DOCSLIB.ORG

A Nonlinear Model of the Shoaling and Refraction of Interfacial Solitary Waves in the Ocean

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Nonlinear Model of the Shoaling and Refraction of Interfacial Solitary Waves in the Ocean 3184 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 A Nonlinear Model of the Shoaling and Refraction of Interfacial Solitary Waves in the Ocean. Part II: Oblique Refraction across a Continental Slope and Propagation over a Seamount JUSTIN SMALL Defence Evaluation Research Agency, Winfrith Technology Centre, Dorchester, United Kingdom (Manuscript received 28 August 2000, in ®nal form 21 March 2001) ABSTRACT The refraction of internal solitary waves can be due to variations in water depth, nonlinearity, strati®cation, or shear. This paper addresses the effects of the ®rst two of these effects. Part I of this paper investigated how the amplitude and phase speed of an interfacial solitary wave varied as the wave propagated across a one- dimensional slope. Important differences between the predictions of Korteweg±de Vries (KdV) and extended KdV (EKdV) theory were found. In particular the EKdV limited the amplitude growth seen in KdV as the wave passed into shallow water, leading to an associated strong retardation in speed. In this second part of the paper these results are applied to two-dimensional bathymetry. The propagation of an oblique planar wave across a wedge-shaped slope is ®rst studied. Whereas the linear internal-wave theory predicts small refraction of the wave front, the nonlinear EKdV model shows that the refraction of large amplitude internal waves is signi®cantly greater. This is explained by the stronger retardation seen in the shoaling experiments. Second, the propagation of a planar internal wave across a seamount is investigated. Here signi®cant differences are found between the KdV and EKdV theory. Linear theory and nonlinear EKdV theory predict a focusing of rays past the seamount, and an eventual caustic. The EKdV predicts that the focusing occurs at shorter ranges than in the linear case, with the range dependent on the wave amplitude. In contrast KdV theory predicts that the focusing and consequent amplitude increase leads to such an increase in speed that the rays start to spread again and the caustic is avoided. It is concluded that nonlinearity has a signi®cant effect on the refraction of interfacial solitary waves. Further, as the strongest refraction occurs in water depths where the EKdV appears to be the more appropriate theory, the EKdV results are considered to be more reliable. 1. Introduction: The refraction of linear and (where H is the water depth for the surface wave case nonlinear waves and h1 and h2 are the upper and lower layer thicknesses in the two-layer case, Dr is the density difference across The oblique incidence of internal wave rays onto the interface, r is the reference density, and g the ac- varying bathymetry may cause refraction of the wave 0 celeration due to gravity). The Boussinesq approxima- fronts, due to the variation of phase speed along the tion, of differences between the layer densities being only wave front. As with surface waves, the internal wave important when multiplied by g, has been assumed here phase speed is dependent on depth (as well as strati®- and will be used because in many open ocean situations cation and current shear) as shown below for the linear the density difference between layers is, in fact, quite case. For both surface waves, and internal waves on an small [O(0.001) of the density value]. The assumption interface whose mean depth does not change with range, of long waves is equivalent to that of shallow water phase speed decreases with decreasing depth. This can be seen simply from the linear phase speed expression waves: that is, the wavelength is signi®cantly longer than the water depth. Clearly c decreases with decreasing H for long surface waves (c ), and long two-layer internal s s while c decreases with decreasing h (assuming h re- waves (c ) (assuming a rigid lid): 0 2 1 0 mains fairly constant), and hence with water depth. g9h Dr For long linear waves the refraction of the wave obeys c 5 ÏgH c 5 1 g95g (1) Snell's law: s 0 1 h /h ! 1 12r 0 sinfrefc ref 5 0 , (2) in in Corresponding author address: Dr. Justin Small, International Pa- sinf c0 ci®c Research Center, School of Ocean and Earth Science and Tech- in ref nology, University of Hawaii at Manoa, 2525 Correa Rd., Honolulu, where f and f refer to the incident and refracted HI 96822. angles, respectively, between the wave direction and the E-mail: [email protected] gradient of bathymetry (Pelinovsky et al. 1994), and Unauthenticated | Downloaded 10/06/21 01:56 PM UTC NOVEMBER 2001 SMALL 3185 TABLE 1. De®nition of acronyms. As the phase speed is dependent on amplitude for DERA Defence Evaluation Research Agency both ®rst- and second order theory, it is clear that the DR Djordevic and Redekopp (1978) change of amplitude across varying bathymetry, the cKdV Cylindrical KdV shoaling effect, is important to the refraction of nonlin- EKdV Extended Korteweg±de Vries ear waves. In Part I the shoaling of solitary waves as JKKD Intermediate long-wave theory of Joseph (1977) they passed on/offshelf over one-dimensional bathym- and Kubota et al. (1978) KdV Korteweg±de Vries etry was examined. Comparisons were made with an- MB Michallat and Bartholemey (1998) alytical results for adiabatic propagation over a slowly NRL Naval Research Laboratory changing slope, and with ®ssion laws for rapidly chang- SAR Synthetic aperture radar ing slopes. The agreement was good. Both ®rst-order SES Shelf Edge Study, a study of oceanographic pro- (vKdV) and extended (vEKdV) nonlinear evolution cesses at the Malin shelf SESAME SES Acoustic Measurement Experiment, a DERA/ models were tested. In the adiabatic KdV case, the sol- NRL study in conjunction with SES itary wave slowly increases in amplitude but retains the TVB Turner and Vanden-Broek (1988) solitary waveform (except for the formation of a long, vEKdV Variable coef®cient EKdV small amplitude tail or ``shelf'' behind it). The phase speed slowly reduces with depth. In contrast, when the bathymetry is rapidly changing, the soliton is seen to in ref split into a number of waves, a process known as soliton cc00and are the corresponding phase speeds of the incident and refracted wave, dependent only on water ®ssion, and the lead wave amplitude is reduced relative depth (assuming the top layer thickness does not to the adiabatic case. change). EKdV dynamics predicted rather different results: un- In this paper the extent by which the linear refraction der adiabatic conditions the solitary wave initially grows effects are modi®ed by the introduction of nonlinear but then in shallow water the amplitude is reduced. This terms is investigated. For instance, as the wave approach- is due to a theoretical upper limit on EKdV amplitude, es shallow water it may shoal and grow in amplitude. given by hc[(27) of Part I], namely that the wave cannot According to ®rst-order nonlinear theory the phase speed depress the interface beyond middepth. This also leads of a wave should increase with increasing amplitude, as to a rapid reduction in wave phase speed. In water of rapidly changing depth, the ®ssion effect of KdV is not follows [Ostrovsky and Stepanyants (1989); see appendix seen, as the shape of the EKdV wave does not drastically B of Part I (Small 2001) of this paper]: change between deep and shallow water (in contrast to ha the signi®cant narrowing of the KdV wave). Offshelf c 5 c 1 0 , (3) 0 3 propagation was also examined. In most cases this lead to the reverse of the onshelf effect, but in cases where where a is a function of the density distribution and h 0 the initial wave was close to the theoretical limit hc, is the amplitude, which leads to the question of whether the wave was so large that the bathymetric slope was this amplitude-dependent effect will cancel the depth- often steeper than the wave, and hence ®ssion of the dependent effect. KdV theory is believed to be relevant wave occurred. for small amplitude waves in many two-layer scenarios From these one-dimensional results it can be seen that (see, e.g., Michallet and Bartholemey 1998, hereafter nonlinearity not only increases the phase speed of the MB); see Table 1 for de®nition of acronyms. internal waves relative to the linear case, but it can also When the wave amplitudes become so large that sec- lead to stronger retardation of the wave (i.e., the ratio ond-order effects become important, a new relationship of phase speed in shallow water to that in deep water between phase speed and amplitude occurs, which was is reduced). This is likely to lead to changes in the two- dealt with explicitly in Part I (appendix B) and which dimensional refractive properties of the waves, as will may be summarized as be discussed in the remainder of this paper, organized as follows. Section 2 recaps on the model developed in c 5 c001 f(h ), (4) Part I and presents the ray spreading algorithm. Section where f is a function of amplitude and density distri- 3 investigates the oblique incidence of internal waves bution and increases with increasing absolute amplitude. on a wedge-shaped continental slope. Section 4 then applies the model to simulate refraction of a planar wave Fully second-order theories exist (see, e.g., Koop and over a seamount. A detailed discussion of the merits of Butler 1981) but in this paper the partially second-order the theoretical basis of the model is presented in the extended KdV (EKdV theory) is used (Holloway et al. discussion, section 5. Section 6 then presents the sum- 1999).
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modi Ed Kdv Equation
    Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modied KdV Equation Lin Huang Hangzhou Dianzi University Nannan Lv ( [email protected] ) Hangzhou Dianzi University School of Science Research Article Keywords: Darboux transformation, Soliton molecule, Positon solution, Rational positon solution, Rogue waves solution Posted Date: May 11th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-434604/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Version of Record: A version of this preprint was published at Nonlinear Dynamics on August 5th, 2021. See the published version at https://doi.org/10.1007/s11071-021-06764-x. Soliton molecules, rational positons and rogue waves for the extended complex modified KdV equation Lin Huang ∗and Nannan Lv† School of Science, Hangzhou Dianzi University, 310018 Zhejiang, China. May 6, 2021 Abstract We consider the integrable extended complex modified Korteweg–de Vries equation, which is generalized modified KdV equation. The first part of the article considers the construction of solutions via the Darboux transformation. We obtain some exact solutions, such as soliton molecules, positon solutions, rational positon solutions, and rogue waves solutions. The second part of the article analyzes the dynamics of rogue waves. By means of the numerical analysis, under the standard decomposition, we divide the rogue waves into three patterns: fundamental pattern, triangular pattern and ring pattern. For the fundamental pattern, we define the length and width of the rogue waves and discuss the effect of different parameters on rogue waves. Keywords: Darboux transformation, Soliton molecule, Positon solution, Rational positon so- lution, Rogue waves solution.
    [Show full text]
  • Andaman Sea by Global Ocean Associates Prepared for the Office of Naval Research - Code 322PO
    An Atlas of Oceanic Internal Solitary Waves (May 2002) The Andaman Sea by Global Ocean Associates Prepared for the Office of Naval Research - Code 322PO Andaman Sea Overview The Andaman Sea is located along the eastern side of the Indian Ocean between the Malay Peninsula and the Andaman and Nicobar Islands (Figure 1). It is a deep-water sea with exits to the Indian Ocean (to the west) and the Strait of Malacca (to the south). Figure 1. Bathymetry of Andaman Sea [Smith and Sandwell, 1997]. 485 An Atlas of Oceanic Internal Solitary Waves (May 2002) The Andaman Sea by Global Ocean Associates Prepared for the Office of Naval Research - Code 322PO Observations As far back as the mid-19th century surface manifestations of solitons have been observed consisting of strong bands of sea surface roughness. These bands were referred to as "ripplings", due to their mistaken association with rip tides. A description of such bands can be found in the book of Maury [1861] published in 1861 and which is quoted in Osborne and Burch [1980]: "In the entrance of the Malacca Straits, near Nicobar and Acheen Islands, and between them and Junkseylon, there are often strong ripplings, particularly in the southwest monsoon; these are alarming to persons unacquainted, for the broken water makes a great noise when the ship is passing through the ripplings at night. In most places ripplings are thought to be produced by strong currents, but here they are frequently seen when there is no perceptible current…so as to produce an error in the course and distance sailed, yet the surface if the water is impelled forward by some undiscovered cause.
    [Show full text]
  • Soliton-Guided Quantum Information Processing
    Chapter 1 Soliton-Guided Quantum Information Processing Ken Steiglitz Computer Science Department Princeton University Princeton NJ 08544 Abstract We describe applications of solitons and soliton collisions to the transport, transfer, and beam-splitting of qubits carried by optical photons. The transport and transfer realize the “flying qubits” necessary for quantum information processing, and the beam-splitting leads, in theory, to an implementation of quantum computing using linear optics. These proposed applications are embedded in a uniform optical fiber and require no special device fabrication, no cooling, and no vacuum. The pioneering papers of Feynman [1] and Deutsch [2] in the 1980s sparked the rapid development of the field of quantum information processing. The theoretical and experimentalprogress has been remarkable, with the development, for example, of quantum error correction and a fast algorithm for factoring, and the exploration of a wide variety of physical implementations. In the latter category, the optical photon as the carrier of a qubit has played an important role in the experimental demonstra- tion of quantum cryptography and other important applications to communications and information processing. At the same time there has been tremendous progress in our understanding of classical nonlinear waves, and, in particular, solitons in op- tical fibers. In this chapter we will explore what role solitons and soliton collisions might play in the development of quantum information processing with optical pho- tons. For a more detailed account, the reader is referred to [3, 4, 5], from which the material in this chapter was drawn. 1 2 Steiglitz 1.1 Photon trapping A pulse traveling down a fiber forms a soliton when the dispersion, which tends to widen the pulse, is counterbalanced by the nonlinear Kerr effect, whereby the elec- tric field changes the index of refraction of the material.
    [Show full text]
  • Mysterious Underwater Waves
    FEATURE | PROF. LEO MAAS, NIOZ ROYAL NETHERLANDS INSTITUTE FOR SEA RESEARCH, THE NETHERLANDS Uncharted Reality? Mysterious Underwater Waves In addition to surface waves generated by wind, tidal forces or, in case of tsunamis, seismic events, there exists a less well-known class of bigger waves under water. These underwater waves (internal waves) are generated by the same forces, but owe their existence to the ocean’s non-uniform density distribution. This paper introduces various underwater waves that appear on interfaces or in continuous density stratification. In uniformly-stratified seas, these waves differ from surface waves in nearly every conceivable aspect, creating underwater ‘storms’ at certain hotspots, which may possibly impact natural and man-made structures, and which may transport heat and material vertically. The ocean’s salt and heat content, which its boundary. While ocean and atmosphere approximately a factor thousand, the ratio of determines its density, vary with depth. are dynamically quite similar, we never see the cross-thermocline density difference to This stable stratification supports underwater such waves in the ocean though, because mean density. Therefore, interfacial waves waves. Due to slight differences in density, the ocean is opaque. Light penetrates a mere attain larger amplitudes (up to hundreds of these waves easily displace fluid parcels hundred metres, leaving the remainder of metres), have longer periods (longer than vertically over hundreds of metres. the ocean literally and figuratively in the dark. 10 minutes, say) and have wavelengths (up Depending on the local rate with which But oceanographic instruments ‘visualise to kilometres) that are short compared to density increases with depth, they propagate the invisible’.
    [Show full text]