Mysterious Underwater Waves
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Internal Gravity Waves: from Instabilities to Turbulence Chantal Staquet, Joël Sommeria
Internal gravity waves: from instabilities to turbulence Chantal Staquet, Joël Sommeria To cite this version: Chantal Staquet, Joël Sommeria. Internal gravity waves: from instabilities to turbulence. Annual Review of Fluid Mechanics, Annual Reviews, 2002, 34, pp.559-593. 10.1146/an- nurev.fluid.34.090601.130953. hal-00264617 HAL Id: hal-00264617 https://hal.archives-ouvertes.fr/hal-00264617 Submitted on 4 Feb 2020 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. Distributed under a Creative Commons Attribution| 4.0 International License INTERNAL GRAVITY WAVES: From Instabilities to Turbulence C. Staquet and J. Sommeria Laboratoire des Ecoulements Geophysiques´ et Industriels, BP 53, 38041 Grenoble Cedex 9, France; e-mail: [email protected], [email protected] Key Words geophysical fluid dynamics, stratified fluids, wave interactions, wave breaking Abstract We review the mechanisms of steepening and breaking for internal gravity waves in a continuous density stratification. After discussing the instability of a plane wave of arbitrary amplitude in an infinite medium at rest, we consider the steep- ening effects of wave reflection on a sloping boundary and propagation in a shear flow. The final process of breaking into small-scale turbulence is then presented. -
Waves on Deep Water, I
Lecture 14: Waves on deep water, I Lecturer: Harvey Segur. Write-up: Adrienne Traxler June 23, 2009 1 Introduction In this lecture we address the question of whether there are stable wave patterns that propagate with permanent (or nearly permanent) form on deep water. The primary tool for this investigation is the nonlinear Schr¨odinger equation (NLS). Below we sketch the derivation of the NLS for deep water waves, and review earlier work on the existence and stability of 1D surface patterns for these waves. The next lecture continues to more recent work on 2D surface patterns and the effect of adding small damping. 2 Derivation of NLS for deep water waves The nonlinear Schr¨odinger equation (NLS) describes the slow evolution of a train or packet of waves with the following assumptions: • the system is conservative (no dissipation) • then the waves are dispersive (wave speed depends on wavenumber) Now examine the subset of these waves with • only small or moderate amplitudes • traveling in nearly the same direction • with nearly the same frequency The derivation sketch follows the by now normal procedure of beginning with the water wave equations, identifying the limit of interest, rescaling the equations to better show that limit, then solving order-by-order. We begin by considering the case of only gravity waves (neglecting surface tension), in the deep water limit (kh → ∞). Here h is the distance between the equilibrium surface height and the (flat) bottom; a is the wave amplitude; η is the displacement of the water surface from the equilibrium level; and φ is the velocity potential, u = ∇φ. -
21 Non-Linear Wave Equations and Solitons
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Open Textbooks 8-2014 21 Non-Linear Wave Equations and Solitons Charles G. Torre Department of Physics, Utah State University, [email protected] Follow this and additional works at: https://digitalcommons.usu.edu/foundation_wave Part of the Physics Commons To read user comments about this document and to leave your own comment, go to https://digitalcommons.usu.edu/foundation_wave/2 Recommended Citation Torre, Charles G., "21 Non-Linear Wave Equations and Solitons" (2014). Foundations of Wave Phenomena. 2. https://digitalcommons.usu.edu/foundation_wave/2 This Book is brought to you for free and open access by the Open Textbooks at DigitalCommons@USU. It has been accepted for inclusion in Foundations of Wave Phenomena by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Foundations of Wave Phenomena, Version 8.2 21. Non-linear Wave Equations and Solitons. In 1834 the Scottish engineer John Scott Russell observed at the Union Canal at Hermiston a well-localized* and unusually stable disturbance in the water that propagated for miles virtually unchanged. The disturbance was stimulated by the sudden stopping of a boat on the canal. He called it a “wave of translation”; we call it a solitary wave. As it happens, a number of relatively complicated – indeed, non-linear – wave equations can exhibit such a phenomenon. Moreover, these solitary wave disturbances will often be stable in the sense that if two or more solitary waves collide then after the collision they will separate and take their original shape. -
Shallow Water Waves and Solitary Waves Article Outline Glossary
Shallow Water Waves and Solitary Waves Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, USA Article Outline Glossary I. Definition of the Subject II. Introduction{Historical Perspective III. Completely Integrable Shallow Water Wave Equations IV. Shallow Water Wave Equations of Geophysical Fluid Dynamics V. Computation of Solitary Wave Solutions VI. Water Wave Experiments and Observations VII. Future Directions VIII. Bibliography Glossary Deep water A surface wave is said to be in deep water if its wavelength is much shorter than the local water depth. Internal wave A internal wave travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface). Internal waves depend on the density-stratification of the fluid. Shallow water A surface wave is said to be in shallow water if its wavelength is much larger than the local water depth. Shallow water waves Shallow water waves correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. Shallow water wave equations Shallow water wave equations are a set of partial differential equations that describe shallow water waves. 1 Solitary wave A solitary wave is a localized gravity wave that maintains its coherence and, hence, its visi- bility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and propagate with constant speed and constant shape. Soliton Solitons are solitary waves that have an elastic scattering property: they retain their shape and speed after colliding with each other. -
Soliton and Some of Its Geophysical/Oceanographic Applications
Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Soliton and Some of its Geophysical/Oceanographic Applications M. Lakshmanan Centre for Nonlinear Dynamics Bharathidasan University Tiruchirappalli – 620024 India Discussion Meeting, ICTS-TIFR 21 July 2016 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Theme Soliton is a counter-intuitive wave entity arising because of a delicate balance between dispersion and nonlinearity. Its major attraction is its localized nature, finite energy and preservation of shape under collision = a remark- ably stable structure. Consequently, it has⇒ ramifications in as wider areas as hydrodynamics, tsunami dynamics, condensed matter, magnetism, particle physics, nonlin- ear optics, cosmology and so on. A brief overview will be presented with emphasis on oceanographic applications and potential problems will be indicated. M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Plan 1 Introduction 2 Historical: (i) Scott-Russel (ii) Zabusky/Kruskal 3 Korteweg-de Vries equation and solitary wave/soliton 4 Other ubiquitous equations 5 Mathematical theory of solitons 6 Geophysical/Oceanographic applications 7 Solitons in higher dimensions 8 Problems and potentialities M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory -
Dissipative Cnoidal Waves (Turing Rolls) and the Soliton Limit in Microring Resonators
Research Article Vol. X, No. X / Febrary 1046 B.C. / OSA Journal 1 Dissipative cnoidal waves (Turing rolls) and the soliton limit in microring resonators ZHEN QI1,*,S HAOKANG WANG1,J OSÉ JARAMILLO-VILLEGAS2,M INGHAO QI3, ANDREW M. WEINER3,G IUSEPPE D’AGUANNO1,T HOMAS F. CARRUTHERS1, AND CURTIS R. MENYUK1 1University of Maryland at Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA 2Technological University of Pereira, Cra. 27 10-02, Pereira, Risaralda 660003, Colombia 3Purdue University, 610 Purdue Mall, West Lafayette, IN 47907, USA *Corresponding author: [email protected] Compiled May 27, 2019 Single solitons are a special limit of more general waveforms commonly referred to as cnoidal waves or Turing rolls. We theoretically and computationally investigate the stability and acces- sibility of cnoidal waves in microresonators. We show that they are robust and, in contrast to single solitons, can be easily and deterministically accessed in most cases. Their bandwidth can be comparable to single solitons, in which limit they are effectively a periodic train of solitons and correspond to a frequency comb with increased power. We comprehensively explore the three-dimensional parameter space that consists of detuning, pump amplitude, and mode cir- cumference in order to determine where stable solutions exist. To carry out this task, we use a unique set of computational tools based on dynamical system theory that allow us to rapidly and accurately determine the stable region for each cnoidal wave periodicity and to find the instabil- ity mechanisms and their time scales. Finally, we focus on the soliton limit, and we show that the stable region for single solitons almost completely overlaps the stable region for both con- tinuous waves and several higher-periodicity cnoidal waves that are effectively multiple soliton trains. -
Internal Waves Study on a Narrow Steep Shelf of the Black Sea Using the Spatial Antenna of Line Temperature Sensors
Journal of Marine Science and Engineering Article Internal Waves Study on a Narrow Steep Shelf of the Black Sea Using the Spatial Antenna of Line Temperature Sensors Andrey Serebryany 1,2,* , Elizaveta Khimchenko 1 , Oleg Popov 3, Dmitriy Denisov 2 and Genrikh Kenigsberger 4 1 Shirshov Institute of Oceanology, Russian Academy of Sciences, 117997 Moscow, Russia; [email protected] 2 Andreyev Acoustics Institute, 117036 Moscow, Russia; [email protected] 3 Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, 119017 Moscow, Russia; [email protected] 4 Institute of Ecology of the Academy of Sciences of Abkhazia, Sukhum 384900, Abkhazia; [email protected] * Correspondence: [email protected] Received: 4 August 2020; Accepted: 19 October 2020; Published: 22 October 2020 Abstract: The results of investigations into internal waves on a narrow steep shelf of the northeastern coast of the Black Sea are presented here. To measure the parameters of internal waves, the spatial antenna of three autonomous line temperature sensors were equipped in the depth range of 17 to 27 m. In observations that lasted for 10 days, near-inertial internal waves with a period close to 17 h and short-period internal waves with periods of 2–8 min, regularly approaching the coast, were revealed. The wave amplitudes were 4–8 m for inertial waves and 0.5–4 m for short-period internal waves. It was determined that most of the short-period internal waves approached from the southeast direction, from Cape Kodor. A large number of short waves reflected from the coast were also recorded. The intensification of short-period waves with inertial periodicity and the belonging of trains of short waves to crests of inertial waves were identified. -
Three-Dimensional Dynamics of Baroclinic Tides Over a Seamount
University of Plymouth PEARL https://pearl.plymouth.ac.uk Faculty of Science and Engineering School of Biological and Marine Sciences 2018-02 Three-dimensional dynamics of baroclinic tides over a seamount Vlasenko, V http://hdl.handle.net/10026.1/10845 10.1002/2017JC013287 Journal of Geophysical Research: Oceans American Geophysical Union All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1002/, Three-dimensional dynamics of baroclinic tides over a seamount 1 1 1 Vasiliy Vlasenko , Nataliya Stashchuk , and W. Alex M. Nimmo-Smith Vasiliy Vlasenko, School of Biological and Marine Sciences, Plymouth University, Plymouth, PL4 8AA, UK ([email protected]) 1School of Biological and Marine Sciences, Plymouth University, Plymouth, PL4 8AA, UK. D R A F T January 17, 2018, 10:45am D R A F T X-2 VLASENKO ET AL.: BAROCLINIC TIDES OVER A SEAMOUNT 1 Abstract. 2 The Massachusetts Institute of Technology general circulation model is used 3 for the analysis of baroclinic tides over Anton Dohrn Seamount (ADS), in 4 the North Atlantic. The model output is validated against in-situ data col- 5 lected during the 136-th cruise of the RRS “James Cook” in May-June 2016. 6 The observational data set includes velocity time series recorded at two moor- 7 ings as well as temperature, salinity and velocity profiles collected at 22 hy- 8 drological stations. -
Internal Wave Generation in a Non-Hydrostatic Wave Model
water Article Internal Wave Generation in a Non-Hydrostatic Wave Model Panagiotis Vasarmidis 1,* , Vasiliki Stratigaki 1 , Tomohiro Suzuki 2,3 , Marcel Zijlema 3 and Peter Troch 1 1 Department of Civil Engineering, Ghent University, Technologiepark 60, 9052 Ghent, Belgium; [email protected] (V.S.); [email protected] (P.T.) 2 Flanders Hydraulics Research, Berchemlei 115, 2140 Antwerp, Belgium; [email protected] 3 Department of Hydraulic Engineering, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands; [email protected] * Correspondence: [email protected]; Tel.: +32-9-264-54-89 Received: 18 April 2019; Accepted: 7 May 2019; Published: 10 May 2019 Abstract: In this work, internal wave generation techniques are developed in an open source non-hydrostatic wave model (Simulating WAves till SHore, SWASH) for accurate generation of regular and irregular long-crested waves. Two different internal wave generation techniques are examined: a source term addition method where additional surface elevation is added to the calculated surface elevation in a specific location in the domain and a spatially distributed source function where a spatially distributed mass is added in the continuity equation. These internal wave generation techniques in combination with numerical wave absorbing sponge layers are proposed as an alternative to the weakly reflective wave generation boundary to avoid re-reflections in case of dispersive and directional waves. The implemented techniques are validated against analytical solutions and experimental data including water surface elevations, orbital velocities, frequency spectra and wave heights. The numerical results show a very good agreement with the analytical solution and the experimental data indicating that SWASH with the addition of the proposed internal wave generation technique can be used to study coastal areas and wave energy converter (WEC) farms even under highly dispersive and directional waves without any spurious reflection from the wave generator. -
Plasmon-Soliton
Plasmon-Soliton Eyal Feigenbaum and Meir Orenstein Department of Electrical Engineering, Technion, Haifa 32000, Israel [email protected] Abstract : Formation of a novel hybrid-vector spatial plasmon-soliton in a Kerr slab embedded in-between metal plates is predicted and analyzed with a modified NLSE, encompassing hybrid vector field characteristics. Assisted by the transverse plasmonic effect, the self trapping dimension of the plasmon-soliton was substantially compressed (compared to the dielectrically cladded slab case) when reducing the slab width. The practical limitation of the Plasmon-soliton size reduction is determined by available nonlinear materials and metal loss. For the extreme reported values of nonlinear index change, we predict soliton with a cross section of 300nm×30nm (average dimension of 100nm). 1 The downscaling of conventional photonics is halted when approaching a transverse dimension of ~ λ/2 ( λ is the wavelength). This limitation can be alleviated by the incorporation of metals, giving rise to surface plasmon polaritons (SPP) [1,2]. Here we analyze a novel configuration where light is transversely tightly guided between two metal layers, and laterally self trapped by a nonlinear Kerr effect to yield a plasmon-soliton. The tight field confinement of this scheme is important for potential excitation of plasmon-solitons by a low power continuous wave optical source, which is further assisted by the expected small group velocity of the plasmon-soliton. In this letter we present for the first time both an analysis of TM spatial solitons, using the Non- linear Schrödinger Equation (NLSE), where the full vectorial field is considered, as well as the prediction and characteristics of SPP based solitons. -
Hydrodynamics V
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON ITYDRODYNAMICS / YOKOHAMA / 7-9 SEPTEMBER 2OOO HYDRODYNAMICSV Theory andApplications Edited by Y.Goda, M.Ikehata& K.Suzuki Yoko ham a Nat i onal Univers ity OFFPRINT IAHR1- S=== @ a' AIRH ICHD2000Local OrganizingCommittee lYokohama I 2000 Modeling of Strongly Nonlinear Internal Gravity Waves WooyoungChoi Theoretical Division and Center for Nonlinear Studies Los Alamos National Laboratory, Los Alamos, NM 87545,USA Abstract being able to account for full nonlinearity of the problem but this simple idea has never been suc- We consider strongly nonlinear internal gravity cessfully realized. Recently Choi and Camassa waves in a multilayer fluid and propose a math- (1996, 1999) have derived various new models for ematical model to describe the time evolution of strongly nonlinear dispersive waves in a simple large amplitude internal waves. Model equations two-layer system, using a systematic asymptotic follow from the original Euler equations under expansion method for a natural small parame- the sole assumption that the waves are long com- ter in the ocean, that is the aspect ratio between pared to the undisturbed thickness of one of the vertical and horizontal length scales. Analytic fluid layers. No small amplitude assumption is and numerical solutions of the new models de- made. Both analytic and numerical solutions of scribe strongly nonlinear phenomena which have the new model exhibit all essential features of been observed but not been explained by using large amplitude internal waves, observed in the any weakly nonlinear models. This indicates the ocean but not captured by the existing weakly importance of strongly nonlinear aspects in the nonlinear models. -
Soliton and Rogue Wave Statistics in Supercontinuum Generation In
Soliton and rogue wave statistics in supercontinuum generation in photonic crystal fibre with two zero dispersion wavelengths Bertrand Kibler, Christophe Finot, John M. Dudley To cite this version: Bertrand Kibler, Christophe Finot, John M. Dudley. Soliton and rogue wave statistics in supercon- tinuum generation in photonic crystal fibre with two zero dispersion wavelengths. The European Physical Journal. Special Topics, EDP Sciences, 2009, 173 (1), pp.289-295. 10.1140/epjst/e2009- 01081-y. hal-00408633 HAL Id: hal-00408633 https://hal.archives-ouvertes.fr/hal-00408633 Submitted on 17 Apr 2010 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. EPJ manuscript No. (will be inserted by the editor) Soliton and rogue wave statistics in supercontinuum generation in photonic crystal ¯bre with two zero dispersion wavelengths Bertrand Kibler,1 Christophe Finot1 and John M. Dudley2 1 Institut Carnot de Bourgogne, UMR 5029 CNRS-Universit¶ede Bourgogne, 9 Av. A. Savary, Dijon, France 2 Institut FEMTO-ST, UMR 6174 CNRS-Universit¶ede Franche-Comt¶e,Route de Gray, Besan»con, France. Abstract. Stochastic numerical simulations are used to study the statistical prop- erties of supercontinuum spectra generated in photonic crystal ¯bre with two zero dispersion wavelengths.