In this paper, a regional study of mode-2 internal
solitary waves (ISWs) at the Pacific coast of Central America is carried out using the seismic reflection method. The observed relationship between
the dimensionless propagation speed and the dimensionless amplitude (DA) of
the mode-2 ISW is analyzed. When DA < 1.18, the dimensionless
propagation speed seems to increase with increasing DA, divided into two parts with different growth rates. When
DA > 1.18, the dimensionless propagation speed increases with
increasing DA at a relatively small growth rate. We
suggest that the influences of seawater depth (submarine topography),
pycnocline depth, and pycnocline thickness on the propagation speed of the
mode-2 ISW in the study area cause the relationship between
dimensionless propagation speed and DA to
diversify. The observed relationship between the dimensionless wavelength
and the DA of the mode-2 ISW is also analyzed. When
DA < 1, the nondimensional wavelengths seem to change from 2.5 to 7
for a fixed nondimensional amplitude. When DA > 1.87, the
dimensionless wavelength increases with increasing DA. Additionally, the seawater depth has a great influence on the
wavelength of the mode-2 ISW in the study area. Overall, the wavelength
increases with increasing seawater depth. As for the vertical structure
of the amplitude of the mode-2 ISW in the study area, we find that it is
affected by the nonlinearity of the ISW and the pycnocline deviation
(especially the downward pycnocline deviation).
Introduction
The amplitude and propagation speed of the mode-1 internal solitary wave
(ISW) are larger than those of the mode-2 ISW. Mode-1 ISWs are more
common in the ocean. In recent years, with the advancement of observation
instruments, mode-2 ISWs in the ocean have been gradually observed, such
as on the New Jersey shelf (Shroyer et al., 2010), in the South China Sea
(Liu et al., 2013; Ramp et al., 2015; Yang et al., 2009), at Georges Bank
(Bogucki et al., 2005), over the Mascarene Ridge in the Indian Ocean (Da Silva
et al., 2011), and on the Australian North West Shelf (Rayson et al.,
2019). Conventional physical oceanography observation and remote-sensing
observation have their limitations: the horizontal resolution of
conventional physical oceanography observation methods (such as mooring) is
low and satellite remote sensing cannot see the ocean interior. Seismic
oceanography (Holbrook et al., 2003; Ruddick et al., 2009; Song et al.,
2021), as a new oceanography survey method, has high spatial resolution
(vertical and horizontal resolution can reach about 10 m). It
can better describe the spatial structure and related characteristics of
mesoscale and small-scale phenomena in the ocean (Biescas et al., 2008,
2010; Fer et al., 2010; Holbrook and Fer, 2005; Holbrook et al., 2013;
Pinheiro et al., 2010; Sallares et al., 2016; Sheen et al., 2009; Tsuji et
al., 2005). Scholars have used the seismic oceanography method to carry out
related studies on the geometry and kinematic characteristics (mainly
related to propagation speed) of ISWs in the South China Sea, the
Mediterranean Sea, and at the Pacific Coast of Central America (Bai et al.,
2017; Fan et al., 2021a, b; Geng et al., 2019; Sun et al., 2019; Tang et
al., 2014, 2018).
At present, research on the mode-2 ISW is mainly based on simulation.
Through simulation, scholars have found that the pycnocline deviation
affects the stability of the mode-2 ISW, and makes the top and bottom
structure of the mode-2 ISW asymmetrical (Carr et al., 2015; Cheng et al.,
2018; Olsthoorn et al., 2013). The instability caused by the pycnocline
deviation mainly appears at the bottom of the mode-2 ISW. It manifests
in the amplitude of the mode-2 ISW peak being smaller than the amplitude
of the trough, because the upper sea layer is thinner than the bottom sea
layer. The wave tail appears similar to a Kelvin–Helmholtz (KH) instability billow and the wave core appears as small-scale overturning motions (Carr et al., 2015; Cheng et al., 2018). Regarding the propagation speed of the mode-2 ISW, through
simulation experiments, scholars found that it increases with increasing amplitude (Maxworthy, 1983; Salloum et al., 2012; Stamp and Jacka, 1995; Terez and Knio, 1998). Brandt and Shipley (2014) simulated the material transport of mode-2 ISWs with large amplitude in the laboratory. They found that when 2a/h2>4 (a is the amplitude of the mode-2 ISW and h2 is the pycnocline thickness; we define the dimensionless amplitude ã=2a/h2 for the convenience of using in the following text), the linear relationship between propagation speed (wavelength) and amplitude is destroyed, i.e., when the amplitude ã≥ 4, the propagation speed increases relatively slowly and the wavelength increases rapidly. The authors believe that the above results are caused by strong internal circulation related to the very large amplitude and the influence of the top and bottom boundaries. Chen et al. (2014) calculated the Korteweg–de Vries (KdV) propagation speed and the fully nonlinear
propagation speed of the ISW as a function of the pycnocline depth and the
pycnocline thickness, respectively. They found that the propagation speed of the
mode-2 ISW increases monotonously with increasing pycnocline depth,
first increasing and then decreasing with the increasing pycnocline
thickness. Carr et al. (2015) found by simulations that the pycnocline
deviation has little effect on the propagation speed, wavelength, and
amplitude of the mode-2 ISW. Maderich et al. (2015) found that for
mode-2 ISWs, when the dimensionless amplitude ã<1, the deep-water weakly nonlinear theory (Benjamin, 1967) can describe the numerical simulation and experimental simulation results well. When ã>1, the wavelength (propagation
speed) increases with the amplitude faster than the results predicted by the
deep-water weakly nonlinear theory. However, the solution of Kozlov and Makarov (1990) can estimate the corresponding wavelength and propagation speed when the amplitude is 1<ã<5 well. Terletska et al. (2016) found that the propagation
speed and amplitude of the mode-2 ISW decreases after passing the step.
Moreover, the closer the mode-2 ISW is to the step in the vertical direction at
the time of incidence, the smaller the propagation speed and amplitude of
the mode-2 ISW are after passing the step. Kurkina et al. (2017) used the Generalized Digital Environmental Model
(GDEM) to find that seawater depth in
the South China Sea is the main controlling factor of the mode-2 ISW
propagation speed, and that the propagation speed increases exponentially with
increasing seawater depth. Deepwell et al. (2019) found by simulation that the relationship between the mode-2 ISW propagation speed and amplitude has a strong quadratic trend. They speculated that this quadratic relationship comes from the
influence of seawater depth (at a smaller seawater depth, the
propagation speed is also lower).
Distribution of multichannel seismic data. The red lines indicate
the positions of the survey lines, and the black filled circles on the red
lines indicate the positions of the observed mode-2 ISWs. The blue lines S1,
S2, and S3 are part of the seismic sections containing the mode-2 ISWs,
which will be displayed in Figs. 3 and 4.
The simulation can reveal the kinematic characteristics of the mode-2
ISW well. However, the actual ocean conditions are often more complicated, which is
manifested in the diversity of controlling factors in the kinematic
process. The observations, including the seismic oceanography method, are also
required to continually provide a basic understanding of the geometry and
kinematic characteristics of the mode-2 ISW. For example, limited by
factors such as lower spatial resolution of the observation methods,
previous scholars have performed less direct observational research on the propagation
speed and wavelength of the mode-2 ISW in the ocean. Furthermore, there is even less
research (including observational research) on the vertical structure of the
mode-2 ISW. The seismic oceanography method has advantages for carrying
out the abovementioned research, due to its higher spatial resolution. The
Pacific coast of Central America (western Nicaragua) has relatively
continuous submarine topography along the coastline, including the
continental shelf and continental slope, with a seawater depth of 100–2000 m
(Fig. 1). At present, there is relatively little research work available on internal
waves in this area. We reprocessed the historical seismic data in this area
and identified a large number of mode-2 ISWs with relatively complete
spatial structures in the region. This discovery is very helpful for carrying
out observational research on the geometry and kinematic characteristics of
mode-2 ISWs. Fan et al. (2021a, b) used the multichannel seismic data
of the survey lines L88 and L76 (cruise EW0412; see Fig. 1 for the survey
line locations) on the Pacific coast of Central America to
report the mode-2 ISWs in this area and study their shoaling features of the
mode-2 ISW in this area, respectively. However, a single survey line can only reveal the
local characteristics of the mode-2 ISW in the study area. A in-depth
understanding of the geometry and kinematic characteristics (mainly related
to propagation speed) of the mode-2 ISW in the study area requires a
regional systematic study. In this work, we reprocessed the seismic data of
the entire study area and identified numerous mode-2 ISWs on multiple
survey lines in the region (the positions of the observed ISWs and the
survey lines they located to are shown by the black filled circles and the red
lines in Fig. 1, respectively). Based on the numerous mode-2 ISWs observed
by multiple survey lines in the study area, this paper will conduct a
regional study on the characteristic parameters. These characteristic
parameters include the pycnocline deviation degree, propagation speed, and
wavelength of the mode-2 ISW, as well as the vertical structure
characteristics of the mode-2 ISW amplitude in the study area.
Data and methods
This paper mainly uses seismic reflection to conduct a regional study on the
mode-2 ISWs at the Pacific coast of Central America. The seismic data of the
cruise EW0412 used in this study were provided by the Marine
Geoscience Data System (MGDS; http://www.marine-geo.org/, last access: 30 March 2022). The cruise EW0412
collected high-resolution multichannel seismic data from the continental
shelf to the continental slope in the coastal areas of the Sandino Forearc
Basin, Costa Rica, Nicaragua, Honduras, and El Salvador (Fulthorpe and
McIntosh, 2014). The seismic acquisition parameters of the cruise EW0412 are
as follows: the sampling interval is 1 ms, each shot gather has 168 traces,
the shot interval is 12.5 m, the trace interval is 12.5 m, and the minimum
offset is 16.65 m. The seawater seismic reflection sections used in this
study were obtained through the following processing processes: defining
geometry, noise attenuation, common midpoint (CMP) sorting, velocity
analysis, normal moveout (NMO), stacking, and post-stack denoising. Previous
studies have demonstrated that seismic reflections generally track isopycnal
surfaces (Holbrok et al., 2013; Krahmann et al., 2009; Sheen et al., 2011).
We believe that the seismic stacked sections (e.g., Figs. 3 and 4) include the
information on density profile. Therefore, we do not provide plots of the
density profile upon which the waves propagate (even in schematic form) in the
following sections.
In this research, we try to use the maximum amplitude (the maximum vertical
displacement of isopycnals) to study the amplitude-related characteristics
of mode-2 ISWs, such as the relationship between propagation speed and
maximum amplitude in Fig. 9. However, the correlativity is not very strong. We also noticed that the amplitude, defined as the maximum vertical
displacement of isopycnals, is used less for quantitatively describing the
amplitude-related characteristics of mode-2 ISWs. Particularly, in mode-2 ISW
simulation research, scholars often use the dimensionless amplitude
ã to quantitatively describe the amplitude-related characteristics of mode-2
ISWs, like the relationship between propagation speed and
dimensionless amplitude (Brandt and Shipley, 2014; Carr et al., 2015). It is
important to point out that in mode-2 ISW simulation research, the dimensionless
amplitude used comes from the three-layer model. However, the mode-2
ISW in the actual ocean has a multilayer structure (multiple continuous
density displacements above and below the mid-depth of the pycnocline). It
is different from the three-layer model used in the simulation experiment to
describe the convex mode-2 ISW. As for the three-layer model, the upper
layer of the convex mode-2 ISW is the peak and the lower layer is the
trough. Because there is almost no work from the previous scholars to define the mode-2 ISW dimensionless amplitude based on the mode-2 ISW in the actual ocean, we attempt to build an equivalent three-layer model to compare our observational results with the simulation results and to quantitatively describe the amplitude-related characteristics of mode-2 ISWs. The equivalent three-layer model
results from the mode-2 ISW with a continuous structure in the actual
ocean. It should be pointed that the equivalent three-layer model is defined
by attempting to analogize with the three-layer model. Therefore, it is
not identical to the three-layer model as shown in Fig. 1 in Brandt and Shipley (2014). We use the equivalent three-layer model to calculate the
equivalent amplitude, the equivalent pycnocline thickness, and the
equivalent wavelength of the mode-2 ISW. Similarly, as the equivalent
three-layer model is defined by seeking to analogize with the
three-layer model, the equivalent amplitude (dimensionless amplitude) is not
completely equivalent to that used by Brandt and Shipley (2014). For the
mode-2 ISW with a multilayer structure, the sum of all ISW peak amplitudes
ap and the sum of all ISW trough amplitudes at are taken as the equivalent peak and trough amplitudes, respectively, of the mode-2 ISW with a three-layer model structure. The equivalent amplitude of the mode-2 ISW with a three-layer model structure is then the average of ap and at. The equivalent pycnocline thickness is calculated by h2=h-ap-at, where h is the seawater thickness affected by the
mode-2 ISW with a multilayer structure. The equivalent wavelength of the
mode-2 ISW with a three-layer model structure is the average of all ISW peak
and trough wavelengths in the multilayer structure. The detailed calculation
process is described in Fan et al. (2021a). This study uses an improved ISW
apparent propagation speed calculation method to calculate the apparent
propagation speed of ISW. This method firstly performs pre-stack migration of
the common-offset gather sections, and then picks the CMP and shotpoint
pairs corresponding to the ISW trough or peak from the pre-stack migration
sections of different offsets with a high signal-to-noise ratio. By fitting
the CMP–shotpoint pairs, we can calculate the apparent propagation speed
and apparent propagation direction of the ISW. The ISW horizontal velocity
can be expressed by the equation as follows:
v=cmp2-cmp1T=cmp2-cmp1(s2-s1)dt,
where cmp1 and cmp2 are the peak or trough positions of the ISW at different times, s1 and s2 are the shot numbers corresponding to cmp1 and cmp2, and dt is the time interval of shots. The detailed calculation process is described in Fan et al. (2021a).
Schematic diagram of the apparent wavelength correction of the
mode-2 ISW. (a) The ISW moves in the same direction as the ship. (b) The ISW moves in the opposite direction to the ship. S1 denotes the self-excitation and self-reception position of the ship at 1/2 amplitude of the ISW at the beginning. S2 denotes the self-excitation and self-reception position of the ship at the peak of the amplitude of the ISW. Vship is the ship speed and Vwater is the ISW propagation speed, λs is the apparent wavelength of the ISW observed by the seismic stacked section, λ is the actual wavelength of the ISW, a is the amplitude of the ISW, and xw is the distance moved by the ISW during the time the ship moves from S1 to S2. The black curve denotes the ISW at the beginning. The red curve denotes the ISW moved xw distance from the starting position. The blue curve denotes the ISW observed on the seismic stacked section.
The wavelength of the mode-2 ISW is usually defined as the half-width at
half-amplitude of the ISW (Carr et al., 2015; Stamp and Jacka, 1995), as
shown by λ in Fig. 2. In a seismic survey, the sound is sent from a
towed source, reflected from aquatic structures, and received by an array of
towed hydrophones with time delays that depends on the geometry of the ray
paths taken. A detailed introduction to seismic principles is provided by
Ruddick et al. (2009). Traditional seismic reflection imaging assumes that
the underground structure is fixed. Since mode-2 ISWs move relatively
fast in the horizontal direction (about 0.5 m s-1) during the seismic
acquisition process, seismic reflection imaging of mode-2 ISWs needs
to consider the influence of the horizontal motion of the ISW. The
wavelength of the mode-2 ISW observed by the seismic reflection method is
the apparent wavelength. The apparent wavelength of the mode-2 ISW is
controlled by the relative motion direction of the ship and the ISW, the
ship speed, and the propagation speed of the ISW. The propagation speed of
the mode-2 ISW (about 0.5 m s-1) is generally lower than the ship speed (about 2.5 m s-1) during seismic acquisition. When correcting the
apparent wavelength of the mode-2 ISW to obtain the actual wavelength, one
distinguishes between two situations: the situation in which the motion direction of the ISW and the
ship are the same, and the situation in which these are opposite to each other, as shown in Fig. 2. When the ISW and the ship
move in the same direction, the wavelength (apparent wavelength) estimated
from the seismic stacked section is larger, i.e., the wavelength
(apparent wavelength λs) of the ISW observed on the seismic
stacked section denoted by the blue curve in Fig. 2a is greater than the
wavelength λ of the actual ISW at the beginning and end,
denoted by the black and red curves in Fig. 2a, respectively. The influence
of the horizontal movement of the ISW should be eliminated. When correcting
the apparent wavelength λs to obtain the actual wavelength
λ, it is necessary to subtract the distance xw moved by the ISW within the seismic acquisition time corresponding to the apparent wavelength distance of the ISW, i.e.,
λ=λs-xw=λs-λsVshipVwater,
where Vship is the ship speed, and Vwater is the propagation speed of the ISW (Fig. 2a).
When the ISW and the ship move in opposite directions, the wavelength
(apparent wavelength) estimated from the seismic stacked section is smaller,
i.e., the wavelength (apparent wavelength λs) of the ISW
observed on the seismic stacked section denoted by the blue curve in Fig. 2b
is smaller than the wavelength λ of the actual ISW at the beginning
and end, denoted by the black and red curves in Fig. 2b, respectively. The
influence of the horizontal movement of the ISW should be eliminated. When
correcting the apparent wavelength λs to obtain the actual
wavelength λ, it is necessary to add the distance xw moved by the ISW within the seismic acquisition time corresponding to the apparent wavelength distance of the ISW, i.e.,
λ=λs+xw=λs+λsVshipVwater,
where Vship is the ship speed, and Vwater is the propagation speed of the ISW (Fig. 2b).
Results and interpretationsTypical sections interpretation and regional distribution
characteristics of the mode-2 ISW
In addition to the survey lines L88 and L76 with mode-2 ISWs observed by Fan
et al. (2021a, b), we also found mode-2 ISWs on many other survey lines
in the study area. Two typical survey lines are L84 and L74 (see the red
lines in Fig. 1 for the locations of these two survey lines). Figure 3 shows
the partial seismic stacked section S1 of the survey line L84 (see the blue
line in Fig. 1 for the location of section S1). We have identified 10
mode-2 ISWs from the seismic section S1 (see Fig. 3 for their positions and
corresponding numbers. ISW1–ISW4 are located at the shelf break and
ISW5–ISW10 are located on the continental shelf) and calculated their
characteristic parameters such as seafloor
depth (seawater depth) H, maximum amplitude (in the vertical direction),
equivalent amplitude a, equivalent pycnocline thickness h2, dimensionless amplitude ã, mid-depths of the pycnocline hc, the degree to which the
mid-depth of the pycnocline deviates from 1/2 seafloor depth Pd,
equivalent wavelength λ, dimensionless wavelength (we define the
dimensionless wavelength λ0=2λ/h2 for the
convenience of using in the following text), and apparent propagation speed
Uc (Table 1). The equivalent wavelength and the dimensionless wavelength in Table 1 have been corrected using Eq. (2) as the ISWs have the same motion direction as the ship, and the ISWs with the large propagation speed estimation error have been corrected using a propagation speed of 0.5 m s-1. The maximum amplitudes of the ISWs ISW1–ISW7 on survey line L84 are all less than 10 m and the maximum amplitudes of ISW8–ISW10 are larger, around 15 m. The ã values (ã=2a/h2) of these 10 mode-2 ISWs on the survey line L84 are all less than 2 (Table 1). We define an ISW whose ã value is less than 2 as a mode-2 ISW with a small amplitude and an ISW whose ã value is larger than 2 as a mode-2 ISW with a large
amplitude. The 10 mode-2 ISWs on the survey line L84 belong to the mode-2
ISWs with small amplitude. The ã values of ISW8, ISW9, and ISW10 are around 1; their amplitudes are relatively large compared to the other small-amplitude mode-2 ISWs. When calculating the Pd values, we find that the pycnocline centers of ISW8, ISW9, and ISW10 are deeper than 1/2 seafloor depths, whereas the pycnocline centers of the other seven mode-2 ISWs are shallower than 1/2 seafloor depths (Table 1). For ISW1, ISW2, and ISW3, the Pd
values are greater than 20 %, which appears as asymmetry of the
waveforms (the asymmetry of the front and rear waveform, and the asymmetry
of the top and bottom waveform). When the Pd value is small, the
waveform of the mode-2 ISW is more symmetrical, such as in ISW8, ISW9, and
ISW10. The waveforms of ISW1, ISW2, and ISW3 at the shelf break are
asymmetrical and their dimensionless wavelengths λ0
(λ0=2λ/h2) are significantly larger than the
λ0 values of the ISWs on the continental shelf which have the
same level of dimensionless amplitudes (ã), for example, the ã value of ISW2 is 0.45 and the value of λ0 is 9.55; the value of ã of ISW7 is
0.42 and the value of λ0 is 3.49). It means that the overall
relationship between dimensionless wavelength λ0 and the
dimensionless amplitude ã is not an absolutely linear correlation (the λ0 increases with the increasing ã). The apparent propagation speeds Uc of the 10 mode-2 ISWs on the survey line L84 are about 0.5 m s-1 and the apparent propagation directions are all shoreward. For ISWs with small apparent propagation speed calculation errors in shallow water (ISW6, ISW7, and ISW9), the Uc does not strictly increase with increasing ã. For example, the ã value of ISW6 is 0.4 and the Uc value is
about 0.58 m s-1, and the ã value of ISW9 is 1.19 and the Uc value is about 0.38 m s-1.
Seismic stacked section S1, with observed mode-2 ISWs on the survey line L84. Arrows and numbers indicate the 10 identified mode-2 ISWs
ISW1–ISW10. The location of the S1 seismic stacked section is shown in Fig. 1. The horizontal axis indicates the distance to the starting point of survey line L84. The survey line L84 acquisition time was from 07:15:14 GMT on 17 December 2004 to 17:26:49 GMT on 17 December 2004.
Characteristic parameters of the 10 mode-2 internal solitary waves
in Survey Line L84.
Note. H, seafloor depths; A, maximum amplitudes;
a, equivalent ISW amplitudes; h2, equivalent pycnocline
thicknesses; hc, the mid-depths of the pycnocline; Pd, the degree to which the mid-depth of the pycnocline deviates from 1/2 seafloor depth; λ, equivalent wavelengths;
Uc, apparent propagation speeds obtained from seismic observation; α, quadratic nonlinear coefficient shown in Eq. (9) and obtained by solving Eq. (6); C, linear phase speed which is obtained
by solving Eq. (6).
Panels (a) and (b) are the seismic stacked sections S2 and S3, respectively, with observed mode-2 ISWs on the survey line L74. The arrows and the numbers indicate the seven identified mode-2 ISWs ISW11–ISW17. The locations of the seismic stack section S2 and S3 are shown in Fig. 1. The horizontal axis indicates the distance to the starting point of survey line L74. The survey line 74 acquisition time was from 06:31:03 GMT on 3 December 2004 to 02:30:01 GMT on 4 December 2004.
Characteristic parameters of the seven mode-2 internal solitary
waves in survey line L74.
Note. H seafloor depths; A, maximum amplitudes; a, equivalent ISW amplitudes; h2, equivalent pycnocline thicknesses; hc, the mid-depths of the pycnocline; Pd, the degree to which the mid-depth of the pycnocline deviates from 1/2 seafloor depth; λ, equivalent wavelengths; Uc, apparent propagation speeds obtained from seismic observation.
Survey line L74 is located southeast of survey line L84 (see Fig. 1 for the specific location). Figure 4 shows the partial
seismic stacked sections (S2 and S3) of survey line L74. We have
identified seven mode-2 ISWs from the seismic sections S2 and S3. Their
positions and corresponding numbers are shown in Fig. 4 and the
characteristic statistical parameters are shown in Table 2. The equivalent wavelength
and the dimensionless wavelength in Table 2 have been corrected using Eq. (2) as the ISWs have the same motion direction as the ship, and the ISWs with
the large propagation speed estimation error have been corrected using a
propagation speed of 0.5 m s-1. The maximum amplitudes of the ISWs
ISW12–ISW17 on the survey line L74 are all less than 10 m, whereas the maximum
amplitude of ISW11 is larger, 13.6 m. The ã values of these seven mode-2 ISWs are all less than 2 (Table 2); they are the mode-2 ISWs with small amplitude. Among them, the amplitude of ISW11 is slightly larger. When calculating the Pd value, we find that the pycnocline centers of the
mode-2 ISWs ISW11–ISW17 are all deeper than 1/2 of the seafloor depths
(Table 2). Except for ISW11 (the bottom reflection event is broken), for
the other six mode-2 ISWs ISW12–ISW17, the Pd values are greater than 15 %. The asymmetry of ISW12 and ISW13 is manifested in that the connection between the top peaks of the ISW and the bottom troughs of the
ISW is not vertical. The pycnocline center of ISW14 deviates from 1/2 of the seafloor depth the most, by 51.5 %. Its asymmetry is manifested in the large difference between the top and bottom waveforms near the
pycnocline center. ISW15, ISW16, and ISW17 are located on the continental
shelf, and their pycnocline deviations are larger. However, their waveforms are
more symmetrical than other ISWs. When the downward pycnocline deviation is
large, the influence of pycnocline deviation on the stability of the mode-2
ISW is more complicated than when the pycnocline deviates upwards, and may be controlled by factors such as wavelength. There is no absolute linear
correlation relationship between the dimensionless wavelengths λ0 and the dimensionless amplitudes ã of the seven mode-2 ISWs on survey line L74 (the λ0 increases with the increasing
ã). For example, the ã values of ISW12 and ISW14 are greater than that of ISW16, but the λ0 value of ISW16 is greater than the λ0 values of ISW12 and ISW14. The apparent
propagation speeds Uc of the seven mode-2 ISWs on survey line L74 are about 0.5 m s-1 and their propagation directions are all
shoreward. For the ISWs in shallow water whose apparent propagation speed
calculation errors are small (ISW12, ISW14, ISW16, and ISW17), the Uc value generally increases with increasing ã.
(a) The time at which the mode-2 ISWs observed in the study area appeared in days. (b) The time at which the mode-2 ISWs observed in the study area appeared in hours. (c) Tracing back the time (in hours) at which internal solitary waves appeared at the continental shelf break in the study area.
(a) Histogram of the dimensionless amplitude of the mode-2 ISW in the study area. (b) Histogram of the propagation speed of the mode-2 ISW in the study area. (c) Histogram of the wavelength of the mode-2 ISW in the study area. The dark blue and red bars denote the ISWs on the survey lines in the SW–NE and NE–SW directions, respectively. (d) Histogram of the maximum amplitude of the mode-2 ISW in the study area.
In addition to survey lines L74 and L84, the mode-2 ISWs also have
sporadic distribution on other survey lines in the area (see the black
filled circles in Fig. 1). We have identified 70 mode-2 ISWs in the study
area. They appeared from 2 to 18 December 2004. On 17 and 18 December 2004, there were more mode-2 ISWs (Fig. 5a): 21 (10 for
survey line L84, 6 for survey line L88, and 5 for survey line L76) and 9 (1
for survey line L72, 5 for survey line L76, and 3 for survey line L103),
respectively. Observe the distribution of the appearance time of mode-2 ISWs
observed in the study area in Fig. 5a (in days). We find that the mode-2
ISWs frequently appeared on the northwest side of the study area in December
2004, and appeared in early and late December. In addition, the spatial
distribution range of the mode-2 ISWs is large, ranging from the continental
slope to the continental shelf (see Figs. 1, 3, and 4). Figure 5b shows the
time at which the mode-2 ISWs observed in the study area appeared in hours.
Combined with Fig. 5a, we can find that from 2 to 8 December
2004, the ISWs appeared at around 12:00 and 00:00 GMT (Greenwich Mean Time) (or 24:00 GMT) in a day. From
10 to 13 December 2004, the ISWs appeared at around 12:00 and
24:00 GMT in a day, and relatively more appeared around 12:00 GMT. From 14 to 18 December 2004, the ISWs appeared at around 12:00 and 00:00 GMT (or
24:00 GMT) in a day, and relatively more appeared around 00:00 GMT (or 24:00 GMT). Survey lines L103, L105, and L107 are perpendicular to the propagation
direction of the mode-2 ISWs in the study area (Fig. 1). Therefore, these
three survey lines are not included in the subsequent statistical analysis
of the mode-2 ISW characteristic parameters. We have counted the
characteristic parameters of 53 mode-2 ISWs in the study area. In these 53
mode-2 ISWs, there are 51 small-amplitude ISWs (ã<2), and there are 40 ISWs with smaller amplitude (ã<1) among these 51 small-amplitude ISWs (Fig. 6a). The
mode-2 ISWs in the study area are dominated by smaller amplitudes (Fig. 6a).
The maximum amplitudes (in the vertical direction) of the mode-2 ISWs mainly
change in the range of 3 to 13 m (Fig. 6d) and the equivalent wavelengths of
most of the mode-2 ISWs are in the order of about 100 m (Fig. 6c, the
equivalent wavelength in the figure has been corrected according to Eqs. 2
and 3). When calculating the propagation speed of the mode-2 ISW, due
to the low signal-to-noise ratio of some survey lines, the calculation
errors of some ISW propagation speeds are relatively large. Therefore, when
analyzing the apparent propagation speed of the mode-2 ISW of the study
area, we only used 26 ISWs with relatively small errors (the error is less
than half of the calculated value). The apparent propagation speeds of the
mode-2 ISWs in the study area are in the order of 0.5 m s-1 (Fig. 6b),
and most of the mode-2 ISWs propagate in the shoreward direction. We have
traced back the time at which each ISW in the study area (mainly the ISWs
located on the continental shelf) appeared at the continental shelf break
using the ISW propagation speed of 0.5 m s-1, as shown in Fig. 5c, in
hours. Combined with Fig. 5a, we find that from 2 to 8 December 2004, the ISWs traced back to the continental shelf break appeared
at around 12:00 and 00:00 GMT (or 24:00 GMT) in a day, and relatively more appeared around 12:00 GMT. From 10 to 13 December 2004, most of the ISWs
traced back to the continental shelf break appeared at around 24:00 GMT (or
00:00 GMT) in a day. From 14 to 18 December 2004, the ISWs traced
back to the continental shelf break appeared at around 12:00 and 24:00 GMT (or 00:00 GMT) in a day. The mode-2 ISWs observed in the study area may be generated by the interaction between the internal tide and the continental shelf
break.
Propagation speed and wavelength characteristics of mode-2 ISWs in the study area
Inspired by the work of Maderich et al. (2015) and Chen et al. (2014), we
calculated the relationships between the dimensionless
propagation speed and the dimensionless amplitude ã, the dimensionless wavelength λ0 and the ã, the propagation speed (Uc) and the maximum amplitude A, the wavelength (λ) and the A, the Uc and the pycnocline depth, and the Uc and the pycnocline thickness. Figure 7 shows
the relationship between the dimensionless propagation speeds (we define the
dimensionless propagation speed Ũ=Uc/C for the convenience of using in the following text) and the dimensionless amplitudes ã of the observed 26
mode-2 ISWs (with relatively small errors) in the study area. When
ã<1.18, it seems that the relationship between the Ũ values and the ã values of the observed mode-2 ISWs in the study area has the trends given by Kozlov and Makarov (1990) and Salloum et al. (2012), respectively; i.e., the Ũ of the mode-2 ISW increases with increasing ã, but with different growth
rates. The fitting effects of Kozlov and Makarov (1990), Salloum et al. (2012), and the segmentation fitting in Fig. 7 are shown in Table 3. The
segmentation fitting computed by ourselves in Fig. 7 can be expressed by the
equation as follows:
Ũ=9.441ã4-27.19ã3+28.14ã2-10.93ã+1.016ã-0.6401.
When ã>1.18, the relationship between the Ũ values and the ã values of the observed mode-2 ISWs in the study area is closer to the result predicted by the deep-water weakly nonlinear theory (Benjamin, 1967). That is, the Ũ of the mode-2 ISW increases with increasing ã at a relatively small growth rate.
The fitting effect of Benjamin (1967) in Fig. 7 is shown in Table 3. Figure 8 shows the relationship between the dimensionless wavelengths λ0 and the dimensionless amplitudes ã of the 32 observed mode-2 ISWs
(there are 13 ISWs on the survey lines in the SW–NE direction, and 19 ISWs
on the survey lines in the NE–SW direction; see Fig. 6c) in the study area.
In Fig. 8, the black and red crosses denote the ISWs on the survey lines in
the SW–NE direction and in the NE–SW direction, respectively. The survey
line in the SW–NE direction is consistent with the movement direction of the
ISWs. Use Eq. (2) to correct the apparent wavelength to obtain the actual
wavelength. The survey line in the NE–SW direction is opposite to the
movement direction of the ISWs. Use Eq. (3) to correct the apparent
wavelength to obtain the actual wavelength. Figure 8 shows the result after
correcting the apparent wavelength of the ISW. When using Eqs. (2) and (3) to correct the apparent wavelength, the propagation speed of the ISW
estimated in Fig. 7 needs to be used. The dimensionless wavelengths
λ0 of the ISWs with a large error in the estimation of the
propagation speed are not shown in Fig. 8. Observing Fig. 8, it can be seen
that when ã<1, the relationship between the λ0 values and the ã values of the observed mode-2 ISWs in the study area is closer to the result predicted by the deep-water weakly nonlinear theory (Benjamin, 1967). However, the λ0 values change from 2.5 to 7 for a fixed ã value. The
fitting effect of Benjamin (1967) in Fig. 8 is shown in Table 4. When
ã>1.87, the relationship between the λ0 values and the ã values of the observed mode-2 ISWs in the study area is closer to the solution of Salloum et al. (2012). That is, the λ0 of the mode-2 ISW increases with increasing ã. The fitting effects of Salloum et al. (2012)
and the segmentation fitting in Fig. 8 are shown in Table 4. The
segmentation fitting computed by ourselves in Fig. 8 can be expressed by the
equation as follows:
λ0=1.865ã+2.066.
When 1 <ã< 1.87, the λ0 values of the observed mode-2 ISWs in the study area are higher than those predicted by the deep-water weakly nonlinear theory (Benjamin, 1967) and Salloum et al. (2012).
Relationship between the dimensionless propagation speeds and
the dimensionless amplitudes of the mode-2 ISWs observed in the study area.
The black and red crosses denote the seismic observation results of the
mode-2 ISWs.
Fitting effects of each curve in Fig. 7 on the observation
points.
ã RangeFitting curveR2Larger than 1.18Benjamin (1967)0.34Smaller than 1.18Kozlov and Makarov (1990)0.67Smaller than 1.18Salloum et al. (2012)less than 0Smaller than 1.18segmentation fitting0.39
Note. For the fitting curve of Kozlov and Makarov (1990), we use the three red cross observation points to compute the R2 value. For the fitting curves of Salloum et al. (2012) and segmentation fitting, we use the black cross observation points, whose Ũ are less than 2, to compute the R2 values.
Relationship between the dimensionless wavelengths and the
dimensionless amplitudes of the mode-2 ISWs observed in the study area. The
black and red crosses denote the ISWs on the survey lines in the SW–NE and NE–SW directions, respectively.
Fitting effects of each curve in Fig. 8 on the observation
points.
ã RangeFitting curveR2Larger than 1.87Salloum et al. (2012)less than 0Larger than 1.87segmentation fitting0.97Smaller than 1Benjamin (1967)less than 0
(a) Relationship between the propagation speeds and the maximum amplitudes of the mode-2 ISWs observed in the study area. (b) Relationship between the wavelengths and the maximum amplitudes of the mode-2 ISWs observed in the study area. The black and red triangles in panel (b) denote the ISWs on the survey lines in the SW–NE and NE–SW directions, respectively.
(a) Relationship between propagation speeds and pycnocline depths of the mode-2 ISWs observed in the study area. (b) Relationship between propagation speeds and pycnocline thicknesses of the mode-2 ISWs observed in the study area. The color-filled circles indicate the dimensionless amplitude.
The relationship between the propagation speeds Uc and the maximum amplitudes A of the mode-2 ISWs observed in the study area is shown in Fig. 9a. The relationship between the wavelengths λ and the maximum amplitudes A is shown in Fig. 9b. It can be seen that Uc and λ of the mode-2 ISW in the study area are less affected by A. There is no obvious linear correlation between Uc and A, or between λ and A (Fig. 9a and b). When the A values are between 6 and 11 m, the range of Uc is relatively large and there is a significant increase in Uc (Fig. 9a). When the A values are between 7 and 13 m, there is a significant increase in wavelength λ (Fig. 9b). The relationship between the propagation speeds Uc and the pycnocline depths hc of the observed mode-2 ISWs in the study area is
shown in Fig. 10a, and the relationship between the propagation speed
Uc and the pycnocline thicknesses h2 is shown in Fig. 10b. As for the observed mode-2 ISWs in the study area, their hc values are mainly concentrated in the range of 40–70 m (Fig. 10a) and their h2 values are mainly concentrated in the range of 10–60 m (Fig. 10b). As with the numerical simulation results of Chen et al. (2014), the Uc values of the
observed mode-2 ISWs in the study area seem to have a trend to increase
slowly with increasing hc and h2 values.
The fitting effects of Chen et al. (2014) in Fig. 10 are shown in Table 5.
The trends mentioned above are not completely monotonous in Fig. 10, as
manifested in the large variation in Uc on the vertical axis.
We postulate that this is due to the fact that other factors (such as seawater
depth) in addition to pycnocline depth hc and pycnocline thickness h2 also affect the propagation speed Uc.
Fitting effects of each curve in Fig. 10 on the observation
points.
figurefitting curveR2Figure 10aChen et al. (2014)less than 0Figure 10bChen et al. (2014)less than 0
Note. For the fitting curve of Chen et al. (2014) in Fig. 10a, we use the observation points whose propagation speeds are less than 0.8 m s-1 and larger than 0.21 m s-1 to compute the R2 value. For the fitting curve of Chen et al. (2014) in Fig. 10b, we use the observation points whose propagation speeds are less than 0.9 m s-1 and pycnocline thicknesses are larger than 40 m to compute the R2 value.
Vertical structure characteristics of the mode-2 ISW amplitude in the study area
The vertical distribution of ISW amplitude (the vertical displacement of
isopycnal) is called its vertical structure. ISWs have different modes
which correspond to different vertical structures (Fliegel and Hunkins,
1975). Previous scholars have used different theoretical models to study the
vertical structure of ISW amplitude (Fliegel and Hunkins, 1975; Vlasenko et
al., 2000; Small and Hornby, 2005). Among them, only Vlasenko et al. (2000)
compared the results of numerical simulation with the results of local
observations. They found that the depths corresponding to the ISW maximum
amplitude (the maximum vertical displacement of isopycnals) given by the two
are in good agreement. At present, there is less work
comparing the theoretical vertical structure of mode-2 ISW amplitude with observed results. This work is conducive to improving our understanding
of the vertical structure of the mode-2 ISW in the ocean (including the
factors that affect the vertical structure). It can also test the validity
and applicability of the theoretical vertical structure to a certain extent.
The seismic oceanographic method has high spatial resolution, and its
clear ISW imaging results are more conducive to the study of vertical
structure. The vertical structure of ISW amplitude is controlled by a
variety of environmental factors. Geng et al. (2019) used the seismic
oceanography method to study the vertical structure of ISW amplitude near
Dongsha Atoll in the South China Sea. They found that when the ISW interacts
intensely with the seafloor, the observed vertical structure of ISW
amplitude may be significantly different from the theoretical result. Gong
et al. (2021) compared the vertical structure of ISW estimated by
theoretical models with the vertical structure of ISW observed by the
seismic oceanography method. They analyzed in detail the factors
affecting the vertical structure of ISW amplitude near Dongsha Atoll in the
South China Sea and found that the vertical structure of ISW is mainly
controlled by nonlinearity. It usually appears that the quadratic nonlinear
coefficients of ISWs that conform to the linear vertical structure function
are small, while the quadratic nonlinear coefficients of ISWs conforming to
the first-order nonlinear vertical structure function are larger. In
addition, topography, ISW amplitude, seawater depth, and background flow may
all affect the vertical structure of ISW amplitude. It appears that larger
seawater depth may weaken the influence of the nonlinearity of the ISW on
the vertical structure, making the vertical structure of ISW more in line
with linear theory. Larger amplitude will make ISW more susceptible to the
influence of topography, which will change the vertical structure. Vlasenko
et al. (2000) observed that the vertical structure of ISW has local extrema, which
they thought to be caused by smaller-scale internal waves. In addition, the
background flow shear also has an important effect on the vertical structure
(Stastna and Lamb, 2002; Liao et al., 2014). Xu et al. (2020) found that the
background flow at the center of the eddy can weaken the amplitude of ISW.
Panels (a)–(j) demonstrate the vertical structure characteristics of the amplitude of the 10 mode-2 ISWs ISW1–ISW10 on survey line L84 as well as the vertical mode function fitting results. The
black circles denote the observed ISWs' amplitudes at different depths. The
blue curves are the linear vertical mode function (nonlinear correction is
not considered) and the red curves are the first-order nonlinear vertical
mode function (nonlinear correction is considered).
Observing the vertical structure of the mode-2 ISW amplitude in the study
area, we find that it follows the following characteristics as a whole: the
amplitude of ISWs in the upper half of the pycnocline decreases with
increasing seawater depth; the amplitude of ISWs in the lower half of the
pycnocline first increases, and then decreases with increasing seawater
depth (see Figs. 11 and 12 in this paper, Fig. 5 of Fan et al., 2021a, and
Fig. 6 of Fan et al., 2021b). Due to the influence of the pycnocline center
deviation on development of the vertical structure of ISW amplitude,
the vertical structure of the mode-2 ISW amplitude in the study area
generally only exhibits a part of the characteristics given by the vertical
mode function. As for the vertical mode function, the amplitude of the ISW
in the upper and lower half of the pycnocline firstly increases and then
decreases with the increasing seawater depth, as shown by the
blue and red curves in Figs. 11 and 12. Since the pycnocline centers of most
of the mode-2 ISWs observed in the study area deviate upwards, the ISW
structure at the top is not as well developed as the ISW structure at the
bottom. Therefore, the amplitude of the ISW in the upper half of the pycnocline
usually decreases with increasing seawater depth. Figure 11 shows the
vertical structures of the amplitude of the 10 mode-2 ISWs ISW1–ISW10 in
survey line L84. The pycnocline centers corresponding to ISW1–ISW7 all
deviate upwards (see the degree to which the mid-depth of the pycnocline
deviates from 1/2 seafloor depth in Table 1; the positive sign indicates
that the pycnocline deviates upward and the negative sign indicates that
the pycnocline deviates downward). Among them, ISW1–ISW4 (Fig. 11a–d) and
ISW7 (Fig. 11g) only have one reflection event in the upper half
of the pycnocline. From ISW6 (Fig. 11f), we can see that the amplitude of
ISWs in the upper half of the pycnocline decreases with increasing
seawater depth. From ISW2 (Fig. 11b), ISW4 (Fig. 11d), ISW5 (Fig. 11e), and
ISW7 (Fig. 11g), it can be seen that the amplitude of ISW in the lower half
of the pycnocline firstly increases and then decreases with the increasing
seawater depth. The pycnocline centers corresponding to ISW8–ISW10 all deviate
slightly downwards (see the degree to which the mid-depth of the
pycnocline deviates from 1/2 seafloor depth in Table 1; the positive sign
indicates that the pycnocline deviates upward and the negative sign
indicates that the pycnocline deviates downward). From ISW8 (Fig. 11h) and
ISW10 (Fig. 11j), it can be seen that the amplitude of the ISW in the upper half
of the pycnocline decreases with increasing seawater depth. Figure 12
shows the vertical structures of the amplitude of the four mode-2 ISWs
(ISW11, ISW12, ISW16, and ISW17) in survey line L74. The pycnocline
centers corresponding to ISW11, ISW12, ISW16, and ISW17 deviate significantly
downwards (see the degree to which the mid-depth of the pycnocline
deviates from 1/2 seafloor depth in Table 2; the positive sign indicates
that the pycnocline deviates upward and the negative sign indicates that
the pycnocline deviates downward). This renders the ISW structure at the top
more developed. From ISW11, ISW12, and ISW17 (Fig. 12a, b, d), it can be
seen that the amplitude of the ISW in the upper half of the pycnocline
first increases and then decreases with increasing seawater depth.
Panels (a)–(d) demonstrate the vertical structure characteristics of the amplitude of the four mode-2 ISWs (ISW11, ISW12, ISW16, and ISW17) on survey line L74 as well as the vertical mode function fitting results. The black circles denote the observed ISWs' amplitudes at different depths. The blue curves are the linear vertical mode function (nonlinear correction is not considered) and the red curves are the first-order nonlinear vertical mode function (nonlinear correction is
considered).
To study the vertical structure of the mode-2 ISW amplitude in more detail
for the study area, we compare the observation result with the
linear vertical mode function (nonlinear correction is not considered, the
blue curves in Figs. 11 and 12) and the first-order nonlinear vertical mode
function (considering nonlinear correction, the red curves in Figs. 11 and
12). The linear vertical mode function can be obtained by solving the
eigenvalue equation that satisfies the Taylor–Goldstein problem (Holloway et
al., 1999):
d2φ(z)dz2+N2(z)C2φ(z)=0φ(0)=φ(-H)=0,
where φ(z) represents the linear vertical mode function, C is the
linear phase speed, and N(z) is the Brunt–Väisälä frequency. We use the temperature and salinity data coming from the Copernicus Marine
Environment Monitoring Service (CMEMS) to compute the Brunt–Väisälä
frequency. The first-order nonlinear vertical mode function is obtained by
adding a nonlinear correction term to the linear vertical mode function
(Lamb and Yan, 1996). It can be expressed by the following equation:
φm(z)=φ(z)+η0T(z),
where η0 is the ISW maximum amplitude in the vertical direction
and T(z) is the first-order nonlinear correction term. T(z) satisfies equations as follows (Grimshaw et al., 2002, 2004):
8d2T(z)dz2+N(z)2C2T(z)=-αCd2φ(z)dz2+32ddzdφzdz29T(0)=T(-H)=0α=3C2∫-H0dφ(z)dz3dz∫-H0dφ(z)dz2dz,
where α is the quadratic nonlinear coefficient. Equation (8) has a
unique solution by adding the restriction condition of T(zmax)=0 (Grimshaw et al., 2002), where zmax represents the depth of
the maximum amplitude of ISW. The detailed calculation process is described
in Gong et al. (2021). The fitting effects of the linear vertical mode
function and the first-order nonlinear vertical mode function in Fig. 11
are shown in Table 6. We comprehensively evaluate the goodness of fit by
the computed R2, the depths corresponding to the maximum amplitude
between the observation results and the fitting results, and the overall
trends between the observation results and the fitting results. Observing
Fig. 11 and Table 6, we find that the overall nonlinearity of the ISWs ISW5
(Fig. 11e) and ISW8 (Fig. 11h) on survey line L84 is relatively strong, and the first-order nonlinear vertical mode function considering nonlinear
correction can be used to better fit the vertical structure of the amplitude
(the red curves in Fig. 11e, h). The nonlinearity is relatively strong at
the bottom of ISW2 (the seawater depth range is 60–80 m in Fig. 11b), the
top of ISW7 (the seawater depth range is 40–60 m in Fig. 11g), and the top
of ISW10 (the seawater depth is about 40 m in Fig. 11j), and the first-order
nonlinear vertical mode function considering nonlinear correction can be
used to better fit the vertical structure of the amplitude (the red curves
in Fig. 11b, g, j). The overall nonlinearity of ISW1 (Fig. 11a), ISW3 (Fig. 11c), ISW6 (Fig. 11f), and ISW9 (Fig. 11i) is relatively weak, and the
linear vertical mode function can be used to better fit the vertical
structure of the amplitude (the blue curves in Fig. 11a, c, f, i). The
nonlinearity is relatively weak at the top of ISW2 (the seawater depth range
is 40–60 m in Fig. 11b), the bottom of ISW7 (the seawater depth range is
60–90 m in Fig. 11g), and the bottom of ISW10 (the seawater depth is below
40 m in Fig. 11j). The linear vertical mode function can be used to better
fit the vertical structure of the amplitude (the blue curves in Fig. 11b, g,
j). The above analysis reflects that the vertical structure of the mode-2
ISW amplitude in the study area is affected by the degree of nonlinearity of
the ISW. The fitting effects of the linear vertical mode function and the
first-order nonlinear vertical mode function in Fig. 12 are shown in Table 7. We comprehensively evaluate the goodness of fit by the computed
R2, the depths corresponding to the maximum amplitude between the
observation results and the fitting results, and the overall trends between
the observation results and the fitting results. Observing Fig. 12 and Table 7, we find that neither the linear vertical mode function (without
considering nonlinear correction) nor the first-order nonlinear vertical
mode function (with consideration of nonlinear correction) can be used to
fit the vertical structure of the amplitude of the ISWs ISW11, ISW12,
ISW16, and ISW17 on L74 well (especially the position of the upper half of the
pycnocline). The ISWs ISW11, ISW12, ISW16, and ISW17 on survey line L74
have a large downward deviation of the pycnocline center (see the degree
to which the mid-depth of the pycnocline deviates from 1/2 seafloor depth in Table 2; the positive sign indicates that the pycnocline deviates upward
and the negative sign indicates that the pycnocline deviates downward). We
have observed the fitting result of the vertical amplitude of the ISW with
the large downward pycnocline deviation on other lines of the study area
(not shown in this article) and found that the fitting result of the
vertical amplitude is usually poorer than that of the ISW corresponding to
the upward deviation of the pycnocline (especially the position of the upper
half of the pycnocline). We believe that when the pycnocline center has a
large downward deviation, the vertical mode function (including the linear
vertical mode function without considering nonlinear correction, and the
first-order nonlinear vertical mode function considering nonlinear
correction) cannot be used to well fit the vertical structure of the mode-2
ISW amplitude in the study area. The above analysis once again reflects that
the pycnocline deviation (especially the downward deviation of the
pycnocline) affects the vertical structure of the mode-2 ISW amplitude in
the study area. In addition, we could not find a good way to fit the
vertical amplitude structure in Fig. 12 based on the basic KdV theory. Another theory may be needed to fit this kind of vertical amplitude structure. We hope this can be solved in future studies.
Fitting effects of each curve in Fig. 11 on the observation
points.
Fitting effects of each curve in Fig. 12 on the observation
points.
ISW#Seawater depthFitting curveR2rangeISW1137–101 mblue curve0.22red curve0.3ISW1230–75 mblue curve0.47red curve0.68ISW1629–78 mblue curve0.46red curve0.25ISW1729–79 mblue curveless than 0red curveless than 0Discussion
Regarding the relationship between the dimensionless propagation speed Ũ and dimensionless amplitude ã of the mode-2 ISW in the study area, as well as the relationship between the dimensionless wavelength λ0 and dimensionless amplitude ã, neither is strictly monotonous in the case of smaller amplitude (ã<1) and both show the characteristics of multi-parameter
controlling (Figs. 7 and 8). For this reason, we analyzed the influence of
seawater depth (seafloor depth) on Ũ and λ0 of the mode-2 ISWs in the study area. The results are shown in Fig. 13a and b,
respectively. Observing Fig. 13a, we find that in shallow seawater
(seafloor depth less than 120 m), the Ũ variation range is small and there are both large-amplitude mode-2 ISWs (ã>2) and small-amplitude mode-2 ISWs (ã<2). In deep seawater (or at the shelf break; seafloor depth is greater than 120 m), the smaller-amplitude mode-2 ISWs (ã< 1; dark blue filled circles in Fig. 13a) have a large Ũ variation range. The maximum Ũ can reach 2.45, and the minimum can reach 0.5. In particular, the
smaller Ũ values are mainly concentrated in the deep seawater, so that in Fig. 7, when ã< 1.18, the relationship between Ũ and ã of the mode-2 ISW seems to have the trend given by Kozlov and Makarov (1990). The sharp decrease in the Ũ values of the mode-2 ISWs with smaller amplitudes in deep seawater
may be caused by collision of the ISWs with the seafloor topography
(including the step) at the shelf break. In addition, from Fig. 10a and b,
it can be seen that on the whole, the pycnocline depths and the pycnocline
thicknesses of the larger-amplitude mode-2 ISWs (ã> 1) are smaller than the pycnocline depths and the pycnocline thicknesses of the smaller-amplitude mode-2 ISWs (ã< 1). Therefore, the propagation speeds of the larger-amplitude mode-2 ISWs (ã> 1) are generally smaller than the propagation
speeds of the smaller-amplitude mode-2 ISWs (ã< 1). In Fig. 7, when ã> 1.18, this makes the relationship between Ũ and ã of the mode-2 ISWs
closer to the result predicted by the deep-water weakly nonlinear theory
(Benjamin, 1967). The above-analyzed influences of seawater depth
(seafloor topography), pycnocline depth, and pycnocline thickness on
the mode-2 ISW propagation speed in the study area have caused the
diversity of the relationship between Ũ and ã: when ã< 1.18, the relationship between the Ũ values and the ã values of the observed mode-2 ISWs
in the study area seems to have the trends given by Kozlov and
Makarov (1990) and Salloum et al. (2012); when ã> 1.18, the relationship between the Ũ values and the ã values of the observed mode-2 ISWs in the study
area is closer to the result predicted by the deep-water weakly nonlinear
theory (Benjamin, 1967).
(a) Relationship between dimensionless propagation speeds and seawater depths of the mode-2 ISWs observed in the study area. The color of the filled circle indicates the dimensionless amplitude. (b) Relationship between dimensionless wavelengths and seawater depths of the mode-2 ISWs observed in the study area. The color of the filled circle indicates the dimensionless amplitude. (c) Relationship between propagation speeds and seawater depths of the mode-2 ISWs observed in the study area. The color of the filled circle indicates the maximum amplitude. (d) Relationship between wavelengths and seawater depths of the mode-2 ISWs observed in the study area. The color of the filled circle indicates the maximum amplitude.
Observing Fig. 13b, we find that the mode-2 ISWs with smaller
amplitudes (ã< 1; dark blue filled circles in Fig. 13b) have a relatively
large variation range of the dimensionless wavelength λ0 in
deep seawater (seafloor depth greater than 120 m). The largest
λ0 can reach up to 9.55 (corresponding to ISW2 on survey
line L84, whose pycnocline deviation is large and waveform is asymmetric), and the smallest λ0 can reach 2.44, so that the λ0 of the vertical axis in Fig. 8 can be reduced to 2.44 when ã< 1. The sharp decrease in the λ0 values of the mode-2 ISWs with
smaller amplitudes (ã< 1) in deep seawater may be caused by collision of
the ISWs with the seafloor topography at the shelf break. The sharp increase
in λ0 values of the mode-2 ISWs with smaller amplitudes in
deep seawater may be related to the waveform asymmetry caused by the
pycnocline deviation.
Figure 13c and d show the relationship between propagation
speed Uc and seawater depth, and between
wavelength λ and seawater depth of the mode-2 ISWs in the study
area, respectively. The color of the filled circles in the figures represents the maximum
amplitude. Observing Fig. 13c, we find that the seawater depth in the study
area has a great influence on the Uc of the mode-2 ISWs. In the shallow seawater area (seawater depth less than 120 m), the Uc range is small. In the deep seawater area (seawater depth greater than 120 m),Uc has a large range. The maximum Uc is 1.52 m s-1 and the minimum Uc is 0.21 m s-1. In Fig. 9a, when the maximum amplitude is between 6 and 11 m, Uc has a large range and there is a significant increase in Uc. The above
phenomenon is controlled by seawater depth, i.e., in the deep
seawater area (seawater depth greater than 120 m), for the ISWs with a
maximum amplitude of 6–11 m, Uc varies widely and the maximum Uc of 1.52 m s-1 appears (Fig. 13c). Observing Fig. 13d, we find
that the seawater depth in the study area has a great influence on the
wavelength λ of the mode-2 ISW. On the whole, the λ of the
ISW increases with increasing seawater depth. For the ISWs with a
maximum amplitude of 7–13 m, considerable parts of the waves are distributed in
the deep seawater area (seawater depth is greater than 120 m), making
their λ values increase significantly. As a result, when the
maximum amplitude is between 7 and 13 m in Fig. 9b, there is a significant
increase in the wavelength λ.
McSweeney et al. (2020a, b) conducted observational studies on the
cross-shore and alongshore evolution characteristics of internal bores near
Point Sal, California. They used the quadratic nonlinear coefficient
α calculated by the KdV theory to characterize the stratification, and found that when the α calculated from the background density is
greater than 0, the waveform of the internal bore becomes steep as the
internal bore passes the site. When the α calculated from the
background density is less than 0, the waveform of the internal bore becomes
rarefied as the internal bore passes the site. Background stratification
affects the evolution of internal bores and the passage of an internal bore
will also change the stratification, which, in turn, affects the evolution of
a subsequent internal bore. They found that the change in α
after the internal bore had passed is positively correlated with the background
α. By analogy with the work of McSweeney et al. (2020a, b), we
calculated the background quadratic nonlinear coefficient α
(corresponding to the stratification before the arrival of the ISW) and the
linear phase speed C at the position of the ISWs in the study area by
solving Eqs. (6) and (9). Because the theoretical vertical structures calculated based on KdV theory cannot well describe the ISWs appearing on the survey line L74 (Fig. 12), we have only calculated α and C at each ISW position on survey line L84. The calculation results are shown in columns 12 and 13 of Table 1, respectively. Observing the calculated α values
in Table 1, we find that the α values of ISW1-ISW4 are all less than
0. And the α values of ISW5-ISW10 are all greater than 0. It
corresponds well to the waveform characteristics of the ISWs in Fig. 3. That
is, for ISW1–ISW4, whose α values are less than 0, their waveforms
are relatively rarefied. For ISW5–ISW10, whose α values are greater
than 0, their waveforms are relatively steep. This indicates that
background stratification has an influence on the shape of the mode-2 ISWs
in the study area. Observing the calculated C values in Table 1, we find that from ISW1 to ISW4, the calculated C values gradually decrease with
decreasing seafloor depths. This is consistent with the observed trend that
the propagation speeds Uc of the ISWs (column 11 of Table 1) also
gradually decrease with decreasing seafloor depths. ISW5 is shallower
than ISW4, but the calculated C and the observed Uc of ISW5 are both greater than those of ISW4. From ISW5 to ISW10, as the seafloor depth
gradually decreases, the calculated C values and the observed ISW Uc values again show an overall decreasing trend. We think the above phenomenon
is caused by background stratification, as ISW1–ISW4 have a similar
background stratification and ISW5–ISW10 have another similar background
stratification. This makes the calculated C values and observed Uc values of SW1–ISW4 decrease with decreasing seafloor depth. The calculated C and observed Uc of ISW5 are greater than those of ISW4. On the whole, the calculated C values and the observed Uc values of ISW5–ISW10 decrease
with decreasing seafloor depth. The above discussion indicates that
background stratification has an influence on the propagation speed of mode-2 ISWs in the study area.
Conclusions
We carried out a regional study of mode-2 ISWs at the Pacific coast of
Central America using the seismic reflection method. Via analysis of
the typical seismic sections L84 and L74, we find that when the degree of
downward pycnocline deviation is large, the influence of pycnocline
deviation on the stability of the mode-2 ISW is more complicated than when
the pycnocline deviates upwards – there are mode-2 ISWs with a large degree
of downward pycnocline deviation, but with a relatively symmetrical
waveform.
The observed relationship between dimensionless propagation speed
Ũ and dimensionless amplitude ã of the mode-2 ISWs in the study area was analyzed. When ã< 1.18, Ũ seems to increase with increasing ã, divided into two parts with different growth rates. When ã> 1.18, Ũ increases with increasing ã at a relatively small growth rate. The observed relationship between dimensionless wavelength λ0 and dimensionless amplitude ã of the mode-2 ISWs in the study area was also analyzed. When ã< 1, λ0 seems to change from 2.5 to 7 for a fixed ã value. When ã> 1.87, λ0 increases with increasing ã. As for the relationships between Ũ and ã, and λ0 and ã of the mode-2 ISWs in the study area, both show the characteristics of multi-parameter controlling. Seawater depth (seafloor topography), pycnocline depth, and
pycnocline thickness all have an influence on the mode-2 ISW propagation speed in
the study area. This causes the diversity of the relationship between Ũ and ã.
The vertical structure of the mode-2 ISW amplitude in the study area is
affected by the degree of nonlinearity of the ISW. The first-order nonlinear vertical mode function considering nonlinear correction can be used to better fit the vertical amplitude structure of the mode-2 ISWs with strong nonlinearity. The first-order nonlinear vertical mode function can also be used to better fit the vertical amplitude structure of the ISW position with strong nonlinearity in the vertical direction. The pycnocline deviation (especially the
downward deviation of the pycnocline) affects the vertical structure of the
mode-2 ISW amplitude in the study area. When the pycnocline center has a
large downward deviation, the vertical mode function cannot be used to well
fit the vertical structure of the mode-2 ISW amplitude in the study area.
Code and data availability
The full seismic data are provided by MGDS (the
Marine Geoscience Data System; http://www.marine-geo.org/), available for academic research at http://www.marine-geo.org/tools/search/entry.php?id=EW0412 (MGDS, 2022). The temperature and salinity data come from CMEMS (Copernicus Marine Environment Monitoring Service, https://resources.marine.copernicus.eu/product-download/GLOBAL_MULTIYEAR_PHY_001_030, login required; CMEMS, 2022).
Author contributions
The concept of this study was developed by HS
and extended upon by all involved. WF implemented the study and
performed the analysis with guidance from HS. YG, SY, and KZ collaborated in discussing the results and composing the manuscript.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Special issue statement
This article is part of the special issue “Nonlinear internal waves”. It is not associated with a conference.
Acknowledgements
We thank the captain, crew, and science party of R/V
Maurice Ewing cruise EW0412 for acquiring the seismic data. We appreciate MGDS and CMEMS for their supporting data used in this study.
Financial support
This research has been supported by the National Natural Science Foundation of China (grant nos. 41976048 and 42176061) and the National Key Research and Development Program of China (grant no. 2018YFC0310000).
Review statement
This paper was edited by Kateryna Terletska and reviewed by two anonymous referees.
ReferencesBai, Y., Song, H., Guan, Y., and Yang, S.: Estimating depth of polarity conversion of shoaling internal solitary waves in the northeastern South China Sea, Cont. Shelf Res., 143, 9–17, 10.1016/j.csr.2017.05.014, 2017.Benjamin, T. B.: Internal waves of permanent form in fluids of great depth,
J. Fluid Mech., 29, 559–592, 10.1017/S002211206700103X, 1967.Biescas, B., Sallarès, V., Pelegrí, J. L., Machín, F.,
Carbonell, R., Buffett, G., Dañobeitia, J. J., and Calahorrano, A.:
Imaging meddy finestructure using multichannel seismic reflection data,
Geophys. Res. Lett., 35, L11609, 10.1029/2008GL033971, 2008.Biescas, B., Armi, L., Sallarès, V., and Gràcia, E.: Seismic imaging
of staircase layers below the Mediterranean Undercurrent, Deep-Sea Res.
Pt. I, 57, 1345–1353, 10.1016/j.dsr.2010.07.001, 2010.Bogucki, D. J., Redekopp, L. G., and Barth, J.: Internal solitary waves in
the Coastal Mixing and Optics 1996 experiment: Multimodal structure and
resuspension, J. Geophys. Res.-Oceans, 110, C02024, 10.1029/2003JC002253, 2005.Brandt, A. and Shipley, K. R.: Laboratory experiments on mass transport by
large amplitude mode-2 internal solitary waves, Phys. Fluids, 26,
046601, 10.1063/1.4869101, 2014.Carr, M., Davies, P. A., and Hoebers, R. P.: Experiments on the structure
and stability of mode-2 internal solitary-like waves propagating on an
offset pycnocline, Phys. Fluids, 27, 046602, 10.1063/1.4916881, 2015.CMEMS: Global Ocean Physics Reanalysis, GLOBAL_MULTIYEAR_PHY_001_030, Copernicus Marine Environment Monitoring Service [data set], https://resources.marine.copernicus.eu/product-download/GLOBAL_MULTIYEAR_PHY_001_030, last access: 30 March 2022.Chen, Z. W., Xie, J., Wang, D., Zhan, J. M., Xu, J., and Cai, S.: Density
stratification influences on generation of different modes internal solitary
waves, J. Geophys. Res.-Oceans, 119, 7029–7046, 10.1002/2014JC010069, 2014.Cheng, M. H., Hsieh, C. M., Hwang, R. R., and Hsu, J. R. C.: Effects of
initial amplitude and pycnocline thickness on the evolution of mode-2
internal solitary waves, Phys. Fluids, 30, 042101, 10.1063/1.5020093, 2018.Da Silva, J. C. B., New, A. L., and Magalhaes, J. M.: On the structure and
propagation of internal solitary waves generated at the Mascarene Plateau in
the Indian Ocean, Deep-Sea Res. Pt. I, 58, 229–240, 10.1016/j.dsr.2010.12.003, 2011.Deepwell, D., Stastna, M., Carr, M., and Davies, P. A.: Wave generation
through the interaction of a mode-2 internal solitary wave and a broad,
isolated ridge, Physical Review Fluids, 4, 094802, 10.1103/PhysRevFluids.4.094802, 2019.Fan, W., Song, H., Gong, Y., Sun, S., Zhang, K., Wu, D., Kuang, Y., and
Yang, S.: The shoaling mode-2 internal solitary waves in the Pacific coast
of Central America investigated by marine seismic survey data, Cont. Shelf Res., 212, 104318, 10.1016/j.csr.2020.104318, 2021a.Fan, W., Song, H., Gong, Y., Zhang, K., and Sun, S.: Seismic oceanography
study of mode-2 internal solitary waves offshore Central America, Chinese
J. Geophys.-Ch., 64, 195–208, 10.6038/cjg2021O0071, 2021b.Fer, I., Nandi, P., Holbrook, W. S., Schmitt, R. W., and Páramo, P.: Seismic imaging of a thermohaline staircase in the western tropical North Atlantic, Ocean Sci., 6, 621–631, 10.5194/os-6-621-2010, 2010.Fliegel, M. and Hunkins, K.: Internal wave dispersion calculated using the
Thomson-Haskell method, J. Phys. Oceanogr., 5, 541–548,
10.1175/1520-0485(1975)005<0541:IWDCUT>2.0.CO;2, 1975.Fulthorpe, C. and McIntosh, K.: Raw Multi-Channel Seismic Shot Data from
the Sandino Basin, offshore Nicaragua, acquired during R/V Maurice Ewing expedition EW0412 (2004), Interdisciplinary Earth Data Alliance (IEDA) [data set], 10.1594/IEDA/309938, 2014.Geng, M., Song, H., Guan, Y., and Bai, Y.: Analyzing amplitudes of internal
solitary waves in the northern South China Sea by use of seismic
oceanography data, Deep-Sea Res. Pt. I, 146, 1–10, 10.1016/j.dsr.2019.02.005, 2019.Gong, Y., Song, H., Zhao, Z., Guan, Y., and Kuang, Y.: On the vertical
structure of internal solitary waves in the northeastern South China Sea,
Deep-Sea Res. Pt. I, 173, 103550, 10.1016/j.dsr.2021.103550, 2021.Grimshaw, R., Pelinovsky, E., and Poloukhina, O.: Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface, Nonlin. Processes Geophys., 9, 221–235, 10.5194/npg-9-221-2002, 2002.Grimshaw, R., Pelinovsky, E., Talipova, T., and Kurkin, A.: Simulation of
the transformation of internal solitary waves on oceanic shelves, J.
Phys. Oceanogr., 34, 2774–2791, 10.1175/JPO2652.1, 2004.Holbrook, W. S. and Fer, I.: Ocean internal wave spectra inferred from
seismic reflection transects, Geophys. Res. Lett., 32, L15604,
10.1029/2005GL023733, 2005.Holbrook, W. S., Páramo, P., Pearse, S., and Schmitt, R. W.:
Thermohaline fine structure in an oceanographic front from seismic
reflection profiling, Science, 301, 821–824, 10.1126/science.1085116, 2003.Holbrook, W. S., Fer, I., Schmitt, R. W., Lizarralde, D., Klymak, J. M.,
Helfrich, L. C., and Kubichek, R.: Estimating oceanic turbulence dissipation
from seismic images, J. Atmos. Ocean. Tech., 30,
1767–1788, 10.1175/JTECH-D-12-00140.1, 2013.Holloway, P. E., Pelinovsky, E., and Talipova, T.: A generalized Korteweg-de
Vries model of internal tide transformation in the coastal zone, J. Geophys. Res.-Oceans, 104, 18333–18350, 10.1029/1999JC900144, 1999.
Kozlov, V. F. and Makarov, V. G.: On a class of stationary gravity currents
with the density jump, Izv. An. SSSR Fiz. Atm.+, 26, 395–402, 1990.Krahmann, G., Papenberg, C., Brandt, P., and Vogt, M.: Evaluation of seismic
reflector slopes with a Yoyo-CTD, Geophys. Res. Lett., 36, L00D02,
10.1029/2009GL038964, 2009.Kurkina, O., Talipova, T., Soomere, T., Giniyatullin, A., and Kurkin, A.: Kinematic parameters of internal waves of the second mode in the South China Sea, Nonlin. Processes Geophys., 24, 645–660, 10.5194/npg-24-645-2017, 2017.Lamb, K. G. and Yan, L.: The evolution of internal wave undular bores:
comparisons of a fully nonlinear numerical model with weakly nonlinear
theory, J. Phys. Oceanogr., 26, 2712–2734, 10.1175/1520-0485(1996)026<2712:TEOIWU>2.0.CO;2, 1996.Liao, G., Xu, X. H., Liang, C., Dong, C., Zhou, B., Ding, T., Huang, W., and
Xu, D.: Analysis of kinematic parameters of internal solitary waves in the
northern South China Sea, Deep-Sea Res. Pt. I, 94, 159–172, 10.1016/j.dsr.2014.10.002, 2014.Liu, A. K., Su, F. C., Hsu, M. K., Kuo, N. J., and Ho, C. R.: Generation and
evolution of mode-two internal waves in the South China Sea, Cont. Shelf Res., 59, 18–27, 10.1016/j.csr.2013.02.009, 2013.Maderich, V., Jung, K. T., Terletska, K., Brovchenko, I., and Talipova, T.:
Incomplete similarity of internal solitary waves with trapped cores, Fluid
Dyn. Res., 47, 035511, 10.1088/0169-5983/47/3/035511, 2015.Maxworthy, T.: Experiments on solitary internal Kelvin waves, J. Fluid Mech., 129, 365–383, 10.1017/S0022112083000816, 1983.McSweeney, J. M., Lerczak, J. A., Barth, J. A., Becherer, J., Colosi, J. A.,
MacKinnon, J. A., MacMahan, J. H., Moum, J. N., Pierce, S. D., and
Waterhouse, A. F.: Observations of shoaling nonlinear internal bores across
the central California inner shelf, J. Phys. Oceanogr., 50,
111–132, 10.1175/JPO-D-19-0125.1, 2020a.McSweeney, J. M., Lerczak, J. A., Barth, J. A., Becherer, J., MacKinnon, J.
A., Waterhouse, A. F., Colosi, J. A., MacMahan, J. H., Feddersen, F.,
Calantoni, J., Simpson, A., Celona, S., Haller, M. C., and Terrill, E.:
Alongshore variability of shoaling internal bores on the inner shelf,
J. Phys. Oceanogr., 50, 2965–2981, 10.1175/JPO-D-20-0090.1, 2020b.MGDS: Raw Multi-Channel Seismic Shot Data from the Sandino Basin, offshore Nicaragua, acquired during R/V Maurice Ewing expedition EW0412 (2004), Marine Geoscience Data System [data set], http://www.marine-geo.org/tools/search/entry.php?id=EW0412, last access: 30 March 2022.Olsthoorn, J., Baglaenko, A., and Stastna, M.: Analysis of asymmetries in propagating mode-2 waves, Nonlin. Processes Geophys., 20, 59–69, 10.5194/npg-20-59-2013, 2013.Pinheiro, L. M., Song, H., Ruddick, B., Dubert, J., Ambar, I., Mustafa, K.,
and Bezerra, R.: Detailed 2-D imaging of the Mediterranean outflow and
meddies off W Iberia from multichannel seismic data, J. Marine Syst., 79, 89–100, 10.1016/j.jmarsys.2009.07.004, 2010.Ramp, S. R., Yang, Y. J., Reeder, D. B., Buijsman, M. C., and Bahr, F. L.: The evolution of mode-2 nonlinear internal waves over the northern Heng-Chun Ridge south of Taiwan, Nonlin. Processes Geophys., 22, 413–431, 10.5194/npg-22-413-2015, 2015.Rayson, M. D., Jones, N. L., and Ivey, G. N.: Observations of
large-amplitude mode-2 nonlinear internal waves on the Australian North West
shelf, J. Phys. Oceanogr., 49, 309–328, 10.1175/JPO-D-18-0097.1, 2019.Ruddick, B., Song, H. B., Dong, C., and Pinheiro, L.: Water column seismic
images as maps of temperature gradient, Oceanography, 22, 192–205,
10.5670/oceanog.2009.19, 2009.Sallares, V., Mojica, J. F., Biescas, B., Klaeschen, D., and Gràcia, E.:
Characterization of the sub-mesoscale energy cascade in the Alboran Sea
thermocline from spectral analysis of high-resolution MCS data, Geophys. Res. Lett., 43, 6461–6468, 10.1002/2016GL069782, 2016.Salloum, M., Knio, O. M., and Brandt, A.: Numerical simulation of mass
transport in internal solitary waves, Phys. Fluids, 24, 016602,
10.1063/1.3676771, 2012.Sheen, K. L., White, N. J., and Hobbs, R. W.: Estimating mixing rates from
seismic images of oceanic structure, Geophys. Res. Lett., 36,
L00D04, 10.1029/2009GL040106, 2009.Sheen, K. L., White, N., Caulfield, C. P., and Hobbs, R. W.: Estimating
geostrophic shear from seismic images of oceanic structure, J. Atmos. Ocean. Tech., 28, 1149–1154, 10.1175/JTECH-D-10-05012.1, 2011.Shroyer, E. L., Moum, J. N., and Nash, J. D.: Mode 2 waves on the
continental shelf: Ephemeral components of the nonlinear internal wavefield,
J. Geophys. Res.-Oceans, 115, C07001, 10.1029/2009JC005605, 2010.Small, R. J. and Hornby, R. P.: A comparison of weakly and fully non-linear
models of the shoaling of a solitary internal wave, Ocean Model., 8,
395–416, 10.1016/j.ocemod.2004.02.002, 2005.Song, H., Chen, J., Pinheiro, L. M., Ruddick, B., Fan, W., Gong, Y., and
Zhang, K.: Progress and prospects of seismic oceanography, Deep-Sea Res. Pt. I, 177, 103631, 10.1016/j.dsr.2021.103631, 2021.Stamp, A. P. and Jacka, M.: Deep-water internal solitaty waves, J. Fluid Mech., 305, 347–371, 10.1017/S0022112095004654, 1995.Stastna, M. and Lamb, K. G.: Large fully nonlinear internal solitary waves:
The effect of background current, Phys. Fluids, 14, 2987–2999,
10.1063/1.1496510, 2002.Sun, S. Q., Zhang, K., and Song, H. B.: Geophysical characteristics of
internal solitary waves near the Strait of Gibraltar in the Mediterranean
Sea, Chinese J. Geophys.-Ch., 62, 2622–2632,
10.6038/cjg2019N0079, 2019.Tang, Q., Wang, C., Wang, D., and Pawlowicz, R.: Seismic, satellite, and
site observations of internal solitary waves in the NE South China Sea,
Sci. Rep., 4, 5374, 10.1038/srep05374, 2014.
Tang, Q., Xu, M., Zheng, C., Xu, X., and Xu, J.: A locally generated
high-mode nonlinear internal wave detected on the shelf of the northern
South China Sea from marine seismic observations, J. Geophys. Res.-Oceans, 123, 1142–1155, 10.1002/2017JC013347, 2018.Terez, D. E. and Knio, O. M.: Numerical simulations of large-amplitude
internal solitary waves, J. Fluid Mech., 362, 53–82, 10.1017/S0022112098008799, 1998.Terletska, K., Jung, K. T., Talipova, T., Maderich, V., Brovchenko, I., and
Grimshaw, R.: Internal breather-like wave generation by the second mode
solitary wave interaction with a step, Phys. Fluids, 28, 116602,
10.1063/1.4967203, 2016.Tsuji, T., Noguchi, T., Niino, H., Matsuoka, T., Nakamura, Y., Tokuyama, H.,
Kuramoto, S. I., and Bangs, N.: Two-dimensional mapping of fine structures in
the Kuroshio Current using seismic reflection data, Geophys. Res. Lett., 32, L14609, 10.1029/2005GL023095, 2005.Vlasenko, V., Brandt, P., and Rubino, A.: Structure of large-amplitude
internal solitary waves, J. Phys. Oceanogr., 30, 2172–2185,
10.1175/1520-0485(2000)030<2172:SOLAIS>2.0.CO;2, 2000.Xu, J., He, Y., Chen, Z., Zhan, H., Wu, Y., Xie, J., Shang, X., Ning, D.,
Fang, W., and Cai, S.: Observations of different effects of an anti-cyclonic
eddy on internal solitary waves in the South China Sea, Prog.
Oceanogr., 188, 102422, 10.1016/j.pocean.2020.102422, 2020.Yang, Y. J., Fang, Y. C., Chang, M. H., Ramp, S. R., Kao, C. C., and Tang, T. Y.: Observations of second baroclinic mode internal solitary waves on the
continental slope of the northern South China Sea, J. Geophys. Res.-Oceans, 114, C10003, 10.1029/2009JC005318, 2009.