Trapped Core Formation within a Shoaling Nonlinear Internal Wave

Transcription

1 APRIL 2012 L I E N E T A L. 511 Trapped Core Formation within a Shoaling Nonlinear Internal Wave REN-CHIEH LIEN, ERIC A. D ASARO, AND FRANK HENYEY Applied Physics Laboratory, University of Washington, Seattle, Washington MING-HUEI CHANG Department of Marine Environmental Informatics, National Taiwan Ocean University, Keelung, Taiwan TSWEN-YUNG TANG Institute of Oceanography, National Taiwan University, Taipei, Taiwan YIING-JANG YANG Department of Marine Science, Naval Academy, Kaohsiung, Taiwan (Manuscript received 14 September 2010, in final form 3 October 2011) ABSTRACT Large-amplitude ( m) nonlinear internal waves (NLIWs) were observed on the continental slope in the northern South China Sea nearly diurnally during the spring tide. The evolution of one NLIW as it propagated up the continental slope is described. The NLIW arrived at the slope as a nearly steady-state solitary depression wave. As it propagated up the slope, the wave propagation speed C decreased dramatically from 2 to 1.3 m s 21, while the maximum along-wave current speed U max remained constant at 2 m s 21.AsU max exceeded C, the NLIW reached its breaking limit and formed a subsurface trapped core with closed streamlines in the coordinate frame of the propagating wave. The trapped core consisted of two counter-rotating vortices feeding a jet within the core. It was highly turbulent with m density overturnings caused by the vortices acting on the background stratification, with an estimated turbulent kinetic energy dissipation rate of O(10 24 )Wkg 21 andaneddydiffusivityofo(10 21 ) m 2 s 21. The core mixed continually with the surrounding water and created a wake of mixed water, observed as an isopycnal salinity anomaly. As the trapped core formed, the NLIW became unsteady and dissipative and broke into a large primary wave and a smaller wave. Although shoaling alone can lead to wave fission, the authors hypothesize that the wave breaking and the trapped core evolution may further trigger the fission process. These processes of wave fission and dissipation continued so that the NLIW evolved from a single deep-water solitary wave as it approached the continental slope into a train of smaller waves on the Dongsha Plateau. Observed properties, including wave width, amplitude, and propagation speed, are reasonably predicted by a fully nonlinear steady-state internal wave model, with better agreement in the deeper water. The agreement of observed and modeled propagation speed is improved when a reasonable vertical profile of background current is included in the model. 1. Introduction Nonlinear internal waves (NLIWs) are often generated over topography, such as submarine ridges or continental shelves or slopes. Because of their intrinsic nonlinearity, they may propagate long distances from the generation site and retain their shape without dispersion: for example, across the deep basin of the South Corresponding author address: Ren-Chieh Lien, Applied Physics Laboratory, University of Washington, 1013 NE 40th Street, Seattle, WA lien@apl.washington.edu China Sea (SCS) (Klymak et al. 2006). Fission (i.e., the disintegration of a single wave into a train of multiple waves) occurs when NLIWs are unsteady or when they encounter a rapid change of environment (e.g., shoaling topography) and become dynamically unbalanced. In the northern SCS, NLIWs have been observed in satellite images (Hsu and Liu 2000; Zhao et al. 2004) and by in situ measurements (Alford et al. 2010; Ramp et al. 2004; Yang et al. 2004; Chang et al. 2006; Lien et al. 2005). Ramp et al. (2004) identify two types of NLIWs, presumably generated via different processes and/or tidal forcing, propagating westward across the northern SCS. However, the exact generation sites and generation DOI: /2011JPO Ó 2012 American Meteorological Society

2 512 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 42 FIG. 1. (a) Experiment location in the SCS. Green lines show NLIW fronts as observed by satellites (Zhao et al. 2004). Dashed lines are waves with a single crest; solid lines are wave groups with multiple crests. (b) Bathymetry along the horizontal line along 21849N shown in (a). Shipboard observations were taken along 21849N and at stations O, E, D, and W (yellow squares). Locations of three sets of ADCP data analyzed by Lien et al. (2005) and Chang et al. (2006) are shown as red dots. The continental shelf, Dongsha Island reef, Dongsha Plateau, continental slope, deep SCS basin, and Luzon Strait are labeled in (b). Water depths of the four stations are labeled. The inset shows the Lagrangian float used in the experiment. It is equipped with a CTD on both ends, a pressure sensor near the center, a SonTek acoustic Doppler velocimeter (ADV) at the bottom, and Iridium and GPS antennas on the top. processes of NLIWs are yet to be determined. The analysis of satellite images by Zhao et al. (2004) suggests that NLIWs in the deep basin of the SCS are often single depression waves and NLIWs on the Dongsha Plateau are often in the form of wave trains (Fig. 1), though occasionally large-amplitude wave trains have been observed in the deep SCS basin (Klymak et al. 2006). Note that NLIWs in the deep thermocline, with small amplitudes, or beneath a rough sea surface may not have distinct surface signatures and evade detection by satellites. In situ observations by Alford et al. (2010) confirm the frequent presence of wave trains on the slope between the deep basin and Dongsha Plateau. Most of the energy of these NLIWs and wave trains is dissipated (Chang et al. 2006; St. Laurent 2008) as they propagate westward across the Dongsha Plateau. Here we describe the direct in situ observations of the fission process on the continental slope between the deep basin and Dongsha Plateau, in support of reports by Zhao et al. (2004) and Alford et al. (2010), and propose that the fission process is a result of an evolution of the trapped core. NLIWs often do not transport significant mass, except when trapped cores are formed (Lamb 2002). Trapped cores within NLIWs have been observed in the atmospheric boundary layer and simulated in numerical models (Clarke et al. 1981; Lamb 2002, 2003; Helfrich and White 2010). Oceanic trapped core NLIWs have been observed in near-bottom elevation NLIWs (Klymak and Moum 2003; Scotti and Pineda 2004). Here, we present direct in situ observations of a shoaling NLIW in the northern SCS as it underwent the formation of a trapped core and then fission into a wave

3 APRIL 2012 L I E N E T A L. 513 FIG. 2. (a) Schematic of streamlines of a NLIW with no trapped core. Negative vorticity is created by baroclinic torque during the downwelling part of the NLIW and positive vorticity created during the upwelling part. (b) Schematic of a NLIW with a surface trapped core. The two black dots are stagnation points. (c) Schematic of a second-mode NLIW with a subsurface trapped core. (d) The observed subsurface trapped core within a first-mode NLIW. The gray contours in (b) (d) encircle the areas where the water velocity in the propagation direction of the NLIW exceeds the propagation speed of the NLIW. Dashed lines in (d) are streamlines extrapolated below the maximum depth of observed streamlines. The gray circles with arrows represent the torque. train. This may be the first time that the properties and evolution of oceanic trapped core waves of depression are described in detail. E. A. D Asaro (1995, unpublished manuscript) observed similar internal solitary waves in Knight Inlet but in less detail. We also show that observed properties, including wave width, amplitude, and propagation speed, are reasonably predicted by a fully nonlinear steady-state internal wave model, with better agreement in the deeper water. The agreement of observed and modeled propagation speed is improved when a reasonable vertical profile of background current is included in the model. The analysis is organized as follows: A basic description of a subsurface trapped core in a NLIW is discussed in section 2. The field experiment and measurement systems are described in section 3, and NLIW observations are presented in section 4. The evolution of one NLIW, including propagation on to the continental slope, trapped core development, wave fission, turbulence mixing, and mass transport are described in section 5. Section 6 compares the observed NLIW with that predicted by the Dubreil Jacotin Long (DJL) equation (or Long s equation) (D Asaro et al. 2007). A summary is given in section Heuristic description of trapped core NLIWs Figure 2 gives a description and the conventional definition of a trapped core NLIW. We work in the frame of reference moving with the wave and assume the motion in this frame is steady, two dimensional, and incompressible. The wave under consideration is moving west (toward the left on the page), so that the water velocity in the wave s frame is mostly positive, toward the east (to the right on the page) (Fig. 2). We first consider the case of no trapped core (Fig. 2a). As the flow enters the wave, denser water is west of lighter water, so there is a negative northward torque on the water, which creates a negative vorticity. As the water leaves the wave, the torque is positive, and the vorticity is returned to zero. A trapped core occurs if the water velocity exceeds the propagation speed of the wave: that is, if it is westward in the wave s frame. A trapped core can be surface or subsurface. If the trapped core reaches the surface, the water velocity at the surface is westward in the reference frame of a westward-propagating wave, so there are two stagnation points (zero velocity by definition) on the surface at the two ends of the core region (Fig. 2b). The water velocity in the lower part of the core is eastward, thus forming a recirculation water body propagating at the same speed as the NLIW. The vorticity inside the core is negative, and its magnitude is larger than it would be if there was no trapped core. Figure 2c describes the subsurface trapped core associated with the second-mode wave. Figure 2d describes the observed subsurface trapped core associated with the first-mode wave. For a subsurface trapped core, the velocities on both the top and bottom of the core region are positive. On the top, the flow is clockwise around the core, and on the bottom it is counterclockwise. The core is

4 514 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 42 bounded by streamlines, and therefore there are stagnation points on the core boundary, within the water. One stagnation point is likely to be near the front of the trapped region, and the other is likely to be at the rear. At the front stagnation point, there is an incoming (i.e., toward the stagnation point) streamline from outside the wave, and perpendicular to it are the two outgoing (i.e., outward from the stagnation point) streamlines bounding the trapped core. There is an incoming streamline inside the core. Similarly, at the rear stagnation point, there are two incoming streamlines bounding the trapped core, an outgoing streamline leaving the wave, and another outgoing streamline entering the trapped region. This last streamline is the same one as the incoming streamline at the front stagnation point, and it divides the trapped region into two eddies. The lower eddy has negative vorticity, as the interior of the wave would have without a trapped core. The upper one has positive vorticity. There are two possible origins for the positive vorticity in the upper part of the trapped core. One origin is the gravitational torque. In two-dimensional inviscid flow, the change rate of vorticity z is expressed as dz dt 5 1 r2$r 3 $P [Müller 2006, Eq. (6.41)]. Making the Boussinesq approximation, only the hydrostatic pressure appears, so this term is due to the gravitational torque. Writing the density as a function of z 2 h, where h is the vertical displacement, and noting that the horizontal density gradient and the horizontal potential density gradient coincide, this equation can be rewritten as dz dt 5 N2 (z 2 h) h x. For example, if there is a light upper layer, that layer may remain above the trapped region. Its streamlines rise as it approaches the wave (i.e., h/ x. 0) and rising streamlines create positive vorticity, because the upper streamlines rise while the deeper ones fall, this is a secondmode wave (Fig. 2c). The other possibility, which is the case for the wave we observed, is that there is a preexisting positive vorticity shear near the surface. The upper part of the trapped region has been created by entraining this positive vorticity water (Fig. 2d). Our observations suggest that both the gravitational torque and the entrainment contribute to the positive vorticity part of the trapped core. This is discussed in section 5e. The configuration yields a streamfunction having the same value on both stagnation points. Suppose this configuration is perturbed slightly to have different values on the two stagnation points. Then the trapped region splits into two trapped vortices with an untrapped flow between them. This untrapped band has a negative velocity in the region between the two vortices and passes above the top eddy and below the bottom eddy. Thus, in some region or regions it is convectively unstable. The unstable flow causes mixing, reducing this untrapped flow and making the two stagnation points approach the same value of the streamfunction. Therefore, the evolving subsurface trapped core might have unstable stagnation points and lead to strong mixing within the trapped core (Fig. 2d). In Lamb s (2002) numerical model studies, as first-mode NLIWs propagate up a slope the particle velocity exceeds propagation speed and a subsurface 1 trapped core is formed with two counter-rotating vortices, similar to Fig. 2d. Lamb defines a breaking limit as occurring when the particle velocity in the direction of NLIW propagation exceeds the wave s propagation speed and identifies this limit as a criterion for the development of a trapped core (areas encircled by gray contours in Figs. 2b d). The analysis here describes development and evolution of a subsurface trapped core formed within a shoaling NLIW in the northern SCS (Fig. 2d). In the steady-state scenario without mixing, the trapped core and the water that it carries travel at the speed of the NLIW without exchange with the outside water. In contrast, the trapped core observed in the SCS continuously exchanged water with its surroundings, mixing this with water previously entrained, thus creating a new water mass within the core and leaving a wake, observed as an isopycnal salinity anomaly. 3. Measurements The experiment was conducted along ;218N between 1188 and 1168E in two legs, and April 2005 (Fig. 1), aboard the Taiwanese R/V Ocean Researcher III. Measurements were taken by shipboard 150-kHz ADCP, shipboard CTD, XBT, Simrad EK500 echo sounder, 9.41-GHz X-band marine radar, expendable current profiler (XCP), expendable CTD (XCTD), and a Lagrangian float. Shipboard ADCP and CTD measurements form the basis of NLIW observations. The shipboard ADCP sent an acoustic ping every 3 s to measure velocity profiles with a bin size of 4 m and a pulse length of 4 m. It recorded 1-min ensembles averaged over 20 pings. The 1 The trapped core that developed within shoaling NLIWs in Lamb s (2002) numerical simulations was in fact a subsurface trapped core, though this is not indicated in the paper (K. G. Lamb 2010, personal communication).

5 APRIL 2012 L I E N E T A L. 515 observation depth range of the ADCP was m. The shipboard CTD was profiled at a vertical speed of ;1 ms 21 from the surface to about 300 m and sampled at 2 Hz yielding ;0.5-m vertical resolution of temperature, salinity, and pressure. During NLIW passages, the shipboard CTD was often operated in a yo-yo mode to maximize temporal spatial resolution. The shipboard Simrad 120-kHz echo sounder sampled at 4 Hz and provided acoustic scattering strength at a vertical resolution of ;1 m. Echo sounder observations provided estimates of NLIW vertical displacement that supplement those from CTD measurements (Chang et al. 2008). A 12-GHz X-band shipboard radar took observations of surface scattering intensity every 1 min, while making four 3608 rotations. It recorded two-dimensional surface scattering intensity at up to a 5-km radius from the ship. Details of X-band radar observations taken during this experiment, as well as the estimation of NLIW propagation speed and direction from radar measurements, are discussed in Chang et al. (2008). At stations O and D (Fig. 1), we deployed XCPs and XCTDs every 2 h for 24 h each between 26 and 28 April to quantify the background tidal current and determine the arrival time of NLIWs. After completing the XCP and XCTD operation at station D, we began the NLIW tracking operation. The primary goal of this experiment was to capture the evolution of NLIWs on the shoaling continental slope. The ship waited at station E (;218N, E; 600-m water depth) for the arrival of a large NLIW from the east. The wave was tracked and measured until it reached station W (;218N, E; 300-m water depth). This operation was conducted twice, on 28 and 29 April. 4. Background conditions and arrival of NLIWs The vertical profile of stratification during these measurements (Fig. 3a) shows a maximum buoyancy frequency, N s 21, at 50-m depth, slightly stronger than the typical spring value but close to the summer value of the climatological stratification compiled by the generalized digital environmental model (Teague et al. 1990) for this area of the SCS (Fig. 3a). The stratification extends to the surface without a clearly defined surface mixed layer. The average flow is southwestward. The zonal flow decreases from ;0.5 m s 21 at the surface down to ;100-m depth and increases to 0.3 m s 21 to 170-m depth (Fig. 3b). The background meridional component of vorticity is positive, z u 0.01 s 21, above 80 m; vanishes between 80- and 120-m depths; and becomes negative, z u s 21, between 120- and 140-m depths (Fig. 3c). Note that the vertical structure of the meridional component of ambient vorticity, FIG. 3. (a) Mean vertical profiles of N 2 observed Apr 2005 (black), from April May climatology (red), and from August climatology (blue). (b) Mean vertical profiles of zonal velocity (black) and meridional velocity (red) observed prior to NLIW arrivals. (c) North (black) and east (red) vorticity computed from (b). Gray shadings in (a) and (b) represent two standard deviations from the mean vertical profiles. positive in the upper layer and negative in the lower layer, is similar to that of a subsurface trapped core (Figs. 2c,d). Lamb (2002) reports that, in the absence of background vorticity, trapped cores are formed in shoaling NLIWs only if stratification increases monotonically toward the sea surface. The presence of a surface mixed layer suppresses the formation of trapped cores within NLIWs. However, a background vorticity with the same sign as the NLIWs favors trapped core formation even in the presence of a surface mixed layer (Lamb 2003). Our vertical stratification observations show no monotonic increase toward the sea surface, poor conditions for trapped core development. However, the background vorticity profile favors subsurface trapped core formation. The NLIWs observed on the Dongsha Plateau may form from baroclinic tides within Luzon Strait (Lien et al. 2005; Niwa and Hibiya 2004; Zhao and Alford 2006; Jan et al. 2008) that propagate from there to our experiment site in 2 3 days. The barotropic springtidepeakoccurredon24april.thestrongest

6 516 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 42 FIG. 4. (a) (d) One NLIW event observed on 30 Apr 2005 is illustrated: (a) density from yoyo CTD profiles, (b) echo intensity from echo sounder, (c) ADCP zonal velocity, and (d) ADCP vertical velocity. Black solid curves in (a) (d) indicate an isopycnal of s u 5 24 kg m 23. The NLIW depresses this isopycnal surface vertically by ;110 m. baroclinic tides and NLIWs arrived at Dongsha on April. The first leg of the experiment, April, took place at 218N, 1178E during a transition from the neap to the beginning of spring tide, and the second leg, April, took place during a transition from the spring to neap tide. During the first leg, only one weak second-mode NLIW was observed. During the second leg, large-amplitude first-mode NLIWs appeared at nearly the same time each day propagating toward the west. Strong surface scattering induced by these large-amplitude NLIWs was observed by marine radar (Chang et al. 2008). During one NLIW event on 30 April (Figs. 4a d), the ship attempted to hold station while yo-yoing the CTD between the surface and ;250 m as the wave passed. The research vessel was carried westward slightly by the wave s strong westward surface current, especially as the NLIW center passed. It took about 45 min for this NLIW to pass the vessel. The s u 5 24 kg m 23 isopycnal was displaced downward by ;110 m, from ;70 to 180 m (Fig. 4a). The flat-bottomed shape of the vertical displacement near the NLIW center probably results from the advection of the ship by the NLIW currents. The isopycnal displacement was also evident in the downward displacement of acoustic backscatter (Fig. 4b) and horizontal velocity (Fig. 4c). Observations from the shipboard ADCP show a strong background eastward zonal current, ;0.5 m s 21, in the upper layer before and after NLIW passage; a westward surface current of ;0.5 m s 21 during; a downward vertical current of ;0.5 m s 21 in front of the NLIW center; and an upward vertical current of ;0.5 m s 21 behind the center (Figs. 4c,d). Vertical displacements, arrival times, and arrival locations of NLIWs over the experiment period are shown in Fig. 5. Vertical displacements were estimated from CTD, Lagrangian float, and echo sounder measurements. NLIWs with vertical displacements greater than 100 m were observed on each day of April. Largeramplitude NLIWs arrived diurnally and smaller waves occurred between the diurnal large-amplitude waves. These are referred to as A waves and B waves, respectively, by Ramp et al. (2004).

7 APRIL 2012 L I E N E T A L. 517 FIG. 5. (top) Bathymetry and (bottom) NLIW amplitudes and longitudes. Wave properties compiled from the EK500 echo sounder (color dots), ADCP, and the Lagrangian float (filled squares). Colors and sizes of symbols represent NLIW amplitudes. Gray curve indicates the ship track. Magenta lines represent the positions of the wave crests. Between 27 and 30 Apr, large-amplitude NLIWs arrived diurnally. 5. Tracking a shoaling NLIW a. Methods A westward-propagating NLIW was tracked for ;9 h beginning at 1420 local time (LT) 28 April. Transiting from ; to ;1178E along N (;50-km distance), we passed the center of the NLIW seven times (positions and times labeled in Fig. 6a) and measured its evolution. A total of 20 XBT casts were made during the second pass, and shipboard yo-yo CTD profiles were made during the third, fourth, fifth, and seventh passes. These CTD, XBT, marine radar, EK500, and shipboard ADCP measurements form the basis of the following analysis. On 27 April, a large-amplitude NLIW passed station D as we performed a 24-h XCP and XCTD time series; this NLIW was not tracked. A NLIW was tracked on 29 April, but few CTD profiles were taken. The focus of this study is therefore on the NLIW measured on 28 April only. The NLIW propagation direction and speed were computed using shipboard marine radar measurements (gray filled dots in Fig. 6b) while (generally 5 10 min) the NLIW was within the range of marine radar, as described by Chang et al. (2008). The propagation direction was determined from the orientation of the band of enhanced surface scattering due to the surface convergence (insets in Fig. 6a). The propagation speed was determined by tracking the speed of the band of enhanced surface scattering, assuming that the distance between the convergence zone (enhanced surface scattering) and the center of the NLIW remains unchanged during the 5 10-min period of radar measurements used to compute wave speed and direction. Although generally a valid assumption, in circumstances when the NLIW undergoes a rapid change in its horizontal structure, the propagation speed of the convergence zone (our estimate) may be different from the propagation speed of the center of the NLIW. The low estimated propagation speed of 1.1 m s 21 at E may indicate such a circumstance. Wave speed is also estimated from a spline fit to the positions and times of the center of NLIWs, determined from ADCP and echo sounder measurements (Fig. 6a), using the propagation direction inferred from shipboard radar measurements. Speed of NLIWs estimated from the radar (gray dots in Fig. 6b) agree well with those estimated from the spline fit (black dots in Fig. 6b), thus providing confidence that the speed and direction are estimated to within an accuracy of about ;0.1 m s 21 and 18. b. Wave structure, speed, and direction At the first encounter, ;1350 LT, the NLIW was asingledepressionwavewithamaximumamplitudeof.150 m. At the third encounter, ;1730 LT, the NLIW had divided into two waves (marine radar images in Fig. 6a). The second wave (open circles in Fig. 6a) had an amplitude about one-third that of the leading wave (black dots in Fig. 6a) and lagged about 2 km behind the leading wave. The separation between the leading and trailing waves increased with time and reached ;3 km by the seventh encounter at ;2250 LT. Thus, the second wave traveled slower than the leading wave, consistent with the expected amplitude dependence of propagation speed of NLIWs. The propagation speed of the leading wave (Fig. 6b) decreased from 2.2 m s 21 at E to;1.2 m s 21 at E, remained at this speed until 1178E, and then decreased below 1 m s 21. The NLIW propagated predominantly westward in the direction 1908 (counterclockwise from the east), with a 208 change near the second encounter. The bottom slope is ;0.01 on the continental slope, with a peak of 0.02 between and E, and is flat on the Dongsha Plateau. The locations of the NLIW center during our seven encounters are marked in Fig. 6d. Formation of the trapped core and wave fission occurred between the second and third encounters, where the bottom slope is the steepest, suggesting that

8 518 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 42 FIG. 6. NLIW evolution on 28 Apr. (a) Circles show positions and times when the center of the NLIW passes the ship, filled circles show positions and times for the leading NLIW, and open circles show positions and times for the smaller wave trailing behind. Insets illustrate the surface signature of NLIWs imaged by the marine radar. Seven encounters with the NLIW center are labeled. (b) Propagation speed of the leading NLIW estimated from marine radar measurements (gray dots with solid line) and from a spline fit to the positions in (a) (black dots with dashed line). (c) Propagation direction of the leading wave counterclockwise from the east estimated by the marine radar (gray dots) and interpolated direction at the NLIW encounters (black dots). (d) Depth along the ship track, nearly identical to the propagation path of the NLIW. The vertical lines along the sloping sea bottom representation indicate the positions of the seven encounters. (e) Zonal component of the bottom slope along the ship track. wave propagation up the steepest slope causes rapid NLIW evolution. c. Formation of a subsurface trapped core The maximum along-wave component of velocity U max compared to the wave propagation speed C (Fig. 7a) shows that, while speed decreased from 2 m s 21 at the first encounter ( E) to 1.25 m s 21 at the third encounter (; E), U max remained constant at 2ms 21. Thus, by the third encounter, U max was much greater than C and the wave exceeded its breaking limit where Lamb (2002) predicts the formation of a trapped core. Closed streamlines (Fig. 8) confirm this prediction. After the third encounter, U max decreased monotonically, whereas C remained nearly constant. Between the fifth and sixth encounters at ;117.18E, C exceeded U max and the trapped core was released. The observed NLIW with a trapped core propagated ;30 km in ;6 h. Note that although the wave was near its breaking limit at the first encounter, the largest convective instability, U max. C, happened at the third encounter when large overturns were observed. Details of trapped core properties observed during the third encounter are discussed in sections 5e and 5f. d. Wave shape The maximum NLIW vertical speed was nearly constant at ;0.5 m s 21 (Fig. 7a). The core of maximum vertical velocity may be deeper than the range of shipboard ADCP measurements at the first two encounters. The maximum vertical displacement, identified from CTD and echo sounder measurements, decreased from ;170 m during the first encounter to less than 140 m at the third (Fig. 7b). We define the NLIW width as that between the maximum downwelling and maximum upwelling L w. It was ;1.1 km during the second and third

9 APRIL 2012 L I E N E T A L. 519 FIG. 7. NLIW evolution on 28 Apr. (a) Maximum zonal velocity (red dots) and vertical speed (stars) estimated from ADCP data and propagation speed estimated from marine radar (blue squares). (b) Vertical displacements of isopycnal initially at 100 m (red dots) and half-amplitude width (black dots and open circles). (c) Depth of maximum zonal velocity (red dots) and bathymetry. (d) The bathymetry in the northern SCS and the red rectangle marks the region shown in (a) (c). encounters and dropped to ;0.8 km in subsequent encounters (Fig. 7b). Because of the depth limit of shipboard ADCP measurements, L w cannot be defined accurately at the first and second encounters. The depth of U max shoals from ;180 m at E to;150 m at 1178E (Fig. 7c). e. Trapped core NLIW properties The along-wave and vertical components of velocity during the third encounter with the NLIW are shown in Figs. 8a,b, respectively. We transform the temporal variation of NLIW observations into the wave s spatial structure using a propagation speed of the leading wave, 1.3 m s 21. In this representation, the wave is propagating from right to left across the page. The maximum along-wave velocity, U max m s 21, is located at ;110-m depth about 150 m behind the NLIW center. The breaking limit, U. C, is between ;90- and 150-m depths and within 0.3 km from the NLIW center. This region is delineated by the magenta contour in all panels of Fig. 8. By definition, the velocity averaged within the trapped core should be identical to the propagation speed of the NLIW. Accordingly, the trapped core region is larger than the region where U. C. The maximum vertical velocity is ;0.5 m s 21, and the horizontal scale (i.e., the horizontal separation between the maximum downwelling and upwelling) is ;1 km. It is likely the depth of the maximum vertical velocity is deeper than the ADCP velocity range, and therefore the true maximum vertical velocity could be greater than 0.5 m s 21. There is a small velocity feature about 500 m behind the center of the leading wave with a maximum alongwave velocity of ;1 ms 21 and a horizontal scale of roughly ;200 m. Note that this feature is distinct from the small wave at 2 km behind the leading wave, shown in Fig. 6a. Because the shipboard ADCP has a 308 beam angle from the vertical, horizontal velocity measurements are spatially averaged within 115 m at 100-m depth and within 230 m at 200-m depth. The observed velocity feature is not well resolved and therefore might be an artifact because of the beam spreading effect. This feature is not discussed further in the following analysis. Several patches of vertical overturning were observed at the center and at the rear of the NLIW; two patches at the rear had vertical scales of tens of meters. The maximum vertical displacement of the NLIW was about 150 m, and most isopycnal surfaces were deeper after the wave s passage. There was an accumulation of water with s u kg m 23 at the rear of the NLIW and s u kg m 23 at the front. The wave-frame streamfunction computation, dz(u 2 C), where z is the vertical coordinate, reveals two counter-rotating circulation vortices (Fig. 8d; see also Fig. 2d). The upper-front clockwise vortex is slightly weaker, with wider streamlines, than the lowerrear counterclockwise vortex; their locations correspond well with the locations of the water masses, s u and 23.6 kg m 23, accumulated in the NLIW center. One CTD profile at the rear of the NLIW core passed directly through the center of the counterclockwise vortex, where the vertical overturning was observed. The observed trapped core had a thickness H tc ; 140 m, extended from 60- to 200-m depth and had an along-stream length L t ; 300 m, estimated from the closed streamlines and isopycnals (Figs. 8c,d). The bottom portion of the trapped core was beyond the shipboard ADCP range, so streamlines were extrapolated (Fig. 2d). Satellite images suggest that NLIW wave crests in this area of the SCS are L 5 O(100 km) (Fig. 1a). Assuming a steady-state trapped core propagating at the same speed as the NLIW, C m s 21, the instantaneous water mass transport within the trapped core is CH tc L ; 18 Sv (1 Sv [ 10 6 m 3 s 21 ). The average transport per day is H tc LL t / s Sv. However, as will be discussed, the observed recirculation feature is not in steady state but evolves in time. When the NLIW reaches the breaking limit U. C, convective instability is the primary mechanism for turbulence mixing. As the trapped core evolves, the strong shear within the NLIW, especially in the two counterrotating vortices, combined with reduced stratification in c 5 Ð z 0

10 520 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 42 FIG. 8. NLIW properties during the third encounter on 28 Apr near E: (a) along-track velocity; (b) vertical velocity; (c) density from yo-yo CTD; (d) streamlines in the wave frame computed from the vertical integration of along-track velocity; (e) reduced shear squared, S2 2 4N2; and (f) cross-track component of relative vorticity zy. Magenta curves encircle the area where the water velocity is greater than the propagation speed of the NLIW. Gray contours in (e) and (f) are isopycnals. The zigzag lines in (c) and (d) represent CTD trajectories. the NLIW center makes this region subject to shear instability. Reduced shear squared (S2 2 4N2), a measure of shear instability (Kunze et al. 1990), is zero or positive throughout the trapped core, indicating shear instability: that is, Richardson number Ri 5 N2/S2, 1/ 4, where N2 is computed from sorted density profiles. Vertical overturning in this same region, as measured by the yo-yo CTD profiles (Fig. 8c), confirms the presence of active mixing. One vertical profile of density taken by the yo-yo CTD downcast through the rear end of the leading wave is shown in Fig. 9e (magenta curve). The Thorpe displacement is computed as the vertical displacement between the isopycnal depth of the observed density profile and the isopycnal depth of the sorted density. Large Thorpe displacements are found between 50and 150-m depths, with a maximum Thorpe displacement of 12.5 m. We further compute the Thorpe scale LT as the root-mean-square of Thorpe displacement. The averaged Thorpe scale between 50- and 150-m depths is 6 m. Previous studies have found close agreement between Thorpe scale and Ozmidov scale, defined as LO 5 «1/2N23/2: for example, LO5 0.8LT (Dillon 1982). Applying the average Thorpe scale of 6 m and the background stratification N ; 0.01 s21, the bulk estimate of the turbulence kinetic energy dissipation rate «; LO2N3 5 (0.8LT)2N3 is W kg21, which is four orders greater than that in the typical open ocean. Recent direct microstructure measurements taken within a similar turbulent NLIW in the same area report the turbulence level «; 1024 W kg21 and eddy diffusivity of 0.1 m2 s21 (St. Laurent et al. 2011). The observed overturning and estimated turbulence dissipation also suggest that the vortices in the trapped core are unstable and that mixing with surrounding waters is expected. One yo-yo CTD upcast taken directly within the trapped core is also shown in Fig. 9e (brown curve). The average Thorpe scale between 50- and 150-m depths of

11 APRIL 2012 L I E N E T A L. 521 FIG. 9. Water mass properties during the third encounter on 28 Apr. (a) Depth time variation of potential density (contours; interval kg m 23 ) and potential temperature (colors). (b) As in (a), but for salinity anomaly relative to the first CTD downcast. The zigzag vertical profiles show the trajectory of the CTD. The two thick black density contours, s u 5 23 and 25 kg m 23, identify the range influenced by the trapped core. (c) Time and positions of yo-yo CTD downcasts (stars) labeled by number and color coded. The filled and open dots represent the leading NLIW and the secondary wave, respectively. The black line represents the NLIW path. (d) Potential temperature salinity plot for selected profiles before (blue), during (red), and after (black) NLIW passage. Thin black lines are isopycnals. Thicker gray lines are trapped core boundaries. (e) As in (d), but for potential density profiles. (f) As in (d), but for salinity anomaly against potential density. (g) As in (d), but for salinity against potential density. Blue, red, and black curves in (d), (f), and (g) represent the first, fifth, and eleventh CTD downcasts, respectively. The colors of curves in (e) correspond to those of CTD tracks shown in (b). Brown curves in (a), (b), and (e) and brown dots in (d) correspond to the CTD upcast at the rear region of the leading wave. this upcast is 55 m. The bulk estimate of «; L O 2 N 3 is Wkg 21. However, because of the possible wake contamination of the yo-yo CTD upcast profile, we question the quality of the Thorpe scale estimate and thereby the estimate of «from this upcast. The transverse component of vorticity, z y 5 z u 2 x w, is dominated by the vertical shear of horizontal velocity. It shows a negative vorticity in the counterclockwise vortex and positive vorticity in the clockwise vortex (Fig. 8f). Most of the upper trapped core has a positive vorticity of s 21, which is about equal to that in the upper region outside the wave, suggesting that this vorticity was entrained into the core when it was formed. A region in the upper right has a larger value of the vorticity, with amaximumof0.05s 21, suggesting that it was generated by the rising streamlines at the back of the wave as it began to overturn. f. Trapped core NLIW effects on the surrounding water mass Figure 9 shows the change in water mass properties due to NLIW passage during the third encounter.

12 522 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 42 Individual CTD downcasts are colored by location within the NLIW (blue indicates ahead, red indicates within, and black indicates behind). Mixing is diagnosed by plotting the isopycnal salinity anomaly relative to the first cast. Before the arrival of the leading wave (blue lines), the salinity anomaly is small (Figs. 9b,f). This is also true of the fourth cast, despite its large ;150-m vertical displacement. After the NLIW passed (black lines), the salinity anomaly is negative near the central isopycnal of the trapped core (23.8 kg m 23 ) and positive on isopycnals above and below this region (Figs. 9b,f). This pattern clearly originates in the region of overturning and mixing within the trapped core. It implies a turbulent diapycnal flux of salt from the trapped core layer into the surrounding layers. The resulting pattern of diapycnal diffusivity is complex, reflecting the detailed but poorly resolved structure of mixing within the trapped core and with a magnitude of roughly 0.1 m 2 s 21. It is comparable to the eddy diffusivity computed as K r 5G«N 22 (Osborn 1980), where the mixing efficiency G50.2 is used. It is also in close agreement with the estimate from direct microstructure measurements (St. Laurent et al. 2011). It is clear that turbulent mixing within the NLIW trapped core yields a wake of mixed water. 6. Fully nonlinear DJL solutions The dynamics of the observed NLIW are investigated further by comparing its structure and propagation speed with steady-state internal solitary wave solutions. Because observed NLIW s isopycnal displacements are comparable to the water depth and to the effective upper-layer depth, they cannot be described adequately by weakly nonlinear models such as the Korteweg de Vries (KdV) equation (Whitham 1974). Fully nonlinear steady-state NLIWs calculated by the Dubreil Jacotin Long (DJL) internal solitary wave model (D Asaro et al. 2007) are more appropriate. Helfrich and White (2010) describe a model for a trapped core NLIW using the DJL solution outside of the trapped core. The flow inside the core satisfies the vorticity streamfunction relation, but the relation between density and streamfunction may be arbitrarily prescribed. Such a model is beyond the scope of this study. Here, we discuss DJL solutions without the trapped core. Two sets of DJL model runs, with and without a background current, are performed. The DJL equation for the vertical displacement of the wave h including a background current is expressed as (Henyey 1999; Stastna and Lamb 2002) = 2 h 1 zu(z 2 h) C 2 U(z 2 h) [ 2 x h 1 (12 2 z h)2 2 1] 1 N2 (z 2 h) [C 2U(z 2 h)] 2h 5 0, where = 2 5 ( 2 x 1 2 z ) and C is the propagation speed of the wave. The background current U, its vertical gradient z U, and the stratification N are evaluated at z 2 h. In the absence of a background current (i.e., U 5 0), the DJL equation is simplified to be a balance between the first and third terms. Vertical profiles of N 2 observed before the wave arrival are low-pass filtered using a Butterworth filter with a half-power vertical scale of 80 m. These vertical profiles are used for DJL model runs. The model background current is constructed as follows: In the upper 164 m, the observed along-track current observed before the arrival of the wave is used. Below 164 m, the maximum depth of shipboard ADCP velocity measurements, a linear vertical profile is arbitrarily assumed between the observed current at 164-m depth and zero at the bottom. The constructed vertical profile of the background current is low-pass filtered using a Butterworth filter with a half-power vertical scale of 80 m. These vertical profiles of low-pass-filtered current are used for DJL model runs. DJL model solutions are compared with observations for the first six wave encounters. The DJL equation is solved, following the method of Turkington et al. (1991), with a slight modification to improve the convergence. The simple extension of the DJL equation to include a background current is given by Henyey (1999) and Stastna and Lamb (2002). Stastna and Lamb proposed a method of solving this extension, based on the Turkington et al. (1991) method, which is used here. The isopycnal displacement from an equilibrium depth of 150 m is used to compare the DJL solution with the observations (Fig. 10a). Estimates of amplitude from 150-m depth and the width of the half amplitude of observations are computed using the combined CTD measurements and EK500 measurements. Providing the vertical profile of background stratification and the vertical profile of background current for the model run including a background current, a set of solutions from the DJL equation are obtained. Two examples of DJL solutions without a background current are shown in Fig. 10a, using the stratification vertical profile taken immediately before the arrival of the wave at the first encounter. One solution (black solid curve) shows good qualitative agreement with observations (red curve); quantitative comparisons follow. Another example (dashed black curve), a conjugate flow solution, obviously

13 APRIL 2012 L I E N E T A L. 523 FIG. 10. Comparison between observed NLIW and DJL solutions. (a) Examples of two DJL solutions (black solid and dashed curves). One solution (black solid curve) more closely matches the observations from the first encounter (red curve). The definitions of wave amplitude h and width of half-amplitude L are shown. (b) Comparison of the observed wave propagation speeds and amplitudes from the 150-m equilibrium depth (color dots) with those predicted by the DJL model without a background current (color dashed curves) and with a background current (color solid curves). (c) Comparison of the observed L and h (color dots) with those predicted by the DJL model without a background current (color dashed curves). (d) Comparison of the observed L and h (color dots) with those predicted by the DJL model with a background current (color solid curves). The DJL solution without a background current at the third encounter is included in (d) for comparison. differs from the observations. This result shows that the observed wave at the first encounter has the spatial structure consistent with one of the possible DJL solutions. For each DJL solution, an amplitude h, displaced from 150-m depth; the width L at half amplitude h/2 (shown in Fig. 10a); and the propagation speed C are computed and their relations are compared with observations. DJL model solutions with and without a background current are distinctly different. The DJL model run with a background current has an amplitude of the conjugate flow: that is, the flow approaching a nearly constant maximum amplitude h with increasing width L, ;10 m larger than that of the same DJL model run without a background current (comparing the blue solid curve and blue dashed curve in Fig. 10d and comparing curves in Figs. 10c,d of the same colors). The DJL model run with a background current opposite the wave propagation yields wave solutions of propagation speeds ;0.3 m s 21 slower than the model run without a background current (comparing solid and dashed curves in Fig. 10b). Comparisons between observations of L versus h and solutions of DJL model runs without a background current are shown in Fig. 10c. The spatial structure of the observed L versus h agrees well with the DJL solutions at the first wave encounter and is significantly different from DJL solutions at later wave encounters. Comparisons between observations of L versus h and solutions of DJL model runs with a background current are shown in Fig. 10d. The DJL model with a background current generally shows a slightly better agreement with observations than the DJL model without a background current. The observed L versus h agrees well with the DJL solutions at the first and third wave encounters. Interestingly, the observation at the second wave encounter falls on the DJL model solution at the

14 524 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 42 first wave encounter (the red dot on the black curve), and the observations at the fourth and fifth wave encounters fall on the DJL model solution at the third wave encounter (green and magenta dots on the blue curve). This result requires further numerical investigation. Presumably, the wave does not respond to the local environment spontaneously. Comparisons between observed C versus h and the solutions of DJL model runs with (solid curves) and without (dashed curves) a background current are shown in Fig. 10b. The DJL model without a background current overestimates the wave propagation speed by ;0.3 m s 21. This deviation is expected because the background current flowing opposite the wave propagation direction is not included in the model. Indeed, the propagation speed of the DJL model with a background current agrees much better with observations. The agreement is much better in shallower water, the third through sixth encounters, where the discrepancy is,0.1 m s 21. Note that the wave propagation speed of the DJL model runs depends strongly on the assumed vertical profile of the background current beyond our observational depth, 164 m. Without observations of the background current at depths greater than 164 m, we cannot provide a more accurate comparison. Overall, the DJL model including a background current predicts the observed spatial structure and the propagation speed better than the DJL model without a background current. 7. Summary Large-amplitude ( m) nonlinear internal waves (NLIWs) were observed by shipboard instruments on the continental slope in the northern SCS during the spring tide. The evolution of one NLIW as it propagated up the slope was measured. The NLIW arrived at the slope as a nearly steady-state solitary depression wave. As it propagated on the slope, the speed C decreased from 2 to 1.3 m s 21, whereas the maximum along-wave current speed U max remained constant at 2ms 21.AsU max exceeded C, the wave reached its breaking limit and formed a subsurface trapped core with closed streamlines in the coordinate frame of the propagating wave. Two counter-rotating vortices fed a jet within the trapped core. The core was highly turbulent with m density overturnings caused by the vortices acting on the background stratification. The core mixed with the surrounding water yielding a wake of mixed water, which was observed as an isopycnal salinity anomaly. Through processes of fission and dissipation, the NLIW broke into a large primary wave and a slower trailing secondary wave. These processes continued on the westward propagation path; the NLIW evolved fromasingledeep-watersolitarywaveasitapproached the continental slope into a train of much smaller waves on the Dongsha Plateau (Alford et al. 2010). Acknowledgments. The authors thank the crew and officers of the R/V Ocean Researcher III for superb ship operation during the experiment. Thanks also to Toshiyuki Hibiya and Naoki Furichi for joining the cruise and providing XCP and ACTD data. Discussions with Kevin Lamb aided the analysis. This work is supported by the National Science Council of Taiwan and by the U.S. Office of Naval Research. REFERENCES Alford, M. H., R.-C. Lien, H. Simmons, J. Klymak, S. Ramp, Y. J. Yang, D. Tang, and M.-H. Chang, 2010: Speed and evolution of nonlinear internal waves transiting the South China Sea. J. Phys. Oceanogr., 40, Chang, M.-H., R.-C. Lien, T. Y. Tang, E. A. D Asaro, and Y. J. Yang, 2006: Energy flux of nonlinear internal waves in northern South China Sea. Geophys. Res. Lett., 33, L03607, doi: /2005gl ,, Y. J. Yang, T. Y. Tang, and J. Wang, 2008: A composite view of surface signatures and interior properties of nonlinear internal waves: Observations and applications. J. Atmos. 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