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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.

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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. ity. When the Froude number is critical

50°N

25°N

0°

25°S

50°S

180° 120°W 60°W 0° 60°E 120°E 180°

Figure 1. The location of nonlinear internal waves observed in 250 m resolution ODISM (Moderate-Resolution Imaging Spectroradiometer) satellite sunglint imagery acquired from August 2002 through May 2004. More than 3,500 nonlinear internal wave occurrences were identified by the survey. Adapted from Jackson (2007)

110 Oceanography | Vol. 25, No. 2 (F = 1), a disturbance on the interface can then evolve into a nonlinear internal Archipelago is one of the principal places will remain stationary and accumulate wave train according to KdV theory. of origin for the nonlinear internal wave energy through resonance. This condi- Figure 2 shows this process schematically. packets that propagate into the Sulu tion can occur when a moving fluid In the figure, the arrows represent the Sea (Apel et al., 1985). The Indonesian encounters a change in depth, with the direction of the overall flow. The patterns seas are home to a wide range of non- Froude number transitioning from sub- above the waveform represent the surface linear internal wave activity due to the critical (F < 1) to supercritical (F > 1), roughness associated with the conver- large number of shallow interisland sills or vice versa. gence zone of the nonlinear internal separating deeper-water basins in the In the context of nonlinear internal wave. Hibiya (1986) modeled this genera- presence of strong currents. Along the wave generation at a sill or bank, the tion as a transient process in response to western side of the Andaman Sea among sloping seabed imparts a vertical com- a time-varying tidal flow. In his model, the Nicobar Islands, lee waves are gener- ponent to the stratified flow, displacing infinitesimal internal waves are continu- ated on both the east and west sides of the pycnocline. The seabed feature also ously generated during the tidal cycle and sills connecting several of the islands. provides the variation in depth necessary constructively superimposed during the Figure 3, a true-color MODIS sunglint to set up the transition zone between supercritical phase of the ebb tide. acquired on April 6, 2004, at 4:25 UTC subcritical and supercritical flow. The Sills and banks are among the prin- (approximately the time when the east/ result is a stationary lee wave of depres- cipal generation sites of large non- west component of the barotropic tidal sion in the pycnocline downstream of linear internal waves in the ocean. The current was close to zero) shows this the sill, which attains a large phase speed Camarinal and Spartel sills are respon- bidirectional generation situation. Wave relative to the moving water in order to sible for the generation of the nonlin- packets 1 and 2 were released from remain stationary relative to the bank. As ear internal waves found at the Strait the eastern side of the sill as the tide the flow slackens and turns subcritical, of Gibraltar (Armi and Farmer, 1988; slackened. These packets will combine the lee wave can advance over the sill and Farmer and Armi, 1988; La Violette and and evolve to resemble Packet 3, which propagate away. This initial disturbance Arnone, 1988). Pearl Bank in the Sulu was generated roughly 12 hours earlier.

a b c

Figure 2. A schematic showing nonlinear internal wave generation via the lee wave mechanism of Maxworthy (1979). (a) As the stratified flow moves over the sill, the initial lee wave develops on the downstream side. (b) As the flow slackens, the lee wave steepens nonlinearly and propagates upstream over the sill. (c) The lee wave evolves into a nonlinear internal wave packet in a manner consistent with the predictions of Korteweg-de Vries theory. Adapted from Haury et al. (1979)

Oceanography | June 2012 111 Packet 4 was released from the western et al. (1986) traced both on- and off-shelf 1982) that is also responsible for the side of the sill during the prior slack propagating waves back to a generation generation of the internal (or baroclinic) tide, approximately six hours before the time at the shelf break during the ebb tide. In its simplest two-layer form, the time of the image. tide. This simultaneous symmetric gen- baroclinic tide comprises a linear inter- In a similar manner, a continen- eration is inconsistent with the lee wave nal wave at tidal period on the density tal shelf break can cause a downward mechanism, but it is consistent with the interface. Baroclinic tidal generation displacement of the pycnocline as the mechanism of internal tide generation occurs in a manner similar to the lee current flows off-shelf. Some nonlinear (Pingree et al., 1983). So while the lee wave mechanism, where vertical dis- internal wave packets observed on the wave mechanism can be useful for gain- placements of the pycnocline are caused continental shelf appear to possess prop- ing insight into the origin of on-shelf by seabed features that induce verti- erties that are consistent with generation nonlinear internal waves, it cannot fully cal components into a tidal flow. The by the lee wave mechanism (Pingree and explain nonlinear internal wave genera- internal tide propagates away from the Mardell, 1985). Huthnance (1981) sug- tion at the shelf break, so other genera- generation zone and may initially have a gested that the lee wave mechanism may tion mechanisms must be considered. sinusoidal shape, but, if sufficiently ener- also be responsible for the generation of getic, it will steepen, break, and evolve off-shelf propagating nonlinear internal Internal Tide Evolution into nonlinear internal waves. waves, where a lee wave formed off the The lee wave is a specific manifestation A number of authors explain the shelf during on-shelf tidal flow is released of the general phenomenon of tide- origin of nonlinear internal waves in as the tide relaxes. However, Pingree topography interaction (see Baines, terms of the evolution of the internal

9°15’N

9°00’N

8°45’N

8°30’N

8°15’N

8°00’N

7°45’N 91°40’E 92°00’E 92°20’E 92°40’E 93°00’E 93°20’E 93°40’E 94°00’E Figure 3. True-color MODIS image acquired on April 6, 2004, at 4:25 UTC showing the bidirectional generation of nonlinear internal waves from the sills among the Nicobar Island chain. Adapted from Jackson (2007)

112 Oceanography | Vol. 25, No. 2 tide, including Holloway (1987), Smyth distance. However, Henyey and Hoering of nonlinear internal waves produced. A and Holloway (1988), Sandstrom et al. (1997) describe a physical mechanism similar result demonstrating the impact (1989), Brickman and Loder (1993), and that could lead to a tendency for trapping of rotation in slowing the development Farmer et al. (2011). The relationship in a scenario where slowly propagating and evolution of nonlinear internal waves between nonlinear internal waves and nonlinear internal waves gain energy was also found by Farmer et al. (2009), the internal tide appears be more subtle from a larger-scale internal bore, while who successfully used the Gerkema than merely a common tide-topography fast-moving nonlinear internal waves (1996) generation model to reproduce interaction. In some locations, nonlinear lose energy. Energetic equilibrium occurs the distinctive features found in mooring internal waves can maintain a fixed close to the linear internal wave phase observations of large nonlinear internal phase relationship with the baroclinic speed, but it should be noted that this waves in the northern South China Sea. tide as both propagate together, with theory was formulated without consider- Following the convention of the day, the nonlinear internal waves apparently ing the effects of rotation. the Gerkema (1996) model represented “trapped” in the troughs of the longer Rotation, as well as nonlinearity, tidal forcing in a two-dimensional slice tidal waves. This relationship has been should be accounted for in models of perpendicular to the continental shelf observed for nonlinear internal waves nonlinear internal wave generation break. It was common practice to include generated at both shelf breaks (Pingree and internal tide disintegration. They only the cross-slope component of tidal and Mardell, 1985; Pingree et al., 1986; were brought together elegantly in the flow, because the along-slope component New and Pingree 1992) and sills (Apel unified nonlinear internal tide model was thought to not contribute towards et al., 1985). Locations also include of Gerkema (1996), which includes inducing vertical motions. However, on places where the nonlinear internal generation of an initial disturbance and many continental margins around the waves and the internal tide remain in subsequent disintegration into nonlinear world, the strongest tidal currents tend phase after three to four tidal cycles internal waves via KdV type physics. The to be aligned along the bathymetry con- (da Silva et al., 2011). effect of rotation is shown to have an tours, with relatively weak cross-slope It is necessary to delve into a little important dispersive influence that could components. Strong internal tide genera- more theory to appreciate the subtleties prevent internal tide disintegration if it tion occurs in regions where strong tides of the relationship between a nonlinear is strong enough to overcome nonlinear- flow perpendicular to the slope of the internal wave and the internal tide. ity. The effects of rotation on the internal seabed. Internal waves can be generated The effect of Earth’s rotation enhances tide increase at higher latitudes as the where depth contours curve to become the phase speed of a two-layer internal Coriolis dispersion increases. perpendicular to the tidal flow, creating a tide (Defant, 1960), but does not affect The Gerkema (1996) unified nonlinear generation “hotspot.” internal solitons, which are typically internal tide generation model success- Sherwin et al. (2002) identified a clas- much shorter in length than the internal fully reproduced the nonlinear internal sic example of this three-dimensional Rossby radius. However, internal solitons wave observations made by Pingree and generation effect off the coast of Portugal, also have an enhanced phase speed, this Mardell (1985) in the Celtic Sea. Coriolis illustrated in Figure 4. The image shows time due to the effect of nonlinearity dispersion prevented disintegration of two packets of nonlinear internal waves (e.g., Osborne and Burch, 1980). It would the internal tide at neap tides but not at on the shelf (packets A and B), with one be remarkable if the very distinct mecha- spring tides, when it was overcome by still over the slope (packet B). A third nisms of Earth’s rotation and nonlinearity increased nonlinearity. The Gerkema packet (packet C) out in the deep ocean produced the same enhancement to the (1996) model also reproduced the appears to be propagating onshore, based wave phase speed. It is therefore likely nonlinear internal waves observed by on the orientation of the SAR wave signa- that the match between these enhance- Halpern (1971) in Massachusetts Bay, ture. The generation site is shown to the ments is only approximate, in which where Coriolis dispersion was not strong south (marked X in Figure 4) where the case nonlinear waves would no longer enough to prevent disintegration of the continental margin bulges out into the appear trapped after a long propagation internal tide, but did reduce the number deep ocean. Internal waves propagated

Oceanography | June 2012 113 offshore into the deep ocean toward the tide, proposing a novel generation Over a flat bottom, linear wave theory north before being refracted onshore to mechanism for nonlinear internal waves predicts the combination of various impact the shelf near 41°N. observed in the central Bay of Biscay. vertical wave modes at tidal frequency This mechanism incorporates the effects that, superposed on each other, create Tidal Beam Generation of continuous stratification, which beams along which the energy propa- New and Pingree (1992) discuss an produce a waveguide that allows inter- gates and the particle velocities inten- interesting connection between non- nal tidal energy to propagate through sify (e.g., Gerkema and Zimmerman, linear internal waves and the internal the whole vertical extent of the ocean. 2008). Energy propagates upward and

3 km –100 41.4°N

20 km −50

−200 −1000–1000−50−2000 −500 −50 Oporto 41.2°N A C B −50 3 2 1 41.0°N

−50

−50 40.8°N −10−500

0 −2000

40.6°N

x Iberia 40.4°N −2000 −200

−500

350.0°E 350.2°E 350.4°E 350.6°E 350.8°E 351.0°E 351.2°E 351.4°E 351.6°E Figure 4. ERS-1 SAR (European Remote Sensing-1 synthetic aperture radar) image acquired August 8, 1994, at 11:20 UTC (orbit 16020, frame 2781). Light regions indicate high backscatter. The “X” denotes the generation site and the orange curves show the refraction of internal waves that propagate onshore to impact the shelf break. In situ measurements of nonlinear internal waves were made at the stations marked 1 to 3. The distance between the leading waves in the packets is approximately 500 m.

114 Oceanography | Vol. 25, No. 2 downward along these internal tidal and New (1991) and New and Pingree the phase speed of vertical modes in a beams, reflecting from the ocean floor (1992). Later, New and da Silva (2002) constantly stratified layer of depth H and ocean surface, in a manner similar reported the coherent two-dimensional (the mode-1 phase speed is NcH/π). The to a laser beam propagating diagonally structure of these nonlinear internal parameter γ therefore reflects the ratio of between two parallel mirrors (or within waves using SAR. the phase speeds of waves on the pycno- an optical fiber). The tidal beams follow New and Pingree coined the term cline and the uniformly stratified lower a curve whose slope changes with depth “local generation” to describe this new layer. Gerkema (2001) put forward a the- due to varying stratification, similar to internal wave generation mechanism. oretical criterion of γ = 0.1 for optimal light propagating through a medium Subsequent observations and modeling excitations of solitons in the pycnocline. with a gradually increasing (or decreas- have confirmed soliton generation in The value of γ = 0.2 appears to be the ing) index of refraction. The curvature is this manner, provided certain require- upper limit for local generation based on greatest where the stratification is stron- ments of the near-surface stratification lab experiments and numerical model- gest (usually near the ocean surface), are met. According to the studies of ing (see, e.g., Grisouard et al., 2011, and and least where the stratification is weak Gerkema (2001) and Akylas et al. (2007), Mercier et al., in press). (usually near the ocean floor). In homo- in a given ocean basin (represented by an Nonlinear internal wave generation geneous stratification, the beams would average depth H, an upper mixed layer of via tidal beams occurs outside the tropics be straight lines reflecting between the thickness d and density ρ, an interfacial (latitudes > 20°) where the stratification ocean floor and the ocean surface. pycnocline of density step Δρ, beneath is relatively moderate. In areas where the In this generation method, the conti- which lies a layer of constant buoyancy beam generation mechanism has been nental shelf edges excite the tidal beams frequency, Nc), the stratification should identified, values of gamma have been where the slope of the bottom topogra- be of intermediate strength (quantified found to fall within this optimum range phy matches the angle of the beam to by the parameter g' = gΔρ/ρ) for the (0.1 ≤ γ ≤ 0.2), including in the Bay of the horizontal. These locations are called internal tidal beam to produce large dis- Biscay (γ = 0.14; Pingree and New, 1991; “critical” slopes, where a downward- placements of the pycnocline that con- New and Pingree, 1992), off the Iberian propagating ray can form, propagate to sequently evolve into nonlinear internal continental shelf in summer (γ = 0.15; the seafloor, reflect upward, and impinge waves. If the pycnocline strength is either Azevedo et al., 2006; da Silva et al., upon the pycnocline a long distance away very small or very large, there is no sig- 2007), and in the Mozambique Channel from the continental shelf. In the early nificant transfer of energy from beams to in winter (γ = 0.17; da Silva et al., 2009). 1990s, Adrian New and Robin Pingree interfacial waves. Local generation of nonlinear internal noted that the location of large solitons in In an attempt to understand and waves is very likely to occur in other the central Bay of Biscay coincided with quantify the way the pycnocline regions, but identification is more dif- the intersection of the thermocline with responds to an impinging internal wave ficult if other generation mechanisms internal tidal beams generated 150 km beam, Gerkema (2001) used linear are also at work. Da Silva et al. (2007) away at the edge of the continental shelf. wave theory and identified a parameter describe the role of SAR images in iden- They suggested that the thermocline in (gamma) to “measure” this interaction. tifying beam-generated solitons. the central bay is excited by tidal energy Gerkema defined gamma by Figure 5 shows an example of local originating from the critical region of the g’d generation of nonlinear internal waves off γ = , (1) continental slope and reflecting from the Nc H the west Iberian shelf. The critical slopes seabed along these beams. The model of where H is the total water depth. The are situated below 1,000 m depth, some New (1988) correctly predicted the loca- parameter γ is a ratio of two phase 50 km to the west of the shelf break. tion where a reflected beam would sur- speeds: the numerator is the phase speed Tidal beams emanate from the criti- face and produce large thermocline oscil- of long interfacial waves in a two-layer cal slopes, one downward and offshore, lations; observational evidence for the system whose upper layer thickness and another upward and inshore. The reflected beam was provided by Pingree is d; the denominator is a measure of offshore-propagating beam reflects from

Oceanography | June 2012 115 the seafloor and impinges the thermo- approximately 30 km and wavelengths flow was perturbed in the vertical by the cline from below some 60 km away from (crest-to-crest length) are up to 1 km presence of a sill or bank and how the its origin (Figure 5b) where it is observed long (Figure 5c and d). For additional Froude number controls the release of in the SAR image propagating offshore details, see da Silva et al. (2007). the nonlinear internal wave. A stratified (Figure 5c). The thermocline at the point flow can also be perturbed by a lateral of impingement is sufficiently mild to Resonant Generation (horizontal) topographic contraction allow the generation of a nonlinear The section above on nonlinear inter- or small-scale undulations in bottom internal wave train of considerable nal wave generation via the lee wave topography, with resulting critical flows dimensions. Crest lengths extend for mechanism discusses how stratified and creation of nonlinear internal waves.

a c 39.5°N

39.0°N

38.5°N Iberian Peninsula

38.0°N

37.5°N

37.0°N

36.5°N –13°E –12°E –11°E –10°E –9°E –8°E b d 0 60 km 0.2 Critical –2000 Bathymetry 0.0

Depth (m) –0.2

–4000 Relative Intensit y –0.4

50 100150 2 4 6 Distance (km) Distance (km) Figure 5. (a) Map of area southwest of the Iberian Peninsula with internal solitary wave crests marked in red based on one Envisat Advanced Synthetic Aperture Radar (ASAR) image in Wide-Swath Mode dated August 4, 2004 (22:27 UTC). (b) Ray-tracing diagram showing internal tide ray paths (in green; emanating from critical topography, in blue) along the black line in part (a). The small black squares in part (a) show where the ray path crosses the near-surface thermocline (taken at a depth of 50 m), and are also marked in part (b). (c) Full-resolution detail of a nonlinear internal wave train observed to propagate toward the west-northwest and believed to be generated by the tidal beam in part (b). (d) Radar backscatter profile showing the wavelength cross section of a nonlinear internal wave train generated by the tidal beam. The narrow rectangle in part (c) represents the cross section from which part (d) was obtained. The square in part (c) is a background backscatter reference used to normalize the radar profile in part (d). Zero in part (d) represents the average unperturbed backscatter of the SAR image in part (c).

116 Oceanography | Vol. 25, No. 2 Lateral contractions are not as common Helfrich (2008) at Race Point Channel only a single train of nonlinear internal in the ocean as sills or banks, and it is in Massachusetts Bay. There, a flat chan- waves are observed each tidal cycle. At much less intuitive to understand why a nel between Race Point and Stellwagen Cape Cod’s Race Point Channel, a pair lateral contraction would result in verti- Bank, with sill depth of just 50 m, con- of internal wave trains is systematically cal displacements of the interface and tains tidal currents that reach super- observed to form during the ebb tide. produce internal waves. But mathemati- critical speeds during the ebb phase of Da Silva and Helfrich (2008) assume the cal models using weakly nonlinear KdV the tide. Da Silva and Helfrich (2008) resonant flow during the ebb phase of the type theory have successfully predicted show that the shear flow in Race Point tide generates both of these wave trains “resonantly generated” upstream- Channel is supercritical in summer, and and speculate that the first of these pairs propagating nonlinear solitary waves in find numerous examples in SAR images of wave trains is generated during the a flat channel of slowly varying lateral of large-amplitude nonlinear internal acceleration phase of the flow passing dimension when the flow is nearly criti- waves generated within the channel, through the supercritical speeds, while cal (e.g., Clarke and Grimshaw, 1994; near the minimum breadth between the second is generated when the flow Redekopp and You, 1995). Generation Stellwagen Bank and the northern tip decelerates from supercritical to subcriti- occurs especially in the presence of a of Cape Cod. These solitons begin with cal speeds (in a similar manner to the vertical shear current. Laboratory experi- short crestlengths (1–3 km) but very river plume mechanism described below, ments (e.g., Melville and Helfrich, 1987; distinct amplitudes, detectable in the except here there is no density front). The Melville and Macomb, 1987) confirm SAR images (see inset in Figure 6). As timing of SAR images seems to be right these findings, but the situation has been the ebb flow slackens, the nonlinear for observing the generation that occurs less-comprehensively studied in the real internal waves are observed to propagate during accelerating and decelerating ocean. When the bottom topography upstream and develop into long-crested flows through the supercritical regime has undulating features whose crests are solitons whose crest lengths exceed (see Figure 11 of da Silva and Helfrich, nearly perpendicular to the flow field, 15 km. These long-lived waves propa- 2008). Both of these nonlinear internal nonlinear internal waves can also appear gate into shallow water and dissipate wave trains propagate upstream into as a result of near-critical flow (see, in Massachusetts Bay (see da Silva and Massachusetts and Cape Cod Bays, but e.g., Pietrzak et al., 1990; Kranenburg Helfrich, 2008, for more details). Their with slightly different propagation direc- et al., 1991; Bogucki et al., 1997; horizontal structure is curved, in the tions, as Figure 6 shows (wave trains A Groeskamp et al., 2011; Stastna, 2011). form of a semicircle, which suggests a and B). Although these observations are This kind of resonant generation of soli- point source at Race Point (see Figure 6). consistent with the theories of resonant tary waves, which theoretically includes A remarkable feature of the nonlinear generation, additional modeling and field mode-2 waves (as described below), is internal waves generated at the chan- measurements are required before a final probably more frequently found in nature nel is the existence of two distinct wave conclusion can be made. than nonlinear waves produced as a trains that appear to be generated dur- result of a lateral contraction. ing the same tidal cycle at effectively Plume Generation In principle, shallow continental the same location (wave trains A and Until this point, this paper has con- shelves where strong tidal currents can B in Figure 6). This generation mecha- sidered pycnocline displacements that reach supercritical speeds are good nism is different from the typical lee evolve into nonlinear internal waves candidates for the resonant generation wave mechanism, which only produces resulting from various forms of tide mechanism of nonlinear internal waves. a single train of internal waves each topography interaction. However, the This mechanism has been measured tidal cycle (Chereskin, 1983). Cummins pycnocline can also be displaced by and reported by Boguscki et al. (1997) et al. (2003) show evidence of nonlinear the interaction between different water on the California continental shelf, with internal waves generated in the super- masses. Plumes, from either a river out- implications for sediment resuspen- critical regime that propagate upstream flow or a throughflow current, have been sion and transport, and by da Silva and in Knight Inlet, British Columbia, but found to produce nonlinear internal

Oceanography | June 2012 117 waves (Nash and Moum, 2005; Matthews due to the plume’s higher horizontal plume, Kilcher and Nash (2010) noted et al., 2011). The plume generation can velocity relative to the ocean water. The how plume remnants from previous tidal be thought of as an example of resonant plume represents a supercritical flow cycles strengthened stratification and generation, with the plume taking on the (F > 1), where an internal wave pro- enhanced the generation of nonlinear role previously played by the topography duced by the displacement would move internal waves. Matthews et al. (2011) (i.e., as the method for imparting vertical more slowly than the plume, so the wave showed how a plume associated with the displacements of the pycnocline). Rivers remains trapped at the plume’s leading Indonesian throughflow could produce discharge fresh water into the ocean edge. As the propagation speed of the “irregular internal waves” at Lombok in tidally modulated pulses (Figure 7) plume decreases (i.e., the flow transitions Strait during certain times of the year that impinge upon the existing coastal from supercritical [F > 1] to subcritical (in addition to the “regular” internal waters. The initial momentum of the [F < 1]), the internal wave separates from waves generated by the tide topog- impinging fresh water plume, and the the plume front and radiates away as a raphy interaction). wind and current conditions at the time freely propagating wave. Two SAR images acquired during the of discharge, determine the plume’s Nash and Moum (2005) documented summer of 2007 (Figure 7) show the character when it enters the ocean. The nonlinear internal wave generation by evolution of nonlinear internal waves impinging plume causes a buildup (or the Columbia River outflow in Oregon. generated by the Columbia River plume. convergence) at the plume’s leading edge, In subsequent observations of this Figure 7A shows a strong river plume

A

215 m –14

–15 Sigma0 (dB) –16 B

0.36 0.72 1.08 Figure 6. A TerraSAR-X image of the northern tip of Cape Cod acquired April 7, 2008, at 22:25 UTC, showing a semicircle-shaped train of internal waves and “nascent solitons” in Race Distance along pro le (km) Point Channel. A pair of packets is generated every tidal cycle. A backscatter profile of the “nascent” solitons is shown at lower right.

118 Oceanography | Vol. 25, No. 2 signature along with the signature of an bottom layer (e.g., Shroyer, 2008). Because, for KdV type theory, mode-2 early-stage plume-generated internal Much of the work on mode-2 solitary waves generally have shorter wave packet leading the plume’s intru- nonlinear internal waves has been theo- length scales compared to mode-1 sion into the Pacific. Figure 7B shows retical or numerical, or involved labora- solitary waves, this result was expected. the more mature signature of a classic tory experiments (see Yang et al., 2009, There have been only a few reports of rank ordered packet (after approximately for an extensive reference list). This body mode-2 wave observations in the ocean five hours of evolution). Both images of work has put forth no fewer than five and none have included direct observa- also show shoreward-propagating possible generation mechanisms. Three tion of the generation process. The waves nonlinear internal waves, generated at of the mechanisms involve a mode-1 have been found to occur around the the shelf break. nonlinear internal wave: a mode-1 wave Mascarene Ridge (Konyaev et al., 1995; propagating onshore and entering the da Silva et al., 2011), on the New Jersey Mode-2 Wave Generation breaking instability stage (Helfrich and shelf (Shroyer et al., 2010), in the South Although first-mode internal waves are Melville, 1986), a mode-1 wave flowing China Sea on the shelf north of the most commonly observed in the ocean, over a sill (Vlasenko and Hutter, 2001), Dongsha coral reef (Yang et al., 2009), higher-mode waveforms also occur and reflection of a mode-1 wave (Chao and at Knight Inlet in British Columbia under the proper conditions. A mode-2 et al., 2006). In addition, more recent (Farmer and Smith, 1980). wave (also known as varicose wave) is results by Stastna (2011) show that Yang et al. (2009) report an extensive best thought of in terms of a three-layer mode-2 resonant generation is possible set of in situ observations of mode-2 stratified ocean. In this configuration, when appropriately small bottom length waves from the upper continental the wave travels as a bulge in the middle scale undulations are chosen as bottom slope of the northern South China Sea. layer, displacing isopycnals upward into topographic features (smaller than for They describe the characteristics of the upper layer and downward into the mode-1 efficient resonant generation). 78 nonlinear internal mode-2 waves

Figure 7. Radarsat-1 SAR images from July 28 and August 7, 2007, showing the evolution of nonlinear internal wave packet generated by the Columbia River plume.

Oceanography | June 2012 119 (20 occurring during the summer and be related to the main thermocline being is that mode-2 waves are subject to a 58 during the winter) observed at a located near mid-depth. This scheme is resonance with mode-1 waves that may mooring deployed at 350 m depth from more in line with the proposed genera- rapidly drain energy from the mode-2 April 29 to July 28, 2005, and again from tion mechanism of (Stastna and Peltier, waves, contributing to the sparsity of November 2, 2005, to February 24, 2006. 2005), but additional study is required to observations. Again, mode-2 wave gen- Although the mooring observations of fully explain the occurrences. eration was not directly observed in this mode-2 waves observed by Yang et al. Shroyer et al. (2010) found the study, but was believed to be the result of (2009) occurred in a region of the conti- mode-2 nonlinear internal waves on tidal forcing near the shelf break or fron- nental shelf, with extensive mode-1 non- the New Jersey shelf to be much more tal intrusions (Shroyer et al., 2010). linear internal wave activity, the relation- ephemeral, observing just eight events Although mode-2 waves are short- ship between the mode-1 and mode-2 during the summer of 2006. Not only did lived on the New Jersey shelf, they have occurrences was not apparent. During the waves appear sporadically in moor- significant lifetimes over the Mascarene the winter, 72% of the mode-2 nonlinear ing records, they also appeared to be Plateau in the Indian Ocean where internal waves were observed without very short-lived; mode-2 packets were mode-2 solitons were identified by a prior mode-1 event. That number only tracked between two moorings sep- means of in situ measurements and dropped to only 10% during the summer arated by a kilometer, and they were not SAR observations (Konyaev et al., 1995; (at a time of extensive mode-1 activity). present at other moorings roughly 10 km da Silva et al., 2011). Figure 8 shows an Yang et al. (2009) believe that the mode-2 distant (along the propagation direc- Envisat ASAR (Advanced SAR) image nonlinear internal waves in winter could tion). One reason for this observation with clear evidence of a mode-2 wave

0.8 a) b) Mode-2 0.6 leading wave

0.4 bright M1 rough M2 0.2 P smooth 0.0 dark

P Relative intensity –0.2

–0.4 Mode-1 tail

–0.6 0 2 4 6 8 10 Distance (km) Figure 8. (a) Envisat ASAR image dated March 28, 2009, at 18:17 UTC of the Mascarene Plateau in the Indian Ocean. Labels M1 and M2 indicate signatures of mode-1 and mode-2 nonlinear internal waves. Profile P is shown in (b), where a mode-2 wave is followed by a tail of shorter-scale mode-1 internal waves (the white square represents an area unperturbed by internal waves from which P is normalized. (b) Radar backscatter profile P normalized with respect to a back- ground area unaffected by nonlinear internal waves. A leading mode-2 solitary wave is identified by a dark signature preceding a bright signature in the direc- tion of propagation (indicated by an arrow). Short-scale nonlinear internal waves follow the mode-2 larger wave in the form of a wave tail.

120 Oceanography | Vol. 25, No. 2 structure, followed by a wave tail of Summary providing the TerraSAR-X imagery short mode-1 waves. In Figure 8a, the The understanding of nonlinear internal under project OCE0056. SAR image intensity of mode-1 waves wave generation has continued to evolve is characterized by bright, enhanced since Maxworthy (1979) proposed the References backscatter preceding dark reduced lee wave generation mechanism. This Akylas, T., and R. Grimshaw. 1992. Solitary internal waves with oscillatory tails. Journal backscatter along the nonlinear internal evolution has been driven by in situ of Fluid Mechanics 242:279–298, http:// wave propagation direction (in agree- studies, the ever-increasing amount of dx.doi.org/10.1017/S0022112092002374. Akylas, T.R., R.H.J. Grimshaw, S.R. Clark, and ment with Alpers, 1985). For mode-2 satellite imagery, and, in more recent A. Tabaei. 2007. Reflecting tidal wave beams solitary wave structures, the polarity of years, the ability to numerically simulate and local generation of solitary waves in the ocean thermocline. Journal of Fluid the SAR signature is reversed (because the workings of the ocean over larger Mechanics 593:297–313, http://dx.doi.org/​ the location of the convergent and diver- areas at high resolution. Important 10.1017/S0022112007008786. gent zones are reversed), and thus a dark new results in nonlinear internal wave Alford, M.H., J.B. Mickett, S. Zhang, P. MacCready, Z. Zhao, and J. Newton. 2012. Internal reduced backscatter crest precedes a research have been published in the last waves on the Washington continental shelf. bright, enhanced backscatter feature in few years, including remote-sensing Oceanography 25(2):66–79, http://dx.doi.org/​ 10.5670/oceanog.2012.43. the propagation direction of the wave. observations of resonant generation Alpers, W. 1985. Theory of radar imaging of Moreover, Figure 8b shows a radar pro- (da Silva and Helfrich, 2008), signifi- internal waves. Nature 314:245–247, http:// dx.doi.org/10.1038/314245a0. file of a mode-2 large wave followed by a cant first time in situ measurements of Apel, J.R., J.R. Holbrook, A.K. Liu, and tail with several mode-1 shorter internal mode-2 waves (Yang et al., 2009; Shroyer J.J. Tsai. 1985. The Sulu Sea internal soli- waves. It should be noted, however, that et al., 2010), and advances in predic- ton experiment. Journal of Physical Oceanography 15(12):1,625–1,651, http:// these mode-2 waves are not truly soli- tion ability (Simmons et al., 2011). But dx.doi.org/10.1175/1520-0485(1985)015​ tary, as mode-2 solitary waves develop the understanding of nonlinear internal <1625:TSSISE>2.0.CO;2. Armi, L., and D.M. Farmer. 1988. The oscillatory tails (of mode-1). These tails waves is still incomplete. The exact gen- flow of Mediterranean water through can drain energy from the main wave, eration mechanism of some nonlinear the Strait of Gibraltar. Progress in Oceanography 21:1–105, http://dx.doi.org/​ as mentioned above, and grow in ampli- internal wave occurrences is still unclear 10.1016/0079-6611(88)90055-9. tude, as demonstrated in Akylas and (see Jackson, 2007; Alford et al., 2012, Azevedo, A., J.C.B. da Silva, and A.L. New. 2006. Grimshaw (1992). Vlasenko et al. (2010) in this issue) and mode-2 generation On the generation and propagation of internal solitary waves in the southern Bay of Biscay. proposed a mechanism through which has not yet been observed in the ocean. Deep Sea Research Part I 53:927–941, http:// these mode-2 waves with mode-1 tails Given their widespread distribution and dx.doi.org/10.1016/j.dsr.2006.01.013. Baines, P.G. 1982. On internal tide generation can be generated. For stable mode-2 the important role they play in connect- models. Deep Sea Research Part A 29:307–338, structures that survive long enough, ing the large-scale tides to smaller-scale http://dx.doi.org/10.1016/0198-0149(82)​ 90098-X. there is a chance that mode-2 solitary turbulence, nonlinear internal waves are Bogucki, D., T. Dickey, and L.G. Redekopp. 1997. waves are “overtaken” by mode-1 soli- expected to remain an important area of Sediment resuspension and mixing by reso- tons generated in previous tidal cycles. study for the future. nantly generated internal solitary waves. Journal of Physical Oceanography 27:1,181–1,196, Zabusky and Kruskal (1965) showed http://dx.doi.org/10.1175/1520-0485(1997)027​ that for weakly nonlinear theory, after a Acknowledgements <1181:SRAMBR>2.0.CO;2. Brickman, D., and J.W. Loder. 1993. Energetics of collision between mode-1 solitons, both Christopher Jackson gratefully acknowl- the internal tide on northern Georges Bank. waves preserve their shapes (but experi- edges the support of the Office of Journal of Physical Oceanography 23:409–424. Chao, S.-Y., P.-T. Shaw, M.-K. Hsu, and Y.J. Yang. ence only a phase shift). For sufficiently Naval Research through contract 2006. Reflection and diffraction of internal nonlinear solitons, this collision may no N0001409C0224. José C.B. da Silva solitary waves by a circular island. Journal of longer be shape preserving, as in the case thanks J.M. Magalhães for help with Oceanography 62:811–823, http://dx.doi.org/​ 10.1007/s10872-006-0100-4. of a faster (mode-1) soliton overtaking the preparation of Figures 4–6 and 8. Chereskin, T.K. 1983. Generation of internal waves a slower (mode-2) solitary wave (see, The authors thank ESA for providing in Massachusetts Bay. Journal of Geophysical Research 88(C4):2,649–2,661, http://dx.doi.org/​ e.g., Akylas and Grimshaw, 1992, and the ERS and Envisat imagery under 10.1029/JC088iC04p02649. Vlasenko et al., 2010). project AOPT-2423 and DLR for

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