Existence of a directional Stokes drift in asymmetrical three dimensional travelling gravity

Description

Existence of a directional Stokes drift in asymmetrical three-dimensional travelling gravity waves Gerard Iooss†, Pavel Plotnikov‡ † IUF, Universite de Nice, Labo J.A.Dieudonne, 06108 Nice Cedex 02, France ‡Lavrentyev Institute of Hydrodynamics, Lavryentyev pr. 15, Novosibirsk 630090, Russia , September 4, 2009 Abstract We consider periodic travelling gravity waves at the surface of an in- finitely deep perfect fluid. The pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non res- onant situation. Defining a couple of amplitudes ?1, ?2 along the basis of wave vectors which satisfy the dispersion relation, following [8], travelling waves exist with an asymptyotic expansion in powers of ?1, ?2, for nearly all pair of angles made by the basic wave vectors with the critical prop- agation direction, and for values of the couple (?21, ?22) in a subset of the plane, with asymptotic full measure at the origin. We prove the remark- able property that on the free surface, observed in the moving frame, the propagation direction of the waves differs from the asymptotic direction taken by fluid particles, by a small angle which is computed. Existence d'une derive directionnelle de Stokes dans les vagues de gravite tri-dimensionnelles asymetriques.

• stokes drift


• propagation direction


• direction de propagation des ondes differe de la direction asymptotique


• dispersion relation


• propagation velocity


• dimensional travelling


• dispersion equation


• vecteurs d'onde de base avec la di- rection critique de propagation

Sujets

• Stokes drift
• Dispersion relation
• Phase velocity

Informations

• Publié par: pefav
• Nombre de lectures: 18
• Langue: English

Extrait

Existence of a directional Stokes drift in asymmetrical threedimensional travelling gravity waves

† ‡ G´erardIooss,PavelPlotnikov † ci,eaLobis´tdeNedonn´e,0J.A.DieuedeC,20x8016eciNanFrceUIrevinU,F ‡ 15, Novosibirsk 630090, RussiaLavrentyev Institute of Hydrodynamics, Lavryentyev pr. [e-mail], [e-mail]

September4,2009

Abstract We consider periodic travelling gravity waves at the surface of an in finitely deep perfect fluid. The pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non res onant situation. Defining a couple of amplitudesε 1 , ε 2 along the basis of wave vectors which satisfy the dispersion relation, following [8], travelling waves exist with an asymptyotic expansion in powers ofε 1 , ε 2 ,for nearly all pair of angles made by the basic wave vectors with the critical prop 2 2 agation direction, and for values of the couple (ε 1 , ε 2 a subset of the) in plane, with asymptotic full measure at the origin. We prove the remark able property that on the free surface, observed in the moving frame, the propagation direction of the waves differs from the asymptotic direction taken by fluid particles, by a small angle which is computed. Existenced’uned´erivedirectionnelledeStokesdanslesvagues degravite´tridimensionnellesasyme´triques.cOnnois`drelese vaguesp´eriodiquesa`lasurfaced’unecouchedefluideparfait,depro fondeurinfinie,soumisea`laseulegravit´e.Lere´seaubidimensionneldes pe´riodesestprisnonsym´etriqueparrapporta`ladirectiondepropaga tionetonsupposenepasˆetredansuncasr´esonant.Onde´finitlecouple d’amplitudesε 1 , ε 2 ellgdontnefiirondedebasequiv´eseedxuevtcuesr’d l’e´quationdedispersion.D’apre`s[8],lesvaguesasym´etriquesexistentet poss`edentunde´veloppementasymptotiqueenpuissancesde(ε 1 , ε 2 ),pour presque tous les angles faits par les vecteurs d’onde de base avec la di 2 2 rection critique de propagation, et pour des valeurs de (ε 1 , ε 2 ) dans un sousensemble du quadrant ayant une mesure asymptotiquement pleine `al’origine.Nousmontronslaproprie´t´eremarquabledanslere´f´erentiel relatif,qu’a`lasurfacelibre,ladirectiondepropagationdesondesdiffe`re de la direction asymptotique prise par les trajectoires des particules de fluide, d’un petit angle qu’on calcule.

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