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Mathematical Modelling of Tsunami Waves Denys Dutykh Mathematical modelling of tsunami waves Denys Dutykh To cite this version: Denys Dutykh. Mathematical modelling of tsunami waves. Mathematics [math]. École normale supérieure de Cachan - ENS Cachan, 2007. English. tel-00194763v1 HAL Id: tel-00194763 https://tel.archives-ouvertes.fr/tel-00194763v1 Submitted on 7 Dec 2007 (v1), last revised 21 Jan 2008 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. No ENSC-2007/76 THESE` DE DOCTORAT DE L’ECOLE´ NORMALE SUPERIEURE´ DE CACHAN Pr´esent´ee par Denys DUTYKH pour obtenir le grade de DOCTEUR DE L’ECOLE´ NORMALE SUPERIEURE´ DE CACHAN Domaine: Math´ematiques Appliqu´ees Sujet de la th`ese: MODELISATION´ MATHEMATIQUE´ DES TSUNAMIS Th`ese pr´esent´ee et soutenue `aCachan le 3 d´ecembre 2007 devant le jury compos´ede: M. Jean-Michel GHIDAGLIA Examinateur M. Jean-Claude SAUT Rapporteur et Pr´esident M. Didier BRESCH Rapporteur M. Costas SYNOLAKIS Examinateur M. Vassilios DOUGALIS Examinateur M. Daniel BOUCHE Membre invit´e M. Fr´ed´eric DIAS Directeur de th`ese Centre de Math´ematiques et de Leurs Applications ENS CACHAN/CNRS/UMR 8536 61, avenue du Pr´esident Wilson, 94235 CACHAN CEDEX (France) To my parents, my sister and A.M. Remerciements Je tiens `aremercier, en tout premier lieu, Fr´ed´eric Dias qui a encadr´ece travail. Il m’a beaucoup soutenu et encourag´eau cours de ces magnifiques ann´ees de recherche. Je lui en suis tr`es profond´ement reconnaissant. Je remercie tout particuli`erement Jean-Michel Ghidaglia qui m’a beaucoup guid´edans mes recherches. Je tiens `asouligner que par sa grande disponibilit´e, j’ai pu b´en´eficier de ses tr`es grandes comp´etences scientifiques tout au long de mes investigations. Enfin, c’est un honneur pour moi qu’il ait accept´ede pr´esider le jury de cette th`ese. J’exprime ma gratitude aux Professeurs Jean-Claude Saut et Didier Bresch, qui ont accept´ed’ˆetre les rapporteurs, ainsi qu’aux Professeurs Costas Synolakis et Vassilios Dou- galis pour leur int´erˆet qu’ils ont port´e`amon travail en ayant accept´ede faire partie du jury. Mes trois ann´ees de th`ese se sont d´eroul´ees dans un contexte scientifique de grande qualit´eau Centre de Math´ematiques et de Leurs Applications de l’ENS Cachan. Je re- mercie son directeur Laurent Desvillettes de m’avoir accueilli et d’avoir cr´ee d’excellentes conditions de travail. Parmi tout le personnel de ce laboratoire je voudrais plus partic- uli`erement remercier Micheline Brunetti pour son aimabilit´eet efficacit´edans le traitement des questions administratives. Je remercie Fr´ed´eric Pascal pour son tutorat p´edagogique. Ses conseils m’ont beau- coup servis lors de trois ans d’enseignement du calcul scientifique aux agr´egatifs de l’ENS Cachan. J’en profite aussi pour remercier tous mes ´el`eves de leur assiduit´eet intelligence. J’exprime mes plus vifs remerciements `aDaniel Bouche de m’avoir accueilli au sein de LRC CEA/CMLA. Ses remarques pertinentes et conseils avis´es, sa gentillesse m’ont beau- coup aid´eet motiv´edurant le DEA MN2MC et plus particuli`erement pendant la r´ealisation de cette th`ese. ii REMERCIEMENTS Parmi tous mes coll`egues de LRC, Christophe Fochesato, Jean-Michel Rovarch et Denis Gueyffier ont occup´eune place tr`es particuli`ere: en plus de leur amiti´e, ils m’ont offert de nombreux conseils amicaux et professionnels. Je vous en remercie tr`es sinc`erement. Parmi les personnes qui ont contribu´e`ama formation, je suis tout particuli`erement reconnaissant envers Vladimir Lamzyuk qui m’a fait d´ecouvrir le monde de la mod´elisation math´ematique et des math´ematiques appliqu´ees en g´en´eral. Il m’a soutenu constamment durant mes ´etudes `al’Universit´eNationale de Dniprop´etrovsk. Mes pens´ees vont ´egalement `atous mes concurrents qui m’ont constamment pouss´e dans mes recherches et m’ont donn´eles forces pour continuer ce travail. Je dis grand merci `aYann, Joachim, Lavinia, Serge, Nanthilde, Thomas, Alexandra et aux autres doctorants du LMT pour leur amiti´eet tous les instants de d´etente partag´es ensemble. Je tiens `aremercier particuli`erement Alexandra Pradeau de nos discussions pas- sionnantes ainsi qu’enrichissantes. Sans ces personnes cette th`ese n’aurait pas eu la mˆeme saveur. Je remercie ma famille et leur d´edie cette th`ese pour leurs encouragements permanents et toutes autres raisons qui s’imposent. Je conclurai en remerciant de tout coeur Anna Maria pour la patience et le soutien sans limite dont elle fait preuve quotidiennement. R´esum´e Cette th`ese est consacr´ee `ala mod´elisation des tsunamis. La vie de ces vagues peut ˆetre conditionnellement divis´ee en trois parties: g´en´eration, propagation et inondation. Dans un premier temps, nous nous int´eressons `ala g´en´eration de ces vagues extrˆemes. Dans cette partie du m´emoire, nous examinons les diff´erentes approches existantes pour la mod´elisation, puis nous en proposons d’autres. La conclusion principale `alaquelle nous sommes arriv´es est que le couplage entre la sismologie et l’hydrodynamique est actuellement assez mal compris. Le deuxi`eme chapitre est d´edi´eessentiellement aux ´equations de Boussinesq qui sont souvent utilis´ees pour mod´eliser la propagation d’un tsunami. Certains auteurs les utilisent mˆeme pour mod´eliser le processus d’inondation (le run-up). Plus pr´ecisement, nous discu- tons de l’importance, de la nature et de l’inclusion des effets dissipatifs dans les mod`eles d’ondes longues. Dans le troisi`eme chapitre, nous changeons de sujet et nous nous tournons vers les ´ecoulements diphasiques. Le but de ce chapitre est de proposer un mod`ele simple et op´erationnel pour la mod´elisation de l’impact d’une vague sur les structures cˆoti`eres. En- suite, nous discutons de la discr´etisation num´erique de ces ´equations avec un sch´ema de type volumes finis sur des maillages non structur´es. Finalement, le m´emoire se termine par un sujet qui devrait ˆetre pr´esent dans tous les manuels classiques d’hydrodynamique mais qui ne l’est pas. Nous parlons des ´ecoulements viscopotentiels. Nous proposons une nouvelle approche simplifi´ee pour les ´ecoulements faiblement visqueux. Nous conservons la simplicit´edes ´ecoulements potentiels tout en ajoutant la dissipation. Dans le cas de la profondeur finie nous incluons un terme correcteur dˆu`ala pr´esence de la couche limite au fond. Cette correction s’av`ere ˆetre non locale en temps. Donc, la couche limite au fond apporte un certain effet de m´emoire `al’´ecoulement. Mots cl´es: Ondes de surface, g´en´eration des tsunamis, ´equations de Boussi- nesq, ´ecoulements diphasiques, ´ecoulements viscopotentiels, volumes finis Abstract This thesis is devoted to tsunami wave modelling. The life of tsunami waves can be conditionally divided into three parts: generation, propagation and inundation (or run- up). In the first part of the manuscript we consider the generation process of such extreme waves. We examine various existing approaches to its modelling. Then we propose a few alternatives. The main conclusion is that the seismology/hydrodynamics coupling is poorly understood at the present time. The second chapter essentially deals with Boussinesq equations which are often used to model tsunami propagation and sometimes even run-up. More precisely, we discuss the importance, nature and inclusion of dissipative effects in long wave models. In the third chapter we slightly change the subject and turn to two-phase flows. The main purpose of this chapter is to propose an operational and simple set of equations in order to model wave impacts on coastal structures. Another important application includes wave sloshing in liquified natural gas carriers. Then, we discuss the numerical discretization of governing equations in the finite volume framework on unstructured meshes. Finally, this thesis deals with a topic which should be present in any textbook on hydro- dynamics but it is not. We mean visco-potential flows. We propose a novel and sufficiently simple approach for weakly viscous flow modelling. We succeeded in keeping the simplicity of the classical potential flow formulation with the addition of viscous effects. In the case of finite depth we derive a correction term due to the presence of the bottom boundary layer. This term is nonlocal in time. Hence, the bottom boundary layer introduces a memory effect to the governing equations. Keywords: Water waves, tsunami generation, Boussinesq equations, two- phase flows, visco-potential flows, finite volumes Contents Remerciements i Contents vii List of Figures xiii List of Tables xv Introduction xvii Propagation of tsunamis ............................ xviii Classical formulation .......................... xx Dimensionless equations ........................ xxi Shallow-water equations ........................ xxi Boussinesq equations .......................... xxiii Korteweg–de Vries equation ...................... xxvi Energy of a tsunami .............................. xxvi Tsunami run-up ................................. xxvii 1 Tsunami generation 1 1.1 Waves generated by a moving bottom ..................... 2 1.1.1 Source model .............................. 4 1.1.2 Volterra’s theory of dislocations .................... 5 1.1.3 Dislocations in elastic half-space .................... 6 1.1.3.1 Finite rectangular source ................... 10 1.1.3.2 Curvilinear fault ....................... 12 1.1.4 Solution in fluid domain ........................ 16 1.1.5 Free-surface elevation .........................
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