1. Introduction
The breaking of internal tides can induce strong current shear and turbulence, which are vital for heat and momentum redistribution (Ansong et al. 2017; Ferrari and Wunsch 2009) and sediment and nutrient transport (Heathershaw et al. 1987; Wilson 2011) in the ocean. Internal tide generation sites are recognized mixing hotspots, and the near-field dissipation mechanisms of internal tides have been widely studied (Nikurashin and Legg 2011; Buijsman et al. 2013; Liu et al. 2019). Away from the internal tide generation sources, ocean mixing over a wide spatial scale is also significantly affected by internal tides due to their radiated energy in long-range propagation (Ansong et al. 2015; Zhao 2014). In hot-spot internal tide generation sites such as the Luzon Strait and the Hawaiian Ridge, the majority of the converted baroclinic energy (approximately 50%–70%; Carter et al. 2008; Jan et al. 2008) radiates away as low-mode internal tides. Of great interest is the ultimate fate of the outgoing energy; yet, where and how the internal tides dissipate in the far-field remain unknown.
The particular regions where internal tides propagate across their critical latitudes1 are believed to be potential destinations where internal tides cascade their energy to smaller scales through parametric subharmonic instability (PSI) (Simmons 2008; MacKinnon et al. 2013a,b; Robertson et al. 2017). PSI is a resonant triad wave interaction that transfers energy from a parent wave to two subharmonic daughter waves. In the governing equations, this resonance is not explicitly resolved as an external force, but is instead caused by parameters that can be periodically modulated and contribute to parametric amplification; thus, it is named parametric subharmonic instability (Liang et al. 2017). PSI-induced daughter waves generally have smaller spatial scales and high wavenumbers, and thus are prone to directly dissipate locally (McComas and Bretherton 1977). In the context of internal tides, PSI can transfer energy from low-mode internal tides to high wavenumber subharmonic waves and is particularly efficient at the internal tide’s critical latitude. The PSI-induced subharmonic waves have the same frequency as the local inertial waves, and they have vanishing vertical velocities and displacements and thus tend to dissipate locally (MacKinnon et al. 2013a,b).
The PSI-induced shear enhancement at the critical latitude has been confirmed by in situ observations (Hibiya and Nagasawa 2004; van Haren 2005; Alford et al. 2007; MacKinnon et al. 2013a,b). For example, MacKinnon et al. (2013a,b) reported elevated vertical diffusivities in the upper ocean by a factor of 2–4 at and below the critical latitude of 28.8°N (for M2). Except for the critical latitudes, PSI has also been observed equatorward of the critical latitudes or near generation sites (Carter and Gregg 2006; Liao et al. 2012; Sun and Pinkel 2013), contributing to local internal tide dissipation (Nikurashin and Legg 2011). In addition, PSI is regarded as an effective mechanism in generating near-inertial internal waves; the near-inertial energy input from PSI can exceed that from wind in some areas (Alford et al. 2007). Overall, observations indicate that PSI acts as an important step in the downscale energy cascade from internal tides to near-inertial internal waves and internal waves at half the tidal frequencies in the ocean.
The South China Sea (SCS) is a marginal sea in the western Pacific, with diurnal tides accounting for the majority of the tidal energy (Fang et al. 1999). Mixing in the SCS is approximately two orders of magnitude greater than that in the western Pacific, and this enhanced mixing is believed to be largely related to the low-mode internal tides from the Luzon Strait (Tian et al. 2009; Yang et al. 2016). The Luzon Strait generates the strongest diurnal internal tides in the world. However, their open-ocean far-field dissipation is less understood in comparison with their generation and near-field dissipation. Energetic diurnal internal tidal beams emanate from the Luzon Strait and travel across the bidiurnal critical latitudes in the central SCS. This unique internal tide geographic position provides an ideal test bed to demonstrate the PSI dissipation mechanism and its implications for mixing in the SCS (Fig. 1). The generated near-inertial energy can propagate downward (Alford 2008) and drive deep thermocline circulation.
Distribution of the instantaneous vertical displacements at the depth of 1000 m for K1 internal tides (dark red–blue colors), superposed with bandpass filtered vertical displacements half of the tidal frequency equatorward of the critical latitude (orange–light blue colors) in the SCS. The red dashed line and arrow denote the location of critical latitude and the main propagation path of K1 internal tide radiating from Luzon Strait, respectively.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
Since the infusive study of MacKinnon and Winters (2005) that reported a catastrophic loss of low-mode M2 internal tide energy near the critical latitude based on an idealized simulation, the growing interest in PSI of tidal constituents has been intrigued over the past dozen. The PSI of diurnal internal tides is less understood because of the leading role of semidiurnal tide in terms of the tidal energetics. The PSI of diurnal internal tides was first reported by Simmons (2008) through idealized numerical predictions. Observational evidence has been reported, mainly in the SCS and adjacent regions (Alford 2008; Chinn et al. 2012; Sun et al. 2011; Xie et al. 2009). Alford (2008) reported the first observation of near-inertial internal waves triggered by diurnal internal tides using shipboard ADCP measurements in the eastern SCS. Using mooring observations, Xie et al. (2009), Sun et al. (2011), and Chinn et al. (2012) observed PSI of diurnal internal tides and energy transfer from diurnal internal tides to near-inertial internal waves (see their mooring locations in Fig. 5b). These observations mainly occurred in the vicinity of critical latitudes and raise the question of whether PSI-induced internal tide dissipation is an important mixing mechanism in the central SCS. To date, observational studies are scarce, and there is a lack of relevant dialog in the literature.
The numerical approach is economical compared to in situ observations. Currently, high-resolution ocean models are able to reproduce internal gravity wave spectra and nonlinear wave–wave interactions in the spectral space (Müller et al. 2015; Savage et al. 2017). Gerkema et al. (2006) studied the generation of M1 internal waves by PSI of M2 internal tides at different latitudes utilizing the MIT general circulation model (MITgcm). Based on the Hallberg Isopycnal Model, Simmons (2008) predicted the potential tidal mixing hotspots along the critical latitude by the PSI of the M2 internal tide. Hazewinkel and Winters (2011, hereinafter HW11) numerically studied different development stages of PSI-induced near-inertial oscillations by 2D idealized simulations. In particular, Ansong et al. (2018, hereinafter A18) investigated the global distribution of both diurnal and semidiurnal internal tides PSI in an ocean general circulation model with the inclusion of atmosphere forcing and horizontally varying stratification. The occurrence of diurnal internal tide PSI in the SCS near the critical latitude can be clearly identified in their global study, however, diurnal internal tide PSI in the SCS has not been investigated in detail. Among the regional numerical studies on the diurnal internal tides in the SCS so far (Jan et al. 2007; Miao et al. 2011; Wang et al. 2016), the diurnal internal tide PSI has not been reported yet.
In this paper, we investigate the mechanism of PSI responsible for the far-field dissipation of Luzon Strait diurnal internal tides in the central SCS with the assessment of the internal wave field equilibration. Disintegration of diurnal internal tide by PSI should include the PSI of major diurnal tidal constituents but exclude the effects of nonlinear wave–wave interactions among the diurnal tidal constituents and associated harmonic waves; however, there are still challenges in isolating nonlinear interactions. To analyze the process of internal tide self-disintegration due to PSI more clearly, here we take single tidal constituent (K1) simulation. Of particular interest is the latitudinal distribution of internal tide induced mixing within the critical latitude range. It is challenging to reproduce the PSI of internal tides exactly as that in the real ocean, but we expect the modeled magnitude and latitudinal dependence of PSI-induced mixing shed new light on this process. Our main goal of this paper is to present a first look at the PSI of the K1 internal tide field in the SCS. We provide numerical evidence for the occurrence of the K1 internal tide PSI. Based on our high-resolution model results, we analyze the internal tide equilibrium state, the influence of PSI on the energy budget, and the latitudinal dependence of dissipation and mixing near the critical latitude. We find that the disintegration of internal tides by PSI may contribute to mixing hot spots in the upper ocean near the critical latitude. The paper is articulated as follows. Section 2 describes the setup of the numerical model. Section 3 presents the basic generation and propagation characteristics of the K1 internal tide in the SCS. Section 4 offers a detailed analysis of the K1 internal tide PSI. Discussions and conclusions are presented in sections 5 and 6, respectively.
2. Model description
We use MITgcm to investigate the disintegration of the low-mode K1 internal tide in the SCS. MITgcm handles the free-surface primitive equation utilizing a finite volume method in space and a third-order Adams–Bashforth time-stepping scheme and makes the nonhydrostatic approximation (Marshall et al. 1997). The model utilizes a partial-step method to represent the rugged topography, which is favorable for internal tide simulation. The model domain covers the entire SCS, from 105° to 123.5°E and from 4° to 23°N (Fig. 1). The horizontal resolution is 1/30°. There are 100 vertical levels distributed unevenly in the water volume, with a higher resolution of 10 m in the upper ocean and gradually decreasing to a coarser resolution of 1200 m in the bottom (Fig. 2d). The model topography is obtained from 30-arc-s bathymetry data of the General Bathymetric Chart of the Oceans (Fig. 3a). In the model, harmonic mixing for horizontal momentum is used to account for the effects of subgrid turbulence. Constant lateral and vertical viscosity coefficients of 10 and 10−4 m2 s−1 and diffusivity coefficients of 10 and 10−5 m2 s−1 are assumed. The KL10 scheme (Klymak and Legg 2010) is used in parameterizing vertical mixing due to breaking internal waves. The bottom friction is parameterized using a quadratic rule with a constant bottom drag coefficient of 0.0025. Nonslip conditions are employed at both lateral and bottom solid boundaries.
(a)–(c) Temperature, salinity, and buoyancy frequency profiles for the initial fields of the simulation. (d) Vertical distributions of model levels are denoted by dots, among which the red dots indicate the halved vertical layers in the sensitive experiment A3.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
(a) Spatial distribution of the depth-integrated period-averaged conversion rate calculated in the 0.5° × 0.5° bins before PSI. Internal tide energy budgets at the Luzon Strait (GW) and the critical latitudinal zone (MW) are also shown (boxes outlined by black dashed lines), numbers out (in) the parentheses indicate the corresponding energetics calculated before (after) PSI. Baroclinic energy fluxes across the region boundaries are denoted by thick black arrows. (b),(c) Amplified views of the distribution of conversion rate (color) and baroclinic energy fluxes in the northern and central SCS. Blue (black) arrows denote the baroclinic energy fluxes before (after) PSI, and the red dashed line denotes the location of the critical latitude. The 200-, 1000-, 2000-, 3000-, and 4000-m isobaths (gray curves) are shown.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
The initial hydrographic fields are assumed to be horizontally homogeneous and vertically stratified. The temperature and salinity profiles are derived by horizontally averaging the annual mean climatology from the World Ocean Atlas 2018 (Garcia et al. 2019) for water depths over 1000 m within the model area (Fig. 2). To map the long-range propagation and disintegration of the K1 internal tide, we consider only tidal forcing at the open boundaries. All the heat and momentum forcings at the surface are set to zero. The model is forced by the K1 barotropic tidal currents from TPXO9 (Egbert and Erofeeva 2002). A flow relaxation scheme is applied at the open boundaries following Carter and Merrifield (2007) to prevent energy artificial reflection.
The runtime of internal tide simulation is generally decided by the propagation time of the first-mode internal tide from generation sites to the farthest boundaries. For the SCS, the models are usually driven for less than 30 days (Jan et al. 2007; Miao et al. 2011; Wang et al. 2016). Because the growth of nonlinear instability is slow, and the propagation speeds of higher vertical modes are lower than the first mode, it may take more time to fully spin up the simulated internal tide field. Inspired by Simmons (2008) and HW11, the model is run for 600 tidal periods (TK1) from the initial state. First, a quasi-steady state is reached at approximately 17 tidal periods; then, PSI begins to develop at approximately 50 tidal periods, and eventually another steady state is reached at approximately 460 tidal periods. The equilibrium state of the simulated internal tide field is discussed in detail in section 4a. The results over tidal periods 30–40 (hereinafter the first steady period) are used for model validation, describing the basic characteristics of the K1 internal tide, and the analysis of the internal tidal field before PSI. The results over tidal periods 500–510 (hereinafter the second steady period) are used for the analysis of the equilibrium internal tidal field with PSI.
We have run several cases to study the effect of model configurations to the internal tide PSI simulation, including the model horizontal and vertical resolution, vertical viscosity (section 5a) and the number of tidal constituents contained in the model (section 5b). The total output data of approximately 104 terabytes have been saved and analyzed.
3. Basic characteristics in the SCS
Before studying the PSI of internal tides in detail and its contributions to internal tide dissipation in the following sections, we first examine the basic characteristics of K1 internal tides in the SCS.
a. Multiple generation sources
b. Long-range propagation across the basin
Low-mode K1 internal tides escape the Luzon Strait, with more energy radiating into the SCS (3.71 GW) than the western Pacific (2.62 GW) (Figs. 3a,b). Satellite observations confirm that the K1 internal tides can propagate through the deep basin and reach the Vietnam coast more than 1600 km away (Zhao 2014). We numerically reproduce the long-range propagation of the K1 internal tide in the SCS. The wave field changes dramatically after the initial days of simulation, and incident internal tidal waves from the Luzon Strait can be clearly identified (Fig. 4). The first mode internal tide wavelengths (phase speeds) are approximately 250–330 km (2.90–3.83 m s−1), consistent with satellite altimetric results. It takes approximately 7–10 days for the mode-1 K1 internal tide to propagate through the entire basin. By comparing the isopycnal displacements between adjacent tidal periods, Fig. 4e shows that internal tidal fields reach a quasi-steady state after approximately 17 days. This is reasonable because more time is needed to allow for higher-mode internal tides to travel across the SCS basin. Snapshots of the isopycnal displacements highlight that the internal tide field features a complex interference pattern produced by multiple generation sites. Along the boundaries of internal tidal waves, active submesoscale phenomena exist, which may be induced by the interaction of tidal beams with bottom topography.
Spatial distribution of the instantaneous vertical isopycnal displacements at 1000 m for the K1 internal tide after (a) 1, (b) 3, (c) 15, and (d) 30 tidal periods. The red line in (d) indicates the main propagation path, and the black plus indicates the intersection point between the main propagation path and critical latitude. (e) The root-mean-square error (RMSE) of the isopycnal displacements at 1000 m between adjacent tidal periods, from the start to 30 tidal periods.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
The K1 internal tides from the Luzon Strait first maintain a nearly westward propagation direction. Near the continental slope at approximately 117°E, the main beam veers toward the equator under the effect of topography and the continuous influence of Earth’s rotation (Zhao 2014), together with a branch directing northwestward to the continental shelf (Fig. 3b). When reaching the CLZ, the baroclinic energy flux is mostly southward, with an incident baroclinic energy flux of 519 MW (Figs. 3a,c). In addition, internal tides generated on the continental slope near Hainan Island also radiate southward into the CLZ (Figs. 3c and 4d) and contribute to the PSI of K1 internal tides.
4. PSI of propagating internal tides
We sketch the horizontal map of K1 internal tides PSI in the form of inertial bandpass-filtered baroclinic velocity variance following Simmons (2008). The model is only forced by the K1 tide, and thus, any inertial signals are believed to come from the nonlinear ocean dynamics of PSI. Figure 5 shows the variance of the full and bandpassed baroclinic velocities (using a fourth-order Butterworth filter with a pass-band of ±30% for the inertial frequency at the critical latitude), which are calculated based on the simulated baroclinic velocity at 100-m depth in the second steady period. The total variance of baroclinic velocity is highest in the northern SCS, consistent with energetic baroclinic motions in the Luzon Strait. More importantly, the bandpass-filtered baroclinic velocity variance mainly appears in the vicinity of critical latitude (CLZ), suggesting the occurrence of PSI. Considering that the SCS is only a marginal sea and the K1 internal tidal beam has a long crest line, PSI-induced near-inertial baroclinic motions nearly cover the central SCS from 109° to 120.5°E. The low baroclinic velocity variance at approximately 114°E may be the effect of island wake (Fig. 3c). The spatial distribution of the K1 internal tide PSI generally agrees with previous observational studies (Alford 2008; Chinn et al. 2012; Sun et al. 2011; Xie et al. 2009). The spatial mismatch between the observation and model results is mainly due to the lack of other diurnal tidal constituents, background current, and atmosphere forcing.
Spatial distribution of the (a) full and (b) inertial bandpass filtered baroclinic velocity variance at 100 m over periods 500–510. The blue dashed line indicates the critical latitude. The box outlined by a black dashed line indicates the range of the critical latitudinal zone. Black circles, crosses, and points indicate the mooring locations where PSI of diurnal internal tide has been observed by Chinn et al. (2012), Sun et al. (2011), and Xie et al. (2009), respectively.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
a. Assessment of the internal tidal field equilibration
An analysis of isopycnal displacements in section 3b shows that the simulated internal tidal field reaches a quasi-steady state after approximately 17 days (Fig. 4e). However, considering that our model runtime is much longer than the majority of previous internal tide simulations (HW11; Jan et al. 2007; Miao et al. 2011; Wang et al. 2016), it is necessary to further assess the equilibration of the internal tidal field during the long simulation period. The time evolution of the baroclinic kinetic energy integrated over the Luzon Strait and CLZ is depicted in Fig. 6 (blue and red lines). The Luzon Strait undergoes rapid baroclinic tidal spinup from the beginning and reaches a quasi-steady state after approximately 17 periods. Then, the region-integrated energy in the Luzon Strait has small fluctuations, but is generally stable. Interestingly, the time evolution of the baroclinic kinetic energy in the CLZ is different. It can be divided into three stages. First, the baroclinic field in the CLZ quickly spins up and reaches a near steady state after approximately 17 periods (steady state I), and the energy in the CLZ is approximately one order of magnitude lower than that in the Luzon Strait. After approximately 50 tidal periods, the region-integrated energy continuously grows with time until period 460. Finally, another equilibrium state is reached after approximately 460 periods (steady state II).
Time evolution of the baroclinic kinetic energy in the Luzon Strait (blue line) and CLZ (red line), as well as the inertial bandpass filtered baroclinic kinetic energy in the CLZ (black line). The gray patches indicate two quasi-steady periods of 30–40 and 500–510, respectively.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
To interpret the three stages of the baroclinic field, we further calculate the time evolution of the bandpassed region-integrated inertial baroclinic kinetic energy in the CLZ (black line in Fig. 6). During the first stage, near-inertial energy barely exists. Then, near-inertial energy grows constantly. The time evolution of region-integrated full and near-inertial baroclinic kinetic energy is almost parallel, suggesting that the continuous growth of baroclinic energy in the CLZ can be attributed to the evolution of PSI. When reaching steady state II, the near-inertial energy dominates the local baroclinic energy. In most internal tide simulations of tens of days, steady state has been achieved and is considered as the equilibrium state (Buijsman et al. 2013; Jan et al. 2007, 2008). But when considering the development of nonlinear wave–wave interactions, this steady stage is not so convincing. More strictly speaking, steady state II is also a quasi-steady state, because the energy level is still developing slowly after period 460. The redistribution of energy in the model from a narrow-band process to broader spectral space is slow, and nonlinear saturation requires very long simulations (HW11). Nevertheless, the growth rate in steady state II is negligibly small compared to the energy evolution over periods 50–460. In this paper, we take steady state II for an equilibrium state when the internal tide PSI is fully developed.
b. Identification of PSI-induced near-inertial signals
To study PSI, we first extract the time series of zonal baroclinic velocity in the upper 500 m at the intersection point of the main propagation path and critical latitude (14.52°N, 114.36°E, the black plus in Fig. 4d) and analyze the properties of the near-inertial and diurnal oscillations. The full baroclinic velocity at the crossing point features high-mode vertical structures (Fig. 7a). However, when removing the near-inertial part, the residual velocities show the properties of low-mode diurnal internal tides (Fig. 7c). The near-inertial velocities are of comparable magnitude to the background diurnal motions and have a period of approximately two days. In addition, near-inertial baroclinic velocities show distinct checkboard patterns, indicating concurrent upward and downward propagating waves (Fig. 7b). These high vertical mode structures and bidirection propagations of the simulated near-inertial velocities are consistent with the classic theory of PSI-induced subharmonic waves (McComas and Bretherton 1977) and in situ observations (Alford 2008; Xie et al. 2016). The characteristic vertical scales of near-inertial motions increase with depth, ranging from approximately 50 m at the near surface to approximately 180 m at a depth of 2000 m (Figs. 7 and 10). They are much closer to the dissipation scale than the background low-mode internal tide. During the second steady period, the near-inertial wave motions are persistent. Figure 7c shows a combination of all other wave signals except for the inertial band, with the diurnal motions dominating the second steady period. To further assess the occurrence of PSI, the different types of oscillations at the crossing point during the first and second steady period are compared in the frequency–wavenumber domain (Fig. 8). During the first steady period, the internal tide oscillations are characterized by horizontal lines of high-energy density patches at the frequency K1 and its harmonics (2K1, 3K1, etc.). During the second steady period, the subharmonic peak at 0.5K1 can be clearly identified, confirming the occurrence of K1 internal tide PSI. Comparisons of the horizontal locations of the high value patches at frequency K1 and 0.5K1 indicate that, the subharmonic waves feature a higher modal structure than the parent waves. Other nonlinear wave–wave interactions besides PSI are also in progress, such as the interaction between the parent waves and daughter waves.
Time series of the (a) full and (b) near-inertial bandpass filtered zonal baroclinic velocities at the crossing point between the main propagation path and critical latitude in the upper 500 m over periods 500–510. (c) Differences between (a) and (b).
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
Vertical wavenumber–frequency spectra of zonal velocities at the crossing point based on hourly output (a) without and (b) with the occurrence of PSI, respectively.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
To investigate the influence of PSI on the internal wave field, we next compare the vertical baroclinic structures along the main propagation path during the two steady periods. Figures 9a and 9b show the instantaneous vertical distribution of zonal baroclinic velocity along the main propagation path. After radiating from the Luzon Strait, internal tides propagate along ray paths, and their amplitudes decrease with propagation distance. Poleward of the critical latitude, the along-section vertical baroclinic velocity structures are nearly the same with and without PSI. However, when PSI is fully developed, the vertical structures of baroclinic velocity within the critical latitude range are quite different from steady state I. The discrepancy of vertical velocity fields between the two steady periods agree very well with the bandpassed inertial velocity field (Fig. 9c), indicating that the background internal tide field is generally stable, and the temporal variability is mainly due to PSI. The near-inertial signals are nearly of the same order as the primary internal tide within the CLZ and are mainly concentrated at the near surface and at a depth range of approximately 200–1500 m. Given that the internal tide energy is mainly within a finite region of the internal tide beam and that the evolution of PSI requires sufficient energy supply, the vertical position of PSI-induced near-inertial waves is in good agreement with the depth ranges where the low-mode internal tide beam intersects with the critical latitude (Fig. 9b). Over depths of 100–200 m, the evolution of near-inertial internal waves may be restrained by strong stratification. Consistent with the linear internal wave theory, the generated near-inertial waves only propagate equatorward to the critical latitude. Nevertheless, the near-inertial signals to the south of the CLZ (south of 13°N) are too weak to significantly affect the background internal tide field.
Internal wave field and properties along the internal tide main propagation path. (a),(b) Instantaneous zonal baroclinic velocity before and after PSI, respectively. (c) Bandpass filtered near-inertial baroclinic velocities after PSI. (d) Shear field after PSI. (e) Squared shear at the model levels over a depth range of 5–1500 m (blue lines), the red line denotes the depth-averaged value. (f) Baroclinic kinetic energy integrated over the upper 1500 m before (green line) and after (magenta line) PSI, respectively.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
Unlike wind-induced inertial waves, PSI can only trigger high-mode waves, indicating high vertical shear (Xie et al. 2009). For illustration, we calculate the vertical and latitudinal distribution of shear along the main propagation path (Figs. 9d,e), and present the along-path period-averaged depth-integrated baroclinic kinetic energy (Fig. 9f). High vertical shears mainly occur in the upper ocean of the CLZ, which are caused by high-mode near-inertial waves. Remarkably, in the far-field of the Luzon Strait, the along-section shear and baroclinic kinetic energy show a dominant peak near and equatorward of the critical latitude. High shear and energy at the critical latitude can lead to enhanced mixing, and the latitudinal dependence of PSI-induced dissipation and mixing are discussed in detail in section 4d.
A major goal of this study is to obtain a clear basin-scale map of the equatorward propagation of PSI-induced daughter waves. Based on the numerical results, we present the full and bandpassed inertial vertical displacements at the depth of 1000 m (Fig. 1). For clarity, the near-inertial vertical displacements are redrawn independently in Fig. 10a. The K1 internal tide emanates from the Luzon Strait and refracts southward to the critical latitude due to Earth’s rotation. Then, PSI occurs, and the generated near-inertial signals nearly fill the central SCS between 10°N and the critical latitude. The amplitudes of near-inertial vertical displacements are one magnitude smaller than the internal tides in the Luzon Strait but are comparable to the local background internal tide in the CLZ. Near the Nansha Islands south of 10°N, the PSI-induced daughter waves dissipate rapidly due to topographic scattering. In addition to the southern SCS, subharmonic signals can be identified in the Luzon Strait and the Celebes Sea (Fig. 10a). In the Luzon Strait, the subharmonic waves are evanescent and can only exist in the vicinity of the internal tide generation site.
Distribution of the instantaneous inertial bandpassed (a) vertical isopycnal displacements at the depth of 1000 m and (b) zonal baroclinic velocity along the main propagation path after PSI is developed.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
The critical latitude acts like a virtual island chain radiating near-inertial waves southward. Along this latitude, several generation sites exist, consistent with positions of intense incident baroclinic energy flux. Similar to the Hawaiian Ridge, the near-inertial wave field is complicated by interference patterns induced by multiple generation sites. Theoretically, given fixed water depth and stratification, internal waves with lower frequencies tend to feature longer wavelengths. The PSI-induced subharmonic waves have a period of approximately two days (Figs. 8b and 9). The horizontal wavelengths of the simulated equatorward propagating near-inertial waves are approximately 50–80 km, which are only approximately a quarter of mode-1 K1 internal tides in the SCS (240–340 km by satellite altimetry observations; Zhao 2014). For illustration, we return to the meridional distribution of near-inertial baroclinic velocity, which is amplified equatorward of the critical latitude in Fig. 10b. The results show that the near-inertial signals are high-mode waves originating from different depths in the upper ocean near the critical latitude. The short wavelengths of the subharmonic waves can be interpreted by their high-mode nature. Although the subharmonic signals can still be identified at a distance of over 350 km from the critical latitude, this spreading range is within the scope of the near field, as the wave ray is flat, and the signals diminish within the first half wavelength. Therefore, we conclude that the generated subharmonic waves are high-vertical-mode motions and dissipate within approximately 3°–4° from 11° to 15°N.
c. Influence of PSI on the baroclinic energy
In this section, we investigate the effect of PSI on the baroclinic field in terms of energy. Based on the numerical results, the region-integrated conversion rate and boundary-integrated baroclinic energy flux are calculated for the Luzon Strait and the CLZ during the two steady periods, respectively (Fig. 3a). In the Luzon Strait, the barotropic to baroclinic converted energy increases by approximately 2% (from 12.90 to 13.15 GW), which may be caused by the local accumulation of PSI-induced evanescent waves. In the CLZ, the percentage gain in local baroclinic energy is significant (38%, from 6.0 to 8.3 MW). However, the absolute value is inappreciable compared to the energetic baroclinic field. Thus, we conclude that PSI has an insignificant impact on local internal wave generation. However, PSI-induced energy transfer may still improve the closer of total tidal energy budgets (diurnal + semidiurnal) in the Luzon Strait.
After the full development of PSI, internal tides in the Luzon Strait radiate slightly more energy into the SCS basin (3.76 vs 3.71 GW). Accordingly, more baroclinic energy reaches the critical latitude after traveling a distance of >1000 km (556 vs 519 MW). A large amount of energy dissipates along the continental slope area during this stage of propagation, as the K1 internal tide features a very long wave crest line (Fig. 4). In contrast to the increase in incident energy, less baroclinic energy leaves the CLZ. By comparing the integrated baroclinic energy along the north and south boundaries of the CLZ in the two steady periods, a net increase of approximately 77 MW energy is trapped within the critical latitude range, roughly 14% of the incident baroclinic energy across the critical latitude. A18 reported a 4% loss of diurnal tidal energy due to PSI in a global simulation. This discrepancy may be caused by the different model configurations of both horizontal resolution and external forcing conditions. In addition, the calculation of baroclinic energy percentage loss is closely depend on the correct assessment of the PSI-induced subharmonic wave field equilibration. In the more realistic simulations of A18, the internal tide PSI may not be able to fully develop if the energy supply is interrupted by the time varying background environment.
Furthermore, the central SCS is not only a region with internal tidal beams across the critical latitude but also a unique region where the internal tide impacts with its critical latitude from the higher latitude. Thus, the PSI-induced daughter waves in the SCS are of the same propagation direction as the parent wave, and the daughter waves that can escape the CLZ may contribute to the equatorward energy flux and lower the baroclinic energy reduction rate. This situation in the SCS is similar to Mendocino, where an M2 internal tide beam crosses its critical latitude toward the equator, and the energy flux loss under the influence of PSI is not significant (Alford et al. 2019). Figures 3b and 3c show the baroclinic energy flux before (blue arrows) and after (black arrows) PSI in the near field of the Luzon Strait and the CLZ. Our results show that poleward of the critical latitude, the baroclinic energy flux is nearly unchanged with and without the occurrence of PSI. Equatorward of the critical latitude, the baroclinic energy flux with a negative meridional component after PSI is generally less than that before PSI. It is also worth noting that approximately one-third of the incident baroclinic energy (161 MW) is consumed when crossing the CLZ before PSI has occurred. This suggests that other processes, such as topographic scattering and energy loss during surface and bottom reflection, may dominate the internal tide dissipation in the CLZ, and PSI is a relatively modest spatially limited internal tide dissipation mechanism.
d. The latitudinal dependence of dissipation and mixing associated with PSI
Model-predicted period-averaged (a),(b) dissipation rate and (c),(d) vertical diffusivity along the main propagation path without [in (a) and (c)] and with [in (b) and (d)] the occurrence of PSI. The red dashed lines denote the location of the critical latitude.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
Before PSI, the along-beam distribution of the energy dissipation rate shows clear ray characteristics originating from the Luzon Strait (Fig. 11a). This implies that energy is mainly dissipated along the internal tide beam. The mixing hotspots are generally concentrated near the rugged topography. In the near field of the Luzon Strait, K1 internal tide induced dissipation and mixing can exceed 10−8 W kg−1 and 10−3 m2 s−1, respectively. In the upper ocean of the far field, the estimated diapycnal diffusivity is generally less than 10−5 m2 s−1. The major difference between the energy dissipation field with and without PSI lies in the upper ocean near the critical latitude, where persistently elevated energy dissipation of O(10−8) W kg−1 due to PSI (Fig. 11b). Accordingly, the vertical diffusivity in this region is approximately O(10−5–10−4) m2 s−1, which is several orders of magnitude larger than the background value in the upper ocean and is comparable to the mixing induced by topographic scattering over 500–1500 m above the bottom (Fig. 11d). Then, we attempt to make comparisons of the simulated PSI-induced mixing with observational studies. Direct observational mixing enhancement near the K1 critical latitude in the upper ocean of the SCS is still lacking, but the magnitude of the PSI-induced near-inertial baroclinic velocity and shear enhancement is consistent with the observations reported by Xie et al. (2009, 2016; 0.05–0.1 m s−1) and Alford (2008; 0.005–0.01 s−1), respectively. Direct microstructure observations are needed to further confirm the mixing enhancement by PSI in the SCS. To the south of 13°N, diapycnal mixing is reduced with the occurrence of PSI, which is in accord with energy conservation.
5. Discussion
a. Sensitivity of the internal tide PSI to model resolution and vertical viscosity
Motivated by HW11 and A18, in this section we carry out a set of numerical experiments to investigate the sensitivity of PSI to the vertical viscosity and model resolution. Except for the control experiment (CTRL), the vertical viscosity is decreased by a factor of 2 in experiment A1, the horizontal resolution is decreased from 1/30° to 1/20° in experiment A2, and the vertical layers are halved in experiment A3 (Fig. 2d), respectively (Table 1).
Summary of the sensitivity experiments.
The temporal evolution of the near-inertial baroclinic kinetic energy integrated in the CLZ is calculated and shown in Fig. 12. The near-inertial baroclinic kinetic energy evolution in the CLZ is similar among the sensitivity experiments. After the first appearance of PSI at the approximately 20–40 periods, the near-inertial baroclinic kinetic energy then grows exponentially and finally reach a quasi-steady state (steady state II in this paper). Discrepancies of the first appearance time of PSI could be possibly caused by the discrepancies of the incident baroclinic energy flux (Simmons 2008). To quantitatively assess the impact of model resolution and vertical viscosity on internal tide PSI, for each experiment, we calculate the region-integrated near-inertial baroclinic kinetic energy averaged over the last 10 periods (590–600) and the baroclinic energy flux reduction due to PSI within the CLZ (Table 1). Results during the last ten periods instead of periods 500–510 are used considering that the time needed to reach steady state II is different among the sensitive experiments. By halving the vertical viscosity in experiment A1, the near-inertial baroclinic kinetic energy over periods 590–600 increases by approximately 9%, since the PSI growth is closely related to dissipation. The baroclinic energy reduction rate also increases by halving vertical viscosity. However, the increase is not as significant as that reported by HW11. Probably because that the propagation directions of the PSI-induced subharmonic waves and the incident internal tides are the same (equatorward) in our case. In the model, the estimated baroclinic energy reduction rate represents an upper bound on the PSI contribution to tidal flux, because atmosphere forcing is excluded and increasing turbulence in the upper ocean may restrict the growth of PSI (A18). Experiment A2 shows that the PSI-induced near-inertial baroclinic kinetic energy in the CLZ greatly decreases when the horizontal model resolution is decreased from 1/30° to 1/20°. The significance of horizontal model resolution in simulating PSI has also been reported by the global study of A18. In experiment A3, which reduces the vertical layers from 100 to 50, the PSI growth rate, the final energy level of the PSI-induced near-inertial waves and the baroclinic energy reduction rate are all affected by the vertical model resolution (Fig. 12). However, it is hard to draw any strong conclusions because the energy levels of the PSI-induced near-inertial baroclinic kinetic energy go seesaw between CTRL and A3. Our model employs a z-level vertical coordinate; changes in vertical levels affect the delineation of bottom topography through the finite volume formulation.
Time evolution of the inertial bandpass filtered baroclinic kinetic energy in the CLZ among the sensitive experiments. The bottom-right insert shows an amplified view of the energy evolution in the experiment A2.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
b. Comparisons of two-constituent and single-constituent simulations
So far, we only present the results of K1 internal tide PSI in the SCS. Similar results have been obtained for O1. In this section, we conducted an additional comparison simulation forced by the two main diurnal tidal constituents (K1 and O1) to analyze the discrepancies of the modeled internal tide PSI between two-constituent and single-constituent simulations. Figure 13 present the 1000-m zonal velocity spectra at a station equatorward of the bidiurnal latitudes (13°N, 115°E; see the station location in Fig. 14) based on simulations of different tidal constituents. To separate the K1 and O1 signals and associated harmonics in the two-constituent simulation, results of longer time series (periods 20–50 and 570–600) are extracted for the spectral analysis to represent the two steady states. Similarly, results of the exactly same time periods in the single-constituent simulations are used for comparison.
Velocity spectra at a station equatorward of the bidiurnal critical latitudes (13°N, 115°E; see the station location in Fig. 14) from the (a),(b) single-constituent simulations (K1 and O1) and (c) two-constituent simulation (K1O1). The black, gray, and yellow dashed vertical lines mark various harmonics (e.g., 0.5O1, O1, 0.5K1, K1, K1O1). (d) Differences between (c) and the sum of (a) and (b). The blue and red lines indicate the results without and with the occurrence of PSI, respectively.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
Shifting of the bidiurnal critical latitudes in the SCS. Colors indicate the relative vorticity from the mean dynamic topography. The red and black lines indicate the critical latitude of K1 and O1 internal tides, respectively. The black plus indicates the station selected for discussions in section 5b.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
Before the occurrence of PSI, the most obvious peaks appear at the forcing frequencies (K1 and O1) and the superharmonics (2K1, 2O1, and K1O1) (blue lines in Figs. 14a–c). During periods 570–600, significant peaks comparable to the forcing frequencies appear at the half-diurnal tidal bands, indicating the occurrence of PSI (red lines in Figs. 14a–c). PSI develops later than the nonlinear interaction of (K1 + K1), (O1 + O1) and (K1 + O1), but the peak heights largely exceed those of superharmonics frequencies. Because of PSI, the peak magnitudes of the forcing frequencies are declined during periods 570–600, comparing to that during periods 50–70. The magnitude of 0.5K1 peak is smaller than that of 0.5O1 due to the longer distance away from the corresponding critical latitudes. For the same reason, the 1.5O1 peak can be clearly identified while the 1.5K1 peak disappears. The internal wave fields in the two-constituent simulation are far more complicated than the single-constituent simulations.
To investigate the effect of tidal constituents included in the model to the PSI simulation, Fig. 13d shows the spectral discrepancies between two-constituent simulation and the sum of the two single-constituent simulations. Without the occurrence of PSI, the peak magnitudes of the forcing frequencies decrease mainly due to the nonlinear interaction between the forcing frequencies. With the occurrence of PSI, the subharmonic peaks decrease while the diurnal frequencies peaks increase. These suggest that the growth of PSI is restricted in the two-constituent simulation, and more energy is retained in the diurnal band. A possible reason is that the nonlinear interactions between the forcing frequencies develop earlier than PSI, they reduce the energy in the forcing band and lead to a restraint of the subsequent self-disintegration of each constituent.
c. Dependence of PSI on the background internal tide field
The energy transfer time scale is a crucial factor of PSI. Our model results suggest that, the PSI of the K1 internal tide near the critical latitude of the SCS needs more than 460 days to fully develop. The real ocean is a mature state, and the ultimate effect of PSI is closely associated with the persistence of the background internal tidal energy, together with the first appearance and full development time of PSI. If the internal tide persistence time is shorter than the first appearance time of PSI, internal tide PSI would not occur. If the internal tide persistence time is longer than the first appearance time, but shorter than the full development time of PSI, then the impact of PSI is reduced. If the internal tide persistence time is longer than the full development time of PSI, with sufficient energy supply, the cross-scale energy transfer may be sustained, and the generated signals can no longer be neglected. PSI-induced daughter waves may gradually accumulate and even reach the same order as the parent wave (Fig. 9b). In the SCS, the internal tides from the Luzon Strait are strong year-round, this explains why the observations of diurnal internal tide PSI so far are mainly concentrated in the SCS. Although PSI does not induce significant energy loss of the background internal tide field, PSI-triggered near-inertial signals and associated mixing elevation still play important roles within the critical latitude range and enrich the spatial pattern of the mixing field in the SCS. In addition, idealized numerical studies suggest that the first appearance and full development time of PSI decrease with increasing background internal tide energy flux (Simmons 2008); thus, the evolution of PSI differs with different internal tide environments. Currently, the critical latitude effects are not yet included in the existing mixing parameterizations. Refinements of model predictions can be expected with consideration of PSI, especially for oceanographic processes that are sensitive to the upper-ocean mixing near critical latitudes.
d. Migration of critical latitudes in the presence of background currents
6. Conclusions
In this study, we reproduce the K1 internal tidal field in the SCS based on MITgcm, a nonhydrostatic three-dimensional high-resolution ocean model, with the main focus on the internal tide PSI process. The model successfully reproduces the multiple generation feature and long-range propagation pattern of the K1 internal tide in the SCS. The model also successfully captures the PSI-induced near-inertial signals near and equatorward of the critical latitude. Our major conclusions are as follows:
The internal tidal dynamics within the critical latitude range in our simulations have two relatively steady states, between which PSI-induced subharmonic waves evolve constantly. With the full development of PSI, the PSI-induced subharmonic waves reach the same order as their parent waves.
The properties of the bandpass filtered near-inertial signals near the critical latitude agree well with the theoretical prediction of PSI-induced subharmonic waves, suggesting the importance of PSI at this latitude.
Based on the numerical results, we present the first basin-scale map of the PSI-induced subharmonic waves in the SCS. The critical latitude acts as a “virtual island chain” radiating near-inertial waves equatorward. The PSI-induced near-inertial waves are dominated by high modes, with vertical scales ranging from approximately 50 m at the near surface to approximately 180 m at a depth of 2000 m.
In the CLZ, an additional 14% of the incident baroclinic energy is lost under the influence of PSI. Additionally, the final energy level of the PSI-induced near-inertial energy and the percentage of energy loss at the critical latitudes differ regionally, and are closely related to the intensity and duration of the background internal tide.
The model horizontal resolution, the vertical eddy viscosity, and the number of tidal constituents contained in the model can affect the internal tide PSI simulation.
The K1 internal tide PSI-induced mixing enhancement mainly occurs in the upper 1500 m, with a magnitude of approximately O(10−5–10−4) m2 s−1. Although PSI is only a relatively modest and spatially limited dissipation mechanism, it is still an important process for the far-field dissipation of the Luzon Strait internal tide.
Acknowledgments
The bathymetric data were obtained from the General Bathymetric Chart of the Oceans (http://www.gebco.net), the temperature and salinity profiles were obtained from the World Ocean Atlas 2018 (https://www.nodc.noaa.gov/OC5/woa18/), the barotropic tidal current velocity used for the boundary condition was obtained from the TPXO9 global inverse tide model (http://volkov.oce.orst.edu/tides/tpxo9_atlas.html). Numerical calculations were performed at the Center for High Performance Computing and System Simulation, Qingdao National Laboratory for Marine Science and Technology. K.L. was funded by the National Natural Science Foundation of China (41906005), the National Basic Research Program of China (2019-JCJQ-ZD-149-00) and the Laboratory for Regional Oceanography and Numerical Modeling, Qingdao National Laboratory for Marine Science and Technology (2019A05).
APPENDIX
Model Verification
(a),(b) Cotidal charts and (c),(d) barotropic energy flux calculated from the simulation and TPXO9, respectively.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
Comparison of the modeled and observed diurnal baroclinic energy fluxes in the Luzon Strait. The station locations and observed energy fluxes of Alford et al. (2011) are indicated as blue circles and arrows, respectively. The black arrows indicate the simulated energy fluxes and colors denote the magnitude.
Citation: Journal of Physical Oceanography 50, 12; 10.1175/JPO-D-19-0320.1
REFERENCES
Alford, M. H., 2008: Observations of parametric subharmonic instability of the diurnal internal tide in the South China Sea. Geophys. Res. Lett., 35, L15602, https://doi.org/10.1029/2008GL034720.
Alford, M. H., J. A. Mackinnon, Z. Zhao, R. Pinkel, J. Klymak, and T. Peacock, 2007: Internal waves across the Pacific. Geophys. Res. Lett., 34, L24601, https://doi.org/10.1029/2007GL031566.
Alford, M. H., and Coauthors, 2011: Energy flux and dissipation in Luzon Strait: Two tales of two ridges. J. Phys. Oceanogr., 41, 2211–2222, https://doi.org/10.1175/JPO-D-11-073.1.
Alford, M. H., H. L. Simmons, O. B. Marques, and J. B. Girton, 2019: Internal tide attenuation in the North Pacific. Geophys. Res. Lett., 46, 8205–8213, https://doi.org/10.1029/2019GL082648.
Ansong, J. K., B. K. Arbic, M. C. Buijsman, J. G. Richman, J. F. Shriver, and A. J. Wallcraft, 2015: Indirect evidence for substantial damping of low-mode internal tides in the open ocean. J. Geophys. Res. Oceans, 120, 6057–6071, https://doi.org/10.1002/2015JC010998.
Ansong, J. K., and Coauthors, 2017: Semidiurnal internal tide energy fluxes and their variability in a Global Ocean Model and moored observations. J. Geophys. Res. Oceans, 122, 1882–1900, https://doi.org/10.1002/2016JC012184.
Ansong, J. K., and Coauthors, 2018: Geographical distribution of diurnal and semidiurnal parametric subharmonic instability in a global ocean circulation model. J. Phys. Oceanogr., 48, 1409–1431, https://doi.org/10.1175/JPO-D-17-0164.1.
Buijsman, M. C., and Coauthors, 2013: Three-dimensional double-ridge internal tide resonance in Luzon Strait. J. Phys. Oceanogr., 44, 850–869, https://doi.org/10.1175/JPO-D-13-024.1.
Carter, G. S., and M. C. Gregg, 2006: Persistent near-diurnal internal waves observed above a site of M2 barotropic-to-baroclinic conversion. J. Phys. Oceanogr., 36, 1136–1147, https://doi.org/10.1175/JPO2884.1.
Carter, G. S., and M. A. Merrifield, 2007: Open boundary conditions for regional tidal simulations. Ocean Modell., 18, 194–209, https://doi.org/10.1016/j.ocemod.2007.04.003.
Carter, G. S., and Coauthors, 2008: Energetics of M2 barotropic-to-baroclinic tidal conversion at the Hawaiian Islands. J. Phys. Oceanogr., 38, 2205–2223, https://doi.org/10.1175/2008JPO3860.1.
Chinn, B. S., J. B. Girton, and M. H. Alford, 2012: Observations of internal waves and parametric subharmonic instability in the Philippines archipelago. J. Geophys. Res., 117, C05019, https://doi.org/10.1029/2011JC007392.
Egbert, G. D., and S. Y. Erofeeva, 2002: Efficient inverse modeling of barotropic ocean tides. J. Atmos. Oceanic Technol., 19, 183–204, https://doi.org/10.1175/1520-0426(2002)019<0183:EIMOBO>2.0.CO;2.
Fang, G., Y. Kwok, K. Yu, and Y. Zhu, 1999: Numerical simulation of principal tidal constituents in the South China Sea, Gulf of Tonkin and Gulf of Thailand. Cont. Shelf Res., 19, 845–869, https://doi.org/10.1016/S0278-4343(99)00002-3.
Ferrari, R., and C. Wunsch, 2009: Ocean circulation kinetic energy: Reservoirs, sources, and sinks. Annu. Rev. Fluid Mech., 41, 253–282, https://doi.org/10.1146/annurev.fluid.40.111406.102139.
Garcia H. E., and Coauthors, 2019: World Ocean Atlas 2018: Product documentation. NOAA/NESDIS, 20 pp., https://odv.awi.de/fileadmin/user_upload/odv/data/WOA18/woa18documentation.pdf.
Gerkema, T., C. Staquet, and P. Bouruet-Aubertot, 2006: Decay of semi-diurnal internal-tide beams due to subharmonic resonance. Geophys. Res. Lett., 33, L08604, https://doi.org/10.1029/2005GL025105.
Hazewinkel, J., and K. Winters, 2011: PSI of the internal tide on a beta-plane: Flux divergence and near-inertial wave propagation. J. Phys. Oceanogr., 41, 1673–1682, https://doi.org/10.1175/2011JPO4605.1.
Heathershaw, A. D., A. L. New, and P. D. Edwards, 1987: Internal tides and sediment transport at the shelf break in the Celtic Sea. Cont. Shelf Res., 7, 485–517, https://doi.org/10.1016/0278-4343(87)90092-6.
Hibiya, T., and M. Nagasawa, 2004: Latitudinal dependence of diapycnal diffusivity in the thermocline estimated using a finescale parameterization. Geophys. Res. Lett., 31, L01301, https://doi.org/10.1029/2003GL017998.
Jan, S., C.-S. Chern, J. Wang, and S.-Y. Chao, 2007: Generation of diurnal K1 internal tide in the Luzon Strait and its influence on surface tide in the South China Sea. J. Geophys. Res., 112, C06019, https://doi.org/10.1029/2006JC004003.
Jan, S., R.-C. Lien, and C.-H. Ting, 2008: Numerical study of baroclinic tides in Luzon Strait. J. Oceanogr., 64, 789–802, https://doi.org/10.1007/s10872-008-0066-5.
Kang, D., and O. Fringer, 2012: Energetics of barotropic and baroclinic tides in the Monterey Bay area. J. Phys. Oceanogr., 42, 272–290, https://doi.org/10.1175/JPO-D-11-039.1.
Klymak, J. M., and S. M. Legg, 2010: A simple mixing scheme for models that resolve breaking internal waves. Ocean Modell., 33, 224–234, https://doi.org/10.1016/j.ocemod.2010.02.005.
Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear. J. Phys. Oceanogr., 15, 544–565, https://doi.org/10.1175/1520-0485(1985)015<0544:NIWPIG>2.0.CO;2.
Kunze, E., L. K. Rosenfeld, G. S. Carter, and M. C. Gregg, 2002: Internal waves in Monterey submarine canyon. J. Phys. Oceanogr., 32, 1890–1913, https://doi.org/10.1175/1520-0485(2002)032<1890:IWIMSC>2.0.CO;2.
Liang, Y., L.-A. Couston, Q. Guo, and M.-R. Alam, 2017: Dominant resonance in parametric subharmonic instability of internal waves. arXiv, https://arxiv.org/abs/1709.06250v2.
Liao, G., Y. Yuan, C. Yang, H. Chen, H. Wang and W. Huang, 2012: Current observations of internal tides and parametric subharmonic instability in Luzon Strait. Atmos. Ocean, 50, 59–76, https://doi.org/10.1080/07055900.2012.742007.
Liu, K., J. Sun, C. Guo, Y. Yang, W. Yu, and Z. Wei, 2019: Seasonal and spatial variations of the M2 internal tide in the Yellow Sea. J. Geophys. Res. Oceans, 124, 1115–1138, https://doi.org/10.1029/2018JC014819.
MacKinnon, J. A., and K. B. Winters, 2005: Subtropical catastrophe: Significant loss of low-mode tidal energy at 28.9°. Geophys. Res. Lett., 32, L15605, https://doi.org/10.1029/2005GL023376.
MacKinnon, J. A., M. H. Alford, R. Pinkel, J. Klymak, and Z. Zhao, 2013a: The latitudinal dependence of shear and mixing in the Pacific transiting the critical latitude for PSI. J. Phys. Oceanogr., 43, 3–16, https://doi.org/10.1175/JPO-D-11-0107.1.
MacKinnon, J. A., M. H. Alford, O. Sun, R. Pinkel, Z. Zhao, and J. Klymak, 2013b: Parametric subharmonic instability of the internal tide at 29°N. J. Phys. Oceanogr., 43, 17–28, https://doi.org/10.1175/JPO-D-11-0108.1.
Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997: A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102, 5753–5766, https://doi.org/10.1029/96JC02775.
McComas, C. H., and F. P. Bretherton, 1977: Resonant interaction of oceanic internal waves. J. Geophys. Res., 82, 1397–1412, https://doi.org/10.1029/JC082i009p01397.
Miao, C., H. Chen, and X. Lü, 2011: An isopycnic-coordinate internal tide model and its application to the South China Sea. Chin. J. Oceanology Limnol., 29, 1339–1356, https://doi.org/10.1007/s00343-011-1023-5.
Müller, M., B. K. Arbic, J. G. Richman, J. F. Shriver, E. L. Kunze, R. B. Scott, A. J. Wallcraft, and L. Zamudio, 2015: Toward an internal gravity wave spectrum in global ocean models. Geophys. Res. Lett., 42, 3474–3481, https://doi.org/10.1002/2015GL063365.
Nikurashin, M., and S. Legg, 2011: A mechanism for local dissipation of internal tides generated at rough topography. J. Phys. Oceanogr., 41, 378–395, https://doi.org/10.1175/2010JPO4522.1.
Pawlowicz, R., R. Beardsley, and S. Lentz, 2002: Classical tidal harmonic analysis including error estimates in MATLAB using T_TIDE. Comput. Geosci., 28, 929–937, https://doi.org/10.1016/S0098-3004(02)00013-4.
Robertson, R., J. Dong, and P. Hartlipp, 2017: Diurnal Critical latitude and the latitude dependence of internal tides, internal waves, and mixing based on Barcoo seamount. J. Geophys. Res. Oceans, 122, 7838–7866, https://doi.org/10.1002/2016JC012591.
Savage, A. C., and Coauthors, 2017: Spectral decomposition of internal gravity wave sea surface height in global models. J. Geophys. Res. Oceans, 122, 7803–7821, https://doi.org/10.1002/2017JC013009.
Simmons, H. L., 2008: Spectral modification and geographic redistribution of the semi-diurnal internal tide. Ocean Modell., 21, 126–138, https://doi.org/10.1016/j.ocemod.2008.01.002.
Sun, L., Q. Zheng, D. Wang, J. Hu, C. K. Tai, and Z. Sun, 2011: A case study of near-inertial oscillation in the South China Sea using mooring observations and satellite altimeter data. J. Oceanogr., 67, 677–687, https://doi.org/10.1007/s10872-011-0081-9.
Sun, O. M., and R. Pinkel, 2013: Subharmonic energy transfer from the semidiurnal internal tide to near-diurnal motions over Kaena Ridge, Hawaii. J. Phys. Oceanogr., 43, 766–789, https://doi.org/10.1175/JPO-D-12-0141.1.
Tian, J., Q. Yang, and W. Zhao, 2009: Enhanced diapycnal mixing in the South China Sea. J. Phys. Oceanogr., 39, 3191–3203, https://doi.org/10.1175/2009JPO3899.1.
van Haren, H., 2005: Tidal and near-inertial peak variations around the diurnal critical latitude. Geophys. Res. Lett., 32, L23611, https://doi.org/10.1029/2005GL024160.
Wang, X., S. Peng, Z. Liu, R. Huang, Y.-K. Qian, and Y. Li, 2016: Tidal mixing in the South China Sea: An estimate based on the internal tide energetics. J. Phys. Oceanogr., 46, 107–124, https://doi.org/10.1175/JPO-D-15-0082.1.
Wilson, C., 2011: Chlorophyll anomalies along the critical latitude at 30°N in the NE Pacific. Geophys. Res. Lett., 38, L15603, https://doi.org/10.1029/2011GL048210.
Xie, X., X. Shang, G. Chen, and L. Sun, 2009: Variations of diurnal and inertial spectral peaks near the bi-diurnal critical latitude. Geophys. Res. Lett., 36, L02606, https://doi.org/10.1029/2008GL036383.
Xie, X., Q. Liu, X. Shang, G. Chen, and D. Wang, 2016: Poleward propagation of parametric subharmonic instability-induced inertial waves. J. Geophys. Res. Oceans, 121, 1881–1895, https://doi.org/10.1002/2015JC011194.
Yang, Q., W. Zhao, X. Liang, and J. Tian, 2016: Three-dimensional distribution of turbulent mixing in the South China Sea. J. Phys. Oceanogr., 46, 769–788, https://doi.org/10.1175/JPO-D-14-0220.1.
Yang, W., T. Hibiya, Y. Tanaka, L. Zhao, and H. Wei, 2018: Modification of parametric subharmonic instability in the presence of background geostrophic currents. Geophys. Res. Lett., 45, 12 957–12 962, https://doi.org/10.1029/2018GL080183.
Zhao, Z., 2014: Internal tide radiation from the Luzon Strait. J. Geophys. Res. Oceans, 119, 5434–5448, https://doi.org/10.1002/2014JC010014.
Zhao, Z., M. H. Alford, J. A. Mackinnon, and R. Pinkel, 2010: Long-range propagation of the semidiurnal internal tide from the Hawaiian ridge. J. Phys. Oceanogr., 40, 713–736, https://doi.org/10.1175/2009JPO4207.1.
The critical latitudes in this paper are defined as the latitudes where the local inertial frequencies equal half of the tidal frequencies: 28.8°, 14.52°, and 13.44°N for M2, K1, and O1 internal tides, respectively.
