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THE Official Magazine of the OCEANOGRAPHY SOCIETY OceThe OFFiciaaL MaganZineog OF the Oceanographyra Spocietyhy CITATION Jackson, C.R., J.C.B. da Silva, and G. Jeans. 2012. The generation of nonlinear internal waves. Oceanography 25(2):108–123, http://dx.doi.org/10.5670/oceanog.2012.46. DOI http://dx.doi.org/10.5670/oceanog.2012.46 COPYRIGHT This article has been published inOceanography , Volume 25, Number 2, a quarterly journal of The Oceanography Society. Copyright 2012 by The Oceanography Society. All rights reserved. USAGE Permission is granted to copy this article for use in teaching and research. Republication, systematic reproduction, or collective redistribution of any portion of this article by photocopy machine, reposting, or other means is permitted only with the approval of The Oceanography Society. Send all correspondence to: [email protected] or The Oceanography Society, PO Box 1931, Rockville, MD 20849-1931, USA. doWNLoaded From http://WWW.tos.org/oceanography SPECIAL ISSUE On InTERNAL WAVES BY CHRISTOPHER R. JACKSON, JOSÉ C.B. Da SILVA, AND GUS JEANS THE GENERATION OF NONLINEAR InTERNAL WAVES ABSTRACT. Nonlinear internal waves are found in many parts of the world ocean. Their widespread distribution is a result of their origin in the barotropic tide and in the variety of ways they can be generated, including by lee waves, tidal beams, resonance, plumes, and the transformation of the internal tide. The differing generation mechanisms and diversity of generation locations and conditions all combine to produce waves that range in scale from a few tens of meters to kilometers, but with all properly described by solitary wave theory. The ability of oceanic nonlinear internal waves to persist for days after generation and the key role internal waves play in connecting large-scale tides to smaller-scale turbulence make them important for understanding the ocean environment. The photo shows the surface manifestations of a train of nonlinear internal waves (visible as alternating bands of smooth and rough water) off the northern tip of Cape Cod as viewed from a whale watcher boat in Massachusetts Bay (near position labeled B in Figure 6). Photograph taken 30 August 2006 (15:30 local time) by J.C.B. da Silva 108 Oceanography | Vol. 25, No. 2 InTRODUCTION phenomena of considerable interest to a its soliton solution. It was the work of Nonlinear internal waves are just one variety of fields, including fluid dynam- Osborne and Burch (1980) in their study type of wave present in the ocean’s ics, plasma physics, condensed mat- of the internal waves in the Andaman interior. Other wave types include ter physics, optics, and biology (Scott, Sea during the 1970s that firmly tied small-amplitude linear internal waves, 2007). As nonlinear internal waves in the properties of solitary waves to baroclinic planetary Rossby waves, and the ocean, they represent a potential the phenomena of oceanic nonlinear the linear baroclinic tide. Each of these hazard to offshore drilling operations internal waves. waves has a distinctive generation pro- (Hyder et al., 2005) and have important Although they propagate along a cess, propagation manner, and propaga- consequences for underwater sound pycnocline in the interior of the ocean, tion timescale. Nonlinear internal waves propagation (e.g., Zhou and Zhang, nonlinear internal waves alter the sea propagate along a pycnocline in the 1991). They represent an important surface roughness, making them readily ocean, the portion of the water column energy transfer mechanism between the detectable in satellite imagery. Currents with a sharp change in density, typically large-scale tides and vertical mixing, and within the nonlinear internal wave pro- the result of differences in temperature they often play a key role in biological duce convergent and divergent zones or salinity. Generation can be achieved primary production (e.g., Sandstrom and on the surface that move in phase with in a variety of ways, some of which Elliott, 1984) by affecting the exchange the internal wave’s subsurface crests and involve the direct displacement of the of heat, nutrients, and other proper- troughs. These zones cause variations in pycnocline and others that involve a ties between the shelf and the open sea surface roughness that appear as dis- more complex conversion of tidal energy ocean (e.g., Huthnance, 1995). In the tinctive features in both synthetic aper- into pycnocline motion. These different atmosphere, solitary waves produce the ture radar (SAR) and optical sunglint mechanisms will be described in detail Morning Glory cloud formations over imagery. With an individual nonlinear in the following sections. northwestern Australia (e.g., Smith internal wave remaining coherent for Nonlinear internal waves are best et al., 1982, Reeder et al., 1995) and up to several days, waves generated on described in terms of a special waveform similar formations over the Mozambique several successive tidal cycles are often called a soliton, or solitary wave (see Channel and the Red Sea (da Silva and visible in the same image. In many Zabusky and Porter, 2010). The soliton Magalhães, 2009; Magalhães et al., 2011. cases, the waves generated in each has, as one of its defining features, the The hydrodynamic soliton was first cycle tend to be grouped together into ability to retain its form over an extended described by J. Scott Russell who, in a “rank ordered” packet, with the larg- time period as it propagates away from 1834, reported the formation of a singu- est wave in the lead, in accordance with its generation site. This persistent coher- lar, unchanging hump on the surface of a KdV type theory. The distance between ence of form is explained by the balance Scottish canal that was generated when a between nonlinear steepening effects and towed barge was brought to a sharp halt. Christopher R. Jackson (goa@ nonhydrostatic dispersion, represented Russell followed the wave for several internalwaveatlas.com) is Chief Scientist, by different terms in the governing equa- miles on horseback, remarking on how Global Ocean Associates, Alexandria, VA, tions. The Korteweg-de Vries (KdV) it retained a constant shape for a surpris- USA. José C.B. da Silva is Senior Scientist, equation, or a variant thereof, is typically ingly long distance. Subsequent labora- Department of Geosciences, Environment used to represent these key physical pro- tory experiments by Russell produced and Spatial Planning and Centro cesses (see Ostrovsky and Stepanyants, the first quantitative description of the Interdisciplinar de Investigação Marinha e 1989; Helfrich and Melville, 2006). The solitary wave (Russell, 1844). In 1895, Ambiental (CIIMAR), University of Porto, nonlinearity term of the KdV equation Diederik Korteweg and Gustav de Vries Portugal, and Guest Investigator, Woods contains the product of the wave ampli- produced a generic equation accounting Hole Oceanographic Institution, Woods tude and its spatial derivative. (to lowest order) for the wave proper- Hole, MA, USA. Gus Jeans is Director, Solitary waves are fascinating ties of dispersion and nonlinearity and Oceanalysis, Wallingford, UK. Oceanography | June 2012 109 leading waves in an individual packet onboard the National Aeronautics and LEE WAVE GENERATION can vary from a few tens of meters to Space Administration’s (NASA’s) Earth Nonlinear internal wave generation at an tens of kilometers, while the separation Observing System satellites Terra and underwater sill or a bank has been tradi- between packets (generated on differ- Aqua (Jackson, 2007). This survey, tionally explained by the lee wave mech- ent tidal cycles) can range from a few using 250 m resolution true-color imag- anism put forth by Maxworthy (1979). kilometers to more than 100 km, with ery and covering 21 months between The mechanism is based on the theory of both depending on the phase speed of August 2002 and May 2004, identified supercritical flows, which uses a param- the waves and the time since generation. more than 3,500 nonlinear internal wave eter called the internal Froude number In situ observations show that the phase occurrences. Limitations in the survey’s to characterize the hydraulic state of a speeds of oceanic nonlinear internal sampling means that this count repre- stratified shear flow. The Froude number –1 waves can vary from less than 0.3 m s sents a significant underestimate of the (F = uf /c) is a dimensionless quantity to more than 3 m s–1, with amplitudes (as number of internal wave occurrences that expresses the ratio of the velocity of measured as a displacement of the pyc- that took place during that period. a fluid (uf ) to the velocity (c) (relative to nocline) ranging from just a few meters The widespread distribution of non- the fluid velocity) of a small-amplitude to in excess of 100 m. linear internal waves is the result of their internal wave on the interface. The Beginning with early photographs origin in barotropic tidal forcing and in velocity (c) is the linear phase speed of taken by astronauts and continuing the different number of ways in which the first vertical internal mode. When through to today’s dedicated Earth they can be generated. The following sec- the Froude number is less than 1 (sub- observing satellites, more than 40 years tions will examine the various generation critical), information can move upstream of orbital imagery shows that non- mechanisms, which include lee waves, (opposite the direction of flow) because linear internal waves commonly occur tidal beams, resonance, plumes, and the the wave velocity is greater than the flow in many locations around the world conversion of the internal tide. The focus velocity. When the Froude number is (Jackson, 2004). Figure 1 presents a is on first mode (mode-1) waves, which greater than 1 (supercritical), informa- map of nonlinear internal wave occur- are the most common and displace the tion can only move downstream (in the rences detected in the sunglint imagery pycnocline downward in deep water, direction of the flow) because the fluid of the Moderate-Resolution Imaging but also includes a discussion of second velocity is greater than the wave veloc- Spectroradiometer (MODIS) sensor mode (mode-2) waves.
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