This study examines the effects of the subtidal circulation on the generation and propagation of the M2 internal tide in the Philippine Sea using a primitive equation model. Barotropic to baroclinic conversion at the Luzon Strait is found to vary due to the background circulation changes over the generation site and the changing influence of remotely generated internal tides from the Mariana Arc. The varying effect of remotely generated waves results from both changing generation energy levels at the Mariana Arc and variability in the propagation of the internal tides across the Philippine Sea. The magnitude and direction of the depth-integrated baroclinic energy fluxes vary temporally, due to a combination of changing generation, propagation, and dissipation. Spatial patterns of internal tide propagation near the Luzon Strait are influenced by the locations of mesoscale eddies to the east and west of the strait. The results provide insight into the mechanisms of variability of the baroclinic tides and highlight the importance of considering both the remotely generated internal tides and the subtidal dynamics to estimate internal tide energetics.
Energy is converted from the surface (barotropic) tide to internal (baroclinic) tides by flow over steep topography causing vertical displacement of the stratified water. While the surface tides are deterministic, the internal tides are influenced by the atmospherically forced subtidal ocean circulation and are highly variable (e.g., Colosi and Munk 2006; Osborne et al. 2011; Zilberman et al. 2011). Barotropic to baroclinic conversion (henceforth referred to as generation) depends on the strength of the barotropic tidal flow, the topographic slope and orientation, the stratification over the topographic feature (Baines 1973, 1974; Hibiya 1986; Holloway and Merrifield 1999; Griffiths and Grimshaw 2007), and remotely generated internal tides altering the local phase and amplitude of the perturbation pressure (Kelly and Nash 2010; Hall and Carter 2011; Zilberman et al. 2011; Powell et al. 2012; Kerry et al. 2013a). The stratification and remote phasing can change significantly on advective time scales associated with the subtidal circulation. Modulation of internal tides has been observed due to the passing of eddies (Park and Watts 2006), low-frequency pycnocline displacement (Mitchum and Chiswell 2000), and changes in the background density field and currents (Colosi and Munk 2006; Osborne et al. 2011). As internal tides propagate, variability can result from advection by background currents (Olbers 1981b; Chavanne et al. 2010b), refraction by a varying background density field (Chuang and Wang 1981; Park and Watts 2006), wave–wave interactions (Müller et al. 1986), topographic scattering (Müller and Xu 1992; Johnston and Merrifield 2003), and nonlinear interactions with eddies (Olbers 1981a; van Haren 2004). Internal tide interactions with mesoscale features such as eddies and fronts can affect tidal energy budgets as the coherent internal tide energy can be transferred into incoherent signals (Rainville and Pinkel 2006; Zaron et al. 2009; Chavanne et al. 2010a).
The Philippine Sea is bounded by Taiwan and the Philippine islands of Luzon and Mindanao to the west and the Mariana Islands to the east (Fig. 1). Internal tides from the two significant generation sites, the Luzon Strait and the Mariana Island Arc, propagate into the region. Some of the most energetic internal tides observed worldwide are generated at the Luzon Strait (Ramp et al. 2004; Alford et al. 2011) and baroclinic tidal energy propagates both westward into the South China Sea (SCS) and eastward into the Philippine Sea, an important region of the North Pacific subtropical gyre. The North Equatorial Current (NEC) bifurcates at the Philippine coast south of Luzon with the majority of the flow turning northward (Qiu and Lukas 1996). The northward current veers westward in the Luzon Strait developing a Loop Current that penetrates into the SCS to varying extents (Yuan et al. 2006; Rudnick et al. 2011), before forming the Kuroshio as it flows northeastward along the east coast of Taiwan. The region is traversed by a band of enhanced eddy kinetic energy centered at 22°N resulting from the baroclinic instability associated with the vertical shear between the eastward, surface-flowing Subtropical Countercurrent (STCC) and the westward subsurface NEC (Qiu 1999; Qiu and Chen 2010). The strong mesoscale variability may result in significant temporal and spatial variability in internal tide generation and propagation in the region.
Understanding internal tide energetics is important for understanding the energy budget of the deep ocean and developing mixing parameterizations for global climate–scale circulation models (Jayne 2009). Our previous work, Kerry et al. (2013a, hereafter referred to as KPC2013), used a numerical model with horizontally uniform stratification to examine the effects of remotely generated M2 internal tides on the Luzon Strait and Mariana Arc generation sites. We found that remote internal tides affected generation by altering both the magnitude and phase of the pressure perturbation at the bottom with respect to the local surface tide and that internal tide generation at the two sites is dynamically connected by propagating low-mode internal tides, despite being thousands of kilometers apart. The background stratification and currents are known to influence internal tides; however, the impact of the oceanic general circulation, and its associated mesoscale eddy field, on internal tides in the global oceans is not well known. Arbic et al. (2010) and Shriver et al. (2012) present an early attempt to model the eddying general circulation and the barotropic and baroclinic tides in a global model and examine the impact of horizontally nonuniform stratification on mean internal tide characteristics. A more thorough understanding of the impact of the subtidal circulation on internal tide generation and propagation, and the resulting temporal and spatial variability of baroclinic tidal energy, is important for improving estimates of the distribution of mixing in the ocean and interpreting observations of internal tides and turbulence.
This paper presents a primitive equation model of the Philippine Sea that simultaneously resolves the general circulation, and its associated mesoscale eddy field, and the M2 barotropic and baroclinic tides to examine the effects of the subtidal flow on the M2 internal tide generated at the Luzon Strait. We present a detailed analysis of the variability in generation and propagation patterns of the internal tides in the Philippine Sea, a region that experiences both energetic baroclinic tides and dynamic mesoscale circulation. In Kerry et al. (2013b, manuscript submitted to J. Phys. Oceanogr.), we examine the role that the subtidal flow plays in dissipating the internal tide energy. We focus on the dominant semidiurnal constituent M2, so as to be able to present a thorough analysis within the scope of this paper. The M2 and K1 tidal constituents are both of approximately equal magnitude in the region (Zu et al. 2008; Alford et al. 2011), and further studies may be interested in investigating the variability of the K1 internal tides in the presence of subtidal circulation. Throughout the paper, when comparing to observations taken over time periods that are too short to separate the semidiurnal constituents M2 and S2, we make comparisons to a model run that includes both semidiurnal constituents. The Philippine Sea is a region of global significance and provides an ideal case study for examining the impact of the subtidal circulation and remotely generated internal tides on internal tide variability, which affects both the magnitude and distribution of mixing processes in the ocean.
2. Model description
a. Model configuration
We use the Regional Ocean Modeling System (ROMS) to simultaneously resolve the eddying general circulation and the principal, lunar, semidiurnal (M2) barotropic and baroclinic tides in the Philippine Sea. ROMS is a free-surface, hydrostatic, primitive equation ocean model solved on a curvilinear grid with a terrain-following vertical coordinate system. A split-explicit time-stepping scheme improves computational efficiency by allowing the barotropic solution to be computed at a much smaller time step than is used for the (slow mode) baroclinic equations. The scheme uses a temporal-averaging filter to ensure the preservation of tracers and momentum and minimize aliasing of unresolved barotropic signals into the baroclinic motions (Shchepetkin and McWilliams 2005). The ROMS computational kernel is described in more detail in Shchepetkin and McWilliams (1998, 2003, 2005).
Although internal tides inherently have a nonhydrostatic component, the hydrostatic approximation remains valid for this study as the internal tide horizontal wavelengths are much greater than the associated scales of vertical displacement. While the baroclinic tides in the SCS have been shown to be highly nonlinear and nonhydrostatic as they steepen on interaction with the continental shelf (Alford et al. 2010; Farmer et al. 2011), we focus on their propagation into the Philippine Sea, where the internal tides travel into deep water and do not form solitons (Buijsman et al. 2010a), and their propagation into the SCS prior to interaction with the shelf. The ROMS model handles nonlinear processes but makes the hydrostatic approximation.
Numerical models have been widely used to make estimates of tidal energy conversion on global (Simmons et al. 2004), ocean basin (Niwa and Hibiya 2001), and regional scales [Niwa and Hibiya (2004) and KPC2013 for the Luzon Strait; Merrifield and Holloway (2002) and Carter et al. (2008) for Hawaii; and Zilberman et al. (2009) for the Mid-Atlantic Ridge], assuming mean, horizontally uniform stratification or typical “summer” and “winter” stratifications [e.g., Jan et al. (2008) for the Luzon Strait]. Arbic et al. (2010) present global simulations with tides and mesoscale flow using the Hybrid Coordinate Ocean Model (HYCOM). All of the above-mentioned regional- and global-scale internal tide studies have used hydrostatic models.
To investigate the variability in the baroclinic tides due to the time-varying circulation and stratification, and the varying influence of remotely generated internal tides, we conduct simulations on an inner grid nested within a larger outer grid (see Fig. 1). The model grid for the outer domain is the same that was used in the KPC2013 simulation encompassing both the Luzon Strait and the Mariana Island Arc. The grid has a variable horizontal resolution with 8 km over most of the domain and a higher zonal resolution of 4.5 km over the Luzon Strait. This allows improved bathymetric resolution while minimizing horizontal pressure gradient errors in the region of steep topography. Some topographic smoothing is applied to alleviate pressure gradient errors, and the method is explained in greater detail in KPC2013. The model is configured with 25 vertical s layers distributed with a higher resolution in the upper 250 m of the water column. The model bathymetry is obtained from the general bathymetric chart of the oceans 1-min gridded bathymetry data generated from combined satellite-derived gravity data and ship depth soundings (IOC et al. 2003).
Subgrid-scale horizontal mixing of momentum and tracers is handled using a harmonic (3-point stencil) mixing operator (Haidvogel and Beckmann 1999), and the viscosity is derived from the horizontal divergence of the deviatory stress tensor (Wajsowicz 1993). The diffusion and viscosity coefficients are scaled by grid size such that less explicit diffusion occurs in the high-resolution (4.5 km) region than in the 8-km region. The Mellor and Yamada (1982) level 2.5, second-moment turbulence closure scheme (MY2.5) is used in parameterizing vertical turbulent mixing of momentum and tracers. The Chapman condition (Chapman 1985) is applied to the free surface at the boundaries, and the Flather condition (Flather 1976) is applied to the barotropic velocity so that barotropic energy is transmitted out of the domain. Baroclinic energy is absorbed at the boundaries using a flow relaxation scheme involving a sponge layer over which viscosity and diffusivity are increased linearly by an order of magnitude from the values applied within the model domain. No internal wave reflections are observed at the model boundaries.
The simulation configured for the outer grid uses boundary and initial conditions from the MERCATOR general ocean circulation model graciously provided by Mercator Océan of France. The atmospheric forcing was provided by the National Centers for Environmental Prediction (NCEP) reanalysis atmospheric model to simulate the atmospherically forced eddying ocean circulation (Kistler et al. 2001). The M2 tides are included by forcing at the four open boundaries with M2 tidal surface elevation and momentum from the global barotropic tidal model provided by the Oregon State University Ocean Topography Experiment (TOPEX)/Poseidon Global Inverse Tide Model (TPXO7.1; Egbert and Erofeeva 2002). The inner grid has identical bathymetry and horizontal and vertical resolution as the outer grid but is bounded between 15.9° and 24.9°N and 116.9° and 136.9°E. The purpose of the nested inner grid is not to provide increased resolution through downscaling of the grid size, but to allow us to investigate scenarios including and excluding remotely generated internal tides from the Mariana Arc. We investigate how the local variability in internal tide generation at the Luzon Strait depends on the locally changing mesoscale circulation and the varying influence of remotely generated internal tides by performing three simulations with the nested inner domain:
Full: The Full simulation includes the influence of baroclinic tides generated at the Mariana Arc under the varying subtidal dynamics. The general ocean circulation and baroclinic tides are applied at the boundaries through nesting with the outer domain. Following the method for nesting tidal models described in Janekovic and Powell (2012), the tidal amplitudes and phases of the barotropic M2 surface elevation and momentum are computed from the outer domain simulation and applied as harmonic forcing to the inner domain simulation. These barotropic components must therefore be removed from the surface elevation and velocity boundary forcings applied as time series to the inner domain. The boundary forcings applied as time series include all flow except the barotropic tides, that is, the general subtidal circulation and the baroclinic tides. Janekovic and Powell (2012) showed that this method of fitting and removing the tidal harmonics from the boundaries and providing the harmonics as separate forcing resulted in significant improvement in the modeled tidal dynamics, as compared to simply prescribing the unmodified boundary conditions to a nested model.
Luzon only: The internal tides generated outside of the inner domain are filtered out in this simulation. The spectral barotropic boundary forcing of M2 surface elevation and momentum that was extracted from the outer domain simulation as in (i) is applied. The boundary conditions taken from the outer domain are low-pass filtered to remove the baroclinic tides.
Constant remote flux: A constant baroclinic energy flux is applied at the eastern boundary. As such, this case excludes generation site variability of internal tides from the Mariana Arc. The spectral barotropic boundary forcing and the low-pass-filtered boundary conditions as in (ii) are used, with a fixed M2 baroclinic energy flux added to the boundary condition specified at the eastern boundary. This constant flux is taken from the M2 tidal simulation of the Mariana Arc for the mean horizontally uniform stratification described in KPC2013.
Each simulation is integrated for 390 days, from 2 December 2009 to 27 December 2010, and the first 30 days are excluded from analysis for barotropic and baroclinic tidal spinup. Analysis is performed for 360 days from 1 January to 27 December 2010.
b. Comparison with observations
We aim to investigate the variability of internal tides in the presence of varying subtidal circulation in our model and, in this section, we present comparisons of the mean flow and the variability with available observations. The mean potential density field to 1000-m depth is shown in Fig. 1 (bottom) for a cross section through the inner domain at 21°N from the Full case simulation (low-pass filtered to remove tides). The pycnocline tilts downward to the east at the Luzon Strait associated with the northward-flowing Kuroshio.
The stratification over the ridges at the Luzon Strait can change significantly depending on the location of the current, influencing internal tide generation (Buijsman et al. 2010b; Jan et al. 2012). The propagation speed of the M2 internal tide (3.5–4.5 m s−1; Niwa and Hibiya 2004; KPC2013) is of the same order of magnitude as the speed of the Kuroshio (0.75–1.5 m s−1), which suggests that the spatial patterns of the M2 internal tide could be significantly altered by the Kuroshio, particularly in the region of the Loop Current where the flow runs both parallel and perpendicular to the direction of propagation of the internal tides. As such, the intrusion of the Kuroshio into the Luzon Strait is the dominant influence on stratification over the internal tide generation sites. In Fig. 2, we examine the mean model structure of velocity and density. Figure 2a shows the northward-flowing Kuroshio along the east coast of Taiwan at 23°N and the associated downward tilt of the pycnocline. The Kuroshio is ~100 km wide, and the mean velocity reaches up to 1 m s−1 with the current strongest in the upper 100 m, which is consistent with the climatological-mean cross sections presented in Rudnick et al. (2011). XBT/expendable conductivity–temperature–depth (XCTD) transects across the Luzon Strait conducted by Gilson and Roemmich (2002) identified two maxima in the northward velocity at 121° and 122.25°E, consistent with Fig. 2b. Nan et al. (2011) identify three typical flow paths of the Kuroshio in the Luzon Strait. The mean cross section in Fig. 2c is consistent with the “looping” path where with the Kuroshio flows into the SCS in the middle part of the Luzon Strait and out in the northern part with velocities of ~45 cm s−1. A cross section of the mean modeled flow along 137°E (from the outer grid simulation, not shown) is consistent with Qiu and Chen (2010, their Fig. 2) based on repeat hydrographic surveys, showing the location of the Subtropical Front, the NEC, and the STCC in the mean flow.
The spatial propagation patterns of internal tides and the decay of baroclinic energy may be influenced by the eddy fields in the SCS and the Philippine Sea. The model simulations are performed without data assimilation and are not expected to correctly represent the temporal and spatial evolution of the mesoscale eddy field. The sea surface height (SSH) variability associated with the mesoscale field over the two regions is compared from the model and from the satellite-derived SSH data, obtained from the Archiving, Validation, and Interpretation of Satellite Oceanographic data (AVISO; www.aviso.oceanobs.com) for 2010. We calculate the variance in SSH at each point in the domain for overlapping 30-day periods, calculated every 10 days, for the Full simulation (low-pass filtered to remove tidal variability) and the AVISO data. The square root of the spatial mean of these variances, taken over each of the SCS and Philippine Sea regions, are presented as time series in Fig. 3. The comparison shows that the model provides a comparable representation of the low-frequency variability in the SSH in both regions.
An overview of the general consistency of the M2 tidal model with sea level observations was presented in KPC2013. The surface expression of the M2 internal tides reaches up to 10 cm near the Luzon Strait and introduces small-scale variability in the barotropic tide SSH amplitudes and phases. Figure 4a shows the mean of the M2 SSH amplitude calculated every 3 days for the year-long Full simulation. The contribution of the internal component to the observed surface tide expression is modulated by the subtidal dynamics, resulting in weak modulation of the total (barotropic plus baroclinic) tidal surface expression. To compare our modeled variability in internal tides with observations, we make use of sea surface elevation data available at a Deep-Ocean Assessment and Reporting of Tsunamis (DART) buoy at 20.949°N, 132.314°E, maintained by the National Oceanic and Atmospheric Administration (NOAA) National Data Buoy Center. We compare the tidal SSH amplitude modulation, calculated over the last five M2 tidal cycles (62 h) every 3 days for the Full simulation. We must consider combined M2 and S2 SSH amplitudes as 62-h averaging does not allow separation of these two tidal constituents from the observations. To do this we conducted a comparison simulation that was identical to the Full case but including all major tidal constituents and calculated the combined M2 and S2 SSH amplitude. Figures 4b and 4c show the mean and standard deviations of the combined M2 and S2 SSH amplitude from the model, respectively. The location of the DART buoy is shown by the black diamonds in the two plots and the mean and standard deviation values from the observations are written next to the diamonds. The mean observed semidiurnal surface expression amplitude is 0.52 m, and the standard deviation is 0.16 m, both of which are within 1 cm of the modeled value at the DART location and compare well with the surrounding model values shown in the plots, showing that the model provides a good representation of the SSH variability at this location. The simulated internal tide energy fluxes from the Full simulation are compared to available observations in section 4a.
3. Internal tide generation
a. Generation variability
Energy is converted from the barotropic tide into baroclinic tidal energy at the Luzon Strait that radiates away as internal tides (this process is referred to as generation). Floor et al. (2011) show the baroclinic tidal energy generation C for a specific tidal frequency θ to be
where is the pressure perturbation at the bottom, and is the vertical component of the barotropic tidal flow. In this work, we assume incompressibility, such that . The pressure perturbation is given by
where is the time-mean quantity. The baroclinicity condition requires a depth-averaged pressure perturbation of zero, such that
Because time snapshots of the model output can lead to undersampling of the tidal amplitudes, we compute the energy fluxes and generation using tidal harmonic components. As such, Eq. (1) can be written in terms of the M2 amplitudes and Greenwich phases of p′(−H) and , as defined in Zilberman et al. (2011), as
where the subscript A refers to the amplitudes.
The generation of a particular tidal flow depends on the topographic slope and the stratification. While the topography is constant on the time scales that we are concerned with, the stratification varies on scales from weeks to months due to the mesoscale circulation, particularly the eddies impinging on the eastern ridges of the Luzon Strait and the intrusion of the Kuroshio into the SCS through the Luzon Strait (the Loop Current). We investigate the generation variability for the three cases (Full, Luzon only, and Constant remote flux) by computing the total generation at the Luzon Strait every 3 days for the year-long simulation. We consider a region around the strait [19°–22.3°N, 120°–122.5°E, shown by the blue dashed line in Fig. 1]) and calculate the area-integrated M2 baroclinic tidal energy generation every 3 days averaged over the last five tidal cycles (62 h) to produce a time series of 120 generation values over the 360-day simulation. The mean, standard deviation, and range of the generation values are given in Table 1. While the mean annual generation is similar over the three cases (and KPC2013), variability is greatest for the Full case and lowest for the Luzon-only case. The Luzon-only case shows the variability in generation at the Luzon Strait due to varying stratification over the generation site without the influence of remotely generated internal tides from the Mariana Arc. The Constant remote flux case results in increased variability in internal tide generation at the Luzon Strait, indicating that the constant internal tide flux alters the generation due to variability in its propagation to the Luzon Strait. The Full case exhibits the largest generation site variability, through the combination of changing stratification over the Luzon Strait generation site, variability in the propagation of fluxes from the Mariana Arc, and varying generation at the Mariana Arc.
The variation in generation for the Full case over the year 2010 is shown in Fig. 5a. The generation ranges from a minimum of 13.01 GW to a maximum of 19.95 GW, with the time-mean generation of 16.21 GW shown by the black dashed line. Using the same model configuration with mean, horizontally uniform stratification, KPC2013 found a total generation for the same region of 16.97 GW (shown by the gray dashed line). It should be noted that there was no time variability in KPC2013, and the value is taken from the final six tidal cycles. All comparisons to KPC2013 in this paper refer to the simulation encompassing both the Luzon Strait and Mariana Arc generation sites. Figure 5b shows the breakdown of generation over the eastern and western ridges at the Luzon Strait for the Full case, with the greatest variability in generation occurring over the eastern ridge where most generation occurs. The mean generation over the eastern ridge is similar to the generation from KPC2013, while it is less over the western ridge. The assumption in KPC2013 of mean, horizontally uniform stratification is not realistic over the Luzon Strait where the stratification is significantly affected by the location of the Kuroshio with its associated thermocline tilted downward toward the east. Jan et al. (2008) investigated the internal tide generation at the Luzon Strait for horizontally uniform typical summer and winter stratifications and found no significant seasonal variations in the internal tide energetics between simulations, although the baroclinic tide propagated about 10% faster in summer than in winter. As such, it appears that the thermocline tilt may be a more important influence on internal tide generation, as well as the influence of remotely generated waves that vary due to the changes in propagation speed. Jan et al. (2012) found that the presence of an idealized Kuroshio in their numerical model caused a significant change in the eastward and westward baroclinic energy fluxes emanating from the Luzon Strait, as compared to the case without background circulation. Buijsman et al. (2012) present a modeling study of internal tides considering the mesoscale circulation in the Southern California Bight and find that the spring–neap variability is modulated by the stratification on the shelf.
The spatial plots of time-mean generation in Figs. 5c and 5e show the main areas where generation occurs in the Full and Luzon-only cases, respectively. The spatial distribution of time-mean generation is similar for the two cases. The majority of generation on the western ridge occurs on its eastern slope due to the steeper topographic slope, while generation is fairly evenly divided between the slopes of the eastern ridge. The standard deviations of generation for the Full case (Fig. 5d) are greater than the Luzon-only case (Fig. 5f) over both ridges. In the Full case, the eastern slope of the eastern ridge experiences the greatest variability and has a significantly higher ratio of standard deviation to mean generation (0.4–0.7) than the other ridge slopes (ratios of 0.2–0.3). The standard deviation of the area-integrated generation energy over the eastern ridge is 1.16 GW, compared to 0.68 GW over the western ridge (Fig. 5b). In comparison, for the Luzon-only case, the standard deviation of area-integrated generation is similar over both ridges, with 0.68 GW on the eastern ridge and 0.63 GW on the western ridge, indicating that the remotely generated internal tides from the Mariana Arc have a significant influence on generation variability over the eastern ridge. The remote internal tides are likely to significantly dissipate before reaching the western ridge and may also reflect off the supercritical eastern ridge.
Changes in internal tide generation can result from changes in both the pressure perturbation at the ocean bottom and the phase difference between this pressure perturbation and the vertical barotropic velocity at the bottom [Eq. (1)]. Local stratification changes affect the bottom pressure perturbation as the barotropic flow is forced over topography. Remotely generated internal tides, which may come from nearby sources (e.g., the adjacent ridge) or far-field sources (e.g., the Mariana Arc), can influence both the pressure perturbation and the phase difference (KPC2013). The time-mean and standard deviations of the components that influence generation variability are shown in Fig. 6 for the Full and Luzon-only cases. The Full case has an increase in variability of the pressure perturbation (Fig. 6c) as compared to the Luzon-only case (Fig. 6d) over both east and west ridge generation sites with the greatest difference occurring over the eastern slope of the east ridge. The variability in the phase difference is also greater in the Full case (Fig. 6g) as compared to the Luzon-only case (Fig. 6h), particularly on the central and northern portions of the eastern slope of the eastern ridge where the majority of the generation on this slope occurs (refer to Fig. 5c).
The influence of the remotely generated waves on internal tide generation depends on the amount of energy that reaches the generation site and the phase of the remotely generated waves upon arrival, affecting the pressure perturbation amplitude and phase, respectively. The energy from remotely generated internal tides may vary as a result of variations in generation, propagation, and dissipation due to the interactions with the background circulation, and their phase upon arrival may be altered by varying propagation speeds. The variability of the remotely generated internal tides from the Mariana Arc has a significant effect on the generation variability at the Luzon Strait in this modeling study.
b. Generation sensitivity analysis
To understand the dominant mechanism of this generation variability, we investigate the sensitivity to changes in both the pressure perturbation and the phase difference between the pressure perturbation and the vertical barotropic velocity at the bottom. From Eq. (4), we can derive the sensitivity of generation with respect to the pressure perturbation amplitude by
The sensitivity to the phase difference term is given by
Both Eqs. (5) and (6) are calculated using the model amplitudes and phases obtained through harmonic analysis, as was done with Eq. (4). Sensitivity to the vertical component of the barotropic tidal flow is not discussed as the magnitude and phase of are invariant.
To quantify how the generation is likely to change due to the typical ocean variability, we normalize the sensitivities by the seasonal standard deviation (σ) of each term; hence, the typical sensitivity to and Δθ is given by and , respectively. Figure 7a shows the area-integrated normalized sensitivities for the Luzon Strait averaged over five tidal cycles, calculated every 3 days for 2010, to the pressure perturbation amplitude and the phase difference. The area-integrated, normalized sensitivity values have units of gigawatts (GW) and signify that if or Δθ were to be changed everywhere in the defined area over the Luzon Strait by one standard deviation, the total area-integrated generation would be altered by the shown quantity.
Generation is dependent both on the pressure perturbation and the phase difference, with the phase difference being the dominant influence for most times. The area-integrated sensitivities over the eastern and western ridge are shown in Fig. 7b. Both the raw (not shown) and normalized sensitivities are greater over the eastern ridge as compared to the western ridge. The time mean of the area-integrated normalized sensitivities (shown in Fig. 7b) over the eastern ridge exceed those over the western ridge by a factor of 1.8 for both and Δθ, and the phase difference is the dominant influence over both ridges at most times. The spatial distributions of the time-mean normalized generation sensitivities to the pressure perturbation and the phase difference are shown in Figs. 7c and 7d, respectively. At the eastern ridge, sensitivity to the phase difference is greatest on the northern portion of the ridge, while sensitivity to the pressure perturbation is greatest in the central region of the eastern slope. At the western ridge, sensitivity to the pressure perturbation is high along the entire eastern slope, while sensitivity to the phase difference is concentrated over the steepest section of the slope. The raw sensitivities have a similar spatial distribution, although the normalized sensitivity to the phase difference is further enhanced over the northern portion of the eastern slope due to the phase difference being highly variable (Fig. 6g). Zilberman et al. (2011) found that at Kaena Ridge, Hawaii, where generation varied by a factor of 2 over a 6-month period, the variability was strongly dominated by changes in the phase of the pressure perturbation, while the pressure perturbation amplitude played a negligible role. In a modeling study, Powell et al. (2012) found that both the pressure perturbation amplitude and the phase difference were equally capable of altering the local generation.
Changes in internal tide generation affect the amount of baroclinic energy available for propagation away from the generation site and for local dissipation. The influence of the subtidal circulation on internal tide propagation is addressed in the following section.
4. Propagation variability
The depth-integrated baroclinic flux is the time-averaged product of the pressure and velocity perturbations over a number of tidal cycles, given by
and represents the radiation of internal tides away from the generation site. The pressure perturbation is given by Eq. (2) and the velocity perturbation is given by
where is the time-mean quantity, and by the baroclinicity condition
Estimating the propagation of baroclinic tidal energy is important to understanding where internal tide energy goes, how it is influenced by the subtidal, and ultimately the locations where the transfer of energy to smaller scales and dissipation occurs.
The variability of the horizontal propagation of the depth-integrated baroclinic energy fluxes in the presence of the subtidal circulation is investigated for fluxes computed every 3 days over the year-long model simulations. As in the generation calculations discussed above, the fluxes are averaged over the last five tidal cycles (62 h) of each 3-day window. The time-mean of the baroclinic energy fluxes for the Full case are shown in Fig. 8a, with the corresponding standard deviation ellipses plotted in Fig. 8b (the flux vectors and ellipses are plotted every three model grid points with only flux values greater than 10 kW m−1 shown). Ellipse centers are plotted at the head of each mean flux arrow, and their major and minor axes represent standard deviations along and across the mean flux vector, respectively. The baroclinic energy fluxes vary in time both in magnitude and direction due to a combination of variability in generation, propagation, and dissipation. The spatial pattern of the mean baroclinic energy fluxes is similar for the Luzon-only case, but flux variability is much less compared to the Full case, particularly in the dominant northwestward- and southeastward-propagating beams. The energy flux analysis presented here focuses on the Full case so as to include the influence of remotely generated internal tides from the Mariana Arc.
The spatial pattern of the mean baroclinic energy fluxes is different from the spatial pattern with mean stratification (see KPC2013, their Fig. 4). The most striking difference is the propagation of fluxes westward into the SCS. In KPC2013, the fluxes are directed due west into the SCS with a small amount of refraction to the north toward the shallow shelf. This contrasts with the mean fluxes with subtidal dynamics that are strongly deflected to the north, resulting in much less internal tide energy traveling westward. This difference is attributed to the spatially varying time-mean stratification and background circulation, in particular the Loop Current. Figure 8c shows the time-mean buoyancy frequency at 250-m depth where a tongue of more highly stratified water associated with the Loop Current penetrates over the Luzon ridge. Chao et al. (2007) show a similar flux pattern for the semidiurnal internal tides from a model that includes the Kuroshio, with two distinct beams propagating into the SCS: one directed westward at 20°N and a second beam farther to the north that is deflected northward to 22°N where it impinges on the shelf.
The northward refraction of the internal tides is consistent with the dispersion relation derived for internal waves by Rainville and Pinkel (2006) in which the phase speed is increased in regions of increased stratification, and internal waves tend to be refracted away from areas of high-phase speed. The Loop Current velocities would also cause refraction of westward internal tides to the north (see Fig. 8d). This was explored by Park and Watts (2006) in the Japan/East Sea, where the northward internal tide beams were observed to be refracted eastward (westward) when a warm (cold) eddy intersected their path. Rainville and Pinkel (2006) also estimated the refraction of an internal tide beam encountering an anticyclonic eddy to the south of Hawaii using the Wentzel–Kramers–Brillouin (WKB) approach and showed that the effect of the mesoscale currents increases with mode number, while stratification affects the refraction of all modes equally, and they showed the beam was refracted toward the anticyclonic eddy.
Zhao et al. (2004) observed internal wave packets in satellite images in the SCS from 1995 to 2001, which they suggest are developed through nonlinear steepening of baroclinic tides generated at the Luzon Strait. The internal wave packets observed in the deep water basin are propagating northwestward (see their Fig. 1), some traveling toward the northern continental shelf similar to the mean modeled fluxes in Fig. 8a, while some are not deflected as far to the north and impinge upon the Dongsha Plateau (between approximately 20.5° and 22°N). Our mean modeled fluxes show two internal tide beams, one directed due west toward the Dongsha Plateau and the other, more dominant beam, directed to the northwest. Most of the internal wave packets are observed on the Dongsha Plateau only once they encounter the shallow shelf and steepen. The internal tides deflected more to the north may simply not be observed in the satellite images if they do not steepen enough. The large number of internal wave packets observed on the Dongsha Plateau from 1995 to 2001 may suggest that the modeled mean westward baroclinic energy flux in the SCS is deflected too much to the north. A comparison of the mean SSH anomaly from the model with satellite-derived SSH data from AVISO over the time period of the satellite observations shows that the SSH anomaly associated with the Loop Current is indeed directed farther to the north in the model. The Loop Current location is highly variable on seasonal and interannual time scales (Yuan et al. 2006; Jia et al. 2010; Nan et al. 2011; Hsin et al. 2012), and it should be noted that the mean propagation of baroclinic energy fluxes in the SCS is related to the mean location of the Loop Current in our model for 2010, which may not be the long-term mean. Specific cases, some of which are discussed below and presented in Fig. 10 (described in greater detail below), show that a strong westward baroclinic energy beam exists intermittently, depending on the mesoscale field at the Luzon Strait. The variability in SSH associated with the mesoscale circulation was shown to be consistent with AVISO observations for 2010 (Fig. 3), providing confidence in our estimates of flux variability.
Observations of baroclinic energy fluxes at the Luzon Strait were presented in Alford et al. (2011). The energy fluxes were calculated from data collected using lowered acoustic Doppler current profiler (LADCP) stations, each lasting 36 h and sampling the entire water column (from 10 m below the surface to approximately 10 m above the bottom). The semidiurnal (combined M2 and S2) energy fluxes were presented, as the 36-h records do not allow for the separation of these two tidal constituents, and the fluxes were corrected for their sample time within the spring–neap cycle. The barotropic S2 is one-third weaker than the M2 in the region; however, to make a valid comparison with these observations, we use results from a simulation that is identical to the Full case, but included all major tidal constituents (also used for the comparison of tidal SSH expression with observations presented in Fig. 4). We calculate the combined M2 and S2 fluxes every 3 days, averaged over 36 h to be consistent with Alford et al. (2011).
Figure 9 shows the time-mean modeled fluxes plotted for every model grid point, the mean fluxes and standard deviation ellipses interpolated onto the observation locations, and the observed fluxes from Alford et al. (2011) for comparison. In Figs. 9a and 9c, the model shows significant variability in both flux magnitude and direction, with the mean flux being close to zero at some of the comparison locations, and the modeled standard deviation ellipses mostly contain the fluxes briefly observed by Alford et al. (2011). In Fig. 9b, most of the model flux variability is in flux amplitude, with less directional variability, as indicated by the elongated shape of the standard deviation ellipses in the direction of the mean. The directions of the Alford et al. (2011) fluxes in this area are consistent with the modeled values, but the observed magnitudes are significantly smaller than the model-mean values. For the most eastern station in Fig. 9b, the observed flux magnitude of 3.9 kW m−1 is smaller than any of the 120 model-calculated fluxes (the minimum of which is 8.7 kW m−1). For the other four stations in Fig. 9b, between 2 and 7 of the 120 model-calculated fluxes are below the observed magnitude at each location. Although the Alford et al. (2011) fluxes in Fig. 9b are mostly within the range of the modeled fluxes, the greater magnitude of the mean modeled fluxes may be a result of model resolution on the bathymetry with the bathymetric smoothing. The modeled ridge extending south from Taiwan (the western ridge) does not reach as far south and is wider than the actual ridge. Figure 9b is actually on top of the ridge rather than at the southern base as it is in the model where the strong fluxes from the eastern ridge are refracted northward (Fig. 8a). Modeled flux magnitudes over the ridge to the north of Fig. 9b are of similar magnitude to the observed fluxes. Understanding the variability in baroclinic tidal energy fluxes is important when interpreting observations taken at a particular time, as seen in this comparison where the modeled fluxes vary significantly. Furthermore, harmonic analysis of time series from long-term measurements (e.g., tide gauges and satellite altimeter data) used to compute fluxes captures only the coherent portion; however, the incoherent, time-varying component may be significant.
To investigate the time-varying effects of mesoscale eddies and the Loop Current in the Luzon Strait on the baroclinic energy fluxes, we present three specific periods from the Full model that were averaged over five tidal cycles around the model times of 28 September, 27 October, and 17 November. These times are shown by the magenta vertical lines in Fig. 5a. Over 3 months, the generation varied significantly and these times capture the minimum and maximums of the oscillation as well as changes in the horizontal propagation patterns due to an eddy separation from the Loop Current. The spatial plots of the three cases are shown in Fig. 10. The first column of Fig. 10 shows the generation. As seen in Fig. 5, area-integrated generation at the Luzon Strait is at a peak for the first and last times and at a low for the middle time. The second column of Fig. 10 shows the depth-integrated baroclinic energy fluxes, and the third column shows the detided SSH anomaly and surface current velocity vectors. Changes in generation energy levels and the Loop Current location are seen to affect the magnitude and direction of the internal tide beams in the model over the 3 months. At the end of September, there is a well-developed Loop Current intruding through the Luzon Strait with anticyclonic circulation to the north against the SCS continental shelf. The baroclinic energy fluxes directed to the north are strong and impinge on the shelf off of southwest Taiwan. A much less energetic westward beam persists to the south. Toward the end of October, the anticyclonic eddy separates from the Loop Current in the SCS and the Loop Current intrusion is greatly reduced. Both westward and northwestward internal tide beams are present containing similar energy. The beams exist on either side of a sea surface low directly to the west of the strait. The northern beam is refracted northward upon interaction with the warm core eddy in the SCS. Over the next month, the Loop Current reforms, and by 17 November, the westward flux propagation is similar to the September case again. The temporal variability of internal tides reaching the continental slope in the northeast SCS was observed by Lee et al. (2012), where low-frequency variability in diurnal and semidiurnal amplitudes was not phase locked with the astronomical forcing, and the incoherent tidal energy accounted for about 75% of the total. Park and Farmer (2013) found that the refraction of westward-propagating internal tides by mesoscale features can explain deviations in observed internal tide arrival times at their measurement location in the western SCS.
Eddies to the east of the Luzon Strait also influence the propagation of the internal tides into the Philippine Sea in the model. For the 28 September case, a strong cold core eddy exists to the east of the Luzon Strait. The energy in the eastward flux beams is fairly even between the northern and southern beams, and both beams are refracted away from the cyclonic eddy. A month later, the eddy weakens and moves closer to the strait. Stratification changes result in reduced generation over both ridges. Again, the northern and southern beams are refracted either side of the cold core eddy. After passing the eddy, the southern beam is then refracted to the north where the velocities associated with the eddy are northward. By 17 November, the cyclonic eddy has almost disappeared and an anticyclonic eddy is present directly east of the Luzon Strait. Generation is greater on the southern end of the eastern slope of the eastern ridge with notably less generation occurring on the northern end, as compared to the September case, and the southern beam of baroclinic tidal energy is significantly stronger than the northern beam.
The mesoscale eddy field has significant effects on the generation and propagation of internal tides generated at the Luzon Strait in the model. Variability in the far field is investigated in Kerry et al. (2013b, manuscript submitted to J. Phys. Oceanogr.) by examining the variability of the meridionally integrated baroclinic energy fluxes across the Philippine Sea.
This study uses a modeling approach to examine the effects of the subtidal flow on the generation and propagation of M2 baroclinic tides in the Philippine Sea. The internal tides vary significantly in time and space, and the modeled SSH variability compares well with observations from a DART buoy within the model domain. Model estimates of internal tide energy fluxes at the Luzon Strait are also consistent with observations by Alford et al. (2011). Mean internal tide energetics differ considerably from the KPC2013 study that assumed a climatological-mean horizontally uniform stratification and omitted subtidal dynamics.
Internal tide generation at the Luzon Strait is affected by local changes in stratification over the generation site and the varying influence of remotely generated internal tides from the Mariana Arc. Generation variability is described by the standard deviation in barotropic to baroclinic conversion at the Luzon Strait calculated every 3 days over year-long simulations. Variability in generation energy in a simulation that did not include baroclinic tides generated at the Mariana Arc (Luzon-only case) is attributed to local changes in the subtidal circulation over the Luzon Strait. A simulation in which a constant baroclinic tidal energy flux was applied to represent internal tides originating from the Mariana Arc (Constant remote case) resulted in a variability that is 1.3 times greater. This result indicates that the constant flux applied has a changing influence due to variability in its propagation across the Philippine Sea. When the internal tide generation at the Mariana Arc was also varying due to the effects of the subtidal circulation (Full case), the generation variability at the Luzon Strait was 1.8 times higher than the case without remotely generated internal tides. The changes in generation energy levels for the Full case result from the combination of changing stratification over the Luzon Strait generation site, variability in the propagation of fluxes from the Mariana Arc, and varying generation at the Mariana Arc.
Changes in the internal tide generation energy result from changes in both the pressure perturbation at the ocean bottom and the phase difference between this pressure perturbation and the vertical barotropic velocity at the bottom. The Full case shows an increase in variability of both of these terms as compared to the Luzon-only case over both east and west ridge generation sites, with the greatest differences occurring over the eastern slope of the eastern ridge. Generation is found to be most sensitive to changes in the phase of the pressure perturbation at the bottom, with the time-mean normalized sensitivities to phase changes at the Luzon Strait being 1.23 times greater than the mean sensitivities to the pressure perturbation amplitude. Sensitivities to both terms are about 1.8 greater over the eastern ridge as compared to the western ridge.
The horizontal propagation of the depth-integrated baroclinic energy fluxes is significantly affected by the subtidal circulation. The modeled intrusion of the Kuroshio Loop Current into the Luzon Strait causes the mean internal tides to be steered more to the north, toward the SCS continental shelf. The mean modeled Loop Current location differs from the AVISO observations for 2010, but the mesoscale variability is consistent between the model and observations. The baroclinic energy fluxes vary in time both in magnitude and direction, due to a combination of variability in generation, propagation, and dissipation. Spatial patterns of internal tide propagation near the Luzon Strait are influenced by the varying spatial distribution of generation energy levels and the locations of mesoscale eddies to the east and west of the strait.
This study shows that both the temporally and spatially varying general ocean circulation and the background internal tide field influence model estimates of baroclinic tides in the Philippine Sea. The results provide insight into the mechanisms of variability of the baroclinic tides and highlight the importance of considering both the remotely generated internal tides and the subtidal dynamics to estimate internal tide energetics. Understanding the spatial and temporal variability of the internal tides is important when interpreting observations of internal tide energetics and ocean mixing and developing parameterizations for climate-scale models. This study represents an important step toward understanding the variability in internal tide energetics in the global oceans, which affects the amount of energy made available for mixing and the locations at which this mixing occurs. Kerry et al. (2013b, manuscript submitted to J. Phys. Oceanogr.) addresses the impact of the subtidal circulation on dissipation of the internal tides in the Philippine Sea and the implications for oceanic mixing.
The authors thank Mercator Océan of France for the model data used to provide the boundary conditions for the ROMS solutions presented. The authors also thank Dr. Ivica Janeković of the Rudjer Bošković Institute for assistance with creating the model grid. Ms. Kerry and Dr. Powell were supported by ONR Grant N00014-09-1-0939. Dr. Carter was supported by ONR Grant N00014-10-1-0334.