is thinner during uprush, this would enhance onshore transport (although this is strongly related to the infiltration/exfiltration during uprush/backwash). Infiltration/exfiltration can influence the effective weight of the sediment, change the thickness of the boundary layer or affect the swash flow asymmetry. The first effect induces a net offshore transport and is dominant for very fine sediments, while the second effect induces a net onshore transport and is dominant for coarser sediments. For very coarse sediment (gravel) swash flow asymmetry becomes dominant which also induces a net onshore sediment transport. In the present study the focus is on sandy beaches with relatively fine sediments for which the offshore transport effect is dominant. The effect of swash-swash interactions is not very well understood. It can either induce onshore or offshore sediment transport. Likewise acceleration (and the horizontal pressure gradient) in the swash zone is a relatively unknown process. Some researchers found it is dominant for onshore transport while others only found an offshore effect. 27 of 142

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43 August Methodology 3.1 Introduction Nowadays the field of morphodynamic computational modelling provides rather sophisticated tools to help better understand the behaviour of dynamic natural sand systems. New knowledge cannot be obtained from the models and therefore morphodynamic modelling will never fully replace field experiments and laboratory research. However, it is a rather cheap and easy (and therefore more and more commonly used) method to help understand coastal systems, and to predict their behaviour. The aim of this chapter is to provide insight into the different modelling approaches used in this thesis. Additionally, the swash zone processes that are taken into account in the approaches are discussed herein. Section 3.2 gives a description of the modelling philosphy used in this thesis followed by a brief description of the XBeach model, and the model approaches used. In the following sections the hydrodynamic modelling approaches are further elaborated. Section 3.3 describes the surf beat approach, section 3.4 the hydrostatic approach and, finally, section 3.5 describes the non-hydrostatic approach. In each of these sections attention is given to the waves and hydrodynamic as well as the sediment transport, and the swash processes. In section 3.6 the bed level updating in the model is described, and at the end of this chapter a brief overview of all approaches and their characteristics is given. 3.2 Modelling approach XBeach The model used in this study is XBeach [Roelvink et al., 29]. XBeach, is an acronym for extreme Beach behaviour model and is developed by IHE-Unesco, Delft University of Technology, Deltares and the University of Miami. The model is designed to simulate nearshore hydrodynamics and morphodynamics, especially during storms or hurricanes, and is able to predict dune erosion, overwash and breaching of dunes and barrier islands. In contrast to most other numerical models, XBeach computes the nearshore water level variations due to the wave motions, and therefore, an actual swash zone is present in the model. This makes the model suitable for detailed modelling of swash zone processes. XBeach is a 2DH depth-averaged numerical model, however, in this thesis only the onedimensional version is used. Therefore, only the (one-dimensional) formulations relevant to this thesis will be discussed from hereon. For further background and the full (twodimensional) equations reference is made to Roelvink et al. [29; 21] and Smit et al. [21] Hydrodynamic approaches In this thesis three different hydrodynamic modelling approaches are used and compared. The first method is the surf beat approach, which is the default approach in the XBeach model. The surf beat approach solves the wave propagation and hydrodynamics on a short wave group time and spatial scale. The main advantage is that it is computationally cheap. 29 of 142

44 August 211 The main disadvantage is that, since all waves are solved on a wave group scale, the individual wave information is lost. Secondly, the hydrostatic approach is used. The main advantage of this approach is that it solves the wave propagation and hydrodynamics on an individual wave scale, therefore being more accurate than the surf beat approach. The main disadvantage is that the model can only be forced on sufficiently shallow water. Finally the recently implemented non-hydrostatic approach is used. The main advantage of this approach is that it solves the waves and hydrodynamics on an individual scale, while it is also possible to force the model in deeper water. The main disadvantage is that the model has not been thoroughly tested yet and is still in development. It is also computionally expensive in comparison with the surf beat aproach Sediment transport formulations In addition, two sediment transport models are used. The first one is the formulation by Van Rijn [1984; 1993; 25; 27] and the second one is a combination of the transport models by Nielsen [1992] and Roelvink & Stive [1989]. The Van Rijn-formula is the default transport model in XBeach, although in the current version of the model a choice can be made for two versions: the adapted formulation by Soulsby-van Rijn [Soulsby, 1997] or the most recent formulation by Van Rijn [27]. The latter is also used in this thesis. The transport models by Nielsen (for bed load transport) and Roelvink & Stive (for suspended load transport) are not present in the model, but are implemented in this study Morphological updating The bed level changes are calculated based on the gradients in sediment transport. Here, the one-dimensional bed update formulation is given: zb fmor qx [3.1] t 1n x where z b is the bed level, f mor is a morphological acceleration factor, n p is the porosity and q x is the sediment transport rate, given by: E hcu C qxxt, Dh s [3.2] x x x where h is the water depth, C is the concentration, u E is the (Eulerian) mean velocity and D s is a sediment diffusion coefficient. Figure 3-1 gives an overview of the structure of the modelling part of this thesis. p 3 of 142

45 August 211 Figure 3-1 Overview of the modelling approaches discussed and used in this thesis. For the hydrodynamic processes the surf beat, hydrostatic or non-hydrostatic approach can be used. For the calculation of the sediment transport rates the transport model by Van Rijn or a combination of the models by Nielsen [1992] and Roelvink & Stive [1989] is used. 3.3 Surf beat approach The surf beat approach is the default XBeach model approach that combines a wave action balance and the nonlinear shallow water equations (NSWE) with a advection-diffusion equation to predict nearshore hydrodynamics, sediment transport and coastal erosion. Due to the interaction between the short wave action balance and the low frequency wave motions in the NSWE, the model is able to accurately simulate surf beat. The surf beat is forced by the wave energy variations in the short wave groups and is mainly responsible for the water reaching the dunes during, for instance, a storm surge. Due to this, XBeach is able to predict dune erosion, overwash and breaching. In the following sections the simulation of waves, hydrodynamics, sediment transport and the specific swash processes will be discussed for the surf beat approach Waves and hydrodynamics In a given wave or water level signal the surf beat approach makes a distinction between high and low frequency waves. A high and a low frequency time series can be put in the model seperately or a total water level elevation time series can be used. In the latter case, XBeach will divide the time series into high and low frequencies by means of a split frequency (usually half of the peak frequency). If only a high frequency wave time series or spectrum is available, XBeach will compute the accompanying long wave water level elevation. High frequency waves The propagation of high frequency waves is simulated with a wave action balance on the scale of the short wave groups. The one-dimensional wave action balance is given by: A ca w g w Dw [3.3] t x 31 of 142

46 August 211 where x is the direction of wave propagation, c g is the short wave group propagation velocity, is the intrinsic wave frequency and D w is the total wave energy dissipation. Wave action (A w ) is defined as the ratio of wave energy and the intrinsic wave frequency: Ewxt, Awxt, xt, [3.4] where E w is the wave energy density. The total wave energy dissipation (D w ) is given by [Roelvink, 1993]: Dw Qb Ew, where n H [3.5] rms 8Ew tanh kh Qb 1exp, Hrms and Hmax Hmax g k where is an empirical constant, E w is the total wave energy, is the water density, g is the gravitational acceleration, is a wave breaking parameter, k is the wave number, h is the local water depth and n is the ratio between the wave group velocity and the individual wave celerity. For the simulation of (broken) wave bores a one-dimensional roller energy balance is given, in which the energy dissipation (D w ) serves as source term: Er cer Dr Dw [3.6] t x where E r is the energy of the roller and D r is the roller energy dissipation, given by [Reniers et al., 24]: 2grEr Dr [3.7] c where c is the wave propagation velocity and r is a factor related to the slope of a breaking wave. Low frequency waves The low frequency wave motions are solved using the nonlinear shallow water equations (NSWE) for continuity and conservation of momentum. The NSWE are only valid for situations where the wave length is significantly larger than the water depth (L>>h), and are given by: L L 2 L E u L u u sx bx Fx u h g 2 t x x h h x h [3.8] L hu t x where u L is the Generalized Lagrangian Mean (GLM) flow velocity (defined as the distance a water particle travels in one wave period divided by the wave period), h is the horizontal viscosity, sx is a wind stress term bx is a bed shear stress term, is the free water surface elevation and F x is the wave force induced by radiation stress. The velocity related to the Eulerian velocity is then given by: u E = u L u S, where u S is the Stokes drift velocity, given by [Philips, 1977]: 32 of 142

47 August 211 u E 2E [3.9] hc S w r From hereon the term NSWE will be used when referring to equation [3.8]. Generation of surf beat The interaction between the wave action balance [3.3] and the NSWE [3.8] is provided by the inclusion of the wave force term (F x ) in the NSWE. For a one-dimensional case the wave force is given by: Sxx, w Sxx, r Fxxt, [3.1] x where S xx,w is the wave-induced radiation stress and S xx,r is the roller induced radiation stress given by: cg 1 Sxx, wx, t Ew2 [3.11] c 2 and xx, r, S x t E [3.12] where c g is the short wave group propagation velocity. The wave energy (E w ) and roller energy (E r ) are computed from respectively the wave action balance (equation [3.3]) and the roller energy balance (equation [3.6]). r Sediment transport In XBeach the sediment transport is predicted using a depth-averaged advection diffusion equation, given by [Galapatti, 1983]: E hc hcu C hceq C Dh s [3.13] t x x x T where C is the depth-averaged sediment concentration, D s is a sediment diffusion coeficient, C eq is the equilibrium sediment concentration, and T s is the adaptation time, given by: h Ts max.5,.2 [3.14] ws where w s is the fall velocity of the sediment particles. The concept of the advection diffusion equation is that sediment will be picked up from the bottom when the local concentration is lower than the equilibrium concentration (underload condition), and sediment will be deposited when the local concentration is higher than the equilibrium concentration (overload condition). The more sediment is in suspension, the more can be transported. The equilibrium sediment concentration depends on the grain characteristics and flow conditions. Presently, for calculating the equilibrium sediment concentration only the Soulsby-Van Rijn [Soulsby, 1997] and Van Rijn [27] methods are implemented in XBeach. The underlying idea of these formulations is that if the sum of the mean flow velocity and the near-bed orbital velocity is larger than a critical velocity value, sediment is transported. Since both formulations are very similar, only the most recent formulation is considered here. The s 33 of 142

48 August 211 equilibrium sediment concentration for bed load (C eq,bed ) and suspended load transport (C eq,sus ) according to Van Rijn [27] is given by: 1.2 D5 1.5 Ceq, bed.15h Me h eq, sus 5 * e where D* is the dimensionless particle size, given by: [3.15] C.12D D M [3.16] g D 13 D * 2 5 [3.17] where is the kinematic viscosity, =( s w )/ s is the relative weight of the sediment and s ( w ) is the sediment (water) density. The mobility parameter (M e ) in the Van Rijn formulas is given by: E 2 2 max u.8 urms,2 ucr, Me [3.18] gd where u e is the (wave-group averaged) velocity and u rms,2 is the near-bed short wave orbital velocity including the contribution of the wave breaking induced turbulence (k b =k b,short +k b,long, for high and low frequency waves), given by [Reniers et al., 24]: 5 u u k [3.19] 2 rms,2 rms turb b turb is a turbulence coefficient estimated to be 1.45 by Van Thiel de Vries [29], while the short wave orbital velocity is calculated by linear wave theory: H rms urms [3.2] T kh 2 sinh and u cr is the critical velocity that has to be exceeded in order to set the sediment into motion. Van Rijn separated the critical velocity in a part for currents (u cr,c ), based on Shields [1936], and a part for waves (u cr,w ), based on Komar & Miller [1975]. The total critical velocity is then given by: u u 1 u [3.21] m cr crc, crw, where is a dimensionless factor (u e / (u e + u rms,2 )) indicating the relative importance of the current and the wave-induced velocity. Finally, to account for the effect of the bed slope a correction factor (1- b m) is applied, where b is a calibration factor, and m is the bed slope Swash processes In Chapter 2 the following processes were found to be dominant for sediment transport in the swash zone: wave skewness, wave asymmetry, turbulence, groundwater flow, swash-swash interactions, acceleration effects, pressure gradients and the presence of the boundary layer. A number of these processes are not (yet) implemented in the model. Flow acceleration effects and pressure gradients are not included for the high frequency wave motions due to the wave group scale resolving of the high frequency waves. For the low frequency waves they are present in the NSWE. They affect the local hydrodynamics and 34 of 142

49 August 211 thereby implicitly the local morphodynamics. However, in the Van Rijn formulations acceleration effects and pressure gradients are not taken into account explicitly. The boundary layer is not present in XBeach due to the depth-averaged approximation used in the model. The remaining processes are included, and a brief model description is given. Wave skewness and asymmetry In the surf beat approach the high frequency waves are solved on a wave group scale, thereby losing all high frequency wave information on an indivual wave scale. For this reason, both (high frequency) wave skewness and asymmetry are included based on a parameterization. In XBeach a choice can be made for two presently implemented methods; a parameterization by Ruessink & van Rijn [manuscript in preparation] or the extended parameterization by Van Thiel de Vries [29]. The latter is required for estimating the bore interval period (T bore, see next section) and is therefore used in this study. To include the wave skewness and asymmetry in the surf beat approach, the Eulerian mean velocity (u E ) in the advection term of the advection diffusion equation [3.13] is replaced by: u AV = u E + u A, where the flow velocity related to wave nonlinearities is given by: u Sk Asu [3.22] A Sk As rms where Sk and As are calibration factors. Van Thiel de Vries [29] proposed the use of a wave shape model by Rienecker & Fenton [1981] to determine the skewness (Sk) and asymmetry (As). In the Rienecker-Fenton model the short wave shape is included in the expression for the near bed short wave velocity (u bed ), represented as the weighted sum of eight sine and cosine functions: 8 i bed i i i1 u wa cos i t (1 w) A sin i t [3.23] where w is a weighting function that affects the wave shape, A i is the amplitude which is a function of the dimensionless wave height (H rms /h) and the dimensionless period (T rep (g/h).5 ) and is the angular wave frequency. The skewness (Sk) factor is then given by: u Sk [3.24] while the asymmetry (As) factor is computed with the same expression replacing u bed by its Hilbert transform. The overbar represents the time mean value and represents the standard deviation of the near bed short wave velocity (u bed ). Van Thiel de Vries [29] found that the weighting (w) in [3.23] is a function of the phase, and found the following relation: w.2719ln.5 [3.25].2933 while the phase itself can be computed for a given skewness (Sk) and asymmetry (As): 1 tan As Sk [3.26] Since the expressions for the skewness and asymmetry are both dependent on the near bed velocity itself (see [3.24]), the Ruessink-Van Rijn expression is used to approximate the Skand As-values in equation [3.26]: 3 bed 3 ubed 35 of 142

50 August Sk cos tanh.64 /.61 log Ur U r 1exp [3.27].79.6 As sin tanh.64 /.61log Ur U 1 exp 2 2 r.35 where U r is the Ursell number (= 3/8 Hk /(kh) 3 ). When the Rienecker-Fenton wave shape model is not used, the results of [3.27] are included in equation [3.22] directly. For the low frequency waves the wave nonlinearities are solved within propagation of the water surface elevation in the NSWE. An extra parameterization is therefore not needed. Turbulence induced by wave breaking In XBeach the high frequency wave breaking induced near-bed turbulence is included in the calculation of the near-bed short wave orbital velocity (equation [3.19]) and can be computed averaged over a wave or over a wave bore. The high frequency wave roller induced turbulence at the water surface is computed as a function of the roller energy dissipation (D r ): k s D r w For the near-bed turbulence (k b,short ) the decay of the turbulence over the water depth has to be taken into account (the turbulence created by the wave bore will be less intense near the bottom). Therefore, the expression is multiplied with an exponential decay factor (=1 / (exp(h / L mix ) 1), where L mix is the mixing length, defined as the thickness of the surface roller). Because of the computation of high frequency waves on a wave group scale in the surf beat approach, the calculated near-bed turbulence is wave averaged. Van Thiel de Vries [29] found better results for a bore averaged approach. The bore averaged short wave turbulence is obtained by multiplying the wave averaged turbulence with the factor T Rep /T bore, where T bore (=H rms /(d/dx max c), where c is the wave celerity) is the bore interval period and T rep is the representive wave period. 2/3 [3.28] For the low frequency wave motions the local turbulence (k long ) is calculated with a long wave turbulence balance, where turbulence is generated by wave bores at the water surface and dissipated at the bed: L klong klongu ksource kdiss [3.29] t x where k source (=g(c+u)) is the source turbulence and k diss (= d (k long ) 3/2 ) is the turbulence dissipation, where d is a calibration factor associated with the stirring of sediment by the near-bed turbulence [Roelvink & Stive, 1989]. The creation of turbulence at the water surface is a function of a user defined critical wave slope (), the wave velocity and the roller thickness (). The roller thickness is updated every time step by calculating the difference between the critical wave front slope and the actual wave front slope. When the wave front slope is steeper than the critical value, the difference will be added to the roller thickness. For a wave front slope gentler than the critical value, the roller thickness will decrease: 36 of 142

51 August 211 max, t t t To account for the vertical decay of the turbulence the same factor is applied as for the high frequency turbulence. crit [3.3] Groundwater flow The effect of water infiltration and groundwater flow in the swash zone is in XBeach implemented by means of a simple model for Darcy flow (laminar flow conditions). The flow velocity is a function of the permeability represented by the hydraulic conductivity (k x ) and the groundwater head gradient: dpgw ugw kx [3.31] dx In order to simulate the interaction between the surface water and groundwater, a vertical flow velocity between the surface water layer and groundwater layer (w) is introduced. For infiltration and exfiltration the vertical velocity is given by: gw zb wex np t [3.32] dp win kz 1 dz where n p is the porosity, k z is the vertical permeability, z b is the bed level and the groundwater surface level ( gw ) is calculated with the ground water continuity equation: dgw dugwhugw w [3.33] dt dx n The vertical flow velocity (w) is then added to the continuity equation (in equation [3.8]), thereby affecting the local hydrodynamics. The vertical velocity is not added to the momentum equation, because it is assumed the vertical flow is a magnitude smaller than the horizontal flow. p Swash-swash interactions For the high frequency wave motions swash-swash interactions are not implemented in XBeach. Due to the wave group scale resolving of the short waves, individual high frequency waves do not have any effect on the waves in front or behind them. For low frequency waves motions the swash-swash interactions are included in the NSWE. When a low frequency swash cycle meets the subsequent swash cycle, both the catch-up and absorb process or the collision process affects (see section 2.6.3) the local hydrodynamics substantially. 3.4 Hydrostatic approach XBeach can be used with just the NSWE and without the wave action balance. In this approach, from hereon referred to as hydrostatic approach, both the high and low frequency 37 of 142

52 August 211 waves are fully resolved within the NSWE. This results in a rather clean computation method where intrawave processes are fully included rather than schematized (e.g. wave asymmetry). One disadvantage of the hydrostatic approach is the shallow water requirement (L>>h or kh<<1), which means wave boundary conditions have to be imposed in shallow water. This approach is therefore mainly applicable in the inner surfzone and the swash zone or when the wave motions are sufficiently long. However, for most practical cases measurements are taken on deeper water (e.g. with a wave buoy), where the hydrostatic approach is not applicable Waves and hydrodynamics The waves and hydrodynamics are fully resolved with the NSWE [3.8], without the wave force term. Since these equations were already treated in section 3.3, they will not be further discussed here Sediment transport In addition to the shallow water requirement there is another disadvantage of the hydrostatic approach, specifically for the present XBeach model. The implemented sediment transport models (Soulsby-Van Rijn [1997] and Van Rijn [27]) both require the near-bed orbital velocity (u rms ) as an input parameter. In the surf beat approach this parameter is computed from the wave action balance (see [3.2]), however, the NSWE do not compute this parameter. Another consideration is that the Van Rijn formulations are not strictly valid for high velocity situations and sheet flow conditions, which are commonly found in the swash zone. Therefore, in the present study, two additional transport models were combined into a new sediment transport model and implemented in the XBeach program code. The first one is the formulation developed by Nielsen [1992]. This formulation is specifically developed for sediment transport in the swash zone, where bed load and sheet flow transport usually dominate due to the high flow velocities and the small water depths (in contrast to the suspended load dominated surf zone). The Nielsen formula does not take into account the presence of pre-suspended sediment and therefore, an additional Bagnold-type transport formulation by Roelvink & Stive [1989] is applied to compute the suspended sediment transports. It is, however, expected that in the swash zone the bed load transport will dominate compared to the suspended load transport. Both transport models and the implementation are discussed next. Nielsen transport formulation for bed load transport The sediment transport formulation by Nielsen [1992] is based on the shear stress based bed load transport model by Meyer-Peter & Müller [1948]. The original sediment transport formulation is given by: 1.5 qc ' t cr [3.34] 38 of 142

53 August 211 where C is equal to eight according to MP&M, cr is the Shields criterion for initiation of motion of sediment (usually.5) and is the Shields parameter, given by [Shields, 1936]: f2.5 u u* ' [3.35] gd gd 5 5 where u* is the shear velocity and f 2.5 is a wave friction factor. Nielsen rewrote the MP&M transport formulation into: s 3 ' sign q t C t t gd u t cr 5 [3.36] and when combining equation [3.35] and equation [3.36], the transport rate is given by: qs Cmax ' t cr,u* t D 5 where C is an empirical constant equal to twelve for fine sand in high flow intensities [Nielsen, 22; 26]. [3.37] For the present study the Nielsen formulation is rewritten into a format so that it can easily be included in the XBeach program code. In order to be able to use the depth-averaged advection diffusion equation the depth-averaged bed equilibrium concentration (C eq,bed ) is computed, given by: D5 ' cr Ceq, bed C [3.38] h Additionally, the wave breaking induced turbulence is taken into account by including the turbulence energy in the Shields parameter: 2 u* k ' b [3.39] g D where the turbulence (k b ) is computed with the low frequency turbulence balance (see [3.29]). To account for boundary layer effects Nielsen [1992] introduced a phase shift ( t ) in the nearbed velocity to calculate the shear stress velocity: 1 u u*.5 f2.5 costusin t [3.4] p t where t is the phase shift between the free stream velocity and the bed shear stress at the (angular) peak frequency p (=2f p ). The wave friction factor (f 2.5 ) corresponding to a bed roughness of 2.5D 5 is given by [Swart, 1974]:.2 2.5D5 f2.5 exp S [3.41] where S is the swash excursion, which is the distance that the water traverses in the oscillating swash motion of run-up and rundown. The coefficients used in the formulation are given by Nielsen [1992] and are based on a number of more recent datasets compared to the data used by Swart [1974]. Nielsen [22] computes the swash excursion as a function of the free stream velocity variance: 2 S Varu t [3.42] p By applying a phase shift the acceleration (u/t) is taken into account in the shear stress velocity calculation (equation [3.4]), while the peak frequency is taken into account to obtain 5 39 of 142

54 August 211 equal dimensions for both terms. The result is that if the velocity (u) is a simple harmonic function, the shear velocity (u*) will be a simple harmonic function shifted forward by t. If u(t) is not a simple harmonic (e.g. a saw tooth shape) the phase shift will result in stronger shear stresses in the more abruptly accelerated part of the velocity signal, thereby enhancing onshore transport. Implementation in XBeach Most of the equations discussed can be implemented in the XBeach program code directly. However, for some parameters an approximation is made. Since XBeach is a time domain model it is difficult to calculate the velocity variance, required for equation [3.42]. Therefore a different expression is used here to approximate the swash excursion. Stockdon et al. [26] proposed a simple formula to calculate vertical swash excursion on dissipative beaches: S.46 HL [3.43] dis where H is the offshore wave height and L is the offshore wave length. By using this formulation it is assumed that the offshore wave conditions do not change during the simulation. However, it is found in the present study that the transport model is not sensitive to (slight) changes in the value of the swash excursion. Therefore, it is assumed equation [3.43] gives a sufficiently accurate approximation of the swash excursion magnitude. The acceleration term in equation [3.4] cannot be obtained directly due to the explicit numerical method in XBeach. It is therfore calculated as the difference between the current velocity and the velocity in the former time step divided by the time step: n1 n1 n u u u t t where the superscript n indicates the time step. [3.44] The instantaneous angular frequency ( p ) in equation [3.4] is rather difficult to determine in the inner surf and swash zone, due to the highly unsteady character of the flow and the transition from high frequency to low frequency waves in these regions. Therefore, an angular frequency constant in time is assumed ( p 2T p where T p is the offshore peak period). Finally, to account for the effect of sediment transport on a slope the Shields parameter is multiplied with a bed slope factor [Fredsøe & Deigaard, 1992]: tan cos 1sign ut ( ) [3.45] tan where is the bed slope and is the friction angle of the sediment. Bagnold-type model for suspended load sediment transport Roelvink & Stive [1989] developed a transport model based on the energetics approach as first proposed by Bagnold [1963; 1966] and later modified by Bailard [1981]. The suspended load transport formulation by Roelvink & Stive is given by: 4 of 142

55 August 211 q s K s t b gn where K s (= s u(t) / w S ) is the dimensionless suspended load transport rate. The local rate of energy dissipation due to turbulence near the bottom induced by wave breaking ( t ) and the local rate of energy dissipation due to bottom friction ( b ) are given by: 3 32 t cf u t and b dkb p [3.46] [3.47] s is an efficiency parameter (which is.1 according to Bagnold [1966] and.25 according to Bailard [1981]), w s is the sediment fall velocity, n p is sediment porosity, k b is the near-bed turbulence (computed with the turbulence balance given by [3.29]), c f is the bed friction coefficient and d is a calibration factor associated with the stirring of sediment by the nearbed turbulence (k b ). Finally, equation [3.46] is rewritten to obtain the equilibrium sediment concentration: s 32 Ceq, sus dkb gw hn s p [3.48] Swash processes In the hydrostatic approach all included processes are found in the calculation of wave propagation and flow in the NSWE. As mentioned before, flow acceleration effects and pressure gradients are solved within the equations but do not have an explicit contribution in the sediment transport formulation by Van Rijn. However, in the formulation by Nielsen the acceleration is taken into account in the expression for the shear stress velocity, equation [3.4], giving it not only an implicit effect via the hydrodynamic but also an explicit effect on the sediment transport rate. Since the hydrostatic approach is still a depth-averaged approach, boundary layer effects are not taken into account. For the remaining processes the implementation is similar to the one described for the low frequency wave motions in the surf beat approach: Wave skewness and asymmetry and swash-swash interactions are included in the NSWE (section 3.3.3). Wave breaking induced turbulence is computed with the turbulence balance given by [3.29]. Infiltration effects and groundwater flow are calculated as described for the low frequency motions in the surf beat approach (section 3.3.3). 3.5 Non-hydrostatic approach Smit et al. [21] implemented a non-hydrostatic module into the existing XBeach model with a numerical scheme based on the work by Stelling & Zijlema [23] and Zijlema & Stelling [28]. It includes a non-hydrostatic pressure term to compensate for the hydrostatic pressure assumption of the NSWE. While the surf beat approach solves the high frequency wave motions on a wave group scale, the non-hydrostatic approach solves, like the hydrostatic approach, both high and low frequency wave motions on an individual wave scale. 41 of 142

56 August 211 The non-hydrostatic module has not been much applied yet. However, the results of Smit [28] and Jacobs [21] show the high potential of this approach. Smit [28] found that when using multiple vertical layers, the non-hydrostatic approach works better due to the better approximation of the linear dispersion relationship. Presently, XBeach is a depthaveraged numerical model, so in this study only the one vertical layer model is used. A numerical model containing the non-hydrostatic approach with multiple layers has recently been developed by Zijlema et al [211]. The main advantage of the non-hydrostatic approach with respect to the surf beat approach is that every wave is solved individually, which makes the model more accurate. However, due to the resolving on an individual wave scale, the model grid size (and thereby also the computational time step) has to be relatively small, making the non-hydrostatic approach computationally expensive, and therefore not suitable for all modelling cases. In addition an extra (Poisson) equation has to be solved every time step to obtain the extra term in the NSWE. With respect to the hydrostatic approach the main advantage is the applicability of the nonhydrostatic approach in deep water. Whereas using the hydrostatic approach requires a wave forcing on shallow water, the non-hydrostatic approach makes it possible to force the model in deeper water. The main disadvantage of the non-hydrostatic versus the hydrostatic approach is the extra (Poisson) equation to be solved, making the non-hydrostatic approach computationally more expensive. In addition, the method has not been much applied yet, making it difficult to assess the ability of the non-hydrostatic approach Waves and hydrodynamics The waves and hydrodynamics are solved in the NSWE including an extra term that serves as a correction for the hydrostatic pressure assumption in the equations. In equation [3.8] the third term on the right hand side of the momentum equation represents the (hydrostatic) pressure term. In reality the pressure consists of a hydrostatic and a dynamic part. When assuming zero atmospheric pressure (p =) and normalizing the pressure by dividing it by the reference density ( w ), the pressure is given by: p p q g z q [3.49] h where p h is the hydrostatic pressure and q is the dynamic pressure. From [3.49] it follows that the variation of the horizontal pressure over the water depth is equal to: p h g [3.5] z Furthermore, pressure changes in time can be neglected due to the incompressible fluid assumption. In XBeach the pressure is depth-averaged and therefore the changes in the depth-averaged hydrostatic pressure over the cross-shore domain can be computed by multiplying the change in water depth (x) with the variation of pressure over water depth (p h z). The total pressure variation to include in the NSWE is then given by: p ph q ph q q g [3.51] x x x z x x x x Whereas in the surf beat and hydrostatic approach the dynamic pressure term (q/x) is assumed to be zero (see [3.8]), it is calculated in the non-hydrostatic approach. The dynamic pressure cannot be computed explicitly due to the incompressible flow assumption. Due to 42 of 142

57 August 211 this assumption there is no equation for the evolution of the dynamic pressure in time, and a Poisson equation is solved every timestep to determine the dynamic pressure Sediment transport As mentioned for the hydrostatic approach (section 3.4), the Van Rijn transport formulation is not strictly applicable for the non-hydrostatic approach due to the lacking of the near-bed orbital velocity (u rms ). Therefore, the combined transport model discussed in section 3.4 seems better applicable for the non-hydrostatic approach Swash processes In the non-hydrostatic approach all included processes are found in the calculation of wave propagation and flow in the NSWE with dynamic pressure term. As mentioned for the hydrostatic approach, flow acceleration effects are explicitly included in the Nielsen formulation, while the pressure gradient only has an implicit effect via the hydrodynamics, even though the dynamic pressure is included. The non-hydrostatic approach in XBeach is also depth-averaged, thereby excluding boundary layer effects in the model. For the remaining processes the implementation is similar to the one described for the low frequency wave motions in the surf beat approach: Wave skewness and asymmetry and swash-swash interactions are included in the extended NSWE (section 3.3.3). Wave breaking induced turbulence is computed with the turbulence balance given by [3.29]. Infiltration effects and groundwater flow are calculated as described for the low frequency motions in the surf beat approach (section 3.3.3). 43 of 142

58 August Overview Table 3-1 gives an overview and the main characteristics of all approaches discussed earlier. Table 3-1 The three approaches summarized with their main characteristics (AB=action balance, NSWE=nonlinear shallow water equations, vr7=van Rijn [27], N92=Nielsen [1992], RS89=Roelvink & Stive [1989]). Surf beat Hydrostatic Non-hydrostatic Hydrodynamics AB / NSWE NSWE (extended) NSWE Sediment transport vr7 vr7 or N92 / RS89 vr7 or N92 / RS89 Swash processes Mainly parameterized Solved in NSWE equations Solved in NSWE equations Advantages Computationally cheap, model boundary on deep water, commonly used Information on individual wave scale, accurate, commonly used Information on individual wave scale, accurate, model boundary on deeper water Disadvantages No information on individual wave scale, less accurate Model boundary on shallow water, computationally more expensive Computationally very expensive, not much used yet 44 of 142

59 August Modelling basic swash processes 4.1 Introduction In this chapter six simple plane beach profile simulation cases are described and analyzed. The cases are characterized by their wave steepness and beach profile slope, leading to a variable Iribarren number (see section 2.4), and include dissipative to intermediate (or semidissipative) beaches and spilling to plunging breaker types. The objective of this chapter is to obtain insight into the simulated swash characteristics for different beach state levels, represented by the Iribarren number Beach state In Figure 4-1 a schematization for the different beach state levels is shown including a scale for the Iribarren number ( ), the Dean number () and the grain size (D). The classification is based on a combination of the findings of Klein [24] and the expression for the Iribarren number and is not completely strict, because the wave conditions are only implicitly included (in the expressions for the Iribarren and Dean number, see section 2.4). However, the figure does give a clear indication of the type of beaches used in the simulation cases here (indicated by the black stars). Since, here, we are interested in sandy beaches, all six cases are located within the lower end of the Iribarren scale. Hence, the six simulations represent only dissipative or semidissipative (intermediate) beach states with spilling to plunging breaker types that occur on sandy beaches. Figure 4-1 Schematization of the different beach state levels based on Iribarren number ( ), Dean number (), grain size (D) and beach face slope ( b). The inclusion of the grain size and beach slope is based on a combination of the findings of Klein [24] and the expression for the Iribarren number but rather disputable due to the missing link with the wave conditions. 45 of 142

60 August Hydrodynamic model approach The objective of this chapter is to obtain insight into the simulated swash characteristics for different beach state levels. Therefore, the surf beat approach is used here, because it is the most commonly used approach included in the XBeach model and it is possible to force the model on deeper water. Additionally, the grid size can be relatively large with respect to the modelling approaches that fully solve the high frequency waves as well. This makes the surf beat approach computationally cheap and thereby applicable as a quick assessment tool Verification method First, the simulations are compared by analyzing time series of wave heights, hydrodynamics and morphodynamics on three locations in the swash zone. The swash zone changes location and dimensions during every swash cycle. However, for simplicity, the swash zone is defined here as the area located between the highest run-up and the lowest run-down level observed during the entire simulation period. The three (relative) locations used for the analysis are chosen based on the percentage of time the points are wet during the simulation. Locations with wetting percentages of 75, 5 and 25 are chosen to represent respectively the lower, middle and upper swash. On these locations short (1s) time series of the wave heights, velocities, concentrations and sediment transports are analyzed and compared for the different Iribarren number cases. Second, a statistical analysis of the wave heights, hydrodynamics and morphodynamics throughout the swash zone is performed. In this statistical analysis the mean values over the cross profile are considered, as well as a measure for the variance in time, from here on referred to as deviation. An upper and a lower deviation value for the model output is defined by sorting all output values from low to high for each cross-shore location. The output obtained during dry periods is excluded from the analysis. The upper (lower) deviation () is then defined as the mean of the highest (lowest) one third of the values: N/3 1 Xi, lower Xk,, t wet [4.1] N/3 k1 N/3 1 X [4.2] Xi, upper j,, t wet N /3 j1 where X j (X k ) represents the highest (lowest) one third of the simulated values for quantity X (e.g. wave height or velocity) on location i. The actual variance (or standard deviation) is not used here to preserve information about the spreading either above or below the mean simulated value. Finally, the actual swash motion (the variation of the shore line on a small timescale) is analyzed and compared for the different simulation cases. This method is considered as a Lagrangian analysis method, because the reference frame (in this case the varying shoreline) moves with the water motion. The time series analysis and the statistical analysis can be considered Eulerian analysis methods, since for both methods the reference frame is on a fixed location (either on one of the three locations in the swash or the entire cross-profile). 46 of 142

61 August Model setup The six simulation cases all have a different Iribarren number, governed by the (offshore) wave steepness (H /L ) and beach profile slope ( b ). Two characteristic (offshore) wave steepness values (.2 for swell and.4 for wind waves) and three characteristic beach slopes (2, 5 and 1 degrees) are included in the simulations Beach profiles A linear beach profile without any foreshore features (e.g. sand bars) is used for all simulation cases, see Figure 4-2. Klein [24] conducted research on a number of beaches in South Brazil and studied the relation between the beach face slope and the grain size. Based on his findings, three characteristic beach slopes are chosen of two, five and ten degrees. According to Klein, these three slopes represent respectively a dissipative, an intermediate and a reflective beach. However, this classification is solely based on the beach slope and not on the wave conditions. Figure 4-1 shows that when the wave conditions are included, the simulation cases contain only dissipative and intermediate beaches. The corresponding grain sizes, given in Table 4-1, are determined with Klein [24, fig. 6.5]. Table 4-1 The three used beach profiles with corresponding grain sizes (D 5) case 1 case 2 case 3 [deg] [%] D 5 [mm] D 9 [mm] Figure 4-2 Schematic profile used in the basic swash simulations, where IWL=initial water level Wave boundary conditions Besides being dependent on the beach slope, the Iribarren number is a function of the (offshore) wave steepness (H /L o ). Therefore, two characteristic wave steepness values are taken into account; a typical swell wave (with H /L =.2) and a typical wind wave condition (H /L =.4). From the wave steepness requirements the remaining parameters are calculated with linear wave theory, while maintaining the wave energy flux (P) forced at the model boundary equal for both wave boundary condition cases. In addition, an extra requirement was formulated for the kh-value, which indicates the relation between the wave length and the water depth. The kh-value was chosen to be equal for all simulations to assure all simulations are forced on equal relative water depth. This has mainly an effect on the actual offshore water depth (h) used in the simulations. To check whether the relative water depth is not too small for the surf beat approach, the ratio (n) between wave group velocity (c g ) and wave celerity (c) is 47 of 142

62 August 211 computed. A value of.78 is found for the current wave boundary conditions, which is small enough for using the surf beat approach. In Table 4-2 an overview is given of the two characteristic wave boundary conditions used in the basic swash simulations. Table 4-2 Offshore model boundary conditions for both wave steepness cases. The parameters are calculated with linear wave theory for a given wave steepness of.2 or.4, an equal wave energy flux (P) and a kh-value of one. boundary conditions swell case wind case wave height H s [m] wave height H rms [m] wave steepness H/L [-].2.4 kh-value [-] waterdepth h [m] wave length L [m] wave number k [-] mean period T m [s] peak period T p [s] peak frequency f p [Hz] energy E [J/m 2 ] wave celerity c [m/s] wave group velocity c g [m/s] energy flux P [J/m/s] Based on the given boundary wave conditions a JONSWAP-spectrum is made from which a wave boundary condition time series, including bound long wave, is generated with a method described by Van Dongeren et al. [23]. The formulations and the origin of this wave boundary condition generation method are further discussed in Appendix B. In Figure 4-3 the variance density spectrum for the swell and for the wind wave boundary condition case is given, showing that the spectrum is clearly located more towards higher frequency ranges for the wind wave case. 48 of 142

63 August 211 variance density [m 2 /Hz] variance density [m 2 /Hz] frequency [Hz] frequency [Hz] Figure 4-3 In the upper panel the variance density spectrum for the swell case (H/L=.2) is given. The peak frequency (f p) is.157 and the significant wave height (H s) is.871m. In the lower panel the variance density spectrum for the wind wave case (H/L=.4) is given. Here, the peak frequency (f p) is.27 and the significant wave height (H s) is 1.1m Iribarren number From the sections above six simulation cases are defined (three beach profiles times two wave boundary conditions). As mentioned in the introduction, the focus is on the effect of a varying beach state level quantified by the Iribarren number. An overview of the simulations, including three spilling breaker and three plunging breaker cases, is given in Table 4-3. Table 4-3 Overview of all simulation cases used for the basic swash simulations Simulation Case [deg] H /L [-] [-] Breakertype spilling spilling spilling plunging plunging plunging Beach state dissip dissip dissip dissip intermed intermed Computational grid and simulation time In the surf beat approach only the low frequency waves are fully resolved, while the high frequency waves are solved on a wave group scale. It is therefore possible to use a rather large grid size in the computational domain. However, to be able to accurately capture the swash processes that occur on a smaller scale, the grid size near the shoreline is chosen to 49 of 142

64 August 211 be smaller. For the simulations, a cross-shore smoothly varying grid is chosen with a grid size of around 5m at the offshore boundary and.2m near the shoreline. In XBeach the time step is computed by the model itself, according to the Courant criterion. The total simulation time is 18s, which is long enough to catch enough swash cycles and short enough to exclude the effect of the tide, and to make the model simulations computationally cheap. To exclude the effect of changing beach geometry on the hydrodynamic and morphodynamic processes, the bed updating is turned off during the simulations. 4.3 Simulation results One of the main characteristics of a more reflective (higher Iribarren number) or more dissipative beach (lower Iribarren number) is the change in the dimensions of the swash zone. A beach with more reflection is relatively steeper, making the water travel a shorter horizontal distance during uprush, hence, making the swash zone smaller. Also the appearance of the swash itself differs: while the swash motion on a dissipative beach mainly consists of wave bores traveling over a relatively long distance, the swash motion on reflective beaches is rather short and intense. In Figure 4-4 the wetting percentage (percentage of the time a location is wet throughout the simulation) is given as a function of the cross-shore distance. The swash zone is defined as the area between zero and a hundred percent wetting. The effect of decreasing swash zone dimensions for increasing Iribarren number is visible, although the swash width increases slightly for a lower wave steepness case (which can be directly related to the larger wave period). In addition, the location of the swash zone changes slightly seawards. This indicates that for lower Iribarren numbers a higher water level set-up is found. This can be explained by considering that on beaches with lower Iribarren numbers more wave breaking is induced, and therefore the gradient in radiation stress increases. Due to the rise in radiation stress gradients, a higher water level set-up can be expected. The stars in Figure 4-4 represent the locations that are considered in the analysis further on. For all simulations three locations with a wetting percentage of 25, 5 and 75% (respectively in the lower, middle and upper swash zone) are considered to study the difference in waves, hydrodynamics and morphodynamics for the six modelling cases. 5 of 142

65 August 211 wet [%] wet [%] wet [%] wet [%] wet [%] wet [%] =.175 =.247 =.437 =.619 =.882 = x[m] Figure 4-4 Wetting percentage for all simulation cases over the cross-shore (where x= is the initial shoreline). The stars (25, 5 and 75%) represent the locations (lower, middle and upper swash) that are analyzed further on Waves High and low frequency waves To accurately analyze the wave propagation in the swash zone a short time series (1s) is studied at the lower, middle and upper swash locations. Since the high and low frequency waves are resolved seperately, they will be discussed seperately here. A short time series of the short wave height is given in Figure 4-5. It can be seen that for the lowest two Iribarren cases (red and blue line) the wave heights show a characteristic asymmetric profile, while for the other cases the waves look quite symmetric. In addition, the wave height magnitude is substantially higher for higher Iribarren numbers, which can be related to the less dissipative character of these cases. XBeach does not compute the long wave height directly. Therefore a long wave height time series is obtained from the (low frequency) water level time series by calculating the difference between the maximum and the minimum water level value between two consecutive zero crossings for each location. From the obtained long wave time series the mean and deviation values are calculated with the method described in section This method, however, does not provide a continuous signal of the long wave height. Therefore, the water depth, which gives a good indication of the behavior of low frequency wave motions, is plotted in Figure 4-6. The water depth shows a behavior similar to the behavior of the short waves. The simulation cases with higher Iribarren numbers show higher magnitudes and a more symmetric profile. In addition, the duration of the swash cycle (time that a point is wet) generally decreases for higher Iribarren numbers. 51 of 142

66 August 211 In Figure 4-7 and Figure 4-8 respectively the time averaged short and long wave height are plotted. The black line represents the mean value while the band around the mean indicates the upper and lower deviation. The colors in the figure represent the wetting percentage and thereby define the location of the swash zone (between zero and a hundred percent). The complete total wave propagation from the offshore boundary through the surf zone to the swash zone is not discussed here, but can be found in Appendix D.1.1 and shows a characteristic propagation profile including wave shoaling, breaking and dissipation. Figure 4-7 shows that while for the lower Iribarren numbers the decrease in short wave height is rather smooth and gentle, while the wave height values for the higher Iribarren numbers are still rather high in the swash zone. Especially in the lower swash an average wave height of.3m can still be observed. Since the wave heights are generally higher and the run-up distance is lower, the decrease of the mean wave height is much stronger for the higher Iribarren number cases (the wave energy has to be dissipated on a shorter distance) with respect to the lower Iribarren number case. It is disputable whether this strong decrease in wave height is realistic, because in the model high frequency waves cannot be reflected, while in nature they can (on relatively steep beaches). In addition, the deviation is larger for higher Iribarren numbers. All this is a consequence of the smaller high frequency wave dissipation on more reflective beaches. 1 lower swash H sig [m].5 1 middle swash H sig [m].5 1 upper swash H sig [m] t [s] Figure 4-5 The significant (short) wave height in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). The colors represent different Iribarren numbers, see also Figure 4-4; red ( =.175), blue ( =.247), green ( =.437), cyan ( =.619), magenta ( =.882) and black ( =1.247). 52 of 142

67 August lower swash h [m].2.4 middle swash h [m].2.4 upper swash h [m] t [s] Figure 4-6 The water depth in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). H short H short H short H short H short H short =.175 =.247 =.437 =.619 =.882 = x [m] Figure 4-7 Time averaged short wave height [m] (black line) as a function of the cross-shore position for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation of 142

68 August H long.5 = H long.5 = H long H long.5.5 =.437 = H long H long.5.5 =.882 = x [m] Figure 4-8 Time averaged long wave height [m] (black line) as a function of the cross-shore position for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation. As shown in Figure 4-8 the effect of varying beach state level has a small effect on the long wave heights. The magnitudes in the swash zone are similar for all comparable locations, while the deviation seems to slightly increase for increasing Iribarren number. Combining Figure 4-7 and Figure 4-8 it can be observed that for higher Iribarren numbers short wave motions become relatively more dominant. This seems obvious (according to section 2.4), and can be explained by the smaller high frequency wave dissipation, but it is good to know that the model is able to reproduce such key concepts. An important measure for the behavior of low frequency swash is the ratio of energy being reflected. In Appendix D.1.2 the incoming and outgoing spectrum at the offshore boundary are shown for all simulation cases. The reflection coefficient (RC) is calculated by dividing the total outgoing (low frequency) energy by the incoming (low frequency) energy. It is expected that for a higher Iribarren number, more energy is reflected and therefore the reflection coefficient will be larger. Table 4-4 gives an overview of the different reflection coefficients. The table shows that more long wave reflection is simulated for increasing bed slope, but there is no clear relation with the Iribarren number. For the wind wave cases (case 2,4 and 6) a coefficient is found comparable in the same order as found in literature [e.g. Battjes et al., 24]. For the swell wave cases, however, relatively high reflection coefficients are found. Since for this case the waves have a lower steepness, it is expected more energy will reflect, but a multiplication of the incoming low frequency energy by six or seven does not seem realistic. It is hypothesized that the waves are too gentle for the (high frequency wave) breaking model, and, therefore, a too large part of the high frequency wave energy is transferred to the nonlinear shallow water equations. The energy in the NSWE, consisting of the original low frequency and the additional high frequency component, gets reflected and travels offshore of 142

69 August 211 Table 4-4 Reflection coefficients for the different simulations cases [-] RC [-] Interaction between the high frequency wave groups and low frequency wave motions An important mechanism in the swash zone is the interaction between the high and low frequency waves (see section 2.4). The high frequency waves are mainly represented by broken bores, but these are not fully resolved within the surf beat approach. There is, however, interaction between the two characteristic wave motions via the radiation stress and the high wave energy is, especially for higher Iribarren numbers, not zero throughout the swash zone. A first indication of the interaction can be given by analyzing how well the high frequency waves are grouped. According to List [1991] this can be evaluated with the wave groupiness factor, given by: 2 GF A [4.3] where is the standard deviation and is the mean of the wave envelope (A) signal. In Figure 4-9 the groupiness factor is plotted as a function of the Iribarren number (interpolated from the six simulation cases) and the cross-shore location. From offshore to onshore the wave groupiness decreases when the waves enter the surf zone and increases again in the swash zone for all Iribarren numbers. At the water line the wave groupiness rapidly goes to zero. The decreasing wave groupiness in the surf zone can be explained by the breaking of the higher waves within a wave group, thus reducing the short wave groupiness. The increase in wave groupiness is, however, less obvious. Considering the behavior of the wave groupiness for varying beach state levels, the main characteristic observed is that the decrease and increase of the wave groupiness over the cross-shore takes place over a shorter distance for the higher Iribarren numbers. It is also observed that the the groupiness factor reaches a lower minimum value (in the surf zone) and a lower maximum value (in the swash zone) when the conditions are more reflective. A possible explanation for the increase of the wave groupiness factor in the swash zone is the influence of the local water depth on the high frequency wave heights. Offshore the wave groups influence the (low frequency) water surface elevation via radiation stress (see section 2.3.2). In the swash zone the water depth decreases and will be dominated by the low frequency wave motions, which now control the (depth limited) short wave height. To illustrate this, the wave groupiness factor is plotted again for all simulation cases in Figure 4-1. In addition, a similar factor is computed and plotted for the long waves, by using the local water depth. Since the long waves do not travel in groups, this factor will be referred to as long wave factor (LF), given by: A h LF [4.4] h where is the standard deviation and is the mean of the water depth (h) signal. Both factors are plotted as a function of the mean water depth to be able to make a better comparison for the different simulation cases. At the offshore boundary the short wave groupiness factor is close to one for all cases (not exactly one, since the waves and wave groups are not regular here), while the long wave 55 of 142

70 August 211 factor is near zero. When the waves approach the coast, the short wave groupiness decreases due to the breaking of the highest waves. At the same time the long wave factor increases due to the decreasing water depth. Just before the swash zone the wave groupiness factor increases again, together with the long wave factor. For very shallow water the groupiness factor rapidly goes to zero, while the long wave factor increases more before it also goes to zero. The figure shows that on shallow water the wave groupiness is still rather high for the higher Iribarren numbers and that the dominance of the low frequency water level variations on the high frequency waves in the swash zone decreases for increasing Iribarren number. Apparently the high frequency waves are still rather grouped on the more reflective beach cases, while the low frequency wave motion dominates less. It can therefore be concluded that for increasing Iribarren number the beach becomes relatively more high frequency wave dominated in the model, which agrees with the literature (see section 2.4). Note that the sudden decrease for both the wave groupiness factor and the long wave factor observed in Figure 4-9 is most likely due to numerical cut-off limitations. In the present simulations, for instance, a cut-off water depth of.1m is used, while the high frequency wave energy is limited by a user defined wave height over water depth ratio ( max ). It is found that especially the cases with the ten degree beach profiles are very sensitive to the latter numerical limiter. To further analyze the interaction between the high and low frequency wave motions, the correlation between the wave groups and the long wave water surface elevation is plotted in Figure 4-11 for all cases. According to theory (see section 2.3.2) the bound long wave is negatively correlated with the short wave groups, when there is no influence of the bottom yet. This is the case on the offshore boundary (the correlation is around minus.5), although the value never reaches the theoretical value of minus one, due to the presence of reflected low frequency waves. Propagating onshore the wave groups and bound long waves loose heir correlation, when the bound long wave is released, but the figure also shows that for low water depths, or in the swash zone, a positive correlation is observed in the simulations. For the more dissipative cases the correlation decreases on relatively deeper water, while a positive correlation of eventually one is also found on relatively deeper water with respect to the simulations with a higher Iribarren number. The clear positive correlation in the inner surf and swash zone (near one) in combination with the wave groupiness and long wave factor development over the cross-shore indicates that the long wave water level variations have a great influence on the high frequency waves at low water depths (in the swash zone). For more dissipative conditions the effect becomes stronger, which is indicated by the positive correlation of nearly one observed until a mean water depth of about.7m for the most dissipative simulation case. The water depth on which the low frequency wave motions dominate, gradually decreases for increasing Iribarren number. 56 of 142

71 August 211 Figure 4-9 Short wave groupiness factor as a function of the cross-shore location (x-axis) and the Iribarren number (y-axis), interpolated over the six simulation cases. H/L=.4 H/L= deg 1.5 =.175 = deg 1.5 =.437 = deg 1.5 =.882 = h mean [m] h mean [m] 1 Figure 4-1 Wave groupiness factor GF (solid line) and long wave factor LF (dashed line) as a function of the mean water depth. [-] h [m] mean 1 =.175 =.247 =.437 =.619 =.882 = Figure 4-11 Correlation between the short wave groups and the long wave water surface elevation for the different simulation cases as a function of the mean water depth. 57 of 142

72 August Hydrodynamics Generalized Lagrangian Mean velocity In Figure 4-12 a short time series of the depth-averaged velocity associated with the low frequency water motions or GLM velocity (u L ) is plotted. The figure shows that the magnitudes, both onshore and offshore, for all Iribarren numbers are rather similar, while the width of the peaks (swash duration) is generally longer for the lower Iribarren number cases (which was also found for the water depths and waves). The velocity signals generally show a characteristic asymmetric profile (like Figure 2-7). Figure 4-13 shows the time averaged GLM velocity over the cross-shore including a band with the lower and upper deviation, given by respectively equation [4.1] and [4.2]. An important observation is that the time mean (depth-averaged) velocity is negative (offshore directed) on every location for all simulations. In addition, the figure shows that for higher Iribarren number the mean velocity magnitude in the swash zone decreases, while the deviation remains similar. Eulerian velocity For the lower Iribarren number cases the Eulerian velocity (depth-averaged velocity minus the high frequency induced Stokes velocity) signal in Figure 4-14 looks rather similar to the one for the GLM velocity. For the lowest two Iribarren numbers (red and blue line) the signal even looks identical (especially in the upper swash), while for the other four cases the onshore velocity peaks are smaller (or sometimes even disappear) and the offshore peaks are substantially larger compared to the GLM velocity, especially for the two cases with the largest Iribarren numbers. This is also observed for the time averaged Eulerian velocity in Figure 4-15, as well as for the deviation. In addition, the relative location with the minimum mean value of the Eulerian velocity (in the inner surf zone) differs from the GLM velocity (in the middle swash). The GLM and Eulerian velocity results look identical for lower Iribarren numbers, while the mean and deviation values increase for the higher Iribarren numbers. The similarity between the Eulerian and GLM velocity for low Iribarren number can be explained by the expression for the Eulerian velocity (u E =u L -u S ), where the Stokes (u S ) velocity is a function of the high frequency wave energy E w. For the low Iribarren number cases more wave energy is dissipated, making the Stokes velocity smaller, while for higher Iribarren cases, more high frequency wave energy is present near the shore, making the Stokes velocity larger, consequently having a larger effect on u E. Orbital velocity The orbital velocity is, amongst others, a function of the short wave height. However, Figure 4-16 shows that for increasing Iribarren number the peak value of the orbital velocity generally stays constant. This is, however, not observed for the mean orbital velocities in Figure 4-17, where both the mean and deviation values increase for higher Iribarren numbers, which seems logical considering the expression for the orbital velocity (equation [3.2]). 58 of 142

73 August 211 u L [ms -1 ] 2-2 lower swash u L [ms -1 ] 2-2 middle swash u L [ms -1 ] 2-2 upper swash t [s] Figure 4-12 The u L -velocity in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). The colors represent different Iribarren numbers (see also Figure 4-4); =.175 (red), =.247 (blue), =.437 (green), =.619 (cyan), =.882 (magenta) and =1.247 (black). 1 u L u L =.175 = u L u L =.437 = u L u L = = x [m] Figure 4-13 Time averaged u L -velocity [m/s] (black line) over the cross-shore for the different cases. The colors represent the wetting percentage while the band represents the deviation computed with equation [4.1] and [4.2] of 142

74 August 211 u E [ms -1 ] 2-2 lower swash u E [ms -1 ] 2-2 middle swash u E [ms -1 ] 2-2 upper swash t [s] Figure 4-14 The u E -velocity in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). The colors represent different Iribarren numbers (see also Figure 4-4); =.175 (red), =.247 (blue), =.437 (green), =.619 (cyan), =.882 (magenta) and =1.247 (black). 1 u E u E =.175 = u E u E =.437 = u E u E = = x [m] Figure 4-15 Time averaged u E -velocity [m/s] (black line) over the cross-shore for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation of 142

75 August 211 u rms [ms -1 ] u rms [ms -1 ] lower swash middle swash u rms [ms -1 ] 2 1 upper swash t [s] Figure 4-16 The orbital velocity in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). The colors represent different Iribarren numbers (see also Figure 4-4); =.175 (red), =.247 (blue), =.437 (green), =.619 (cyan), =.882 (magenta) and =1.247 (black) = = = = = = x [m] u rms u rms u rms u rms u rms u rms Figure 4-17 Time averaged orbital velocity [m/s] (black line) over the cross-shore for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation. 61 of 142

76 August Concentration and sediment transport Concentration Figure 4-18 shows a short time series of the depth-averaged sediment concentration at the three locations for all simulation cases. The peaks for the lowest four Iribarren number cases (red, blue, green and cyan) seem to have a similar magnitude, while the magnitude for the two higher Iribarren cases is substantially lower. There seems to be a maximum cut-off value for all cases, but it is not clear where this would originate from. The time averaged concentration (Figure 4-19), as well as the deviation, decrease for increasing Iribarren numbers. Looking back to the hydrodynamic time series, the concentration seems to be inversely dependent on the water depth. There is no clear relation with the velocity magnitudes. However, the concentration peaks are higher when the swash events have a larger period. Sediment transport In Figure 4-2 the bed load transport time series show large offshore directed peaks for the higher Iribarren numbers. The magnitude of the two highest Iribarren number cases (pink and black) is substantially higher compared to the magnitude of the two lowest Iribarren number cases (red and blue). In addition, the lower Iribarren number cases show some onshore peaks in the lower and middle swash, that are in magnitude comparable to the offshore peaks. For the highest Iribarren numbers the onshore peaks are hardly visible, being an order of magnitude smaller than the offshore peaks. Figure 4-21 gives the same image with a low mean bed load transport for low Iribarren numbers while the higher Iribarren numbers give substantially higher sediment transport magnitudes in the swash zone. In addition, the deviation is much larger for the higher Iribarren numbers. From Figure 4-22, where the suspended load time series is shown, it can be observed that the lower Iribarren numbers (red and blue) induce onshore and offshore suspended load transport peaks that are similar in magnitude. For the higher Iribarren numbers (green, cyan, magenta and black) there is only a very small onshore peak or, in some cases, no peak visible. The offshore peaks for these cases are, similar to the bed load transport, relatively large, especially for the case with the highest Iribarren number. Figure 4-23 shows the time averaged suspended load transport over the cross-shore. It can be observed that the mean suspended transport magnitudes and deviation increase for increasing Iribarren number. For both sediment transport modes a substantial increase in offshore sediment transport for higher Iribarren number is observed in the simulations. Note that the suspended load transport is in general an order of magnitude larger than the bed load transport in the swash zone, and is therefore the dominant transport mode. This is in contrast with the findings in literature (Chapter 2). However, the dominance of either bed or suspended load transport in the swash zone depends on many factor and can even vary over one tide [Horn & Mason, 1994]. Looking back to the hydrodynamic results and knowing that both bed load and transport are a power function of the Eulerian velocity (see section 3.3.2), it is apparent that the observed higher transport are a direct result of the larger offshore directed Eulerian 62 of 142

77 August 211 velocities. For both transport modes the highest time averaged mean offshore peaks are observed on an equal cross-shore location as the Eulerian velocity. The effect of the concentration and the orbital velocity visible in the results for the sediment transport. lower swash 15 C [gl -1 ] middle swash C [gl -1 ] upper swash C [gl -1 ] t [s] Figure 4-18 The concentration in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). The colors represent different Iribarren numbers (see also Figure 4-4); =.175 (red), =.247 (blue), =.437 (green), =.619 (cyan), =.882 (magenta) and =1.247 (black) = =.247 C C C C C C =.437 =.619 =.882 = x [m] Figure 4-19 Time averaged concentration [g/l] (black line) over the cross-shore for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation of 142

78 August 211 S bed [m 2 s -1 ] S bed [m 2 s -1 ] lower swash middle swash S bed [m 2 s -1 ] upper swash t [s] Figure 4-2 The bed load transport in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). The colors represent different Iribarren numbers (see also Figure 4-4); =.175 (red), =.247 (blue), =.437 (green), =.619 (cyan), =.882 (magenta) and =1.247 (black). 1-5 = S bed S bed S bed S bed S bed S bed =.247 =.437 =.619 = = x [m] Figure 4-21 Time averaged bed load transport [1-4 m 2 s -1 ] (black line) over the cross-shore for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation of 142

79 August 211 x 1-3 lower swash S sus [m 2 s -1 ] -1-2 x 1-3 middle swash S sus [m 2 s -1 ] -1-2 x 1-3 upper swash S sus [m 2 s -1 ] t [s] Figure 4-22 The suspended load transport in time (period of 1s) for three locations in the swash zone (25, 5 and 75% wetting percentage). The colors represent different Iribarren numbers (see also Figure 4-4); =.175 (red), =.247 (blue), =.437 (green), =.619 (cyan), =.882 (magenta) and =1.247 (black). 1-5 = S sus S sus S sus S sus S sus S sus =.247 =.437 =.619 =.882 = x [m] Figure 4-23 Time averaged suspended load transport [1-3 m 2 s -1 ] (black line) over the cross-shore for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation Morphological response The simulations described do not include bed level updating, and therefore the model does not give direct information on the morphological response of the beach profile for the different 65 of 142

80 August 211 cases. To give an indication of the (initial) bed level changes that can be expected as a result of the current simulations, the sediment transport rate is plotted in Figure The transport rate is computed as the spatial derivative of the mean total transport. A positive (negative) transport load corresponds to accretion (erosion). The figure shows that the mean transport rates alternate, especially for the higher Iribarren numbers. It is expected that the actual erosion/accretion pattern will be smoother when bed level updating is taken into account. A clear relation is visible between the transport rate magnitudes and Iribarren number: a higher Iribarren number gives higher transport rates and therefore more erosion/sedimentation. It can also be concluded that for all cases, generally only erosion takes place in the swash zone (eventhough this is not clearly visible in the figure for the lower Iribarren cases). S/x S/x S/x S/x S/x S/x =.175 =.247 =.437 =.619 =.882 = x [m] Figure 4-24 Time averaged total load transport rate [1-4 ms -1 ] (black line) over the cross-shore for the different cases. The colors represent the wetting percentage while the band gives an indication of the deviation Swash motions In the above analysis a Eulerian method (where the reference frame is fixed) was used to obtain insight in the swash processes in the model. Since the swash zone is very dynamic and changes location and dimensions constantly, a Langragian approach (where the reference frame moves with the swash motion) is applied here to obtain more insight. A first indication can be given by considering the shoreline variation (or swash excursion) in time. Due to the presence of a slope in the bottom the shoreline variation can be divided into a horizontal and a vertical motion. Here, we only consider the vertical swash excursion, which can be easily related to the horizontal or actual swash excursion because of the linear profiles used in the simulations. In Appendix D.2 the vertical swash excursion in time is plotted for all simulation cases, showing a general increase in the variance for increasing Iribarren number. In Figure 4-4 it was shown that the horizontal run-up distance decreases for higher Iribarren numbers, while the figure in the appendix shows an opposite effect for the (mean) vertical run-up. It is expected that the different observation can be fully related to the bed slope. 66 of 142

81 August 211 Taking the bed slope into account, a larger (actual) run-up distance is found for the lower Iribarren number cases, as observed in Figure 4-4. Following the method of Stockdon et al. [26], a significant swash excursion height (S sig ) is computed for all cases: S [4.5] sig 4 S sig where is the standard deviation of the vertical swash excursion in time. In Figure 4-25 a variance density spectrum is plotted for all simulation cases based on the swash excursion. The significant swash excursion heights are indicated within the figure. It is observed that the energy is only located on the lower end of the frequency band, especially for the lower Iribarren number cases. This means that there is no high frequency swash present in the simulations, which is logical because in the surf beat approach the high frequency swash motions are not solved for. The lower Iribarren number beaches only contain very low frequency swash energy, while the higher Iribarren beaches contain some more higher frequency energy (which is of course still not considered high frequency wave energy). The higher frequency energy found for the higher Iribarren beaches can be related to low frequency wave dissipation for the lower Iribarren number cases, or could be the result of an increase in the (low frequency) swash-swash interactions. According to the calculations of the vertical swash excursion, there is clear relation with the Iribarren number. A higher (offshore) Iribarren number results in a higher vertical swash excursion. Since only low frequency wave motions are taken into account here, it is expected that this effect is solely due to more low frequency wave energy dissipation on dissipative beaches. E(f) E(f) E(f) E(f) E(f) E(f) =.175, S sig =.487 =.247, S sig =.5337 =.437, S sig =.6222 =.619, S sig = =.882, S sig = = 1.247, S sig = frequency [Hz] Figure 4-25 Variance density spectrum of the vertical swash excursion for the six simulation cases. 67 of 142

82 August 211 From the significant swash excursion height the extreme (two percent) run-up elevation is computed as the sum of the time averaged water level elevation at the shore line and half of the significant swash excursion height [Stockdon et al., 26]: R2% [4.6] 2 The simulated extreme run-up elevation values are compared to a commonly used empirical parameterization that is a function of the Iribarren number, proposed by Holman [1986]: R2%.83.2 [4.7] H where H is the wave height measured at 18m water depth, but here taken as the wave height at the offshore model boundary. In Table 4-5 the results predicted by the empirical formula are compared with the model results. Both the model and the empirical relation show a gradual increase in the extreme run-up elevation. For the lower Iribarren numbers the runup values are similar, while for more reflective conditions the model gives a lower value with respect to the empirical formula. The reason for this is that for these cases relatively more high frequency energy is present in the swash zone, which is not represented by the model due to the use of the wave action balance for high frequency wave motions. For the simulation cases with a wind wave condition ( =.175,.437 and.882) the difference between the model and the empirical formula is substantially larger. The same reasoning can be given here since the relatively more higher frequency wave energy is not taken into account in the simulated swash motions. S sig Table 4-5 Values for the extreme (2%) run-up elevation predicted by the model and by a commonly used emperical formula proposed by Holman [1986]. =.175 =.247 =.437 =.619 =.882 = R 2%,Model R 2%,Formula Discussion Grain diameter In the above simulations a grain size distribution constant in cross-shore direction was used (defined by a constant D 5 and a constant D 9 ). Research, however, showed that the grain size distribution can vary over the cross profile and that the larger grains are often found in the swash zone [e.g. Reniers et al., in preparation]. Therefore additional simulations are conducted where the median diameter (D 5 ) is assumed to be equal to the original ninety percent diameter (D 9 ), while the latter is increased with ten percent. In Appendix D.3 a comparison is made for the mean bed and suspended load transport for all simulation cases. In addition, the time series for one case ( =.437) is compared to the time series of the original simulation for that case. From the figures it can be concluded that changing the grain diameter slightly, has only minor effect on the sediment transport that can be neglected. 68 of 142

83 August Sediment equilibrium concentration According to the expression for the sediment equilibrium concentration (see equation [3.15] to [3.21]), the simulated sediment transport is a function of the orbital velocity (u rms ) and the Eulerian velocity (u E ). The first is associated with the stirring of the sediment, while the latter is mainly associated with the current transporting the sediment grains. Both velocity quantities have been analyzed in section 4.3.2, and based on the results, it is expected that the Eulerian velocity has the largest relative contribution to the simulated sediment transport. To confirm this observation, the sediment equilibrium concentration (which is the governing parameter for sediment transport, see section 3.3.2) is re-calculated by hand, where either the orbital velocity or the Eulerian velocity is set to zero. The results of these calculations are plotted in Figure 4-26 and show that the orbital velocity is dominant offshore, while the Eulerian velocity is dominant further onshore, especially in the swash zone. Based on these results and the findings earlier in this chapter, it can be stated that in XBeach the Eulerian velocity is the most important velocity parameter for the simulation of sediment transport in the swash zone. Note that in all simulations the sediment equilibrium concentration is (slightly) higher near the shore when the orbital velocity is not taken into account, while further offshore the sediment equilibrium concentration without the mean velocity contribution is slightly higher in contrast with the simulations including both mean and orbital velocity. At first sight this seems odd, especially according to equation [3.15] to [3.18]. However, this effect can be explained by the expression for the critical velocity (equation [3.21]), which included a factor for the relative importance of the current and wave-induced velocity, which is a function of the mean and orbital velocity. 69 of 142

84 August = =.247 C eq [m 3 /m 3 ] C eq [m 3 /m 3 ] x[m] x[m] = =.619 C eq [m 3 /m 3 ] C eq [m 3 /m 3 ] x[m] x[m] = = C eq [m 3 /m 3 ] C eq [m 3 /m 3 ] x[m] x[m] Figure 4-26 Mean sediment equilibrium concentration over the cross-shore (C eq) simulated by the model (black dashed line) and calculated by hand (blue line) for the different Iribarren numbers. The red line represents the equilibrium concentration without the contribution of the Eulerian velocity (u E ), while the green line represents the equilibrium concentration without the contribution of the orbital velocity (u rms). 4.5 Conclusions The simulations presented in this chapter include six cases with different Iribarren numbers, providing insight into the simulated swash characteristics for different beach state levels. Based on the results described earlier a number of conclusions are drawn. It should be noted that these linear profile simulations do not represent natural (non-linear) beach profiles, and the results should therefore be interpreted carefully. 7 of 142

85 August 211 An important first observation is that for more reflective (higher Iribarren number) cases, a smaller swash period and a smaller swash zone width is found, which is also found in literature (see section 2.4). As a result of the smaller high frequency wave energy dissipation, high frequency waves become more dominant with respect to the low frequency wave motions and in the model this has mainly an effect on the Eulerian velocity (u E ). While the velocity associated with the low frequency wave motions (u L ) does not seem to be affected by the changing beach state level, the Eulerian velocity shows substantially larger offshore peaks for the more reflective cases. This effect is a direct result of the fact that the Eulerian velocity is a function of the Stokes velocity (u E =u L -u S ), which is a function of the high frequency wave energy. The simulated orbital velocities also increase for more reflective conditions, but this effect is less obvious. Another finding described in this chapter is that the long wave motions are dominated by the short wave groups offshore, while the opposite effect takes place in the inner surf and swash zone. For increasing Iribarren number the low frequency wave dominance on the high frequency waves in the swash zone decreases. Considering the morphodynamics the effect of changing beach state level is not that apparent for the simulated concentrations. The concentrations generally even decrease for increasing Iribarren numbers. It is, however, found that the concentration is related to the swash period, which is shorter for the more reflective beach cases. The sediment transport magnitudes substantially increase for increasing Iribarren numbers and it was found that this is due to the increase in the Eulerian velocity. The expected morphological response, represented by the transport rate, shows solely a net offshore transport effect for all cases, resulting in erosion of the swash zone. From the bed and suspended load transport time series it is concluded that this is mainly due to the lacking or underestimation of onshore directed transport peaks. This effect increases for reflective conditions, which can be directly related to the observed transport magnitudes. The simulated run-up elevation shows a good agreement with the run-up elevation predicted with an empirical formula for the swell wave cases. For the cases with a higher wave steepness, the empirical formula gives substantially higher predictions. This can be related to the relatively more high frequency wave energy present in the wave conditions. Since the model only considers the low frequency swash motion, the run-up values are underestimated. Thus, based on the results from the simulation cases presented in this chapter it can be stated that at beaches with more reflective wave conditions more high frequency wave energy is present near the coast resulting in a higher Stokes velocity. The higher Stokes velocity is responsible for the large offshore directed Eulerian velocities, leading eventually to a larger sediment transport and a larger transport rate. Even though no field or laboratory data is used to validate the simulation results, it is most likely that for higher Iribarren numbers the contribution of the short waves (which are solved less accurate in the action balance) is overpredicted in offshore direction, compared to the contribution of the long waves (which are fully resolved in the shallow water equations), leading to the observed effects described above. Hence, it is questionable whether the concept of the Stokes drift and undertow implemented in the model is still applicable in the swash zone, especially on reflective beaches. 71 of 142

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87 August Modelling additional swash processes 5.1 Introduction In Chapter 4 insight into the simulated swash characteristics for different beach state levels was obtained. It was concluded that a larger transport (rate) is obtained for a higher Iribarren number, mainly due to the higher relative contribution of the high frequency waves, which are solved in the wave action balance. In Chapter 2 a number of processes were indicated to be dominant in the swash zone, while in Chapter 3 the implementation of the swash processes presently included in the XBeach model was described. The objective of the current chapter is to study the effect of including the swash processes on the simulation cases described in Chapter 4. Wave asymmetry, wave skewness, high frequency turbulence, low frequency turbulence and groundwater flow are presently included in the XBeach model, either within the nonlinear shallow water equations (NSWE), or as a parameterization for the high frequency wave motions (see also Chapter 3). The effect of these processes will be studied in this chapter. The remaining processes indicated in Chapter 2 are either included implicitly (acceleration and horizontal pressure gradients), or not present in the model (boundary layer effects) and will therefore not be discussed here. Finally, the non-hydrostatic approach is used in this chapter to study whether the fully resolving of the high frequency wave motions has a substantial effect on the model results. 5.2 Swash processes in the surf beat approach It was found that the effect of the different swash processes on the simulated hydrodynamics and/or morphodynamics is generally rather small with respect to the original simulations described in Chapter 4. It is difficult to visualize these effects in figures similar to the ones used before. Therefore, to obtain a clear insight in the effects of the different processes, all simulations are conducted with bed updating taken into account. Comparing the simulated bed level changes over the total simulation period (18s) was found to be most clarifying. Therefore, for an analysis of the effect on the velocities, concentrations and/or transports reference is made to Appendix D.4, while the effect on the bed level changes as a result of including swash processes are plotted here. Note that the figures do not contain the actual bed level changes, but the effect on the bed level changes, relative to the original simulations. In Figure 5-1 the bed level changes over the simulation period are shown for all (original) simulation cases. In the following sections the effects of the swash processes on the model result will be analyzed by comparing the bed level changes to the bed level changes in the original simulations (as shown in Figure 5-1). All original simulations show erosion in the swash zone, and based on the transport gradients found in Chapter 4 (see Figure 4-24), this was already expected. 73 of 142

88 August 211 z b z b z b z b z b z b = = = = = = x[m] Figure 5-1 Total bed level changes [m] in the swash zone for all simulation cases presented in Chapter Wave skewness and asymmetry As described in Chapter 3, the effects of wave skewness and asymmetry are represented by a velocity term (u A ) that is added to the Eulerian mean velocity term in the advection diffusion equation. The velocity associated with wave nonlinearity is then given by: A Sk As rms u Sk Asu [5.1] where the skewness (Sk) and asymmetry (As) are computed with a Rienecker-Fenton wave shape model (see section 3.3.3). In Chapter 4 the calibration coefficients Sk and As were set to zero. Here, the calibration coefficients are varied to obtain insight in the effect of including wave skewness or asymmetry. From literature it is expected that both skewness and asymmetry have a substantial effect on the sediment transport in the swash zone. Whether the effect is net onshore or offshore depends on the level of nonlinearity (see section 2.6.1) and is not beforehand clear for the present simulations. Since wave asymmetry and skewness are only included in the advection diffusion equation, they do not have any effect on the hydrodynamics (only implicitly via the varying bed levels, but it is assumed this can be neglected). Therefore, only morphodynamic parameters are compared in Appendix D.4.1 and D.4.2. The figures in the appendix show the effect of wave asymmetry and skewness on the concentration, suspended load transport and bed load transport. The figures in the appendix show that including wave asymmetry has a net onshore transport effect. This effect gradually increases for increasing asymmetry calibration factor. The effect of the wave asymmetry is more dominant for higher Iribarren numbers, which can be related to the more high frequency wave energy present in the swash zone. This effect is also observed in the effect on the bed level changes, plotted in Figure 5-2, where the swash zone erodes less for all cases relative to the original simulations, and the effect is stronger for higher Iribarren number cases. 74 of 142

89 August 211 The wave skewness shows a similar but smaller effect on the results. There does not seem to be any effect on the concentration, while the bed and suspended load transport decrease for increasing wave skewness calibration factor. The bed level changes (Figure 5-3) show a similar effect, but the magnitude is clearly smaller with respect to wave asymmetry. As for the wave asymmetry, the effect of including wave skewness is stronger for the higher Iribarren number cases due to the more high frequency wave energy present near the shore. z b z b z b z b z b z b =.175 =.247 =.437 =.619 =.882 = x[m] Figure 5-2 The effect of wave asymmetry on the bed level changes [m], relative to the bed level changes for a simulation without additional swash processes taken into account (Figure 5-1). The asymmetry calibration factor is set to.1 (solid line),.3 (dotted line) and.5 (dashed line). z b =.175 z b z b z b z b =.247 =.437 =.619 =.882 z b = x[m] Figure 5-3 The effect of wave skewness on the bed level changes [m], relative to the bed level changes for a simulation without additional swash processes taken into account (Figure 5-1). The skewness calibration factor is set to.1 (solid line),.3 (dotted line) and.5 (dashed line). 75 of 142

90 August High frequency turbulence In Chapter 3 the implementation of the high frequency turbulence was described, indicating that the calculated near bed turbulence is mainly a function of the roller energy dissipation and the local water depth. The high frequency turbulence can either be included averaged over a wave period or over a bore period. Both methods are used here to verify the effect on the results. Since, according to literature, turbulence is mainly associated with the stirring of the sediment and is more dominant during uprush (section 2.6.2) it is expected that including turbulence in the simulations will have a net onshore effect. The figures in Appendix D.4.3 show the effect of including the high frequency turbulence methods on the concentration, suspended load transport and bed load transport, while in Figure 5-4 the effect of including short wave turbulence on the bed level changes is shown. Including high frequency turbulence has a net offshore effect on the results for all simulation cases. For the wave averaged method a minor net offshore effect is observed, while the bore averaged method gives substantial larger transport and consequently bed level changes in the swash zone, especially for the more reflective beach cases. Hence, including high frequency turbulence does not have the effect on the model results that was expected from the literature study. Especially the higher Iribarren number cases in combination with the bore averaged method show a great (net offshore) effect on the simulation results, which does not seem realistic based on the findings in literature. The high frequency turbulence is a direct function of the roller energy dissipation (see equation [3.28]), and increases the sediment equilibrium concentration via the expression for the orbital velocity (u rms,2 ) and the mobility parameter (M e ), see also section The increasing concentration for an equally offshore dominant mean velocity, results in a larger offshore directed sediment transport rate. The increasing transport effect for higher Iribarren numbers is most likely due to the less steady character of wave dissipation on a more reflective beach. While on a dissipative beach the wave energy dissipates more gentle, the wave energy decreases over a shorter distance on the more reflective beach profiles. It is hypothesized that the sudden decrease in wave energy, directly affects the locally dissipated roller energy (since there is more high frequency wave energy on a shorter cross-shore domain), leading to a larger (high frequency) turbulence. Whereas, the wave averaged method has a relative small offshore effect on the short wave turbulence, the bore averaged method has a great effect. The difference in both methods is the multiplication factor (T rep /T bore ), which seems to be relatively large in the current simulations. The bore interval period (T bore ) is a function of the short wave height, the wave celerity and a maximum water level gradient. The method was developed for surf zone sediment transport, and based on the current results it is concluded that this method does not work satisfactory for the swash zone, and should therefore not be used. 76 of 142

91 August 211 z b z b z b z b z b =.175 =.247 =.437 =.619 =.882 z b = x[m] Figure 5-4 The effect of wave averaged (solid line) and bore averaged (dashed line) short wave turbulence on the bed level changes [m], relative to the bed level changes for a simulation without additional swash processes taken into account (Figure 5-1) Low frequency turbulence Chapter 3 described the implementation of the low frequency turbulence by a turbulence balance. It is expected that including the long wave turbulence has most effect on the simulation cases with the lowest Iribarren numbers, where the relative dominance of low frequency wave motions in swash zone is larger and more lower frequency waves will break. As for the short wave turbulence, it is expected that based on literature (section 2.6.2) long wave turbulence will have a net onshore effect. In the figures in Appendix D.4.4 the effect of including the low frequency turbulence on the concentration, suspended load transport and bed load transport is shown. The effect of the long wave turbulence is visible for as well the concentrations as the bed load and suspended load transport. In contrast to the simulated high frequency turbulence, low frequency turbulence does promote onshore directed transport. The effect, however, is stronger for more reflective conditions, which is even more apparent from Figure 5-5, where the bed level changes relative to the original simulations are shown. The increase for higher Iribarren numbers is most likely due to the shorter swash periods, increasing the local rate of change of the water level (/t) (see Figure 4-12). The larger rate of change of the water level results in larger turbulence source term (since the source term is a function of the roller thickness which is a function of the rate of change of the water level, see section 3.3.3). 77 of 142

92 August 211 z b z b z b z b z b z b =.175 =.247 =.437 =.619 =.882 = x[m] Figure 5-5 The effect of including the long wave turbulence on the bed level changes [m], relative to the bed level changes for a simulation without additional swash processes taken into account (Figure 5-1) Groundwater flow In Chapter 2 it was found that the effect of infiltration of water in the bed and the subsequent groundwater flow can influence the morphodynamic processes in the swash zone. It was found that for finer sediments this leads to a net offshore effect, while for larger grains the effect is expected to be onshore transport enhancing (section 2.6.5). An important parameter considering groundwater flow is the permeability of the bed, represented by the hydraulic conductivity (K). The hydraulic conductivity depends on multiple variables (e.g. grain characteristics and porosity) and a large number of different formulations have been proposed in literature. López de San Román Blanco [23] identified the following expressions as most commonly used: Hazen [1911] defined the hydraulic conductivity as a function of the ten percent grain diameter (D 1 ) and an empirical constant C (with a value between.1 and.15): 2 K C D 1 [5.2] Kozeny [1927] included the porosity (n p ) in his formulation and used a typical particle size d and an empirical constant C with a value of around five: 2 3 Cdn p K [5.3] 2 1 n In the Kozeny-Carman formulation [Carman, 1937], the empirical constant C in the Kozeny formulation is replaced by a factor consisting of the water density ( w ), the gravitational acceleration (g) and the dynamic viscosity (): 3 2 w g n dm K [5.4] 1 n 2 18 p 78 of 142

93 August 211 Krumbein & Monk [1942] developed an even more sophisticated expression by including the distribution of grain diameters given by the standard deviation of the function (where =- log 2 (d) ): 2 K 76dw exp 1.31 [5.5] For the current simulation cases there is no information available about the grain diameter distribution and therefore only the Hazen, the Kozeny and the Kozeny- Carman formulation are considered from here on. The hydraulic conductivity values obtained from the different formulations for different median grain diameters are given in Figure 5-6 (where in the Hazen formulation D 1 is assumed to be.9d 5 ). From the figure it is clear that all formulation give similar results, but that the Hazen formulation is very dependent on the chosen empirical constant (and most likely also on the choice for D 1 ). Since it has the most physical background, it was chosen to use the Kozeny-Carman formulation for the calculation of the hydraulic conductivity for the simulations. In reality, there is often a difference between the horizontal and vertical permeability, for instance when cracks are present in the bed. However, for the current simulations the permeability is assumed to be equal for both directions. In Table 5-1 the hydraulic conductivity values required for the simulations with groundwater flow are given. K [m/s] D [mm] 5 Figure 5-6 Hydraulic conductivity (K) as a function of the median grain diameter (D 5) for the Hazen formulation (dashed line), Kozeny-Carman formulation (solid line), while the grey area indicates the results for the Kozeny formulation for the different C-values. The black circles indicate the values used in the current study, see Table 5-1. Table 5-1 Values for the hydraulic conductivity per beach slope simulation case [deg] D 5 [mm] K x [m/s] K z [m/s] Because for the simulations with a higher Iribarren number a higher hydraulic conductivity is used, it is expected that the effect will be larger for these cases, which is also what is expected based on the findings in literature (section 2.6.5). Since groundwater flow also has effect on the local hydrodynamics, the effect of including groundwater flow on the Eulerian and GLM velocity, as well as the concentration, suspended load transport and bed load transport is shown in Appendix D.4.5. The figures show that the GLM velocity is slightly increased for the more dissipative cases, while the velocity clearly decreases for the more reflective cases. The effect is less clear for the Eulerian velocities, however the concentration and bed load and suspended load transport show a substantial decrease for higher Iribarren numbers. Figure 5-7 shows the effect of the groundwater flow on the bed level changes. As expected from the findings in literature and the figures in the appendix, the effect increases for higher Iribarren numbers. For all simulation cases there is a net onshore effect. Out of the three 79 of 142

94 August 211 effects of infiltration and exfiltration described in section (effective weight effect, effect on boundary layer thickness and the effect on the swash flow asymmetry), only the effect on the swash flow asymmetry is taken into account in the model. Based on the results it can be concluded that the groundwater flow module substantially reduces the backwash strength, leading to less erosion in the swash zone. In literature, it was found that the effect on swash flow asymmetry only has effect for grain sizes larger than 1.5mm. This is not the case for the current simulations (for the steepest beach profiles a D 5 of.68mm was used), and it is therefore most likely that the groundwater flow effect is overpredicted. It was found from Appendix D.4.5 that the effect of the groundwater flow especially clear in the results for the GLM velocity. The mean and standard deviation of the GLM velocity over the cross-shore are therefore plotted in Figure 5-8. A clear effect on the (mean) velocities near the shore line can be observed, especially for the higher Iribarren number cases. These observations strengthen the hypothesis that the groundwater flow module affects the GLM velocity by reducing the backwash velocity (and thereby increasing the swash flow asymmetry), and becomes more dominant for more reflective beaches (which is related to the characteristic grain size). z b z b z b z b z b =.175 =.247 =.437 =.619 =.882 z b.2.1 = x[m] Figure 5-7 The effect of including groundwater flow on the bed level changes [m], relative to the bed level changes for a simulation without additional swash processes taken into account (Figure 5-1). 8 of 142

95 August 211 ul ul ul ul ul ul x [m] ul ul ul ul ul ul =.175 =.247 =.437 =.619 =.882 = x [m] Figure 5-8 The effect of the groundwater flow module (red line) on the time averaged GLM velocity (left panels) and the standard deviation of the GLM velocity (right panels) for all simulation cases, compared to the original results (blue line) Swash processes combined Based on the insight obtained from the simulations with the individual swash processes taken into account, an extra set of simulations is conducted with a combination of the different processes. The wave asymmetry is found to have a large effect on the model results and is therefore a wave asymmetry calibration factor of.3 is used. Wave skewness was found to be less dominant and therefore the default value (.1) for the calibration factor is used. Low frequency turbulence and groundwater flow are both taken into account and high frequency turlbulence is taken into account via the wave averaged method, since the bore averaged method gave unrealistically high offshore transport values. The results of this optimal swash simulation settings are shown in Figure 5-9, where the actual total bed level changes are plotted. The figure shows less erosion in the swash zone for all cases, but the effect of the processes is smaller than expected based on the analysis for the individual swash processes. This becomes even more apparent when looking to change in bed level changes for including the additional swash processes, see Figure 5-1. The effect is in the same order as the effect of including solely long wave turbulence, groundwater flow or wave asymmetry and can therefore not be predicted by simply adding the effects for including the individual swash processes. 81 of 142

96 August 211 z b z b z b z b z b =.175 =.247 =.437 =.619 =.882 z b = x[m] Figure 5-9 The actual bed level changes [m] for the simulations without (dashed line) and with (solid line) the additional swash processes taken into account. z b z b z b z b z b =.175 =.247 =.437 =.619 =.882 z b.2.1 = x[m] Figure 5-1 Effect of including multiple swash processes on the bed level changes [m], relative to the bed level changes for a simulation without additional swash processes taken into account. 5.3 The non-hydrostatic approach All simulations presented before were conducted with the surf beat approach, that only solves the high frequency waves on a wave group scale. Here, the non-hydrostatic approach is applied on the original simulations described in Chapter of 142

97 August Boundary conditions To obtain a wave boundary condition time series the same method was used as for the surf beat approach simulations, which is described in section The only difference is that the high and low frequency wave motions are combined to get a total water surface elevation time series. For the non-hydrostatic approach a depth-averaged velocity time series has to be provided at the boundary as well. The depth-averaged velocity signal is computed from the water level signal by using linear wave theory: z 1 cosh( k( h z)) u() t t dz h [5.6] sinh( kh) zh where h is the water depth, is the water surface elevation, k is the wave number and z is the vertical position Wave propagation Since the non-hydrostatic approach represents a different method of modelling, first the total wave propagation over the cross-shore is considered. In Figure 5-11 the propagation of the high and low frequency waves is plotted for both modelling approaches. It can be observed that, especially for the more dissipative cases, the high frequency waves tend to break substantially more onshore for the non-hydrostatic approach. This effect seems to be minor for the higher Iribarren number cases, however, the maximum high frequency wave height reached (due to shoaling) is clearly larger than for the surf beat approach. The low frequency wave motions show a generally more equal propagation signal, except for the most reflective case where the low frequency wave heights are substantially lower over the entire crossshore domain. For the case with the highest Iribarren number (1.247) the long wave height shows a big difference for both approaches offshore. Apparently the surf beat approach includes a substantial larger amount of reflected waves. It can be stated that the high frequency waves are solved rather cheaply in the surf beat approach, and can therefore not be used as an accurate verification for the high frequency non-hydrostatic wave heights. However, it was found that the process of wave breaking is very sensitive in the non-hydrostatic approach. By tuning the viscosity parameter slightly better results were obtained. 83 of 142

98 August H [m].6 =.175 H [m].6 = x [m] x [m] H [m].6 =.437 H [m].6 = x [m] x [m] H [m].6 =.882 H [m].6 = x [m] x [m] Figure 5-11 High frequency (solid line) and low frequency (dashed line) wave propagation for the surf beat approach (blue) and the non-hydrostatic approach (red) for the six simulation cases Run-up Due to the full resolving of both low and high frequency waves it is expected that the run-up is higher when applying the non-hydrostatic approach. In Figure 5-12 the wetting percentages for all simulations are compared to the surf beat approach, and show that the run-up indeed increases for all simulation cases. A change in the run-down limit location is not observed, and due to the higher run-up, the swash width is larger for all cases. 84 of 142

99 August 211 In section an extreme run-up parameter was calculated from the simulations and compared to the result of an empirical formulation based on the Iribarren number. Since for the non-hydrostatic approach the high frequency wave motions are also taken into account, it is expected this will also have an effect on the extreme run-up value. In Figure 5-13 the runup values are plotted for the surf beat approach, the non-hydrostatic approach and the empirical relation. The non-hydrostatic approach gives a slight improvement for the more reflective cases, but gives higher values for the lower Iribarren number cases. wet [%] wet [%] wet [%] wet [%] wet [%] wet [%] =.175 =.247 =.437 =.619 = = x[m] Figure 5-12 Wetting percentages as a function of the cross-shore location for all simulation cases for the surf beat approach (dashed line) and the non-hydrostatic approach. 1 R 2% [m] [-] Figure 5-13 Extreme (2%) run-up elevation predicted with the empirical relation by Holman [1986] (black circles) and simulated with the surf beat (red squares) and non-hydrostatic (blue stars) modelling approach Mean velocity In Chapter 4 it was found that the Eulerian (mean) velocity is predominantly responsible for the bed and suspended load transport in the model. Therefore, the mean velocity is compared for both modelling approaches. The morphodynamics are not taken into account here, because the non-hydrostatic approach is not implemented in the XBeach (yet) in such a way that it can realistically predict sediment transport. Based on the findings in Chapter 4 it is 85 of 142

100 August 211 assumed that the mean velocity gives a good indication of the expected morphodynamic behavior of the model. Since the non-hydrostatic approach fully resolves the high frequency waves, it is interesting to see whether this has any substantial effect on the mean velocities and thereby eventually the sediment transport. For the surf beat approach the Eulerian velocity is considered, since it contains the high frequency waves induced Stokes velocity. For the non-hydrostatic approach the high frequency waves are included in the (modified) NSWE, and therefore also in the Generalized Lagrangian Mean (GLM) velocity. Hence, the Eulerian velocity simulated with the surf beat approach is compared to the GLM velocity simulated with the non-hydrostatic approach, see Figure The figure shows that the effect of using the non-hydrostatic approach has a minor effect on the mean velocities. For the more dissipative cases a slight increase in the offshore directed mean velocities is observed, while the higher Iribarren numbers show a slight decrease in mean velocity. 86 of 142

101 August u mean [m/s] -.4 =.175 u mean [m/s] -.4 = x [m] x [m] u mean [m/s] -.4 =.437 u mean [m/s] -.4 = x [m] x [m] u mean [m/s] -.4 =.882 u mean [m/s] -.4 = x [m] x [m] Figure 5-14 Time averaged Eulerian velocity as a function of the cross-shore distance for the surf beat approach (blue) and the time averaged GLM velocity as a function of the cross-shore distance for the non-hydrostatic approach (red) for the different simulation cases. 5.4 Discussion Combined swash processes The effect of applying the combined swash processes in section was found to be rather small. The results show less erosion, but it was expected that the sediment transport could be 87 of 142

102 August 211 influenced until such an extent that the net transport direction is offshore. It is likely that for example a further increasing wave asymmetry and wave skewness calibration factor would affect the transport to a larger extend than observed here. The question then rises how large the effect of these parameterized processes may become to be still considered realistic Non-hydrostatic approach The non-hydrostatic approach is used to verify whether the model results improve in the swash zone for different beach state levels. It is, however, a recently developed modelling method, and has not been used much yet. The simulation results show that the process of wave breaking seems to be underpredicted, althoug this cannot be confirmed with data. The modelling approach is still in development and it is expected that the model will be further updated and extended in future. However, an important remaining limitation of the nonhydrostatic approach in XBeach compared to other non-hydrostatic models (e.g. SWASH) is the depth-averaged approximation in the model. The lacking of multiple vertical computional layers in the model could have a large effect at locations where complex processes (like wave breaking) occur. 5.5 Conclusions In this chapter a number of swash processes that are implemented in the XBeach model were discussed and their effect on the model results were tested. It was found that especially wave asymmetry, low frequency turbulence and groundwater flow have a great (onshore transport promoting) effect on the simulation results. High frequency turbulence does have a net offshore effect, especially with the bore averaged method, which is in contrast with literature. All swash processes have the largest effect on the higher Iribarren number beaches, which can eventually be related to the more high frequency wave energy present near the shore. The six simulation cases are also conducted with the non-hydrostatic approach to compare the effect of the fully resolving high frequency waves. The mean velocities are slightly affected by the non-hydrostatic approach, giving larger values for the lower Iribarren number cases and smaller values for the higher Iribarren number cases. It was, however, observed that the high frequency wave breaking within the non-hydrostatic approach is not captured well enough. It is hypothesized that this is due to the depth-averaged assumption in the XBeach model. 88 of 142

103 August Field case: Le Truc Vert 6.1 Introduction From Chapter 4 and 5 insight was obtained regarding the simulated swash characteristics for different beach state levels and the effect of including several swash processes on the simulated hydrodynamics and morphodynamics. In this chapter a field dataset is used to verify whether XBeach is able to accurately predict the hydrodynamics and morphodynamics observed during the experiment. The dataset used in the analysis is obtained during the spring of 28 at the beach of Le Truc Vert, France. The experiment was specifically aimed at obtaining measurements for the swash zone and was part of the larger ECORS project, where a large number of (international) universities and agencies took part in [Sénéchal et al., 28]. For a detailed description of the Truc Vert swash zone experiment, reference is made to Masselink et al. [29] and Blenkinsopp et al. [211]. The beach of Le Truc Vert is located on the south-west Atlantic (Gironde) coast, at about the same latitude as the city of Bordeaux, see Figure 6-1. The beach is subject to a semidiurnal tide with a mean range of 3.2 meter, increasing to 4.3 meter during spring tide. The wave climate is characterized by swell waves approaching from the Atlantic Ocean and locally generated wind waves with an average significant wave height of 1.3 meter and a significant period of 7.6 seconds. During storms waves can reach up to 7 meters with a 2 seconds period. The dominant wave direction during summer is West-northwest, while waves coming from the North-northwest are dominant during winter [De Melo Apoluceno et al., 22]. Figure 6-1 Location of Le Truc Vert Beach [Google Earth; Sénéchal et al., 28] 89 of 142

104 August Data description Le Truc Vert 8 swash experiment During the 17-day-long field experiment, that took place from March 19 th until April 4 th 28, a large swash zone dataset was obtained by measuring water levels, bed level elevations and flow velocities during several tidal cycles on a swash time scale. Instrumentation Detailed descriptions about the instrumentation used can be found in Turner et al. [28] and Blenkinsopp et al. [211]. Here, only the instrumentation relevant for the analysis is described. The data was collected with a number of sensors mounted on a large scaffold rig that extended over the entire swash zone during most of the high tides. These sensors measured either the bed level (when the location was dry) or the water level (when the location was wet). At four locations below the scaffold rig mini electromagnetic current meters (EMCMs) were installed to measure swash flow velocities. At approximately 2 m offshore of the scaffold rig an acoustic Doppler velocimeter (ADV) was installed together with a pressure transducer (PT) to measure respectively flow velocity and water level elevation in the (inner-) surf zone. All measurements were conducted with a sampling frequency of 4 Hz [Blenkinsopp et al., 211]. In Figure 6-2 the main rig and the ADV/PT in the (inner) surf zone can be seen. Figure 6-2 The measurement rig applied at Le Truc Vert, France in the spring of 28 [Blenkinsopp et al., 211]. The main rig contains a number of sensors that measure water levels (when the bed is wet) or bed levels (when the bed is dry). The flow velocity is measured with four mini electromagnetic current meters (EMCMs) located directly below the main rig. In the top mid of the picture the ADV/PT can be seen, that is used to obtain water level and velocity signal in the (inner) surf zone. 9 of 142