DOCSLIB.ORG

Rogue Waves and Their Generating Mechanisms in Different Physical Contexts

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Physics Reports 528 (2013) 47–89 Contents lists available at SciVerse ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Rogue waves and their generating mechanisms in different physical contexts M. Onorato a,b, S. Residori c,∗, U. Bortolozzo c, A. Montina d, F.T. Arecchi e,f a Dipartimento di Fisica Generale, Università degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy b INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy c INLN, Université de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles, 06560 Valbonne, France d Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada e Dipartimento di Fisica, Università di Firenze, Italy f CNR-INO, largo E. Fermi 6, 50125 Firenze, Italy article info a b s t r a c t Article history: Rogue waves is the name given by oceanographers to isolated large amplitude waves, Accepted 19 December 2012 that occur more frequently than expected for normal, Gaussian distributed, statistical Available online 14 March 2013 events. Rogue waves are ubiquitous in nature and appear in a variety of different contexts. editor: G.I. Stegeman Besides water waves, they have been recently reported in liquid Helium, in nonlinear optics, microwave cavities, etc. The first part of the review is dedicated to rogue waves in the oceans and to their laboratory counterpart with experiments performed in water basins. Most of the work and interpretation of the experimental results will be based on the nonlinear Schrödinger equation, an universal model, that rules the dynamics of weakly nonlinear, narrow band surface gravity waves. Then, we present examples of rogue waves occurring in different physical contexts and we discuss the related anomalous statistics of the wave amplitude, which deviates from the Gaussian behavior that were expected for random waves. The third part of the review is dedicated to optical rogue waves, with examples taken from the supercontinuum generation in photonic crystal fibers, laser fiber systems and two-dimensional spatiotemporal systems. In particular, the extreme waves observed in a two-dimensional spatially extended optical cavity allow us to introduce a description based on two essential conditions for the generation of rogue waves: nonlinear coupling and nonlocal coupling. The first requirement is needed in order to introduce an elementary size, such as that of the solitons or breathers, whereas the second requirement implies inhomogeneity, a mechanism needed to produce the events of mutual collisions and mutual amplification between the elementary solitons or wavepackets. The concepts of ``granularity'' and ``inhomogeneity'' as joint generators of optical rogue waves are introduced on the basis of a linear experiment. By extending these concepts to other systems, rogue waves can be classified as phenomena occurring in the presence of many uncorrelated ``grains'' of activity inhomogeneously distributed in large spatial domains, the ``grains'' being of linear or nonlinear origin, as in the case of wavepackets or solitons. ' 2013 Elsevier B.V. All rights reserved. Contents 1. Introduction............................................................................................................................................................................................. 48 2. Introduction to the statistical properties of ocean waves.................................................................................................................... 49 ∗ Corresponding author. Tel.: +33 0492967317. E-mail address: [email protected] (S. Residori). 0370-1573/$ – see front matter ' 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physrep.2013.03.001 48 M. Onorato et al. / Physics Reports 528 (2013) 47–89 2.1. The wave spectrum..................................................................................................................................................................... 49 2.2. Rayleigh distribution for the wave envelope: linear theory.................................................................................................... 50 3. Experimental evidence of rogue waves in the ocean ........................................................................................................................... 52 4. Nonlinearity and rogue waves in one horizontal dimension............................................................................................................... 53 4.1. The nonlinear Schrödinger equation for water waves in infinite water depth ...................................................................... 54 4.1.1. The NLS in nondimensional form: the Benjamin–Feir-Index ................................................................................... 54 4.2. The modulational instability...................................................................................................................................................... 55 4.3. Breather solutions....................................................................................................................................................................... 55 4.3.1. Relation between the breather solutions and standard solitons.............................................................................. 57 4.3.2. Higher order breathers................................................................................................................................................ 57 4.3.3. Experimental/numerical verification of the breather solutions in wave tanks....................................................... 57 4.4. Finite depth effects ..................................................................................................................................................................... 59 4.5. The fully nonlinear approach in one horizontal dimension..................................................................................................... 60 4.6. Rogue waves in random sea states ............................................................................................................................................ 61 4.7. Some experimental results in wave tank laboratories for 1-dimensional propagation ........................................................ 61 5. Kurtosis and skewness of the surface elevation.................................................................................................................................... 63 5.1. The contribution to the kurtosis from free modes ................................................................................................................... 63 5.2. The contribution to deviations from Gaussian statistics by bound modes............................................................................. 64 6. Rogue waves in two horizontal dimensions.......................................................................................................................................... 64 7. The role of inhomogeneity: the Alber–Saffman approach ................................................................................................................... 66 8. Rogue waves in crossing seas................................................................................................................................................................. 67 9. Rogue waves and currents...................................................................................................................................................................... 68 10. Rogue waves in other physical systems ................................................................................................................................................ 68 10.1. Wave turbulence and rogue waves in superfluid Helium........................................................................................................ 69 10.2. Microwave experiments: two-dimensional rogue waves in the linear regime ..................................................................... 70 10.3. Extreme events in parametrically driven capillary waves....................................................................................................... 70 10.4. Rogue waves in plasmas............................................................................................................................................................. 71 11. Rogue waves in optics............................................................................................................................................................................. 72 12. Optical rogue waves in fiber supercontinuum generation................................................................................................................... 73 12.1. The 1D generalized nonlinear Schrödinger equation............................................................................................................... 74 12.1.1. Numerical simulations ................................................................................................................................................ 75 12.1.2. Statistical properties.................................................................................................................................................... 75 12.2. Optical rogue waves in continuous pumped supercontinuum................................................................................................ 75 12.3. Optical rogue waves
Recommended publications
  • Acoustic Nonlinearity in Dispersive Solids

    Acoustic Nonlinearity in Dispersive Solids

    ACOUSTIC NONLINEARITY IN DISPERSIVE SOLIDS John H. Cantrell and William T. Yost NASA Langley Research Center Mail Stop 231 Hampton, VA 23665-5225 INTRODUCTION It is well known that the interatomic anharmonicity of the propagation medium gives rise to the generation of harmonics of an initially sinusoidal acoustic waveform. It is, perhaps, less well appreciated that the anharmonicity also gives rise to the generation of a static or "dc" component of the propagating waveform. We have previously shown [1,2] that the static component is intrinsically linked to the acoustic (Boussinesq) radiation stress in the material. For nondispersive solids theory predicts that a propagating gated continuous waveform (acoustic toneburst) generates a static displacement pulse having the shape of a right-angled triangle, the slope of which is linearly proportional to the magnitude and sign of the acoustic nonlinearity parameter of the propagation medium. Such static displacement pulses have been experimentally verified in single crystal silicon [3] and germanium [4]. The purpose of the present investigation is to consider the effects of dispersion on the generation of the static acoustic wave component. It is well known that an initial disturbance in media which have both sufficiently large dispersion and nonlinearity can evolve into a series of solitary waves or solitons [5]. We consider here that an acoustic tone burst may be modeled as a modulated continuous waveform and that the generated initial static displacement pulse may be viewed as a modulation-confined disturbance. In media with sufficiently large dispersion and nonlinearity the static displacement pulse may be expected to evolve into a series of modulation solitons.
  • High-Throughput Biological Cell Classification Featuring Real-Time Optical Data Compression

    High-Throughput Biological Cell Classification Featuring Real-Time Optical Data Compression

    Conference on Information Sciences and Systems, CISS 2015, Baltimore, MD, USA, March 18-20 2015 High-throughput Biological Cell Classification Featuring Real-time Optical Data Compression Bahram Jalali*, Ata Mahjoubfar, and Claire L. Chen Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA Plenary Talk *Email: [email protected] Abstract—High throughput real-time instruments are needed the current gold standard in high-speed fluorescence imaging to acquire large data sets for detection and classification of rare [16]. events. Enabled by the photonic time stretch digitizer, a new class of instruments with record throughputs have led to the discovery Producing data rates as high as one tera bit per second, of optical rogue waves [1], detection of rare cancer cells [2], and these real-time instruments pose a big data challenge that the highest analog-to-digital conversion performance ever overwhelms even the most advanced computers [17]. Driven achieved [3]. Featuring continuous operation at 100 million by the necessity of solving this problem, we have recently frames per second and shutter speed of less than a nanosecond, introduced and demonstrated a categorically new data the time stretch camera is ideally suited for screening of blood compression technology [18, 19]. The so called Anamorphic and other biological samples. It has enabled detection of breast (warped) Stretch Transform is a physics based data processing cancer cells in blood with record, one-in-a-million, sensitivity [2]. technique that aims to mitigate the big data problem in real- Owing to their high real-time throughput, instruments produce a time instruments, in digital imaging, and beyond [17-21].
  • Nearshore Current Pattern and Rip Current Occurrence at Jungmun Beach, Jeju by Numerical Computation 1. Introduction

    Nearshore Current Pattern and Rip Current Occurrence at Jungmun Beach, Jeju by Numerical Computation 1. Introduction

    한국항해항만학회지 제41권 제2호 J. Navig. Port Res. Vol. 41, No. 2 : 55-62, April 2017 (ISSN:1598-5725(Print)/ISSN:2093-8470(Online)) DOI : http://dx.doi.org/10.5394/KINPR.2017.41.2.55 Nearshore Current Pattern and Rip Current Occurrence at Jungmun Beach, Jeju by Numerical Computation Seung-Hyun An*․†Nam-Hyeong Kim *Coastal, Harbor and Disaster Prevention Research Institute, Sekwang Engineering Consultants Co., LTD. †Department of Civil Engineering, J eju National University, Jeju 63243, Korea Abstract : A nearshore current or a wave-induced current is an important phenomenon in a nearshore zone, which is composed of longshore, cross-shore, and rip currents. The nearshore current is closely related to the occurrence of coastal accidents by beachgoers. A considerable number of coastal accidents by beachgoers involving the rip current have been reported at Jungmun Beach. However, in studies and observations of the nearshore current of Jungmun Beach, understanding of the rip current pattern remains unclear. In this study, a scientific approach is taken to understand the nearshore current and the rip current patterns at Jungmun Beach by numerical computation for year of 2015. From results of numerical computation, the occurrence and spatial characteristics of the rip current, and the similarities between the rip current and incident wave conditions are analyzed. The primary results of this study reveal that the rip currents are frequently generated at Jungmun Beach, especially in the western parts of the beach, and that the rip currents often occur with a wave breaking height of around 0.5 ~ 0.7 m, a wave period of around 6 ~ 8 seconds, and a breaking angle of around 0 ~ 15 degrees.
  • Dissipative Rogue Waves Generated by Chaotic Pulse Bunching in a Mode-Locked Laser

    Dissipative Rogue Waves Generated by Chaotic Pulse Bunching in a Mode-Locked Laser

    week ending PRL 108, 233901 (2012) PHYSICAL REVIEW LETTERS 8 JUNE 2012 Dissipative Rogue Waves Generated by Chaotic Pulse Bunching in a Mode-Locked Laser C. Lecaplain,1 Ph. Grelu,1 J. M. Soto-Crespo,2 and N. Akhmediev3 1Laboratoire Interdisciplinaire Carnot de Bourgogne, U.M.R. 6303 C.N.R.S., Universite´ de Bourgogne, 9 avenue A. Savary, BP 47870 Dijon Cedex 21078, France 2Instituto de O´ ptica, C.S.I.C., Serrano 121, 28006 Madrid, Spain 3Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia (Received 14 December 2011; published 5 June 2012) Rare events of extremely high optical intensity are experimentally recorded at the output of a mode- locked fiber laser that operates in a strongly dissipative regime of chaotic multiple-pulse generation. The probability distribution of these intensity fluctuations, which highly depend on the cavity parameters, features a long-tailed distribution. Recorded intensity fluctuations result from the ceaseless relative motion and nonlinear interaction of pulses within a temporally localized multisoliton phase. DOI: 10.1103/PhysRevLett.108.233901 PACS numbers: 42.65.Tg, 47.20.Ky Waves of extreme amplitude, or ‘‘rogue waves,’’ have satisfying the criteria of rogue waves [17,18]. In order to been the subject of intense research activity during the past strongly interact, the multiplicity of pulses should be con- decade [1,2]. Initiated first in the context of oceanic waves, fined within a relatively tight temporal domain. The two where the induced stakes and consequences are the most predictions also stressed the dominant impact of dissipa- critical, the notion of ‘‘rogue waves’’ or ‘‘freak waves’’ is tive effects on the involved nonlinear dynamics.
  • The Universal Route to Rogue Waves

    The Universal Route to Rogue Waves

    PHYSICAL REVIEW X 9, 041057 (2019) Featured in Physics Experimental Evidence of Hydrodynamic Instantons: The Universal Route to Rogue Waves Giovanni Dematteis ,1,2 Tobias Grafke,3 Miguel Onorato ,2,4 and Eric Vanden-Eijnden5 1Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy 2Dipartimento di Fisica, Universit`a degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy 3Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 4INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy 5Courant Institute, New York University, 251 Mercer Street, New York, New York 10012, USA (Received 3 July 2019; revised manuscript received 2 October 2019; published 18 December 2019) A statistical theory of rogue waves is proposed and tested against experimental data collected in a long water tank where random waves with different degrees of nonlinearity are mechanically generated and free to propagate along the flume. Strong evidence is given that the rogue waves observed in the tank are hydrodynamic instantons, that is, saddle point configurations of the action associated with the stochastic model of the wave system. As shown here, these hydrodynamic instantons are complex spatiotemporal wave field configurations which can be defined using the mathematical framework of large deviation theory and calculated via tailored numerical methods. These results indicate that the instantons describe equally well rogue waves created by simple linear superposition (in weakly nonlinear conditions) or by nonlinear focusing (in strongly nonlinear conditions), paving the way for the development of a unified explanation to rogue wave formation. DOI: 10.1103/PhysRevX.9.041057 Subject Areas: Fluid Dynamics, Nonlinear Dynamics, Statistical Physics I.
  • CHAPTER 100 Another Approach to Longshore Current Evaluation M.A

    CHAPTER 100 Another Approach to Longshore Current Evaluation M.A

    CHAPTER 100 Another approach to longshore current evaluation M.A. Losada, A. Sanchez-Arcilla and C. Vidal A simple model to predict the longshore current velocity at the breaker line on a beach with oblique wave incidence, is presented. The model balances driving and resistance terms (gradients of radiation and turbulent Reynolds stresses and bottom friction) to get a general expression for the velocity. This equation shows explicitely the influence of Iribarren's parameter on longshore current generation. It has been tested with field and laboratory data, obtaining a reasonable fit to measured values. The resulting (predictive) model is expected to be valid for any type of breakers though the calibration has been mainly done for spilling and plunging types, due to the scarcity of results for other breakers. 1.- INTRODUCTION Longshore currents in the surf zone have been acceptably modelled using the radiation stress concept (Longuet-Higgins,1970). The longshore-trust (Nw/m ) due to an oblique wave approach, given by the gradient of the radiation stress, is balanced (in stationary and longshore-uniform conditions) by bottom friction and horizontal mixing (Bowen, 1969), (Longuet-Higgins, 1970). The gradient of the radiation stress is evaluated using sinusoidal theory (as a first approximation for slowly varying depths) and turns out to be proportional to the local rate of energy dissipation, D (joules/(m x sec)), regardless of its origin. Inside the surf zone a significant fraction of D comes from wave breaking because the turbulence associated to the breaking process is responsible for most of the dissipated energy. Bottom friction plays a minor role in this context, being important only in special cases (e.g.
  • Chapter 187 Wave Breaking and Induced Nearshore

    Chapter 187 Wave Breaking and Induced Nearshore

    CHAPTER 187 WAVE BREAKING AND INDUCED NEARSHORE CIRCULATIONS Ole R. S0rensen f, Hemming A. Schafferf, Per A. Madsenf, and Rolf DeigaardJ ABSTRACT Wave breaking and wave induced currents are studied in two horizontal dimen- sions. The wave-current motion is modelled using a set of Boussinesq type equations including the effects of wave breaking and runup. This is done with- out the traditional splitting of the phenomenon into a wave problem and a current problem. Wave breaking is included using the surface roller concept and runup is simulated by use of a modified slot-technique. In the environment of a plane sloping beach two different situations are studied. One is the case of a rip channel and the other concerns a detached breakwater parallel to the shoreline. In both situations wave driven currents are generated and circulation cells appear. This and many other nearshore physical phenomena are described qualitatively correct in the simulations. 1. INTRODUCTION State of the art models for wave driven currents are based on a splitting of the phenomenon into a linear wave problem, described e.g. by the mild-slope equation, and a current problem, described by depth-integrated flow equations including radiation stress. Wave-current interaction effects such as current re- fraction and wave blocking may be included in these systems by successive and iterative model executions. A more direct approach to the problem would be the use of a time do- main Boussinesq model which automatically includes the combined effects of wave-wave and wave-current interaction in shallow water without the need for explicit formulations of the radiation stress.
  • A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal

    A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal

    A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling Martin O. Paulsen Henrik Kalisch · g Abstract In this article we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg-de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a gently sloping beach. The transformation of these waves along the slope can be described using the shoaling equations, a set of three nonlinear equations in three unknowns: the wave height H, the set-downη ¯ and the elliptic parameter m. We define a numerical algorithm for the efficient solution of the shoaling equations, and we verify our shoaling formulation by comparing with experimental data from two sets of experiments as well as shoaling curves obtained in previous works. Keywords: Surface waves KdV equation Conservation laws Radiation stress Set-down · · · · 1 Introduction In the present work, the development of surface waves across a gently sloping bottom is in view. Our main goal is the prediction of the wave height and the set-down of a periodic wave as it enters an area of shallower depth. This problem is known as the shoaling problem and has a long history, with contributions from a large number of authors going back to Green and Boussinesq. In order to reduce the problem to the essential factors, we make a number of simplifying as- sumptions. First, the bottom slope is small enough to allow the waves to continuously adjust to the changing depth without altering their basic shape or breaking up.
  • Optical Rogue Wave in Random Distributed Feedback Fiber Laser Jiangming Xu1,2, Jian Wu1,2, Jun Ye1, Pu Zhou1,2,*, Hanwei Zhang1,2, Jiaxin Song1, & Jinyong Leng1,2

    Optical Rogue Wave in Random Distributed Feedback Fiber Laser Jiangming Xu1,2, Jian Wu1,2, Jun Ye1, Pu Zhou1,2,*, Hanwei Zhang1,2, Jiaxin Song1, & Jinyong Leng1,2

    Optical rogue wave in random distributed feedback fiber laser Jiangming Xu1,2, Jian Wu1,2, Jun Ye1, Pu Zhou1,2,*, Hanwei Zhang1,2, Jiaxin Song1, & Jinyong Leng1,2 The famous demonstration of optical rogue wave (RW)-rarely and unexpectedly event with extremely high intensity-had opened a flourishing time for temporal statistic investigation as a powerful tool to reveal the fundamental physics in different laser scenarios. However, up to now, optical RW behavior with temporally localized structure has yet not been presented in random fiber laser (RFL) characterized with mirrorless open cavity, whose feedback arises from distinctive distributed multiple scattering. Here, thanks to the participation of sustained and crucial stimulated Brillouin scattering (SBS) process, experimental explorations of optical RW are done in the highly-skewed transient intensity of an incoherently-pumped standard-telecom-fiber-constructed RFL. Furthermore, threshold-like beating peak behavior can also been resolved in the radiofrequency spectroscopy. Bringing the concept of optical RW to RFL domain without fixed cavity may greatly extend our comprehension of the rich and complex kinetics such as photon propagation and localization in disordered amplifying media with multiple scattering. 1 College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073, China . 2 Hunan Provincial Collaborative Innovation Center of High Power Fiber Laser, Changsha 410073, China. J. X. and J. W. contributed equally to this paper. Correspondence and requests for materials should be addressed to P. Z. (email: [email protected]). The concept of random laser employing multiple scattering of photons in an amplifying disordered material to achieve coherent light has attracted increasing attention due to its special features, such as cavity-free, structural simplicity, and promising applications in imaging, medical diagnostics, and other scientific or industrial fields [1-3].
  • Optical Rogue Waves in Integrable Turbulence

    Optical Rogue Waves in Integrable Turbulence

    Optical Rogue Waves in Integrable Turbulence Pierre Suret et Stéphane Randoux Postdoc : F. Gustave PhD : R. El Koussaifi, A. Tikan, P. Walczak (2016) Collaborators : Fast Measurement : S. Bielawski, C. Szwaj, C. Evain (Phlam) Oceanography : M. Onorato (Turin, Italie) Integrable Systems : G. El (Loughbourough, UK) 1. Context 2. Experiments 3. Theoretical approach Optical Rogue Waves in Wave Turbulence Integrable Turbulence ü Waves ubiquitous phenomena in Physics ü Dispersion ü Nonlinearity ü Often, randomness plays a significant role Wave turbulence = nonlinear dispersive random waves hydrodynamics, oceanography, optics, mechanics, plasma physics… Optical Rogue Waves in Nonlinear Statistical OpticsIntegrable Turbulence Statistical Optics (linear) ü XXth century Random / partially coherent light (spatial or temporal fluctuation, coherence / spectrum…) Nonlinear Statistical (fiber) Optics ü 1960s Lasers / nonlinear optics ü 1990s Supercontinuum (ps, cw) : nonlinear + incoherent A.Kudlinski and A. Mussot, Opt. Lett., 33, (2008) J. M. Dudley et al., Opt. Express 17, (2009) ü 2000s Challenge : statistical description of nonlinear random waves A. Picozzi et al., Optical wave turbulence : Toward a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics Physics Reports, (2014) Optical Rogue Waves in “Turbulence” in Optical fibersIntegrable Turbulence Optical fibers : tabletop laboratories of hydrodynamical phenomena ü Turbulence E.G. Turitsyna et al., « The laminar–turbulent transition in a fibre laser », Nat. Photonics,
  • Time Stretch and Its Applications Ata Mahjoubfar1,2, Dmitry V

    Time Stretch and Its Applications Ata Mahjoubfar1,2, Dmitry V

    REVIEW ARTICLE PUBLISHED ONLINE: 1 JUNE 2017 | DOI: 10.1038/NPHOTON.2017.76 Time stretch and its applications Ata Mahjoubfar1,2, Dmitry V. Churkin3,4,5, Stéphane Barland6, Neil Broderick7, Sergei K. Turitsyn3,5 and Bahram Jalali1,2,8,9* Observing non-repetitive and statistically rare signals that occur on short timescales requires fast real-time measurements that exceed the speed, precision and record length of conventional digitizers. Photonic time stretch is a data acquisition method that overcomes the speed limitations of electronic digitizers and enables continuous ultrafast single-shot spectroscopy, imaging, reflectometry, terahertz and other measurements at refresh rates reaching billions of frames per second with non-stop recording spanning trillions of consecutive frames. The technology has opened a new frontier in measurement science unveiling transient phenomena in nonlinear dynamics such as optical rogue waves and soliton molecules, and in relativistic electron bunching. It has also created a new class of instruments that have been integrated with artificial intelligence for sensing and biomedical diagnostics. We review the fundamental principles and applications of this emerging field for continuous phase and amplitude characterization at extremely high repetition rates via time-stretch spectral interferometry. he Shannon–Hartley theorem provides a tool to quantify the in instrumentation have led to new science. We anticipate that a new information content of analog data. The theorem prescribes that type of instrument that performs time-resolved single-shot measure- Tunpredictability — the element of surprise — is the property that ments of ultrafast events and that makes continuous measurements defines information. No information is carried by a periodic signal, over long timescales spanning trillions of pulses will unveil previously such as a sinusoidal wave, because its evolution is entirely predictable.
  • The Three-Dimensional Current and Surface Wave Equations

    The Three-Dimensional Current and Surface Wave Equations

    1978 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 The Three-Dimensional Current and Surface Wave Equations GEORGE MELLOR Princeton University, Princeton, New Jersey (Manuscript received 26 April 2002, in ®nal form 4 March 2003) ABSTRACT Surface wave equations appropriate to three-dimensional ocean models apparently have not been presented in the literature. It is the intent of this paper to correct that de®ciency. Thus, expressions for vertically dependent radiation stresses and a de®nition of the Doppler velocity for a vertically dependent current ®eld are obtained. Other quantities such as vertically dependent surface pressure forcing are derived for inclusion in the momentum and wave energy equations. The equations include terms that represent the production of turbulence energy by currents and waves. These results are a necessary precursor for three-dimensional ocean models that handle surface waves together with wind- and buoyancy-driven currents. Although the third dimension has been added here, the analysis is based on the assumption that the depth dependence of wave motions is provided by linear theory, an assumption that is the basis of much of the wave literature. 1. Introduction pressure so that their results differ from mine and, after vertical integration, differ from the corresponding terms In the last several decades there has been signi®cant in Phillips (1977). They did not address three-dimen- progress in the understanding and numerical modeling sional effects on the wave energy equation. of ocean surface waves (e.g., Longue-Higgins 1953; To contrast the developments in this paper with con- Phillips 1977; WAMDI Group 1988; Komen et al.