Large-Amplitude Internal Wave Transformation into Shallow Water

Gregory Sinnett aUniversity of California, Irvine, Irvine, California

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https://orcid.org/0000-0002-5617-5366
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Steven R. Ramp bSoliton Ocean Services LLC, Carmel Valley, California

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Yiing J. Yang cInstitute of Oceanography, National Taiwan University, Taipei, Taiwan

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Ming-Huei Chang cInstitute of Oceanography, National Taiwan University, Taipei, Taiwan

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Sen Jan cInstitute of Oceanography, National Taiwan University, Taipei, Taiwan

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Kristen A. Davis aUniversity of California, Irvine, Irvine, California

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Abstract

Large-amplitude internal solitary wave (ISW) shoaling, breaking, and run-up was tracked continuously by a dense and rapidly sampling array spanning depths from 500 m to shore near Dongsha Atoll in the South China Sea. Incident ISW amplitudes ranged between 78 and 146 m with propagation speeds between 1.40 and 2.38 m s−1. The ratio between wave amplitude and a critical amplitude A0 controlled breaking type and was related to wave speed cp and depth. Fissioning ISWs generated larger trailing elevation waves when the thermocline was deep and evolved into onshore propagating bores in depths near 100 m. Collapsing ISWs contained significant mixing and little upslope bore propagation. Bores contained significant onshore near-bottom kinetic and potential energy flux and significant offshore rundown and relaxation phases before and after the bore front passage, respectively. Bores on the shallow forereef drove bottom temperature variation in excess of 10°C and near-bottom cross-shore currents in excess of 0.4 m s−1. Bores decelerated upslope, consistent with upslope two-layer gravity current theory, though run-up extent Xr was offshore of the predicted gravity current location. Background stratification affected the bore run-up, with Xr farther offshore when the Korteweg–de Vries nonlinearity coefficient α was negative. Fronts associated with the shoaling local internal tide, but equal in magnitude to the soliton-generated bores, were observed onshore of 20-m depth.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: Gregory Sinnett, gsinnett@gmail.com; Kristen Davis, davis@uci.edu

Abstract

Large-amplitude internal solitary wave (ISW) shoaling, breaking, and run-up was tracked continuously by a dense and rapidly sampling array spanning depths from 500 m to shore near Dongsha Atoll in the South China Sea. Incident ISW amplitudes ranged between 78 and 146 m with propagation speeds between 1.40 and 2.38 m s−1. The ratio between wave amplitude and a critical amplitude A0 controlled breaking type and was related to wave speed cp and depth. Fissioning ISWs generated larger trailing elevation waves when the thermocline was deep and evolved into onshore propagating bores in depths near 100 m. Collapsing ISWs contained significant mixing and little upslope bore propagation. Bores contained significant onshore near-bottom kinetic and potential energy flux and significant offshore rundown and relaxation phases before and after the bore front passage, respectively. Bores on the shallow forereef drove bottom temperature variation in excess of 10°C and near-bottom cross-shore currents in excess of 0.4 m s−1. Bores decelerated upslope, consistent with upslope two-layer gravity current theory, though run-up extent Xr was offshore of the predicted gravity current location. Background stratification affected the bore run-up, with Xr farther offshore when the Korteweg–de Vries nonlinearity coefficient α was negative. Fronts associated with the shoaling local internal tide, but equal in magnitude to the soliton-generated bores, were observed onshore of 20-m depth.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding authors: Gregory Sinnett, gsinnett@gmail.com; Kristen Davis, davis@uci.edu

1. Introduction

Internal waves (isopycnal oscillations) are a ubiquitous and important feature throughout the ocean which mix and transport larvae, sediment, and nutrients (Leichter et al. 1996; Pineda 1999; Quaresma et al. 2007; Omand et al. 2011). The South China Sea contains some of the largest internal solitary waves (ISW) in the world, and their generation and propagation have been well studied in recent decades (Guo and Chen 2014). Large-amplitude internal waves in the South China Sea are generated by the barotropic (surface) tide in the Luzon Strait, which forces mode-1 semidiurnal and diurnal internal tides that can then steepen into westward propagating solitons (Vlasenko et al. 2012; Alford et al. 2015). Separate generation sites in the Luzon Strait create “a-type” waves (formed near a southern source) and “b-type” waves (formed from a northern source). A-type waves typically have larger amplitudes and traverse deeper bathymetry, affecting their subsequent propagation (Ramp et al. 2019). ISW transit across the South China Sea can be affected by the Kuroshio Current (Jan et al. 2012; Xu et al. 2021), rotation, nonlinearity (Farmer et al. 2009), and stratification (Ramp et al. 2010). Additionally, strong turbulence has been observed both at the generation site (Alford et al. 2015) and within the soliton’s trailing edge in water roughly 300 m deep (Chang et al. 2021), highlighting the mixing potential these waves have to broad regions of the upper ocean.

Large-amplitude ISW begin to shoal and break with significant dissipation near the continental shelf in depths greater than 200 m (Fu et al. 2012). Here, the ISW depression wave approaches a critical depth where polarity reverses (e.g., Shroyer et al. 2009) and the depression ISW may become an elevation ISW or scatter into higher-frequency modes (Helfrich et al. 1984). ISW breaking has been observed on the continental shelf of the South China Sea (e.g., Ramp et al. 2004; Duda et al. 2004; Fu et al. 2012, 2016), while nearby observations supported by a nonlinear nonhydrostatic model indicate solitons may occasionally undergo a fissioning breaking mechanism (Bai et al. 2019), generating a trailing train of high-frequency elevation waves.

Broken internal waves often evolve into bottom-trapped onshore propagating dense bores (e.g., Bourgault et al. 2008; Davis and Monismith 2011; Walter et al. 2014a; Sinnett et al. 2018; Jones et al. 2020). Bores propagating up the Dongsha Atoll east forereef (facing the incoming large-amplitude solitons) have been tracked from 24-m depth to the reef crest (Davis et al. 2020) where they cool the bottom environment and increase nitrate flux (Reid et al. 2019), significantly lowering the risk of bleaching (Safaie et al. 2018). Bore propagation and reflection on the Dongsha east forereef was shown to be sensitive to changes in bathymetric slope and pycnocline depth (Davis et al. 2020), with some bores propagating into the surfzone and onto the reef crest under favorable conditions (Reid et al. 2019). Elsewhere, changes in background stratification (represented by the nonlinearity coefficient α in the Korteweg–de Vries equation) have been shown to steepen or relax a shallow bore front (McSweeney et al. 2020), and offshore stratification and thermocline depth affected bore strength and structure (Walter et al. 2014b).

Although large-amplitude soliton generation, propagation, shoaling, breaking, and run-up have been observed at each stage in the South China Sea, no observations have tracked these events through the entire process. A field campaign was conducted from May to June 2019 to observe internal wave transition from their offshore large-amplitude soliton expression all the way to run-up termination. An overview of this experiment and aspects of the incident wave field are detailed in Ramp et al. (2022).

Here, key aspects of the wave field transition are quantified by tracking single events. The effect individual wave properties and background conditions have on shoaling, breaking, and ultimate onshore expression are quantified using observations from a dense and rapidly sampling instrument array spanning depths from 500 m to shore. Experiment details are given in Ramp et al. (2022) and summarized in section 2. Some of the first observations tracking soliton events from 500-m depth to run-up termination are presented in section 3. Aspects controlling key elements of the shoaling process are discussed in section 4, and the experiment is summarized in section 5.

2. Experiment

a. Location

Dongsha Atoll is a nearly circular coral reef roughly 28 km in diameter located at the edge of the continental shelf in the South China Sea (Fig. 1a). This region is well known for the large-amplitude internal tides generated at the Luzon Strait between Taiwan and the Philippines (Guo and Chen 2014). These waves travel westward, becoming large-amplitude ISW before reaching the continental shelf and eventually Dongsha Atoll. The eastern side of the atoll (where westward propagating internal waves break) is the focus of this study (Fig. 1b) and contains a moderate slope (0.03 < s < 0.07) from the surfzone to depth h ≈ 500 m, with the exception between h ≈ 25 and 50 m where the slope is steeper (s ≈ 0.15). Coordinate axes are defined with +x extending to the east offshore and +y in the alongshore northward direction and +z upward, such that internal wave propagation is generally in the −x direction.

Summertime ocean conditions near Dongsha Atoll are modulated by monsoonal winds which typically drive northeasterly surface currents (Song 2011). A mixed semidiurnal barotropic tide caused surface elevation to vary by as much as ±0.5 m during the deployment and drove primarily alongshore currents at the study site. The average thermocline depth at h = 500 m (approximated here as the depth of the 19°C isotherm) was z = −112 m during the observational period.

b. Instrumentation

For a 10-day period from 16 to 26 May 2019 the Dongsha east reef crest was heavily instrumented to observe internal wave transition and shoaling. Readers are referred to Ramp et al. (2022) for a detailed experimental description and instrument list. The subset of instruments included in this analysis include moored T-chains and ADCPs at h = 500, 300, and 100 m (Fig. 1b), and further onshore at h = 50, 20, and 10 m (Fig. 1c). Moorings are referred to by their depth (e.g., M50, referring to the mooring near h = 50 m) for convenience. At each location, temperature (resolution between 20 and 1.5 m), water velocity, and pressure (to monitor mooring blowdown) were observed. Instruments deeper than h = 50 m sampled at 1-min intervals, while those at h = 50 m and shallower sampled at 1 Hz, with the exception of four SBE37/39 instruments which sampled at 1 min intervals.

Additionally, a distributed temperature sensing (DTS) cable was deployed along the bottom, originating from the east reef flat and spanning the east forereef to roughly h = 20 m before continuing roughly along isobaths and extending to h ≈ 50 m (Fig. 1c). DTS systems use pulsed laser light along a fiber optic cable to determine temperature along the cable length. This DTS system sampled temperature every 0.25 m along the 5-km cable length every 30 s. The system included a station (located at the coordinate origin) which contained the power supply, a Silixa XT-DTS system, and two calibration baths with RBR SoloT instruments (accuracy 0.002°C). Each calibration bath contained a reference 10-m section of coiled cable and was maintained at a warm (ambient) temperature and cold (ice bath) temperature as conditions allowed following Sinnett et al. (2020). An additional 10-m calibration coil was deployed near h = 10 m starting 2445 m along the cable from the DTS instrument. The DTS cable location on the bottom between the DTS station and h = 25 m were surveyed twice by divers during the deployment.

c. Data processing

The fastest common instrument sample rate across the 108 independently sampling instruments was 1 min. Instruments that sampled faster than 1 min were averaged, and those that sampled at 1-min intervals were interpolated to conform to the same 1-min time grid. Vertical linear interpolation was applied for comparison between moorings at consistent depths.

Strong currents sometimes caused moorings to deflect from vertical. Pressure sensors near the top of the mooring allowed for linear instrument depth correction along the mooring line. Moorings rarely experienced deflections greater than 45°. However, the flotation on M50 leaked slowly, causing a gradual decrease in buoyant resistance to strong currents, requiring analysis of this mooring to be restricted to times before 26 May. During the analysis period between 16 and 26 May, the M50 ADCP pitch never exceeded 28°. ADCP velocities were quality controlled and corrected for beam spreading effects following Scotti et al. (2005) using methods described in Chang et al. (2011).

DTS temperature data were calibrated using three calibration coils and two additional validation thermistors. The calibration coil prepared at the end of the cable was severed during the deployment, so calibration was performed following the method of Sinnett et al. (2020) with constant DTS calibration parameter γ. Light spatial and temporal filtering with cutoffs at 1.5 m and 3 min reduced the signal RMSE by up to 30% with expected temperature error less than 0.1°C over the entire cable length (Sinnett et al. 2020). These filtering parameters were selected to retain the highest resolution in space and time as possible, while reducing RMSE to acceptable levels for resolving onshore propagating internal wave bores.

3. Results

Eight a-type waves and seven b-type waves were observed transiting toward Dongsha Atoll between 16 and 25 May 2019 (Fig. 2). This section quantifies key shoaling and breaking processes observed during the passage of these waves. Section 3a details characteristics related to the westward propagating depression waves in 500–300-m-deep water. Section 3b addresses the shoaling transition observed in water depths between 300 and 100 m. Section 3c addresses the bore run-up phase observed in water depths shallower than 50 m to termination. Throughout this section and the following discussion, example events 2, 10, 13, and 16 (see Fig. 2a) are highlighted to contrast aspects of the deep water and shoaling transition phase, and events 13 and 16 are further highlighted to contrast aspects of the run-up and termination.

Fig. 1.
Fig. 1.

Progressively zoomed maps of the experiment location showing: (a) the South China Sea and Dongsha Atoll within the westward propagating path of large-amplitude internal waves originating from the Luzon Strait between Taiwan and the Philippines. (b) Dongsha Atoll and moorings M500, M300, and M100 in water 500, 300, and 100 m deep, respectively. (c) The east reef crest and east forereef showing locations of the DTS station and cable (white line) as well as moorings M50, M20, and M10 in water 50, 20, and 10 m deep, respectively.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

Fig. 2.
Fig. 2.

Soliton arrivals at the 500-m isobath (dots) numbered sequentially and grouped by wave type (colors) with reference to (a) the local barotropic tide as observed at the 20 m pressure sensor and (b) the local baroclinic tidal displacement represented by the 3-h low-pass-filtered and linearly detrended 19°C isotherm elevation in 500-m depth. Note a-type events (black dots) conveniently correspond to odd numbers, and b-type events (red dots) correspond to even numbers. Event 12 (missing) was observed, though the mooring in 300-m depth was undergoing an instrument switch-over at the time and the event is thus excluded from further analysis.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

a. Depression wave characteristics

1) Soliton arrivals

Soliton arrivals were modulated by the barotropic tidal cycle and ocean conditions during transit from the generation region in the Luzon strait (e.g., Ramp et al. 2019) with a-type wave arrivals occurring near the low phase of the barotropic tide and b-type waves arriving near the high phase of the barotropic tide during the observational period (Fig. 2a). Waves are numbered sequentially in time and hereafter referred to by these numbers for convenience. The local internal (baroclinic) tide, here represented by the low-pass-filtered displacement of the 19°C isotherm, is out of phase with the barotropic tide. During the study period, a-type wave arrivals at Dongsha Atoll generally coincided with periods where the 19°C isotherm was positively displaced (shallower), and b-type wave arrivals coincide with periods where the 19°C isotherm is negatively displaced (Fig. 2b).

Wave amplitudes at M500 are defined as the maximum 19°C isotherm displacement from the pre-event 30-min average depth. Amplitudes ranged between 78 m < A500 < 146 m with propagation speeds 1.40 m s−1 < cp < 2.38 m s−1. These waves of depression caused surface temperatures (z ≈ 40 m) to warm rapidly as the wave passed, with dT/dt as high as 0.067°C s−1, occasionally warming surface waters by as much as 9.9°C in roughly 4 min. Readers are referred to Ramp et al. (2019, and references therein) for a detailed description of these waves in depths greater than 500 m. Hereafter, internal wave and transformation characteristics onshore of the 500-m isobath are explored.

2) Internal wave shoaling

As ISW approach the shallow sloping bathymetry east of Dongsha Atoll, they begin to shoal and break. This process occurs in a variety of ways depending on both background conditions and properties of the soliton itself. Temperature evolution during the passage of four example internal wave events (10, 2, 16, and 13, in Fig. 2a) highlight a range of offshore transformations and subsequent onshore expressions. In 500-m depth (first column, Fig. 3), the leading waves of the example wave packets have amplitudes A500 = 136, 109, 104, and 90 m, respectively. Each event arrival is preceded by a relatively quiet and stratified water column. Isotherms are then deflected sharply downward as the soliton passes, with notable deflections observed nearly throughout the entire water column in each case.

Breaking internal waves are often classified by the internal Iribarren number,
Ir=sA/Lw,
where s is the bathymetric slope, A is the wave amplitude, and Lw is the half of the wavelength (e.g., Aghsaee et al. 2010). The shallow and slowly varying bathymetric slope (s = 0.021 near h = 500 m) and large soliton amplitude resulted in small Ir, which often corresponds to fission-type breaking (Aghsaee et al. 2010). Slope was effectively reduced farther by any off-axis wave propagation angle, usually not more than 17° from shore normal (Ramp et al. 2022). During the measurement period, Iribarren number ranged from 0.066 < Ir < 0.093, with the example events in Fig. 3 having Ir = 0.086, 0.072, 0.078, and 0.073 (from top to bottom). Characteristic fission-type isotherm displacements are observed at M500 trailing the leading soliton depression (column 1, Fig. 3). However, the magnitude and number of trailing depression waves varies between the example events (and others, not shown).
The leading wave shape at M500 resembled a soliton solution to the KdV equation with
η=η0sech2(x/l)(m),
where x/l is a normalized length scale. Here, the cross-shore distance x is estimated from the mooring observation time series and the observed propagation speed cp. The scaling length l is the horizontal distance between the location of the 19°C isotherm maximum displacement ηmax and the location corresponding to ηmax sech2(π/2). Typically, isotherm elevations were stable before the event passage (x/l < −π) and then closely followed a sech2 profile from −π/2 < x/l < π/2 (Fig. 4a). Characteristic fissioning oscillations were observed behind the soliton. For example, the 19°C isotherm relative displacement near x/l = π was sometimes below and sometimes above the prewave elevation η/η0 = 0. For the four example events (10, 2, 16, and 13) at x/lπ, the 19°C isotherm had returned to −0.25 < η/η0 < 0.19 (Fig. 4b) before oscillating at various amplitudes and frequencies. The relative amplitude and width of the trailing elevation pulse (formed behind the primary depression) varied at M500, with the trailing peak to trough amplitude ranging between 0.92η0 during event 16 to 0.29η0 during event 10.
Fig. 3.
Fig. 3.

Temperature (color bar) vs scaled depth z/h and distance cpt at mooring locations corresponding to the 500-, 300-, 100-, 50-, 20-, and 10-m isobaths (columns). Rows correspond to events 10, 2 16, and 13 on 21, 17, 24, and 23 May, respectively. Temperature is contoured at 2°C. Note the expanded color bar limits to resolve colder water at the 500- and 300-m-deep locations.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

Fig. 4.
Fig. 4.

Displacements of the 19°C isotherm at h = 500 m for (a) event 13 (blue) compared with the sech2 KdV soliton (black), and (b) four wave profiles corresponding to event numbers in Fig. 2a (legend, colors) compared with the sech2 KdV soliton (black), all scaled by KdV amplitude η0.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

The large-amplitude internal solitons observed at this location were often near or above a critical amplitude A0 defined as
A0=12cph(cph104ν)0.12(m),
which depends on propagation speed cp as well as water depth h and kinematic viscosity ν (Diamessis and Redekopp 2006). Waves with amplitudes above A0 likely produce an adverse pressure gradient and “global instability” in the bottom boundary layer trailing the wave, leading to elevated turbulence and possible sediment resuspension. Observed propagation speeds between M500 and M300 during example events 10, 2, 16, and 13 were cp = 2.06, 2.02, 1.83, and 1.88 m s−1, respectively. All but three waves observed transiting the 500-m isobath (dots, Fig. 2) had amplitudes at or above A0, with A500/A0 ranging between 0.78 and 1.6. In all cases, A300/h300 > A500/h500, though wave amplitudes evolved differently. The example events in Fig. 3 had A500/A0 of 1.24, 1.02, 1.05 and 0.90 (from top to bottom) with A500A300 of 46, −8, 10, and −7 m, respectively.

Background conditions encountered by the shoaling solitons varied at M300. Cross-shore barotropic current ranged between 0.043ms1<U¯<0.171ms1 during the fifteen observed leading ISW events, with the example events in Fig. 3 having U¯=0.004,0.043,0.124,and0.043ms1 (from top to bottom). Background stratification at M300 varied with the phase of the internal tide (Fig. 2b) and is also known to vary by season, due to synoptic warming events and mesoscale eddies (Ramp et al. 2010). The hourly averaged maximum squared buoyancy frequency before soliton arrivals at the 300 m isobath ranged between 3.6×104s2<Nmax2<1.83×103s2 at associated thermocline depths between 32 m < zth < 117 m. Nmax2 was 1.24 × 10−3 s−2, 1.83 × 10−3 s−2, 4.6 × 10−4 s−2, and 6.3 × 10−4 s−2 at zth depths of 117, 66, 42, and 32 m for the example events in Fig. 3 (from top to bottom).

Mixing and fissioning dynamics varied behind the soliton passage at M300. For example, very little mixing (vertical isotherm spreading) and minimal isotherm oscillations were observed behind event 13 (second column, bottom, Fig. 3). Both isotherm depths and near bottom temperature remained constant after event 13, with minimal fissioning trailing isotherm oscillations. Event 10 also had few trailing oscillations, but mixing behind the event caused the surface mixed layer to spread vertically by as much as 120 m, cooling the surface slightly and warming the near bottom by 3.5°C in 1 h. Events 2 and 16 both retained aspects of the trailing fission oscillations at M300, though event 16 widened the mixed layer by 105 m and caused near-bottom temperature to rise 3.8°C in 1 h. Dynamics related to the fissioning waveform are discussed in section 4.

b. Transition

1) Breaking and bore formation

In a two-layer (strong stratification) case, shoaling begins when the internal wave trough depth is greater than half the water column, so that (A + zth)/h = 1/2 (e.g., Scotti et al. 2008). At M500, this metric was always less than 1/2, indicating the waves were not yet undergoing significant shoaling. The critical point where many large-amplitude waves of depression either dissipate or transition to elevation waves on the Dongsha slope has been shown to lie between h = 300 and 100 m (Fu et al. 2012; Ramp et al. 2022). During this experiment, (A + zth)/h was always greater than 1/2 at h = 300 m, indicating all waves were likely shoaling at this location.

As the internal tidal phase commonly caused isopycnals to be deeper than average during b-type events (Fig. 2b), the b-type wave arrivals commonly coincided with a warmer and less stratified water column onshore of h = 100 m during the study period (Fig. 3). Onshore of h = 100 m, remnants of the b-type leading depression wave were evident as a top to bottom warming with a baroclinic velocity profile containing an onshore surface and offshore bottom velocity component. The back of the depression wave steepened into an elevation wave containing cold water and a baroclinic velocity profile marked by onshore bottom and offshore surface velocity. The trailing elevation wave properties were related to the fissioning dynamics, as broader and denser elevation waves were associated with more elevated isotherm displacements during the first fissioning elevation wave (cf. b-type event elevation waves at M100 near cpt100 = 0 in Fig. 3 column 3 with η/η0 displacements near x/l = π in Fig. 4b).

A-type events typically arrived when the internal tide phase caused the pycnocline to be shallower than average (Fig. 2b), and propagated into a colder water column at h = 100 m (e.g., Fig. 3, bottom row). Average a-type wave amplitude was larger, however, such that shoaling onset likely occurred in water only 30 m shallower on average than b-type waves. This difference represents only 3.5% of the total shoaling distance observed and is not expected to significantly affect results onshore of M100. A-type wave events were characterized by a weakening depression wave pulse that still generated a strong baroclinic velocity signature (onshore near the surface and offshore at depth). At M100, the back of the depression wave was typically not as steep as the elevation wave formed from b-type events (Fig. 3). The cooler elevation wave behind the a-type event always contained a strong baroclinic velocity signature, similar to the b-type events.

The temperature time series was filtered (zero-phase fourth-order Butterworth) to isolate the rapid internal wave driven temperature variation from the slower internal tide oscillation. Each event’s resulting high-pass-filtered bottom temperature magnitude ΔTb is defined here as the maximum filtered temperature difference during a 20 min window centered at the event arrival. At M100, ΔTb varied between 3.07°C < ΔTb < 12.26°C with mean ΔTb = 8.19°C.

Mean event temperature magnitude at M50 was ΔTb = 7.42°C and ranged between 2.99°C < ΔTb < 12.04°C (blue, Fig. 5), though temperature evolution during a-type and b-type events was different. After elevation waves formed behind b-type events (typically between h = 300 and 100 m), their temperature signature was observed upslope at M50 as a slowing cold water bore of decreasing width as determined by propagation speed (cf. middle columns of events 10, 2, and 16, Fig. 3). The cold bores of these b-type events originated from the steepening trailing depression face and subsequent fissioning wave, and propagated through warmer, less stratified water on the shallow shelf as modified by the warmer preceding depression wave remnants (e.g., event 16 from h = 300 to 50 m, Fig. 3). The minimum bottom temperature at h = 50 m during the seven observed b-type events was on average 5.68°C colder than the background temperature an hour before the cold bore arrival, and 7.40°C colder than conditions directly preceding the cold bore.

Fig. 5.
Fig. 5.

Histogram of event bottom temperature magnitude scaled by the mean bottom temperature magnitude at all locations, ΔTb/ΔTb¯. Bin edges are indicated by the top axis, and colors correspond to observations at depths h = 100 m (yellow), h = 50 m (green), h = 20 m (red), h = 10 m (blue). The scaled mean ΔTb/ΔTb¯ at each location is indicated by a dot with respect to the top axis.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

A-type events also generated upslope propagating cold bores; however, these formed primarily from the steepening and breaking backside of the initial depression event, with subsequent trailing fission events forming far behind the initial depression (e.g., event 13, Fig. 3). Consequently, although the average a-type bore bottom temperature reduction was similar to the b-type reduction (6.90°C compared to 7.40°C), the a-type bore temperature was closer to the ambient bottom temperature an hour prior (2.35°C difference compared to 5.68°C).

ISW packet onshore propagation speed derived from arrival times between M50 and M20, ranged between 0.17 cm s−1 < cp < 0.59 cm s−1. There was no statistically significant cp difference between a-type and b-type events propagating between M50 and M20. Breaking events, now cold bores at these shallow depths, propagated upslope with a baroclinic velocity ub here defined as the 1 min averaged baroclinic velocity 3.5 m above bottom. Upslope bottom baroclinic velocity ranged between 0.80ms1<ub<0.25ms1 and mean |ub|/cp=1.2, indicating these events were highly nonlinear.

The elevation wave temperature signature at M50 ranged between 2.99°C < ΔTb < 12.04°C (green, Fig. 5) and was not well correlated with offshore wave amplitude or velocity. For example, the largest of the four example events in Fig. 3 (event 10, A500 = 136 m) had the weakest temperature signature at M50, ΔTb = 4.84°C and average ub. Event 11 generated the largest temperature variation at M50 (ΔTb = 12.04°C) and the strongest near-bed baroclinic velocity ub=0.80ms1, yet this a-type wave amplitude at both M500 and M300 were slightly below average. The apparent mismatch between offshore internal wave amplitude and nearshore elevation wave ΔTb is related to the transition between depression and elevation waves and is addressed in section 4.

2) Shoaled wave energetics

The event energy flux is estimated at M50 and M20 to quantify consequences of the offshore shoaling and breaking processes and the transition to the resulting upslope bore evolution. Vertically integrated pseudoenergy flux Ef is defined for negligible diffusivity and viscosity effects (Lamb and Nguyen 2009; Venayagamoorthy and Fringer 2005) as
Ef=(KEf+APEf+Wf)(Wm1).
Here, onshore available kinetic energy flux KEf, potential energy flux APEf, and linear energy flux Wf due to the pressure perturbation are
KEf=h0uEkdzAPEf=h0uEadzWf=h0updz
where the baroclinic cross-shore velocity component u=uu¯, the kinetic energy density is Ek, available potential energy density is Ea, and the perturbation pressure is p′. Primed terms vertically integrate to zero, assuring that the baroclinic condition is satisfied (e.g., Kunze et al. 2002).
Kinetic energy density is
Ek=12ρ0(u2+υ2+w2)(Jm3),
where primed terms are the baroclinic horizontal velocities, w is the vertical velocity, and ρ0 = 1024 kg m−3 is a reference density. At M50, instantaneous mid water column Ek peaked above 500 J m−3 during three a-type events (11, 13, and 15). A-type waves generated both the strongest and weakest Ek at M50 with mean and standard deviation 412 and 182 J m−3, while b-type waves generated Ek with mean and standard deviation 368 and 46 J m−3. Near the bed, peak Ek was approximately 5 times larger for a-type events than for b-type events at M50 and M20. This difference may be related to the contrasting bore shapes onshore of M50 (cf. steepening depression a-type event 13 with the characteristic b-type events 10, 2, and 16, Fig. 3).
The positive-definite available potential energy density is defined here for density perturbations of length scale ζ following, e.g., Kang and Fringer (2010) as
Ea=zζzg[ ρ(z)ρr(z) ]dz(Jm3),
where ρr is the far-field reference density defined as the sorted hourly averaged density 1.5–0.5 h before the event front arrival. At M50, instantaneous Ea peaked above 500 J m−3 during the same three a-type events (11, 13, and 15) that generated the largest Ek. A-type event Ea was greatest prior to the passage of the steepening back of the wave as warm buoyant water was pushed down by the residual depression. B-type events, especially those arriving during the low internal tidal phase (Fig. 2b), had peak Ea coincident with the bore arrival as cold water was elevated in the water column. Example event 16 (Fig. 3) had maximum Ea = 320 J m−3 at z = −17 m, 90 m behind the front face of the bore.
The perturbation pressure is
p=ppr(Jm3),
where pr is a reference pressure profile defined by the reference density ρr. At M50, instantaneous p′ peaked above 60 J m−3 during a-type event 5, with a-type p′ mean and standard deviation 43 and 19 J m−3. Example event 16 had the largest b-type instantaneous p′ = 54 J m−3. B-type events had slightly smaller mean and standard deviation of 38 and 15 J m−3, with events arriving when the 19°C isotherm displacement was near zero containing the lowest instantaneous p′ (e.g., events 2, 4, 6, and 8, Fig. 2b).

The vertically integrated pseudoenergy flux terms [Eq. (5)] at M50 were largely balanced between kinetic energy and potential energy. At the bore’s leading face, KEf ≈ 2APEf and KEf ≈ 33Wf on average. The smaller pressure work contribution at M50 (relative to APE and KE) are in contrast to observed ISW energetics at h = 500 m on the Dongsha plateau, where pressure work contributes roughly 3/4 of the total energy budget (Lien et al. 2014). By M20, Ef was nearly equally divided between KEf and APEf with a negligible contribution from Wf.

c. Bore run-up

1) Background conditions and bore characteristics

Large-amplitude internal wave shoaling depends on higher-order nonlinearities not captured by the standard KdV equation. Farther onshore, shoaling internal waves can generate bores if (A + zth)/h > 1/2 (as was always the case by h = 300 m during this study) and their onshore evolution is related to the background stratification conditions (Scotti et al. 2008). In depths shallower than 50 m, the onshore propagating cold bore became the dominant internal wave feature. Background stratification conditions were successfully related to characteristics of the bore shape and propagation through the nonlinearity coefficient α (McSweeney et al. 2020), where
α=32h0c02(ϕz)3dzh0c0(ϕz)2dz.

Here, the vertical structure function ϕ is found from the temperature profile. Both background conditions and bore run-up was observed on the steep sloping reef crest at M20 and M10 as well as by the DTS cable spanning the slope from the surfzone h ≈ 1 m past the 20-m mooring.

Background stratification varied with the internal tide, oscillating between a well mixed and generally warm water column during the low internal tidal phase, to a stratified water column during the high internal tidal phase. Thus, the buoyancy frequency at M20 varied between N ≈ 0.016 s−1 (weakly stratified) before bore arrivals preceding a low internal tide (events 8 and 10, see Fig. 2b) and N ≈ 0.083 s−1 (moderately stratified) before bore arrivals preceding a high internal tide (events 13 and 15, see Fig. 2b).

As bores propagate into shallow water, the background stratification acts to steepen or dissipate the bore front. Bore steepening or dissipation has been related to the sign or relative magnitude of α in continuously stratified shallow water (e.g., McSweeney et al. 2020) and in two layer analogs (e.g., Scotti et al. 2008). Here, α at h = 50 m varied from −0.049 (likely dissipative) to 0.026 (likely steepening). Although the 19°C isotherm was positively displaced whenever α > 0.002, the internal tidal phase was not significantly correlated with α, emphasizing the importance of the internal tide as well as other factors controlling thermal properties on the shallow forereef, such as mesoscale eddies, air–sea flux and seasonal stratification differences (Ramp et al. 2010).

Key bore run-up features at M50, M20, and M10 are detailed with velocity and temperature profiles centered on the bore arrival (Fig. 6). Here, example events 16 (b-type) and 13 (a-type) are compared as each occurred during the spring barotropic tidal phase and had large-amplitude expressions (η = 95 and 98 m, respectively) at M300. However, event 16 occurred during the lowest internal tidal phase, while event 13 arrived during the highest internal tidal phase in the record (Fig. 2b). Although the event 16 and 13 u′ observations at M50 (Figs. 6a,d) contain rundown (offshore directed ub) before each event, the b-type event 16 rundown was much stronger and persisted until the bore front arrival.

Fig. 6.
Fig. 6.

Cross-shore baroclinic velocity u′ (color) vs scaled depth z/h and distance cpt at the 50-, 20-, and 10-m isobaths during (a)–(c) b-type event 16 and (d)–(f) a-type event 13. Positive velocity is offshore, and the density contour (black) interval is 0.5 kg m−3 with ρ = 1029 kg m−3 highlighted for reference (bold black).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

The nonlinearity coefficient α was near zero prior to event 16 (indicating either steepening or dissipating conditions may exist), yet onshore directed u′ occupied more than half the water column, with | ub |/cp>1 and a fast surface return flow, indicating a highly nonlinear bore (Fig. 6a). The bore thrust cold dense water from within 5 m of the bottom (z/h = 0.9) to within 10 m of the surface. Thus, the top of the bore (and strongest density gradient) is contained within the offshore directed return flow, causing the top of the bore to become heavily sheared. Additionally, there is a strong surface divergence near the bore front, and a strong surface convergence at the rear (commonly associated with surface boils and slicks, respectively). These conditions were common to large b-type events arriving near the low internal tide.

In contrast, the a-type event 13 rundown at M50 slowed and stagnated before the arrival of the fast-moving onshore core, mostly confined to the lower 40% of the water column. Again, | ub |/cp>1, though α = 0.011 indicating a preference for steepening bore fronts. Isopycnals were not yet as steep at M50 during event 13 as during event 16 (cf. Figs. 6a,d). The onshore surge is also more than twice as wide, containing a significantly larger volume of cold dense water even though the top of the bore (and highest density gradient) was contained within the lower 50% of the water column and still within the onshore directed surge. Despite the strong baroclinic signature, the surface divergence zone near the front of the steepening bore front is comparatively weak.

Farther onshore at M20, the b-type example 16 still had a steep face (Fig. 6b), though the rundown present at M50 is less significant. Onshore ub=0.33ms1 (2/3 the magnitude compared to M50) and the onshore propagating bore width had decreased by half. A significant relaxation phase (offshore directed near-bottom drainage after the wave) caused ub>0.30ms1. Isopycnal displacements associated with the example 16 bore passage were still deflected more than half the water column at M20 (lines, Fig. 6b), and the top of the bore was still heavily sheared. As at M50, strong surface convergence and divergence zones exist near the front and rear of event 16.

The a-type example 13 had a steepening face at M20 (cf. near-bottom isopycnals in Figs. 6d,e) and a longer period of onshore ub with peak flow near the bore face as large as ub=0.25ms1. The onshore u′ component of example event 13 occupied a similar percentage of the water column at M20 as it did at M50 (Fig. 6e). There is evidence of a recirculation region within the bore head when cpt20 > 0.25 (offshore directed flow near the bed, Fig. 6e). Weak surface divergence and strong convergence zones exist near the bore front and rear, respectively.

Both example events 16 and 13 were observable near the reef crest at M10 (Figs. 6c,f). The b-type event 16 maintained the steep isopycnal front slope, though the bore height occupied less than half the water column. Baroclinic velocity associated with the onshore bore propagation continued to weaken (ub=0.15ms1), though |ub|/cp1. The bore width associated with the onshore ub narrowed, preceding a bottom intensified offshore return flow. Surface divergence and convergence zones were still observed at M10 during event 16 near the bore front and trailing edge, respectively.

The a-type example event 13 bore height (indicated by the u′ = 0 and maximum pycnocline location) still occupied roughly half the water column at M10 (Fig. 6f). However, other bore features were more similar to the b-type example in shallow water. The isopycnals at the front of event 13 had steepened by M10, similar to the b-type example 16. Also, onshore ub=0.18ms1, and | ub |/cp1. The significantly narrower a-type event 16 bore width was similar to the b-type event 13 width by h = 10 m, and surface divergence and convergence zones were similarly observed at the surface near the front and back of the bore.

2) Fine-scale bottom observations

The DTS fiber-optic cable continuously observed the forereef bottom temperature, capturing details of the bore run-up bottom expression with 1.5-m spatial and 3-min temporal resolution. This spatial and temporal resolution is well suited to capture the evolution of onshore propagating bore fronts. However, the very high-frequency (≈5-min period) internal oscillations observed in shallow water with DTS instruments elsewhere (e.g., Lucas and Pinkel 2022) may not be completely resolved. Here, we focus on the bore-front evolution and other features of similar temperature magnitudes and time scales.

Bottom DTS temperature is shown for two example 2-h periods centered on event 13 (Fig. 7a) and event 16 (Fig. 7b). Before the event 13 arrival, bottom temperature was cool offshore of M20. Beginning around 10:30 UTC, bottom temperature warmed as the a-type event approached and warm surface water was pulled down by the depression wave rundown (e.g., Fig. 3). Bottom temperature along the reef slope during this phase was roughly uniform, though slightly warmer bottom water was observed down the slope h > 20 m around 1045 UTC than just offshore of the surfzone (h = 2 m) indicative of the strong vertical advection from the depression wave remnant. The steepening trailing bore arrived at the h = 20 m location at 1108 UTC (black triangle, Fig. 7a) causing bottom temperature to drop 8°C in 2 min. The instantaneous vertically integrated energy flux [Eq. (4)] at M20 during run-up was nearly balanced between APEf and KEf, with little contribution from Wf [black, Fig. 7a(1)].

The event 13 bore front (Tb ≈ 19°C), propagated upslope past M20, where water temperature T > 24°C before the event arrival. The wave front (black line, Fig. 7a) is defined when |dT/dt| > 0.01°C and passes M10 (black dot, Fig. 7a) before terminating at xdts = 2266 m or approximately 86 m from the surfzone at the run-up extent Xr. During run-up, observed Ef [Eq. (4)] at M10 contained an onshore (negative) APEf component and weaker KEf and Wf components [Fig. 7a(2)]. Bottom temperature was nearly unaffected by the bore onshore of Xr during this event. At least two b-type events (14 and 16) propagated into the surfzone where breaking waves advected the cold bore water over the reef crest at velocities near 0.2 m s−1, close to the average tidal and wave-driven cross-reef velocities observed at this location previously (Reid et al. 2020).

By contrast, the b-type event 16 propagated through warmer offshore water (Fig. 7b). Beginning just before 0245 UTC, bottom temperature gradually warmed, with warmer waters slowly advected downslope until the abrupt arrival of the bore front at M20 near 0320 UTC (black triangle, Fig. 7b). Instantaneous Ef at M20 during run-up [black, Fig. 7b(1)] was much weaker for event 16 compared to event 13.

The event 16 wave front persisted past M10 with weaker onshore APEf [black, Fig. 7b(2)] than event 13. The temperature drop caused by the bore front, ΔTb, was 2°C less for event 16 than for event 13; however, the bore propagated 74 m further upslope, terminating near 0400 UTC at Xr = 12 m. Though the forereef water temperature was cooler after event 16, the cool water persisted for roughly the same amount of time before either receding or mixing. Offshore-directed APEf and KEf during the relaxation phase (red, Fig. 7) was largely similar.

Interesting temperature features were observed where the forereef slope steepens at h = 25 m, the site of previously observed wave reflections (Davis et al. 2020). During event 13, the densest water was retained below the steep drop-off while a significant portion of the bore continued onshore around 1115 UTC (Fig. 7a). Similarly, beginning near 0340 UTC, event 16 contained an offshore warming signature starting near M20 proceeding offshore. The receding bottom cold water signature at depths greater than 20 m occurred simultaneously with the onshore propagating cold bore in depths shallower than 20 m. In addition to the primary event 13 bore front (black line, Fig. 7a), several high-frequency and short-duration “mini-bores” are faintly observable after the passage of the bore front onshore of M20 around 1130 UTC. High-frequency mini-bores continued to arrive during the relaxation phase of event 13 (after 1140 UTC, Fig. 7a); however, the dominant bottom temperature feature was a general warming beginning first near Xr and extending offshore.

Although background conditions and the wave properties themselves have much to do with the onshore evolution of the residual broken soliton, bathymetry controls much of the relaxation dynamics. The DTS cable was resting on the bottom and was occasionally in the coral groove, resulting in persistent localized cold spots (Fig. 7a for example, near xDTS = 2300 and 2390 m, though there are several others). During the relaxation phase, cold water advected up-slope by the bore then preferentially drains off the slope in the grooves, causing colder temperatures (in this case as much as 3°C) to persist nearly an hour after the water on the adjacent spur has warmed. The offshore directed bottom velocity during the relaxation phase also commonly generated the warm feature seen during event 13 beginning at 1145 UTC and event 16 beginning at 0335 UTC near M20 (xDTS = 2638 m, Figs. 7a,b). As cool dense water drains over the steep drop-off at this location, strong mixing may entrain warmer surface water.

4. Discussion

a. Controls on soliton shoaling

Soliton shoaling and breaking observed here is influenced by both thermocline depth zth and the adverse pressure gradient. A nonlinear and nonhydrostatic internal wave model suggests soliton fissioning will have an elevated pycnocline behind the leading wave and large-amplitude trailing elevation waves when zth is deep (Bai et al. 2019). Thermocline depth zth ranged between 32 m < zth < 116 m near where wave shoaling and transformations occur at M300, and the trailing wave amplitude relationship to zth was evident among several waves with otherwise similar characteristics.

For example, events 2 and 16 both contained near critical amplitudes at M500 (A500/A0 = 1.02 and 1.05, respectively), though both the strength and depth of maximum stratification was different. Example event 2 (Fig. 3 and green line in Fig. 4b) propagated into strong stratification, N2 = 0.043 s−1, and a deeper than average thermocline depth, zth = 65 m. The resulting fissioning at M300 contained a broad trailing elevation wave with many large-amplitude trailing fission elevation waves (Figs. 3 and 4b). Event 16, however, (Fig. 3 and orange line in Fig. 4b) propagated into weak stratification, N2 = 0.022 s−1, and a shallower than average thermocline depth zth = 40 m. The resulting fissioning at M300 still contained a broad and elevated trailing thermocline, though the trailing fission elevation waves were smaller in number and magnitude (Figs. 3 and 4b). Despite the weaker trailing fissioning events during event 16, the subsequent onshore bore expression was large (see, e.g., Fig. 3), suggesting the isotherm displacement amplitude at the trailing face of the depression wave affects bore formation, while the coherence of the trailing fission events is less consequential.

In addition to background thermocline depth and stratification, wave properties themselves affect fissioning and the resulting upslope propagating bore. For example, the strong stratification (N = 0.035 s−1) and deep thermocline (zth = 116 m) before event 10 are likely conditions for strong fissioning, similar to event 2. However, the mixing behind the event 10 soliton passage at M300 with little fission wave displacement (see Fig. 3) indicate that the wave was breaking without significant fissioning despite the favorable background conditions. A key difference between event 10 and others is the high A500/A0 = 1.24 ratio, indicating that this event likely had a strong adverse pressure gradient (Diamessis and Redekopp 2006) and may have been a collapsing breaker rather than a fissioning breaker, despite the shallow sloping bathymetry and background conditions (Aghsaee et al. 2010). Collapsing breakers contain high mixing at the back of the wave, as shear instabilities and overturning are possible (e.g., Chang et al. 2021; Chang 2021).

b. Controls on bore run-up extent Xr

Onshore of M50, the soliton expression is no longer a coherent depression wave, but rather a deteriorating depression signature followed by a trailing onshore propagating dense bore (section 3c). Internal bore run-up on the shallow forereef slope is a dynamic event, with near-bed velocities above 40 cm s−1 and large rapid temperature changes sometimes above 10°C. Bore run-up extent Xr is a primary factor affecting which portions of reef are exposed to deep water masses and significant ΔTb, potentially affecting the reef crest and lagoon (Reid et al. 2019).

DTS observations track the bore cross-shore location versus time (black line, Fig. 7a) and are used to find the bore front upslope deceleration df and subsequent Xr. Upslope front deceleration df was estimated following Sinnett et al. (2018) by fitting the observed front position to a quadratic such that
Δx=df2(Δt2)+cf0Δt,
where cf0 is the initial front speed. All fronts were fit with high skill (r2 > 0.97) yielding observed upslope deceleration 2.5 × 10−5 m s−2 < df < 14.2 × 10−5 m s−2.
Two-layer theory has been used extensively to describe gravity current run-up in both laboratory (e.g., Wallace and Wilkinson 1988; Helfrich 1992; Marleau et al. 2014) and numerical studies (e.g., Arthur and Fringer 2014). Here, two layer stratification is adapted to continuously stratified observed conditions following Sinnett et al. (2018) with deceleration,
dgc=12gszbh˜(1zbh˜),
dependent on the bedslope s, and reduced gravity g′, equivalent two-layer bore height above bottom zb, and tidal water depth h˜ all evaluated near M20 along the DTS cable track. Vertical temperature observations at M20 define the change in potential energy [Eq. (8)] associated with the bore front and are used to find zb in Eq. (11). Resulting dgc estimates agree well with the observed run-up deceleration (Fig. 8) with best fit linear slope of 0.75 and r2 = 0.84, implying bore fronts on this section of the forereef behave similarly to an upslope propagating gravity current.
Fig. 7.
Fig. 7.

Temperature (color bar) sampled along the DTS fiber-optic cable length xDTS indicated in Fig. 1c (positive offshore) vs time during (a) event 13 and (b) event 16. The wave front is defined when |dT/dt| > 0.01°C s−1 (black line). Approximate depths corresponding to M20, M10, and the surfzone are indicated for reference as dotted lines. Locations and times when the wave front was observed at M20 and M10 are indicated by a black triangle and circle, respectively. Data are removed (vertical white stripes) when the DTS signal was below the noise threshold. Maximum vertically integrated kinetic, potential, and linear energy flux (from left to right) are shown during the run-up (black) and relaxation phase (red) for (a1) event 13 at 20-m depth and (a2) event 13 at 10-m depth (note the change of scale). Onshore directed flux is negative. (b1),(b2) Energies for the corresponding event 16 at M20 and M10 are shown with the same scale.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

Fig. 8.
Fig. 8.

Two-layer gravity current upslope deceleration dgc vs observed event front deceleration df for events propagating at least 30 m past the 20-m isobath. The 1:1 line is indicated for reference.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

The observed Xr is near but never exactly the predicted extent based on initial speed and deceleration at M20, ranging between 85% and 96% the theoretical Xr. Run-up extent derived from two layer theory is relatively consistent, ranging only 140 m; however, the observed run-up extent varies between just outside the surfzone to 600 m offshore. Offshore amplitude η0 and propagation speed cp affect the soliton shoaling and breaking regimes, but are not directly correlated with the resulting onshore bore run-up extent Xr (all r2 < 0.05).

Rather, Xr is primarily affected by two factors: 1) the fissioning and breaking conditions that define the isopycnal displacement and temperature gradient at the steepening bore front [section3a(2)], and 2) the background conditions through which the bore front propagates (section 3c). The first sets the initial bore conditions, while the second is represented by the background α and influences the bore onshore evolution (Fig. 9). When α > 0, the steep front is more likely to maintain its shape (section 3c and Fig. 6). Steepening conditions were most likely to occur during a high internal tidal phase (black dots in Fig. 9). Conversely, fronts propagating when α < 0 are less likely to maintain their shape and may disintegrate before they reach their maximum predicted Xr.

Fig. 9.
Fig. 9.

Event run-up distance from the surfzone Xr (positive offshore) vs KdV nonlinearity parameter α at the 50-m isobath and phase of the internal tide at the time of event arrival at h = 50 m (color). Note, three events are clustered near (α = 0.002, Xr = 20 m) and two events are clustered near (α50 = −0.003, Xr = 220 m); N = 15.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

Bores propagating through a high internal tide (mostly a-type waves, Fig. 2b) generally encountered positive α and propagated farther (black dots in Fig. 9). When the internal tidal was near mid- or low phase, α was generally negative and bores disintegrated earlier. The shoaling onset location may contribute to this effect as well, since two layer theory predicts the average a-type wave to begin shoaling in water roughly 30 m shallower than average observed b-type waves. Exact shoaling onset locations are unknown, though all waves began shoaling between the 500- and 300-m isobaths.

Two exceptions to the relationship between α and Xr, associated with events 14 and 16, occurred when the internal tide was near its minimum (Fig. 2b). These waves created some of the largest isotherm elevations in the trailing fission elevation wave (yellow line in Fig. 4b), maintained significant energy after breaking and reached the surfzone (blue dots near Xr = 0, Fig. 9) despite propagating through a neutral α while disintegrating upslope (Figs. 6a–c). Events 14 and 16 emphasize the complex interplay between the fission characteristics and nearshore bore run-up conditions.

All observed bores had their largest upslope (negative) u′ near the bed at the bore front (e.g., Fig. 6). This velocity profile is consistent with a backward overturning bolus, despite the high Fr and large amplitude more typical of a top breaking turbulent surge (Moore et al. 2016). Bore generation between 100 m < h < 300 m (Fig. 3) may not initially cause backward overturning bores, though their subsequent interaction with the relaxation from previous bores or the dissipating depression wave (e.g., Fig. 6a; Davis et al. 2020) may eventually resemble a backward overturning bolus. Additionally, the steep bathymetric feature at h ≈ 25 m may disrupt otherwise consistent upslope bore propagation properties. Wave reflection and vertical mixing has been observed at this site previously Davis et al. (2020) and again here (e.g., Fig. 7a).

c. Internal tidal fronts

Large-amplitude soliton shoaling, breaking, and run-up causes significant temperature variability and generates strong currents in water depths shallower than 50 m (e.g., Fig. 6). However, the shoaling local internal tide can develop similar onshore propagating temperature and velocity signatures which are entirely decoupled from the offshore soliton arrivals.

A locally generated internal wave front was observed at 0200 UTC 19 May, after the soliton event 4 arrival around 1830 UTC 18 May but before the event 5 arrival near 0620 UTC 19 May. During this period the internal tide had predominantly semidiurnal oscillations (cf. 17–20 May with 22–27 May in Fig. 2b), and the 19°C isotherm was shoaling higher during the wave front arrival. The rising internal tidal is visible in Fig. 10a as depth of the 19°C isotherm at M300 increases from its low of z = −170 m near 2330 UTC to z = −67 m near 0250 UTC. Although there are isotherm oscillations with amplitude ≈10 m, no significant soliton events (with amplitude ≈90 m, e.g., Fig. 3, column 1) exist during this time at M300. Despite the absence of an offshore soliton, a strong temperature front associated with the shoaling internal tide was observed by the DTS onshore of M20 (Fig. 10b).

As the internal tide front approached the nearshore, strong near bottom temperature gradients observed in depths h < 20 m with dT/dt > −0.039°C caused bottom temperature to fall 7.73°C in 3.5 min (Fig. 10b). Baroclinic velocities similar to those associated with soliton run-up were observed at M20 (not shown), which produced onshore bottom ub magnitude greater than 0.32 m s−1. Similar to soliton waves discussed in section 3c, the densest portion of the initial front was reflected away from the steep drop-off at M20. The offshore propagating warmer bottom layer observed near M20 after 0200 UTC (Fig. 10b) was associated with the rundown and relaxation of the primary and subsequent events which occurred near the steep drop-off. Successive high-frequency (T ≈ 20 min) waves arrived after the initial internal tidal front, and propagated as individual decelerating bores throughout the tidal phase. Eventually, the internal tide and the associated high-frequency bores affected water just outside the surfzone (Xr = 70 m), with cold water persisting in coral grooves throughout the high internal tidal phase. Although the abrupt temperature front and associated baroclinic velocity were observed during both soliton run-up and the shoaling internal tide, persistent high-frequency bore arrivals more commonly occurred during the high phase of the internal tide.

Bores generated from internal soliton wave breaking, and fronts associated with the internal tide both advect cold dense water up the reef slope. The continuous in time and space near-bottom DTS temperature (e.g., Figs. 7a and 10b) allow for a unique estimation of the potential energy change due to the displacement of dense fluid up the forereef slope by an internal bore. The integrated potential energy change along the DTS track from z = −20 m to zxr is
PEDTS=z=20z=zxr[ρ(z,tf)ρ(z,t0)]gzdz,
defined near the bottom (z′ = 1 m above the bottom) with temperature-derived density difference between the front arrival, tf and reference time before the arrival t0 along the cable track. Integration is made with the trapezoidal rule, and dz ≈ 0.25 m with small variation depending on the local bathymetric slope. The resulting PEDTS represents the cumulative magnitude and spatial extent of a bore’s modification to the water properties in the shallow forereef environment.

Integrated potential energy along the DTS track ranged between 0 and 585 J m−2 (Fig. 11). Note, events with Xr > 458 m (Fig. 9) were not observed at M20. Solitons with similar amplitudes at M300 had a wide range of PEDTS as they were affected differently by the fissioning [section 3a(2)] and background characteristics (section 3c) unique to each event. However, very large-amplitude solitons (A300 > h/3) were not likely to produce significant PEDTS in h < 20 m (Fig. 11). Internal tidal fronts (e.g., Fig. 10b) had no offshore amplitude, but did generate PEDTS comparable to the large-amplitude internal solitons (shaded range, Fig. 11). Thus, the local internal tide modulates both the extreme upslope propagation of bores from broken internal solitons, and also the waveguide permitting locally generated propagating fronts of roughly equal magnitude.

Fig. 10.
Fig. 10.

Temperature (color bar) (a) at M300 vs depth and time, and (b) on the forereef as in Fig. 7 vs distance along the DTS cable with the same time axis. Note, the time series in (a) is expanded to show the offshore conditions 2.5 h before (b). The front arrival at M20 (black triangle), M10 (black dot), and the surfzone location (dotted) are indicated for reference.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

Fig. 11.
Fig. 11.

Integrated potential energy PEDTS between h = 20 m and Xr vs soliton amplitude at h = 300 m, A300 for 11 events observed by the DTS. Minimum and maximum PEDTS observations of the shoaling internal tide (shaded) are indicated for reference.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0273.1

5. Summary

Observations from the eastern side of the Dongsha Atoll between 16 and 26 May 2019 provide a comprehensive perspective of large-amplitude internal wave transformation from 500 m depth to the shore. Internal solitary waves with amplitudes ranging between 78 and 146 m propagated through a variety of background conditions. As solitons approached h = 300 m, they began to transform. The large-amplitude waves began to fission over the shallow slope (low Ir) with the background thermocline depth zth affecting the amplitude of the first fissioning wave. Solitons passing through a background stratification with deeper zth had larger trailing elevation fission wave amplitudes, which became bores farther onshore. However, large solitons with amplitudes above a critical amplitude A0 became collapsing breakers (rather than fissioning) and had significant offshore mixing behind the soliton. These waves were less likely to form coherent bores in h < 50 m.

Consequences of the breaking process and nearshore background conditions are evident in the energetics at h = 50 m. A range of bore magnitudes and resulting energy flux signatures were observed. Narrow, bottom-confined bores usually resulted from waves with a small trailing fission amplitude and contained a minimal energy signature. Larger bores developed a strong baroclinic energy signature. Some of the largest contained rundown signatures before the bore front arrival and a strong bottom relaxation phase after the event. Offshore near surface potential energy flux occurred in some of the largest waves, as the top of the bore protrudes into the offshore directed baroclinic current.

Conditions on the shallow forereef where h < 20 m were significantly affected by the shorewoard propagating bores, as onshore ub magnitude can exceed 0.4 m s−1 and ΔTb can exceed 10°C. During this phase of the run-up, bores decelerated at rates near those predicted for a two-layer upslope gravity current. Observed run-up extent Xr was 85%–96% that predicted from gravity current deceleration. Background stratification, quantified here using the nonlinearity coefficient α, was related to Xr, with α > 0 corresponding to run-up farthest up the reef. When α was negative, the propagating bore front was less likely to maintain its shape and the bore disintegrated earlier. When the internal tidal phase was high, locally generated fronts intersected the shallow reef slope. Fronts not associated with offshore solitons, but with roughly equivalent ub and ΔTb, were observed during high internal tidal phases. These locally generated internal tidal fronts affected the shallow forereef similarly to bores generated from offshore solitons.

This experiment tracked large-amplitude solitons from deep water all the way to shore, and highlights the complex interactions and cascading effects throughout breaking and run-up. Offshore soliton properties alone are not a good predictor of eventual conditions on the forereef. Rather, those wave properties together with background conditions, set shoaling and breaking conditions. Resulting bores contain a variety of energetics, and propagate upslope where they are in turn affected by local background conditions. Findings from these coherently tracked events, from their offshore soliton expression to their onshore run-up extent, contextualize other more localized observations.

Acknowledgments.

The authors would like to acknowledge the support of the Dongsha Atoll Research Station (DARS) and the Dongsha Atoll Marine National Park, as well as the researchers and crew aboard the R/V OR2 whose efforts made this research possible. Authors would also like to thank G. Lohmann (WHOI), E. Reid (UC, Irvine), E. Pawlak (UCSD), S. Merrigan (UC, Irvine), and K.H. Fu (National Sun Yat-sen University) for diving and logistical support. F. Feddersen, S. Monismith, and C. Sladek provided equipment support. The authors would also like to thank J. McSweeney, J. Rogers, O. Fringer, A. Saffie, S. Tyler J. Selker, C. Walter, and two anonymous reviewers for helpful comments which improved both the science and manuscript. This work and K. Davis and G. Sinnett were supported by NSF Grant NSF-OCE1753317. S. Ramp was supported by the U. S. Office of Naval Research under Grants N00014-19-1-2686 (322MM) for the field work and N00014-20-1-2727 (322PO) for the data analysis. Profs. Y. J. Yang, M.-H. Chang, and S. Jan were supported by the Taiwanese Ministry of Science and Technology (MOST).

Data availability statement.

Data used to produce the analysis and figures contained in this manuscript are available at https://doi.org/10.5072/zenodo.962769 and by request.

REFERENCES

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Save
  • Aghsaee, P., L. Boegman, and K. G. Lamb, 2010: Breaking of shoaling internal solitary waves. J. Fluid Mech., 659, 289317, https://doi.org/10.1017/S002211201000248X.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Alford, M. H., and Coauthors, 2015: The formation and fate of internal waves in the South China Sea. Nature, 521, 6569, https://doi.org/10.1038/nature14399.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Arthur, R. S., and O. B. Fringer, 2014: The dynamics of breaking internal solitary waves on slopes. J. Fluid Mech., 761, 360398, https://doi.org/10.1017/jfm.2014.641.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bai, X., Z. Liu, Q. Zheng, J. Hu, K. G. Lamb, and S. Cai, 2019: Fission of shoaling internal waves on the northeastern shelf of the South China Sea. J. Geophys. Res. Oceans, 124, 45294545, https://doi.org/10.1029/2018JC014437.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bourgault, D., D. E. Kelley, and P. S. Galbraith, 2008: Turbulence and boluses on an internal beach. J. Mar. Res., 66, 563588, https://doi.org/10.1357/002224008787536835.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chang, M.-H., 2021: Marginal instability within internal solitary waves. Geophys. Res. Lett., 48, e2021GL092616, https://doi.org/10.1029/2021GL092616.

  • Chang, M.-H., R.-C. Lien, Y. J. Yang, and T. Y. Tang, 2011: Nonlinear internal wave properties estimated with moored ADCP measurements. J. Atmos. Oceanic Technol., 28, 802815, https://doi.org/10.1175/2010JTECHO814.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chang, M.-H., and Coauthors, 2021: Direct measurements reveal instabilities and turbulence within large amplitude internal solitary waves beneath the ocean. Commun. Earth Environ., 2, 15, https://doi.org/10.1038/s43247-020-00083-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davis, K. A., and S. G. Monismith, 2011: The modification of bottom boundary layer turbulence and mixing by internal waves shoaling on a barrier reef. J. Phys. Oceanogr., 41, 22232241, https://doi.org/10.1175/2011JPO4344.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davis, K. A., R. S. Arthur, E. C. Reid, J. S. Rogers, O. B. Fringer, T. M. DeCarlo, and A. L. Cohen, 2020: Fate of internal waves on a shallow shelf. J. Geophys. Res. Oceans, 125, e2019JC015377, https://doi.org/10.1029/2019JC015377.

    • Crossref
    • Export Citation
  • Diamessis, P. J., and L. G. Redekopp, 2006: Numerical investigation of solitary internal wave-induced global instability in shallow water benthic boundary layers. J. Phys. Oceanogr., 36, 784812, https://doi.org/10.1175/JPO2900.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Duda, T. F., J. F. Lynch, J. D. Irish, R. C. Beardsley, S. R. Ramp, C.-S. Chiu, T. Y. Tang, and Y.-J. Yang, 2004: Internal tide and nonlinear internal wave behavior at the continental slope in the northern South China Sea. IEEE J. Oceanic Eng., 29, 11051130, https://doi.org/10.1109/JOE.2004.836998.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Farmer, D., Q. Li, and J.-H. Park, 2009: Internal wave observations in the South China Sea: The role of rotation and non-linearity. Atmos.–Ocean, 47, 267280, https://doi.org/10.3137/OC313.2009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fu, K.-H., Y.-H. Wang, L. St. Laurent, H. Simmons, and D.-P. Wang, 2012: Shoaling of large-amplitude nonlinear internal waves at Dongsha Atoll in the northern South China Sea. Cont. Shelf Res., 37, 17, https://doi.org/10.1016/j.csr.2012.01.010.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fu, K.-H., Y.-H. Wang, C.-P. Lee, and I.-H. Lee, 2016: The deformation of shoaling internal waves observed at the Dongsha Atoll in the northern South China Sea. Coastal Eng. J., 58, 1650001, https://doi.org/10.1142/S0578563416500017.

    • Crossref
    • Export Citation
  • Guo, C., and X. Chen, 2014: A review of internal solitary wave dynamics in the northern South China Sea. Prog. Oceanogr., 121, 723, https://doi.org/10.1016/j.pocean.2013.04.002.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Helfrich, K. R., 1992: Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech., 243, 133154, https://doi.org/10.1017/S0022112092002660.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Helfrich, K. R., W. K. Melville, and J. R. Miles, 1984: On interfacial solitary waves over slowly varying topography. J. Fluid Mech., 149, 305317, https://doi.org/10.1017/S0022112084002664.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jan, S., C.-S. Chern, J. Wang, and M.-D. Chiou, 2012: Generation and propagation of baroclinic tides modified by the Kuroshio in the Luzon Strait. J. Geophys. Res., 117, C02019, https://doi.org/10.1029/2011JC007229.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jones, N. L., G. N. Ivey, M. D. Rayson, and S. M. Kelly, 2020: Mixing driven by breaking nonlinear internal waves. Geophys. Res. Lett., 47, e2020GL089591, https://doi.org/10.1029/2020GL089591.

    • Search Google Scholar
    • Export Citation
  • Kang, D., and O. Fringer, 2010: On the calculation of available potential energy in internal wave fields. J. Phys. Oceanogr., 40, 25392545, https://doi.org/10.1175/2010JPO4497.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg, 2002: Internal waves in Monterey Submarine Canyon. J. Phys. Oceanogr., 32, 18901913, https://doi.org/10.1175/1520-0485(2002)032<1890:IWIMSC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lamb, K. G., and V. T. Nguyen, 2009: Calculating energy flux in internal solitary waves with an application to reflectance. J. Phys. Oceanogr., 39, 559580, https://doi.org/10.1175/2008JPO3882.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Leichter, J. J., S. R. Wing, S. L. Miller, and M. W. Denny, 1996: Pulsed delivery of subthermocline water to Conch Reef (Florida Keys) by internal tidal bores. Limnol. Oceanogr., 41, 14901501, https://doi.org/10.4319/lo.1996.41.7.1490.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lien, R.-C., F. Henyey, B. Ma, and Y. J. Yang, 2014: Large-amplitude internal solitary waves observed in the northern South China Sea: Properties and energetics. J. Phys. Oceanogr., 44, 10951115, https://doi.org/10.1175/JPO-D-13-088.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lucas, A. J., and R. Pinkel, 2022: Observations of coherent transverse wakes in shoaling nonlinear internal waves. J. Phys. Oceanogr., 52, 12771293 https://doi.org/10.1175/JPO-D-21-0059.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marleau, L. J., M. R. Flynn, and B. R. Sutherland, 2014: Gravity currents propagating up a slope. Phys. Fluids, 26, 046605, https://doi.org/10.1063/1.4872222.

    • Crossref
    • Export Citation
  • McSweeney, J. M., and Coauthors, 2020: Observations of shoaling nonlinear internal bores across the central California inner shelf. J. Phys. Oceanogr., 50, 111132, https://doi.org/10.1175/JPO-D-19-0125.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moore, C. D., J. R. Koseff, and E. L. Hult, 2016: Characteristics of bolus formation and propagation from breaking internal waves on shelf slopes. J. Fluid Mech., 791, 260283, https://doi.org/10.1017/jfm.2016.58.

    • Crossref
    • Search Google Scholar
    • Export Citation
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    • Crossref
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  • Fig. 1.

    Progressively zoomed maps of the experiment location showing: (a) the South China Sea and Dongsha Atoll within the westward propagating path of large-amplitude internal waves originating from the Luzon Strait between Taiwan and the Philippines. (b) Dongsha Atoll and moorings M500, M300, and M100 in water 500, 300, and 100 m deep, respectively. (c) The east reef crest and east forereef showing locations of the DTS station and cable (white line) as well as moorings M50, M20, and M10 in water 50, 20, and 10 m deep, respectively.

  • Fig. 2.

    Soliton arrivals at the 500-m isobath (dots) numbered sequentially and grouped by wave type (colors) with reference to (a) the local barotropic tide as observed at the 20 m pressure sensor and (b) the local baroclinic tidal displacement represented by the 3-h low-pass-filtered and linearly detrended 19°C isotherm elevation in 500-m depth. Note a-type events (black dots) conveniently correspond to odd numbers, and b-type events (red dots) correspond to even numbers. Event 12 (missing) was observed, though the mooring in 300-m depth was undergoing an instrument switch-over at the time and the event is thus excluded from further analysis.

  • Fig. 3.

    Temperature (color bar) vs scaled depth z/h and distance cpt at mooring locations corresponding to the 500-, 300-, 100-, 50-, 20-, and 10-m isobaths (columns). Rows correspond to events 10, 2 16, and 13 on 21, 17, 24, and 23 May, respectively. Temperature is contoured at 2°C. Note the expanded color bar limits to resolve colder water at the 500- and 300-m-deep locations.

  • Fig. 4.

    Displacements of the 19°C isotherm at h = 500 m for (a) event 13 (blue) compared with the sech2 KdV soliton (black), and (b) four wave profiles corresponding to event numbers in Fig. 2a (legend, colors) compared with the sech2 KdV soliton (black), all scaled by KdV amplitude η0.

  • Fig. 5.

    Histogram of event bottom temperature magnitude scaled by the mean bottom temperature magnitude at all locations, ΔTb/ΔTb¯. Bin edges are indicated by the top axis, and colors correspond to observations at depths h = 100 m (yellow), h = 50 m (green), h = 20 m (red), h = 10 m (blue). The scaled mean ΔTb/ΔTb¯ at each location is indicated by a dot with respect to the top axis.

  • Fig. 6.

    Cross-shore baroclinic velocity u′ (color) vs scaled depth z/h and distance cpt at the 50-, 20-, and 10-m isobaths during (a)–(c) b-type event 16 and (d)–(f) a-type event 13. Positive velocity is offshore, and the density contour (black) interval is 0.5 kg m−3 with ρ = 1029 kg m−3 highlighted for reference (bold black).

  • Fig. 7.

    Temperature (color bar) sampled along the DTS fiber-optic cable length xDTS indicated in Fig. 1c (positive offshore) vs time during (a) event 13 and (b) event 16. The wave front is defined when |dT/dt| > 0.01°C s−1 (black line). Approximate depths corresponding to M20, M10, and the surfzone are indicated for reference as dotted lines.