Caractérisation des vagues côtières et littorales appliquée aux énergies marines renouvelables : focus sur le déferlement et la variabilité spatiale du champ de vagues

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le déferlement et la variabilité spatiale du champ de vagues

Audrey Varing

To cite this version:

Audrey Varing. Caractérisation des vagues côtières et littorales appliquée aux énergies marines re- nouvelables : focus sur le déferlement et la variabilité spatiale du champ de vagues. Océanographie.

Université de Bretagne occidentale - Brest, 2019. Français. �NNT : 2019BRES0105�. �tel-03004667�

T HESE DE DOCTORAT DE

L'UNIVERSITE

DE BRETAGNE OCCIDENTALE C OMUE U NIVERSITE B RETAGNE L OIRE

E COLE D OCTORALE N ° 598 Sciences de la Mer et du Littoral

Spécialité : Océanographie physique et Environnement

Wave characterization for coastal and nearshore marine renewable energy applications: focus on wave breaking and spatial variability of the wave field.

Thèse présentée et soutenue à Plouzané, le 19 décembre 2019.

Unité de recherche : France Energies Marines, Laboratoire Géosciences Océan (UMR 6538-IUEM-UBO) Par

Audrey VARING

Composition du Jury :

Président

Nadia SENECHAL Enseignant-chercheur

Université Bordeaux 1- UMR 5805 Phillipe BONNETON Directeur de recherche CNRS

Université Bordeaux 1- UMR 5805 Luc HAMM Directeur technique

Artelia Eau & Environnement René GARELLO Professeur

IMT Lab-STICC UMR 6285

Jochen HORTSMANN Head of Department Radar Hydrography Helmholtz-Zentrum Geesthacht

Directeur de thèse

Christophe DELACOURT Professeur UBO

Laboratoire Géosciences Océan UMR 6538 Encadrant scientifique

Jean-François FILIPOT Directeur scientifique France Energies Marines Invité

France FLOC’H Enseignant-chercheur UBO

Rapporteurs avant soutenance :

Phillipe BONNETON

Directeur de recherche CNRS Université Bordeaux 1 - UMR 5805

Luc HAMM

Directeur technique

Artelia Eau & Environnement

I want to express my gratitude to my scientific supervisor Jean-François Filipot.

Without his expertise, understanding, generous guidance and support, this thesis would not have been achieved.

I am thankful to my thesis supervisor Christophe Delacourt for giving the right advice at the right time.

The research presented in this thesis was made possible by the financial support of France Energies Marines and Région Bretagne, as part of a thesis scholarship within the DiMe project. In the context of this project, I have been gifted to work with a panel of researchers and this collaborative work has been an extremely rewarding experience.

I want to thank Stephan Grilli for sharing his knowledge and experience with his model during many occasions and especially during a stay at the University of Rhode Island.

Many thanks to Volker Roeber who guided me on the use of his model and other wave subjects. No one believes me when I explain I had a great working time during my stay at the University of Hawaii, but for sure, I will remember this experience for a long time.

I am very grateful to Matthias Delpey for the support provided during my thesis work. His help has been precious regarding the BOSZ model and wave discussions.

I would also like to thank Gilles Guitton and Dr. Fab from OceanDataLab for the valuable help regarding satellite data.

In the course of my three years as a PhD student at France Energies Marine, I have been fortunate to interact with a large number of people who at some point contributed to the work presented in this thesis : Marissa Yates-Michelin, Bertrand Michard, France Floc’h, Christophe Maisondieu, Fabien Leckler, Guillaume Dodet.

Special thanks to my colleagues at France Energies Marines. Without Rui, my

computer would have not survived these three years and at some point would have

gone out the window ! To my office mates and all the others at France Energies

Marines who have made the time here lovely.

Acknowledgements 3

1 General introduction 6

1.1 Context . . . . 6

1.2 Marine Renewable Energies technologies . . . . 8

1.2.1 Tidal energy . . . . 8

1.2.2 Stream or tidal current energy . . . . 9

1.2.3 Wave energy . . . 10

1.2.4 Offshore wind energy . . . 11

1.2.5 Ocean thermal energy . . . 12

1.2.6 Osmotic energy . . . 13

1.3 Wave processes in coastal and nearshore waters . . . 14

1.3.1 Ocean waves definition . . . 14

1.3.2 Coastal and nearshore wave processes . . . 15

1.4 Wave observation means . . . 24

1.4.1 In situ . . . 24

1.4.2 Remote sensing . . . 25

1.5 Wave modeling tools . . . 28

1.5.1 Phase-averaged models . . . 28

1.5.2 Phase-resolving models . . . 29

1.5.3 The BOSZ model . . . 33

1.6 Study objectives and content . . . 35

2 Numerical study of wave breaking initiation 36 2.1 Introduction . . . 36

2.1.1 Generalities about wave breaking . . . 36

2.1.2 Extreme and breaking waves effects on Marine Renewable En- ergy structures . . . 37

2.1.3 Wave breaking initiation criteria . . . 39

2.1.4 Numerical investigations of wave breaking initiation . . . 42

2.2.2 Definition of a new kinematic breaking criterion . . . 48

2.2.3 The FNPF model and some definitions . . . 49

2.2.4 Solitary wave propagating over a plane beach . . . 51

2.2.5 Solitary wave propagating over a submerged bar . . . 56

2.2.6 Periodic wave propagating over a submerged bar . . . 61

2.2.7 Discussion and conclusions . . . 64

2.3 Afterword . . . 66

2.3.1 Wave breaking signature in the water column . . . 66

2.3.2 How to improve the parameterization of wave breaking in Boussinesq-type models ? . . . 68

3 Wave characterization for MRE applications : focus on wave re- fraction and satellite observation mean 70 3.1 Introduction . . . 70

3.1.1 Wave energy resource assessment . . . 71

3.1.2 Need for refined wave characterization . . . 72

3.1.3 Wave characterization with a phase-resolving model . . . 73

3.2 Sea state spatial variability in coastal waters for Marine Renewable Energy applications: insights from numerical wave models and inno- vative satellite observations . . . 76

3.2.1 Introduction . . . 77

3.2.2 Models and observations . . . 81

3.2.3 Case study . . . 83

3.2.4 Methodologies for deriving wave amplitude map from SAR images . . . 91

3.2.5 Conclusion . . . 107

4 Conclusions and perspectives 110 A BOSZ model governing equations and numerical implementation 114 A.1 Governing Equations . . . 114

A.2 Wave breaking . . . 115 B Journées Hydrodynamiques 2018 - Conference paper 118 C FNPF model governing equations and numerical implementation 129 D Convergence of the FNPF model results as a function of the dis-

cretization 131

E EWTEC 2017 - Conference paper 135

Bibliography 144

1

GENERAL INTRODUCTION

1.1 Context

French ambition is to reach 32% of energy production based on renewable sources

by 2030. In order to contribute to this energetic mix, Marine Renewable Energy

(MRE) technologies need to increase their competitiveness in the near to medium-

term future. The MRE sector has a large potential, however, there are still some

technological obstacles regarding the resource characterization and the design of

MRE systems. An insufficient knowledge of the hostile environment where MRE

systems are deployed and more specifically in the case of extreme sea states and

intense wave breaking, leads to an over or under design of the systems and affects

the CAPEX (capital expenditure) and OPEX (operational expenditure) of the MRE

projects. When designing MRE structures for a specific site, the industry has to an-

alyze the Ultimate Limit State (ULS) of the structure, i.e. the maximum expected

response that the machine will experience over its lifetime (Veritas, 2014; IEC, 2009;

Veritas, 2008). In the assessment of the ULS, the effects of energetic steep or break- ing waves are thought to be responsible for “considerable uncertainties” (Chella et al., 2012).

The present thesis is part of the DIME project, led by France Energies Marines, that aims at improving the knowledge of storm wave conditions with a focus on breaking waves in the coastal and nearshore areas targeted for the deployment of some of the MRE systems (fixed and floating offshore wind turbine, wave energy converters and tidal turbines). The overall objective of this project is to contribute to the diminution of the Levelized Cost of Energy (LCOE) thanks to a diminution of the CAPEX cost trough improvement of MRE design but also the OPEX cost trough a better resource knowledge for monitoring and maintenance operations. It is also expected that the project contributes to the evolution of MRE design standards with the support of Bureau Veritas and in particular Veritas (2015a) and Veritas (2015b).

This project enables cooperation between a large consortium composed of France Energies Marines (leader), Shom, Rivages Pro Tech (SUEZ Eau France), Ifremer, ...

see the list of partners in Figure 1.1.

Figure 1.1: Logos of the DIME project consortium

1.2 Marine Renewable Energies technologies

All MRE systems face the same problematic, they have to deal with the harsh marine environment (storm, corrosion) and the impact cost it has on the achievement of MRE structures. Energy that can be harvested from the ocean mainly comes from six sources, described below.

1.2.1 Tidal energy

The tide is the variation in sea level, which results in intense currents at some

specific spots, due to the combined gravitational action of the moon and sun. One

important advantage of this energy source compared to the wind and wave energy

is its highly predictable character: tidal water levels and currents can be accurately

forecasted centuries ahead. There are two ways of exploiting tidal power: ’stream

energy’ (see after) and ’tidal power’. The latter exploits the potential energy created

by the cyclic rise and fall of sea level, that runs turbines enclosed in a tidal dam. The worldwide exploitable potential tidal energy has been estimated between 300 to 800 TWh/year (Mosseri and Jeandel, 2013). There is a limited number of places in the world where the local geography results in particularly large tidal ranges thank to resonance between the tidal wave and the coastal geometry. In France, this energy is quite famous thanks to the La Rance tidal power plant (Figure 1.2) which is one of the largest dam with a power capacity rated at 240MW.

Figure 1.2: La Rance tidal power station in France. ©Tswgb

1.2.2 Stream or tidal current energy

The tidal water level variations also generates kinetic energy trough tidal cur- rents. Since it is a predictable and intermittent source, this kinetic energy can be exploited by tidal stream turbines. Potential sites require specific conditions, e.g.

straits, capes, narrows, ..., where increased speeds are observed. For example, in

France, Alderney Race, Fromveur Strait, Raz de Sein are the most promising sites

in terms of tidal energy production (e.g. Guillou et al., 2016). The current from

river also presents an interest. A wide range of technological concepts is presently

being developed. Sabella D10 tidal stream turbine, deployed in the Fromveur Strait

in June 2015, is the first machine that produced electricity to the french electric grid

Figure 1.3: SABELLA D10 tidal stream turbine being installed close to Ushant Island in the Fromveur Strait, France. ©Sabella

and provided tidal energy electron to the Ushant island inhabitants (Figure 1.3).

1.2.3 Wave energy

The energy of the wind blowing over the water is transferred to the sea, giving rise to wind waves, then called swell waves once they have left their generation area.

The waves and especially swell can travel and carry kinetic and potential energies across oceans. This energy can be harnessed by Wave Energy Converters (WEC) either offshore, nearshore or onshore which is brought to shore via subsea cables when needed. Ocean wave energy productions has an estimated annual exploitable potential of 23000 TWh/year (Mosseri and Jeandel, 2013). Figure 1.4 illustrates the annual mean wave power density computed from a global wave model (Tolman et al., 2009).

This resource is irregular over time, varies as a function of the synoptic situa- tions (daily/weekly basis) and of course with the seasons (e.g. Cornett et al., 2008;

Reguero et al., 2015). In coastal areas, where most WEC should be deployed, the

wave resource may also be highly heterogeneous (in space) in presence of complex

sea bottoms. At this time, there is more than a hundred projects being developed

Figure 1.4: Annual mean wave power density (colour) and annual mean best direc- tion (arrows) computed by Gunn and Stock-Williams (2012) based on the NOAA WaveWatch III (WW3) global wave model (Tolman et al., 2009).

in the world. Only a limited number has reached pre-commercial stages (for ex- ample Pelamis machine in Figure 1.5). Babarit (2016) provides a comprehensive description of wave energy converters history and technologies. This MRE sector is in constant evolution and there is still a lot to do to produce reliable technology which will result in cost effective wave energy generation.

1.2.4 Offshore wind energy

Electricity is produced by turbines which harness energy from the wind blowing over stretches of sea and the electricity is carried to shore by undersea cables. The

"dry" part of the offshore wind turbine is very similar to those found onshore, which has speed up the industrial deployment of this device at sea. The most favorable areas are in shallow waters, typically less than 50m depth for fixed offshore wind turbines. Large wind turbines farm installed in northern Europe (North Sea, Baltic Sea) already provide electricity.

Floating wind turbines moored to the seabed are also being developed (e.g.

Figure 1.6), since they present fewer constraints in terms of depth limitations, in-

teraction with the socio-economics coastal actors and they further provide access to

Figure 1.5: Pelamis machine in Aguçadoura, Portugal, in 2014 before the financial collapse of the company. ©Pelamis

greater wind resources.

The French government recently announced the development of offshore wind energy projects with an objective of 1GW per year by 2024. Three tenders have been initiated. Already 8 commercial farms with fixed wind turbines (Fécamp, Courseulles-sur-Mer, Saint-Brieuc, Saint-Nazaire, Dieppe-Le Tréport, îles d’Yeu- Noirmoutier, Dunkerque and another site still under discussion, producing more than 3.5GW in total) and 4 pilot farms (Groix-Belle-Ile-en-mer, Leucate, Gruissan- Port La Nouvelle, Faraman) with floating wind turbines (3 to 4 wind turbines of about 6 MW each) are under development.

1.2.5 Ocean thermal energy

The theory behind ocean thermal energy conversion (OTEC) is to use a tem- perature difference between deep water (1000m, cold waters) and the surface (warm waters) to generate electricity. In tropical areas, this vertical gradient of temper- ature can reach 20° to 25° over typically 1000m. At the world scale this resource could potentially generates about 10000 TWh/year (Mosseri and Jeandel, 2013).

Many projects are currently being developed (Figure 1.7). Note that this type of

Figure 1.6: Floating wind turbine Floatgen installed at SEM-REV test site in France.

©Ideol

energy can be exploited for other purposes than the production of electricity (for ex- ample, cooling for direct air-conditioning, desalinization of seawater, aquaculture).

In France, the overseas departments La Réunion and La Martinique are potential areas for OTEC.

Figure 1.7: OTEC power plants being developed. (IEA-OES, 2018)

1.2.6 Osmotic energy

When a river flows into the sea, it releases a large amount of energy due to the

difference in salinity concentrations. A way of recovering this energy is based on

osmosis, through the harvest of the pressure gradient across a membrane separating two fluids with different salinity (Figure 1.8). This technology is the least advanced.

The technical-economic feasibility is currently being studied.

Figure 1.8: Operating scheme of an osmotic power plant. ©Statkraft Pure Energy

1.3 Wave processes in coastal and nearshore waters

1.3.1 Ocean waves definition

Ocean waves are wind-generated surface gravity waves with a characteristic length of the order of a few dozen to several hundred meters. The typical parameters used to characterise the wave dynamic are (Figure 1.9) :

- the sea surface elevation η,

- the height H from crest to through, - the wave amplitude a = H/2 ,

- the wave length L or the wave number k = 2π/L, - the period T ,

- the water depth d .

Figure 1.9: Illustration identifying the sea surface elevation η , the wave height H , the wave amplitude a , the wave length L and the water depth d .

1.3.2 Coastal and nearshore wave processes

In the context of our study, coastal waters are waters shallow enough to affect the waves while nearshore waters encompass the area where waves are breaking due to the interaction with the bottom. The coastal and nearshore zones are regions of complex coupled phenomena in the interplay of hydrodynamics, sediment transport and morphodynamics. Wave transformations in the coastal and nearshore regions due to bottom topography changes, result in strong variation in space of the wave height, length and direction, distortion of the wave profile and finally wave breaking that may drive intense currents. It is visually spectacular and a scientifically and technically concerning phenomena. They are of serious importance regarding coastal erosion, flooding risk, desing of coastal structures and as far as we are concerned wave energy resource and design conditions of MRE systems.

Linear and non-linear dynamical processes change the characteristics of the wave field (e.g. Figure 1.10): refraction, diffraction, reflection, shoaling, non-linear wave- wave interactions and wave breaking are typical manifestations of these transforma- tions (e.g. Massel, 1989; Hamm et al., 1993; Svendsen, 2006; Masselink et al., 2014).

These processes constantly interact and occur at different time and space scales.

Particular emphasis will be put on the wave breaking and refraction processes in

the chapters 2 and 3.

SHOALING WAVE BREAKING

DIFFRACTION

REFRACTION

REFLECTION

Figure 1.10: Identification of the main coastal and nearshore processes from a sky view of "Le bout du monde marseillais", Les Goudes, Marseille. Photo taken from an aircraft. ©J.F. Filipot

1.3.2.1 Shoaling

Wave shoaling is the variation of wave amplitude in their direction of propagation due to depth-induced changes of the group velocity in that direction. A harmonic wave, propagating over a fixed seabed topography with gentle slopes and no currents, retains its frequency, but, since the dispersion relationship remains valid,

ω

= gk tanh(kd) (1.1)

with g the gravitational constant and ω the angular frequency defined as ω = 2πf . The wave length will decrease, if the depth decreases and the phase speed will correspondingly decrease:

c = r g

k tanh(kd) (1.2)

Initially, the group velocity c

increases slightly, but then it also decreases.

c

= 1 2

1 + 2kd sinh(2kd)

r g

k tanh(kd) (1.3)

As the wave propagates into shallower water, the phase speed approaches the group velocity and the wave becomes less and less dispersive (phase speed becomes less dependent on frequency). Both the phase speed and the group velocity approach zero at the waterline. Such variations in the group velocity cause variations in local wave energy E and hence in amplitude a .

To illustrate this, let’s consider a monochromatic wave in a stationary wave field, no dissipation by breaking or bottom friction and neglecting non-linear energy transfer (Figure 1.11). Therefore, if we know the wave conditions in the deeper area (wave field index 1), we can determine its transformation to a shallower area (wave field index 2). Wave energy can only enter the volume G through plane 1 and leave through plane 2 at rate P

b and P

b per unit time, respectively, where P = Ec

is the energy transport per unit crest length and b is the distance between the two lateral sides of the volume G . Because of the conservation of energy, P

b = P

b :

P

b = P

b → [Ec

]

= [Ec

]

→ 1

2 ρga

c

= 1

2 ρga

c

(1.4) so that the amplitude in plane 2 can be obtained from the amplitude in plane 1 :

a

= r c

c

a

(1.5)

The effect of shoaling is initially to decrease but then to increase the amplitude, up to infinity at the coastline. This theory obviously breaks down long before that. In reality, other processes such as refraction and wave breaking or bottom friction limit the shoaling over an arbitrary seabed topography.

1.3.2.2 Refraction

A common observation at the seashore is that approaching waves have crest lines nearly parallel to the shore. This phenomenon is known as wave refraction.

Refraction is the turning of waves in their direction of propagation due to depth- or

current-induced changes of the phase speed in the lateral direction (i.e. along the

Figure 1.11: A wave approaching a straight coastline at normal incidence under stationary conditions. Under such conditions in the absence of wave generation or dissipation, the wave energy leaving volume G through plane 2 is equal to the wave energy entering volume G through plane 1. (Figure 7.2 of Holthuijsen (2007))

wave crest).

When a wave approaches a sloping beach obliquely, the crest line spans various depths, the part closer to shore travels over a shallow bottom while the outer portion travels over deeper water. The phase speed (dependant on the depth, see Equation (1.2)) along the crest thus varies and the crest in deeper water moves faster than it does in shallow water, making the wave bending and aligning the wave crest progressively with the shoreline.

For the specific case of a sea bottom with parallel isobath, the angle θ taken between the wave propagation direction and the normal to the depth contours, can be expressed following the Snel’s Law :

sin θ

c = constant (1.6)

The wave direction at any location in the shallow water (SW) can therefore be obtained from the deep water (DW) wave direction :

sin θ

= c

c

sin θ

(1.7)

Thus when a wave approaches the shoreline, where the phase speed c = 0, the wave direction will be θ = 0.

To determine the effect of refraction on the wave amplitude, let’s consider the situation in Figure 1.12 with oblique incidence. The energy between two rays is conserved in a stationary situation and in absence of dissipative processes, we have then P

b

= P

b

:

P

b

= P

b

→ [Ec

]

b

= [Ec

]

b

→ 1

2 ρga

c

= 1

2 ρga

c

b

b

(1.8) so :

a

= r c

c

r b

c

a

(1.9)

The wave amplitude grows when the two rays bend and approach each other.

Figure 1.12: Under stationary conditions, in the absence of wave generation or dissipation, the wave energy leaving volume G through plane 2 is equal to the wave energy entering volume G though plane 1. (Figure 7.8 of Holthuijsen (2007))

1.3.2.3 Diffraction

Diffraction is the turning of waves towards areas with lower amplitudes due

to amplitude changes along the crest. Diffraction is particularly strong along the

geometric shadow line of obstacles such as islands, headlands and breakwaters but

also above shoals.

For refraction, the linear theory was used, here in the presence of strong am- plitude gradients, this theory is not valid. If we consider these rapid variations of amplitude in the Laplace equation, the phase speed and group velocity become:

c

= c

(1 + δa)

(1.10)

c

= c

(1 + δa)

(1.11) where δa is the diffraction parameter that measures the rate of change of the am- plitude (curvature in space) :

δa = ∇

a

k

a (1.12)

1.3.2.4 Reflection

The coast to which the waves propagate will very probably reflect waves to some degree. A vertical cliff may virtually reflect 100 % of the incoming wave energy, whereas a gentle beach will barely reflect energy since most of the energy would be dissipated by wave breaking before reaching the shoreline, see for example Figure 1.10.

1.3.2.5 Dissipation by bottom friction

When the water depth becomes less than one half of the wave length, the wave

orbital motions extend to the bottom resulting in energy dissipation via various

linear or non-linear processes. Shemdin et al. (1978) identified and examined four

dissipation processes: wave orbital velocity friction on the bed, percolation of the

water particles through the bottom substrate, wave-induced motion of a soft muddy

bottom, and scattering of surface waves by bottom irregularities. Friction dominates

in sandy and hard substrate conditions commonly found along the world’s coast-

lines. Several expressions are available for cases with or without mean currents (e.g.

Hasselmann and Collins, 1968; Madsen et al., 1988; Tolman, 1992).

1.3.2.6 Non-linear triad interactions

Observations of shoaling wave fields indicate distortions of the wave profile, that precede wave breaking. They are typical manifestation of the non-linear effects in the nearshore region. The triad-interactions dominate the non-linear energy transfer among frequency components in coastal waters with kd < 1 . During shoaling, waves evolve from a slightly peaked, nearly sinusoidal shape in deep water to a shape char- acterized by a steeper crest and flatter trough. The wave looses its quasi-symmetry due to energy transfers from the spectral peak to super-harmonics frequency com- ponents, whose phase relation is no longer random (Elgar and Guza, 1985).

Figure 1.13: Breaking wave at Deolen, France. ©Maelle Nexer.

1.3.2.7 Depth-induced wave breaking

During shoaling, due to non-linear interactions, the waves steepen and ultimately becomes unstable and breaking occurs. Wave breaking is generally the most impor- tant sink of energy for wind-generated waves.

Depth-induced wave breaking often drives the hydrodynamics in the nearshore

zone and more specifically in the surf zone. It is the most non-linear process af-

fecting waves in nearshore waters. It controls wave setup (e.g. Longuet-Higgins and

Stewart, 1964), drives long-shore currents, rip-currents and undertow (e.g. Longuet-

Higgins and Stewart, 1964; Longuet-Higgins, 1970a,b; Svendsen, 1984; MacMahan et al., 2006) and is involved in the generation (or release) of infragravity waves (e.g.

Symonds et al., 1982; Battjes et al., 2004; De Bakker et al., 2014). Its knowledge is also essential for various scientific and engineering applications, such as sediment transport, impact on infrastructure, and global aerosol mass transfer.

A continuum of breaker shapes occurs in nature. Thought, three main breaker types are commonly recognized (Figure 1.14). Spilling breakers are associated with gentle beach gradients and steep incident waves (large wave height relative to the wave length). The wave shape as a whole remains initially more or less symmetric.

A gradual peaking of the wave is observed until the crest becomes unstable, resulting in a gentle forward spilling of the crest with the appearance of aerated water near the top of the wave, which subsequently spreads down over the front face. Surging breakers are found on steep beaches with low steepness waves. The front face and crest of the wave remain relatively smooth and the wave slides directly up the beach without breaking. In surging breakers, a large portion of the incident wave energy is reflected at the beach. Plunging breakers tend to occur on steeper beaches than spilling breakers, with waves of intermediate steepness. They are characterized by a steepening and overturning of the front face and the formation of a jet that plunges into the water ahead of the wave.

Dimensionless empirical parameters have been developed to predict the breaker types. For example, Galvin (1968) classified breaker types on plane impermeable beaches using the surf similarity parameter ξ

:

ξ

= tanβ q

L0

(1.13)

where H

is the deep-water wave height, L

= gT

/2 is the deep-water wave length, g is the gravitational acceleration, T the wave period and tan β the beach slope.

Breaker types are also commonly classified in terms of the breaking wave height

H

using the parameter called Iribarren number after Iribarren and Nogales (1949)

ξ

:

ξ

= tanβ q

L0

(1.14)

According to Battjes (1974), for ξ

< 0.4, breaking wave are generally of spilling type while ξ

> 2, corresponds to surging breaking. When ξ

= 0.4 − 2, plunging breakers prevail.

Figure 1.14: Three types of breaking waves include (A) plunging breakers, (B) spilling breakers, and (C) surging breakers. ©Byron Inouye

Significant efforts have been made the last decades to obtain a better under-

standing of the wave dynamics in the coastal and nearshore areas for various coastal

activities including the deployment of MRE converters. The following sections de-

scribe the means that are used to measure and model these waves.

1.4 Wave observation means

A range of different methods are available for measuring waves, including buoys, acoustic dopplers, satellites and radars.

1.4.1 In situ

In situ instruments may be located at the sea surface (e.g. a floating surface buoy), or below the sea surface (e.g. a pressure sensor mounted on a frame at the sea bottom). Most of these instruments are used to acquire time records of the up-and-down motion of the surface at one location.

Shallow-water wave-measuring devices include pressure sensors, which estimate the sea surface elevation from measurements of the fluid pressure near the sea bot- tom. The reconstruction of the wave field from bottom pressure measurements is a difficult mathematical problem. The wave field is commonly reconstructed by means of a transfer function based on linear wave theory (e.g. Guza and Thornton, 1980; Bishop and Donelan, 1987). However, in very shallow water, the non-linearity effects increase. The linear transfer function is no longer suitable and a non-linear reconstruction method is required (Bonneton and Lannes, 2017).

Most wave measurements in deep and intermediate water depths are made by

moored buoys. The buoy closely follow the motion of the water particles by float-

ing at the sea surface. Heave-pitch-roll buoys contain accelerometers that measure

the acceleration of the water in three directions. The measured accelerations are

converted into estimates of the wave energy spectrum or can be integrated twice

to obtain the sea-surface elevation as a function of time. Depending on the type of

instrument and on the presence of currents, the horizontal position of the wave buoy

is not fixed but nearly follows the wave orbital motion. This later property may be

a problem as it partly filters out the free surface non-linearity. First wave buoy

prototypes were developed by Longuet-Higgins et al. (1963); Cartwright and Smith

(1964). More recently, Brown (2014) developed a wave measurement buoy aimed at

capturing the characteristic signature of a breaking wave impact (Figure 1.15).

Figure 1.15: A sequence of screen captures from the video of an incipient wave breaking. The wave buoy is shown: 1) floating in the preceding wave trough, 2) rising up the front face of the wave, 3) just prior to cresting the wave, 4) passing over the wave crest, 5) diving into the rear face of the wave, and 6) regaining equilibrium after the wave has passed. Figure 5.8 of (Brown, 2014)

The above pressure sensors and wave buoys are the most popular instruments used to observe waves. Other in situ techniques exist such as wave gauges (widely used in the laboratory), PUV sensors or current profilers (ADCP) (give access to the frequency-direction wave energy spectrum).

In situ observations have limitations regarding the instrument itself (e.g. a buoy floating at the sea surface may swerve around or capsize in a very steep wave) and they are sensitive to the aggressive marine environment (e.g. mechanical impacts, marine fouling and corrosion, storm waves). Also, in situ sensors only measure wave conditions at a single location. Accurate wave measurements over very large areas can only be performed by complex and expensive arrays of synchronized sensors or remote sensing.

1.4.2 Remote sensing

Instruments that are mounted above the water surface on a fixed or moving plat-

form are called remote sensing instruments. The platform may be an observation

Figure 1.16: SAR roughness image of Mer d’Iroise, France on January 22, 2019 06:31:31 (UTC) from the Ocean Virtual Laboratory https://ovl.oceandatalab.com/

tower at sea, a ship, an airplane or a satellite. Remote sensing are generally not sensitive to the marine environment but they may be sensitive to the atmospheric environment (e.g. rain, clouds, water droplet, condensation). Radar is one of the most useful tools for obtaining ocean information using the technologies of remote sensing. The most widely accepted ocean sensing radars include, but are not lim- ited to, the high-frequency surface wave and sky wave radars, microwave nautical radar, and laser radar (LIDAR). These “ocean radars” are able to provide sea sur- face information such as wind, wave, current, hard target, and bathymetry with different spatial and temporal resolutions. The most important operational differ- ence from in situ techniques is that large areas can be covered, more particularly with satellites. Another advantages of the remote sensing techniques, in the case of satellite mounted instruments, is that they record information even in the most intense storms (if the instrument is not sensitive to rain or clouds of course).

Remote sensing of the ocean surface by satellites has been possible for about

30 years. Two types of instruments measure ocean waves. Altimeters determine the

distance between the satellite and the ocean surface by measuring the time required

for a radar pulse to bounce off the sea surface and return to the satellite. The time difference between the pulses that reflect from wave crests and those that reflect from wave troughs provides a rough measure of total wave energy. This measure is usually converted to significant wave height (e.g. Fedor and Brown, 1982; Young, 1994; Queffeulou, 2004). The second type of satellite instrument used for measuring ocean waves is the Synthetic Aperture Radar (SAR). SAR systems have a side- looking imaging geometry and are based on a pulsed radar installed on a platform with a forward movement. The radar system transmits electromagnetic pulses with high power and receives the echoes of the backscattered signal in a sequential way. It provides high-resolution two dimensional images independent from daylight, cloud coverage and weather conditions (e.g. Tomiyasu, 1978; Curlander and McDonough, 1991; Oliver and Quegan, 2004; Moreira et al., 2013). Extraction of wave height, wind speed and wave period from the satellite altimeters and directional wave spectra from the SAR are reviewed in Krogstad and Barstow (1999). Satellite data are used in numerical wave prediction programs, either for model tuning and validation or for direct assimilation (e.g. Bauer et al., 1996; Breivik and Reistad, 1994; Chu et al., 2004; Aouf et al., 2006).

Many efforts have been made to develop stereo-video measurements and demon-

strate their capabilities to measure time-space wave properties accurately (e.g. Bene-

tazzo, 2006; Ardhuin et al., 2010; Gallego et al., 2011; De Vries et al., 2011; Schwen-

deman and Thomson, 2017). Stereo-video systems reconstruct the evolution in time

three-dimensional water surface waves. Recently, Filipot et al. (2019) presented

waves data set recorded from stereo-video during storm conditions in the vicinity of

La Jument lighthouse (Figure 1.17) and demonstrated the ability of this system to

capture valuable information of the wave properties in storm conditions.

Figure 1.17: Examples of stereo video three-dimensional surface elevation map z(iR, jR, t) plot over its corresponding right camera bitmap image in the pixel reference system. The colorbar shows the sea surface elevation and ranges from -15 m (blue) to +15 m (red) above the mean sea level computed at the time of the record. The sequence of images shows an extreme breaking event (corresponding wave height:

19.0 m). (Figure 6 of Filipot et al. (2019))

1.5 Wave modeling tools

A wide range of numerical models is available for simulating surface wave pro- cesses, based on different physical formulations, assumptions, and numerical frame- works. For an extensive state of the art regarding wave modeling see WISE Group (2007). Wave models can be divided into two major categories: phase-resolving models and phase-averaged spectral models.

1.5.1 Phase-averaged models

Phase-averaged spectral wave models are widely used around the world for nu-

merous applications from operational wave predictions at global and regional scales

to site-specific coastal engineering applications. These models provide a phase-

averaged description of the generation and propagation of ocean waves subjected

to wind forcing. They compute the evolution of the wave spectrum in space and time. With a discretization of the surface elevation spectrum E over frequencies f and directions θ, the spectral models solve the wave energy balance equation (Gelci et al., 1957):

dE(f, θ)

dt = S (1.15)

where d/dt denotes material derivative and S the parameterization of the wind gen- eration, dissipation by breaking, bottom friction, and quadruplet and triplet non- linear wave interactions (e.g. Tolman, 2008; Komen et al., 1996; Zijlema, 2010). The oceanographic community has developed many spectral wave models, and the most recent are the third-generation version of these models. The most popular third gen- eration spectral wave models are WAM (WAMDI Group, 1988), WAVEWATCHIII (Tolman, 1991), SWAN (Booij et al., 1999), TOMAWAC (Benoit et al., 1997). They have mostly been designed for the open ocean, but efforts have been made to ex- tend their capabilities and implementations to shallow water (Eldeberky, 1996; Ris et al., 1999; Filipot and Ardhuin, 2012). The SWAN model (Booij et al., 1999) was especially designed to account for the more complex physical processes that take place in intermediate and shallow waters. These models still present limitations to accurately recreate extreme wave events (generally underestimation) (Stopa and Cheung, 2014; Rusu et al., 2017).

1.5.2 Phase-resolving models

The alternative approach of phase-resolving models for shallow water waves are

usually based on Boussinesq-type (BT) equations (e.g. Peregrine, 1967; Freilich and

Guza, 1984; Madsen and Sørensen, 1992; Liu and Losada, 2002), or on the mild-slope

equation (e.g. Berkhoff, 1972). These models reconstruct the sea surface elevation in

space and time and account for the horizontal and vertical flow velocity. Processes

such as refraction, diffraction, and, in some models, also triad and quadruplet wave-

wave interactions are intrinsically represented. Dissipation processes such as bottom

friction and depth-induced wave breaking can be parameterized. Wave generation

Figure 1.18: Map of the significant wave height along with the wave spectrum at Pierre Noire wave buoy on January 22, 2019 10:00:00 (UTC+2) provided by WW3 model from MARC project, http://marc.ifremer.fr/

by wind is not accounted for in these models. The wave propagation must be computed on a grid with a resolution finer than the wave length which makes this type of models computationally expensive and limited to small domain ( O (1km)) and short duration wave events ( O (1h)).

Nonetheless, thanks to their ability in capturing key coastal and nearshore wave

processes, numerical models based on BT equations have become an indispensable

tool in coastal engineering (Rusu and Soares, 2012a). The standard BT equations for

variable water depths were first derived by Peregrine (1967), but they were limited to

weakly non-linear and weakly dispersive shallow water flows. Madsen et al. (1991)

and Nwogu (1993) extended these standard BT equations to obtain a practical

tool to simulate the non-linear transformation of irregular, multi-directional waves

in water of varying depth prior to wave breaking. The BT equations of Nwogu

(1993) consist in a continuity equation and two momentum equations in the x and

y directions as :

η

+ [(d + η) u]

+ [(d + η) v]

+

z

2 − d

6

d (u

+ v

) +

z

+ d 2

d ((hu)

+ (hv)

)

x

+ z

2 − d

6

d (v

+ u

) +

z

+ d 2

d ((hv)

+ (hu)

)

y

= 0 (1.16)

u

+ uu

+ vu

+ gη

+ z

2 (u

+ v

) + z

[(hu

)

+ (hv

)

] + τ

= 0 (1.17) v

+ vv

+ uv

+ gη

+ z

2 (v

+ u

) + z

[(hv

)

+ (hu

)

] + τ

= 0 (1.18) where the subscripts (x, y) and t denote partial derivatives with respect to space and time, g is gravitational acceleration, and (τ

, τ

) is the bottom shear stress. The reference depth z

is the vertical position within the still water column, d , at which the velocities (u, v) are evaluated. It is just below mid depth and varies linearly with the bathymetry. The governing equations exhibit good dispersion accuracy up to kd ≈ π ; for kd > π ( k = 2π/L ) the error increases positively, i.e. the wave length is then overestimated. The reference depth, z

, has a significant impact on the properties of the governing equations. So far, most studies have used values around z

= − 0.53d as initially proposed by Nwogu (1993).

For a comprehensive review of BT models see Brocchini (2013).

The most advanced BT models can thus be applied from deep to shallow waters and also to highly non-linear waves. Therefore, they are well suited for studying wave propagation.

One shortcoming of the traditional BT models lies in the treatment of wave breaking.

BT equations are unable to describe the overturning of a breaking wave and the inherent turbulence generation. BT models require two mechanisms to simulate the breaking process numerically. The first one is an initiation (and termination) mechanism and the second one is an energy dissipation mechanism. Therefore researchers have developed semi-empirical approaches.

Among the dissipation formulations, there are the surface roller model (e.g. Svendsen, 1984; Sørensen et al., 1998; Schäffer et al., 1993; Madsen et al., 1997a,b), the eddy viscosity model (Zelt, 1991; Karambas and Koutitas, 1992; Wei et al., 1995; Kennedy et al., 2000;

Chen et al., 2000; Roeber et al., 2010, e.g.), the vorticity model Svendsen et al. (1996);

Veeramony and Svendsen (1998) and the deactivation of the dispersive terms in the region

where breaking occurs (e.g. Tonelli and Petti, 2009; Roeber and Cheung, 2012; Tissier et al., 2012; Bonneton et al., 2011a,b). These dissipation mechanisms are locally activated in the surf zone when an initiation criterion is triggered.

The wave breaking formulation count different type of breaking criteria. Geometric criteria are principally based on the wave steepness ( H/L or ka ), the relative wave height ( H/d ), the angle of the wave front θ (Figure 1.19 for illustration). The wave steepness threshold is typically ka = 0.44 (Stokes, 1880; Miche, 1944b) in deep water. The relative wave height (e.g. McCowan, 1894; Tonelli and Petti, 2009) is used for example in the BT model FUNWAVE (Shi et al., 2011), when H/d > 0.8 breaking occurs. The angle of the wave front has been widely used in BT models (e.g. Schäffer et al., 1993; Madsen et al., 1997a; Sørensen et al., 1998; Cienfuegos et al., 2010; Tissier et al., 2012). With this parameter a surface roller is defined and two parameters control the initiation and termination of breaking (Figure 1.19). Breaking starts when the front slope of the wave is above the angle φ

, breaking ends when the angle is below φ

. Schäffer et al. (1993) defined that the critical parameters (φ

, φ

) = (20 ° , 10 ° ) provide a good agreement when compared with observations of a regular wave train above a beach of slope 1:40 and an irregular sea state propagating on a barred beach. Madsen et al. (1997a) gives another set of parameters, (φ

, φ

) = (14 ° , 7 ° ) for a spilling wave breaking over a barred beach.

Sørensen et al. (1998) improved the precision of dispersion in their BT model and adjusted the breaking parameters as (φ

, φ

) = (32 ° , 10 ° ) . These thresholds have to be calibrated depending on the bathymetry and the breaker type. The breaking criteria of Kennedy et al. (2000); Utku (1999); D’Alessandro and Tomasicchio (2008) used in BT models can be classified as kinematic criteria. Kennedy et al. (2000) defines a criterion based on the normal speed of the free surface elevation, Utku (1999) uses a relative Froude number and D’Alessandro and Tomasicchio (2008) combines the two previous criteria. These criteria have been used in other studies (e.g. Chen et al., 2000; Okamoto and Basco, 2006) that provide different threshold values of the calibration parameters. Kazolea et al. (2014) combines a geometric and a kinematic criteria. These criteria require calibration and are not universally applicable.

In this thesis (Chapter 3), the BT model BOSZ (Boussinesq Ocean & Surf Zone model

of Roeber (2010)) will be used, a short description is therefore provided in the next section.

Figure 1.19: Relation between the angle of the wave front and the critical conditions for a breaking wave according to Schäffer et al. (1993). (Figure 1 of Okamoto and Basco (2006)).

1.5.3 The BOSZ model

The Boussinesq Ocean & Surf Zone model, BOSZ (Roeber, 2010), is a computational tool to solve the dispersive and hydrostatic water wave and flow problems. The governing equations are based of a modified conserved-variable formulation of Nwogu’s BT equations.

From a numerical point of view, the model uses a finite volume scheme on a cartesian grid.

A Godunov-type scheme integrates the evolution variables in time. BOSZ equations retain

terms of the order of O(ε, µ

), where ε and µ define the non-linearity and the frequency

dispersion parameters respectively. BOSZ model uses the reference depth z

= − 0.55d

that improves the nonlinear energy transfer (shoaling properties) of high frequency spectral

components over irregular bathymetry (Simarro et al., 2013). BOSZ considers an approach

to deactivate the dispersion terms to allow the Riemann solver to describe the breaking

wave as a bore or hydraulic jump. The BT equations locally and momentarily reduce

to the non-linear shallow-water equations for shock-related hydraulic processes, while the

solution in the rest of the domain remains dispersive. Since the flow is flux dominated near

the wave breaker, local deactivation of the dispersion terms does not introduce an abrupt

change to the overall flow characteristics. BOSZ can account for time- and space-varying

wave spectra as well as for changing tide and storm surge levels along the boundaries. An

efficient parallelization scheme makes the model applicable to large computational domains

on the order of millions of cells. The governing equations and wave breaking formulation

are reminded in appendix A and details of the numerical formulation can be found in Roeber (2010), Roeber et al. (2010) and Roeber and Cheung (2012).

BOSZ was originally applied to energetic waves in a fringing reef environment. Roeber and Cheung (2012) provides comprehensive validation of BOSZ with laboratory and field data corresponding to both continental and tropical island conditions. The BOSZ model has been applied to continental shelf conditions in Brittany (France) in Filipot et al. (2013) and Varing et al. (2017) (e.g. Figure 1.20). BOSZ has proven to be a stable and accurate model for irregular bathymetry locations and extreme wave conditions (Li et al., 2014).

The model is currently used for scientific and teaching purposes in various universities across Europe, Japan, and the United States.

Figure 1.20: Snapshot of the sea surface elevation from BOSZ model close to Esqui-

bien, France on December 12, 2013 13:00 (local time). The bottom panel presents

the sea surface elevation corresponding to the black square in the top panel, around

a breakwater.

1.6 Study objectives and content

The aim of the study presented in this thesis is twofold.

First, despite a great number of studies, fundamental questions about wave breaking remains unsolved, especially about what determines whether, when and where a wave will break. Although one of the initial objective was to provide an improved wave breaking parameterization for phase-resolving wave models, we felt and addressed a need for a better understanding of the wave breaking process initiation. This work is described in Chapter 2.

Second, the knowledge of the spatial variability of the wave field in coastal areas is mostly obtained from numerical wave models. The accuracy of the models are typically tested against cross-shore arrays of sensors while the longshore wave field variability is often disregarded. In the study presented in Chapter 3, we show that different numerical modeling approaches may lead to significantly different spatial wave height and power patterns. This drove us to define a methodology to extract wave information from satellite SAR images for comparison with wave models output.

Conclusions and perspectives are finally exposed in Chapter 4.

2

NUMERICAL STUDY OF WAVE BREAKING INITIATION

2.1 Introduction

As mentioned in the general introduction wave breaking needs to be taken into con- sideration for the design of MRE structures and for accurate wave modeling, however our current understanding of why, how and when it happens is still limited. This first study aims at providing a better understanding of the wave breaking initiation process. In the course of this work, we propose a new definition of the kinematic wave breaking criterion, that is found to be more accurate than the conventional one.

2.1.1 Generalities about wave breaking

Over the years, a large number of articles have been published dealing with wave

breaking (e.g. Galvin, 1968; Peregrine, 1983; Bonmarin, 1989; Banner and Peregrine, 1993;

Melville, 1996; Grilli et al., 1997; Babanin et al., 2001; Bonneton, 2007; Babanin et al., 2011; Perlin et al., 2013; Sutherland and Melville, 2015; Lubin and Chanson, 2017). Wave breaking usually happens on a very fast time scale, lasting only a fraction of a wave period (e.g. Bonmarin, 1989; Rapp and Melville, 1990; Barthelemy et al., 2018).

Wave breaking research has been split between the deep and shallow water environ- ments. In the present work, focus is given to coastal and nearshore environments, although deep water studies will be mentioned since wave breaking in deep and shallow water shares many similarities. Waves probably break due to group modulations in deep and interme- diate water (e.g. Song and Banner, 2002; Banner et al., 2000; Babanin et al., 2001; Banner et al., 2002), bottom induced shoaling in shallow water (e.g. Thornton and Guza, 1983;

Kirby and Derakhti, 2019) or wave current interaction (that may occur in any water depth) (e.g. Chawla and Kirby, 2002; Romero et al., 2017).

A comprehensive understanding of the wave breaking process and its many causes and consequences is out of reach at this point. One of the open issues surrounding wave breaking onset is its definition. Many wave researchers would probably agree with the following statement by the authors of Banner and Peregrine (1993): "It is generally recognized that an individual wave breaking event usually starts when water particles near a wave crest develop a velocity in the wave propagation direction sufficiently large for them to fall down the front of the wave". The breaking wave community also use other definitions of wave breaking initiation, including: when the wave front slope reach a certain limit, when overturning starts, at the genesis of turbulence, when foams appear. A definition that would most likely reconciliate everybody, is the one given by Mark Buckley at the B’Waves workshop (Marseille, 2018) : "In the end, breaking starts when something happens".

2.1.2 Extreme and breaking waves effects on Marine Renew- able Energy structures

Extreme wave conditions and breaking waves are a major cause of concern for MRE

developers. The knowledge of the extreme conditions through the 50 or 100-year return

periods wave is a very necessary aspect to determine the suitability of a prospective loca-

tion for the deployment of MRE devices. So far, statistical methods are used to estimate

the extreme wave heights such as a Peaks-Over-Threshold approach associated to a distri-

bution ( e.g. Generalized Pareto Distribution, Poisson distribution) (e.g. Coles et al., 2001;

Hamm et al., 2010; Mazas and Hamm, 2011; Agarwal et al., 2013; Viselli et al., 2015). MRE engineers indeed require accurate sea state characterization to model the loading and dy- namic response of the structures in non-linear waves. In coastal and nearshore waters (and possibly in intermediate and deep water as well), these 50 or 100-year return periods are likely to break due to their interaction with the bottom. Such extreme waves under breaking conditions and their associated impact loads are challenging to model because the breaking process is a strongly non-linear phenomenon.

In this regards, there has been considerable research on wave loading of offshore oil and

gas structures but this has been for relatively deep water (e.g. Tickell, 1977; Wirsching,

1984; Chaudhury and Dover, 1985; Bea et al., 1999; Tromans et al., 1991). The predic-

tion of hydrodynamic loads on structures in the ocean are dependent on the velocity and

acceleration of the fluid particles, and in breaking waves these are far greater than those

found in non-breaking waves of the same height (Melville and Rapp, 1988). The tradi-

tional engineering methods to estimate the orbital velocities under steep or breaking wave

of a marine structures generally relies on Airy linear theory (Airy, 1841) or on the Stream

Function theory (Dean, 1965), the latter being valid for regular irrotational waves over a

flat bottom. Design criteria need to be adapted for coastal and nearshore areas where MRE

systems - such as fixed or floating wind turbines or wave energy converters - are most likely

to be deployed. Luck and Benoit (2005) found that breaking wave forces on a monopile

could be up to three times the forces due to non-breaking waves, in intermediate water

depth. Another interesting question concerns the stages of breaking where the maximum

loads occurs. Experimental (e.g. Oumeraci et al., 1993; Wienke and Oumeraci, 2005) and

numerical work (e.g. Hildebrandt and Schlurmann, 2012) showed that the impact load was

very sensitive to wave shape and reported the maximum force on the cylinder at the point

of overturning. Chella et al. (2019) recently proposed a numerical investigation of wave

impact pressures and kinematics where the wave shape is a required information. Informa-

tion regarding individual waves shape and orbital fluid velocities distributions are needed

for the purpose of engineering design or device tuning, for any MRE systems submitted to

waves, either fixed or floating or even bottom fixed, where waves can be felt in the water

column.

Observations of wave loads on marine structures is complex, expensive, with high risk of instruments loss. With the objective of designing an experimental protocol to be deployed on offshore wind turbine, Filipot et al. (2019) presents an ambitious field experiments from an offshore lighthouse (Figure 2.1) and collects valuable observations on synchronized stereo-video reconstructed storm waves and the induced structural response (see also Figure 1.17).

Figure 2.1: Photos of an extreme wave breaking against the Jument lighthouse in Finistère, France. To appreciate the scale of the incoming breaking wave, the lighthouse and its substructure are respectively 48m and 14m. ©Air, Vide et Eau.

2.1.3 Wave breaking initiation criteria

As part of the project DIME, an objective of this thesis is to improve the parameter- ization of wave breaking in the Boussinesq-type (BT) model BOSZ described in section 1.5.3. BT models assume a single-valued free surface and can not reproduces the crest overturning inherent to breaking waves, therefore they can not reproduce breaking waves.

As already mentioned in the general introduction, in such models, two steps are required

to model wave breaking, an initiation formulation and a dissipation formulation. The

present study focuses exclusively on the initiation of wave breaking. There exists a variety

of wave breaking initiation mechanisms dedicated to BT models (see section 1.5.2) but

none of them seems universally applicable. To improve wave breaking parameterization in

the model BOSZ, we decided to increase our understanding of the wave breaking initiation

process.

Many breaking onset criteria have been proposed for a variety of practical applications based on theoretical studies, numerical simulations, laboratory experiments, or field obser- vations (e.g. Tian et al., 2010; Kazolea et al., 2014). Wave breaking criteria are typically classified into three categories: geometric, kinematic, and dynamic. An extensive review regarding wave breaking criteria is given in Perlin et al. (2013). In the following, emphasis is given to the analysis of the kinematic criterion. The other types of criteria are described in section 2.2.1.

The most widely accepted definition of wave breaking is an instability that develops near the wave crest when the orbital velocity approaches the phase speed. If a particle velocity exceeds the phase speed, the wave is breaking. This definition gives rise to the conventional kinematic criterion based on the ratio between the horizontal wave particle velocity u and the wave phase speed c , wave breaking occurs when u/c ≥ 1 (Longuet- Higgins, 1969). Because an irregular wave is not of permanent form, unsteady and rapidly evolving, it is difficult to define or measure u and c (Banner and Peregrine, 1993). In the literature, the examination of the kinematic criterion is performed at the crest, defined as the maximum surface elevation of the wave.

Waves kinematic has been studied using different imaging techniques during the last two decades (e.g. Perlin et al., 1996; Peirson, 1997; Jessup et al., 1997; Chang and Liu, 1998; Banner and Peirson, 1998; Stansell and MacFarlane, 2002; Melville and Matusov, 2002; Banner and Peirson, 2007; Tian et al., 2010; Shemer, 2013; Itay and Liberzon, 2017).

Techniques such as particle tracking velocimetry (PIV) (e.g. Chang and Liu, 1998) (Fig- ure 2.2), particle tracking velocimetry (PTV) (e.g. Itay and Liberzon, 2017) or thermal image velocimetry (TIV) (e.g. Saket et al., 2017) have been used to measure the fluid particle velocities in the crest.

Different values of the conventional kinematic criterion have been measured depending

on the experimental configurations. Itay and Liberzon (2017) found that the ratio between

the crest particle velocity ( u

) and the phase speed ranges between 0.7 and 1.05 including

error bars (according to their Figure 5) at the inception of spilling breaking waves. Perlin

et al. (1996) measured u

/c = 0.74 just prior to the wave crest overturning of deep-water

plunging breaker. They found that the largest captured particle speed is situated at the

tip of the overturning jet, with a horizontal velocity 30% greater than the phase speed.

Figure 2.2: Example of a velocity field under a breaking wave observed from PIV measurements. (Figure 1 of Chang and Liu (1998))

Chang and Liu (1998) reported u

/c = 0.86 prior to wave breaking and u

/c = 1.07 when the particle velocities at the crest tip became almost horizontal for monochromatic break- ing wave in shallow water. Stansell and MacFarlane (2002) experimentally investigated the conventional kinematic criterion using PIV measurements and using three different methods to determine c . They found u

/c to be a maximum of 0.81 for plunging breakers and 0.95 for spilling breakers. From infrared imagery, Saket et al. (2017) study shows that for marginal breaking waves in unsteady deep water wave groups, u

/c is consistently less than unity i.e. 0.84 ± 0.016.

Overall, the conventional kinematic criterion investigation shows that a wave will break

when the ratio between the horizontal fluid velocity at the crest and the crest speed u

/c , is

in the range 0.7 and 1.3 , this range is fairly broad. These type of experiments in controlled

laboratory remains complex and errors regarding the breaking criterion threshold can be

relatively large (Itay and Liberzon, 2017) considering the accuracy of the measurements.

Relying on human decision regarding the exact moment of the inception of breaking com- plicates the experimental research. Measuring the kinematics of a breaking wave in the field is even more challenging, given the random nature of breaking occurrence. Through the last decades, progresses have been made in numerical studies.

2.1.4 Numerical investigations of wave breaking initiation

One of the first numerical studies of breaking waves was completed by Longuet-Higgins and Cokelet (1976) using the boundary integral equation method (BIEM), based on po- tential flow theory. Their computations were limited to a periodic domain in deep water.

Similar methods were adopted to consider cases of finite depth and various types of break- ers (e.g. Vinje and Brevig, 1981; New et al., 1985). More recent two-dimensional models can accommodate both arbitrary waves and complex bottom topography (e.g. Grilli et al., 1989; Cointe, 1990; Grilli and Subramanya, 1996). These models are often referred to as numerical wave tanks (NWT). Boundary element methods (BEM) are efficient for repre- senting wave propagation and overturning until the wave surface reconnects (e.g. Grilli and Subramanya, 1996; Guyenne and Grilli, 2006) (e.g. Figure 2.3). The volume of fluid method (VoF) also proved to accurately simulate wave breaking (Chen et al., 1999). Mod- els based on this method are capable of representing the wave beyond the point at which the jet impact the surface (Figure 2.4) and can therefore account for complex phenomenon such as splash-up, gas entrainment, energy dissipation, that are out of the scope in the present thesis.

Recently, Barthelemy et al. (2018) studied wave packets in deep and intermediate water generated with the three-dimensional (3D) extension of the two-dimensional (2D) fully non- linear potential flow model of Grilli et al. (1989) (Grilli et al., 2001). Barthelemy et al.

(2018) defines a dynamic wave breaking onset criterion B , based on the ratio of the local

energy flux to the energy density, normalized by the local crest speed magnitude; at the

free surface, this criterion reduces to the conventional kinematic criterion, B

= u

/c . This

study shows numerically that when B

exceeds the threshold 0.85 − 0.86 , wave breaking will

inevitably occur within a fraction of a wave period. This criterion is not strictly speaking

a breaking onset criterion, but is a precursor to breaking onset criterion, an imminence

breaking criterion. Seiffert et al. (2017); Saket et al. (2017, 2018) investigated the breaking

Figure 2.3: Example of shoaling and breaking solitary waves of initial height H

= 0.20 over a over a gentle 1:35 slope. (Figure 4 of Grilli and Subramanya (1996))

parameter of Barthelemy et al. (2018) in comparison to laboratory measurements. They found it to be robust for deep and intermediate water wave groups. Furthermore, Derakhti et al. (2018) showed that the time derivative of B

near the breaking onset was highly correlated with the wave breaking-induced dissipation.

Figure 2.4: Example of snapshots of time sequence of a plunging breaking waves

simulated from the volume of fluid method. (Figure 4 of Chen et al. (1999))