1. Introduction
Nonlinear internal solitary waves (ISWs) observed in the northern South China Sea (SCS) (Orr and Mignerey 2003; Ramp et al. 2004; Yang et al. 2009; Li et al. 2009; Zhao and Alford 2006) are formed by conversion from internal tides generated in the Luzon Strait or are formed directly in the Luzon Strait (Ramp et al. 2004; Zhao et al. 2004; Lien et al. 2005; Klymak et al. 2006). From satellite images, waves of a single depression are observed in the deep basin, and waves of multiple depressions are observed on the Dongsha Plateau between the continental slope and continental shelf (Zhao et al. 2004). ISW propagation speed decreases during shoaling on the continental slope east of Dongsha Plateau (hereafter also referred to as Dongsha slope) (Lien et al. 2012; Alford et al. 2010), and most of the ISW energy dissipates on the plateau (Chang et al. 2006). On the continental shelf west of Dongsha Plateau, ISWs are much weaker (Xu et al. 2010), although strong turbulence is associated with them (St. Laurent 2008).
The rapid evolution of one ISW with a recirculation-trapped core formation was observed shoaling on the Dongsha slope (Lien et al. 2012). As the wave shoaled, the wave speed decreased, but the current speed remained relatively steady. As the breaking limit was reached (Lamb 2003), that is, the wave particle speed exceeds the wave propagation speed, a trapped core was formed within the ISW. Measurements of velocity and density were limited to the upper 200 m of the 500-m-deep ocean; therefore, the wave energetics could not be quantified.
Theory and numerical models show that ISWs break due to shear instability or convective instability (Lamb and Farmer 2011). Kelvin–Helmholtz (KH) billows are found where the gradient Richardson number Ri = N2/S2 < 0.11 in ISWs (Lamb and Farmer 2011; Barad and Fringer 2010). Moum et al. (2003) report shear instability at the interface and within the upper layer of ISWs.
Numerical models of shoaling ISWs show that convective instability occurs when the particle speed exceeds the wave propagation speed (Vlasenko and Hutter 2002; Lamb 2002, 2003; Helfrich and White 2010). Elevation ISWs with a trapped core formation have been observed near the ocean bottom on the Oregon Shelf (Moum et al. 2003) and near the Massachusetts coast (Scotti and Pineda 2004), and a depression ISW with a trapped core was observed on the Dongsha slope in the northern SCS (Lien et al. 2012). A numerical model study to simulate Kelvin–Helmoltz billows within an ISW observed on the Oregon Shelf (Moum et al. 2003) concludes that the observed ISW had a surface-trapped core (Lamb and Farmer 2011).
A numerical model study (Vlasenko and Hutter 2002) of ISW convective instability shows that as an ISW propagates on a sloping bottom a subsurface jet is generated during the breaking event, producing a horizontal velocity maximum beneath the surface, instead of at the sea surface. An empirical breaking criterion was found depending on the normalized maximum vertical displacement 
Direct observations of breaking ISW energetics are rare because of the intermittency of the breaking process. Furthermore, full-depth velocity and density measurements with high vertical and temporal resolution are needed to capture the convective instabilities of a breaking event. ISWs break on the upper Dongsha slope in the northern SCS regularly during the period of strong internal tides, providing an ideal opportunity to study the properties and energetics of breaking ISWs and comparing these observations with theoretical, model, and laboratory results.
Here, we describe the experiment to observe large-amplitude ISWs on the upper Dongsha slope (section 2) and review the theoretical framework of ISW energetics (section 3). In section 4, we present properties and energetics of five observed ISWs and then detail one breaking ISW (section 5) describing the time evolution of available potential energy (APE), kinetic energy (KE), energy fluxes, and vertical overturns within the wave core. Steady-state fully nonlinear ISWs are simulated using the Dubriel–Jacotin–Long (DJL) model and compared with an observed ISW (section 6).
2. Experiment
One surface mooring (448-m depth) and one subsurface mooring (500-m depth) were deployed on the continental slope about 6 km apart (Fig. 1). Next to each a bottom pressure sensor mooring was deployed (not shown). The mooring positions were chosen to capture breaking ISWs based on previous observations (Lien et al. 2012).
(a) Bathymetry of the northern SCS, (b) the water depth along 21°N, and (c) configurations of surface and subsurface moorings. (d) The inset shows the map of the northern SCS. The red box in (d) marks the area in (a) and Dongsha Island is labeled in (a). The yellow triangle and brown dot in (a),(b) represent the surface and the subsurface moorings, respectively. The red curve (covered by the white dots) and blue curve labels the ship track survey of ISWs on 3 and 4 June, respectively. White dots represent the locations where the ISW center is identified by shipboard echo sounder and ADCP velocity measurements.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
The surface mooring was equipped with one upward-looking 600-kHz ADCP at 30-m depth, one upward-looking 75-kHz ADCP at 450-m depth, one downward-looking 1200-kHz ADCP at 452-m depth, 14 Sea-Bird Electronics (SBE) 37 CTD sensors, and three temperature loggers (Fig. 1c). Unfortunately, the mooring landed on a bottom shallower than planned so both 75- and 1200-kHz ADCPs did not provide useable data. The 600-kHz ADCP pinged at 1 s and recorded 1-min ensemble averages with a bin size of 0.75 m.
The subsurface mooring was equipped with one 75-kHz ADCP at 10 m above the bottom, 10 SBE39 temperature sensors, and three SBE37 CTD sensors. The 75-kHz ADCP pinged and recorded at 1-min intervals with a vertical resolution of 8 m. All the SBE37 and SBE39 sensors sampled at 10 s, and the temperature loggers sampled at 20 s. Despite the strong buoyancy of the subsurface mooring, ISWs dragged it down by as much as 95 m; therefore, wave cores in the upper ocean were not measured well.
In this study, we analyze ISWs at the surface mooring site using the combined velocity measurements from the 75-kHz ADCP at the subsurface mooring and 600-kHz ADCP at the surface mooring, with a time lag determined by the arrival times of the wave on the two moorings, and CTD measurements taken on the surface mooring.
Both moorings were deployed from the Taiwanese research vessel (R/V) Ocean Researcher 1 (OR1) on 30 May 2011. The subsurface mooring was recovered by R/V Ocean Researcher 3 (OR3) on 3 June 2011. The bottom pressure sensor mooring next to the subsurface mooring was recovered on 4 June, but that next to the surface mooring could not be recovered. The observation period 31 May–4 June was chosen because of the strong spring tide forcing.
On 3 June, R/V OR3 tracked one large-amplitude ISW westward from 117.46° to 117.08°E for nearly 6 h (Fig. 1). The R/V OR3 overtook the ISW then waited for the wave to pass. This sampling strategy was repeated for a total of nine encounters with the same ISW. Shipboard ADCP and yoyo-CTD measurements were taken on the seventh and ninth encounters with the wave between the surface and 200- and 300-m depths, and fixed-depth CTD measurements at 75 m were taken on the fifth encounter. On 4 June, R/V OR3 first measured an ISW east of the mooring positions. Later, R/V OR3 stayed within the wave west of the surface mooring for more than 2 h. ISW properties, energetics, and instabilities are studied using the combined shipboard and mooring measurements.
3. Energetics of internal solitary waves
4. Large-amplitude internal solitary waves: Properties and energetics
a. Properties
Temperature, salinity, and potential density observations on the surface and subsurface moorings reveal clear semidiurnal and diurnal internal tides as well as ISWs (Fig. 2). The vertical displacement of internal tides shows a nonlinear nonsinusoidal form. The amplitude of the vertical displacement of internal tides is on the order of 50 m, with the maximum at middepth, suggesting the dominance of the low-mode internal tides, consistent with previous findings (Alford et al. 2010). A layer of salty water occurs on 2–4 June at 100–200-m depth. A similar salty water mass was observed in the wake of the trapped core ISW described in Lien et al. (2012), apparently resulting from horizontal advection.
Observations of (a) temperature, (b) salinity, (c) potential density on the surface mooring, and observations of (d) zonal velocity, (e) meridional velocity, and (f) vertical velocity from combined measurements by the upward-looking 600-kHz ADCP on the surface mooring and the upward-looking 75-kHz ADCP on the subsurface mooring. The horizontal white stripes in panels (d)–(f) indicate the measurement gap between the 75- and 600-kHz ADCPs.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
From 31 May to 4 June, five large-amplitude ISWs arrived at the moorings at almost the same time each day (Fig. 3). They arrived at the subsurface mooring first and at the surface mooring about 1 h later. Ramp et al. (2004) categorize ISWs in the SCS as A and B waves. The A waves arrive at the same location nearly the same time every day, whereas B waves arrive later each day. A waves typically have larger amplitudes than B waves. Our observed ISWs are consistent with A waves.
(a)–(e) Contour plots of temperature, (f)–(i) along-wave velocity, and (j)–(m) vertical velocity of five ISWs observed (from top to bottom) on 31 May–4 June. Thick black curves represent the isopycnal of the maximum vertical displacement. The magenta curves in (h),(i) are contours of the wave propagation velocity. Within the magenta contours, the current velocity is greater than the wave propagation velocity, indicating convective instability. The brown dots in (d),(e) represent the near-bottom gravitational instability events behind the waves.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
The wave propagation direction is computed based on the direction of the horizontal velocity associated with the ISWs (Chang et al. 2011). ISWs propagate predominantly westward. On 3–4 June, shipboard radar observations also provided the wave propagation directions, which agree with those computed from the horizontal velocity.
The wave propagation speed is computed by three methods. 1) The averaged propagation speed between the surface and subsurface moorings 
ISW wave speed decreases rapidly from 2.2 m s−1 at 117.275°E (the subsurface mooring site) to 1.82 m s−1 at 117.225°E (the surface mooring site) on 3 June (Fig. 4). The wave speed near the surface mooring site on 4 June is 1.60 m s−1. Propagation speed decreases to its minimum (1.6 m s−1 on 3 June and 1.5 m s−1 on 4 June) between the subsurface and surface moorings, about 1–2 km east of the surface mooring.
ISW propagation speed and direction estimated using the orientation and propagation speed of the intensified surface scattering associated with the surface convergence of ISWs. (a)–(c) Location of intensified surface scattering as a function of longitude and time, propagation speed of ISW, and propagation direction (counterclockwise from the east) of ISW on 3 June (black) and on 4 June (red). The vertical blue lines mark the positions of subsurface at 117.275°E and surface at 117.225°E moorings. The blue horizontal bar represents the approximate width of an ISW centering at the surface mooring. The propagation speed and direction averaged within the wave width are 1.82 m s−1 at 170° on 3 June and 1.60 m s−1 at 180° on 4 June.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
In the following analysis, the wave propagation speed at the surface mooring site Cw is assumed as 
Properties of internal solitary waves: the average propagation speed 
ISW maximum vertical displacement, computed by following the fluctuations of isopycnals, increases from 106 m on 31 May to 173 m on 4 June (Figs. 3a–e; Table 1). The initial depth of the isopycnal undergoing the maximum vertical displacement, however, decreases from ~90-m depth 31 May–2 June to 38-m depth on 4 June. That the stronger wave has shallower initial depth is consistent with results of numerical model simulations (Lamb and Nguyen 2009). Vertical overturns above the maximum vertical displacement near the center of the wave were observed on 2–4 June (Figs. 3c–e, white isotherm contours) and near the bottom on 3–4 June (Figs. 3d,e, brown dots). Isopycnal fields show similar vertical overturns.
In the following analysis, we rotate the horizontal velocity to the along-wave component u and the cross-wave component υ. The maximum westward along-wave velocity umax increases from 1.05 m s−1 on 1 June to 2.15 m s−1 on 3 June (Figs. 3f–i). From 31 May to 1 June, umax decreases slightly. On 2–3 June, umax exceeds Cw (Figs. 3h,i, within the magenta contours), indicating the convective instability. Our ADCP velocity measurements have a standard error of 0.07 m s−1, and the estimate of the wave propagation speed from radar measurements has a standard error of 0.04 m s−1. The observed umax > Cw on 2–3 June is significant to the 95% confidence interval (Table 1). The observed wave breaking is consistent with the vertical overturns observed within the wave core. One notable feature of umax is its presence at ~100-m depth, except for the ISW observed on 1 June. This feature is different from a theoretical mode-1 depression ISW. The subsurface umax observation is neither due to the combination of the background shear and ISW (Figs. 3f–i) nor to the beam spreading effect, because both the shipboard downward-looking ADCP measurements (section 5) and upward-looking mooring ADCP measurements reveal the same feature.
Numerical model simulations conclude that a subsurface jet plunges into the wave core during the wave-breaking process (Vlasenko and Hutter 2002). This may explain the subsurface velocity maximum feature observed on 2–3 June. The observed ISWs on 31 May and 1 June are close to the breaking limit (i.e., umax ~ Cw). The umax occurs below the surface on 31 May, but at the sea surface on 1 June. It is possible that the plunging jet is a transient feature for small-amplitude waves on 31 May and 1 June. The maximum vertical velocity increases from 0.23 m s−1 on 31 May to 0.71 m s−1 on 3 June (Figs. 3j–m).
On 2–4 June the maximum vertical displacement at the center of the waves reaches 210-m depth, nearly half the water depth (Fig. 5a). Numerical model simulations (Vlasenko and Hutter 2002) and theoretical DJL model solutions conclude that the wave reaches a conjugate flow limit or breaks when its maximum displacement reaches half the water depth. Laboratory experiments (Helfrich 1992) suggest that the wave breaking occurs when the normalized maximal vertical displacement 
Properties of five internal solitary waves on 31 May (black curves), 1 June (red), 2 June (blue), 3 June (brown), and 4 June (green). Vertical profiles of (a) vertical displacement at the center of the wave, (b) vertical displacement at the center of the wave normalized by the lower-layer thickness (H – Hm), (c) along-wave velocity at the center of the wave, (d) along-wave velocity at the center of the wave normalized by the wave propagation speed, (e) the averaged vertical velocity at the maximum upwelling and downwelling, and (f) the vertical velocity normalized by the maximum along-wave velocity.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
Breaking limit as a function of normalized maximum vertical displacement 
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
b. Energetics
Vertical profiles of energetics are integrated over the propagation direction (Fig. 7). APE is strong at the middepths, and zero at the two end boundaries. The APE maximum increases from 0.95 MJ m−2 on 31 May to 2.35 MJ m−2 on 4 June; the vertically integrated APE increases from 121 to 365 MJ m−1, a factor of 4 increase in 5 days. The ISW KE is computed by subtracting the background KE vertical profile averaged between 20 and 30 min before the center of the wave from the total KE. KE is strong in two layers, separated by a low value at ~200-m depth, that is, a feature for mode-1 ISWs. The KE should be even stronger on 4 June, but there are no velocity measurements for confirmation. The depth-integrated KE is weakest, 72 MJ m−1, on 1 June and strongest, 254 MJ m−1, on 3 June. The vertically integrated APE exceeds that of KE (Table 1), which is a distinct feature of shoaling ISWs. Lamb and Nguyen (2009) perform numerical model simulations of shoaling ISWs and report that KE is converted to APE as the ISW shoals. The APE/KE ratio reaches its maximum value during wave breaking.
Wave propagation direction integration of (a) APE, (b) KE, (c) total energy, (d) advection of APE, (e) advection of KE, (f) pressure work, and (g) total energy flux. Vertical profiles of waves on 31 May–4 June are represented by the black, red, blue, brown, and green curves, respectively.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
The APE flux [u(APE) in (4)] integrated over the propagation direction is a strong negative between 50- and 200-m depths and a weak positive in the lower layer. Note that the integrated energy fluxes are computed to minimize the measurement errors and the effect of unsteadiness when evaluating (10), but they have no physical meaning. The advection of KE is also mostly negative between 50- and 200-m depths, and a small positive in the lower layer. The velocity–pressure work 〈up〉 is the strongest flux term. In their numerical simulations, Lamb and Nguyen (2009) also find that APE flux exceeds KE flux, and the velocity–pressure work is the strongest flux component, similar to our observations. The sum of vertically integrated APE and KE fluxes is 467, 380, 486, and 862 MW for waves on 31 May and 1–3 June, respectively. The energy flux to energy ratio yields respective wave speed estimates of 1.9, 1.7, 2.2, and 1.6 m s−1, mostly within 15% of the observed speeds. The observed APE and KE are at least three orders of magnitude greater than those for ISWs observed on the Oregon Shelf (Moum et al. 2007a).
5. Detailed descriptions of convectively unstable ISWs observed on 3 and 4 June
a. Shipboard and mooring observations
The combined shipboard and mooring observations on 3 and 4 June measured ISW properties and evolution. On 3 June, R/V OR3 tracked the wave for nearly 5 h from 117.47° to 117.05°E and for about 42 km. The ship encountered the wave nine times (Fig. 8a). Shipboard 75-kHz ADCP and 38-kHz echo sounder measured the wave at each encounter. Yoyo-CTD measurements were taken on the seventh and ninth encounters and during the fifth the ship stayed on a station 4 km north of the surface mooring to take CTD measurements at 75-m depth.
Echo sounder measurements of an ISW taken on 3 June as the R/V OR3 tracked the wave. (a) The ship track as a function of time and longitude. The red dots indicate the time and longitude of the center of the wave. (b) The black curve and black open squares show the wave propagation speed computed from the positions and times of the wave center, and the brown curve with the red dots represents the maximum along-wave velocity. The black dot shows the estimate of wave propagation speed computed using the difference of arrival time on the subsurface and surface moorings. Two brown squares show the wave propagation speed computed using the shipboard radar. The blue curve shows the propagation speed of a similar wave observed at the same location in 2005 (Lien et al. 2012). (c) The echo sounder images during eight encounters with the wave. (d) The echo sounder during the fifth encounter when the ship was maintained on station about 4 km north of the surface mooring. The two vertical lines in (a),(b) mark the positions of the surface mooring (TC1) and the subsurface mooring (TC2).
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
The shipboard radar measured wave propagation 170° counterclockwise from the east. The wave speed, calculated using the arrival times of the center of the wave between consecutive passes, decreases from 2.4 m s−1 at about 117.4°E to ~2 m s−1 between 117.3° and 117.25°E to 1.5 m s−1 west of 117.15°E, which is consistent with our estimates based on radar images (Fig. 4).
During the first encounter with the wave in the deep water, ~117.45°E, umax is less than the wave propagation speed (Fig. 8b). During the second encounter, umax increases to nearly equal the wave propagation speed. During the following encounters, both umax and C decrease, but the umax decrease lags C. During the third, fifth, and ninth encounters, umax exceeds C, indicating wave breaking. During the fifth encounter, shipboard ADCP measures umax at 2.2 m s−1 at about 100-m depth, identical to mooring measurements (Fig. 5c). The agreement of subsurface umax from both shipboard downward-looking ADCP and mooring upward-looking ADCP suggests it is not a beam spreading effect. The observed propagation speed is much greater than that of a trapped core ISW similarly observed in 2005 (Lien et al. 2012), but the westward-decreasing trend is similar.
Shipboard echo sounder measurements show a O(10 m) high scattering layer at 380 m extending eastward from 117.2° to 117.45°E (nearly 25 km; Fig. 8c). This high scattering layer appears as sediment resuspension at about 117.2°E and is advected by the eastward velocity of the ISW in the lower layer. The maximum eastward velocity in the lower layer of the ISW varies from 1 to 2 m s−1 (Table 1; Fig. 5). During the period of ISW passage, O(10 min), the sediment could be advected by O(1 km). Because the strong scattering layer extends more than 20 km, there must be local, in addition to horizontal, sediment resuspension. The echo sounder measurements taken during the fifth encounter with the wave when the ship was on station recorded ~100-m-thick strong scattering off the bottom with its intensity enhanced after ISW passage (Fig. 8d). Bogucki et al. (2005) suggest that vortex formation behind the wave could be the mechanism for sediment resuspension. Sand dunes with amplitude of 16 m have been observed on the Dongsha slope and are likely formed by large-amplitude ISWs (Reeder et al. 2010). Similar sand dunes with amplitudes of order of tens of meters were measured by echo sounder between 117.4° and 117.45°E (Fig. 8c).
Mooring observations of temperature, salinity, and potential density on 3 June exhibit vertical overturns within the wave core (Fig. 9). Temperature and salinity fluctuations measured from the shipboard CTD at ~75-m depth agree well with those measured by CTD on the mooring. Both exhibit high-frequency variations within the core of the wave and its wake.
Wave observed on 3 June. Contour plots of (a) temperature, (b) salinity, (e) potential density, (f) along-wave velocity, (g) across-wave velocity, and (h) vertical velocity. (c),(d) Time series of temperature and salinity at ~75-m depth observed by the shipboard CTD (red curves) and by the mooring (blue curves). The thick black dashed curve in (b) represents the depth of the shipboard CTD. The thick black solid curves in (e),(f) depict the isopycnal of the maximum vertical displacement. The two white dots in (e) mark the time and depth of the half amplitude of the maximum displacement. The magenta contour in (f) represents the contour of the wave propagation velocity. The horizontal red lines on the right of (a),(b) indicate the depths of CTD measurements on the surface mooring.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
b. Vertical profiles from mooring measurements
Vertical profiles of temperature, salinity, and potential density in the upper 100 m after the wave passage suggest turbulent mixing (Fig. 10). The water is warmer (0.2°C) and fresher (Figs. 10a,b) within the upper 5 m in the center of the wave; these anomalies are not observed before or after the wave. This warm fresh layer may be carried by the wave. Within the center of the wave, vertical overturns of 102 m are observed lasting for an order of minutes (Fig. 10c); these are well resolved by mooring CTD measurements at 7-s sampling interval. The vertical stratification in the upper 100 m is reduced after the wave passes. The stratification within the wave core is distinctly different than that outside the wave. Negative buoyancy squared is present between 50- and 120-m depths, corresponding to the vertical overturn (Fig. 10d).
ISW observed on 3 June at the surface mooring. Vertical profiles of (a) temperature, (b) salinity, (c) potential density, (d) buoyancy frequency squared, (e) linear eigen-modal structure computed from upstream background stratification, (f) vertical velocity, (g) along-wave velocity, (h) cross-wave velocity, (i) vertical shear of along-wave velocity, (j) vertical shear of cross-wave velocity, (k) total vertical shear squared of horizontal velocity, and (l) gradient Richardson number. The inset (m) in (a) shows the temperature profiles in the upper 30 m, and (n) in (c) the potential density in the upper 150 m. In (a)–(d) and (g)–(l), black curves represent the upstream profiles (defined as 20–30 min before the center of the wave), red curves represent the profiles near the center (defined as 1 min within the center of the wave), and the blue curves represent the downstream profiles (defined as 20–30 min after the center of the wave). The linear eigen-modal structures (e) of the mode-1 vertical velocity (black curve) and mode-1 horizontal velocity (red curve) are computed using the upstream stratification [black curve in (d)]. The black and red curves in (f) represent vertical profiles of maximum downwelling and maximum upwelling, respectively. The vertical solid line in (g) marks the wave propagation velocity. The vertical dashed line and solid line in (l) represent Ri = ¼ and 0.11, respectively.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
Vertical profiles of downwelling and upwelling show maxima occurring around 150-m depth (Fig. 10f). Note that the upwelling and downwelling vertical profiles are not symmetric. The vertical profile of maximum upwelling in Fig. 10f was taken at 0430 UTC (Fig. 9h), which was within the core of the wave upper layer where turbulent overturns were observed. Vertical profiles of u before and after the wave are similar; within the wave core u > Cw between 60- and 120-m depths (Fig. 10g). At the center of the wave, u nearly vanishes close to the sea surface, suggesting the flow reversal associated with the convective breaking reaches nearly to the surface. Further study is needed to confirm this striking feature. Vertical profiles of cross-wave velocity are similar before, during, and after the wave at nearly all depths, except between the 200- and 350-m depths where a northward jet of up to 0.5 m s−1 was measured. This may represent the three-dimensional wave shape but is beyond the scope of this study.
The vertical shear of the horizontal velocity shows two primary peaks centering at 50- and 200-m depths, dominated by the along-wave component, at the center of the wave (Figs. 10i–k). The gradient Richardson number 
c. Shear and convective instabilities
During the ninth encounter with the wave on 3 June, yoyo-CTD measurements were taken from the ship at about 117.08°E. The wave umax increases from 1 m s−1 on the eighth encounter to 1.49 m s−1 on the ninth, which is greater than the propagation speed (1.32–1.45 m s−1; Figs. 8 and 4). Gravitational instability (N2 < 0) and shear instability (Ri < 0.25) are present within and before the center of the wave mostly within the core of the wave (Fig. 11a). In the major portion of the core, Ri is less than 0.11, a condition favorable for KH billows in numerical model simulations (Lamb and Farmer 2011). Moum et al. (2003) report that the shear instability of ISWs observed on the Oregon Shelf occurs at the interface and the vertical scale is measured insufficiently by the shipboard ADCP. Here, shear instability is also likely to occur at vertical scales smaller than the shipboard ADCP measurements.
ISW properties on the ninth encounter with the wave on 3 June from shipboard observations. Contours of (a) along-wave velocity, (b) cross-wave velocity, and (c) vertical velocity. The depth–time track of the shipboard CTD is shown in each panel. White and brown dots in (a) show the depth and time of Ri < 0.25 and Ri < 0.11, respectively. Brown dots in (b) indicate the gravitational instability (i.e., N2 < 0).
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
Shipboard ADCP and CTD measurements on 4 June reveal ISW gravitational and shear instabilities northwest of the surface mooring (Figs. 12a,b, brown dots). The R/V OR3 stayed ahead of the center of the wave in the downwelling region and took yoyo-CTD measurements for nearly 3 h during the wave’s westward propagation (Fig. 12). The umax is 1.90 m s−1, greater than the wave propagation speed of 1.6–1.8 m s−1 estimated from shipboard radar measurements (Fig. 4b). CTD profiles 2–10 (of 16 total) show apparent vertical overturns (Fig. 12d). CTD profiles 6–7, separated by 4–8 min, show nearly identical vertical overturns greater than 100 m between 25- and 137-m depths (Figs. 12e,f). The rms Thorpe scale LT (Dillon 1982) is 52 m within the two overturn patches. The N2 computed from the sorted densities is 2 × 10−5 s−2 and 1.8 × 10−5 s−2 for profiles 6 and 7, respectively. The turbulent kinetic energy dissipation rate within these two overturn patches is computed as 
Observations within the wave core on 4 June from shipboard measurements. Contour plots of (a) along-wave velocity, (b) cross-wave velocity, and (c) vertical velocity. (d) Vertical profiles of density of different CTD casts. The density profiles of casts (e) 6 and (f) 7 (black and red curves). The gray and magenta curves in (e),(f) are sorted density. Casts 6 and 7 are labeled in (b). Both show nearly identical vertical overturns between 25- and 136-m depths. The two vertical profiles are taken about 4–8 min apart. The dissipation rate is close to 10−3 W kg−1. The white contour in (a) shows the wave speed estimated from shipboard radar images. White and brown dots in (a) show the depth and time of Ri < 0.25 and Ri < 0.11, respectively. Brown dots in (b) show the depth and time of gravitational instability (i.e., N2 < 0).
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
d. Temporal variation of energetics
The depth-integrated APE and KE for the wave observed on 3 June are comparable and are the strongest at the center of the wave (Fig. 13). However, the distribution of APE has broader tails than the KE. The along-wave and depth-integrated APE and KE are 299 and 254 MJ m−1, respectively. The excess APE to KE is due to the shoaling process and is the strongest when the wave breaking commences (Lamb and Nguyen 2009). The time rate of change of depth-integrated APE is positive before the wave and negative after the wave and is comparable to the depth-integrated buoyancy flux (Fig. 13c). The KE is converted to APE before the center of the wave and vice versa after the center of the wave. After the wave passage, 1 kJ m−2 of APE is left in the wake, which likely results from the wave breaking (Fig. 13d).
Time variation of ISW energy on 3 June observed by surface mooring instruments. Time series of (a),(b) the depth-integrated and the propagation direction and depth-integrated APE (red), KE (black), and the total energy E (brown), respectively; (b) the propagation direction and depth-integrated APE (red), KE (black), and E (brown); (c) the time rate change of APE (red) and the depth-integrated buoyancy flux (black); and (d) the depth-integrated APE plotted in logarithm scale.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
The pressure perturbation associated with the wave consists of the hydrostatic, nonhydrostatic, and surface displacement components (section 3). In a comparison of vertical integrations of these components (Fig. 14a), the hydrostatic pressure perturbation is the largest negative at the center of the wave due to the negative density perturbation of the depression wave. The nonhydrostatic pressure and pressure perturbation due to the surface displacement are mostly positive.
ISW properties from 3 June observed on the mooring. Vertically integrated time variations of (a),(b) hydrostatic pressure and velocity–pressure work of hydrostatic pressure (black), nonhydrostatic pressure (red), and external pressure due to surface elevation disturbance (blue), respectively; and in (b) the total velocity–pressure work (brown). (c) Horizontal advection of KE (black), horizontal advection of APE (red), the total velocity–pressure work (blue), and the total flux (the sum of the above three components) (brown). (d) The propagation direction integration of (c).
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
The depth-integrated velocity–pressure work consists of the strong westward hydrostatic component and the small eastward nonhydrostatic and surface displacement components (Fig. 14b). The total energy flux includes the advection of KE, the advection of APE, and the velocity–pressure work. The velocity–pressure work is the largest component in the energy flux; the advection of APE is small, and KE advection is the smallest. All components are westward in the wave. At the center of the wave, the total energy flux reaches −1.25 MW m−1. The vertically integrated energy flux in a steady solitary wave is the wave speed times the vertically integrated energy density. To minimize the effects of unsteadiness and measurement error, we integrate the energy density and energy flux over the propagation direction. The integrated energy flux is −862 MW, which yields a wave speed estimate of 1.6 m s−1. This is 88% of the observed wave speed averaged within the wave width (1.82 m s−1), and agrees with the wave speed 1.6 m s−1 measured 1 km east of the surface mooring (Fig. 4).
6. Fully nonlinear DJL solutions
Fully nonlinear steady-state ISWs described by the DJL internal solitary wave model (Lamb 2003) are studied and compared with the ISW observed on 3 June. Model results are not expected to agree well with observations because the model’s assumed steady state is clearly different from our observations. The goals for the DJL model study are to determine the effect of background shear on the steady-state fully nonlinear internal waves and to compare properties and energetics of the steady-state DJL model results with observations. The differences may be attributed to the shoaling breaking process and turbulent dissipation.
Vertical profiles of N2 and current observed before the wave arrival on 3 June are low-pass filtered using a Butterworth filter with a half-power vertical scale of 80 m (Fig. 10d). Note that the x and z dependences of the fully nonlinear wave described in (11) are not separable. 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 (2002) propose a method of solving this extension, based on the Turkington et al. (1991) method, which is used here.
DJL solutions of the same observed maximum vertical displacement ηmax = 155 m are studied. The modeled DJL wave speed is 1.9 m s−1, and the equilibrium depth of the maximum vertical displacement ~40 m are in close agreement with the observations, showing little effect by the background shear (Table 2).
Results of DJL model simulations initialized with properties of the ISW observed on 3 June, with and without the background shear. The term Hm is the initial depth of the isopycnal undergoing maximum vertical displacement, Zηmax is the depth of the isopycnal at its maximum vertical displacement, L1/2 is the width of the wave at half of the maximum vertical displacement, and T1/2 is the elapsed time of the wave at half of the maximum vertical displacement.
The width of the half amplitude is 867 m for the model wave without the shear and is 760 m with the shear, narrower than the observed width, but the horizontal velocity of the DJL wave including the effect of the background shear is in close agreement with the observation (Table 2). It exceeds the propagation speed, indicating convective instability (Fig. 15). The streamline shows a surface-trapped core (not shown). The model wave without the background shear, however, does not have a trapped core. The vertical velocity of the model wave is not affected significantly by the background shear and is about 78% of the observed vertical velocity. The KE of the DJL wave is 1.15 times the APE; greater KE than APE is a feature for typical ISWs. This is in distinct contrast to the observed wave with KE about 0.82 of the APE, which results from the shoaling process. The DJL model assumes steady state and therefore does not include such nonsteady processes. The total energy of the DJL wave is in good agreement with the observation (Fig. 15e). Overall, the most significant effect by including the background current is the presence of a surface-trapped core.
DJL model simulations based on ISW properties observed on 3 June, including the effect of background shear. Contour plots of (a) potential density, (b) vertical displacement, (c) along-wave velocity, (d) vertical velocity, and (e) the vertical integration of KE (black), APE (red), and total energy (brown). The blue dashed curve in (e) shows the observed total energy of the ISW on 3 June. The magenta curve in (c) shows the contour of the wave propagation speed.
Citation: Journal of Physical Oceanography 44, 4; 10.1175/JPO-D-13-088.1
7. Summary
Five large-amplitude mode-1 depression ISWs were observed on the Dongsha slope during a spring tide using both shipboard and mooring ADCP and CTD measurements. Wave properties and energetics are summarized as follows:
Along-wave velocity of ~2 m s−1, vertical velocity of ~0.7 m s−1, and maximum vertical displacement of 167 m, 0.37 of the total depth, were observed. The along-wave velocity shows the salient feature of a subsurface maximum.
The maximum vertical displacement normalized by the lower-layer thickness (H – Hm) reaches 0.4 for breaking waves, in agreement with laboratory experiment results (Helfrich 1992).
At least three waves reach the breaking limit umax > C, indicating convective instability. Vertical overturns of ~100 m occur within the core of the wave and last for 48 min, with turbulent kinetic energy dissipation rate estimated as large as 1.5 × 10−4 W kg−1.
Within the major portion of the wave core and at the interface, Ri < 0.11, the critical Richardson number when KH billows are found in numerical model simulations.
All observed waves had greater APE than KE likely due to the conversion of KE to APE during the shoaling process, in agreement with numerical model simulations (Lamb and Nguyen 2009). The maximum energy per unit crest meter is 365 MJ m−1 for APE on 4 June and 254 MJ m−1 for KE on 3 June.
The energy flux is dominated by the pressure work. The horizontal advection of KE, the horizontal advection of APE, and the pressure work are 7%, 16%, and 77%, respectively, of the total energy flux. This is in contrast to observations on the Oregon Shelf (Moum et al. 2003), where advection of KE is greater than the pressure work.
On average, C〈E〉z ~ 〈Fx〉z, suggesting that although the turbulent dissipation is significant, it is small compared with the energy flux and the time rate change of energy in the energy budget.
Results of DJL model simulations with and without the background current predict the observed wave speed, equilibrium depth, maximum zonal velocity, and total energy. The simulations fail to predict APE greater than KE and the subsurface maximum of horizontal velocity, which is expected because these properties are features for shoaling ISWs, a process not included in the steady-state DJL model. The significant effect of including the background shear in the DJL model is the formation of a surface-trapped core.
Here, energetics of ISWs observed in the SCS with and without convective instability are presented for the first time. The observed ISW energy in the SCS is more than three decades greater than that observed on the Oregon Shelf (Moum et al. 2003) and in Massachusetts Bay (Scotti and Pineda 2004). Results of this analysis should provide reference values for future research employing model simulations and laboratory experiments of shoaling ISWs.
Acknowledgments
This work is supported by the Office of Naval Research (N00014-09-1-0279). We appreciate the constructive comments from reviewer Kevin Lamb and another anonymous reviewer. Their comments have improved this paper substantially. We thank the crews on the Taiwanese R/V Ocean Researcher 1 and R/V Ocean Researcher 3 for deploying and recovering the moorings and conducting shipboard ADCP and CTD surveys. We thank Drs. David Tang and Sen Jan of National Taiwan University and Dr. Ming-Huei Chang of National Taiwan Ocean University for their collaboration on this joint program and for arranging the cruise. Mooring operation and design by Mr. Wen-Hua Her is greatly appreciated. We thank Dr. Rhuo-Sheng Tseng of National Sun-Yat-Sen University for the MicroRider data and resulting comparison to our estimated turbulent kinetic energy dissipation.
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