Numerical simulation of freak wave events

Vincent-Xavier JUMELESIM 2006, option Génie Marin

August 29, 2006

THESISpresented for the degree of

MASTER OF SCIENCE

University of OsloPO Box 1053, Blindern

NO-0316 OsloNorway

September 2005

I want first to thanks both Pr. C. Kharif who provide methis internship and Pr. J. Grue who welcomed me in Oslo.

I also thanks the complete hydrodynamic lab team who takecare of me during my stay.

I also thanks Véronique Le Corvec for her proofreading andher useful commentaries.

Abstract

This report deals with extreme wave phenomena. Exploration of the classicalwave theories are made, both on the theoratical approach and on the statisticalone. The first one shows wave generation phenomenon using only Euler’s equationfor a perfect fluid and gravity. On the other one, the statistical approach providesus with more real observations. Both models fail to explain some rare (or notso rare ?) events: freak waves. Then we defined what is a freak wave and someof the explanations that are given. Exploration on the non linear Schrödingerequation, which is known to give birth to gigantic waves is then the path taken.This equation could be easily derived from Euler’s equations. Numerical solutionof this equation are provided in the last chapter. Finally, the third part dealswith spectral methods and how they are used to compute very easily non linearinteraction for waves. Last chapter provides also results on this. In fact, the lastchapter is devoted to the results obtained, either on solving NLS, either on thecomputation of surface waves.

Résumé

Ce rapport traite du phenomene des vagues géantes. La premiere partie est consa-crée aux théories classiques, que se soit l’approche analytique ou l’approche sta-tistique. Celles-ci sont mises en défaut par l’existence de vagues géantes. Aprèsavoir défini ce qu’est une vague extrême, on explore deux des mécanismes pouvantconduire à la formation de telles vagues. Le premier est l’équation non-linéairede Schrödinger, qui est connue pour avoir une solution de la forme d’un soliton.Cette équation non-linéaire a la même source que l’équation de Stokes pour lesvagues. Quelques résultats numériques sont commentés dans le dernier chapitre.Enfin, la troisième partie traite d’une méthode nouvelle pour le calcul de vaguesnon-linéaires en utilisant une méthode spectrale. Le dernier chapitre est dediéaux résultats et à leur analyse, que ce soit pour NLS ou pour le calcul des vaguesnon-linéaires.

Contents

Introduction 5

1 Presentation of the University of Oslo 91.1 University of Oslo . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.2 Long term plan . . . . . . . . . . . . . . . . . . . . . . . . 101.1.3 Organisation of the University . . . . . . . . . . . . . . . . 14

1.2 The Faculty of Mathematics and Natural Sciences . . . . . . . . . 171.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.3 Department of Mathematics . . . . . . . . . . . . . . . . . 18

1.3 Hydrodynamics laboratory . . . . . . . . . . . . . . . . . . . . . . 19

2 Classic waves in deep water 212.1 The governing equation . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Stokes waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Theoretical background . . . . . . . . . . . . . . . . . . . . 232.3 Probabilistic approach . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Weaknesses of these models . . . . . . . . . . . . . . . . . . . . . 29

2.4.1 Events not taken into account . . . . . . . . . . . . . . . . 292.4.2 First explanations . . . . . . . . . . . . . . . . . . . . . . . 322.4.3 A need for a complete explanation . . . . . . . . . . . . . 33

3 Nonlinear Schrödinger equation 363.1 Derivation of the NLS equation . . . . . . . . . . . . . . . . . . . 36

3.1.1 Concept of a wave train . . . . . . . . . . . . . . . . . . . 363.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.3 Heuristic derivation of the NLS equation . . . . . . . . . . 38

3.2 Time integration of this equation . . . . . . . . . . . . . . . . . . 393.2.1 Analytic solution . . . . . . . . . . . . . . . . . . . . . . . 393.2.2 Numerical integration of NLS . . . . . . . . . . . . . . . . 40

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4 Spectral methods 434.1 Three dimensional waves simulation . . . . . . . . . . . . . . . . . 44

4.1.1 Obtention of V . . . . . . . . . . . . . . . . . . . . . . . . 454.1.2 Reformulation of the boundary integrals . . . . . . . . . . 46

4.2 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.1 Analytical part of the equation . . . . . . . . . . . . . . . 484.2.2 Linear problem . . . . . . . . . . . . . . . . . . . . . . . . 494.2.3 Non linear problem . . . . . . . . . . . . . . . . . . . . . . 50

5 Numerical solution 515.1 Stokes wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Non linear Schrödinger equation . . . . . . . . . . . . . . . . . . . 52

5.2.1 Reasons to use this integration scheme . . . . . . . . . . . 525.3 Wave simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.2 Linear simulation . . . . . . . . . . . . . . . . . . . . . . . 62

Conclusion 67

A Differentiation using FFT 68A.1 Analytic background . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.1.1 Mathematical background on Fourier transform . . . . . . 68A.1.2 Fourier transform of the derivative of a function . . . . . . 69

A.2 Numerical computation . . . . . . . . . . . . . . . . . . . . . . . . 69A.3 The FFT algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.3.1 Transformation into a periodic function . . . . . . . . . . . 70

B Useful information about Fourier transform 72B.1 Calculation of Fourier transform of one over R . . . . . . . . . . . 72B.2 Calculation of Fourier transform of hyperbolic secant . . . . . . . 74

C Green function history 75

D KdV equation 77D.1 Yielding the linear problem . . . . . . . . . . . . . . . . . . . . . 77D.2 Solution of the linear problem . . . . . . . . . . . . . . . . . . . . 78

Bibliography 80

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List of Figures

2.1 Finite depth water . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Hyperbolic tangent . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 3D simulation of a second order stoke wave . . . . . . . . . . . . . 252.4 Wave spectra of a fully developed sea for different wind speeds . . 282.5 Wave spectra of a developing sea for different fetches. . . . . . . . 292.6 A freak wave over a ship . . . . . . . . . . . . . . . . . . . . . . . 302.7 Draupner wave record . . . . . . . . . . . . . . . . . . . . . . . . 312.8 Agulhas current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.9 Analytic breather . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 The propagation of the soliton . . . . . . . . . . . . . . . . . . . . 41

5.1 Rk4 scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Matlab time integrator . . . . . . . . . . . . . . . . . . . . . . . 555.3 FFT obtained by RK4 scheme . . . . . . . . . . . . . . . . . . . . 565.4 FFT obtained by Matlab . . . . . . . . . . . . . . . . . . . . . . 575.5 Comparison between the two methods . . . . . . . . . . . . . . . 585.6 RK4 integrator for NLS using a sine wave . . . . . . . . . . . . . 595.7 FFT of the sine wave . . . . . . . . . . . . . . . . . . . . . . . . . 605.8 Sine wave with a hyperbolic sechant . . . . . . . . . . . . . . . . . 615.9 Propagation of a linear sine wave . . . . . . . . . . . . . . . . . . 635.10 Propagation of a linear sine wave (3D) . . . . . . . . . . . . . . . 645.11 Aliasing effect in the computation of product . . . . . . . . . . . . 655.12 Zeros padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.1 Wrong derivative of the hyperbolic secant . . . . . . . . . . . . . . 70A.2 Derivative of the hyperbolic secant . . . . . . . . . . . . . . . . . 71

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List of Tables

1.1 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 Physical parameters and dimension . . . . . . . . . . . . . . . . . 212.2 Notations used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . . . 262.4 Two classical spectra . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1 Parameters and variables . . . . . . . . . . . . . . . . . . . . . . . 62

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Introduction

Motivations In order to complete my courses and obtain my engineer degreefrom my school: École Supérieure d’Ingénieurs de Marseille (ESIM), I was to havean internship. As I was also attending a Master Degree in fluids mechanics withthe University Paul Cézanne of Marseille, I also was to be in a laboratory or atleast in a university. Pr. Ch. Kharif had made a proposition to work in Norwaywith a colleague of him, Pr. J. Grue. We quickly find a internship subject whichwas “numerical modeling of freak waves”. I shall presents the University of Osloand in particular my department in a first chapter.

Fluids mechanics have always been of great interest for me. Since I have beginto study mechanics, I was wondering about this. This field of mechanics seems tobe much more complex than it appears. Everyone experienced his basics, such aswater flowing in a river or waves on the sea. Visualisations of the phenomena areeasy to find, but not as easy to understand. Some explanation even go againstintuition. Studies began in the 16th century with Newton and where followed bygreat scientists as Euler. But by now, only some few case have been completlyexplained. For the most part, we must rely on numeric solver, for no ones knowto solve the general Laplacien equation. While using those numerical recipes, wecould approximate the reality. This should be all physicist goal, not to matchthe exact solutions, which is amthematcians jobs, but to present models that fitrather well the observations. In this particular field, one could easily observe,qualify and quantify the phenomena, and confront with his model.

Wave research history Fluids mechanics start with the study of potentialflow: We may pass on Bernoulli’s work and jump directly to one of the greatscientist of the 18th century: Leonhard Euler. This equation are now known tobe a incomplete form of the Navier-Stokes equation, which are more complete.These Navier Stokes equation describe motion of a fluid, without any assumptionon it. It has been one of the greatest contribution to fluids mechanics in the 19thcentury. Since now, no one have been able to solve directly this set of equations.However, some exploration have been made on particular case of Navier–Stokesequations. Stokes have also been one of the first to derived equation that couldexplain Ocean surface waves. First works were conducted on the propagationof wave and not some much on the generation of the wave. We could see for

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instance the work of the engineer John Scott Russell in the middle of the 19thcentury. At the end of the 19th century, a great contributors to hydrodynamicsand wave theories was fond in Lord Rayleigh. Korteweg and de Vries (1895) alsoderived the famous equation which applies to shallow water. This equation wasto explain the phenomena observed by the engineer John Scott Russell on theUnion canal in 1834, when he followed this solitary wave:

[. . . ] rolled forward with great velocity, assuming the form of a largesolitary elevation, a rounded, smooth and well-defined heap of wa-ter, which continued its course along the channel apparently withoutchange of form or diminution of speed. [. . . ] I have called the Waveof Translation. Russel (1844)

We must then wait the years after the Second World War to find a newinterested in hydrodynamics and wave study. In the 1950s, Pierson (1952) startedto work on the storm generated ocean surface waves. His biggest contributionwas achieved in 1964 when Moskowitz and him just proposed a Wave spectrathat could be used by oceanographers to represents the wave distribution on theocean. Bretherton (1964) explored the resonant interactions between waves. Inthe same time, many contributions where made to the theories, as the ones byWhitham (1965) about the use of a Lagrangian or Whitham (1967) on non–lineardispersion of water waves. Gardner et al. (1967) also discover a new method tosolve Korteweg–de Vries equation: the inverse scattering method. Benjamin andFeir (1967) discussed about the disintegration of wave trains in deep water. Ayear after Zakharov (1968) discussed the stability of wave of finite amplitudein deep water. A little bit after, Zakharov and Shabat (1972) used the inversescattering method with the non–linear schrödinger equation. In this time, thepoint was to focus on instabilities of fully developed forms. Scientists where onlydealing with weakly non-linear theories.

In the 1970’, Norwegians conduct a huge experimental campain: Joint NorthSea Wave Atmosphere Program (JONSWAP) to get a spectrum that fitted theNorth Sea, where oil exploration was beginning.

With oil exploration came a new field in hydrodynamics which is fluid bodyinteraction. In this time, people were interested into evaluating the interactionthat could exist between wave and body, mostly oildrilling platform. Since theseplatforms mostly consisted of sets of cylinders, most of the research where con-ducted with this particular shape.

In this time, research about non-linear phenomena where not as popular.However, we must acknowledge the work by Yuen and Lake (1982). Dysthe(1979) modified the non–linear Schrödinger equation when it applied in deepwater. We must then wait the Draupner event to get some more developmenton non–linear waves. After this event, we could see papers by Trulsen et al.(1997,1998,1999).

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Freak waves It is common for mid-ocean storm waves to reach 7 metres (23feet) in height, and in extreme conditions such waves can reach heights of 15metres (50 feet). However, for centuries maritime lore told of the existence ofvastly more massive waves – veritable monsters up to 30 metres (100 feet) inheight (approximately the height of a 12-story building) – that could appearwithout warning in mid-ocean, against the prevailing current and wave direction,and often in perfectly clear weather. Such waves were said to consist of an almostvertical wall of water preceded by a trough so deep that it was referred to as a“hole in the sea”; a ship encountering a wave of such magnitude would be unlikelyto survive the tremendous pressures of up to 100 tonnes/m2 (980 kPa) exertedby the weight of the breaking water, and would almost certainly be sunk in amatter of seconds. Usual ship design allows for rounded storm waves up to 15 mand pressures around 15 tonnes/m2 (147 kPa) without damage, and somewhatmore if some deformation is allowed for, which is about a wave of twenty metres.

Scientists long dismissed such stories, asserting that mathematical modelsindicated that ocean waves of greater than 15 metres in height were likely to berare “once in 10,000 years” events. However, satellite imaging has in recent yearsconfirmed that waves of up to 30 metres in height are much more common thanmathematical probability would predict based on a linear model of wave size. Inaddition, pressure readings from buoys moored in the Gulf of Mexico at the timeof Hurricane Katrina also indicate the presence of such large waves at the timeof the storm. In fact, they seem to occur in all of the world’s oceans many timesevery year. This has caused a re-examination of the reason for their existence, aswell as reconsideration of the implications for ocean-going ship design.

These localized freak waves are not the same as tsunami or megatsunami.Tsunami are displacement waves which travel at high speed and are more orless unnoticeable in deep water; they only become dangerous as they approachthe shoreline. In the deep sea, tsunami do not represent a threat to shipping.Megatsunami are also rare events, but only arise in confined spaces, such asinlets and river valleys. Freak waves, by contrast, are localized short-lived waterphenomena that most frequently occur far out to sea.

For both scientists and engineers, freak waves is a new phenomenon. Even ifthe approach is not the same, this new problem interests both of them. Engineersand naval architects are mostly concerned with the loads and the interaction withmarine structure, whereas scientists search a description of the phenomenon andcauses which lead to it. This phenomenon have been taken into account onlyrecently because of the lack of evidences about it. In fact, the first real evidencewas found the 1st of January, 1995 on the Draupner platform on the NorthSea. Even now, reliable information about such events are poor an incomplete.For more information about it read Modeling freak waves from the North SeaSlunyaev et al. (2005) and Focusing of Nonlinear Wave Groups in Deep WaterKharif et al. (2001). Different explanations are given about the creation of suchgigantic waves. Tsunamis are created by underwater earthquake, freak waves on

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the south-east coast of Africa are due to the Agulhas current or saptio-temporalfocusing in shallow water; but there is no such current or earthquake in areasuch as the North Sea. So another explanation might be found. Since the 19thcentury, we have known that solitary waves could exist. It had then been provedthat non-linear theory of water waves in deep water leads to the existence ofsolitary waves. The solutions is the classic “breathers” which amplitude can betwice than other waves (this is the formal definition of a freak wave). This non-linear theory leads to the non-linear cubic Schrödinger equation or even morecomplex for real wind waves. Confrontation between numerical modeling andreality shows that this equation is a good model if the steepness of the waves isnot too large

In this report I will present classical water waves theory focusing on the twomain approach and its weaknesses. Then I shall present the non-linear cubicSchrödinger equation, and I shall solve it. A quick overview of Korteweg equationis in the appendix. After while, I shall presents a code implementation to simulateevolution of the free-surface in a three dimensional numerical tank. Eventually, Iwill discuss the numerical methods, its implementation and the results obtained.

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Chapter 1

Presentation of the University ofOslo

1.1 University of Oslo

1.1.1 HistoryAs a satellite of the modest Danish kingdom until early in the 19th century,Norway was understandably not among the first places to found a university.The larger kingdoms of medieval Europe – Italy, France, England – establishedtheir first universities as early as the 12th century. The Scandinavian monarchies,Sweden and Denmark at the time, opened their first universities at Uppsala in1477 and at Copenhagen in 1479, respectively. 1100

In 1811, the Danish monarch King Frederick VI granted Norway the right tofound its own university. Originally, the campus was to be located in Kongsberg –a regional city 55 kilometres south-west of Oslo. By 1812, the authorities decidedon Oslo instead. Today, almost 40,000 people work and study at the University ofOslo, while the current population of Kongsberg hovers around 22,000 people. Inretrospect, you could say things turned out for the better, both for the Universityand for Kongsberg.

The Royal Frederick University (Det Kongelige FrederiksUniversitet) opened in Oslo in 1813. At the start, conditionswere meagre; classes were held in rented buildings. There were,however, only 17 students and 6 teachers at that time – basicallyless than the smallest department of the smallest faculty today.1813 Philosopher and PM Niels Treschow was among the firstprofessors at The University of Oslo.

The early modest period persisted for almost half a century,before the University had its own building complex. By 1852,

the first university buildings on Oslo’s main boulevard, Karl Johans gate (thenSlotsveien), stood completed. Nonetheless, the pace of growth among the stu-

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dent body was so dramatic that, by the 1900s, there was again a scarcity ofinstructional room.

In 1923, university administrators and the Norwegian government agreed todevelop a new campus just outside of the downtown area in a neighbourhoodcalled Blindern. Because of an economic recession, however, the plans wereshelved.

Only in 1931 could the first institutes move tothe new campus. Shortly thereafter, in 1939, theuniversity changed is name to the University ofOslo or Universitas Osloensis. In the 1960s theGreater Blindern area was developed. This area istoday associated by most people as the centre of theUniversity of Oslo. The administration, welfare,humanities, social sciences and natural sciencesbuildings are located here. The student body in-creased rapidly during this period, approaching the20,000-student watermark by the 1970s. This num-ber remained stable for a while, but new growth in

recent years has pushed the figure even higher. Today there are approx. 30,000students at the University of Oslo, down from almost 40,000 a couple of years ago.The problems with overcrowded instruction rooms has been reduced somewhatover the past few years...

Initially, the University had four faculties: Theology, Law,Medicine and Philosophy. In 1861 the Faculty of Philosophy wassplit into The Faculty of Arts and The Faculty of Mathematicsand Natural Science. Norway?s College of Dentistry became apart of the university in 1959, renamed The Faculty of Dentistry.In 1963 The Faculty of Social Sciences was established, and thelatest addition has been The Faculty of Education in 1996. Inmost informational material, the faculties are presented in this chronological or-der. Besides the faculties, the university operates a number of independent cen-tres and affiliated units.

1.1.2 Long term planThe University of Oslo is in a community in transition:

• New demands in knowledge and training require overall renewal, more pro-ficiency in selected areas and a reassessment of curricular priorities.

• Norway’s politics regarding knowledge capital and the university’s missionguidelines are under transition. These changes are challenging the univer-sity’s curricular profile, fund-raising strategy and management.

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• Competition for researchers, students and economic resources is increasing.This requires the University of Oslo to offer working conditions and envi-ronments that are attractive enough to support future recruitment efforts.

• Instructional facilities have a decisive effect on study quality, contentmentand successful execution. The need to renovate increases as the variety ofsubject offerings and competition for students increase.

• The internationalisation of the intellectual arena in terms of research, ed-ucation and employment is in full swing. This compels the university toaddress the quality of both research and instruction and the need for do-mestic and international relationships.

• The need for growth and higher quality within the Norwegian researchcommunity is clear. This is dependent upon better financing, more concen-tration on promising fields and the education of new researchers who canpromote these developments.

• The development of a information-based economy in Norway is progressingtoo slowly. This challenges the University of Oslo to step up its valuecreation and develop new methods of co-operation.

• Developments within information and communication technologies placenew demands on the university’s proficiency in these fields and its instruc-tion of students via relevant courses and study opportunities.

• Weak recruitment of qualified teachers threatens the university’s standingas a pillar of the information society. This is a social problem that theUniversity of Oslo is affected by as an organ that trains teachers; thus, theuniversity must contribute to the problem’s solution.

• Demands for reorganisation threaten the university’s core values and criticalcapability. As a result, the university must both renew and actively promotethose values that the university and community jointly decide are importantto maintain.

Prioritised Goals 2000 à 2004:

1. In order to remain a research institution of high international standard theuniversity will:

• Work to create an economic zone that enables the university to offerbetter conditions for researchers within their research community

• Stimulate the development of both long-established and trend-settingresearch environments that have the disciplinary resources to be lead-ers within and outside of Norway

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• Increase the recruitment of internationally renowned researchers fromNorway and abroad and offer competitive terms of employment inorder to retain the best researchers

2. In order to remain an attractive instructional atmosphere for both studentsand employees with the same standards of the best universities of Europethe university will:

• Readdress study offers and teaching facilities to promote diligence,varied instructional and exam formats, increased use of computerswith new teaching methods, international contact and comprehensiverecreational opportunities for students and staff

• Develop strong instructional programs within adult and continuingeducation

3. To be an open and initiative-taking educational and research partner theuniversity will:

• Actively partake in international joint projects that stimulate the de-velopment of the university and strengthen international recruitment

• Actively contribute to value creation in the community and work toimprove the interplay between the Norwegian business world and theuniversity

• Co-operate with the national school system to develop a strong pro-gram of teacher training and strengthen recruitment in fields that arerelevant to the university

4. To increase flexibility, creativity and quality within the organisation theuniversity will:

• Create attractive and competitive working environments for the em-ployees and students of the university, as they are our most importantresource

• Strengthen, improve and invigorate disciplinary leadership at all levels• Continue to develop the organisation of institutional activities

5. To improve the general morale of students and employees and to create goodmeeting spaces for Oslo’s residents, public authorities and the Universityof Oslo’s guests the university will:

• Preserve the university’s properties around their own strengths andincrease the standard of the infrastructure as a basis for studies andresearch

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• Develop new university properties and the bonds between the different,geographically-separated campuses of the university

• Actively exploit the university’s library, museums and collections tosupport internal, national and international academic and cultural ac-tivities

The University of Oslo will heighten its quality by:

• Re-evaluating the university’s application of resources in specific subjectsand the entire curriculum. Future evaluations will give greater disciplinaryconcentration and a stronger profile. This shall be attained via reorgan-isation, re-evaluation of resources and successful delegation of tasks on anational level.

• Directing a more active financing and budgeting program. The goal is toincrease the economic zone in order to invest in quality and strengthenfund-raising for research.

• Establishing and allocating an annual budget of roughly NOK 100 millionto specified actions that promote quality. The budget will be financed byresources for reorganisation and quality-control from the Church, Educationand Research Department and the Norwegian Research Council, as well asin-house capital for independent allocation. Budgeting will address resourceneeds by:

1. Reallocation to create more intensive instruction programs and thedevelopment of IT-based instruction

2. Strengthening of long-established and newly developed communitiesand reorganisation as a result of the changes in departmental resourceallocation

3. Actions that improve the working conditions for employees and insurefuture recruitment

• Identifying and support new research areas and centres of excellence withinresearch and enter joint projects with the Norwegian Research Council tosupport these areas.

• Incorporating IT-based instruction and change examination formats as partof a more comprehensive approach towards instruction.

• Stimulating higher concentration in study areas by, among other things,offering programs for the "100

• Differentiating and continuing to develop the academic leadership and in-crease the leader’s possibilities to stimulate his/her personnel and activities.

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• Continuing to develop human resources through new incentives and profi-ciency requirements by recruitment, which, in turn, strengthens the univer-sity’s collective abilities.

• Executing an international evaluation of central parts of the Universityof Oslo’s activities in order to realise both our strong and weak pointsqualitatively. The purpose should be to rationalise future quality control.

• Partaking in an international evaluation of researcher instruction in Norway.

• Reconsidering the university’s disciplinary organisation, such as the facultystructure. The goal is to increase the chances of interdisciplinary projects,synergy within a field and greater flexibility to strengthen or reduce certainactivities.

• Developing new means of co-operation with schools, independent institu-tions and the business world in order to strengthen the collective approachto projects within research, education and proficiency development.

1.1.3 Organisation of the UniversityInstruction

Most of the instruction at the University of Oslo takes place in the eight primaryfaculties, which are each divided into departments. The academic year is dividedinto two semesters: an autumn semester and a spring semester.

Some courses are professional, designed around a rigid curriculum to qual-ify for an exclusive professional title (doctor, attorney-at-law, psychologist, etc.).Other courses are divided into sub-courses, where the student may choose how farto pursue a particular line of study. The periods of study range from half-semestercourses to PhD degrees; the latter require a final thesis that demonstrates pro-found knowledge of the subject area, based on many years of study.

Student life at the University of Oslo is a rich field. Students participate to agreat extent in the operation of the University of Oslo through the Student Par-liament, regularly publish newspapers and magazines and operate a wide varietyof organisations and clubs. More detailed presentations of student offerings, bothinside and outside the classroom, can be found here.

The Faculties are divided into:

• The Faculty of Theology offers courses to qualify for the ministry, coursesin Christianity, etc.

• The Faculty of Law offers courses in law, criminology, etc.

• The Faculty of Medicine is primarily a medical school, but also offers coursesin other health-related subjects.

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• The Faculty of Arts covers the humanities: language, history, philosophy,culture, media, etc.

• The Faculty of Dentistry offers dentistry, dental orthopaedics and othercourses relating to oral health.

• The Faculty of Social Sciences offers social studies at all levels, from politicalscience to psychology, interspersed with sociology, statistics and other fields.

• The Faculty of Education offers courses in pedagogy.

• The Faculty of Mathematics and Natural Sciences is the university largestfaculty, offering courses in subjects such as mathematics, chemistry, phar-macology, geology, physics, etc.

Research

Although many regard the university as a centre of instruction and training,research and the dissemination of research results is an equally important partof the University of Oslo’s activities. The principle is that the “teachers” shallbe researchers sharing their first-hand knowledge with students through lectures.That is why the staff is considered scientific personnel.

Research can be categorised as either appliedresearch or basic research. Basic research pursuesnew knowledge simply for its own sake; in this in-stance, any use or application of the sought-afterknowledge is hitherto undetermined. In contrast,applied research seeks knowledge with a predefinedgoal or use in mind. Both types of research are con-ducted at the University of Oslo. Applied research,in particular, has a strong place in the interdisci-plinary centres.

In 1969, Ragnar Frisch and Odd Hassel becamethe first Norwegian researchers to receive the No-bel Prize. Frisch was awarded the very first NobelPrize in economics, while Hassel received the prize

in chemistry. They were later joined by Ivar Giæver, who won the Nobel Prizein physics in 1973. He has a professorship at the University of Oslo, although hehas worked mostly in the U.S. In 1989, Frisch’s former student Trygve Haavelmowon the Nobel Prize in economics.

Fridtjof Nansen, a professor of oceanography and zoology at the University ofOslo, was awarded the Nobel Peace Prize in 1922, albeit more for his humanitar-ian efforts than his research (for more about Nansen, see the Nobel Institute).

15

A host of researchers at the University of Oslo have earned other forms ofinternational acclaim. For example, the work of Svein Rosseland, an astrophysi-cist at the University of Oslo, caught the attention of the charitable RockefellerFoundation, which financed the construction of the Department of TheoreticalAstrophysics at the university in his honour. Today the building bears his name.

Research can only make a difference when discoveries are shared with theacademic community and beyond. Consequently, the University of Oslo widelycommunicates its research results. While Apollon is the leading name amonguniversity research publications, the University of Oslo publishes numerous otherperiodicals with contents ranging from environmental dangers to human rights.

The University of Oslo’s many museums, among other educational units, havemade a bold foray into cyberspace. The university’s Viking Ship Museum wasjust one of the last institutes to go on-line. You can also find The Natural HistoryMuseums and the Ethnographic Museum.

Dissemination of science and research

“Never an Ivory Tower”, resounds a review of the book The History of the Univer-sity of Oslo (Historien om Universitetet i Oslo, in Norwegian). Myths describingthe university as an ivory tower of the intellectual elite have little connectionto today’s academic reality: Many researchers at the University of Oslo take anactive part in socio-political debate through different media channels. The uni-versity’s academicians also relate their knowledge and opinions through booksand lectures - or via the university’s research pages on the World Wide Web. Atthe University of Oslo campus, one can also find researchers holding both plannedand impromptu discussions at academic conferences large and small. Many ofthese events are open to the public and arranged annually, for instance a day forNorway’s public school teachers to brush up on the latest research in their field,a day for high school students at the University and the Arts Faculty’s HumanityDays festival.

A fundamental mission for the university museums isto disseminate research findings to the world outside of theuniversity. The public can visit museum installations, takepart in guided tours, bring children to specially-arrangedyouth days and listen to lectures. The university’s muse-ums have educational and entertaining web-sites as well.One of the museums, the Viking Ship Museum on Bygdøy,is Norway’s most popular museum. The University of Osloalso has a special periodical dedicated to bringing researchinto the public discussion space: Apollon. Apollon is pub-lished both as a paper magazine and electronically.

Dissemination of research findings is one of the univer-sity’s three primary objectives. On the 25th of January

16

2000, the senate of the University of Oslo ratified the following definition: “Whenwe say dissemination of research, we mean that researchers disseminate scientificconclusions, procedures and hypotheses about their specialised field of study topeople outside of this field; researchers are also obligated to take part in publicdebate with arguments derived from their research.”

1.2 The Faculty of Mathematics and NaturalSciences

1.2.1 OverviewThe Faculty of Mathematics and Natural Sciences was established in 1861, and isone of eight faculties at the University of Oslo. It is the largest educational andbasic research science faculty in Norway. More than 625 scientists, 5000 studentsand 350 technical and administrative staff members contribute to the pursuit ofscientific knowledge and keep the research standard at a high international level.

During the last decades there has been a rapid development within mathe-matics and natural sciences. These disciplines form the fundament of our under-standing of nature and of applied disciplines like information technology, genetics,materials and micro technology. The natural sciences affects our view of the worldand the way we think - they form the premises for technological development,and consequently for our material wealth and standard of living.

Being part of this development, today’s Faculty of Mathematics and NaturalSciences at the University of Oslo combines a strong research program, solidinstruction and a good relationship with the political and business worlds in itsinvestigations into a wide range of fields. Environmental science, IT, petroleumand astrophysics are areas of particular excellence at the faculty.

Research activities at the faculty span a broad diversity of subjects. Dividedinto nine departments and three Centres of Excellence, the faculty addressesfundamental problems of theory and practise in mathematics, physics, chem-istry, informatics, earth and life sciences. With a special focus on basic researchwithin these fields, the faculty provides leading-edge scientific competence to itsstudents, its industrial research and development (R&D) partners and to otheracademic institutions in all parts of the world.

Two Nobel prizes and participation in global research projects like ESA’s (Eu-ropean Space Administration) and NASA’s "Solar and Heliospheric Observatory"are testament to the faculty’s strength in scientific research.

17

DepartmentsTheoretical astrophysicsMolecular biosiencesPharmacyPhysicsGeosciencesInformaticsChemistryMathematics

Centres of ExcellencePhysics of Geological processesMathematics for applicationEcological and Evolutionary Synthesis

Others units

Table 1.1: Organisation

1.2.2 Organisation

1.2.3 Department of MathematicsThe department of mathematics is divided in three very distinct part dependingon the field.

The first section deals with general mathematics. In there study are held onabstract subject such as topology, conics, algebra, etc. First year student startwith some of this general courses, among other. Even if this department dealswith pure mathematics, there is still some applied mathematics that are made inthis department.

The second section is devoted to statistics and economics. Even if somemathematicians say that is not mathematics, there is still research to be made.Statistics are a fast way to represent and study a huge population just by lookingat a quite reduced sample. Thus it allows informations to be get not on thenetire population. But we know also, that this statistic representation are notcompletely reliable. Therefore, research in this field is continued. Economicsdeals with partial differential equation to model the behaviour of incomes andso on. Since we want more accurate predications, this field have also a greatimportance. These two field are heavily linked to the man in the street life, morethan the previous.

The last section of the mathematics department, how surprising it could be, isapplied mechanics. Mostly research and study are made about fluids mechanics,but some also deals with solids mechanics. The prevalence of fluids mechanics isdue to the presence of a quite well known hydrodynamics laboratory. Even then,

18

the research in solod mechanics are deeply linked with marine engineering.Since the point is not to present the complete mathematics department, I will

then focus on one of its items: the fluid mechanics section and more over on thehydrodynamic laboratory.

1.3 Hydrodynamics laboratoryDue to oil resources in the North Sea, Norwegiansare very interested in studying hydrodynamics, bothfor water wave theory and wave–body interaction.Since the seventies, most of the research have beenconducted in this fields. Oil exploration and produc-tion have created a huge need of theories. In orderto validate a theory, it is common to make some ex-periments. In hydrodynamics, this experiments areconducted in water tank. The Oslo hydrodynamicslaboratory have two water thanks that are currentlyused for breaking of surface waves and internal waves.This kind of wave tank is suited for unidirectionalflows. Studies have been conducted about hydrody-namics effects on cylinder, immersed or not etc. Thistank is also used to study turbulent flows. In orderto acquire results about turbulence, a massive use ofPIV methods is done. PIV stands for Particle ImageVelocimetry and is a method to calculate the veloc-ity field in a turbulent flow. Even if I did not use

this methods, some information could be find about in Grue et al. (2004). Themain ideas of this method is to track seed in the flow that is illuminated by alaser. High speed camera will record the motion of the illuminated seed. Thenby analysing the different frames, one is able to get the evolution of the velocityvector field. This is only a quick introduction to the theory.

By now, the increasing power of computer and research made to use it, suchas the spectral methods (present in chapter 4) allow scientists and engineers touse numerical water tanks. Then a comparison is made between the numericaltank, the real water tank and real evidences to see if the numeric results fit theobservations or not.

19

20

Chapter 2

Classic waves in deep water

This is a classic derivation, made by Airy and Stokes, that leads to the classicwater waves theory. This approach is still used for the description of the simplewave fields.In physics, it is very common to make some assumptions and to use some nota-tions, to simplify the writing of the equations. As I will be also using some

Symbol Physical quantity dimension1

g Acceleration due to gravity L · T−2

ρ Water density M · L−3

P Pression2 M · L−1 · T−2

p Normalized pression2 (p = P/ρ) L2 · T−2

k = (k1, k2) Wave number vector L−1

k0 Wave number of the carrier L−1

a Wave amplitude Lω Frequency T−1

ω0 Frequency of the carrier T−1

x = (x, y) Horizontal coordinates Lz Vertical upward coordinate Lt Time T

η(x, y, t) Elevation of the free surface Lφ(x, y, z, t) Velocity potential L2 · T−1

Table 2.1: Physical parameters and dimension

other convenient notations, I am giving a summary of its. Vector will be mostlywritten in bold, when it is not obvious that it is a vector (∇f is always a vector).As well I shall denote by a “ˆ” the Fourier transform of a quantity (F f = f)

1M: mass; L: length; T: time2with the subscript 0, it means atmospherical pression

21

Operator Meaning∆ ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2

∇ (∂/∂x, ∂/∂y, ∂/∂z)4= equal by definition

× curl, i.e.

xyz

×

x′

y′

z′

=

yz′ − zy′

zx′ − xz′

xy′ − yx′

· scalar product, i.e.

xyz

·

x′

y′

z′

= xx′ + yy′ + zz′

f = F f Fourier transform of ffn = F f Fast Fourier transform of fn

Table 2.2: Notations used

2.1 The governing equationUsing the well known Euler equations for an incompressible, inviscid fluid, as-suming the flow is irrotational at t = 0, we obtain

∆φ = 0 −∞ < z < η(x, y, t), (2.1)φt +

12(∇φ)2 + gz = p

ηt +∇φ · ∇η − φz = 0

z = η(x, y, t), (2.2)

φ(z) → 0 z → −∞ (2.3)

where ∆ = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 and ∇ = (∂/∂x, ∂/∂y, ∂/∂z). p denotesthe pression P at the free surface, normalized by ρ (see table 2.1). It is provedthat if the flow is irrotational at t = 0, for an incompressible and inviscid fluidthen it remains irrotational. One shall notice that in ηt + ∇φ · ∇η − φz = 0,we used the horizontal gradient. Using this equation and if we consider a smalldisturbance (i.e. z ≈ η) we can linearize the equations. Thus, we obtain

φt + gη = pηt − φz = 0

z = 0, (2.4)

as new boundary condition for this problem. We obviously also used the fact that∇φ = O (φ) and ∇η = O (η) to reduce the previous set of equation to this one.This set of equation could be re-written in one equation, assuming that pt = 0:

φtt + gφz = 0 on z = 0. (2.5)

This yields the following linear problem:

∆φ = 0, −∞ < z < 0, (2.6)φtt + gφz = 0, z = 0, (2.7)

φ(z) → 0, z → −∞ (2.8)

22

In these equations, some assumptions have been made. In fact, it comes from thefact the flow is supposed incompressible (∂ρ/∂t = 0) and inviscid (the dynamicviscosity ν = 0). At the free surface, we could also take the pressure P as eitherzero or balanced by the surface tension. We are going to neglect the capillarityeffect. Since gravity waves deals with wavelength about 1 meter, it is then greaterthan the length order of capillarity effects, which is about the millimeter for water.

These equation are here written for infinite deep water, but if we assume thatthe bottom is at the −h depth, the equation 2.8 just transform into

φ(z) = 0, z = −h (2.9)

ηz

∆φ = 0

Figure 2.1: Finite depth water

2.2 Stokes waves

2.2.1 Theoretical backgroundFrom these governing equation, we shall derive some simpler equation, whichare known as Stokes waves. From the equation 2.4, we could solve this set ofdifferential equations using variable separation method (i.e. we want φ(x, z, t) =f(x, t)× g(z)) for equation 2.8. Thus, the set

φ(x, z, t) = agωekz sin(kx− ωt)

η(x, t) = a cos(kx− ωt)(2.10)

could be a solution. This solution φ must also fulfill ∆φ = 0.

∆φ = φxx + φzz = −agωekzk2 sin(kx− ωt) +

ag

ωk2ekz sin(kx− ωt) = 0 (2.11)

23

In this equation, we use only the deep-water dispersion relation: ω2 = gk. Actu-ally, if we use the finite depth water, we should find that the dispersion relationis

ω2 = gk tanh(kh) (2.12)

This figure 2.2 shows how quick tanh(x)1).

Figure 2.2: Hyperbolic tangent

tanh(x)4=

sinh(x)

cosh(x)=ex − e−x

ex + e−x. (2.13)

One must remind that if x 1, then obviously ex e−x and we therefore findthat if kh 1. Using then

limx→∞

tanh (x) = 1 (2.14)

then we could use the infinite depth water dispersion relation:

ω2 = gk. (2.15)

24

In fact, Stokes has been further in his derivation and he find what we call secondorder stokes wave:

η(x, t) = a cos(kx− ωt) +1

2ka2 cos 2(kx− ωt) (2.16)

and the velocity potential:

φ(x, z, t) = (ωa/k)ekz sin(kx− ωt) (2.17)

where k is such as k = (k, 0) and ω is linked to k and a by the following dispersionequation:

ω =√gk

(1 +

1

2k2a2

)(2.18)

This is taken from Yuen and Lake (1982). These equations come mainly fromlooking the comportment of the hyperbolic tangent when kh 1. It is impor-tant to notice that this development describe the free surface elevation for a smallsteepness of the waves. The steepness of the waves is controlled by the ak pa-rameter which should be small (i.e. ak 1). So in our case, we shall first give avalue for k and then calculate a the wave amplitude in respect of this condition.This stokes wave could even be extended to a third and a fifth order, but these

Figure 2.3: 3D simulation of a second order stoke wave

25

formulation are not very used, because it exists some more accurate ones3.Either, we are not going to explore any shallow water theory (kh 1), which

leads to a simpler linear dispersion relation: ω = k√gh, and for a non linear

dispersion relation: ω =√ghk(1 − 1

6k2h2). This approach would leads us to

Korteweg-De Vries equation, which is not the purpose in this report.There is some dimensionless parameters used in water-wave theory.

Table 2.3: Dimensionless parametersParameter Expression Dimension MeaningSteepness ak a: amplitude ak < 0.14 for non-breaking

k: wave numberDepth kh h: water depth kh > 1 for deep water theory

k: wave number

2.3 Probabilistic approachA second wayof approaching the waves on the sea is given by oceanographers andengineers: They used a probabilistic approach, as the JONSWAP4 spectrum. Buthow do we get to the use of this spectrum.Using Kharif and Pelinovsky (2003), Iwill explain it.

Just looking at the classical dispersion relation, or onto the surface of thesea, it is obvious that the waves do not have the same velocity and even, thatthe speed depend upon the frequency. Let just assume that all the waves arepure sinusoid. They come from every direction, with various amplitude, variousfrequency and not in the same time. If we only in only one direction, we shallwrite:

η =∞∑i=0

∞∑j=0

∞∑n=0

ai sin(kjx− ωjt− φn), (2.19)

where ω2j = gkjη is the sea level eleveation with a zero mean level (〈η〉 = 0. We

may also assume that this wave field can be considered as as stationnary randomnormal. That is to say the process is gaussian and the probability distributionis:

f (η) =1√2πσ

exp

(− η2

2σ2

), (2.20)

where σ2 is the variance computed from the spectrum, S (ω)

σ2 =⟨η2⟩

=

∫ ∞

0

S (ω) dω. (2.21)

3Engineers and oceanographers use also wave spectrum such as JONSWAP spectrum4Joint North Sea Wave Atmosphere Program

26

As I said before, the most used spectrum are JONSWAP, for the North Seaand the Pierson Moskowitz. Specrta are used in the every day calculation by

S (ω) = ag2ω−5 exp[−β (ω0/ω)4] ω0 = g/U

U = wind speed at h = 19.5m β = 0.74

S (f) = αg2 (2π)−4 f−5 exp

[−5

4

(f0f

)4]γ

exp

»− (f−f0)2

sσf20

–α = 0.076

(gXU2

10

)−0.22

f0 = 3.5(gXU2

10

)−0.33gU10

σ =

0.07 f < f0

0.09 f0 < f

Table 2.4: Two classical spectra

enginneers (see for instance a software such as Deeplines from Principia). Thefirst of this spectra is the Pierson Moskowitz spectrum. This one was proposedin the sixties and could be used in almost every sea, although it exists somemore precise and more reliable spectrum for each sea. One of this attempt wasconducted by the norwegians in the seventies and ended on the definition ofthe JONSWAP spectrum. One must be careful with this spectrum, if to becompared with the Pierson Moskowitz, because they used f = 2πω instead of ωand U10 instead of U19.5. Uh is the wind speed h meters above the sea level. Thefactors and parameters depends on the experimental campains that have beenconducted5. Neither Pierson Moskowitz nor JONSWAP take into account thewind direction. The figures6 2.4 and 2.5 provide us with a comparison betweenthe two spectra. Then, assuming the wind spectrum is also narrow, we coulddefine the cumulative probability function of the wave heights trough a Rayleighdistribution:

P (H) = exp

(−H2

8σ2

)(2.22)

Thus P (H) is the probability that wave heights exceed the level H. When weare dealing with wind waves, the significant wave height is denoted HS, which isdefined as the average of the third of the highest waves in time series. HS couldbe written in this way:(

3√

2π erfc(√

ln 3 + 2√

2 ln 3))

σ∼= 4σ, (2.23)

where erfc is the complementary error function:

erfc(x) =2√π

∫ +∞

0

e−t2

dt.

5this is the reason why it seems a little bit strange6Both figures come from

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_04.htm

27

Figure 2.4: Wave spectra of a fully developed sea for different wind speedsaccording to Moskowitz (1964).

Using the Rayleigh distribution 2.22 and the equation 2.23, we obtain an expres-sion of the probability which depens only on the significant wave height:

P (H) = exp

(−2H2

H2S

)(2.24)

If we suppose that waves with Hf > 2HS could exist, we may want to estimatethe probability, using 2.24. The probability is therefore P (2HS) = 0.000336.Other way can be explored if we choose Hmax in a group of N waves with theprobability P (Hmax) = 1/N , we should find that

Hmax∼=

√lnN

2HS. (2.25)

Since HS is always more or less the same, Hmax will increase with N , which isproportional to the duration of the recording. Oceonagraphers mostly used the3 hours duration, which is caractersitic of most events at sea.

Waves are also created by the wind. On a calm day, surface tension balancesthe effects of the wind and the waves cannot grow up. But if the wind speed

28

Figure 2.5: Wave spectra of a developing sea for different fetches measured atJONSWAP. From Hasselmann et al. (1973).

is up to 4 knots (2 Beaufort), waves appear. According to Lawton (2001), ahurricane-force wind blowing for one hour will generate 17 700 km further 4.2height waves. After 24 hours, that would grow up to 14 meters. And this couldleads to a 21 meters water wall. This phenomenon is cause by spatial focusing(see 2.4.3).

2.4 Weaknesses of these models

2.4.1 Events not taken into accountBut these model do not take into account some real phenomena such as giganticwaves or even solitary waves. Evidences of such waves were found very early, bythe engineer John Scott Russell on the Union canal in 1834, when he followedthis solitary wave:

I was observing the motion of a boat which was rapidly drawnalong a narrow channel by a pair of horses, when the boat suddenly

29

stopped - not so the mass of water in the channel which it had putin motion; it accumulated round the prow of the vessel in a stateof violent agitation, then suddenly leaving it behind, rolled forwardwith great velocity, assuming the form of a large solitary elevation,a rounded, smooth and well-defined heap of water, which continuedits course along the channel apparently without change of form ordiminution of speed. I followed it on horseback, and overtook it stillrolling on at a rate of some eight or nine miles an hour, preserving itsoriginal figure some thirty feet long and a foot to a foot and a half inheight. Its height gradually diminished, and after a chase of one ortwo miles I lost it in the windings of the channel. Such, in the monthof August 1834, was my first chance interview with that singular andbeautiful phenomenon which I have called the Wave of Translation.

J. Scott Russell. Report on waves, Fourteenth meeting of theBritish Association for the Advancement of Science, 1844.

Then in 1895, Korteweg and De Vries derived the famous KdV equation (Ap-pendix D: ∂η

∂t+c(1 + 3η

2h

)∂η∂x

+ ch2

6∂3η∂x3 = 0), which explains the appearance of such

a wave in shallow water. On the sea, mariners presented evidences of this kind ofwaves. But since 1995 and the Draupner wave, the scientist were not very keen

Figure 2.6: A freak wave over a ship Olagnon (2002)

on accepting theses kind of waves, for they did not fitted the classical theories.

30

The 1st of January, scientists recorded a unusual wave elevation on the Draupnerplatform in Norway. Figure 2.7 is the recorded wave elevation measured this day.In this case, the mean surface elevation was around 8-10 meters, and a unique 24

Figure 2.7: Draupner wave record

meters high wave just rushed onto the platform. This evidence of the reality ofthis wave leads the hydrodynamics onto new path. They just discovered that thelinearized equation failed to explained these waves. After this event, a definitionwas given for this waves: a freak wave is a wave which amplitude (or elevation)is as musch as twice of the mean amplitude (or elevation). In the case of thisDraupner wave, it is obvious that they had deal with a freak wave.

• 1943, North Atlantic. Cruise liner Queen Elizabeth is hit by two massivesuccessive waves. The bride window 28 meters above the water line col-lapsed.

• 1995, North Atlantic. The QE2 encounters a hurricane. The events havebeen recorded by Captain Ronal Warwick.

• 1998, North Atlantic. Schiehallion, a BP flotation production platform ishit by a wave 18 meters above the waterline.

More evidences could be find on Olagnon (2002) web page or in Lawton (2001).

31

2.4.2 First explanationsSince this is a new phenomena, scientists wanted to find the causes and moreover,they wanted to put it into a new model. First noticeable events were submarineearthquake, which produce a gigantic wave. This wave propagate on kilometersin the sea, and when it reach a beach (i.e. when the depth decrease), the wavebreaks. This phenomenon is rather well known for the recent events in Malaysia.In this case, the generation of the wave is mostly done by the earthquake itself.Then, since the water wave propagation is mostly a non dispersive phenomenon,this wave keeps its energy and lost it only when it break on the beach. That whythey are so dangerous.A BBC program (BBC2 (2002)) provides us with some evidence and some re-constitution of freak waves events. Some of the events are known to be on thesouth-east coast of Africa (see figure 2.8). In this area 20 vessels have been struck

Figure 2.8: Agulhas current

by waves off the South African coast since 1990. All the ships had been at theedge of the Agulhas Current, the meeting point of two opposing flows mixingwarm Indian Ocean water with a colder Atlantic flow. Radar surveillance bysatellite confirmed that wave height at the edge of this current could grow wellbeyond the linear model’s predictions, especially if the wind direction opposed

32

the current flow. So, the problem was apparently solved. Vessels just had toavoid some strong current area under some special weather conditions. This wasvery interesting for both naval architects and insurance. But then it was not theonly explanation. A short time after this discovers, two vessels designed for cruis-ing into the south Atlantic ocean encounter this gigantic waves in an area wherethere is no current. As well, no strong current is known in the neighbourhood ofthe Draupner platform. Clearly, there was another effect investigators needed tofind.

Even more information could be find about this very subject in Lawton (2001).Stating clearly that this rogue waves are not seafearing myth as mermaids orkraken, he give some evidences of this amazing waves. Witnessed by shipmateLijour on his Esso tanker in 1980, he had the opportunity to grab his camera andgive the oceanographers a proof of the existence of such waves.

2.4.3 A need for a complete explanationOne of the main issue about freak waves is to know how a wave field could leadto such a gigantic wave. Haver and Andersen (2000) ask the question about freakwaves:

Are they extremly rare realizations of a typical slightly non-Gaussianpopulation? – or – Are they typical realizations from are a rare stronlynon-Gaussian population?

Faced with the failure of the standard theory, scientists tried to use non–linearapproach. This kind of interaction, widely known as chaos, may be the cause ofthis problem. For this section, I am using information provided by Kharif andPelinovsky (2003) and by Trulsen (2005)

Linear mechanisms

Spatio-temporal focusing Every one at sea have seen this phenomenon: Shortwaves are slower than long waves. So then just suppose that the short onescome before the long ones. Obviously, the long waves will overtake theshort waves. In some case the waves merge at some fixed location. This isthe so-called superposition phenomenon. Since this phenomenon is due toreal wind wave, the mechanism is not reliable, and this event is rare. Whenit appears, it do not last very long.

Wave-current interaction According to Kharif and Pelinovsky (2003), we mayconsider a sea surface current which flows with the horizontal velocityU (x, y). We may introduce a correction to the dispersion relation:

ω = Ω (k) + ~k · ~U (x, y) ; Ω =√gk. (2.26)

33

I f we suppose the current steady and only propagating in the x direction,against the waves, we may admit that it exists one point x0 where the wavesare blocked. This could be write:

cgr =dω

dk=

1

2

√g

k+ U (x0) = 0. (2.27)

In this process, the wave number k increases. The wave amplitude A couldthen be deduced from

∂

∂t

(A2

Ω

)+∇ ·

(cgrA

2

Ω

)= 0. (2.28)

Thus the wave amplitude derived at the blocking point is

AcA0

∼(

Ω

dU/dx

)1/6

, (2.29)

where A0 is the amplitude of the incident wave and Ac the amplitude ofthe crest.

Non-linear theories

Althought the main mechanisms accounted for freak waves formation are de-scribed within the framework of linear theory, we should investigate onto non-linear effect. They could be divised into two types: weakly non-linear and strongnon-linear. The first one is only a non-linear correction on the equation, whereasthe second is in fact completely different. About the weakly non-linear effect, thepoint is to know if it will destroy the freak waves or, on the contrary, reinforcethe phenomenon.

Weakly non–linear models use linearized Euler’s equation but a non–lineardispersion relation (i.e. ω2 6= gk).

Weakly non-linear rogue waves When dealing with deep water, one of themost used equation is the non-linear Schrödinger equation (see section 3,page 36). This lead to the Benjamin–Feir instability. Solution of the theNLS equation could also be the so-called breather. This breather couldbe either periodic on time, either periodic in direction, either both. Hereis a representation of the analytic breather. Figure 2.9, known to be thePeregrine breather, is a representation of a unitary analytic breather (i.e.a0 = 1) using this:

a (x, t) = a0 exp(iω0t)

[1− 4 (1 + 2iω0t)

1 + 16k20x

2 + 4ω20t

2

]. (2.30)

In this case, even if the initial amplitude is a0 = 1, the max height of thebreather is a = 3. Looking to the classical definition of a freak wave, we can

34

Figure 2.9: Analytic breather

say that breather are even rogue waves (a rogue wave is a wave which heightis more than twice the significant height HS). In the case of this non-linearapproach, using the non-linear dispersion relation, the system of equation2.3 becomes elliptic instead of hyperbolic. Interpretation and resolutionare therfore much more complex. However, equation such as Korteweg deVries7 or NLS do not change if A is transformed into A∗, where A∗ denotesthe complex conjugate. This is also equivalent to change x → −x andt → −t. Then we shall only consider a Cauchy problem with a singularinitial data. If this initial profile (Gaussian or even Dirac δ function evolvesinto a classical wave field, by reversing time and space scale, this wave fieldcould evolve into a freak wave.

Fully non-linear models Only non-linear mathematical models have been de-velopped to handle this problem. One of them is the Fructus et al. (2005)model . These models could also include wave current interaction, or evenother interaction, such as the one with wind or with the bottom, see Cla-mond et al. (2005).

7This equation implies that we are dealing with shallow water

35

Chapter 3

Nonlinear Schrödinger equation

We may assume in this chapter that we are dealing with the linear form ofEuler’s equations. However, we may use a non–linear dispersion relation. Thecombination of these two assumptions will lead to the Non Linear Schrödingerequation. This is also the reason why this theory is called weakly non linear.Only the dispersion relation is non linear whereas the next chapter will deal withfully non–linear equation.

This equation is used in very different fields of mechanics, but we are usingit in hydrodynamics. Interests to this equation come mainly from the fact thatthe classical theory failed to explained some phenomena such as the “new yearwave” (see 2.4).The main non linear effect on waves are described by the non-linear Schrödingerequation (see Yuen and Lake (1982)), hereby referenced as NLS.

How do we obtain this equation, this is a point I shall explain. There are twomain approach to derives the NLS equation: heuristic or more theoretical ones.Since the heuristic approach is maybe the easier to understand, I shall not use itin this report. Anyway, this approach could be find out from the complete andformal one.

3.1 Derivation of the NLS equationActually, to understand where does this equation come from, we shall introducethe wave train concept

3.1.1 Concept of a wave trainThis concept includes the wave concept and try to make him more general. Inthe waves that are described by the equation 2.16, we do not allow the mainparameters (i.e. wave frequency ω, wave amplitude a and wave vector k) tochange. We now allow these parameters to vary; we even assume that their

36

variations are slow, compare to the time and length scale of the wave, we maynow be a little more clear about this concept. We introduce a phase function θ onwhich depends either η and φ. This function is linked to both the wave numberand the wave frequency by:

θt = −ω, ∇θ = k (3.1)

One can now use the variational principle of the Lagrangian (Yuen and Lake,1982). This yields either the energy equation(

a2)t+(Cga

2)x

= 0, (3.2)

where Cg is the group velocity, define as

Cg =∂ω

∂k(3.3)

and a is the wave amplitude such as in η = a cos(kx− ωt). Following Yuen andLake (1982), the variational principle yields also the nonlinear dispersion relation:

ω =√gk(1 + (1/2)k2a2 + axx/(8k

2a))

(3.4)

Since the wave propagation is a conservative phenomena, we shall use θxt = θtx,which can be easily written as

kt + ωx = 0. (3.5)

Equations 3.2, 3.4 and 3.5 form a complete set of equations for the tree parametersa, k and ω.

3.1.2 PropertiesWe shall here restrain to 2D deep water waves, propagating in the x direction.z represents the depth. This considerations, however not truly representative ofthe nature of the real ocean, are sufficient for a first approach of the phenomena.Thus, it allow us to avoid to complexity of a tri-dimensional derivation an put allour effort onto the non linear effect. In a further section, we will give additionalinformation to extend the equation to a real tri-dimensional simulation.

Moreover, we have made some strong approximation (ak 1) about theweakness of the non linearity, in order to keep a low leading order in our equa-tion. But even with this, non linear effect are still visible. Using the generalassumption made about these wave trains, we could write the frequency ω andthe wave number k as a sum of to terms, one which do not vary and a secondwhich is varying. This approach is very common in fluid mechanics (acoustic orturbulence). There we should write

ω = ω0 + ω′ (3.6)k = k0 + k′ (3.7)

37

It is obvious that ω0 and k0 are given by the initial condition and that ω′ ω0

and k′ k0. Using now the phase function which have been introduced before,we can split it in to as well:

θ = θ0 + θ′ (3.8)

One very easily find that

θ0 = k0x− ω0t (3.9)θ′t = −ω′ θ′x = k′ (3.10)

Using this new expressions for θ into the generals equations 3.2, 3.4 and 3.5coming from the wave train concept, we obtain:

at +ω0

2k0

ax −ω0

8k20

(θ′xx + 2θ′xax) = 0, (3.11)

θt +ω0

2k0

θ′x −ω0

8k20

[θ′2x +

axxa

]+

1

2ω0k

20a

2 = 0. (3.12)

The introduction of the complex envelop A = a exp(iθ′) allow us to write thesetwo equation in a comprehensive one for A:

i

[At +

ω0

2k0

Ax

]− ω0

8k20

Axx −1

2ω0k

20 |A|

2A = 0. (3.13)

3.1.3 Heuristic derivation of the NLS equationAs this previous derivation is a bit complex, here is a heuristic derivation for NLS(which was used by Kharif in his 2006 lecture for the master). This derivationuse the weakly non linear dispersion relation 3.4 which comes from the Stokessolution. We still assume that ak 1. We allow k to vary slowly around a meanvalue k = k0 + k′, k′ k0. Using the non-linear dispersion relation (3.4), wetherefore obtain

ω′ − ω0

2k0

k′ +ω0

8k20

k′2 − 1

2ω0k

20a

2 = 0. (3.14)

This derivation comes from Yuen and Lake (1982). Using formal correspondencethat comes from exponential differentiation, it yields the following formal opera-tors:

− iω′ → ∂/∂t, ik′ → ∂/∂x. (3.15)

Using these two equations, we can create an operator:

P = i∂

∂t+ i

ω0

2k0

∂

∂x− ω0

8k20

∂2

∂x2− 1

2ω0k

20a

2. (3.16)

This should be applied to the complex wave envelope A = aeiθ. This is aneasiest way to understand where does the different term of the NLS equation

38

come from. It is then obvious to see that they all come from the non-lineardispersion relation1.

3.2 Time integration of this equationIn a first time, we shall only deal with the linear part of NLS. Then we shallresolve the complete equation. Since the new variable chose does not affect thecomplete equation, we shall keep them even for the complete equation.

3.2.1 Analytic solutionLinear Part

If we restrain our study to the linear part of the NLS equation:

i

[At +

ω0

2k0

Ax

]− ω0

8k20

Axx = 0, (3.17)

which can be solve either analytically or using numerical methods. This we allowus to compare the results. Using part of the method described in Yuen and Lake(1982) for steady solution, we shall introduce

X = x− ω0

2k0

t.

One can now rewrite the (3.17) according to a new function B(X, t) = A(x −ω0

2k0t, t) which gives us

iBt =ω0

8k20

BXX . (3.18)

This is a very simple equation which can easily be solved analytically. But, wewill go a little bit further, using Fourier transform. We shall note B(k, t) =F B(x, t). We shall then use the following implicit notations:

f = f(k, t)

f = f(x, t)

Thus, Eq. 3.18 now readsBt =

ω0

i8k20

k2B. (3.19)

which is even more simpler. Actually this equation is a first order ordinarydifferential equation, which solution is

B(k, t) = B0 exp

(−i ω0

8k20

k2t

)(3.20)

1Same heuristic derivation process could be easily applied to derive Korteweg-de Vriesequation in shallow water (using limited development of the hyperbolic tangent).

39

Given the initial profile of the carrier, which is

A0 = A(x, t = 0) = a0sech(√

2εa0k20x) , (3.21)

and the fact that the Fourier transform of a hyperbolic secant is a hyperbolicsecant (see Appendix B.2), we are able to write A. We may also use the fact thatthe initial equation is written with respect to x, which have been translated togive X = x− ω0

2k0t. Using the initial profile defined by 3.21, we therefore obtain:

B(k, t) = a0π√

2εa0k20

sech

(π

2√

2εa0k20

k

)exp

(−i ω0

8k20

k2t

)exp−i ω0

2k0

kt (3.22)

This equation could be written in a more comprehensive form:

B(k, t) =π√2εk2

0

sech

(πk

2√

2εa0k20

)exp

(−i ω0

8k20

k2t− iω0

2k0

kt

)(3.23)

Complete equation

The non linear part of this equation is a dispersive non linear term:−12ω0k

20 |A|

2A.This non linear term is of a great importance in the equation. Either on thephysical point of view, because it could yield some dispersion (negative term) oron the computational point of view, because solution of the linear part of theequation could be found very easily.

In this report, we are not to solve this complete equation, and we will onlyuse the classic shape of the hyperbolic secant. We can now plot the analyticsolution given by Yuen and Lake (1982): A = A(x, t) = a0sech(

√2εa0k

20(x −

ω0/(2k0)t))e−1/2iω0k2

0a20t for different values of the time. This figure should be a

reference for the numerical analysis. On this figure, we could see the evolutionof the initial soliton (parametrized by a hyperbolic secant). The shape remainedunchanged and propagate without decaying. This was the initial observationmade by the engineer John Scott Russell. Other results are provided in section5.2, page 54.

3.2.2 Numerical integration of NLSAll the numerical results have been put together in the last chapter 5. We aresolving the equation 3.19. This equation is a simple differential equation thatcould be written on the form y′ = f(y, t), where y′ = Bt and

f(y, t) = f(B) =1

i

ω0

8k20

k2B, (3.24)

40

Figure 3.1: The propagation of the soliton

which is very simple. In order to perform this calculation, a fourth order Runge-Kutta scheme is used:

yi+1/2 = yi + ∆t2 f(yi, ti)

yi+1/2 = yi + ∆t2 f(yi+1/2, ti+1/2)

yi+1 = yi + ∆t2 f(yi+1/2, ti+1/2)

yi+1 = yi + ∆t6

(f(yi, ti) + 2f(yi+1/2, ti+1/2) + 2f(yi+1/2, ti+1/2) + f(yi+1, ti+1)

) .

(3.25)This scheme could also be used with the non-linear equation. There is no

assumption made on the form of f . This functional depends only on a functionand the time. It does not exist any counter indication to use a non linear f . Inthis case, f becomes

f(y, t) = f(B) =1

i

ω0

8k20

k2B − 1

2ω0k0F

|B|2B

. (3.26)

Note that in this case, we must compute the FFT of the product at each stepof the integrator, which requires much more time. The implementation of theRunge and Kutta scheme must be done carefully.

Both implementations have been made, see Chapter 5

41

This chapter just show a first approach of the freak wave phenomenon in deepwater. NLS equation just that if such a wave exists, it will propagate withoutchanging shape (assuming that we are in a non-viscous fluid). This is already aresult of interest.

42

Chapter 4

Spectral methods

As it appears in the two previous examples, we are using Fourier transformsfor the most part of the calculation. In fact, this massive use of these trans-formations come from spectral and pseudospectral methods. I shall then give aquick explanation about this spectral and pseudo spectral methods. As it is saidin Fornberg (1998), these methods are well used in solving partial differentialequations (PDEs) that cannot be solved numerically. Most of the time, the firsttaught which arise for solving PDEs is using those finite element or finite volumemethods. This methods are well suited for complex geometry, when no simplifi-cation could be made. But if you work in some very simple geometry as box orsphere, spectral methods offer either superior accuracy and cost efficiency.

Finite difference methods are closely related to both finite difference and fi-nite volume methods. These methods use a very local approach of the prob-lem: the derivatives are calculated locally with a n-order approximation (suchas du(xk)/dx = [u(xk + h)− u(xk − h)] /2h where h is a space grid, k = 1 . . . n).This approach is close to one of the definition of the derivative:

du

dx

∣∣∣∣xk

= limh→0

u(xk + h)− u(xk)

h.

This method gives also exact results for low order polynomial. But one mightalways be very careful with derivation when it is not performed numerically,because it could lead to very strange results as shown in appendix A.

On the contrary, spectral methods are based on a global view of the problem.The function we want to find is approached over the entire domain by a sum ofsmooth function:

u(x) =N∑k=0

akφk(x),

where the φk are functions such as trigonometric functions. In the present time,these spectral methods are used in turbulence modeling, weather prediction, non-linear waves. . . Fornberg (1998, pp. 103-140) provides us with a rather good

43

example using the well known Korteweg-de Vries equation1:

ut + cux + αuux + βuxxx = 0. (4.1)

I will not develop this example in this report, since I will mostly be dealing withthe cubic nonlinear Schrödinger equation1:

iut + uxx + 4a |u|2 u = 0, (4.2)

once normalized. The other example I will use is based on the JCP paper byFructus et al. (2005) which is using spectral method for computing the free surfaceelevation. This article is in fact using pseudo spectral methods. We call itpseudo spectral, because some of the computations are made in the spectralspace, whereas other computation are made in the real space. For the sake ofsimplicity, when the calculation is performed in the spectral space, we focus onthe global contributions, whereas the other focus on local contributions.

4.1 Three dimensional waves simulationThe main point of this section is to give an insight for the computation of fullynonlinear waves in tri-dimensional. The main algorithm is provide by what canbe found on different papers Clamond and Grue (2001); Clamond et al. (2006);Fructus et al. (2005). The aim is just to provide the reader with a full com-prehension of the main part of this code: the use of the Green function and itsimplication when we are in the Fourier Space. The main point that gives itsspeed to this method is the fact that we are able to split the Green integral infour new integral which are either Fourier transform or integral with a kernel inO(R−2). This decaying is given by the use of the third dimension in the Greenfunction.Following Zakharov (1968), one of the interesting points of Fructus et al. (2005);Clamond and Grue (2001) is to rewrite the Euler equations at the free surface,using the tilde notation to denotes the quantities on y = η. Therefore, theyobtained

u =∇φ− V∇η + (∇η ×∇φ)×∇η

1 + |∇η|2, v =

V +∇η · ∇φ1 + |∇η|2

(4.3)

where V = ∂φ/∂n√

1 + |∇η|2 and ~n is the unitary outward normal. Thus Eulerequations now reads

ηt − V = 0, (4.4)

φt + gη +1

2u · ∇φ− 1

2vV = 0, (4.5)

1These are the mathematical form of the equations.

44

4.1.1 Obtention of VSince V is the main unknown in equations 4.4 and 4.5, we want to compute it.In fact, the Laplace equation ∆φ = 0 is solved by the means of a Green function(See appendix C) ∫

S

1

r

∂φ′

∂n′dS ′ = 2πφ+

∫S

φ′∂

∂n

(1

r

)dS ′ (4.6)

In this equation φ = φ(x, t), φ′ = φ(x′, t), r2 = R2 + |z′ − z|2. R is thedistance between the field point x′ and the source point x R = |x′ − x| with theconvenient notation

x = (x1, x2)

and ∫• dx′ ≡

∫• dx′1dx

′2

Clamond and Grue introduced in Clamond and Grue (2001): D =[η(x′, t)− η(x, t)] /R = [η′ − η] /R . The elementary surface can be expressedby

dS ′ =

√1 + |∇′η′|2dx′1dx′2 . (4.7)

Using this notation, it’s easy to write:∫S

1

r

∂φ′

∂n′dS ′ =

∫S

1

r

∂φ′

∂n′

√1 + |∇′η′|2dx′

=

∫S

1

rV ′dx′

=

∫S

V ′(R2 + |z′ − z|2

)1/2 dx′

=

∫S

V ′

R(1 + |η′−η|2

R2

)1/2dx′

=

∫S

V ′R−1

(1 +D2)1/2dx′

45

∫S

φ′∂

∂n′

(1

r

)dS ′ =

∫S

φ′∂

∂n′

(1

R (1 +D2)1/2

)dS ′

=

∫S

φ′

[1

(1 +D2)1/2

∂R−1

∂n′+R−1 ∂

∂n′

(1

(1 +D2)1/2

)]dS ′

=

∫S

φ′

[1

(1 +D2)1/2

∂

∂n′

(1

R

)− 1

R

D

(1 +D2)3/2

∂D

∂n′

]dS ′

=

∫S

φ′R ∂∂n′

(R−1) (1 +D2)−D ∂D∂n′

R (1 +D2)3/2dS ′

We shall now use the definition of a derivative with respect to ~n:

∂f

∂~n= ~∇f · ~n.

Finally, we find that∫S

φ′∂

∂n′

(1

r

)dS ′ =

∫S

φ′ (R · ∇′η′ − η′ + η)

(1 +D2)3/2

dx′

R3(4.8)

4.1.2 Reformulation of the boundary integralsIn our case, we suppose the water depth infinite, so we might consider the fol-lowing equation, deriving from the previous ones.∫

V ′

(1 +D2)−1/2

dx′

R= 2πφ+

∫φ′(R · ∇′η′ − η′ + η)

(1 +D2)−3/2

dx′

R3(4.9)

These integral are going to be rewrote using the fact that

R · ∇′η′

R3− η′ − η

R3= −∇′ ·

[(η′ − η)∇′ 1

R

]. (4.10)

The interest of this reformulation is to keep the lowest order as convolution andthen the remaining integrals will have fast-decaying kernels. This will allows usto perform truncated integration and even in a first time, neglect the influence ofthis terms. Then using the Gauss theorem, we could rewrite equation 4.9. Thistheorem reads ∫∫∫

V

(∇ · F )dV =

∫∫∂V

F · dS, (4.11)

where V is a subset of Rn and has a piecewise smooth boundary ∂V whichoutwards normal is dS. This theorem is therefore used with n = 2, so ∂V is aclosed line. Gauss theorem is used to obtain that:∫∫ ∞

−∞∇′ ·

[φ′(η′ − η)∇′ 1

R

]dx′ = 0. (4.12)

46

To obtain a more interesting way of writing V ′, we obviously used the classicalcomputation

V ′R−1

(1 +D2)−1/2= V ′R−1 −

(V ′R−1 − V ′R−1

(1 +D2)−1/2

). (4.13)

We keep on the left hand side only the term in V ′R−1 and on the right hand side,only the terms containing the rest. This yields to a implicit scheme, but someof the calculation will be explicit. According to Fructus et al. (2005), we coulddecompose V = V1 + V2 + V3 + V4, where:∫

V ′1R

−1dx′ = 2πφ (4.14)∫V ′

2R−1dx′ =

∫(η′ − η)∇′φ′ · ∇′R−1dx′ (4.15)∫

V ′3R

−1dx′ =

∫φ′[1− (1 +D2)−3/2

]∇′ · [(η′ − η)∇′R−1]dx′ (4.16)∫

V ′4R

−1dx′ =

∫φ′[1− (1 +D2)−1/2

]dx′ (4.17)

V3 & V4 are computed by the means of local integrals with fast decaying kernels.In a first approach, we can neglect their influence upon the final result. Inversionof equations 4.14 and 4.15 provides us with a nice formulas of computations, ifyou exploit the fact that

FR−1

= 2πk−1e−ik·x

′, (4.18)

see Appendix B for more information about Fourier transform and the derivationof this result.

V1 = F−1kφ

(4.19)

V2 = F−1−kF ηV1 − ikF

η∇φ

(4.20)

In fact, this come from rewriting 1− (1 +D2)−1/2 into the sum

1− (1 +D2)−1/2 =1

2D2 −

[(1 +D2)−1/2 − 1 +

1

2D2

](4.21)

This will allow us to neglect the local contribution in a first time. We could findout different way of writing this sum which will contribute to the the neglectingof the integrals.

47

4.2 Time integrationEquations 4.4 and 4.5 allows us to integrate η and φ. Using Fourier transform,we may write it as

ηt = V

φt = 12F vV − 1

2F

u · ∇φ− gη

. (4.22)

In fact, we should extract the linear part of this equation. Thus, it is moreinteresting to write it in this way: kηt = kφ+ k

(V − V1

)φt = −gη + 1

2FvV − u · ∇φ

. (4.23)

Using then the linear dispersion relation for deep water (ω =√gh), we may write

the system 4.23 as matrix equation:(kηkωgφ

)t

+

[0 −ωω 0

](kηkωgφ

)=

k(V − V1

)kω2g

FvV − u · ∇φ

. (4.24)

We can now use a more comprehensive form as in Fructus et al. (2005); Clamondet al. (2006). Using obvious notation, we may consider the following differentialequation:

Ft + AF = N , (4.25)where A is 2× 2 matrix.

Note that N contains all the non-linear term. So if we neglect them, it isequivalent to write N = 0. As well, some other terms can be added as damp-ing pressure, capillarity effects or vertical velocity, if we consider an immaterialsurface. This is a simple first order differential integration, and we may solve itanalytically. This kind of solution may be always prefered, when possible, for thesake of accuracy. Analytical integration of such a simple integration is very easyto compute and the results will be much more interessting.

The first and most used way to deal with this equation is to integrate thewhole equation numerically. This method have some drawback because we hadsome numerical errors due to the integration of the linear terms of the equation. Ifthese errors are important, they even can hide the non–lineart effects. Therefore,we will first integrate the linear part.

4.2.1 Analytical part of the equationIntegration in Fructus et al. (2005) is written as

F (k, t) = exp[A(t− t0)

]G(k, t), (4.26)

48

where the matrix exponentiation is not defined. However, we could use the classiclimited development of the exp function:

exp(x) = 1 + x+x2

2+x3

6+ · · ·+ xn

n !+ · · · (4.27)

Since A is a square matrix, we could calculate its square and its cube. I willperform this calculation:[

0 −ωω 0

]2

=

[−ω2 0

0 −ω2

](4.28)[

0 −ωω 0

]3

=

[0 ω3

−ω3 0

](4.29)

This could be easily extended to the following result:

∀n ∈ N[

0 −ωω 0

]2n

=

[(−ω)2n 0

0 (−ω)2n

](4.30)

∀n ∈ N[

0 −ωω 0

]2n+1

=

[0 ω2n+1

−ω2n+1 0

](4.31)

Using this and substituting x by A in equation 4.27, we obtain for exp(At)[1− (ωt)2

2+ · · ·+ (−1)n(ωt)2n

(2n) !−ωt+ (ωt)3

6+ · · ·+ (−1)n(ωt)2n+1

(2n+1) !

ωt− (ωt)3

6+ · · ·+ (−1)n+1(ωt)2n+1

(2n+1) !1− (ωt)2

2+ · · ·+ (−1)n(ωt)2n

(2n) !

]. (4.32)

So now, we could write in a very more comprehensive form

exp(At) =

[cos(ωt) sin(−ωt)sin(ωt) cos(ωt)

], (4.33)

which is an easiest way for performing the computation. This way of writingthis is also given directly by Clamond et al. (2006). This equation is then solvedwith the use of a forth order Runge and Kutta scheme. The use of this schemeis discussed in 5.2.1.

4.2.2 Linear problemIf we study only the linear problem, neglecting also V 2, we may obtain fromequation 4.24 a very simple expression for η and φ. This expression is written inthe Fourier space, but the translation to the real space is easily performed. Thus,we have

η = η0 cosωt+ω

gφ0 sinωt (4.34)

φ = φ0 cosωt− g

ωη0 sinωt (4.35)

49

4.2.3 Non linear problemAliasing could be a problem when dealing with product in physical space (seeCanuto et al. (1987) for more information about it). Therefore, we want tocompute desaliased product. In this case, dealiasing will only be performed onthe V2 term. This dealiasing technic is only used for the calculation of productssuch as FFT f g. Here is the scheme used.

fpad = FFT−1 FFT f, 2Ngpad = FFT−1 FFT g, 2N

⇒ FFT f g → FFT fpad gpad, N (4.36)

Both f and g are numeric functions of N discret points. We may assume thatFFT is a function defined as following2:

• if one argument, which sould be an array, then it computes the FFT of thisarray

• if two arguments are given (first array, second integer)

– first argument array– truncate the length if N is smaller than the length of the array– pads with trailing zeros if N is greater than the length of the array

Using this method, we avoid aliasing. It is obvious that the final results havethe same size as before. This dealiasing is performed either on the F η V1 andFη∇φ

which are both present in V2 which are both present in V2 (equation

4.20).

2This a simplification of the real Matlab FFT function.

50

Chapter 5

Numerical solution

5.1 Stokes waveTrying to solve linear problem and using a first order plain Stokes wave (i.e. φ =ag/ωek0z sin(k0x− ωt)). Since we know φ, we want to know also the derivativesinvolved. We easily obtain that

φt|z=0 = ag cos(k0x− ωt) (5.1)

φz =agk0

ωek0z sin(k0x− ωt) (5.2)

The aim is to solve this problem using spectral methods (See 4, page 43). Inthis case, this set of equation is only true on the free surface, i.e. z = η or ifwe consider only a small disturbance z = 0. In the spectral space, the equationsread:

φt

∣∣∣z=0

= age−iωtF cos(kx) (5.3)

φz =agk

ωekze−iωtF sin(kx) (5.4)

We just used the fact that F f(x− u) = e−ikuF f(x) . Then, we must com-pute F cos(kx) . Since we are dealing with Fourier Transform in Matlab, thefunction are sampled on a finite number of point. Thus, f(x) becomes f(xn) = fn.As well in this case, Matlab uses Fast Fourier Transform (FFT) which are de-fined by:

fk = FFT(f) =N∑j=1

fje−i2π/N(j−1)(k−1) (5.5)

fn = FFT−1(f) =1

N

N∑k=1

fkei2π/N(j−1)(k−1), (5.6)

51

whereas Fourier Transform is defined by:

f(k) = F (f) =

∫ ∞

−∞f(x)e−ikxdx (5.7)

f(x) = F−1(f) =1

2π

∫ ∞

−∞f(k)eikxdx. (5.8)

In the case of periodic function, the length L should be a period. This avoidthe formation of Dirac δ function, which is the normal transform of a sine or acosines.

5.2 Non linear Schrödinger equationUsing the classic Runge-Kutta integration scheme as described in section 3.2.2,equation 3.25. In fact, this scheme is more interesting than first order scheme,because it look forward. The points used for the computation of the next stepare not only the previous point. In fact, it create some intermediary point whichdo not appears at the end.

5.2.1 Reasons to use this integration schemeIn the note about time integrators for wave simulations Clamond et al. (2006),They give some explanation. Although it is not my aim to explore this in thisreport, I will give some reasons for the choice of this scheme.The first point is that each time we could perform a analytic integration, thiswill be done, for the sake of accuracy. The reasons for this choice are:

• This problem requires a scheme with a high stability.

• RK scheme allows larger step, so even if the computation is longer for eachstep, it does not increase the global time of integration.

52

If we write this scheme in a more algorithmic way, we shall have:

while t<Tfdydt = ifft(i ∗ (omega/8/k2. ∗ Kx.2 + omega/2/k. ∗ Kx). ∗ fft(y));p1 = dydt;t = t + dti/2;y1 = y + dti/2 ∗ dydt;dydt = ifft(i ∗ (omega/8/k2. ∗ Kx.2 + omega/2/k. ∗ Kx). ∗ fft(y1));p2 = dydt;y2 = y + dti/2 ∗ dydt;dydt = ifft(i ∗ (omega/8/k2. ∗ Kx.2 + omega/2/k. ∗ Kx). ∗ fft(y2));p3 = dydt;y3 = y + dti/2 ∗ dydt;t = t + dti/2;dydt = ifft(i ∗ (omega/8/k2. ∗ Kx.2 + omega/2/k. ∗ Kx). ∗ fft(y3));p4 = dydt;y = y + dti/6 ∗ (p1 + 2 ∗ p2 + 2 ∗ p3 + p4);

endAs we know the Fourier transform of the initial condition, we could compute

all in the results in the Fourier space. Then, if we compute the Fourier transformof the analytic solution, we will be able to compare the two of them. In this case,we eventually got back into the real space to compare the two results. But wecould have also done the comparison in the spectral space.

53

Figure 5.1: Analytic solution (blue plain) and RK4 scheme (red dashed)

To obtain these results, the initial hyperbolic secant have been extended tentimes both on left and right, in order to deal with a periodic, or at least pseudo-periodic function. So the true initial function is of the form:

A0(x) =10∑

i=−10

sech(aεk2(x− iLx

2)), (5.9)

where L is chosen to have only one secant in the interval [0; 2π], centered on π.This little trick have allowed us to have a correct time integration in the Fourierspace. Thus, we can even go back in the physical space. An explanation of thiscould be find in AThis integration have been performed over 75 periods Tp = 2π

ω0which is quite long

time. In this case, the propagation is done in the direction −x. We can alsonotice that the numerical scheme as some dispersion, for the amplitude is lessthan the analytical one.

In order to validate my own implementation of the Runge and Kutta scheme,I compared it to the Matlab embedded time integrator. I have solved the sameproblem by either way and thus obtained the same results:

54

Figure 5.2: Same comparison as figure 5.1: analytic solution (blue plain) andMatlab embedded time integrator (red dashed)

This computation have been performed using the Matlab ode23t solver.As appendix A.1.2 show that calculation should be performed over an extendedperiodic function, I used the same function (i.e. equation 5.9). As well, we havea decaying of the amplitude when the equation is solved numerically. As well forthis we could have stop on the Fourier transform (computed by FFT).

55

Figure 5.3: FFT obtained by RK4 scheme

56

Figure 5.4: FFT obtained by Matlab

Comparison between the FFT of the two results (Matlab embedded timeintegrator or my own RK4 scheme) provides us with the same results both in thespectral space (figures 5.4 and 5.3). Impression is also reinforced by the compar-ison made in real space between the two computations, as shown on figure 5.5.Since the amplitudes are exactly the same, this is a proof that my implementationof the Runge and Kutta scheme is correct. This scheme will be massively usedin Section 4, so I want it to be accurate and fast.

57

Figure 5.5: Comparison between the two methods

These results were provided using a hyperbolic secant as initial condition.Using a different initial condition (i.e. a sine wave), we may obtain some differentresults: figure 5.6

Due to some Matlab approximation in the computation of FFT, it is advis-able to force it to be real. This explains the lines under the curve.

58

Figure 5.6: RK4 integrator for NLS using a sine wave

Since the integration is performed in the Fourier space, we easily get theresults (see figure 5.7), and we obtain two Dirac δ function, which is the Fouriertransform of a sine function. The real space result do not exactly look like a sinefunction, but figure 5.7 prove it quite well. Since the results with my own codeproved good accordance with the one obtained by using the Matlab embeddedtime integrator, I will not present the sine wave comptation performed usingMatlab integrator.

In this case, it is none that obvious that we have a sine wave, due to Matlaberrors. To ensured this, looking the FFT of this signal will prove that we have asine function.

59

Figure 5.7: FFT of the sine wave

This FFT just prove that the previous signal was indeed a sine wave. Wecould also combine the wave field ans the hyperbolic sechant into a more realisticpoint of view. Thus the hyperbolic secant is indeed a freak, because its height istwice the wave height.

60

Figure 5.8: Sine wave with a hyperbolic sechant

The little decreasing in height is only due to round off errors in Matlab andnot to the physics of the process involved.

5.3 Wave simulationThis section will provides you with some results obtained with the code whichis described in Clamond and Grue (2001); Fructus et al. (2005). The theorybeneath this code is mainly described in 4, so I will not give more informationabout it. Although, before discussing the results, it seems to be interesting tohave the code in an algorithmic way.

5.3.1 AlgorithmThis section deals only with the algorithm and not with its implementation inMatlab.

61

Algorithm:

cefinition of the domain (real and spectral)

choosing the wave number and deducing the wave period

choosing some initial condition eta0 and phi0

while t<Tfcalculation of the inner variablesintegration in timecalculation of eta and phiincrementation

end while

plotting the outputWhen dealing with this kind of algorithm, it is common to define and documentthe input, the output and the inner variables This is the complete set of

Name Variable used CommentsInput

Length of the real domain Lx,Ly Used to define thespectral spaceNumber of points Nx,Ny

Gravity gDuration of the simulation Tf Number of periods

Time increment dtiOutput

Free surface elevation eta PlottedVelocity potential phi Not plotted

Inner variablesWave amplitude a Deduced from kHorizontal speed u(:,:,1),u(:,:,2)

Vertical speed vη derivatives neta(:,:,1),neta(:,:,2)φ derivatives nphi(:,:,1),nphi(:,:,2)

Normal velocity V

Table 5.1: Parameters and variables

parameters is used by the code. We could have added some more paramters, tocontrol more precisely the time integration or the local integrations as in Fructuset al. (2005).

5.3.2 Linear simulationIf we use only the linear part of the equation, it is very easy to find out someresults.

62

Figure 5.9: Propagation of a linear sine wave

As well, in a 3d representation, we obtain:

63

Figure 5.10: Propagation of a linear sine wave (3D)

We shall then compute something much non–linear, that is to say, includingV2. In this case, we use completly the spectral methods. It is at this point thatcomputation problems arise. The principal problem with spectral methods, isthat if something is going wrong at any point, the method will not provide anyresult. This drawback is also an advantage for it prevents from wrong analysis.Thus computations work and are correct or the do not work.

The first problem to arise, in my opinion was due to aliasing effect (Canutoet al. (1987)):

64

Figure 5.11: Aliasing effect in the computation of product

This kind of oscillation aroud the curve, which seems to be a modulationis typical of aliasing effects. Aliasing usually appears in convolution products.Since there is no explicit convolution or deconvolution, it comes from the FFTof a product. Thus the problem comes from the two products that are involedin V2 To avoid this effect, the “trick” is to use zeros-padding. The point ofzeros-padding is to add point in the sample.

65

Figure 5.12: Zeros padding

On this figure, blue crosses on the blue curve correspond to the initial paddingof the function, whereas the red circles correspond to the function after zeropadding. This correspond to a resample of the initial function with twice thenumber of point.

In a more mathematical form, this will then give

fpad = FFT−1 FFT f, 2Ngpad = FFT−1 FFT g, 2N

(5.10)

⇒ FFT f g = FFT fpad gpad, N (5.11)

Since the results did not improve after this modification, it is clearly someother unsuspected effects. Thus a reimplementation of the Runge and Kuttascheme have begun. Since this was done in the very end, it was not completedand therefore not tested.

66

Conclusion

General conclusions The background needed to understand the complete sub-jetc is not as extended as one might first think. Few equations and only basics influids mechanics is required. This is not as well true for the mathematics involvedin it. Even if we did not explore this waay in this report, there are some ana-lytic results to equations such KdV or NLS. But these solutions needs a completemathematical backgroud to be discussed.

As well for numerical methods, a completly new method have been offered.This is different from finite difference methods for its accuracy stability and speed.Even with the increasing power of the computer and calculator, finite differencemethods remains to be very slow. In addition, these methods provides more roundoff errors than this new method. Even if it appears quite simple to understand,the method is not that simple to implement in a code. But, it is easily tunable,and one could include almost all effects.

Hoping that this models becomes more common in marine engineering andhydrodynamics, it will help to a better comprehension of oceans. It will bealso a great achievment in the design of offshore structures, such as oil drillingplatform. As well for ship design this will add some new construction rules but itwill prevents disparition such as the München in 1972, from which only a lifeboathave been find.

Since this fully non–linear wave simulation have also been computed witha “moving” bottom, it could aslo be used in the computation of tsunami andmegatsunami as those which have break on Java island.

Personnal conclusion On a personnal point of view, I have experienced whata research fellow position is. My time was splitted between reading scienttificpapers, to understand the theories, making my own derivations from what I wasreading and the implementation of the algorithm I was reading about. It asalways a deception to see that the derivations are not easy at i tseems in thearticle or the implementation of the algorithm do not provides results from thevery first time. It is also a deception, because there is no results to provideproving that the theory and the algorithm are right. Therefore, I was forced toexplained why the implementation did not work properly.

67

Appendix A

Differentiation using FFT

A.1 Analytic background

A.1.1 Mathematical background on Fourier transformGiven f a real function such as f : x 7→ f(x) fulfilling this three sufficientconditions:

1. f and its derivative f ′ are piecewise continuous in every finite interval[−L;L];

2.∫∞−∞ |f(x)| dx converges;

3. f(x) is replaced by 12f(x+) + f(x−) if x is a point of discontinuity.

This information about Fourier transform are taken from Spiegel (1968) We cannow define the Fourier transform F = F f of the function f as

F (k) = F f(x) =

∫ ∞

−∞f(x)e−ikxdx (A.1)

In this definition, we have used the e−ikx kernel, but some times, other kernelsare employed, including or not a 2π factor or using i instead of −i. This coulddone little changes when you are to use the derivative of a Fourier transform.As the application F is bijective, we could define the inverse Fourier transform,noted F−1 which is

f(x) = F−1 F (k) =1

2π

∫ ∞

−∞f(x)eikxdx (A.2)

68

A.1.2 Fourier transform of the derivative of a functionOne of the interesting thing about the Fourier transform is that, once you are inthe Fourier space, you can easily take the derivative of a function. Let me demon-strate it. We suppose that f and F form a Fourier transform pair, respectivelyof the variable x and k. We want to evaluate F

∂f∂x

. Thus, we have,

F

∂f

∂x

4=

∫ ∞

−∞

∂f

∂x(x)e−ikxdx

=[f(x)e−ikx

]∞−∞ −

∫ ∞

−∞f(x)

∂e−ikx

∂xdx

= −∫ ∞

−∞−ikf(x)e−ikxdx,

where 4= means is equal to by definition.

And finally, we obtain:

F

∂f

∂x

= ikF f , (A.3)

which is a very simple form. The Fourier transform converts in a simple multipli-cation a derivative. This is very interesting for numerical computations. Usingthis derivative for functions could be calculated using

∂f

∂x= F−1 ikF f . (A.4)

For multiple derivative, it is easy to fond that

∂nf

∂xn= F−1 (ik)n F f . (A.5)

As well, integration is easily computed using:∫ ∞

−∞f(x)dx = F−1

1

ikF f

. (A.6)

A.2 Numerical computation

A.3 The FFT algorithmThe aim of this paper is neither to present in an extensive way nor to discuss overthe different FFT algorithm. We are just using them as well and it is interestingto know that computation are the fastest when the number of point is a powerof two. This algorithm are implemented in most scientific coding libraries andsoftwares. One of the main point to be known is that the theory behind FFT

69

assume that the data are periodic.But for some purpose, we may have some infinite periodic function (i.e. functionwhich period is infinity). In this case, the FFT algorithm could provide us withstrange results, especially when we want to compute the derivative of a functionby FFT, using eq. A.4. In this case, we can see that the derivative is correctly

Figure A.1: Derivative of the hyperbolic secant, with Gibbs phenomenon on theboundaries

computed in the center of the discrete domain, but not on the borders. If we wantto use this derivative in a iterative code, we must have a perfect accordance to thederivative of the function, everywhere in the discrete domain. This is obviouslynot the case. So how to avoid this?

A.3.1 Transformation into a periodic functionIn fact, we must consider a "truly" periodic function, or at least a function whichis periodic in an interval. Since infinity cannot be reached using numerics, weknow that if f is a infinite period, we can transform it into a periodic function,considering a new function f such as:

f(x) =∞∑

j=−∞

f(x− jL). (A.7)

70

It is now obvious that f is L− periodic.In fact, since we are dealing with numerics, we shall reduce the scope of this sum.Depending on the decaying of the function, we must choose a number of periodto be repeated on left and right. Then, this allow us to compute the derivative ofa function with a very fair accuracy. We shall now consider a numeric functionf which is

f(xi) =N∑

j=−N

f(xi − jL). (A.8)

On the last graph, it is interesting ton notice that the maximal is 10−14, which

Figure A.2: Derivative of the hyperbolic secant

is very close to the computer precision (in this case, using Matlab, we haveeps = 2.2204 · 10−16). Here N = 10 is sufficient to have a fair enough derivativeof the function.This quite simple example show us a method for dealing with no periodic functionwhich have a sufficiently decaying.

71

Appendix B

Useful information about Fouriertransform

Firstly, we shall notice that

F f(x− u) =

∫f(x− u)e−ikxdx

=

∫f(x)e−ik(x+u)dx

= e−iku∫

f(x)e−ikxdx

F f(x− u) = e−ikuF f(x) (B.1)

This equation is as well usable for 2-D Fourier transforms.

B.1 Calculation of Fourier transform of one overR

Using equation B.1, we may may write and evaluate

F

1

R

= e−ik·x

′∫ ∞

−∞

∫ ∞

−∞

1√x2 + y2

e−ikxxe−ikyydxdy (B.2)

Firstly, we shall change (x, y) into (r, φ), using polar notations

x = r cosφ, y = r sinφ ⇒ dxdy = rdrdφ

Therefore, (B.2) becomes

F

1

R

= e−ik·x

′∫ ∞

0

∫ 2π

0

1√(r cosφ)2 + (r sinφ)2

e−ikxr cosφe−ikyr sinφrdrdφ

72

F

1

R

= e−ik·x

′∫ ∞

0

∫ 2π

0

e−ikxr cosφe−ikyr sinφdrdφ (B.3)

We shall now write, using as well polar notations

kx = k cosψ, ky = k sinψ where k = |k|

Then (B.3) now reads

F

1

R

= e−ik·x

′∫ ∞

0

∫ 2π

0

e−irk(cosφ cosψ+sinφ sinψ)drdφ

F

1

R

= e−ik·x

′∫ ∞

0

∫ 2π

0

e−irk cos(φ−ψ)drdφ (B.4)

Using nowθ = φ− ψ ⇒ dφ = dθ

and multiplying inside by and dividing outside k, we obtain

F

1

R

=

e−ik·x′

k

∫ ∞

0

∫ 2π

0

e−irk cos θkdrdθ (B.5)

Since we are dividing by k, we must assume that it is non null. But if k = 0,we are not anymore in the field of propagation of a wave. Nevertheless, this casewill be explain more precisely when it arise. We should now introduce

kr = ρ ⇒ kdr = dρ

The precedent integral (B.5) transform into

F

1

R

=

e−ik·x′

k

∫ ∞

0

∫ 2π

0

e−iρ cos θdρdθ (B.6)

We can now see that the integral in (B.6) does not depend anymore on k. In factit is just a number. Tables as Ryshik and Gradstein (1957) could provide us witha value: ∫ ∞

0

∫ 2π

0

e−iρ cos θdρdθ = 2π (B.7)

But however, we could try to find out this value. So, we must introduce a newfunction f which is defined by

f :

R+ → Rρ 7→

∫ 2π

0e−iρ cos θdθ.

(B.8)

One should notice that f(0) = 2π. In fact, there are series of function which arevery close to this function. If we just have a look to Bessel functions of the firstkind Jp and how they could be expressed using integrals, we may found that

Jn(z)4=

1

2π

∫ π

−πe−niθ+iz sin θdθ, (B.9)

73

where 4= means is equal to by definition.

Since in our case n = 0, the changing of θ = θ′ − π2

and the periodicity of thesine function, we can write B.7 merely as an integral of Bessel function. Thuswe shall now integrate and show that∫ ∞

0

J0(z)dz = 1 (B.10)

because f = 2πJ0. A combination of Bessel and Trigonometric function pro-vides us with a new integration formula:∫ ∞

0

Jp(αx) cos βxdx =cos(p arcsin β

α

)√α2 − β2

, (B.11)

β < α; <(p) > −1.

So if we choose p = 0, α = 1 and β = 0, we are still fulfilling the condition andB.11 now reads ∫ ∞

0

J0(x)dx =cos(0 arcsin 0

1

)√

12 − 02= 1 (B.12)

Eventually, we found that

F

1

R

= 2π

e2iπk·x′

k(B.13)

B.2 Calculation of Fourier transform of hyper-bolic secant

Since the hyperbolic secant is a well known function, which decays quickly(sech x → 0 when x → ±∞) we can compute analytically its Fourier trans-form. In fact, the analysis have been already made and can be found in tables asRyshik and Gradstein (1957). Thus we found that

F sech(ax) =π

asech

(πk

2a

)(B.14)

Thus we can say that the Fourier Transform of a hyperbolic secant is a hyperbolicsecant. This will allow us to work in the spectral space.

74

Appendix C

Green function history

Since the computation of the velocity V is performed using the Green functionstheory, it is worth to have a quick insight on this theory. The original Greenproblem came from the quite new to him electromagnetic theory. This was in histime one of the largest field theory employed to describe the effects of electricityand magnetism. He wanted to examine the solution of∇2u = −f within a volumeV and with some strict boundary conditions on S. The brilliant idea of Greenwas to deal with a global view and not a local view. In order to develop histheory, he first prove what is now known as Green theorem:∫∫∫

V

(ϕ∇2χ− χ∇2ϕ)dV =

∫∫S

(ϕ∇χ− χ∇ϕ) · ndS, (C.1)

where n represents the outward pointing normal. Applying this to the originalproblem which is

∇2g(r− r0) = −4πδ(r− r0), (C.2)in addition with the introduction of a small ball about the r0 point, Green ob-tained ∫∫∫

V

(g∇2u)dV +

∫∫S

g∇u · ndS

=

∫∫∫V

(u∇2g)dV +

∫∫S

u∇g · ndS − 4πu(r0)

(C.3)

because the surface of the ball reduce to 4πu(r0) as the radius tends to zero.Both g and u must satisfy the boundary conditions. So he found

u(r) =1

4π

∫∫S

u∇g · ndS, (C.4)

when f = 0 (Laplace’s equation) for any r within S. He called u the value of u onthe boundary S. The existence of g was proved by the reality of his problem: itwas the electric potential, and he was able to measure it. So Green had proved its

75

existence. Since Green did not give a name to this function g he had introduced,Riemann gave it the name of "Green’s function". Neumann studies in 2D foundthat the equivalent of this function was no more describe by a singularity of theform 1/ |r− r0| but by one of the form log(1/ |r− r0|). Thus this new "Green’sfunction" was used in more and more physics field such as for solving the heatequation.Then we must wait for Laurent Schwartz and his theory of distributions.

76

Appendix D

Korteweg-de Vries equation

D.1 Yielding the linear problemIf we neglect the stress on the free surface and the tension effects, the dispersionrelation (linking ω to k) could be easily derived into:

ω =√gk tanh(kh) (D.1)

where g is the gravity, k the wave number and h is the depth. To outline spatiotemporal focalisation, we examine the following equation for ω:

∂ω

∂t+ cgr

∂ω

∂x= 0 (D.2)

where cgr is the group velocity. Under these conditions, equation D.2 could besolved, assuming that the initial condition is given by:

ω(x, 0) = f(x) (D.3)

where f (x) is a function modelling the initial wave field at t = 0. Solutionsof D.2 are classic for this wave propagation problem. This resolution use thecaracteristic method used in Pr. Kharif courses (Lecture 4).

ω(x, t) = f(x− cgr(ω)t) (D.4)

If we assume to be in shallow waater (i.e. kh 1), a limited development of thehyperbolic tangent gives an approximation of the group velocity c0

ω =√ghk(1− 1

6k2h2) (D.5)

ω = c0k − βk3 (D.6)

Second order development is sufficent to reveal the non linear effects. The groupvelocity c0 comes from D.2. In this case, it is shown that c0 =

√gh. Using some

77

formal writing, we can now link dispersion relation and derivative.

∂

∂t→ −iω ⇒ ω → i

∂

∂t∂

∂x→ ik ⇒ k → −i ∂

∂x

Using this formalism with the equation D.6, one may find:

i∂

∂t= −ic0

∂

∂x− iβ

∂3

∂x3(D.7)

Then it is sufficient to apply equation D.7to the free surface elevation η.

∂η

∂t+ c

∂η

∂x+ch2

6

∂3η

∂x3= 0 (D.8)

This equation D.8is known as linear Korteweg-de Vries. This appraoch showsthat a giant wave evolves into a classic wave field. Since D.8 is un variant by thetransformation of t into −t amd x into −x, as shown below.

∂η

∂t+ c

∂η

∂x+ch2

6

∂3η

∂x3= 0

∂η

∂(−t)+ c

∂η

∂(−x)+ch2

6

∂3η

∂(−x)3= 0

−∂η∂t− c

∂η

∂x− ch2

6

∂3η

∂x3= 0

∂η

∂t+ c

∂η

∂x+ch2

6

∂3η

∂x3= 0

Under these conditions, the problem is solvable.

D.2 Solution of the linear problemA classical solution for this kind of problems uses spectral methods. The equa-tions are therefore written into FOurir space. Thus η (x, t) is conveniently written

η(x, t) =

∫η(k)ei(ωt−kx)dk (D.9)

where η agreesη(k) =

1

2π

∫η(x, 0)eikxdk. (D.10)

η(k) is the Fourier spectrum of the initial time. This approximation allows thenthe resolution of equation D.8. In a very simple model, freak wave is a dirac of

78

finite height η(x, 0) = Qδ(x− x0). But a gaussian shape, or a square hyperbolicsecant is much closer to the reality of the phenomenon. For the sake of simplicity,we chose a Gaussian impulse:

η(x, 0) = A0e−K2x2 (D.11)

where A0 is the initial amplitude and K−1 is the initial width of the freak wave.Analytic resolution have been performed in Pelinovsky et al. (2000) and thus give

η(x, t) =A0

K(3βt)1/3e

h(12βtK2)

−1“x−c0t+(12βtK2)

−1”i

×Ai[(3βt)−1/3

(x− c0t+

(12βtK4

)−1/2)] (D.12)

where β = ch2/6 is a dispersion coefficient and Ai (x) is Airy’s function. EquationD.12 therefore describes how a Gaussian impusle evolves into a random wave field.Using then the formalism, we can deduce that a random wave field could evolveinto a freak wave.

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