Modelling Sediment Transport in the Swash Zone

Modelling Sediment Transport in the Swash Zone

Title

Modelling Sediment Transport in the Swash Zone Client

Pages

Delft University of Technology

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

aug. 2011 A. A. van Rooijen

State

final

Initials Review

J.S.M. van Thiel de Vries

Initials Approval

T. Schilperoort

Initials

August 2011

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, 2011


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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

Modelling Sediment Transport in the Swash Zone / A. A. van Rooijen / August 19, 2011 This research was carried out at: Deltares Rotterdamseweg 185 2600 MH Delft The Netherlands and Rosenstiel School of Marine and Atmospheric Science – University of Miami 4600 Rickenbacker Causeway Miami, FL 33149 U.S.A. Arie Arnold van Rooijen © 2011. 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.

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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, 2009


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‘Modelling Sediment Transport in the Swash Zone’

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 [2007] 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.


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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.

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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.


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Contents Abstract Acknowledgements 1 Introduction 1.1 Background 1.2 Problem description 1.3 Research question 1.4 Objectives 1.5 Thesis outline

v vii 1 1 1 2 2 3

2 Swash zone physics 2.1 Introduction 2.2 Coastal terminology 2.3 Waves and wave-induced swash 2.4 Beach states and swash motions 2.5 Sediment transport 2.6 Hydrodynamics and morphodynamics in the swash zone 2.7 Morphological response of the swash zone 2.8 Numerical modelling and the swash zone 2.9 Conclusions

5 5 5 7 15 16 17 24 25 26

3 Methodology 3.1 Introduction 3.2 Modelling approach 3.3 Surf beat approach 3.4 Hydrostatic approach 3.5 Non-hydrostatic approach 3.6 Overview

29 29 29 31 37 41 44

4 Modelling basic swash processes 4.1 Introduction 4.2 Model setup 4.3 Simulation results 4.4 Discussion 4.5 Conclusions

45 45 47 50 68 70

5 Modelling additional swash processes 5.1 Introduction 5.2 Swash processes in the surf beat approach 5.3 The non-hydrostatic approach 5.4 Discussion 5.5 Conclusions

73 73 73 82 87 88

6 Field case: Le Truc Vert 6.1 Introduction 6.2 Data description

89 89 90


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6.3 6.4 6.5 6.6 6.7

Model setup Hydrodynamic results Morphodynamic results Discussion and model sensitivity Conclusions

97 103 111 120 127

7 Conclusions and recommendations 7.1 Conclusions 7.2 Recommendations

129 129 131

Bibliography

133

List of Abbreviations and Symbols

139

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1 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).


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Figure 1-1 An example of field measurements in the swash zone at Santa Rosa Island, USA [Houser & Barrett, 2009]

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.

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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.


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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.

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2 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, 2002; Bosboom & Stive, 2010]. 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, 2002]. The zones within the nearshore region most relevant for the present study will be discussed next.

2.2.1

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 (cg) decreases. Before breaking, the wave energy flux is conserved within propagating waves [e.g. Holthuijsen, 2007]: P Ecg constant [2.1] where E is the wave energy and cg is the wave (group) velocity. From the energy flux conservation balance it follows that the wave energy will increase for decreasing propagation Modelling Sediment Transport in the Swash Zone

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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.

2.2.2

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, 2006], 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, 2006]. 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

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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, 2006; Bakhtyar et al., 2009]. 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 0.1 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 2.3.3.

2.3.1

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 0.25 and 30 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.


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4

2

z s [m]

0

-2

-4

-6 High frequency wave motions Short wave group envelope -8

0

1000

2000

3000 x [m]

4000

5000

6000

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.

2.3.2

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 0.003 to 0.03 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 [1950] 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

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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 xmomentum 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:

S xx

ux ux

[2.2]

pwave dz

h

where is the water surface elevation, h is the water depth, is the water density, ux is the velocity in x-direction, pwave 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 xx

n

1

2

[2.3]

E nE

where E is the wave energy and the ratio between wave group velocity and individual wave celerity (n) is given by:

n

1 2kh 1 2 sinh 2kh

[2.4]

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:

Fx

dS xx dx

[2.5]

Finally, from the wave-induced force the resulting water level gradient can be computed :

Fx

dS xx dx

gh

d dx

g h0

d dx

[2.6]

where g is the gravitational acceleration, is the water level variation, h0 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 dx

3 dE 2 gh dx

[2.7]

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) 180 degrees out of phase with the bound long wave. Maximum shortwave values correspond with minimum bound long wave values and vice versa. In practice,


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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]

0

-2

-4

-6 High frequency wave motions Mean water surface elevation -8

0

1000

2000

3000 x [m]

4000

5000

6000

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., 2009].

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., 2007]). 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 10 of 142


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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 ky 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 2-4.

2.3.3

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., 2009]. 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 2-5.


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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., 2007]) and reflected low frequency waves in the swash zone [Short, 1999; Masselink & Puleo, 2006]. 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]

High freq. swash Subharm. swash Low freq. swash

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Frequency [Hz] 0.07 – 0.2 0.03 – 0.07 0.003 – 0.03

Period [s] 5 – 15 15 – 30 30 – 300


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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.


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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.

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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

2.4.1

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.

2.4.2

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:

HB wS T Modelling Sediment Transport in the Swash Zone

[2.9]

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where HB is the breaker wave height, wS 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 6.

2.4.3

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: 0

tan H 0 / L0

[2.10]

where is the beach slope, H0 is the deep water wave height and L0 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 0.5 or smaller) can be expected.

2.4.4

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, 2006; Miles et al., 2006]. 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.

2.5.1

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

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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.

2.5.2

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, 2010]. 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.

2.5.3

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., 2009]. Due to the large amount of formulations, only three formulations will be described in Chapter 3.

2.6

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., 2000]. 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, 2002; Masselink & Puleo, 2006].


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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., 2009], wave skewness and asymmetry [Grasso et al., 2011], turbulence due to wave breaking [Puleo et al., 2000; Longo et al., 2002], swash-swash interaction [Erikson et al., 2005; Blenkinsopp et al., 2011], boundary layer flow and shear stress [Barnes & Baldock, 2010] and infiltration/exfiltration [Li & Barry, 2000; Butt et al., 2001; Horn, 2006; Steenhauer, 2010]. In addition, there can be a longshore sediment transport component present in the swash zone [Elfrink & Baldock, 2002]. In the present study only the cross-shore processes, which are usually predominant, are considered and will be discussed.

2.6.1

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., 2009]. 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, 2007]. 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., 2011]. 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., 2002]. Another wave nonlinearity that plays a role in sediment transport is wave asymmetry [e.g. Grasso et al., 2011]. Wave asymmetry is the occurrence of waves in saw-tooth shapes with a steep wave front and a gentler wave back [Bosboom & Stive, 2010]. 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. [2011] 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.

2.6.2

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


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that is present within it [Masselink & Puleo, 2006]. Turbulence is the highest frequency motion in the swash zone, and can originate from generally three different sources [Aagaard & Hughes, 2006]: 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., 2004]. Puleo & Butt [2006] and Masselink & Puleo [2006] 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. [2000] 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., 2002]. It can, however, be stated that turbulence is responsible for lifting large volumes of sediment especially during uprush [Bakhtyar et al., 2009]. 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).

2.6.3

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., 2005]. 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. [2005] 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. [2011] concluded that swash-swash interaction induces larger transport rates, either onshore or offshore.


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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., 2005].

2.6.4

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. [2003] 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 [2006] 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.

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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.

2.6.5

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, 2002]. Groundwater flow below the beach surface has a significant influence on the swash zone morphology [e.g. Bakhtyar et al., 2009], and according to Li & Barry [2000] 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, 2002]. 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., 2001] found that infiltration has a stabilizing effect on the bed, while exfiltration destabilized the bed (see Figure 2-10 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., 2001]. 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 10 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


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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 2-10. 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 [2001] 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 (Vi) is more than two percent of the swash uprush volume (Vu). 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 (Vi > 0.02Vu) can therefore be translated into a critical grain size of D50=1.5 mm [Masselink & Li, 2001]. This threshold value indicates that the swash asymmetry effect of infiltration only takes place on gravel beaches with a D50 >1.5mm and not on sandy beaches where grain sizes are usually smaller than 1mm.

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Figure 2-10 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., 2001].

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. [2001] 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 0.45 and 0.58 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 (0.58 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 Effective weight effect Boundary layer thickness effect Swash flow asymmetry


Grain size [mm] 0.58 >1.5

Transport effect Offshore Onshore Onshore

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2.7

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]:

Z berm

0.125H b5 8 gT 2

38

[2.11]

where Hb 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, 2002], 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].

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Figure 2-11 Beach cusp on Praia do Moçambique, Florianópolis, Santa Catarina, Brazil (photo by A.A. van Rooijen, 2010)

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.

2.8.1

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., 2008].

2.8.2

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 Modelling Sediment Transport in the Swash Zone

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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, 2006]. 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., 2009]. 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., 2011]. 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., 2010].

2.8.3

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 [2008] and Karambas [2006].

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


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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.


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3 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

3.2.1

Modelling approach

XBeach The model used in this study is XBeach [Roelvink et al., 2009]. 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. [2009; 2010] and Smit et al. [2010].

3.2.2

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.


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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.

3.2.3

Sediment transport formulations In addition, two sediment transport models are used. The first one is the formulation by Van Rijn [1984; 1993; 2005; 2007] 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 [2007]. 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.

3.2.4

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 t

f mor qx 1 np x

[3.1]

0

where zb is the bed level, fmor is a morphological acceleration factor, np is the porosity and qx is the sediment transport rate, given by:

qx x, t

hCu E x

x

Ds h

C x

[3.2]

where h is the water depth, C is the concentration, uE is the (Eulerian) mean velocity and Ds is a sediment diffusion coefficient. Figure 3-1 gives an overview of the structure of the modelling part of this thesis.

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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.

3.3.1

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:

Aw t


cg Aw

Dw

[3.3]

x

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where x is the direction of wave propagation, cg is the short wave group propagation velocity, is the intrinsic wave frequency and Dw is the total wave energy dissipation. Wave action (Aw) is defined as the ratio of wave energy and the intrinsic wave frequency:

Ew x , t x, t

Aw x, t

[3.4]

where Ew is the wave energy density. The total wave energy dissipation (Dw) is given by [Roelvink, 1993]:

Dw Qb

Qb Ew , where H rms H max

1 exp

n

8 Ew and H max g

, H rms

tanh kh k

[3.5]

where is an empirical constant, Ew 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 (Dw) serves as source term:

Er t

cEr x

Dr

[3.6]

Dw

where Er is the energy of the roller and Dr is the roller energy dissipation, given by [Reniers et al., 2004]:

2g

Dr

r

Er

[3.7]

c

where c is the wave propagation velocity and wave.

r

is a factor related to the slope of a breaking

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:

uL t t

uL x

uL hu x

uL x2

sx

E bx

h

h

2 h

g

x

Fx h

L

[3.8]

0

where uL 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 Fx is the wave force induced by radiation stress. The velocity related to the Eulerian velocity is then given by: uE = uL – uS, where uS is the Stokes drift velocity, given by [Philips, 1977]:

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Ew 2 Er hc

uS

[3.9]

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 (Fx) in the NSWE. For a one-dimensional case the wave force is given by:

S xx , w

Fx x, t

S xx ,r

[3.10]

x

where Sxx,w is the wave-induced radiation stress and Sxx,r is the roller induced radiation stress given by:

S xx, w x, t

Ew 2

cg c

1 2

[3.11]

and

S xx, r x, t

[3.12]

Er

where cg is the short wave group propagation velocity. The wave energy (Ew) and roller energy (Er) are computed from respectively the wave action balance (equation [3.3]) and the roller energy balance (equation [3.6]).

3.3.2

Sediment transport In XBeach the sediment transport is predicted using a depth-averaged advection diffusion equation, given by [Galapatti, 1983]:

hC t

hCu E x

x

Ds h

C x

h Ceq C Ts

[3.13]

where C is the depth-averaged sediment concentration, Ds is a sediment diffusion coeficient, Ceq is the equilibrium sediment concentration, and Ts is the adaptation time, given by:

Ts

max 0.05

h , 0.2 ws

[3.14]

where ws 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 [2007] 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


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equilibrium sediment concentration for bed load (Ceq,bed) and suspended load transport (Ceq,sus) according to Van Rijn [2007] is given by:

D 0.015h 50 h

Ceq,bed Ceq , sus

0.012 D50 D*

1.2

[3.15]

M e1.5 0.6

[3.16]

M e2.4

where D* is the dimensionless particle size, given by:

D*

13

g

[3.17]

D50

2

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 (Me) in the Van Rijn formulas is given by:

max

uE

2

0.8urms ,2

Me

2

ucr , 0 [3.18]

gD50

where ue is the (wave-group averaged) velocity and urms,2 is the near-bed short wave orbital velocity including the contribution of the wave breaking induced turbulence (kb=kb,short+kb,long, for high and low frequency waves), given by [Reniers et al., 2004]:

urms,2

2 urms

[3.19]

k

turb b

turb is a turbulence coefficient estimated to be 1.45 by Van Thiel de Vries [2009], while the short wave orbital velocity is calculated by linear wave theory:

urms

H rms

[3.20]

2Tm sinh kh

and ucr 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 (ucr,c), based on Shields [1936], and a part for waves (ucr,w), based on Komar & Miller [1975]. The total critical velocity is then given by: [3.21] ucr ucr ,c 1 ucr , w where is a dimensionless factor ( ue / (ue + urms,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 (1b is a calibration factor, and m is the bed slope.

3.3.3

b m)

is applied, where

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

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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 [2009]. The latter is required for estimating the bore interval period (Tbore, 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 (uE) in the advection term of the advection diffusion equation [3.13] is replaced by: uAV = uE + uA, where the flow velocity related to wave nonlinearities is given by: uA [3.22] Sk Sk As As u rms where Sk and As are calibration factors. Van Thiel de Vries [2009] 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 (ubed), represented as the weighted sum of eight sine and cosine functions: i 8

ubed

wAi cos i t

[3.23]

(1 w) Ai sin i t

i 1

where w is a weighting function that affects the wave shape, Ai is the amplitude which is a function of the dimensionless wave height (Hrms/h) and the dimensionless period (Trep(g/h)0.5) and is the angular wave frequency. The skewness (Sk) factor is then given by: 3 ubed

Sk

[3.24]

3 ubed

while the asymmetry (As) factor is computed with the same expression replacing ubed by its Hilbert transform. The overbar represents the time mean value and represents the standard deviation of the near bed short wave velocity (ubed). Van Thiel de Vries [2009] found that the weighting (w) in [3.23] is a function of the phase , and found the following relation:

w 0.2719 ln

1.8642 0.2933

[3.25]

0.5

while the phase itself can be computed for a given skewness (Sk) and asymmetry (As):

tan

1

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]:


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0.79 cos 0.61 log U r 1 exp 0.35 0.79 sin As 0.61 log U r 1 exp 0.35

Sk

2

2

tanh 0.64 / U r 0.60 [3.27]

2

2

tanh 0.64 / U r

0.60

where Ur 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 (Dr):

Dr

ks

2/3

[3.28]

w

For the near-bed turbulence (kb,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 / Lmix)–1), where Lmix 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 [2009] 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 TRep/Tbore, where Tbore (=Hrms /(d /dxmaxc), where c is the wave celerity) is the bore interval period and Trep is the representive wave period. For the low frequency wave motions the local turbulence (klong) is calculated with a long wave turbulence balance, where turbulence is generated by wave bores at the water surface and dissipated at the bed:

klong

klong u L

t

x

ksource kdiss

[3.29]

where ksource (= g (c+u)) is the source turbulence and kdiss (= d(klong)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


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max

t

,0

t

t

[3.30] crit

To account for the vertical decay of the turbulence the same factor is applied as for the high frequency turbulence.

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 (kx) and the groundwater head gradient:

u gw

dpgw

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:

wex win

zb

gw

t kz

np [3.32]

dp 1 dz

where np is the porosity, kz is the vertical permeability, zb is the bed level and the groundwater surface level ( gw) is calculated with the ground water continuity equation:

d

gw

dt

du gw hugw dx

w np

[3.33]

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.

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


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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