1. Introduction
Breaking of internal waves is one of the major energy sources for abyssal diapycnal mixing (Munk and Wunsch 1998), which provides 2.1 TW of energy to maintain the global overturning circulation and stratification. Within the internal wave bands, near-inertial waves (NIWs) occupy the predominant energy and shear variance in the ocean (Ferrari and Wunsch 2009), with wind forcing as the major energy source (Alford et al. 2016). In addition to wind energy input at the sea surface, parametric subharmonic instability (PSI), a nonlinear triad interaction wherein larger-scale, low-mode internal tides (ITs) transfer energy to a pair of smaller-scale, high-mode “daughter waves” with opposite wave vectors, is considered as an effective mechanism in generating subharmonic NIWs (McComas and Bretherton 1977; Müller et al. 1986).
Theoretical studies initially derived a rather long interaction time scale of O(100 days) based on the random phase assumption (Olbers and Pomphrey 1981), implying that PSI might not be detectable in the ocean. However, recent numerical studies suggested that PSI could take place over a much shorter time scale (e.g., Hibiya et al. 1998, 2002; Furuichi et al. 2005; MacKinnon and Winters 2005; Simmons 2008; Hazewinkel and Winters 2011; Onuki and Hibiya 2015). MacKinnon and Winters (2005) demonstrated that the energy transfer from low-mode M2 ITs to high-mode subharmonic NIWs occurred in the time scale of O(10 days), and pointed out that this downscale energy transfer was highly latitude dependent; it was most efficient at the M2 critical latitude and sharply dropped as ITs propagated away from the critical latitude. Based on a global simulation forced by both diurnal and semidiurnal ITs, Ansong et al. (2018) showed a strong phase coherence between ITs and subharmonic NIWs concentrating around the diurnal and semidiurnal critical latitudes over the whole ocean column, with the strongest energy transfer occurring in the upper ocean due to the intensified stratification and internal tidal energy. Through modeling experiments, Liu and Zhao (2020) also reported that the diurnal ITs, which radiated from the Luzon Strait and propagated southwestward into the South China Sea, would undergo PSI between 13° and 15°N and transfer energy to subharmonic NIWs, contributing to a locally elevated mixing.
Motivated by the numerical results, a series of in situ observations have been carried out to explore the occurrence of PSI and its role in transferring energy from ITs to subharmonic NIWs. At the M2 critical latitude, based on observations from the Internal Waves Across the Pacific (IWAP) experiment, Alford et al. (2007) reported the vertically standing patterns of near-inertial horizontal velocity and shear that were attributed to the superposition of equally upward- and downward-propagating PSI-generated NIWs. Recent observations from the Hawaiian Ocean Mixing Experiment (HOME) revealed that PSI of M2 ITs occurred not only at the critical latitude (MacKinnon et al. 2013), but also at sites hundreds of kilometers equatorward (Carter and Gregg 2006; Frajka-Williams et al. 2006; Rainville and Pinkel 2006; Sun and Pinkel 2013). By applying the bicoherence method, which distinguished whether the waves were nonlinearly coupled or independently excited, Carter and Gregg (2006) demonstrated a statistically significant phase locking between M2 ITs and M2 subharmonic waves, suggesting PSI as a candidate for transferring energy from M2 to M2/2. Furthermore, MacKinnon et al. (2013) and Sun and Pinkel (2013) simplified the nonlinear terms to estimate the PSI energy transfer rate utilizing mooring observation at a single site, and the calculated PSI energy transfer rates (∼10−9 W kg−1) were roughly in accordance with the local dissipation rates, showing the importance of PSI in determining the local turbulent mixing. In contrast to the semidiurnal ITs, the PSI of diurnal ITs was relatively less explored mostly owing to the lack of long-term in situ observations. The few existing studies were based on cruise surveys or short-term mooring observations (e.g., Alford 2008; Xie et al. 2009; Chinn et al. 2012). For instance, based on shipboard observations in the South China Sea, Alford (2008) reported two narrow shear peaks dominated by the PSI-generated NIWs, which were much more narrowly confined around the O1 and K1 critical latitudes when compared with that around the M2 critical latitude.
At the critical latitudes where the subharmonic frequency is equal to the local inertial frequency, the group velocity of PSI-generated NIWs vanishes. This results in an accumulation of high-mode near-inertial energy and shear that are dissipated locally and, consequently, leads to the elevated diapycnal mixing (S. Wang et al. 2021). In recent decades, enhanced mixing at the critical latitudes have been detected at all depths over the global ocean (Kunze et al. 2006; Tian et al. 2006; Hibiya and Nagasawa 2004; Hu et al. 2023). Extracting the M2 IT signals from the satellite altimetry, Tian et al. (2006) demonstrated that the diapycnal mixing at the M2 critical latitude in the upper ocean is one order of magnitude larger than the equatorial oceans. Using microstructure profiler observations, Hibiya et al. (2007) also reported a PSI-induced “mixing hotspot” in thermocline at the M2 critical latitude, with the diapycnal diffusivity (10−4 m2 s−1) one order of magnitude larger than the background (10−5 m2 s−1) in the North Pacific. In a global survey of abyssal mixing, Kunze et al. (2006) documented elevated diffusivities near the M2 critical latitude below 4500 m. In addition, PSI-generated NIWs could nonlinearly couple with ITs and transfer energy to superharmonic waves, further contributing to diapycnal mixing (Simmons 2008). Therefore, through transferring energy from low-mode ITs to high-mode subharmonic NIWs and superharmonic waves, PSI fuels diapycnal mixing at the critical latitudes and shapes its global pattern.
Theoretically, the resonance of PSI is perfectly tuned at the critical latitudes where the frequency of subharmonic waves is exactly inertial and hence leads to a vanished vertical group velocity, allowing their exponential buildup (Young et al. 2008). While in the realistic ocean, multiscale dynamical motions, such as large-scale circulations and mesoscale eddies, can significantly shift the resonance frequency from local inertial frequency f0 to the effective Coriolis frequency feff (feff = f0 + ζ/2, where ζ is the relative vorticity; Kunze 1985), and thus modulate the PSI efficiency remarkably (Yang et al. 2018, 2020; Dong et al. 2019). Based on idealized numerical experiments, Yang et al. (2018) showed that PSI could occur poleward of the critical latitudes when negative relative vorticity was present, suggesting that PSI efficiency could be significantly influenced by the varying ζ. The model results were subsequently confirmed by a 2-month mooring observation on the East China Sea shelf slope (Yang et al. 2020), which demonstrated that the PSI energy transfer rate could be 6–9 times as large when the Kuroshio swung feff close to the M2 subharmonic frequency. In the abyssal ocean of the South China Sea, Hu et al. (2023) also reported the occurrence of PSI poleward of the K1 critical latitude due to the negative ζ related to strong subinertial flows.
The northwest Pacific, as predicted by numerical simulations in the literature (e.g., Xu et al. 2021), features the largest tidal energy conversion among global oceans (totally ∼193 GW for M2 and S2 and ∼101 GW for K1 and O1), with nearly one-half of the global diurnal baroclinic energy being converted therein (Müller 2013). Diurnal ITs generated from the Luzon Strait (∼11.7 GW for K1) and the Talaud-Halmahera Passage (∼8.9 GW K1) radiate a long distance into the northwest Pacific (Y. Wang et al. 2021; Zhao et al. 2021) and encounter each other roughly between 12° and 16°N in the western Mariana basin, potentially fueling PSI around the diurnal critical latitudes. Nevertheless, the PSI of diurnal ITs in the northwest Pacific has been less explored due to the lack of long-term in situ observations. Based on mooring observations from the Northwestern Pacific Eddies, Internal Waves, and Mixing Experiments (NPEIM) during 2015/16, which deployed 17 moorings along 143°E from 0° to 22°N (Fig. 1a), Zhang et al. (2018) first revealed three peaks of finescale turbulent shear and diapycnal diffusivity in the upper ocean, and the one located at 14°N was largely attributed to the occurrence of PSI. However, a detailed understanding of the basic characteristics of PSI of diurnal ITs, such as its occurrence and behavior over the whole water column, and the energy transfer rate between ITs and subharmonic NIWs, is still lacking. In addition, the relatively vigorous mesoscale eddies in this region as well as the North Equatorial Current may potentially shift feff and change the PSI efficiency, however, their modulatory role in PSI remains unclear. Therefore, utilizing the full-depth observations of the three moorings (at 12°, 14°, and 16°N) from the NPEIM project (Figs. 1a,b), this study characterizes the occurrence and behavior of PSI of diurnal ITs from a full-depth view, and then examines the influence of PSI on shear instability in the northwest Pacific. In addition, the modulation of PSI efficiency by mesoscale eddies and shear instability is also explored.
Mooring locations and observed horizontal kinetic energy of near-inertial waves, diurnal internal tides, and eddies: (a) topography and locations of the mooring sites (red stars indicate the positions of three moorings used in this study), (b) schematic diagram of the configuration of the mooring, and mooring-observed (c) NIKE, (d) D1KE, and (e) EKE, averaged between 100 and 400 m (red) and at 2000-m depth (blue).
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
The remainder of this study is organized as follows. We first introduce observations in section 2. The evidence of PSI occurrence via bicoherence analysis, and a causal analysis newly introduced in PSI studies is presented in section 3. The modulation of PSI efficiency by mesoscale eddies is described in section 4. In section 5, the influence of PSI on shear instability is discussed. The results are summarized in section 6.
2. Data
a. Mooring observations
During 2015–16, the NPEIM project was conducted to investigate the characteristics of eddies, internal waves, mixing, and their interactions in the tropical–extratropical northwest Pacific (Zhang et al. 2018), with 17 full-depth moorings deployed along 143°E in November 2015 and then recovered in March 2017 (lasting 13–15 months). The moorings were set up at every 2° between 0° and 16°N, every 1° between 16° and 20°N, and every 0.5° between 20° and 22°N, respectively (Fig. 1a). In the upper ocean, all moorings were equipped with one or two 75 kHz acoustic Doppler current profilers (ADCPs; looking upward or downward) and temperature chains comprising several CTDs and dozens of SBE temperature loggers to record current velocities and temperatures above 1000 m. The ADCPs were deployed at about 500 m (only one upward-looking ADCP at 14°–19°N), measuring horizontal velocities half-hourly with a 16-m vertical resolution (Fig. 1b). The temperature chains sampled temperature every 5 min, with depth intervals being 20 m above 300 m, 40 m between 300 and 600 m, and 100 m below 600 m. In the abyss, velocities and temperatures were measured by RCMs and CTDs at 1000, 2000, 3000, and 4000 m. The observed raw velocities and temperatures were first averaged hourly and then linearly interpolated to fixed 5 m vertical bins. More detailed information about the NPEIM moorings can be found in Zhang et al. (2018).
In the tropical–extratropical upper ocean along 143°E, two regions with elevated near-inertial kinetic energy (NIKE) were detected by these mooring observations (Fig. 1c; Zhang et al. 2018): one is located between 20° and 22°N (the southern eddy-rich zone; Qiu 1999) where the energetic anticyclonic eddies enhanced NIKE via the inertial chimney effect (Kunze 1985), the other one appeared around 14°N that was largely attributed to the PSI mechanism (Zhang et al. 2018). The kinetic energy of diurnal ITs (D1KE), with multiple energy sources, generally propagated southward and exhibited an overall decreasing trend from north to south (Fig. 1d). Focusing on the occurrence and behavior of PSI of diurnal ITs, observations from mooring M14 (14.08°N), which was located between the O1 (71 km equatorward) and K1 (49 km poleward) critical latitudes, was mainly utilized in this study. Observations from M12 (11.99°N) and M16 (15.88°N), which were 161 km southward of O1 and 151 km northward of K1 critical latitudes, respectively, were also used for comparison. Owing to the backscattering from the sea surface, the valid depth ranges of velocity measurements in the upper ocean were 85–425, 85–440, and 85–520 m for M12, M14, and M16 (300–880, 60–890, and 150–995 m for temperature), respectively. In the present study, we referred to the depth range above 500 m as the upper layer to differentiate from the abyssal ocean below 1000 m. In the abyssal ocean, velocities were measured by RCMs at 1000 m (M14), 2000 m (M12, M14, M16), 3000 m (M12, M14), and 4000 m (M12, M14), and temperatures were measured by CTDs at 2000 m (M12, M14, M16) and 4000 m (M12, M14). Information of the sensor depths is presented in Table S1 in the online supplemental material. During the observational period, although under similar wind-forcing conditions (Zhang et al. 2018), the averaged NIKE at M14 was 38% and 64% larger than that of M12 in the upper and abyssal ocean at 2000 m, respectively and the averaged NIKE at M14 was 61% and 8% larger than that of M16 in the upper and abyssal ocean at 2000 m, respectively, whereas the strengths of D1KE were comparable (Figs. 1c,d). Meanwhile, the relatively high eddy kinetic energy (EKE; 20–120-day bandpass filtered) at M14 in the upper and abyssal ocean (Fig. 1e) implies the relatively active mesoscale activities during the observational period, potentially modulating the PSI behavior.
b. Observed multiscale motions
The observed raw velocities contain signals from multiscale motions, including the westward North Equatorial Current (NEC), mesoscale eddies, and internal waves. Thus, a high-pass filter with a cutoff period of 3 days is applied to extract the currents and vertical displacements of isotherms η of internal waves, and a low-pass filter with a cutoff period of 15 days is also applied to extract signals of subinertial flows at M14 (Fig. 2). Considering the dominant westward flow in the zonal velocity by the prevailing NEC, depth–time maps of the low-pass filtered meridional velocities υ are depicted in Figs. 2a–f to highlight signals of mesoscale eddies. In the upper layer, a 2-month cycle is visible (with amplitude of ∼0.2 m s−1), suggesting the frequent passages of mesoscale eddies. For example, a cyclonic eddy surrounded the mooring site during 19 May–19 June, shifting the velocity from southward to northward and heaving up the isotherms, whose modulatory role in PSI behavior will be discussed in section 4.
Meridional velocity and isothermal displacement in subinertial (15-day low-pass) and internal wave (3-day high-pass) bands at M14: (a) depth–time map of the subinertial meridional velocity; (b) depth–time map of subinertial isothermal displacement at M14; subinertial meridional velocity at (c) 1000, (d) 2000, (e) 3000, and (f) 4000 m; and (g)–(l) as in (a)–(f), but for internal waves during 17 May–14 Jun.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
Figures 2g–l are the zoom-in view of high-passed meridional velocities during 17 May–14 June, and display signals of observed internal waves at M14, including a mixture of NIWs (∼0.5 cpd), diurnal (∼1 cpd), and semidiurnal tides (∼2 cpd), as reported in Zhang et al. (2018). The maximum amplitude of internal waves reaches 0.3 m s−1 above 500 m and reduces to 0.1 m s−1 at 1000 m and to 0.05 m s−1 at 4000 m. In contrast to the highly sheared meridional currents, the high-passed η features a more likely low-mode vertical structure, possibly dominated by internal tides given the fact that the potential energy of NIWs is relatively weak.
The power spectral density (PSD) of horizontal velocities is also estimated (Fig. 3). At the three moorings, the internal wave bands are dominated by NIWs, diurnal and semidiurnal tides in both upper and abyssal ocean, with NIWs featuring a relatively broader spectrum that has been typically observed in the ocean (Alford et al. 2016). At M14, the energy variance at the inertial frequency f is larger than that at M12 and M16 over the full depth, as shown in Fig. 1c. Considering the similar wind-forcing conditions at the three mooring sites during the observational period (Zhang et al. 2018), we expect that the elevated NIKE at M14 is likely to be associated with PSI of diurnal ITs. Note that, at superharmonic frequencies, such as D1 + f and D2 + f, the highest energy peak also appears in a broadband pattern as that of NIWs at M14, which may be generated from the nonlinear interaction between ITs and NIWs (Guan et al. 2014). To ensure that these superharmonic waves are not caused by the “fine-structure contamination” related to tidal advection (Alford 2001a), a parallel, semi-Lagrangian analysis (the velocities are linearly interpolated along a reference frame of isotherms) is conducted following Carter and Gregg (2006), and the results still show significantly elevated spectral peaks at the superharmonic frequencies (Fig. S1 in the online supplemental material).
Power spectral density of meridional velocity at the three moorings: (a) upper-layer-averaged (85–440 m) and (b) 2000-m-depth velocity spectra at M12 (blue solid line), M14 (red solid line), and M16 (black solid line), respectively. The vertical dashed lines indicate the local inertial frequency (f), O1, K1, O1 + f, M2, S2, and M2 + f frequencies; green vertical lines mark the 95% confidence interval at f.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
At the three moorings, energy peaks of dominant waves across the internal wave bands are well separated; thus, a fourth-order Butterworth filter is applied to extract the NIWs and ITs. Velocities within the semidiurnal, diurnal, and near-inertial bands are bandpass filtered with frequencies of (1/10–1/14) cph, (1/20–1/27) cph, and (0.8–1.3) f. In addition, the baroclinic tidal velocity is estimated by further subtracting the barotropic tidal velocity predicted by the TOPEX/Poseidon global tidal model (Egbert and Ray 2000) from filtered tidal velocity in observations. Meanwhile, to provide estimations of PSI energy transfer rate in meaningful physical units, the velocity and shear used here are not Wentzel–Kramers–Brillouin (WKB) scaled (Leaman and Sanford 1975), as that in Yang et al. (2020).
3. PSI in the upper and abyssal ocean
The elevated NIKE at M14 around the diurnal critical latitudes is naturally related to the occurrence of PSI. In this section, the widely used bicoherence analysis (e.g., Mccomas and Briscoe 1980; Carter and Gregg 2006; Chinn et al. 2012; Sun and Pinkel 2013), and a causality analysis method newly introduced in PSI analysis, are employed to examine the occurrence and behavior of PSI. The energy transfer rate from D1 ITs to NIWs via PSI is also estimated.
a. Diurnal internal tides and near-inertial waves
The mooring-observed internal waves are dominated by a combination of NIWs, diurnal and semidiurnal tides (Fig. 3). Figure 4 shows the filtered velocities of D1 ITs and NIWs at M14 in the upper and abyssal ocean during 20 January–6 March 2016. There are approximately three fortnightly cycles of spring-neap D1 ITs with spring tides arriving around 27 January and 14 and 29 February (Fig. 4a). Overall, the D1 ITs are visibly dominated by a low-mode structure. However, during 28 January–6 February, the D1 ITs feature a more complex vertical structure and irregular spring-neap cycle, likely because of the superposition of the diurnal tidal beams originated from the Luzon Strait and the Talaud-Halmahera Passage (Y. Wang et al. 2021).
Observed meridional velocity and kinetic energy of D1 ITs and NIWs at M14: depth–time maps of meridional velocity of (a) D1 ITs and (b) NIWs (green lines represent the envelope curves of the velocity) and time series of D1KE (black lines) and NIKE (red lines) (c) averaged in the upper layer (85–440 m) and (d) at 2000 m depth during 20 Jan–6 Mar. In (c) and (d), R is the correlation coefficient between the two time series at the 95% confidence level.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
The observed NIWs have relatively shorter vertical wavelengths when compared with D1 ITs (Fig. 4b). Sometimes, NIWs are characterized by a typical “checkerboard” structure (for instance, NIWs between 300 and 400 m during 20 January–4 February, and 150–250 m during 24 February–6 March), possibly due to a superposition of comparable upward- and downward-propagating NIWs; such a structure is consistent with the PSI-generated NIWs as reported in the literature (e.g., Alford et al. 2007; MacKinnon et al. 2013). Overall, the temporal variations of NIWs and NIKE appear to roughly follow D1 ITs, with elevated NIKE appearing during the spring tides both in the upper and abyssal ocean (Figs. 4c,d). The observed NIKE correlates fairly well with D1KE with a correlation coefficient reaching 0.43 (0.52) at the 95% confidence level in the upper (abyssal at 2000 m) layer. Such correlation between D1KE and NIKE has also been documented by Rainville and Pinkel (2006), who attributed a similar phenomenon to the PSI mechanism, although at the M2 critical latitude.
b. Decomposition of near-inertial waves
In contrast to wind-generated NIWs, which typically propagate energy downward from the sea surface, theoretically NIWs generated by PSI are expected to have comparable up- and down-going energy components. As shown in Fig. 5, despite lying in similar wind-forcing conditions during 19 May–14 June at the three mooring sites, the observed NIWs at M14 have strikingly different vertical structures when compared with those observed at M12 and M16. At M12 and M16, the phase of NIWs mainly propagates upward (thus with energy propagating downward), implying the wind generation mechanism. However, NIWs at M14 feature concurrent upward and downward components (“checkerboard” pattern), especially between 200 and 440 m from 25 May to 7 June.
Decomposition of NIWs: (top) meridional component of the near-inertial velocity and decomposed NIWs with (middle) upward phase (CW; downward energy) and (bottom) downward phase (CCW; upward energy) propagation at (a)–(c) M12, (d)–(f) M14, and (g)–(i) M16.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
We thus decompose the NIWs into clockwise (CW) and counterclockwise (CCW) components by taking the positive and negative quadrants of the vertical Fourier transforms of (u + iυ) at each point in time, following MacKinnon et al. (2013). According to the linear internal wave polarization relations, a sense of CW rotation with increasing depth represents downward energy (upward phase) propagation, whereas CCW rotation represents upward energy (downward phase) propagation (Leaman 1976). At M12 and M16, the sense of phase propagation is dominated by CW rotation (Figs. 5b,h), while both upward- and downward-propagating waves are present in roughly equal parts at M14 (Figs. 5e,f). During this period, the ratio of CCW to CW components is about 0.92 at M14, twice larger than that at M12 and M16, but similar to that observed at the M2 critical latitude (e.g., Alford et al. 2007; MacKinnon et al. 2013). The roughly comparable upward- and downward-propagation characteristic of NIWs observed at M14 is consistent with the prediction based on the classical theory of PSI (McComas and Bretherton 1977). However, this does not mean that the wind generation mechanism is negligible at M14. Although the wind forcing is relatively weak and PSI tends to dominate the generation of NIWs during the period in Fig. 5, the downward energy is slightly larger than the upward components, implying wind generation mechanism is still at play during this period. As will be shown later (see Fig. 8), the wind energy input to NIWs can exceed PSI during the passages of four typhoons.
c. Bicoherence analysis
Assuming that all depth bins are within the same wave, the degrees of freedom will not increase; thus ndof is independent at each depth. To obtain the same ndof at the three moorings, all time series from 14 January 2016 to 14 January 2017 are chosen for bispectral estimations here. The time series are divided into 50%-overlapping 10-day windows (∼10 diurnal tidal periods and 5 local inertial periods) with Hamming window applied. Considering that the phase and amplitude of NIWs change significantly about every 2 days, we estimate ndof = 2 × (366/2) = 366. Benefitting from the longer-lasting observational period, the ndof in this study is much larger than that in previous studies (e.g., Carter and Gregg 2006; Sun and Pinkel 2013), thus providing smaller significance thresholds of 0.13 (at 95% level), 0.11 (at 90% level), and 0.09 (at 80% level).
Depth-averaged bispectrum and bicoherence in the upper ocean at the three moorings are shown in Figs. 6a–f. According to Sun and Pinkel (2013), the inputs of the bispectrum have referred to variables in PSI energy transfer equations, hence a positive real part of the bispectrum is associated with the positive energy transfer. At M14, the depth-averaged bispectrum has a positive peak centered at (ω1, ω2, ω1 + ω2) = (−D1/2, −D1/2, −D1) ≈ (−f14, −f14, −D1) with a significant bicoherence at the 95% level (the negative frequencies here denote CW rotation with time), indicating a positive energy transfer from D1 ITs to subharmonic NIWs (Figs. 6b,e). In contrast, the bispectra at M12 and M16 are one order of magnitude smaller than that at M14, with bicoherences failing to be significant at (−D1/2, −D1/2) (Figs. 6d,f). From 85 to 440 m, the bicoherence at each depth at M14 is generally larger than that at M12 and M16, and mostly significant at the 95% level (Fig. 6g). Consequently, the significant bicoherence and positive bispectrum value at (−D1/2, −D1/2) here, suggest not only a confident phase locking between the PSI triads, but also a positive energy transfer from D1 ITs to NIWs at M14. At M16, observations show a much weaker bicoherence than that at M14, implying the absence of significant PSI. Though theoretically, PSI could occur equatorward of the critical latitudes, and previous observations (e.g., Alford et al. 2007; Sun and Pinkel 2013) have also reported its occurrence equatorward of the M2 critical latitude at the generation sites of semidiurnal ITs, here, the observed PSI signals at M12 are not as significant as those at M14, which may be attributed to the relatively weak D1 ITs.
Bispectral estimations in the upper layer: (a)–(c) the real part of the depth-averaged bispectrum and (d)–(f) bicoherence at (top) M12, (middle) M14, and (bottom) M16. Black lines indicate the local inertial frequency (−f, −f), and red lines indicate the subharmonic frequency of diurnal ITs (−D1/2, −D1/2). (g) Vertical profiles of bicoherence at (−D1/2, −D1/2, −D1) at M12 (blue solid line and dots), M14 (red solid line and dots), and M16 (black solid line and dots). Vertical lines indicate the 80%, 90%, and 95% significance levels.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
In the abyssal ocean, PSI of diurnal ITs is rarely reported due to the lack of long-term in situ observations, although elevated mixing has been reported around the M2 critical latitude by observational profiles from cruise surveys (Kunze et al. 2006). Here we explore the phase locking between NIWs and D1 ITs at the three moorings below 1000 m. The RCMs on the mooring system can only observe the horizontal velocity at single depths (1000, 2000, 3000, 4000 m), and thus cannot estimate the current shear. Therefore, the current shear in Eqs. (2) and (3) is replaced by velocity to calculate bispectrum and bicoherence in the abyss following Carter and Gregg (2006) and Hu et al. (2023).
As in Figs. 6a–f, but for 2000-m depth.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
d. Causality analysis
Aside from the extensively used bicoherence to diagnose the occurrence of PSI in the literature (e.g., MacKinnon et al. 2013), here we introduce a new method—causality analysis, to explore the cause of enhanced NIWs at M14 and address the relative importance of wind and PSI in generating NIWs.
Causality analysis is a useful method for quantitatively evaluating the drive and feedback of causal relations between time series (Granger 1969), which has attracted enormous interest in various disciplines such as neuroscience (e.g., Pereda et al. 2005), finance (e.g., Marschinski and Kantz 2002), and climate science (e.g., Wang et al. 2004). Within the casual relationship, the strength of causality is measured as the time rate of information flowing from one time series to another (information flow). Based on the concept that causality is a real physical notion attached to information flow, the Liang–Kleeman information flow, rigorously derived from Shannon entropy (Liang and Kleeman 2005; Liang 2016, 2018, 2021), is proposed to provide a measure of causation in dynamical systems. Since the inception, it has been successfully applied to Earth science, such as climate change (Vannitsem et al. 2019; Stips et al. 2016) and ocean–atmosphere interactions (Liang et al. 2021). For instance, Stips et al. (2016) employed this method to demonstrate that greenhouse gases are the main causal drivers of anthropogenic global warming. Although established under the assumption of linear stochastic dynamical systems, the information flow remains invariant upon arbitrary nonlinear transformations (Liang 2016), allowing us to detect the causal structure between parent waves (D1 ITs) and subharmonic waves (NIWs) within the PSI resonant interaction.
The depth-averaged (85–440 m) information flow in the upper ocean is shown in Fig. 8a. Overall,
Results of causality analysis at M14: (a) time series of the information flow from D1KE to NIKE (
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
e. The PSI energy transfer rate
Depth–time maps of the calculated
Calculated energy transfer rates of PSI: depth–time maps of the PSI energy transfer rates at (a) M12, (b) M14, and (c) M16, and (d) annually averaged energy transfer profiles at M12 (blue solid line), M14 (red solid line), and M16 (black solid line).
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
Overall, the energy transfer profile at M14 is one order of magnitude larger than that of M12 and M16 (Fig. 9d), with the net value of 1.5 × 10−10 W kg−1, suggesting an effective PSI energy transfer at M14. Strong 〈
The estimated 〈
4. Modulation by mesoscale eddies
Mooring M14 is located between the O1 and K1 critical latitudes, thus the passages of mesoscale eddies may swing feff to either O1 (anticyclonic eddy) or K1 (cyclonic eddy) subharmonic frequencies, and theoretically can both enhance the PSI efficiency. Two typical events are examined in detail here (Fig. 10), the first one has very weak ζ(feff ≈ f0 = f14) during 11–18 April, and the second one has a cyclonic eddy passage over M14 during 9–16 May, shifting feff toward K1 subharmonic frequency (feff ≈ K1/2). During the first event, the observed total, upward and downward NIKE are all very weak due to the weak WW (mostly less than 2 mW m−2) (Figs. 11a–e). In contrast, during the second event (with feff ≈ K1/2), although featuring the similarly weak WW and thus weak NIKE in the upper 200 m, both upward and downward NIKE are significantly enhanced by a factor of 3 and 5 between 200 and 400 m, which we expect to be associated with PSI (Figs. 11f–i). In this case, the enhanced
The effective Coriolis frequency feff around the mooring observation sites modulated by the passages of mesoscale eddies: the spatial distribution of feff/f0 on (a) 15 Apr 2016 and (b) 11 May 2016. The geostrophic velocity data are downloaded from CMEMS. The red stars indicate mooring positions; blue and red dashed lines indicate the O1 and K1 critical latitudes, respectively.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
Comparisons of NIKE and PSI energy transfer rate
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
To statistically investigate the modulatory role of mesoscale eddies in PSI efficiency over a longer period, observations from 26 November 2015 to 1 August 2016 are focused on due to the lack of contamination by typhoon- and wind-enhanced NIWs. The total, downward, and upward NIKE are composed into bins of 0.005 feff/f14 and are then averaged to show their dependence on feff (Fig. 12). Here, we find a near-symmetrical structure of NIKE centered on the local inertial frequency (f14) with three energy peaks emerging around frequencies at O1/2 (1.8 J m−3), K1/2 (1.8 J m−3), and 1.01f14 (1.7 J m−3) (Fig. 12a). Among the three peaks, the one appearing at the frequency that is ∼1% blue-shifted from f14 is dominated by downward energy propagation (Figs. 12b,c), which is typically reported in the literature for wind-generated NIWs (e.g., Alford et al. 2016). On the contrary, peaks near O1/2 and K1/2 frequencies seem to have more comparable upward and downward components. In terms of the downward NIKE, it reaches 1.1 J m−3 around subharmonic frequencies, while the magnitudes of upward NIKE are 0.61 and 0.54 J m−3 near O1 and K1 subharmonic frequencies, respectively. Vertically, the elevated NIKE mostly occurs between 200 and 400 m when feff approaches O1 or K1 subharmonic frequencies, consistent with the depth range of efficient PSI energy transfer rate discussed in section 3e. Notably, here the calculation of feff is based on surface geostrophic velocities. Though ζ induced by baroclinic mesoscale eddies weakens with depth, likely reflecting uncertainties in our composite analysis, our results indeed show the elevated NIKE near O1 and K1 subharmonic frequencies, especially around K1/2 frequency within cyclonic eddies. Such findings are rarely documented in previous studies and will be discussed in detail together with the PSI-enhanced shear instability events in section 5b. Additionally, it should be noted that we may oversimplify this question by just thinking about the modulation of effective Coriolis frequency by the mesoscale vorticity. Actually, the submesoscale motions and spatial gradient of mesoscale vorticity may potentially play important roles in modulating PSI and NIWs dynamics as well. Unfortunately, due to the limited observations here, we can hardly examine their potential effects. However, since the elevated NIKE and PSI efficiency observed in this study mostly appear between 200 and 400 m, far below the depth of active submesoscale motions [upper 150 m documented in Zhang et al. (2021)], we think the modulatory effect by relative vorticities observed between 200 and 400 m are more subject to the effect of mesoscale eddies.
The modulatory role of mesoscale eddies in PSI by shifting feff at M14: composite maps of (a) total NIKE at corresponding feff/f14 and the (b) downward (CW) and (c) upward (CCW) NIKE. White, blue, and pink dashed lines indicate the local inertial (f14), O1/2 and K1/2 frequencies, respectively. Green solid lines show the dependence of depth-averaged (200–400 m) NIKE on feff/f14.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
Case study and composite analysis indicate that passages of both anticyclonic and cyclonic eddies can elevate PSI efficiency at M14 by shifting feff, and the enhancement of NIKE is found to be concentrated in a narrow band around the diurnal critical latitudes (O1:13.44° ± 0.2°N; K1:14.52° ± 0.2°N). This result based on long-term mooring observations is consistent with the findings in Alford (2008), who analyzed latitudinal sections from a cruise survey in South China Sea and reported that the shear peaks at the O1 and K1 critical latitudes were also confined within ∼0.2° of the critical latitudes. In contrast, Yang et al. (2020) revealed that the large ζ induced by Kuroshio could shift the feff towards the M2 subharmonic frequency, leading to the occurrence of PSI at 25.4°N, ∼378 km equatorward of the M2 critical latitude. Therefore, the modulatory role of mesoscale eddies at 14°N seems to be more sensitive around the diurnal critical latitudes, probably due to the relatively weaker D1 ITs here. In other regions with strong internal tides, it remains uncertain whether PSI can take place far away from the diurnal critical latitudes, thus further observation is needed.
Occasionally, the influence of mesoscale eddies can extend to the ocean bottom (Zhang et al. 2013, 2016), hence they may also modulate PSI efficiency in the abyss. As shown in Fig. S2 in the online supplemental material, an anticyclonic (31 March–5 April) and a cyclonic eddy (22–31 April) passed by M14 during 31 March–5 May. Correspondingly, the subinertial currents and associated kinetic energy were enhanced during the two periods; meanwhile, concurrently strengthened NIKE was also observed at 3000 m, implying the potential modulatory role of mesoscale eddies in abyssal PSI. However, calculating abyssal ζ is challenging due to the far distance between M12 and M16 (∼432 km). Therefore, it is hard to discern whether feff was shifted to O1/2 or K1/2 frequencies at that time. Furthermore, although concurrent increase of subinertial kinetic energy during eddy passages was detected, the elevated subinertial kinetic energy may also be associated with the propagation of topographic Rossby waves, which is difficult to disentangle due to the limited observations here.
5. Discussion
The PSI-generated NIWs with high-mode vertical structure play an important role in elevating shear instability and triggering mixing hotspots near the M2 critical latitude (e.g., Hibiya et al. 2007). In this section, we compare the potential shear instability events at the three moorings to explore the influence of PSI on mixing and also examine the potential modulation by the passages of mesoscale eddies around the diurnal critical latitudes.
a. The influence of PSI on shear instability
The squared buoyancy frequency N2 and vertical shear S2 are shown in Figs. S3 and S4 in the online supplemental material. Overall, N2 exhibits a similar vertical structure with no significant difference in magnitudes among the three moorings, whereas S2 at M14 is much stronger than that at M12 and M16. Especially between 250 and 400 m, the averaged S2 is 1.3 and 1.6 times as large as that at M12 and M16 (Fig. S4 in the online supplemental material), leading to the smallest Ri at M14 (Fig. 13). Here, we use a critical bulk Richardson number of 1/4 to identify the occurrence of potential shear instability events (Ansong et al. 2018). Although S2 is surface intensified, the potential instability events (with Ri < ¼) at the three moorings are mainly observed below 200 m owing to the strong stratification in the upper thermocline. In comparison with M12 and M16, the potential instability events at M14 occupy a wider range of depth with a longer duration, as indicated in Figs. 13a–c (green patches). At M14, the most frequent occurrence of potential instability events is intermittently located between 250 and 400 m (Fig. 13d) where efficient PSI occurs, with the highest probability of 0.03, which is 2–6 times that of M12 and M16.
The Ri at the three moorings: depth–time maps of Ri and isotherms (blue contours) at (a) M12, (b) M14, and (c) M16. The potential shear instability events with Ri < 1/4 are marked with green colors. The contour interval of isotherms is 3°C. (d) Depth-dependent occurrence probability of Ri < ¼ at M12 (blue solid line), M14 (red solid line), and M16 (black solid line). The magenta solid line indicates time-averaged (before 1 Aug) PSI energy transfer rate at M14.
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
By decomposing S2 into four energetic components, we calculate the contributions of each shear component to the total shear and their correlations with the inverse Richardson number (Ri−1) to examine the relative importance of each component on S2 and shear instability at M14. The filtered wave components are near-inertial (NI S2), diurnal tidal (D1 S2), semidiurnal tidal (D2 S2), subinertial (SI S2), and the superharmonic (D1 + f) [(D1 + f)S2, bandpassed by 0.85–1.15(D1 + f)] waves. Overall, the NI S2 dominates S2 and thus shear instability in the upper layer with the largest contribution of ∼40% and it has the highest correlation coefficient with Ri−1 (∼0.4 at the 95% confidence level) (Fig. S5 in the online supplemental material). As expected, D1 S2 contributes little to S2 because of its low-mode vertical structure. Interestingly, at M14, within the depth range of efficient PSI occurrence (330–360 m), the contribution of superharmonic waves (D1 + f), which are generated by the nonlinear interactions between NIWs and D1 ITs (also shown in Fig. 2), ranks second to total S2, only after NIWs. For instance, at 340 m, (D1 + f)S2 is comparable to NI S2, especially from 15 January to 15 February, (D1 + f)S2 even exceeds NI S2 by a factor of 3 and becomes the main driver of shear instability (Fig. S6 in the online supplemental material).
At 14°N, both PSI- and wind-generated NIWs can nonlinearly couple with background diurnal ITs, transferring energy to superharmonic waves such as D1 + f and contributing to the enhanced shear. The accumulation of the subharmonic and superharmonic shear drives Ri below ¼, leading to the enhanced shear instability and potentially turbulent dissipation between 250 and 400 m. Unlike most discussions of abyssal tidal dissipation over rough topography (e.g., Nikurashin and Legg 2011) and surface-intensified turbulence triggered by wind (e.g., Alford 2001b; Le Boyer and Alford 2021), this instability associated with PSI occurs in the subsurface between 250 and 400 m. Therefore, PSI may play an essential role in enhancing vertical mixing and redistributing energy in the ocean interior around the critical latitudes.
b. Modulation of shear instability by mesoscale eddies
In this section, we examine whether the mesoscale eddies also modulate PSI-induced NI S2 and thus influence shear instability. As Fig. 12, results for the composite analysis focusing on the NI S2 and the probability of shear instability events (ratio of Ri < ¼) at the corresponding feff/f14 are shown in Fig. 14. Consistent with NIKE in section 4, the NI S2 also has three peaks appearing near the local inertial (f14), O1/2 and K1/2 frequencies. The elevated NI S2 with a blue-shifted peak frequency at 1.01 f14 tends to be surface intensified, implying an energy source from the wind energy input. In contrast, the NI S2 around O1 and K1 subharmonic frequencies is 10%–30% larger than that of f14, mainly occurring below 150 m and consistent with the occurrence of efficient PSI (Fig. 14a). Because of the strong stratification in the upper thermocline, the potential instability events emerge at a deeper depth range (250–400 m) than NI S2. However, the largest occurring frequencies of potential instability events appear around the same frequency bands as NIKE and NI S2 (Fig. 14b), suggesting that mesoscale eddies can indeed influence shear instability via modulating PSI efficiency.
The modulatory role of mesoscale eddies in shear instability by shifting feff at M14: composite maps of (a) NI S2 at corresponding feff/f14 and (b) the probability of shear instability events. White, blue, and pink dashed lines indicate the local inertial (f14), O1/2, and K1/2 frequencies, respectively. Green solid lines show the dependence of depth-averaged (200–400 m) NI S2 [in (a)] and probability of shear instability events on feff/f14 [in (b)].
Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0055.1
In general, the enhancement of NIWs and diapycnal mixing within anticyclonic eddies is widely reported in previous studies (e.g., Joyce et al. 2013; Liu et al. 2017; Whalen et al. 2018). This is because anticyclonic eddies with negative ζ tend to trap NIWs (known as the “inertial chimney effect”), thereby triggering enhanced diapycnal mixing inside. However, cyclonic eddies with positive ζ are expected to expel NIWs, leading to weaker NIWs (e.g., Kunze 1985; Zhang et al. 2018). Here, peaks of NIKE, NI S2, and shear instability are documented within both anticyclonic (feff ≈ O1/2) and cyclonic (feff ≈ K1/2) eddies based on long-term mooring observation located between the diurnal critical latitudes (M14). Considering the significant phase locking between the PSI triads and the causal relation from D1 ITs to NIWs, as shown in section 3, we attribute this enhancement of NIWs mainly to PSI when anticyclonic and cyclonic eddies shift feff toward O1/2 and K1/2. This mechanism differs from the inertial chimney effect of anticyclonic eddies. Notably, the abnormal enhancement due to enhanced PSI efficiency within cyclonic eddies is rarely reported in the literature (Alford et al. 2016), and we expect this modulatory role of cyclonic eddies in PSI to occur equatorward of the M2 critical latitude, which merits further exploration.
6. Conclusions
In comparison with the extensively explored PSI of M2 ITs, PSI near O1 and K1 critical latitudes is less understood due to the lack of long-term in situ observations. By utilizing more than yearlong observations from three full-depth moorings (at 12°, 14°, and 16°N, along 143°E) deployed during the NPEIM project, this study examines characteristics of PSI of diurnal ITs in the northwest Pacific. The mooring-observed horizontal velocity is dominated by energetic multiscale motions, including the prevailing westward NEC, mesoscale eddies, and internal waves. The latter include a combination of NIWs, D1 ITs, and D2 ITs. Around the diurnal critical latitudes, the observed NIKE at M14 features an energy peak both in the upper and abyssal ocean, much larger than that at M12 and M16. In addition, the temporal variation of NIKE correlates fairly well with D1KE both in the upper (R = 0.43) and abyssal (R = 0.52) ocean at the 95% confidence level. Furthermore, the observed NIWs at M12 and M16 feature a predominantly upward phase (downward energy) propagation, implying a major energy source from wind. However, NIWs observed at M14 display a “checkerboard” structure, indicating comparable upward- and downward-propagating components. Such a structure is commonly recognized as a typical pattern of PSI-generated NIWs based on observations around the M2 critical latitude (e.g., Alford et al. 2007).
Bicoherence analysis, a method to distinguish between nonlinearly coupled waves and independently excited waves, is employed to explore the phase locking between NIWs and D1 ITs. At M14, there is a consistent, statistically bicoherent sense of phase between NIWs and D1 ITs both in the upper and abyssal ocean, while no significant phase relationships are detected at M12 and M16, providing evidence for the occurrence of PSI at 14°N. To further address the relative importance of wind and PSI in generating NIWs at M14, the causality analysis method, a quantitative measure of evaluating cause and effect in terms of the information flow, is introduced in PSI analysis for the first time. The calculated
At 14°N, NIWs associated with PSI tend to dominate vertical shear at 14°N, contributing to elevated shear instability around the diurnal critical latitudes. When compared with M12 and M16, the occurrence of potential instability events at M14 is much more frequent (approximately 2–6 times larger than M12 and M16), implying the important role that PSI plays in triggering elevated mixing around the critical latitudes. Besides, we also reveal that superharmonic waves such as D1 + f, which are generated by the nonlinear coupling between the background D1 ITs and NIWs, contribute second to the total shear at 14°N. Especially between 330 and 360 m, the role of D1 + f in triggering shear instability is comparable to or even larger than that of NIWs.
Located between the O1 and K1 critical latitudes, PSI behavior at M14 is found to be significantly modulated by the passages of mesoscale eddies, which shift feff close to or away from the subharmonic frequencies. During the event when a cyclonic eddy shifts feff ≈ K1/2, the NIKE (both upward and downward components) and energy transfer rate are significantly enhanced when compared with the event without eddy activity (feff ≈ f14). Furthermore, the composite maps of total, upward and downward NIKE show that the energy peaks emerge near O1 (feff ≈ O1/2) and K1 (feff ≈ K1/2) subharmonic frequencies, with a magnitude 50% larger than the background. As in NIKE, the remarkable enhancement of NI S2 and shear instability also occurs near the O1 and K1 subharmonic frequencies. Due to the significant phase locking and causal relationship between D1 ITs and NIWs, this enhancement is mainly attributed to the elevated PSI efficiency when the passages of mesoscale eddies with negative and positive ζ shift feff close to the O1/2 and K1/2 frequencies. Particularly, the observed abnormally enhanced NIKE, NI S2, and shear instability within cyclonic eddies are rarely reported in the literature and differ from the understanding that cyclonic eddies tend to expel NIWs (Kunze 1985). As compared with previous studies based on relatively shorter observations (e.g., 2 months) as in Yang et al. (2020), this study utilizes yearlong in situ observations and provides a more robust and systematic evidence of the modulatory role of mesoscale eddies in PSI behavior, especially the enhanced NIWs within cyclonic eddies.
Based on the observations from the NPEIM project, this study examines the characteristics of PSI of diurnal ITs in the northwest Pacific. The occurrence of PSI at M14 is proved by bicoherence analysis and causality analysis. The PSI-generated NIWs subsequently trigger and enhance the shear instability and potentially diapycnal mixing near the diurnal critical latitudes. Furthermore, the efficiency of PSI and the associated shear instability are modulated by both anticyclonic and cyclonic eddies. Considering the important role that latitude-dependent diapycnal diffusivity plays in ocean and climate models (Jochum 2009), processes associated with PSI as well as the modulatory effects of mesoscale eddies should be considered to improve parameterizations of the localized mixing around critical latitudes, and ultimately incorporated into the ocean and climate models.
Acknowledgments.
We thank the captain and crew of the R/V Dongfanghong2 for deploying and recovering the moorings. This study was supported by the National Natural Science Foundation of China (92258301, 91958205, 41876011), the National Key Research and Development Program (2022YFC3104304, 2022YFC3105003), the Fundamental Research Funds for the Central Universities (202001013129, 1901013184), and the Hainan Province Science and Technology Special Fund (Grant ZDYF2021SHFZ265).
Data availability statement.
The geostrophic velocity, IPRC Argo product, and wind data used in this study were obtained from the websites of CMEMS (https://data.marine.copernicus.eu/products), the Asia–Pacific Data-Research Center (APDRC; http://apdrc.soest.hawaii.edu/), and ECMWF (https://cds.climate.copernicus.eu/cdsapp#!/dataset/), respectively. The mooring data used in this study are available from the corresponding author upon reasonable request. All figures in this paper were plotted in MATLAB.
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