1. Introduction
The Luzon Strait (LS) is the only deep-water passage linking the western Pacific Ocean and the South China Sea (SCS). The LS is a hot spot for oceanographic research owing to its abundant dynamic marine processes. Energetic internal tides are one of the important dynamic processes and are generated by robust barotropic tides propagating across the two steep submarine ridges (Buijsman et al. 2012). These internal tides originate from the LS and transfer approximately 7.0–9.4 GW and 6.0–8.7 GW of energy into the SCS and the western Pacific Ocean, respectively (Alford et al. 2011; Jan et al. 2008; Xu et al. 2016). In the SCS, energetic internal tides are an important source of deep-sea mixing (Wang et al. 2016) and play a significant role in driving SCS circulation (Quan and Xue 2019; Xie et al. 2018). Additionally, westward-propagated internal tides steepen nonlinearly and break to generate large-amplitude internal solitary waves (Li and Farmer 2011; Zhao et al. 2004b), which can be observed in satellite images (Fig. 1a) (Bai et al. 2014; Li et al. 2008). Notably, the appearance of internal solitary waves is more closely related to semidiurnal than diurnal internal tides due to the modulation of Earth’s rotation (Li and Farmer 2011).
(a) Snapshot SSH fields induced by mode-1 M2 internal tides (background color indicates the UTC phase time amplitude) and signatures of internal solitary waves (black lines; Zhao et al. 2004a) west of the Luzon Strait. (b) Location map of the mooring array. The PIESs (C13–C40) and mooring stations are indicated by magenta dots (the lost C19 station is shown as a black dot) and red pentagrams, respectively. (c) Schematic diagram of two mooring configurations. The depths of the corresponding instruments (upward- and downward-looking 75-kHz ADCP, RCMs, and temperature chain) are marked.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
The propagation features of semidiurnal internal tides generated from the LS have been characterized by numerical models (Alford et al. 2015; Jan et al. 2008; Kerry et al. 2013; Niwa and Hibiya 2004; Xu et al. 2016) and satellite observations (Zhao 2014, 2020). The semidiurnal internal tides are dominated by M2 internal tides, which propagate into the SCS as two beams: an energetic northwestward beam north of 20°N and a weak southwestward beam south of 20°N. These M2 internal tides are also generated at the continental slope of the southern Taiwan Strait and propagate southward (Jan et al. 2008; Xu et al. 2016; Zhao 2020). Zhao (2020) reported that the superposition of the northwestward and southward beams could generate an interference pattern that shows significant spatial variations in its magnitude and direction of energy flux. As these two beams propagate in nearly opposite directions near the LS, they are expected to superimpose on each other and form standing waves. So far, no studies have focused on this phenomenon west of the LS.
Field observations have revealed standing waves generated by internal tides in various regions in global oceans (Alford et al. 2006; Eich et al. 2004; Johnston et al. 2015; Martini et al. 2007; Nash et al. 2006; Rainville et al. 2010; Rayson et al. 2012, 2021, 2011). Using mooring observations, Alford et al. (2006), Eich et al. (2004), and Martini et al. (2007) reported partial standing internal tides in Mamala Bay on the south coast of Oahu, which were composed of converging free waves from the east and west. Nash et al. (2006) calculated the ratios between the baroclinic horizontal kinetic energy and available potential energy based on moorings at five stations across Kaena Ridge. They found that the ratios in this section satisfied the spatial features of standing waves. The standing-wave pattern was also reported at the Australian North West Shelf (Rayson et al. 2012, 2021, 2011), where the nodes and antinodes vary in position due to seasonally varying stratification. Based on glider data, Johnston et al. (2015) identified standing internal tides in the Tasman Sea, produced by the Tasmanian slope reflecting the incident internal tides emanating from the Macquarie Ridge. The potential energy of these standing internal tides showed an antinode-node structure, whereas the kinetic energy followed an opposite pattern.
The advent of satellite altimetry measurements has provided a new method for exploring sea surface height (SSH) variations induced by internal tides. Previous studies have revealed that the phase lag of progressive waves increased in the direction of propagation, whereas standing waves were characterized by nearly constant phase lags, with 180° jumps at nodal points where the amplitude drops to zero (Ray and Mitchum 1996, 1997; Zhao and Alford 2009). Therefore, the spatial amplitude and phase obtained from satellite observations can be used to determine the presence of standing waves. Satellite observations typically provide gridded datasets of internal tides, which can be used to separate the components that contribute to standing wave formation. For example, Zhao and Alford (2009) separated standing waves into two components propagating southward and northward in the central North Pacific Ocean. In the present study, satellite observations of internal tides were used to confirm the presence of standing waves and investigate the components of internal tides contributing to these standing waves.
This study discovered a standing wave comprising the M2 internal tides west of the LS by a large-scale observation array, and its temporal and spatial features were investigated. The rest of this study is structured in the following manner. Section 2 details pressure-recording inverted echo sounder (PIES) data, satellite altimeter measurements, and the methods employed in this study. Section 3 describes the spatial features of standing waves and their modulation by an anticyclonic eddy. Section 4 summarizes our findings and presents a discussion of this study.
2. Data and methods
a. Field observation data
From June 2018 to August 2019, 28 PIESs and two moorings were deployed west of the LS (Zhao et al. 2023; Zheng et al. 2022a,b), almost covering the west entrance of the LS, where energetic internal tides and internal solitary waves were recorded (Fig. 1a). C19 was lost due to technical issues. Except for C16, C17, and C32, which obtained incomplete data on the round-trip acoustic travel time τ, all other stations obtained complete data for τ and bottom pressure Pbot during the entire observational period. Following the same procedure as Kennelly et al. (2007), Park and Watts (2006), and Zhao et al. (2018), τ (24 pings per hour) and Pbot (six times per hour) were processed into hourly samples.
Two moorings equipped with a thermistor chain were deployed near C21 and C27 (Fig. 1c). The temperature was recorded at 15 depths (450–2950 m) and 37 depths (400–2200 m) for M01 and M02, respectively, with a temporal sampling interval of 10 min. The data were linearly interpolated to a uniform 5-m sample interval and were also processed into hourly samples [TM01(z, t) and TM02(z, t)].
Using the gravest empirical mode (GEM) technique, Zhao et al. (2023) derived a three-dimensional daily time series of velocity U, temperature T, and salinity S field datasets from τ with statistical information from historical hydrographic temperature and salinity profiles. These datasets, referred to as GEM datasets, were in agreement with ADCP and CTD observations and were utilized to calculate the phase speed of the internal tide.
b. Satellite altimeter products
c. Vertical displacement
To obtain the isopycnal vertical displacement of the semidiurnal internal tides at each station, the conversion relation from the internal tide–induced acoustic travel time τIT to the thermocline vertical displacement was used (Li et al. 2009). The τIT was obtained after bandpassing the adjusted travel time τIT_ad with the cutoff periods of 11.5 and 13.1 h; τIT_ad was calculated by removing barotropic tide-related travel time τBT from the observed τ (Zhao et al. 2018). The conversion ratio is 24.0 m ms−1 between the isopycnal vertical displacement to the τIT (Li et al. 2009, 2016; Park and Farmer 2013; Ramp et al. 2019; Zhao et al. 2018). Specifically, an increase or decrease of 1 ms in τIT corresponds to elevation or depression, respectively, of the thermocline by 24 m at the depth of the maximum vertical mode-1 eigenfunction Φ1. The vertical displacements of mode-1 semidiurnal internal tides (ITobs) at each station are shown in Fig. 2.
(a) Comparison between the vertical displacements of semidiurnal ITs obtained from (a) C21 and the thermistor chain at M01 and (b) C27 and M02. (c) Amplitudes and (d) phases of tidal constituents obtained from C21 and thermistor chain at M01. (e),(f) As in (c) and (d), but for C27 and M02.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
To validate the accuracy of ITobs, the independent thermistor chains of M01 and M02 deployed near C21 and C27 were used to calculate the isothermal vertical displacement of the mode-1 semidiurnal internal tides (Fig. 1b). Considering the temperature chain data of M01 [TM01(z, t)] as an example, the vertical displacement η at M01 was calculated using the relation η(z, t) = T′(z, t)/Tz(z) following Alford (2003); where T′(z, t) is the bandpass-filtered (11.5–13.1 h) result based on TM01(z, t) and Tz(z) is the vertical temperature gradient calculated from the mean of the entire observational time series. Note that η consists of orthogonal vertical modes and can be expressed as
Comparisons between the vertical displacements calculated from the PIESs (C21 and C27) and thermistor chains (M01 and M02) are shown in Fig. 2. The time series from PIESs and thermistor chains showed similar spring–neap cycles (Figs. 2a,b). However, the vertical displacement of the daily variation was slightly different due to the PIESs and mooring stations being located approximately 12 km apart. The harmonic constants of the four semidiurnal internal tides calculated from the vertical displacement time series using harmonic analysis (Pawlowicz et al. 2002) were in good agreement (Figs. 2c–k). The data from the PIESs and thermistor chains showed that M2 was dominant in semidiurnal internal tides. In the northern portion of the study area, the amplitude of the M2 internal tide was 14.6 and 14.7 m for C21 and M01, respectively, and the phase of the M2 internal tide was 350.1° and 359.6° for C21 and M01, respectively. In the southern portion of the study area, the amplitude of the M2 internal tide was 12.3 and 13.9 m for C27 and M02, respectively, and the phase of the M2 internal tide was 162.2° and 164.7° for C27 and M02, respectively. This study focused on the features of the M2 internal tides. The amplitude and phase derived from PIESs and moorings were consistent in the northern and southern regions of the study area. Therefore, the derivation method from the round-trip travel time was feasible, and the harmonic constituents of the M2 internal tides obtained from the PIESs were reliable at each station.
d. Two-dimensional bandpass filter
Because only the mode-1 M2 internal tide was used, the mode-2 component of MIOST-IT needed to be removed using the 2D bandpass filtering method supplied by Zhao et al. (2019). Briefly, the 2D wavenumber spectrum (Fig. 3b) was calculated using the internal tide–induced SSH field (Fig. 3a). The magenta cycle indicates the theoretical mode-1 M2 horizontal wavenumber K with a value of 7.93 × 10−6 m−1 calculated using K = (ω2 − f2)1/2/C1, where ω and f are the tidal and inertial frequencies, and C1 indicates the averaged mode-1 horizontal phase speed in the study area derived from normal mode analysis. According to the wavenumber spectrum, the mode-1 component is converted back to physical space by applying the inverse Fourier transform to the spatial wavenumber spectrum filtering, with a cutoff band of 0.8K and 1.25K (Zhao 2020).
The M2 internal tide–induced SSH field and its horizontal 2D wavenumber spectrum: (a) Snapshot SSH field of M2 internal tide from raw MIOST-IT. (b) 2D wavenumber spectrum using the SSH data from (a). The magenta-lined circle indicates the theoretical wavenumber, and the blue-lined circles indicate the filter’s cutoff wavenumbers, which are 0.8 and 1.25 times the theoretical wavenumber. (c) The mode-1 components of M2 internal tide after 2D bandpass filtering. The black line indicates the isobathic contour at 200 m.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
The mode-1 M2 internal tides are shown in Fig. 3c, and the spatially filtered field was smooth. Ubelmann et al. (2022) provided the mode-1 M2 internal tides around the LS (Fig. 7a in their study), which are consistent with the results in Fig. 3c. Therefore, the mode-1 M2 internal tides were extracted from raw MIOST-IT, and are hereinafter designated as MIOST-IT.
e. Directional Fourier filter
The directional Fourier filter (DFF) method separates standing internal tidal signals into different propagation directions (Gong et al. 2021). This is a more general version of the Hilbert transform filter (Mercier et al. 2008), which involves taking a 2D Fourier transform of the complex spatial internal tidal amplitude, filtering out the desired horizontal wavenumbers, and then applying the inverse Fourier transform to obtain the internal tides in the expected direction (Rayson et al. 2021). Note that this method differs from the 2D plane wave fit method described by Zhao et al. (2016), which requires a theoretical horizontal wavenumber. In addition, this method is suited to a gridded internal tidal dataset.
The DFF method has been validated in other studies (Gong et al. 2021; Rayson et al. 2021; Wang et al. 2021), and we also applied this method to the internal tidal dataset (Fig. 4a) provided by Zhao (2020) and obtained the westward component (Fig. 4c), which agreed with the results obtained using the 2D plane wave fit method (Fig. 4b) in Zhao (2020).
Comparison of the method to resolve multiple internal tides using the 2D Plane Wave Fit method and the DFF: (a) The spatial features of M2 internal tide west of the LS supplied by Zhao (2020). The westward component of the M2 internal tide was extracted using (b) the 2D Plane Wave Fit method [data provided by Zhao (2020)] and (c) the DFF method. The mapping colors indicate internal tide–induced amplitudes at the sea surface, and the corresponding isophase is denoted by magenta lines. The black line indicates the isobathic contour at 200 m.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
3. Results
a. Features of standing waves
The amplitudes of the predominant M2 internal tide at each station (green bars in Fig. 5) were obtained by applying harmonic analysis to the full records of semidiurnal internal tidal displacements. The amplitudes of M2 were discontinuously variable from a spatial perspective, with several small-amplitude (<∼10 m) stations (i.e., C15, C20, C23, C26, C34, and C38–C40) sporadically distributed among all observed stations (Fig. 5). The small amplitudes observed west of the study area (C38–C40) can be explained by the fact that M2 internal tides are generated in the LS, and the mode-1 components gradually decreased from east to west with their westward propagation. Additionally, C26 and C34 were located between the two beams of M2 (Alford et al. 2015; Zhao 2020), where the internal tide was weak. However, the amplitudes of M2 at stations C15, C20, and C23 were significantly smaller than the neighboring stations north of 20°N. For instance, the amplitude of C15 was 9.2 m, while those of C14 and C17 were 21.6 and 22.7 m, respectively. Similarly, the amplitude of C20 was 10.1 m, while that of C21 was 14.5 m; the amplitude of C23 was 8.8 m, while those of C22 and C24 were 17.1 and 18.2 m, respectively. According to the temporally varying amplitudes of M2 internal tides (magenta lines in Fig. 5), significant intraseasonal variations were observed. Therefore, the spatial characteristics of the average amplitudes mentioned above can change. Specifically, north of 20°N, these intraseasonal variations resulted in larger amplitudes at the small-amplitude station and smaller amplitudes at the large-amplitude station. For example, in December, the monthly mean amplitude at C20 was 14.6 m, while that at C21 was 11.4 m. However, their respective mean values during the observation period were 10.1 and 14.5 m. This is discussed further in section 3c.
Amplitude of M2 internal tide at each station (m). Green bars indicate the amplitudes from all observation records based on harmonic analysis, and the value (m) is labeled at the top of the bar. Magenta lines indicate the temporally varying tidal amplitudes from harmonic analysis for the respective observations with a 28-day window. Different shaded colors represent different seasons.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
This spatial variation of amplitudes was also observed from the satellites (Fig. 6a). The amplitudes demonstrated significantly low- and high-value stripes north of 20°N, and the small-amplitude stations (i.e., C15, C20, and C23) were located on the low-value stripes. The average theoretical wavelength λ in the study area (20°–22.5°N, 118.5°–120.8°E) was approximately 134 km according to normal mode analysis. The distances between the two high-value and two low-value stripes were approximately equal to one-half of the wavelength. As reported in the central North Pacific (Zhao 2016; Zhao and Alford 2009), Australian North West Shelf (Rayson et al. 2021), and Tasman Sea (Johnston et al. 2015), the half-wavelength features in amplitude were related to standing waves. Low-value and high-value stripes in this pattern corresponded to nodes and antinodes of the standing wave, respectively.
The amplitudes and phases of M2 internal tide presented by MIOST-IT and PIES observations: (a) The mapping color and filled circles (both size and color) represent the amplitudes of M2 internal tide based on MIOST-IT and PIES observations. (b) The mapping color and filled-circle color represent the phase of M2 internal tide; black contours indicate isophases with an interval of 30°. The magenta line is an auxiliary line passing through stations C15, C22, and C30. (c) The amplitudes of each station were derived from MIOST-IT (red line) and PIES observations (blue line). The green shadings indicate stations near the continental slope. The magenta shadings indicate that the two datasets agreed well at these stations. (d) As in (c), but for phase comparison.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
The phase pattern of MIOST-IT is shown in Fig. 6b and is consistent with the PIES observations. According to the variation of isophases, a southwestward propagation beam was located south of 19.5°N, whereas the internal tide propagated mainly westward north of 19.5°N. In addition, the phase pattern exhibited complicated multiwave interference features because of the nonmonotonic phase variations (Fig. 6b).
The amplitudes and phases of MIOST-IT at most stations were consistent with the PIES observations, as shown in Figs. 6c and 6d. However, comparisons at the stations (i.e., C13, C20, C29, and C36) near the continental slope show weak consistency, which may be attributed to the limitation of altimetry in filtering out the barotropic and internal tide signals in shallow water (Rayson et al. 2021). However, the amplitudes and phases of C14–C17, C21–C24, C30–C33, and C37–C38 were highly consistent north of 20°N, where the observed amplitudes exhibited significant spatial variation. The root-mean-square errors (RMSEs) of amplitude and phase at these stations were 1.4 m and 14.0°, respectively. As a result, PIESs and MIOST-IT showed discontinuous variations in the M2 internal tide north of 20°N, with significant spatial variation in amplitude.
The amplitudes and phases of the PIES observations and MIOST-IT along the auxiliary line are shown in Fig. 7 to demonstrate that the spatially varying amplitudes resulted from standing waves. The amplitude exhibited three complete crests and four troughs along this auxiliary line (Fig. 7a). The average distance between the wave crests was 75.9 km, and that between troughs was 73.0 km. Notably, the estimated average wavelength of the M2 internal tide near the auxiliary line is ∼146 km, approximately double the average distance between the wave crests or troughs, consistent with the previous description that the amplitude demonstrates half-wavelength modulation. The wave crests and troughs corresponded to the antinodes and nodes of the standing waves, respectively. Additionally, along this auxiliary line, the phase changed nonmonotonically. It exhibited a 180° jump at the wave node at 40 and 100 km, which is also consistent with the typical features of standing waves (Fig. 7b). The spatial amplitude and phase of the standing waves were also observed at three PIES stations on the auxiliary line.
(a) Amplitude and (b) phase of the M2 internal tide along the auxiliary line (Figs. 6a,b) from MIOST-IT. The auxiliary line close to the slope is set as the coordinate origin; the corresponding longitude along the auxiliary line is labeled on the top. The magenta circles indicate the amplitudes and phases derived from PIESs at C15, C22, and C30. (c) Amplitude and (d) phase of a perfect standing wave (black line) superimposed from northwestward-propating (NW; thick red line) and southeastward-propagating (SE; thin blue line) waves along the auxiliary line when they have equal amplitudes (10 m). (e),(f) As in (c) and (d), but for a partial standing wave superimposed from two waves of unequal amplitudes (8 and 10 m). Wave antinodes (orange squares) and nodes (purple triangles) are indicated.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
Two hypothetical M2 internal tides with wavelengths of 146 km, propagating in opposite directions (southeast and northwest) along the auxiliary line, are superimposed on each other to form standing waves. When two waves show the same amplitudes (Fig. 7c), a perfect (100%) standing wave is formed, which has zero amplitude (Fig. 7c, black line) at the wave node and twice the amplitude at the wave antinode. The phase (Fig. 7d, black line) shifts 180° at the nodes and is a piecewise constant between the nodes. In the case of unequal amplitudes (Fig. 7e), a partial standing wave is formed, which demonstrates half-wavelength modulation of the amplitude (Fig. 7e, black line) and a nonmonotonic increase of the phase (Fig. 7f, black line). The amplitude of the southeastward component (blue line) is 80% of that of the northwestward component (red line), which is defined as an 80% partial standing wave (Hall et al. 2017). The larger the number, the closer the partial standing wave is to a perfect standing wave (100%). The smaller the number, the closer it is to a progressive wave (0%).
The observed half-wavelength oscillations in amplitude (Fig. 7a) and phase (Fig. 7b) were consistent with the superposition of two internal tides propagating in opposite directions. According to the phase variations along the auxiliary line, the wave tended to be a perfect standing wave from 0 to 150 km. On the other hand, the wave was a partial standing wave from 150 to 280 km. A more detailed description is presented in the next section.
b. Compositions of standing waves
The DFF method was used to separate the M2 internal tide field into the northwest (Fig. 8a, filter band of 90°–180° from the east) and southeast (Fig. 8b, filter band of 270°–360°) propagating components based on MIOST-IT. The amplitude of the northwestward-propagating M2 (NW) component was up to 10 m near the LS, and became weaker as it moved westward (Fig. 8a). The southeastward-propagating M2 (SE) component, which originates from the continental slope of the southern Taiwan Strait, was weaker than that of the NW. The SE beam also appeared in several simulations (Jan et al. 2008; Xu et al. 2016) and satellite observations (Zhao 2020).
The snapshot of the (a) northwestward and (b) southeastward components of the M2 internal tide. Background color indicates the amplitude time phase; black contours indicate isophase lines with an interval of 40°; magenta points represent C15, C22, and C30 stations; and red and blue lines indicate auxiliary lines in Figs. 6a and 6b. The mapping color indicates the (c) amplitude and (d) phase of the M2 internal tide obtained by superimposing the northwestward and southeastward components. Scatters in (c) and (d) represent the amplitudes (color and size) and phase (color) observed by PIES observations. The black lines in (c) indicate two auxiliary lines (i.e., LC20 and LC21) across C20 and C21, respectively. Black contours in (d) indicate isophase lines with an interval of 30°.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
The amplitude and phase of the superposition of the two beams are presented in Figs. 8c and 8d, respectively. The amplitude pattern (Fig. 8c) showed a more pronounced striped feature with half-wavelength variations than the spatial amplitude of the observed standing wave (Fig. 6a). This superimposed amplitude pattern further confirms that the small-amplitude stations (C15, C20, and C23) are on the nodal stripe of the standing waves superimposed by NW and SE. The large-amplitude stations (C13–C14 and C21–C22) were on the antinode stripe. Therefore, the spatially varying amplitudes of M2 internal tides at these stations were observed. The superimposed phase pattern was generally consistent with the phases observed by the PIESs, and propagated northwest (Fig. 8d). This finding indicates that the standing wave is a partial standing wave, the NW component is dominant, and the propagation direction of the standing wave is consistent with that of the dominant component (Zhao et al. 2016).
The amplitude (Fig. 9a) and phase (Fig. 9b) of the NW and SE components along the auxiliary line were calculated to further characterize the composition of the standing waves. The amplitude of the NW decreased gradually from east to west, whereas that of the SE did not change significantly (Fig. 9a). Meanwhile, the phases of these two components propagated in opposite directions (Fig. 9b). As the amplitude of the NW was larger than that of the SE, the superposition created a partial standing wave. However, the amplitude difference between the NW and SE increased gradually from west to east along the auxiliary line. Within 100 km from the coordinate origin (Fig. 9a), the average amplitude of the SE was 85% that of the NW, indicating a partial standing wave (85%). However, since the amplitudes of SE and NW were very close, the partial standing wave tended to resemble a perfect standing wave. The amplitudes of the first (approximately 35 km) and second (approximately 108 km) nodes were only 2.2 and 2.4 m, respectively, and the phase presented a nearly 180° difference. When the distance exceeded 200 km, the amplitude of the SE was 69% of that of the NW, and the superposition of the two waves generated a partial standing wave (69%). In conclusion, the amplitudes of the SE and NW were close near the continental slope, resulting in a nearly perfect standing wave; whereas the NW was significantly larger than the SE near the LS, causing a partial standing wave.
(a) Amplitude and (b) phase of northwestward-propagating (red) and southeastward-propagating (blue) M2 internal tides along the auxiliary line. (c) Amplitude and (d) phase of standing wave superposed from northwestward- and southeastward-propagating M2 internal tides along the auxiliary line (green). Those from observations from MIOST-IT (black line) and PIESs (magenta circles) are also indicated; they are described in Figs. 7a and 7b.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
The superposition of the two components (Figs. 9c,d, green line) was generally consistent with the observations (Figs. 9c,d, black line) along the auxiliary line, indicating that the standing waves observed by MIOST-IT and PIESs were superimposed by NW and SE. Between 110–160 km and 250–280 km, the amplitudes from observations were greater than that from superposition, indicating the possible influence of internal tides in other propagation directions.
c. Modulation of the standing wave by an anticyclonic eddy
Based on the observations of PIES and satellite altimeter, the amplitude of the half-wavelength modulation can be attributed to the formation of standing waves. Previous studies have demonstrated that standing waves can be transformed into progressive waves (Hall et al. 2014; Zhao et al. 2012), or their wave nodes can shift (Rayson et al. 2012, 2021) due to varying stratification. To investigate the temporal variation in the standing wave west of the LS, the time-varying amplitudes of the M2 internal tides for all stations were calculated (magenta lines in Fig. 5). Focusing on the region with standing waves (north of 20°N), the seasonal variability of amplitudes was insignificant, as was the impact of seasonal stratification on standing-wave nodes. Although the intraseasonal variations in amplitude were significant, the variations among the stations were different. Notably, the time-varying amplitudes at C20 and C21 were exactly opposite, and the correlation coefficient was −0.73, even though the distance between the two stations was very small (32 km).
Based on the two parallel auxiliary lines (LC20 and LC21) passing through C20 and C21 (Fig. 8c), the relative positions of the two stations on the standing wave were obtained. C20 was located only 0.1 km from the wave node, while C21 was between the node and antinode, i.e., 17.7 km from the wave node and 20.0 km from the antinode. By projecting C20 parallel to LC21, the positions of two stations relative to LC21 were on two sides of the wave node (Fig. 10a). The amplitude of C21 was higher than that of C20 under normal conditions, which is consistent with the average amplitude observed by PIES and MIOST-IT at these two stations. However, if the wave node moved toward C21, the wave node was closer to C21 and farther from C20, and the amplitude of C21 decreased, whereas that of C20 increased (Fig. 10b). Therefore, shifts in wave nodes (standing waves) can cause opposite amplitude changes at stations located on either side of the node.
(a) The amplitudes observed at C20 (blue asterisks) and C21 (magenta asterisks) under normal conditions. The green solid line indicates the amplitudes of the M2 internal tide along LC21 in Fig. 8c. Positions of C20 (blue dashed line) and C21 (magenta dashed line) relative to LC21 are indicated. (b) The amplitudes observed at C20 (blue triangles) and C21 (magenta triangles) after the wave node shifts. The green dashed line indicates an 11-km southeastward movement of the solid green line, indicating a southeastward shift of the standing-wave node. Others in (b) are the same as (a). (c) Temporal amplitudes of M2 obtained from harmonic analysis with 28-day windows at C20 (blue) and C21 (magenta). (d) Averaged sea level anomaly around C20 and C21 from the Copernicus-Marine Environment Monitoring Service. Pink shadings indicate the periods when anticyclonic eddies covered the region around C20 and C21.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
The time-varying amplitudes of C20 and C21 were opposite, and the amplitude of C21 was mostly larger than that of C20 (Fig. 10c). However, during the three shadow periods, the C20 amplitude increased, whereas that of C21 decreased, so that the C20 amplitude was similar or even greater than C21. According to the above analysis, the opposite change in the amplitude at the two stations during the three time periods may have been caused by the wave node shifting toward C21. It is noted that according to the average sea level anomaly (SLA) around C20 and C21, anticyclonic eddies are found in all three periods (Fig. 10d).
To demonstrate the impact of the anticyclonic eddy, the period from 1 to 20 December 2018 was selected, during which the amplitude of C20 was much greater than that of C21. As shown in Fig. 11a, the region where standing waves were observed was covered by an anticyclonic eddy. A general standing wave spatial feature of the M2 internal tidal with spatially varying amplitudes, still existed during this period as shown in Fig. 11b. However, the amplitude of C20 was larger than that of C21, which is contrary to the average state (Fig. 8c). In addition, the amplitude of C13 was smaller than the average value, whereas the amplitudes of C14–C15 and C22–C23 were larger than their average values.
The influence of anticyclonic eddy on the standing wave: (a) Averaged SLA and geostrophic current from 1 to 20 Dec 2018. (b) The M2 amplitude obtained by superposing the NW and SE components (background color) under the normal condition and applying harmonic analysis to the PIES observations from 1 to 20 Dec 2018 (both size and color of the filled circles). Red and blue lines respectively indicate antinodes and nodes of the standing wave. PIES stations in (b) are indicated by magenta dots in (a). (c) Schematic diagram of the anticyclonic eddy affecting standing-wave nodes. The black circle denotes the anticyclonic eddy, and thin brown lines indicate SE and NW components of the M2 internal tide. Propagation direction without (dashed brown arrows) and with (dashed purple arrows) the influence of the eddy is indicated. Thick brown and purple lines indicate the superimposed standing-wave node without and with the influence of the eddy, respectively. (d) Schematic diagram of deflection of antinodes and nodes. Filled circles and lines are the same as in (b). Arrows indicate the deflection direction of the wave antinodes and nodes, and the black circle indicates the anticyclonic eddy. The black-outlined box in (a) is enlarged in (b)–(d).
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
Previous studies have shown that the wavefront of internal tides deflect in a clockwise direction after entering an anticyclonic eddy (Huang et al. 2018). The schematic in Fig. 11c illustrates that two counterpropagating internal tides (SE and NW; thin brown line) were superimposed to form a standing-wave node (thick brown line) under the normal condition without the influence of the eddy. When an anticyclonic eddy appears, the SE and NW components deflect clockwise after entering the eddy (thin purple line), generating a standing-wave node different from the normal condition (thick purple line). This suggests that the clockwise deflection of the standing-wave node modified by the anticyclonic eddy may cause the opposite change in the C20 and C21 amplitudes.
The positions of the wave nodes and antinodes under the normal condition were obtained based on the average amplitude of the M2 internal tide superposed by the NW and SE components (Fig. 11b; red and blue lines). From 1 to 20 December 2018, the standing wave area was dominated by the anticyclonic eddy, causing clockwise deflection of the standing wave antinodes and nodes (Fig. 11d). After clockwise deflection of the spatial nodes and antinodes, the amplitude of C20 decreased as the node moved away, while the amplitude of C21 increased as the node approached. Consequently, the amplitudes of C20 and C21 exhibited opposite variations during this period. In addition, the amplitude of C13 decreased because it was closer to the node and the amplitude of C22 increased because it was closer to the antinode (Fig. 11d). Similar phenomena also occurred during the other two anticyclonic eddy periods.
To demonstrate that the wave node shift was modified by mesoscale flows, the phase speed was calculated by solving the Taylor–Goldstein equation with time-varying background current and stratification during the eddy period (Huang et al. 2018). The three-dimensional daily time series of U, T, and S fields from the GEM datasets were used, and the theoretical phase speed was calculated based on the average GEM stratification during the observation period, without considering the background current. As shown in Fig. 12a, eddy-induced stratification increased the phase speed in the eddy area, with a mean value of approximately 0.08 m s−1. However, when only considering the eddy-induced background current (Fig. 12b), the westward phase speed in the northern and southern portions of the eddy appeared to decrease (i.e., with a mean value of approximately 0.19 m s−1) and increase (i.e., with a mean value of approximately 0.18 m s−1), respectively, owing to the superimposed background current (Chen et al. 2022; Huang et al. 2018; Park and Watts 2006). Notably, the influence of the background current was opposite for the eastward phase speed. Therefore, the SE and NW beams into the eddy were deflected clockwise because of the different phase speeds in the northern and southern portions of the eddy, which is consistent with previous descriptions. As shown in Fig. 12c, when considering both eddy-induced stratification and background current, the spatial variation of the phase speed is similar to that when only considering the background current. Consequently, the eddy-induced background current mainly modulates the wave node shift rather than the eddy-induced stratification variation.
Phase speed variations when only considering the eddy-induced (a) stratification, (b) background current, and (c) both eddy-induced stratification and background current. The mapping color indicates the phase speed variation from the reference value, calculated based on the average GEM stratification during the observation period, without considering the background current. Black lines represent zero contours. The colored lines and arrows in (c) are the same as those in Fig. 11d.
Citation: Journal of Physical Oceanography 53, 9; 10.1175/JPO-D-23-0043.1
When an anticyclonic eddy dominated the region, the nodes of the standing wave deflected clockwise, further modifying the amplitude of the standing wave. The effect of the cyclonic eddy was not analyzed owing to the scarcity of cyclonic eddy occurrences in the region where standing waves were observed during our observation (i.e., the SLA values during the observation are positive in Fig. 10d). However, the effects of eddies on the characteristics of standing waves still require quantitative descriptions, as the deflection angle of the wave nodes is unknown. In addition, internal tides propagating in other directions may also affect standing wave characteristics. Further studies, including numerical simulations, are needed to reveal the detailed effects of eddies and internal tides in other directions.
4. Summary and discussion
The standing wave features of the mode-1 M2 internal tide west of the Luzon Strait were characterized based on a large-scale observation array, together with satellite altimeter measurements (the MIOST-IT dataset) in this study. These two datasets are complementary, the PIESs provide a high-resolution temporal view of standing waves, whereas satellite altimeters provide broad spatial coverage. Both of them capture the mode-1 internal tides, and are used to investigate the temporal variation and formation mechanism of the standing waves, respectively.
The standing waves exhibited nonmonotonic spatial phase shifts and half-wavelength amplitude modulation. According to MIOST-IT, the standing wave results from interference between the northwestward and southeastward beams from the LS and the slope of the southern Taiwan Strait, respectively. The amplitudes of the northwestward beam were significantly larger than those of the southeastward beam near the LS, resulting in a partial standing wave. Near the slope of the southern Taiwan Strait, the amplitudes of the northwestward beam were slightly larger, and the superimposed wave tended to be a perfect standing wave. When an anticyclonic eddy dominated the region, the nodes and antinodes of the standing wave deflected clockwise under the influence of the eddy-induced background, further modifying the intraseasonal variation in the amplitude of the standing wave. C20 and C21 are located on two sides of the node, and when the node deflected clockwise, the amplitude of C20 increased, whereas that of C21 decreased, resulting in opposite amplitude variations at the two stations during the three anticyclonic eddy events.
This study reported the standing wave features west of the LS for the first time. Due to the previous lack of large-scale field observations, standing wave characteristics have been challenging to identify. Satellite observations conducted by Zhao (2020) captured this standing wave, but they referred to it as interference of M2 internal tides and its features were not investigated further.
Based on 20-yr averaged field data from satellite observations and 1-yr averaged field data from PIES observations, we concluded that standing waves persist west of the LS. This is attributed to the energetic barotropic tides in the LS and the southern slope of the Taiwan Strait (Zhao 2020). These energetic barotropic tides generate stable internal tidal beams that propagate northwest and southeast, and their superposition results in the formation of standing waves. Notably, this differs from the principles mentioned in previous studies conducted in other regions, where standing waves were formed by the superposition of incident and reflected waves (Hall et al. 2017; Nash et al. 2004).
During the observation period, three strong anticyclone eddies affected the positions of the standing-wave nodes and antinodes without destroying them. The first anticyclone eddy in August was slightly weaker than the strongest eddies historically observed in this region, further highlighting the persistent presence of the standing-wave pattern. Additionally, further analysis using numerical simulations is required to determine the strength of eddies that can disrupt the standing-wave pattern, and to investigate how energy is exchanged between eddies and standing waves (Dunphy and Lamb 2014; Löb et al. 2020; Savage et al. 2020).
The presence of standing waves can impact ocean mixing. According to Alford et al. (2006), standing waves increase low-mode shear, leading to instability and favoring increased ocean mixing. Previous studies observed enhanced turbulent mixing on the western side of the LS relative to the eastern side in the western Pacific region (Tian et al. 2009; Yang et al. 2016). This difference was attributed to “energy trapped” resulting from the reflection of diurnal internal tides in the northern SCS (Yang et al. 2016). However, Wang et al. (2021) calculated a small amount of energy reflection from diurnal internal tides in the northern SCS, suggesting a limited contribution to mixing. Therefore, it is speculated that semidiurnal standing waves may play a significant role in driving strong mixing in the northern SCS. Considering that enhanced mixing caused by standing waves leads to reduced internal tide amplitudes, while the standing waves themselves create high internal tide amplitudes, it is worth conducting further research on the impact of standing waves on the nonlinear steepening of internal tides west of the LS.
Acknowledgments.
This study was sponsored by the National Natural Science Foundation of China (Grants 41920104006 and 41906024), the Scientific Research Fund of Second Institute of Oceanography, MNR (Grant JZ2001), the Project of State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography (SOEDZZ2106 and SOEDZZ2207), the Oceanic Interdisciplinary Program of Shanghai Jiao Tong University (Project SL2021MS021), the Innovation Group Project of Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) (311020004), and the Global Climate Changes and Air-sea Interaction Program (GASI-02-PAC-ST-Wwin).
Data availability statement.
The bathymetry data were obtained from ETOPO1 (https://doi.org/10.7289/V5C8276M). The World Ocean Atlas 2018 is produced and made available by NOAA/National Centers for Environmental Information (https://accession.nodc.noaa.gov/NCEI-WOA18). MIOST-IT products were obtained online (ftp://ftp-access.aviso.altimetry.fr/). The internal tidal dataset provided by Zhao (2020) is available online (https://doi.org/10.6084/m9.figshare.12811586). Surface geostrophic currents and sea-level anomaly data were obtained from the Copernicus-Marine Environment Monitoring Service (http://marine.copernicus.eu/). The Taylor–Goldstein equation was calculated using a MATLAB script (http://salty.oce.orst.edu/wave_analysis/SSF_index.html; Smyth et al. 2011). The analyzed data are available online (https://figshare.com/articles/dataset/Scripts_and_data_of_the_internal_tides/23552763).
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