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Analyzing amplitudes of internal solitary waves in the northern South China Sea by use of seismic oceanography data
Author’s Accepted Manuscript Analyzing amplitudes of internal solitary waves in the northern South China Sea by use of seismic oceanography data Minghui Geng, Haibin Song, Yongxian Guan, Yang Bai www.elsevier.com
PII: DOI: Reference:
S0967-0637(18)30192-4 https://doi.org/10.1016/j.dsr.2019.02.005 DSRI3009
To appear in: Deep-Sea Research Part I Received date: 19 September 2017 Revised date: 10 February 2019 Accepted date: 22 February 2019 Cite this article as: Minghui Geng, Haibin Song, Yongxian Guan and Yang Bai, Analyzing amplitudes of internal solitary waves in the northern South China Sea by use of seismic oceanography data, Deep-Sea Research Part I, https://doi.org/10.1016/j.dsr.2019.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analyzing amplitudes of internal solitary waves in the northern South China Sea by use of seismic oceanography data Minghui Genga,b, Haibin Songa,*, Yongxian Guanb, Yang Baia a State Key laboratory of Marine Geology, College of Ocean and Earth Science, Tongji University, Shanghai 200092, China b Key Laboratory of Marine Mineral Resources, Ministry of Land and Resources, Guangzhou Marine Geological Survey, China Geological Survey, Guangzhou 510760, China * Corresponding author. Email: [email protected]. Address: School of Ocean and Earth Science, Tongji University, 1239 Siping Road, Yangpu District, Shanghai, 200092, P. R. China
Abstract: In the northern South China Sea, numerous multichannel seismic reflection sections are used to identify internal solitary wave (ISW) packets and extract wave amplitudes and corresponding water depths. The analyzed data show that these depression ISWs occur on the upper continental slope at water depths between 263 m and 740 m, with maximum amplitudes ranging from 35 m to 128 m. Our results, in conjunction with previous studies, suggest that the maximum amplitudes of the ISWs on the northern South China Sea continental slope are highly correlated with seafloor depths and that they have a logarithmic function relationship. The maximum amplitudes decrease with decreasing water depths. Interactions between the ISWs and the seafloor play a crucial role in decreasing ISW maximum amplitudes, especially in shallow areas. In addition, we compare the observed vertical amplitude distribution with theoretical results and find that they are concordant. The “bottom depth” (H) in the boundary conditions of eigenfunctions represents the extension depth of the ISW rather than the actual seafloor depth.
Here, the ISW extension depth is where the ISW amplitude becomes zero and the seafloor depth is just under the ISW. If the ISW interacts intensely with the seafloor, its observed vertical amplitude distribution may exhibit prominent differences from the theoretical result. Keywords: Internal solitary wave; amplitudes; water depths; eigenfunctions; northern South China Sea; seismic oceanography
1. Introduction Large-amplitude internal solitary waves (ISWs) occur frequently in the northern South China Sea (SCS), where these waves can have maximum amplitudes in excess of 200 m (Ramp et al., 2004; Klymak et al., 2006; Cai et al., 2012; Fu et al., 2012; Lien et al., 2014; Alford et al., 2010, 2015; Huang et al., 2016). Given the large number of ISWs in the northern SCS, this region has become a target area for research on large-amplitude ISWs. Over the past decade, field studies have been conducted in this area that have provided a comprehensive understanding of the generation, propagation, shoaling and breaking of ISWs (Klymak and Moum, 2003; Orr and Mignerey, 2003; Yang et al., 2004; Cheng and Hsu, 2010; Cai et al., 2012; Fu et al., 2012). The propagation of large-amplitude ISWs brings about a strong seawater convergence and sudden wave-induced currents, thus affecting the mixing process and marine ecosystems and threatening the safety of subsea oil and gas drilling operations (Jeans and Sherwin, 2001; Cai et al., 2003; Hyder et al., 2005, Alford et al., 2015; Cai et al., 2012). Previous studies of ISWs in the northern SCS have used various methods to derive the amplitudes of waves, including in situ mooring observations (Vlasenko et al., 2000; Alford et al., 2010; Lien et al., 2014), remote sensing techniques (Chen et al., 2011; Zha et al., 2012; Qian et al.,
2015), and numerical simulations (Vlasenko et al., 2000; Vlasenko et al., 2010; Guo and Chen, 2012; Rubino et al., 2001), and each method naturally has unique advantages and disadvantages. The in situ mooring observations can provide a detailed ISW amplitude distribution. The remote sensing techniques can provide a wealth of data for inferring ISW amplitudes, but the results have accuracy issues. The numerical simulation can trace the changes in an ISW amplitude during its evolution. More recently, multichannel seismic data have been used for the extraction of ISW amplitudes (Tang et al., 2014 and 2015; Bai et al., 2017). The method can map the thermohaline structure of water columns with an extensive spatiotemporal coverage, thus characterizing the internal wave structure; it also offers the possibility for gathering more vertical and horizontal information on ISW amplitudes in the northern SCS for further analysis. Here, a single ISW’s amplitudes at different water depths are extracted, known as the vertical distribution of the ISW amplitudes. The maximum amplitude changes versus seafloor depth are evaluated for different ISWs, which is called the horizontal distribution of ISW amplitudes. The ISW’s steady depression shape results from the balance between nonlinear and dispersive effects (Osborne and Burch, 1980), and it commonly exhibits depression waveforms in the deep basin (Alford et al., 2010). The large amplitude usually indicates great wave nonlinearity, and therefore, the ISWs are also called nonlinear internal waves. Amplitude is one of the most important parameters of the ISWs since it is of great significance to the global instability in the wave-induced boundary layer for the onset of sediment resuspension (Bogucki et al., 2005). Amplitude is also important due to it being one of the major factors determining the ISWs evolution on continental slopes (Vlasenko and Hutter, 2002; Guo and Chen, 2012; Bai et al., 2017). For maximum amplitudes and the horizontal distribution of the ISWs in the northern SCS, Alford
et al. (2010) detected 14 ISWs spanning from the Luzon Strait to the upper continental slope in the northern SCS. The wave amplitudes show a sharp increase as they move on to the upper continental slope because the decrease in the seafloor depth results in the amplification of the nonlinearity and the reduction of the nonhydrostatic dispersion effect of the ISWs (Alford et al., 2010). However, the studies performed to date do not provide enough information on the horizontal distribution of the ISW amplitude versus seafloor depth in shallow areas, and we know little about the seafloor’s impacts on ISW amplitude variations in their following inshore propagations. In previous studies, different models were used to investigate the vertical distribution of the single ISW amplitudes versus water depths (Fliegel and Hunkins, 1975; Vlasenko et al., 2000; Small and Hornby, 2005). Of these, only Vlasenko et al (2000) have compared the results of numerical simulations with in situ measurements, and the comparison shows a good agreement with the depth of maximum amplitude. However, there are several observed local maxima not captured by the models that are supposed to result from the superposition of different subscale ISWs (Vlasenko et al., 2000). In this study, our overall aims are to utilize numerous observed seismic oceanography sections and understand the vertical and horizontal distribution of ISW amplitudes as a function of either water or seafloor depths on the upper continental slope and shelf in the northern SCS. Section 2 describes our data acquisition process, how to extract ISW amplitudes from stacked and common offset gathers, and how to obtain theoretical results using eigenfunctions. The main results are then presented in section 3. These include 14 depression ISWs and the relationship between their maximum amplitudes and the corresponding seafloor depths. We also explain the
difference between the observed and theoretical vertical amplitude distribution of the single ISW amplitudes versus water depths.
2. Data and Method The study area is located in the northern SCS, across the shelf and upper slope area, at water depths of 100 m to 1000 m near Dongsha Island (Fig. 1). In this area, a set of 2D multichannel reflection seismic profiles were acquired by Guangzhou Marine Geological Survey (GMGS) in summer 2009 by using a 6000 m long streamer with 480 channels (trace interval 12.5 m and CMP interval 6.25 m), an air gun source with a total volume of 5080 in3 (1 in 3 =16.39 cm3), and a shot interval of 25 m. The sampling interval is 2 ms, and the dominant frequency of the seismic wavelet is 35 Hz, indicating a vertical resolution of approximately 10 m (average sound velocity 1500 ms-1).
Figure 1. Seismic sections L1 to L11 (red lines), S1 to S3 (black lines), and corresponding Argo
data A1 to A3 (black points). S1 and A1 were recorded on Jul 7th, S2 and A2 were recorded on Sep 20th and 21st, respectively, and S3 and A3 were recorded on Sep 19th. The seismic sections labeled as both L1 to L11 and S1 to S3 are used to evaluate the horizontal distributions of ISW maximum amplitudes versus the seafloor; the S1 to S3 sections are also used to extract the vertical distribution of individual ISW amplitudes versus water depths since they can provide extensively detailed ISW structures.
The multichannel reflection seismic dataset has been reprocessed using Seismic Unix to obtain seismic oceanography sections. Each section represents a “snapshot” of seawater column features. The data processing method for seismic oceanography is similar to the conventional processing of multichannel seismic data used in oil and gas exploration (e.g., Ruddick et al., 2009). The stacked sections of this dataset cannot provide all reflectors for the seawater column, as their shallow parts (water depth < 150 m, 1500 m/s acoustic velocity is used for time-to-depth conversion of the seismic sections) have been muted as part of the normal move out (NMO) correction. We use common offset gathers to provide the ISWs’ amplitudes in shallow water (between 50 m and 150 m). A detailed description of the method has been presented by Bai et al. (2017). For instance, the stacked section (Fig. 2a) and the common offset gather section (Fig. 2b) of L1 are used to extract the ISW amplitudes versus water depth (Fig. 2d). The Argo data are available at the Argo information center, and they are obtained in the corresponding time-space domain with the ISWs (Fig. 1). These corresponding hydrographic data are used to calculate the theoretical ISW amplitudes versus the water depths. In addition, bathymetric data in Fig. 1 are provided by the General Bathymetric Chart of the Oceans. The
profiles are generated with a spatial resolution of 1’ (approx. 1.85 km). Several nonlinear equations have been formulated for describing the evolution of ISWs in oceans, including the Korteweg–de Vries (KdV) equation, Boussinesq equation, Peregrine equation, Bona–Chen equation, shallow water wave equations and several others (Triki et al., 2013). The KdV type model is the most popular weakly nonlinear theory to govern the wave dynamics on the continental margin (Osborne and Burch, 1980; Duda et al., 2004; Liu et al., 2005; Liu et al., 2013). Then ISWs vertical structures can be determined by the solution of eigenfunctions presented by Fliegel and Hunkins (1975). The eigenfunctions
( ) satisfy the Taylor-Golstein Boundary Value
Problem (Fliegel and Hunkins, 1975; Liu et al., 1985; Pelinovksy et al., 1995; Apel et al., 2006; Zha et al., 2012): ( )
( ) where and
+ (
( )
)
( )
( ) ( )
( ) represents an ISW’s vertical amplitude distribution, C0 is the linear long wave speed, ( (𝑔 𝜌)/(𝜌
))1/ is the Brunt-Väisälä or buoyancy frequency.
is the bottom
depth. Eq. 1 is the governing differential equation with the boundary conditions Eq. 2. Based on Eq. 2, the top and bottom are hypothesized as rigid and the ISW amplitude at the boundaries is zero (Fliegel and Hunkins, 1975). Eq. 1 can be solved by the Thomson-Haskell method. A detailed solution is presented by Fliegel and Hunkins (1975) and Zha et al. (2012).
Figure 2. (a) Stacked section, (b) common offset section, (c) picked shallow ISW reflections after NMO corrections and ISW’s amplitude definition, and (d) vertical amplitude distribution versus water depths of L1. The red lines indicate the picks of the ISW reflection events. The amplitudes are combined from stacked and common offset sections. 3. Result We have identified a total of 14 ISW packets on the seismic sections L1 to L11 and S1 to S3,
and the method described in section 2 is utilized to get the waves’ amplitudes versus water depths. Fig. 2a shows a single-wave packet containing one typical soliton with an oscillating tail. The red lines are picked reflection events delineating a typical depression wave identified on the stacked section of L1, whereas the shallow part of the wave is presented as a common offset gather section in Fig. 2b. These two sections are combined to determine the L1 wave amplitudes versus water depths. As shown in Fig. 2c, we define the mean depth of the front and rear of the soliton as the water depth. The distance between the mean depth and the lowest point depth of the same reflection event is defined as the wave amplitude. Fig. 2d shows that the L1 ISW amplitudes first increase sharply and reach their maxima of 128 m in the middle layer at a depth of 142 m, and then decrease slowly with increasing water depths. The L1 ISW belongs to the mode-1 wave since its amplitude curve only has one extreme point. Figs. 3 and 4 similarly depict two ISW packets composed of a group of solitons. Each one has a dominant wave with the maximum amplitude. The dominant soliton is selected for our analysis as it may be the leading wave or others, which is indicated by the black arrow on the seismic sections. As shown in Fig. 5, all depression ISWs belong to the mode-1 wave because their vertical amplitude distributions only have one extreme point. Theoretically, the mode-1 ISW amplitudes should increase first and then decrease with the increasing water depth. The shape of the vertical distribution should be similar to Fig. 2d. We select the distribution’s extreme as the ISW maximum amplitude. However, the vertical amplitude distributions of L4, L5, L7, and L9 all exhibit monotone decreases versus water depth. In these cases, we still select the curve maximum as the ISW maximum amplitude because this is the clearest, most objective information provided
by the seismic data since they are unable to provide further shallow depth information for the ISW. As shown in Table 1, these L1-L11 depression ISWs are located at water depths between 263 m and 597 m on the northern SCS upper continental slope, and their maximum amplitudes range from 60 m to 128 m.
Figure 3. Stacked section of L4. The selected ISW is indicated by the black arrow. The yellow line represents the seafloor.
Figure 4. Stacked section of L6. The selected ISW is indicated by the black arrow. The yellow line
represents the seafloor.
Figure 5. The observed vertical amplitude distributions of L2-L11 ISW amplitudes versus water depths. The shaded rectangles indicate the area below the seafloor depths.
Seismic Line
ISW maximum amplitude /m
Seafloor depth /m
L1
128
510
L2
61
442
L3
78
318
L4
104
331
L5
82
341
L6
60
402
L7
70
300
L8
97
597
L9
90
365
L10
69
348
L11
66
263
S1
35
345
S2
91
740
S3
72
405
Table 1. Maximum amplitudes and corresponding seafloor depths of ISWs identified on sections L1 to L11 and S1 to S3.
We select 3 mode-1 ISWs to compare between the observed and theoretical vertical profiles of ISW amplitudes. These ISWs are labeled S1 to S3 and are shown in Figs. 8 to 10. All three ISWs exhibit abundant reflection events on the seismic sections, meaning we can extract detailed amplitudes. The major difference between them is their waveforms near the seafloor. The displacements of the S1 ISW reflectors first increase, then gradually decrease with the water depths, and finally just show a flat reflection event near the seafloor (Fig. 6). The amplitude of the S2 ISW decreases to zero at a distance from the seafloor. As shown in Fig. 7, the seafloor depth
below the S2 ISW is 730 m, whereas there are only flat reflection events below a water depth of 600 m. The S3 ISW extends all the way to the seafloor. As shown in Fig. 8, the reflection event at the ISW bottom is undulating and not parallel to the seafloor as the ideal example shown in Fig. 6. The ISW appears to have its bottom “cut” by the seafloor. For the S1 ISW, the corresponding seafloor depth is 354 m. We use corresponding hydrographic data (A1, Fig. 6c) to obtain the theoretical vertical amplitudes versus water depths of the S1 ISW (Fig. 9a). The black line represents the observed results, and the vertical amplitude distribution only has one extreme point, thus indicating that the S1 ISW is a mode-1 wave. The theoretical amplitude distribution (red line in Fig. 9a) from the eigenfunctions shows that the mode-1 ISW amplitude starts from zero at the sea surface, then positively correlates with the water depth and reaches the maximum (calibrated by the observed maximum amplitude) at a certain depth; after that depth is reached, the amplitude begins to decrease, and is finally reduced to zero at the seafloor. Although the amplitude at shallower depths is unavailable due to the already described data limitations, we still find a concordance between the vertical distributions of the observed S1 ISW amplitudes extracted from our seismic oceanography section and theoretical values from the eigenfunctions. This makes the S1 ISW an ideal case. For the S2 ISW, the hydrographic data of point A2 are shown in Fig. 7c and have temporal and geographical proximities with the seismic section of S2. We use these corresponding hydrographic data in Eq. 2 and obtain two theoretical amplitude distributions by using a different “bottom depth” (H) of the ISW extension depth at 600 m and the seafloor depth at 730 m. Fig. 9b clearly shows that the theoretical amplitude distribution using the ISW’s extension depth of 600 m is more consistent with the observed values.
Fig. 8 shows a case of the ISW’s amplitude having nonzero values near the seafloor. The S3 ISW amplitude does not decrease to zero close to the seafloor. The corresponding seafloor depth is 400 m, and the corresponding hydrographic data are shown in Fig. 8c. As shown in Fig. 9c, the vertical amplitude distribution of the S3 ISW amplitudes shows a similar trend as the S1 ISWs. As the water depth increases, the wave amplitude increases first and then decreases. Its theoretical and observed amplitude distributions trends are approximately similar, and their basic distribution is consistent. However, compared to the previous two cases, there are prominent differences: the observed amplitude distribution is generally offset upward from the theoretical one; and the depth of the maximum observed amplitude is 26 m less than the theoretical value.
Figure 6. (a) Stacked and (b) common offset gather sections of the ISW identified in profile S1, and (c) temperature, salinity, and the calculated density and Brunt-Väisälä frequency profiles of A1 points. Refer to Fig. 1 for location and collecting date. The amplitude at the seafloor is zero (indicated by the red arrow).
Figure 7. (a) Stacked and (b) common offset gather sections of the profiled ISW identified on S2, and (c) the temperature, salinity, and the calculated density and Brunt-Väisälä frequency profiles of the A2 points. Refer to Fig. 1 for location and collecting date. The extension depth is shallower than the seafloor.
Figure 8. (a) Stacked and (b) common offset gather sections of profile S3 ISW, and (c) the temperature, salinity, and the calculated density and Brunt-Väisälä frequency profiles of the A3 points. Refer to Fig. 1 for location and collecting date. The ISWs are at a nonzero amplitude near the seafloor.
Figure 9. Comparison of the observed and theoretical vertical amplitude distributions of S1 (a), S2 (b), and S3 (c) ISW amplitudes versus water depths. The black lines represent the observed data, and the red lines represent the theoretical amplitude distribution. In panel (b), the red line represents the theoretical amplitude distribution acquired using the ISW extension depth (H = 600 m), and the blue line represents the theoretical amplitude distribution acquired using seafloor depth (H =730 m).
4. Discussion 4.1 The horizontal distribution of the ISWs maximum amplitudes versus seafloor depths Fig. 10 presents all samples’ amplitudes versus water depth, including the elevation and transition waves in the study area parameterized by Bai et al. (2017) and the results from previous studies (e.g., Klymak and Moum, 2003; Orr and Mignerey, 2003; Cheng and Hsu, 2010; Fu et al., 2012). As shown in Tables 1 and 2, the observed ISWs are located at water depths between 87 m and 740 m on the northern SCS continental slope and shelf and have maximum amplitudes ranging from 13 m to 128 m. In the deep ocean, the ISWs’ maximum amplitudes are commonly large and widely distributed. These ISWs westward-propagating away from the Luzon strait arise from the interactions between the tidal flow and topography at the generation site, and they are not affected by deep topography variations (Liu et al., 1998; Zhao et al., 2004; Alford et al., 2010). The magnitude of their amplitudes mainly depends on their energy source and the structure of ocean stratification, exhibiting cyclical variations (Warn-Varnas et al., 2010; Xie et al., 2014). Once these ISWs enter the shallow northern SCS continental margin, they are strongly affected by the
shoaling topography. As the seafloor gets shallower, the friction between the ISW and the seafloor will bring about large amounts of energy dissipation (Chang et al., 2006; St. Laurent, 2008). Especially for large-amplitude ISWs, intense interactions with the seafloor occur. Their amplitudes will decrease more rapidly than those of smaller amplitude ISWs; hence, the distribution range of the maximum amplitudes will concentrate as the seafloor depth gets shallower. As shown in Fig. 10, the upper envelope is the connection line of these ISWs’ maximum amplitudes, and it indicates the maximum amplitude of the allowed ISWs at a certain seafloor depth. Due to limited data, the upper envelope can change if more data are added. The ISW maximum amplitude is highly correlated to the seafloor depth on the northern SCS margin. We have found a highly correlated logarithmic function relationship between all ISWs’ maximum amplitudes and water depths: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒
7.47 ∙ ln( .3375 ∙ 𝑒𝑝𝑡ℎ 87 ≤ 𝑒𝑝𝑡ℎ ≤ 74
7.3 48)
(3)
Elevation ISWs: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒 .3593 ∙ 𝑒𝑝𝑡ℎ 5. 4 𝑒𝑝𝑡ℎ ∈ [93.5 3 . ] Transition ISWs: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒 .63 ∙ 𝑒𝑝𝑡ℎ 45.85 9 𝑒𝑝𝑡ℎ ∈ [ 36.5 .4] Depression ISWs: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒 . 7 6 ∙ 𝑒𝑝𝑡ℎ + 49.4 6 𝑒𝑝𝑡ℎ ∈ [ 63 74 ]
(4)
where ln is the log base e (natural logarithm), and the water depths range from 87 to 740 m. Accounting for the different types of ISWs, three linear correlations Eq. 4 are found between the ISWs maximum amplitudes and water depths. The depth domains of definition are discrete due to the limited data samplings. Inferred from Eq. 4, the linear coefficient for the elevation waves in shallow water is the largest one, whereas the smallest coefficient is valid for the deepest depression waves. With a decrease in seafloor depth, the decrease rate of wave maximum amplitude will accelerate. This suggests that in a shallow area, the seafloor depth will play a more important role as a constraint for the maximum amplitudes than in deep waters. Using the
maximum amplitudes-seafloor depths relationship determined by Eqs. 3 and 4, the maximum ISW amplitudes can thus be predicted. As presented by Alford et al. (2010), from the Luzon Strait to the continental slope, a decrease in seafloor depth results in the amplification of the nonlinearity and the reduction of the nonhydrostatic dispersion effect of the ISWs, and the wave amplitudes show a sharp increase on the upper continental slope. However, our research demonstrates that from the upper continental slope to the shelf, the shoaling topography is likely to have opposite effects on wave amplitude variations and the ISW maximum amplitudes decrease as the seafloor is getting shallower. Hence, in the northern SCS, the largest-amplitude ISW will occur around the upper continental slope.
Data sources
ISW types
Maximum amplitude /m
Seafloor depth /m
Bai et al.
Depression
71.3
304.2
Bai et al.
Elevation
49.5
126.6
Bai et al.
Elevation
48.5
124.4
Bai et al.
Elevation
14.7
86.6
Bai et al.
Elevation
12.7
98.5
Bai et al.
Transition
58.4
179.3
Bai et al.
Transition
92.7
214.1
Bai et al.
Transition
51.0
150.4
Orr and Mignerey
Elevation
45.4
117.2
Orr and Mignerey
Transition
53
169.8
Orr and Mignerey
Transition
54.1
152
Orr and Mignerey
Transition
45.1
135.8
Klymak and Moum
Elevation
29
116
Cheng and Hsu
Elevation
62.8
127.3
Cheng and Hsu
Transition
88
221.7
Fu et al.
Elevation
50.2
124.2
Fu et al.
Elevation
41.5
102.1
Fu et al.
Transition
72.2
173.6
Table 2. ISW maximum amplitude and corresponding seafloor depth from previous studies (e.g., Klymak and Moum, 2003; Orr and Mignerey, 2003; Cheng and Hsu, 2010; Fu et al., 2012; Bai et al., 2017)
Figure 10. Horizontal distribution of ISW maximum amplitudes versus seafloor depth on the northern SCS continental slope. The triangles represent elevation ISWs, the circles represent transition ISWs, and the squares represent depression ISWs. The dashed black lines are the upper envelopes. The fine solid black line is the fitted amplitude distributions for all ISWs, and the bold solid black lines are the fitted curves for the depression, transition, and elevation ISWs.
4.2 Comparison between the observed and theoretical vertical amplitude distributions of single ISW amplitudes versus water depths As described in section 3, the comparisons between the observed and theoretical vertical amplitude distributions of the S1, S2, and S3 ISWs show very different results. The S1 ISW amplitude happens to be zero at the seafloor, and its observed and theoretical values are in strong concordance. The S2 wave amplitude decreases to zero above the seafloor. Its theoretical vertical amplitude distribution calculated using the ISW’s extension depth is noticeably more consistent with the observed result than using the corresponding seafloor depth. As observed for the S3 ISW, its bottom appears to be “cut” by the seafloor, and there are prominent differences between the theoretical and observed results. The theoretical vertical distribution of single ISW amplitudes is defined by the eigenfunctions
( ), which satisfy the Taylor-Golstein Boundary Value Problem (Eqs. 1 and 2).
Fliegel and Hunkins (1975) proposed that this governing differential equation (Eq. 1) is valid for frictionless motion in a stratified fluid with a rigid top and bottom. Although we supplement common offset gather sections to delineate the ISWs’ shallow parts, the amplitudes shallower than 50 m deep are still not obtained due to the NMO correction processing step. We shall assume real ocean conditions with a rigid top and that the ISW amplitude at the sea surface is zero. In addition, the boundary condition presented by Eq. 2 requires the wave amplitude at the bottom to be zero. The S2 ISW comparison result indicates that the “bottom” represents the ISW extension depth rather than seafloor. It is possible that some ISWs such as S1 happen to extend to the seafloor. As an ISW comes into shallow waters, the interaction between the ISW and the seafloor will be strong and will lead to changes in the wave’s vertical amplitude distribution. The seafloor in this
case is no longer a rigid bottom. We propose that the seafloor depth does not equal the bottom boundary depth used in the eigenfunction calculation. The seafloor topography may have a huge impact on the shoaling ISWs’ amplitudes in the shallow area, especially when estimating their vertical amplitude distributions.
5. Conclusions In this paper, we use seismic oceanography data to extract ISW amplitudes and their corresponding water depths on the upper continental slope and shelf in the northern SCS. We emphasize the horizontal and vertical distributions of the ISW amplitudes versus water depths. The results suggest the following: (1) An ISW’s maximum amplitude is highly correlated to the seafloor depth. In general, in the northern SCS at a seafloor depth of 87 to 740 m, the shallower the seafloor depth, the smaller is the ISW maximum amplitude. A highly correlated logarithmic function relationship is found between all types of ISW maximum amplitudes and seafloor depths. For existing depression, transition, and elevation ISWs, three different linear correlations exist between the ISWs’ maximum amplitudes and the seafloor depth. The elevation waves have the largest linear coefficient, indicating that decreasing water depths have the strongest impact on them. (2) In general, the observed vertical distribution of ISW amplitudes versus water depth shows a strong concordance with the theoretical result. The eigenfunctions can accurately describe the vertical distribution of the ISW amplitudes versus water depths. However, the eigenfunctions cannot govern the full depth range, and the bottom depth in the boundary conditions should be the ISW extension depth rather than the seafloor depth. When estimating the vertical amplitude
distributions of the ISWs that interact with the seafloor on the continental slope, the observed amplitude distribution may not be perfectly consistent with the theoretical result because there are other ocean processes existing in the area. Thus, we have determined that topography plays an important role in an ISW’s evolution when it comes into a shallow area. The interaction between the seafloor and the ISW will alter the wave’s structure, and release a large amount of energy, which will erode the seafloor, strengthen water mixing, and pose a safety hazard to oceanic engineering. The mixing caused by the ISW dissipation can drive nutrients from the deep water into the surface layer and further the local marine ecosystems.
Funding: This work was supported by the National Natural Science Foundation of China (Grant nos. 41576047, 41576054) and the project for CAS Interdisciplinary Innovation Team.
Acknowledgements Bathymetry data are provided by General Bathymetric Chart of the Oceans (GEBCO, http://www.gebco.net/). Argo data are available from the Argo information center (http://www.jcommops.org/argo). The two free open source software, Seismic Unix (SU, http://www.cwp.mines.edu/cwpcodes/) and Generic Mapping Tools (GMT, http://gmt.soest.hawaii.edu/), are appreciated for supporting the data processing and mapping. The helpful suggestions of four anonymous reviewers are gratefully acknowledged.
References Alford, M. H., Lien, R. C., Simmons, H., Klymak, J., Ramp, S., Yang, Y. J., Tang, D., Chang, M. H., 2010. Speed and evolution of nonlinear internal waves transiting the South China Sea. Journal of Physical Oceanography, 40, 1338-1355. http://dx.doi.org/10.1175/2010JPO4388.1. Alford, M. H., Peacock, T., MacKinnon, J. A., Nash, J. D., Buijsman, M. C., Centuroni, L. R., Chao, S. Y., Chang, M. H., Farmer, D. M., Fringer, O. B., Fu, K. H., Gallacher, P. C., Graber, H. C., Helfrich, K. R., Jachec, S. M., Jackson, C. R., Klymak, J. M., Ko, D. S., Jan, S., Johnston, T. M. S., Legg, S., Lee, I. H., Lien, R. C., Mercier, M. J., Moum, J. N., Musgrave, R., Park, J. H., Pickering, A. I., Pinkel, R., Rainville, L., Ramp, S. R., Rudnick, D. L., Sarkar, S., Scotti, A., Simmons, H. L., St. Laurent, L. C., Venayagamoorthy, S. K., Wang, Y. H., Wang, J., Yang, Y. J., Paluszkiewicz, T., Tang T. Y. , 2015. The formation and fate of internal waves in the South China Sea. Nature, 521, 65-69. http://dx.doi.org/10.1038/nature14399. Apel, J. R., Ostrovsky, L. A., Stepanyants, Y. A., Lynch, J. F., 2006. Internal solitons in the ocean. Technical Report, Woods Hole Oceanography Institution. https://darchive.mblwhoilibrary.org/handle/1912/1070. Bai, Y., Song, H. B., Guan, Y. X., Yang, S. X., 2017. Estimating depth of polarity conversion of shoaling internal solitary waves in the northeastern South China Sea. Continental Shelf Research,143, 9-17. https://doi.org/10.1016/j.csr.2017.05.014. Bogucki, D. J., Redekopp, L. G., Barth, J., 2005. Internal solitary waves in the Coastal Mixing and Optics 1996 experiment: Multimodal structure and resuspension. Journal of Geophysical Research Oceans, 110, C02024. https://doi.org/10.1029/2003JC002253. Cai, S. Q., Long, X. M., Gan, Z. J., 2003. A method to estimate the forces exerted by internal
solitons on cylindrical piles. Ocean Engineering, 33 (5), 673- 689. https://doi.org/10.1016/S0029-8018(02)00038-0. Cai, S. Q., Xie, J. S., He, J. L., 2012. An overview of internal solitary waves in the South China Sea. Surveys in Geophysics, 33(5), 927-943. http://dx.doi.org/10.1007/s10712-012-9176-0. Chang, M. H., Lien, R. C., Tang, T.Y., D’Asaro, E. A., Yang, Y. J., 2006. Energy flux of nonlinear internal waves in northern South China Sea. Geophysical Research Letters, 33, L03607. http://dx.doi.org/10.1029/2005GL025196. Chen, G. Y., Su, F. C., Wang, C. M., Liu, C. T., Tseng, R. S., 2011. Derivation of internal solitary wave amplitude in the South China Sea deep basin from satellite images. Journal of Oceanography, 67(6), 689–697. http://dx.doi.org/10.1007/s10872-011-0073-9. Cheng, M. H., Hsu, J. R. C., 2010. Laboratory experiments on depression interfacial solitary waves over a trapezoidal obstacle with horizontal plateau. Ocean Engineering, 37(8-9), 800-818. http://dx.doi.org/10.1016/j.oceaneng.2010.02.016. Duda, T. F., Lynch, J. F., Irish, J. D., Beardsley, R. C., Ramp, S. R., Chiu, C. S., Tang, T. Y., Yang, Y. J., 2004. Internal tide and nonlinear internal wave behavior at the continental slope in the northern South China Sea. IEEE Journal of Oceanic Engineering, 29(4), 1105-1130. http://dx.doi.org/10.1109/JOE.2004.836998 Fliegel, M., Hunkins, K., 1975. Internal wave dispersion calculated using the Thomson-Haskell method. Journal of Physical Oceanography, 5(3), 541-548. http://dx.doi.org/10.1175/1520-0485(1975)005<0541:IWDCUT>2.0.CO;2. Fu, K. H., Wang, Y. H., Laurent, L. S., Simmons, H., Wang, D. P., 2012. Shoaling of large-amplitude nonlinear internal waves at Dongsha Atoll in the northern South China Sea.
Continental Shelf Research, 37(37), 1-7. http://dx.doi.org/10.1016/j.csr.2012.01.010. Guo, C., Chen, X., 2012. Numerical investigation of large amplitude second mode internal solitary waves over a slope-shelf topography. Ocean Modelling, 42(1), 80-91. http://dx.doi.org/10.1016/j.ocemod.2011.11.003. Huang, X. D., Chen, Z. H., Zhao, W., Zhang, Z. W., Zhou, C., Yang, Q. X. Tian, J. W., 2016. An extreme internal solitary wave event observed in the northern South China Sea. Scientific Reports, 6, 30041. http://dx.doi.org/10.1038/srep30041. Hyder, P., Jeans, D. R. G., Cauquil, E., Nerzic, R., 2005. Observations and predictability of internal solitons in the northern Andaman Sea. Applied Ocean Research, 27(1), 1-11. http://dx.doi.org/10.1016/j.apor.2005.07.001. Jeans, D. R. G., Sherwin, T. J., 2001. The evolution and energetics of large amplitude nonlinear internal waves on the Portuguese shelf. Journal of Marine Research, 59(3), 327-353. http://dx.doi.org/10.1357/002224001762842235. Lien, R. C., Henyey, F., Ma, B., Yang, Y. J., 2014. Large-amplitude internal solitary waves observed in the northern South China Sea: Properties and energetics. Journal of Physical Oceanography, 44 (4), 1095-1115. http://dx.doi.org/10.1175/JPO-D-13-088.1. Liu, A. K., Holbrook, J. R., Apel, J. R., 1985. Nonlinear internal wave evolution in the Sulu Sea. Journal of Physical Oceanography, 15 (12), 1613-1624. http://dx.doi.org/10.1175/1520-0485(1985)015<1613:NIWEIT>2.0.CO;2. Liu, A. K., Su, F. C., Hsu, M. K., Kuo, N. J., Ho, C. R., 2013. Generation and evolution of mode-two internal waves in the South China Sea. Continental Shelf Research, 59(59), 18-27. http://dx.doi.org/10.1016/j.csr.2013.02.009.
Liu, A. K., Zhao, Y. H., Hsu, M. K., 2005. Nonlinear Internal Wave Study in the South China Sea. Advances In Engineering Mechanics – Reflections And Outlooks: In Honor of Theodore Y-T Wu, 297-313. Klymak, J. M., Moum, J. N., 2003. Internal solitary waves of elevation advancing on a shoaling shelf. Geophysical Research Letters, 30(20), 2045. http://dx.doi.org/10.1029/2003gl017706. Klymak, J. M., Pinkel, R., Liu, C. T., Liu, A. K., David, L., 2006. Prototypical solitons in the South China Sea. Geophysical Research Letters, 33, L11607. http://dx.doi.org/10.1029/2006GL025932. Orr, M. H., Mignerey, P. C., 2003. Nonlinear internal waves in the South China Sea: Observation of the conversion of depression internal waves to elevation internal waves. Journal of Geophysical Research-Oceans, 108(C3), 3064. http://dx.doi.org/10.1029/2001jc001163. Osborne, A. R., Burch, T. L., 1980. Internal solitons in the Andaman Sea. Science, 208(2), 451-460. http://dx.doi.org/10.1126/science.208.4443.451. Pelinovksy, E., Tlipova, T., Ivanov, V., 1995. Estimations of the nonlinear properties of the internal wave field off the Israel coast. Nonlinear Processes in Geophysics, 2 (2), 80-88. http://dx.doi.org/10.5194/npg-2-80-1995. Qian, H. B., Huang, X. D., Tian, J. W., Zhao, W., 2015. Shoaling of the internal solitary waves over the continental shelf of the northern South China Sea. Acta Oceanologica Sinica, 34(9), 35-42. http://dx.doi.org/10.1007/s13131-015-0734-4. Ramp, S. R., Tang, T. Y., Duda, T. F., Lynch, J. F., Liu, A. K., Chiu, C. S., Bahr, F. L., Kim, H. R., Yang, Y. J., 2004. Internal solitons in the northeastern South China Sea. Part I: sources and deep water propagation. IEEE Journal of Oceanic Engineering, 29(4), 1157-1181.
http://dx.doi.org/10.1109/JOE.2004.840839 Rubino, A., Brandt, P., Weigle, R., 2001. On the dynamics of internal waves in a nonlinear, weakly nonhydrostatic three-layer ocean. Journal of Geophysical Research Oceans, 106(C11), 26899-26916. http://dx.doi.org/10.1029/2001JC000958 Ruddick, B., Song, H. B., Dong, C. Z., Pinheiro, L., 2009. Water column seismic images as maps of temperature gradient. Oceanography, 22(1), 192-205. http://dx.doi.org/10.5670/oceanog.2009.19. Small, R. J., Hornby, R. P., 2005. A comparison of weakly and fully non-linear models of the shoaling of a solitary internal wave. Ocean Modelling, 8(4), 395-416. https://doi.org/10.1016/j.ocemod.2004.02.002. St. Laurent, L., 2008. Turbulent dissipation on the margins of the South China Sea. Geophysical Research Letters, 35, L23615. https://doi.org/10.1029/2008GL035520. Tang, Q. S., Wang, C. X., Wang, D. X., Pawlowicz, R., 2014. Seismic, satellite, and site observations of internal solitary waves in the NE South China Sea. Scientific Reports, 4, 5374. http://dx.doi.org/10.1038/srep05374. Tang, Q. S., Hobbs, R., Wang, D. X., Sun, L. T., Zheng, C., Li, J. B., Dong, C. Z., 2015. Marine seismic observation of internal solitary wave packets in the northeast South China Sea. Journal of Geophysical Research: Oceans, 120(12), 8487-8503. http://dx.doi.org/10.1002/2015JC011362. Triki, H., Milovic, D., Biswas, A., 2013. Solitary waves and shock waves of the KdV6 equation. Ocean Engineering, 73(8), 119-125. http://dx.doi.org/10.1016/j.oceaneng.2013.09.001. Vlasenko, V., Brandt, P., Rubino, A., 2000. Structure of Large-Amplitude Internal Solitary Waves.
Journal of Physical Oceanography, 30(9), 2172-2185. http://dx.doi.org/10.1175/1520-0485(2000)030<2172:SOLAIS>2.0.CO;2. Vlasenko, V., Hutter, K., 2002. Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography. Journal of Physical Oceanography, 32, 1779-1793. http://dx.doi.org/10.1175/1520-0485(2002)032<1779:NEOTBO>2.0.CO;2. Vlasenko, V., Stashchuk, N., Guo, C., Chen, X., 2010. Multimodal structure of baroclinic tides in the South China Sea. Nonlinear Processes in Geophysics, 17(5), 529-543. http://dx.doi.org/10.5194/npg-17-529-2010. Warn-Varnas, A., Hawkins, J., Lamb, K. G., Piacsek, S., Chin-Bing, S., King, D., Burgos, G., 2010. Solitary wave generation dynamics at Luzon Strait. Ocean Modelling, 31(1–2), 9-27. http://dx.doi.org/10.1016/j.ocemod.2009.08.002 Xie, J. S., Chen, Z. W., Xu, J. X., Cai, S. Q., 2014. Effect of vertical stratification on characteristics and energy of nonlinear internal solitary waves from a numerical model. Communications in Nonlinear Science and Numerical Simulation, 19(10), 3539-3555. http://dx.doi.org/10.1016/j.cnsns.2014.03.007. Yang, Y. J., Tang, T. Y., Chang, M. H., Liu, A. K., Hsu, M. K., Ramp, S. R., 2004. Solitons northeast of Tung-Sha Island during the ASIAEX pilot studies. IEEE Journal of Oceanic Engineering, 29(4), 1182-1199. http://dx.doi.org/10.1109/JOE.2004.841424. Zha, G. Z., Lin, M. S., Shen, H., He, Y. J., Lv, H. B., 2012. Extraction of internal wave amplitude from nautical X-Band radar observations. Chinese Journal of Oceanology and Limnology, 30(3), 497-505. http://dx.doi.org/10.1007/s00343-012-1113-z. Zhao, Z. X., Klemas, V., Zheng, Q. A., Yan, X. H., 2004. Remote sensing evidence for baroclinic
tide origin of internal solitary waves in the northeastern South China Sea. Geophysical Research Letters, 31(6), L06302. http://dx.doi.org/10.1029/2003GL019077.
Highlights
Fourteen individual internal solitary waves have been identified in the northern South China Sea from multi-channel seismic data, with maximum amplitudes ranging from 35 to 128 m and corresponding water depths from 263 to 740 m. A logarithmic correlation between maximum amplitude and corresponding depth has been observed. The individual ISW's amplitude distribution over depth can be best described by the solution of eigenfunctions.
PII: DOI: Reference:
S0967-0637(18)30192-4 https://doi.org/10.1016/j.dsr.2019.02.005 DSRI3009
To appear in: Deep-Sea Research Part I Received date: 19 September 2017 Revised date: 10 February 2019 Accepted date: 22 February 2019 Cite this article as: Minghui Geng, Haibin Song, Yongxian Guan and Yang Bai, Analyzing amplitudes of internal solitary waves in the northern South China Sea by use of seismic oceanography data, Deep-Sea Research Part I, https://doi.org/10.1016/j.dsr.2019.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analyzing amplitudes of internal solitary waves in the northern South China Sea by use of seismic oceanography data Minghui Genga,b, Haibin Songa,*, Yongxian Guanb, Yang Baia a State Key laboratory of Marine Geology, College of Ocean and Earth Science, Tongji University, Shanghai 200092, China b Key Laboratory of Marine Mineral Resources, Ministry of Land and Resources, Guangzhou Marine Geological Survey, China Geological Survey, Guangzhou 510760, China * Corresponding author. Email: [email protected]. Address: School of Ocean and Earth Science, Tongji University, 1239 Siping Road, Yangpu District, Shanghai, 200092, P. R. China
Abstract: In the northern South China Sea, numerous multichannel seismic reflection sections are used to identify internal solitary wave (ISW) packets and extract wave amplitudes and corresponding water depths. The analyzed data show that these depression ISWs occur on the upper continental slope at water depths between 263 m and 740 m, with maximum amplitudes ranging from 35 m to 128 m. Our results, in conjunction with previous studies, suggest that the maximum amplitudes of the ISWs on the northern South China Sea continental slope are highly correlated with seafloor depths and that they have a logarithmic function relationship. The maximum amplitudes decrease with decreasing water depths. Interactions between the ISWs and the seafloor play a crucial role in decreasing ISW maximum amplitudes, especially in shallow areas. In addition, we compare the observed vertical amplitude distribution with theoretical results and find that they are concordant. The “bottom depth” (H) in the boundary conditions of eigenfunctions represents the extension depth of the ISW rather than the actual seafloor depth.
Here, the ISW extension depth is where the ISW amplitude becomes zero and the seafloor depth is just under the ISW. If the ISW interacts intensely with the seafloor, its observed vertical amplitude distribution may exhibit prominent differences from the theoretical result. Keywords: Internal solitary wave; amplitudes; water depths; eigenfunctions; northern South China Sea; seismic oceanography
1. Introduction Large-amplitude internal solitary waves (ISWs) occur frequently in the northern South China Sea (SCS), where these waves can have maximum amplitudes in excess of 200 m (Ramp et al., 2004; Klymak et al., 2006; Cai et al., 2012; Fu et al., 2012; Lien et al., 2014; Alford et al., 2010, 2015; Huang et al., 2016). Given the large number of ISWs in the northern SCS, this region has become a target area for research on large-amplitude ISWs. Over the past decade, field studies have been conducted in this area that have provided a comprehensive understanding of the generation, propagation, shoaling and breaking of ISWs (Klymak and Moum, 2003; Orr and Mignerey, 2003; Yang et al., 2004; Cheng and Hsu, 2010; Cai et al., 2012; Fu et al., 2012). The propagation of large-amplitude ISWs brings about a strong seawater convergence and sudden wave-induced currents, thus affecting the mixing process and marine ecosystems and threatening the safety of subsea oil and gas drilling operations (Jeans and Sherwin, 2001; Cai et al., 2003; Hyder et al., 2005, Alford et al., 2015; Cai et al., 2012). Previous studies of ISWs in the northern SCS have used various methods to derive the amplitudes of waves, including in situ mooring observations (Vlasenko et al., 2000; Alford et al., 2010; Lien et al., 2014), remote sensing techniques (Chen et al., 2011; Zha et al., 2012; Qian et al.,
2015), and numerical simulations (Vlasenko et al., 2000; Vlasenko et al., 2010; Guo and Chen, 2012; Rubino et al., 2001), and each method naturally has unique advantages and disadvantages. The in situ mooring observations can provide a detailed ISW amplitude distribution. The remote sensing techniques can provide a wealth of data for inferring ISW amplitudes, but the results have accuracy issues. The numerical simulation can trace the changes in an ISW amplitude during its evolution. More recently, multichannel seismic data have been used for the extraction of ISW amplitudes (Tang et al., 2014 and 2015; Bai et al., 2017). The method can map the thermohaline structure of water columns with an extensive spatiotemporal coverage, thus characterizing the internal wave structure; it also offers the possibility for gathering more vertical and horizontal information on ISW amplitudes in the northern SCS for further analysis. Here, a single ISW’s amplitudes at different water depths are extracted, known as the vertical distribution of the ISW amplitudes. The maximum amplitude changes versus seafloor depth are evaluated for different ISWs, which is called the horizontal distribution of ISW amplitudes. The ISW’s steady depression shape results from the balance between nonlinear and dispersive effects (Osborne and Burch, 1980), and it commonly exhibits depression waveforms in the deep basin (Alford et al., 2010). The large amplitude usually indicates great wave nonlinearity, and therefore, the ISWs are also called nonlinear internal waves. Amplitude is one of the most important parameters of the ISWs since it is of great significance to the global instability in the wave-induced boundary layer for the onset of sediment resuspension (Bogucki et al., 2005). Amplitude is also important due to it being one of the major factors determining the ISWs evolution on continental slopes (Vlasenko and Hutter, 2002; Guo and Chen, 2012; Bai et al., 2017). For maximum amplitudes and the horizontal distribution of the ISWs in the northern SCS, Alford
et al. (2010) detected 14 ISWs spanning from the Luzon Strait to the upper continental slope in the northern SCS. The wave amplitudes show a sharp increase as they move on to the upper continental slope because the decrease in the seafloor depth results in the amplification of the nonlinearity and the reduction of the nonhydrostatic dispersion effect of the ISWs (Alford et al., 2010). However, the studies performed to date do not provide enough information on the horizontal distribution of the ISW amplitude versus seafloor depth in shallow areas, and we know little about the seafloor’s impacts on ISW amplitude variations in their following inshore propagations. In previous studies, different models were used to investigate the vertical distribution of the single ISW amplitudes versus water depths (Fliegel and Hunkins, 1975; Vlasenko et al., 2000; Small and Hornby, 2005). Of these, only Vlasenko et al (2000) have compared the results of numerical simulations with in situ measurements, and the comparison shows a good agreement with the depth of maximum amplitude. However, there are several observed local maxima not captured by the models that are supposed to result from the superposition of different subscale ISWs (Vlasenko et al., 2000). In this study, our overall aims are to utilize numerous observed seismic oceanography sections and understand the vertical and horizontal distribution of ISW amplitudes as a function of either water or seafloor depths on the upper continental slope and shelf in the northern SCS. Section 2 describes our data acquisition process, how to extract ISW amplitudes from stacked and common offset gathers, and how to obtain theoretical results using eigenfunctions. The main results are then presented in section 3. These include 14 depression ISWs and the relationship between their maximum amplitudes and the corresponding seafloor depths. We also explain the
difference between the observed and theoretical vertical amplitude distribution of the single ISW amplitudes versus water depths.
2. Data and Method The study area is located in the northern SCS, across the shelf and upper slope area, at water depths of 100 m to 1000 m near Dongsha Island (Fig. 1). In this area, a set of 2D multichannel reflection seismic profiles were acquired by Guangzhou Marine Geological Survey (GMGS) in summer 2009 by using a 6000 m long streamer with 480 channels (trace interval 12.5 m and CMP interval 6.25 m), an air gun source with a total volume of 5080 in3 (1 in 3 =16.39 cm3), and a shot interval of 25 m. The sampling interval is 2 ms, and the dominant frequency of the seismic wavelet is 35 Hz, indicating a vertical resolution of approximately 10 m (average sound velocity 1500 ms-1).
Figure 1. Seismic sections L1 to L11 (red lines), S1 to S3 (black lines), and corresponding Argo
data A1 to A3 (black points). S1 and A1 were recorded on Jul 7th, S2 and A2 were recorded on Sep 20th and 21st, respectively, and S3 and A3 were recorded on Sep 19th. The seismic sections labeled as both L1 to L11 and S1 to S3 are used to evaluate the horizontal distributions of ISW maximum amplitudes versus the seafloor; the S1 to S3 sections are also used to extract the vertical distribution of individual ISW amplitudes versus water depths since they can provide extensively detailed ISW structures.
The multichannel reflection seismic dataset has been reprocessed using Seismic Unix to obtain seismic oceanography sections. Each section represents a “snapshot” of seawater column features. The data processing method for seismic oceanography is similar to the conventional processing of multichannel seismic data used in oil and gas exploration (e.g., Ruddick et al., 2009). The stacked sections of this dataset cannot provide all reflectors for the seawater column, as their shallow parts (water depth < 150 m, 1500 m/s acoustic velocity is used for time-to-depth conversion of the seismic sections) have been muted as part of the normal move out (NMO) correction. We use common offset gathers to provide the ISWs’ amplitudes in shallow water (between 50 m and 150 m). A detailed description of the method has been presented by Bai et al. (2017). For instance, the stacked section (Fig. 2a) and the common offset gather section (Fig. 2b) of L1 are used to extract the ISW amplitudes versus water depth (Fig. 2d). The Argo data are available at the Argo information center, and they are obtained in the corresponding time-space domain with the ISWs (Fig. 1). These corresponding hydrographic data are used to calculate the theoretical ISW amplitudes versus the water depths. In addition, bathymetric data in Fig. 1 are provided by the General Bathymetric Chart of the Oceans. The
profiles are generated with a spatial resolution of 1’ (approx. 1.85 km). Several nonlinear equations have been formulated for describing the evolution of ISWs in oceans, including the Korteweg–de Vries (KdV) equation, Boussinesq equation, Peregrine equation, Bona–Chen equation, shallow water wave equations and several others (Triki et al., 2013). The KdV type model is the most popular weakly nonlinear theory to govern the wave dynamics on the continental margin (Osborne and Burch, 1980; Duda et al., 2004; Liu et al., 2005; Liu et al., 2013). Then ISWs vertical structures can be determined by the solution of eigenfunctions presented by Fliegel and Hunkins (1975). The eigenfunctions
( ) satisfy the Taylor-Golstein Boundary Value
Problem (Fliegel and Hunkins, 1975; Liu et al., 1985; Pelinovksy et al., 1995; Apel et al., 2006; Zha et al., 2012): ( )
( ) where and
+ (
( )
)
( )
( ) ( )
( ) represents an ISW’s vertical amplitude distribution, C0 is the linear long wave speed, ( (𝑔 𝜌)/(𝜌
))1/ is the Brunt-Väisälä or buoyancy frequency.
is the bottom
depth. Eq. 1 is the governing differential equation with the boundary conditions Eq. 2. Based on Eq. 2, the top and bottom are hypothesized as rigid and the ISW amplitude at the boundaries is zero (Fliegel and Hunkins, 1975). Eq. 1 can be solved by the Thomson-Haskell method. A detailed solution is presented by Fliegel and Hunkins (1975) and Zha et al. (2012).
Figure 2. (a) Stacked section, (b) common offset section, (c) picked shallow ISW reflections after NMO corrections and ISW’s amplitude definition, and (d) vertical amplitude distribution versus water depths of L1. The red lines indicate the picks of the ISW reflection events. The amplitudes are combined from stacked and common offset sections. 3. Result We have identified a total of 14 ISW packets on the seismic sections L1 to L11 and S1 to S3,
and the method described in section 2 is utilized to get the waves’ amplitudes versus water depths. Fig. 2a shows a single-wave packet containing one typical soliton with an oscillating tail. The red lines are picked reflection events delineating a typical depression wave identified on the stacked section of L1, whereas the shallow part of the wave is presented as a common offset gather section in Fig. 2b. These two sections are combined to determine the L1 wave amplitudes versus water depths. As shown in Fig. 2c, we define the mean depth of the front and rear of the soliton as the water depth. The distance between the mean depth and the lowest point depth of the same reflection event is defined as the wave amplitude. Fig. 2d shows that the L1 ISW amplitudes first increase sharply and reach their maxima of 128 m in the middle layer at a depth of 142 m, and then decrease slowly with increasing water depths. The L1 ISW belongs to the mode-1 wave since its amplitude curve only has one extreme point. Figs. 3 and 4 similarly depict two ISW packets composed of a group of solitons. Each one has a dominant wave with the maximum amplitude. The dominant soliton is selected for our analysis as it may be the leading wave or others, which is indicated by the black arrow on the seismic sections. As shown in Fig. 5, all depression ISWs belong to the mode-1 wave because their vertical amplitude distributions only have one extreme point. Theoretically, the mode-1 ISW amplitudes should increase first and then decrease with the increasing water depth. The shape of the vertical distribution should be similar to Fig. 2d. We select the distribution’s extreme as the ISW maximum amplitude. However, the vertical amplitude distributions of L4, L5, L7, and L9 all exhibit monotone decreases versus water depth. In these cases, we still select the curve maximum as the ISW maximum amplitude because this is the clearest, most objective information provided
by the seismic data since they are unable to provide further shallow depth information for the ISW. As shown in Table 1, these L1-L11 depression ISWs are located at water depths between 263 m and 597 m on the northern SCS upper continental slope, and their maximum amplitudes range from 60 m to 128 m.
Figure 3. Stacked section of L4. The selected ISW is indicated by the black arrow. The yellow line represents the seafloor.
Figure 4. Stacked section of L6. The selected ISW is indicated by the black arrow. The yellow line
represents the seafloor.
Figure 5. The observed vertical amplitude distributions of L2-L11 ISW amplitudes versus water depths. The shaded rectangles indicate the area below the seafloor depths.
Seismic Line
ISW maximum amplitude /m
Seafloor depth /m
L1
128
510
L2
61
442
L3
78
318
L4
104
331
L5
82
341
L6
60
402
L7
70
300
L8
97
597
L9
90
365
L10
69
348
L11
66
263
S1
35
345
S2
91
740
S3
72
405
Table 1. Maximum amplitudes and corresponding seafloor depths of ISWs identified on sections L1 to L11 and S1 to S3.
We select 3 mode-1 ISWs to compare between the observed and theoretical vertical profiles of ISW amplitudes. These ISWs are labeled S1 to S3 and are shown in Figs. 8 to 10. All three ISWs exhibit abundant reflection events on the seismic sections, meaning we can extract detailed amplitudes. The major difference between them is their waveforms near the seafloor. The displacements of the S1 ISW reflectors first increase, then gradually decrease with the water depths, and finally just show a flat reflection event near the seafloor (Fig. 6). The amplitude of the S2 ISW decreases to zero at a distance from the seafloor. As shown in Fig. 7, the seafloor depth
below the S2 ISW is 730 m, whereas there are only flat reflection events below a water depth of 600 m. The S3 ISW extends all the way to the seafloor. As shown in Fig. 8, the reflection event at the ISW bottom is undulating and not parallel to the seafloor as the ideal example shown in Fig. 6. The ISW appears to have its bottom “cut” by the seafloor. For the S1 ISW, the corresponding seafloor depth is 354 m. We use corresponding hydrographic data (A1, Fig. 6c) to obtain the theoretical vertical amplitudes versus water depths of the S1 ISW (Fig. 9a). The black line represents the observed results, and the vertical amplitude distribution only has one extreme point, thus indicating that the S1 ISW is a mode-1 wave. The theoretical amplitude distribution (red line in Fig. 9a) from the eigenfunctions shows that the mode-1 ISW amplitude starts from zero at the sea surface, then positively correlates with the water depth and reaches the maximum (calibrated by the observed maximum amplitude) at a certain depth; after that depth is reached, the amplitude begins to decrease, and is finally reduced to zero at the seafloor. Although the amplitude at shallower depths is unavailable due to the already described data limitations, we still find a concordance between the vertical distributions of the observed S1 ISW amplitudes extracted from our seismic oceanography section and theoretical values from the eigenfunctions. This makes the S1 ISW an ideal case. For the S2 ISW, the hydrographic data of point A2 are shown in Fig. 7c and have temporal and geographical proximities with the seismic section of S2. We use these corresponding hydrographic data in Eq. 2 and obtain two theoretical amplitude distributions by using a different “bottom depth” (H) of the ISW extension depth at 600 m and the seafloor depth at 730 m. Fig. 9b clearly shows that the theoretical amplitude distribution using the ISW’s extension depth of 600 m is more consistent with the observed values.
Fig. 8 shows a case of the ISW’s amplitude having nonzero values near the seafloor. The S3 ISW amplitude does not decrease to zero close to the seafloor. The corresponding seafloor depth is 400 m, and the corresponding hydrographic data are shown in Fig. 8c. As shown in Fig. 9c, the vertical amplitude distribution of the S3 ISW amplitudes shows a similar trend as the S1 ISWs. As the water depth increases, the wave amplitude increases first and then decreases. Its theoretical and observed amplitude distributions trends are approximately similar, and their basic distribution is consistent. However, compared to the previous two cases, there are prominent differences: the observed amplitude distribution is generally offset upward from the theoretical one; and the depth of the maximum observed amplitude is 26 m less than the theoretical value.
Figure 6. (a) Stacked and (b) common offset gather sections of the ISW identified in profile S1, and (c) temperature, salinity, and the calculated density and Brunt-Väisälä frequency profiles of A1 points. Refer to Fig. 1 for location and collecting date. The amplitude at the seafloor is zero (indicated by the red arrow).
Figure 7. (a) Stacked and (b) common offset gather sections of the profiled ISW identified on S2, and (c) the temperature, salinity, and the calculated density and Brunt-Väisälä frequency profiles of the A2 points. Refer to Fig. 1 for location and collecting date. The extension depth is shallower than the seafloor.
Figure 8. (a) Stacked and (b) common offset gather sections of profile S3 ISW, and (c) the temperature, salinity, and the calculated density and Brunt-Väisälä frequency profiles of the A3 points. Refer to Fig. 1 for location and collecting date. The ISWs are at a nonzero amplitude near the seafloor.
Figure 9. Comparison of the observed and theoretical vertical amplitude distributions of S1 (a), S2 (b), and S3 (c) ISW amplitudes versus water depths. The black lines represent the observed data, and the red lines represent the theoretical amplitude distribution. In panel (b), the red line represents the theoretical amplitude distribution acquired using the ISW extension depth (H = 600 m), and the blue line represents the theoretical amplitude distribution acquired using seafloor depth (H =730 m).
4. Discussion 4.1 The horizontal distribution of the ISWs maximum amplitudes versus seafloor depths Fig. 10 presents all samples’ amplitudes versus water depth, including the elevation and transition waves in the study area parameterized by Bai et al. (2017) and the results from previous studies (e.g., Klymak and Moum, 2003; Orr and Mignerey, 2003; Cheng and Hsu, 2010; Fu et al., 2012). As shown in Tables 1 and 2, the observed ISWs are located at water depths between 87 m and 740 m on the northern SCS continental slope and shelf and have maximum amplitudes ranging from 13 m to 128 m. In the deep ocean, the ISWs’ maximum amplitudes are commonly large and widely distributed. These ISWs westward-propagating away from the Luzon strait arise from the interactions between the tidal flow and topography at the generation site, and they are not affected by deep topography variations (Liu et al., 1998; Zhao et al., 2004; Alford et al., 2010). The magnitude of their amplitudes mainly depends on their energy source and the structure of ocean stratification, exhibiting cyclical variations (Warn-Varnas et al., 2010; Xie et al., 2014). Once these ISWs enter the shallow northern SCS continental margin, they are strongly affected by the
shoaling topography. As the seafloor gets shallower, the friction between the ISW and the seafloor will bring about large amounts of energy dissipation (Chang et al., 2006; St. Laurent, 2008). Especially for large-amplitude ISWs, intense interactions with the seafloor occur. Their amplitudes will decrease more rapidly than those of smaller amplitude ISWs; hence, the distribution range of the maximum amplitudes will concentrate as the seafloor depth gets shallower. As shown in Fig. 10, the upper envelope is the connection line of these ISWs’ maximum amplitudes, and it indicates the maximum amplitude of the allowed ISWs at a certain seafloor depth. Due to limited data, the upper envelope can change if more data are added. The ISW maximum amplitude is highly correlated to the seafloor depth on the northern SCS margin. We have found a highly correlated logarithmic function relationship between all ISWs’ maximum amplitudes and water depths: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒
7.47 ∙ ln( .3375 ∙ 𝑒𝑝𝑡ℎ 87 ≤ 𝑒𝑝𝑡ℎ ≤ 74
7.3 48)
(3)
Elevation ISWs: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒 .3593 ∙ 𝑒𝑝𝑡ℎ 5. 4 𝑒𝑝𝑡ℎ ∈ [93.5 3 . ] Transition ISWs: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒 .63 ∙ 𝑒𝑝𝑡ℎ 45.85 9 𝑒𝑝𝑡ℎ ∈ [ 36.5 .4] Depression ISWs: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢 𝑒 . 7 6 ∙ 𝑒𝑝𝑡ℎ + 49.4 6 𝑒𝑝𝑡ℎ ∈ [ 63 74 ]
(4)
where ln is the log base e (natural logarithm), and the water depths range from 87 to 740 m. Accounting for the different types of ISWs, three linear correlations Eq. 4 are found between the ISWs maximum amplitudes and water depths. The depth domains of definition are discrete due to the limited data samplings. Inferred from Eq. 4, the linear coefficient for the elevation waves in shallow water is the largest one, whereas the smallest coefficient is valid for the deepest depression waves. With a decrease in seafloor depth, the decrease rate of wave maximum amplitude will accelerate. This suggests that in a shallow area, the seafloor depth will play a more important role as a constraint for the maximum amplitudes than in deep waters. Using the
maximum amplitudes-seafloor depths relationship determined by Eqs. 3 and 4, the maximum ISW amplitudes can thus be predicted. As presented by Alford et al. (2010), from the Luzon Strait to the continental slope, a decrease in seafloor depth results in the amplification of the nonlinearity and the reduction of the nonhydrostatic dispersion effect of the ISWs, and the wave amplitudes show a sharp increase on the upper continental slope. However, our research demonstrates that from the upper continental slope to the shelf, the shoaling topography is likely to have opposite effects on wave amplitude variations and the ISW maximum amplitudes decrease as the seafloor is getting shallower. Hence, in the northern SCS, the largest-amplitude ISW will occur around the upper continental slope.
Data sources
ISW types
Maximum amplitude /m
Seafloor depth /m
Bai et al.
Depression
71.3
304.2
Bai et al.
Elevation
49.5
126.6
Bai et al.
Elevation
48.5
124.4
Bai et al.
Elevation
14.7
86.6
Bai et al.
Elevation
12.7
98.5
Bai et al.
Transition
58.4
179.3
Bai et al.
Transition
92.7
214.1
Bai et al.
Transition
51.0
150.4
Orr and Mignerey
Elevation
45.4
117.2
Orr and Mignerey
Transition
53
169.8
Orr and Mignerey
Transition
54.1
152
Orr and Mignerey
Transition
45.1
135.8
Klymak and Moum
Elevation
29
116
Cheng and Hsu
Elevation
62.8
127.3
Cheng and Hsu
Transition
88
221.7
Fu et al.
Elevation
50.2
124.2
Fu et al.
Elevation
41.5
102.1
Fu et al.
Transition
72.2
173.6
Table 2. ISW maximum amplitude and corresponding seafloor depth from previous studies (e.g., Klymak and Moum, 2003; Orr and Mignerey, 2003; Cheng and Hsu, 2010; Fu et al., 2012; Bai et al., 2017)
Figure 10. Horizontal distribution of ISW maximum amplitudes versus seafloor depth on the northern SCS continental slope. The triangles represent elevation ISWs, the circles represent transition ISWs, and the squares represent depression ISWs. The dashed black lines are the upper envelopes. The fine solid black line is the fitted amplitude distributions for all ISWs, and the bold solid black lines are the fitted curves for the depression, transition, and elevation ISWs.
4.2 Comparison between the observed and theoretical vertical amplitude distributions of single ISW amplitudes versus water depths As described in section 3, the comparisons between the observed and theoretical vertical amplitude distributions of the S1, S2, and S3 ISWs show very different results. The S1 ISW amplitude happens to be zero at the seafloor, and its observed and theoretical values are in strong concordance. The S2 wave amplitude decreases to zero above the seafloor. Its theoretical vertical amplitude distribution calculated using the ISW’s extension depth is noticeably more consistent with the observed result than using the corresponding seafloor depth. As observed for the S3 ISW, its bottom appears to be “cut” by the seafloor, and there are prominent differences between the theoretical and observed results. The theoretical vertical distribution of single ISW amplitudes is defined by the eigenfunctions
( ), which satisfy the Taylor-Golstein Boundary Value Problem (Eqs. 1 and 2).
Fliegel and Hunkins (1975) proposed that this governing differential equation (Eq. 1) is valid for frictionless motion in a stratified fluid with a rigid top and bottom. Although we supplement common offset gather sections to delineate the ISWs’ shallow parts, the amplitudes shallower than 50 m deep are still not obtained due to the NMO correction processing step. We shall assume real ocean conditions with a rigid top and that the ISW amplitude at the sea surface is zero. In addition, the boundary condition presented by Eq. 2 requires the wave amplitude at the bottom to be zero. The S2 ISW comparison result indicates that the “bottom” represents the ISW extension depth rather than seafloor. It is possible that some ISWs such as S1 happen to extend to the seafloor. As an ISW comes into shallow waters, the interaction between the ISW and the seafloor will be strong and will lead to changes in the wave’s vertical amplitude distribution. The seafloor in this
case is no longer a rigid bottom. We propose that the seafloor depth does not equal the bottom boundary depth used in the eigenfunction calculation. The seafloor topography may have a huge impact on the shoaling ISWs’ amplitudes in the shallow area, especially when estimating their vertical amplitude distributions.
5. Conclusions In this paper, we use seismic oceanography data to extract ISW amplitudes and their corresponding water depths on the upper continental slope and shelf in the northern SCS. We emphasize the horizontal and vertical distributions of the ISW amplitudes versus water depths. The results suggest the following: (1) An ISW’s maximum amplitude is highly correlated to the seafloor depth. In general, in the northern SCS at a seafloor depth of 87 to 740 m, the shallower the seafloor depth, the smaller is the ISW maximum amplitude. A highly correlated logarithmic function relationship is found between all types of ISW maximum amplitudes and seafloor depths. For existing depression, transition, and elevation ISWs, three different linear correlations exist between the ISWs’ maximum amplitudes and the seafloor depth. The elevation waves have the largest linear coefficient, indicating that decreasing water depths have the strongest impact on them. (2) In general, the observed vertical distribution of ISW amplitudes versus water depth shows a strong concordance with the theoretical result. The eigenfunctions can accurately describe the vertical distribution of the ISW amplitudes versus water depths. However, the eigenfunctions cannot govern the full depth range, and the bottom depth in the boundary conditions should be the ISW extension depth rather than the seafloor depth. When estimating the vertical amplitude
distributions of the ISWs that interact with the seafloor on the continental slope, the observed amplitude distribution may not be perfectly consistent with the theoretical result because there are other ocean processes existing in the area. Thus, we have determined that topography plays an important role in an ISW’s evolution when it comes into a shallow area. The interaction between the seafloor and the ISW will alter the wave’s structure, and release a large amount of energy, which will erode the seafloor, strengthen water mixing, and pose a safety hazard to oceanic engineering. The mixing caused by the ISW dissipation can drive nutrients from the deep water into the surface layer and further the local marine ecosystems.
Funding: This work was supported by the National Natural Science Foundation of China (Grant nos. 41576047, 41576054) and the project for CAS Interdisciplinary Innovation Team.
Acknowledgements Bathymetry data are provided by General Bathymetric Chart of the Oceans (GEBCO, http://www.gebco.net/). Argo data are available from the Argo information center (http://www.jcommops.org/argo). The two free open source software, Seismic Unix (SU, http://www.cwp.mines.edu/cwpcodes/) and Generic Mapping Tools (GMT, http://gmt.soest.hawaii.edu/), are appreciated for supporting the data processing and mapping. The helpful suggestions of four anonymous reviewers are gratefully acknowledged.
References Alford, M. H., Lien, R. C., Simmons, H., Klymak, J., Ramp, S., Yang, Y. J., Tang, D., Chang, M. H., 2010. Speed and evolution of nonlinear internal waves transiting the South China Sea. Journal of Physical Oceanography, 40, 1338-1355. http://dx.doi.org/10.1175/2010JPO4388.1. Alford, M. H., Peacock, T., MacKinnon, J. A., Nash, J. D., Buijsman, M. C., Centuroni, L. R., Chao, S. Y., Chang, M. H., Farmer, D. M., Fringer, O. B., Fu, K. H., Gallacher, P. C., Graber, H. C., Helfrich, K. R., Jachec, S. M., Jackson, C. R., Klymak, J. M., Ko, D. S., Jan, S., Johnston, T. M. S., Legg, S., Lee, I. H., Lien, R. C., Mercier, M. J., Moum, J. N., Musgrave, R., Park, J. H., Pickering, A. I., Pinkel, R., Rainville, L., Ramp, S. R., Rudnick, D. L., Sarkar, S., Scotti, A., Simmons, H. L., St. Laurent, L. C., Venayagamoorthy, S. K., Wang, Y. H., Wang, J., Yang, Y. J., Paluszkiewicz, T., Tang T. Y. , 2015. The formation and fate of internal waves in the South China Sea. Nature, 521, 65-69. http://dx.doi.org/10.1038/nature14399. Apel, J. R., Ostrovsky, L. A., Stepanyants, Y. A., Lynch, J. F., 2006. Internal solitons in the ocean. Technical Report, Woods Hole Oceanography Institution. https://darchive.mblwhoilibrary.org/handle/1912/1070. Bai, Y., Song, H. B., Guan, Y. X., Yang, S. X., 2017. Estimating depth of polarity conversion of shoaling internal solitary waves in the northeastern South China Sea. Continental Shelf Research,143, 9-17. https://doi.org/10.1016/j.csr.2017.05.014. Bogucki, D. J., Redekopp, L. G., Barth, J., 2005. Internal solitary waves in the Coastal Mixing and Optics 1996 experiment: Multimodal structure and resuspension. Journal of Geophysical Research Oceans, 110, C02024. https://doi.org/10.1029/2003JC002253. Cai, S. Q., Long, X. M., Gan, Z. J., 2003. A method to estimate the forces exerted by internal
solitons on cylindrical piles. Ocean Engineering, 33 (5), 673- 689. https://doi.org/10.1016/S0029-8018(02)00038-0. Cai, S. Q., Xie, J. S., He, J. L., 2012. An overview of internal solitary waves in the South China Sea. Surveys in Geophysics, 33(5), 927-943. http://dx.doi.org/10.1007/s10712-012-9176-0. Chang, M. H., Lien, R. C., Tang, T.Y., D’Asaro, E. A., Yang, Y. J., 2006. Energy flux of nonlinear internal waves in northern South China Sea. Geophysical Research Letters, 33, L03607. http://dx.doi.org/10.1029/2005GL025196. Chen, G. Y., Su, F. C., Wang, C. M., Liu, C. T., Tseng, R. S., 2011. Derivation of internal solitary wave amplitude in the South China Sea deep basin from satellite images. Journal of Oceanography, 67(6), 689–697. http://dx.doi.org/10.1007/s10872-011-0073-9. Cheng, M. H., Hsu, J. R. C., 2010. Laboratory experiments on depression interfacial solitary waves over a trapezoidal obstacle with horizontal plateau. Ocean Engineering, 37(8-9), 800-818. http://dx.doi.org/10.1016/j.oceaneng.2010.02.016. Duda, T. F., Lynch, J. F., Irish, J. D., Beardsley, R. C., Ramp, S. R., Chiu, C. S., Tang, T. Y., Yang, Y. J., 2004. Internal tide and nonlinear internal wave behavior at the continental slope in the northern South China Sea. IEEE Journal of Oceanic Engineering, 29(4), 1105-1130. http://dx.doi.org/10.1109/JOE.2004.836998 Fliegel, M., Hunkins, K., 1975. Internal wave dispersion calculated using the Thomson-Haskell method. Journal of Physical Oceanography, 5(3), 541-548. http://dx.doi.org/10.1175/1520-0485(1975)005<0541:IWDCUT>2.0.CO;2. Fu, K. H., Wang, Y. H., Laurent, L. S., Simmons, H., Wang, D. P., 2012. Shoaling of large-amplitude nonlinear internal waves at Dongsha Atoll in the northern South China Sea.
Continental Shelf Research, 37(37), 1-7. http://dx.doi.org/10.1016/j.csr.2012.01.010. Guo, C., Chen, X., 2012. Numerical investigation of large amplitude second mode internal solitary waves over a slope-shelf topography. Ocean Modelling, 42(1), 80-91. http://dx.doi.org/10.1016/j.ocemod.2011.11.003. Huang, X. D., Chen, Z. H., Zhao, W., Zhang, Z. W., Zhou, C., Yang, Q. X. Tian, J. W., 2016. An extreme internal solitary wave event observed in the northern South China Sea. Scientific Reports, 6, 30041. http://dx.doi.org/10.1038/srep30041. Hyder, P., Jeans, D. R. G., Cauquil, E., Nerzic, R., 2005. Observations and predictability of internal solitons in the northern Andaman Sea. Applied Ocean Research, 27(1), 1-11. http://dx.doi.org/10.1016/j.apor.2005.07.001. Jeans, D. R. G., Sherwin, T. J., 2001. The evolution and energetics of large amplitude nonlinear internal waves on the Portuguese shelf. Journal of Marine Research, 59(3), 327-353. http://dx.doi.org/10.1357/002224001762842235. Lien, R. C., Henyey, F., Ma, B., Yang, Y. J., 2014. Large-amplitude internal solitary waves observed in the northern South China Sea: Properties and energetics. Journal of Physical Oceanography, 44 (4), 1095-1115. http://dx.doi.org/10.1175/JPO-D-13-088.1. Liu, A. K., Holbrook, J. R., Apel, J. R., 1985. Nonlinear internal wave evolution in the Sulu Sea. Journal of Physical Oceanography, 15 (12), 1613-1624. http://dx.doi.org/10.1175/1520-0485(1985)015<1613:NIWEIT>2.0.CO;2. Liu, A. K., Su, F. C., Hsu, M. K., Kuo, N. J., Ho, C. R., 2013. Generation and evolution of mode-two internal waves in the South China Sea. Continental Shelf Research, 59(59), 18-27. http://dx.doi.org/10.1016/j.csr.2013.02.009.
Liu, A. K., Zhao, Y. H., Hsu, M. K., 2005. Nonlinear Internal Wave Study in the South China Sea. Advances In Engineering Mechanics – Reflections And Outlooks: In Honor of Theodore Y-T Wu, 297-313. Klymak, J. M., Moum, J. N., 2003. Internal solitary waves of elevation advancing on a shoaling shelf. Geophysical Research Letters, 30(20), 2045. http://dx.doi.org/10.1029/2003gl017706. Klymak, J. M., Pinkel, R., Liu, C. T., Liu, A. K., David, L., 2006. Prototypical solitons in the South China Sea. Geophysical Research Letters, 33, L11607. http://dx.doi.org/10.1029/2006GL025932. Orr, M. H., Mignerey, P. C., 2003. Nonlinear internal waves in the South China Sea: Observation of the conversion of depression internal waves to elevation internal waves. Journal of Geophysical Research-Oceans, 108(C3), 3064. http://dx.doi.org/10.1029/2001jc001163. Osborne, A. R., Burch, T. L., 1980. Internal solitons in the Andaman Sea. Science, 208(2), 451-460. http://dx.doi.org/10.1126/science.208.4443.451. Pelinovksy, E., Tlipova, T., Ivanov, V., 1995. Estimations of the nonlinear properties of the internal wave field off the Israel coast. Nonlinear Processes in Geophysics, 2 (2), 80-88. http://dx.doi.org/10.5194/npg-2-80-1995. Qian, H. B., Huang, X. D., Tian, J. W., Zhao, W., 2015. Shoaling of the internal solitary waves over the continental shelf of the northern South China Sea. Acta Oceanologica Sinica, 34(9), 35-42. http://dx.doi.org/10.1007/s13131-015-0734-4. Ramp, S. R., Tang, T. Y., Duda, T. F., Lynch, J. F., Liu, A. K., Chiu, C. S., Bahr, F. L., Kim, H. R., Yang, Y. J., 2004. Internal solitons in the northeastern South China Sea. Part I: sources and deep water propagation. IEEE Journal of Oceanic Engineering, 29(4), 1157-1181.
http://dx.doi.org/10.1109/JOE.2004.840839 Rubino, A., Brandt, P., Weigle, R., 2001. On the dynamics of internal waves in a nonlinear, weakly nonhydrostatic three-layer ocean. Journal of Geophysical Research Oceans, 106(C11), 26899-26916. http://dx.doi.org/10.1029/2001JC000958 Ruddick, B., Song, H. B., Dong, C. Z., Pinheiro, L., 2009. Water column seismic images as maps of temperature gradient. Oceanography, 22(1), 192-205. http://dx.doi.org/10.5670/oceanog.2009.19. Small, R. J., Hornby, R. P., 2005. A comparison of weakly and fully non-linear models of the shoaling of a solitary internal wave. Ocean Modelling, 8(4), 395-416. https://doi.org/10.1016/j.ocemod.2004.02.002. St. Laurent, L., 2008. Turbulent dissipation on the margins of the South China Sea. Geophysical Research Letters, 35, L23615. https://doi.org/10.1029/2008GL035520. Tang, Q. S., Wang, C. X., Wang, D. X., Pawlowicz, R., 2014. Seismic, satellite, and site observations of internal solitary waves in the NE South China Sea. Scientific Reports, 4, 5374. http://dx.doi.org/10.1038/srep05374. Tang, Q. S., Hobbs, R., Wang, D. X., Sun, L. T., Zheng, C., Li, J. B., Dong, C. Z., 2015. Marine seismic observation of internal solitary wave packets in the northeast South China Sea. Journal of Geophysical Research: Oceans, 120(12), 8487-8503. http://dx.doi.org/10.1002/2015JC011362. Triki, H., Milovic, D., Biswas, A., 2013. Solitary waves and shock waves of the KdV6 equation. Ocean Engineering, 73(8), 119-125. http://dx.doi.org/10.1016/j.oceaneng.2013.09.001. Vlasenko, V., Brandt, P., Rubino, A., 2000. Structure of Large-Amplitude Internal Solitary Waves.
Journal of Physical Oceanography, 30(9), 2172-2185. http://dx.doi.org/10.1175/1520-0485(2000)030<2172:SOLAIS>2.0.CO;2. Vlasenko, V., Hutter, K., 2002. Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography. Journal of Physical Oceanography, 32, 1779-1793. http://dx.doi.org/10.1175/1520-0485(2002)032<1779:NEOTBO>2.0.CO;2. Vlasenko, V., Stashchuk, N., Guo, C., Chen, X., 2010. Multimodal structure of baroclinic tides in the South China Sea. Nonlinear Processes in Geophysics, 17(5), 529-543. http://dx.doi.org/10.5194/npg-17-529-2010. Warn-Varnas, A., Hawkins, J., Lamb, K. G., Piacsek, S., Chin-Bing, S., King, D., Burgos, G., 2010. Solitary wave generation dynamics at Luzon Strait. Ocean Modelling, 31(1–2), 9-27. http://dx.doi.org/10.1016/j.ocemod.2009.08.002 Xie, J. S., Chen, Z. W., Xu, J. X., Cai, S. Q., 2014. Effect of vertical stratification on characteristics and energy of nonlinear internal solitary waves from a numerical model. Communications in Nonlinear Science and Numerical Simulation, 19(10), 3539-3555. http://dx.doi.org/10.1016/j.cnsns.2014.03.007. Yang, Y. J., Tang, T. Y., Chang, M. H., Liu, A. K., Hsu, M. K., Ramp, S. R., 2004. Solitons northeast of Tung-Sha Island during the ASIAEX pilot studies. IEEE Journal of Oceanic Engineering, 29(4), 1182-1199. http://dx.doi.org/10.1109/JOE.2004.841424. Zha, G. Z., Lin, M. S., Shen, H., He, Y. J., Lv, H. B., 2012. Extraction of internal wave amplitude from nautical X-Band radar observations. Chinese Journal of Oceanology and Limnology, 30(3), 497-505. http://dx.doi.org/10.1007/s00343-012-1113-z. Zhao, Z. X., Klemas, V., Zheng, Q. A., Yan, X. H., 2004. Remote sensing evidence for baroclinic
tide origin of internal solitary waves in the northeastern South China Sea. Geophysical Research Letters, 31(6), L06302. http://dx.doi.org/10.1029/2003GL019077.
Highlights
Fourteen individual internal solitary waves have been identified in the northern South China Sea from multi-channel seismic data, with maximum amplitudes ranging from 35 to 128 m and corresponding water depths from 263 to 740 m. A logarithmic correlation between maximum amplitude and corresponding depth has been observed. The individual ISW's amplitude distribution over depth can be best described by the solution of eigenfunctions.