The vertical isopycnal displacements associated with internal waves generated by the barotropic tidal currents in the vicinity of Luzon Strait are estimated using measurements collected by autonomous underwater gliders. Nearly 23,000 profiles from Seagliders and Spray gliders, collected during 29 different missions since 2007, are used to estimate the amplitude and phase of the linear semidiurnal and diurnal internal waves in this energetic region, particularly in the previously poorly sampled area near the eastern ridge and on the Pacific side of Luzon Strait. The mean and variability of the internal wave field in the upper 1000 m of the water column are described. The phase progression of internal waves as they propagate away from their generation sites is captured directly. The glider-based observations are used to map the mode-1 semidiurnal and diurnal internal wave fields, providing the baroclinic energy flux over a roughly 600 km × 800 km region based strictly on in situ observations.
 Luzon Strait, located between the Philippines and Taiwan, is the primary passage between the western Pacific and the South China Sea (Figure 1). Two meridional ridges extend across the strait and are known to be the generation location of energetic internal waves, created by strong barotropic tidal currents flowing across abrupt topography [Chao et al., 2007; Jan et al., 2008; Alford et al., 2011]. In particular, large nonlinear internal waves have been detected and tracked from Luzon Strait, through the South China Sea, and onto the continental shelf near the Dongshua Plateau [Ramp et al., 2004; Lien et al., 2005; St. Laurent, 2008; Alford et al., 2010; Farmer et al., 2011].
 The majority of the observations of linear and nonlinear internal tides near Luzon Strait have been located on the South China Sea (western) side of Luzon Strait [Ramp and Tang, 2011]. The extent to which tidal internal waves propagate toward the Pacific remains an open question [Jackson, 2009; Buijsman et al., 2010], but numerical models suggest that large internal tides are present on both sides of Luzon Strait, carrying a significant energy flux [Niwa and Hibiya, 2004]. The observations presented here are part of an effort to determine the spatial structure of internal waves in this region. A companion paper on high-frequency internal waves [Rudnick et al., 2013], showed an asymmetry on either side of the Luzon Strait, with energy levels decaying from the strait eastward into the Pacific. This paper completes this description by focusing on tidal internal waves.
 More generally, tidal currents flowing over topography lead to barotropic to baroclinic conversion. A fraction of this energy dissipates locally [e.g., Klymak et al., 2006]. As generation becomes increasingly nonlinear (stronger forcing, more obstructive topography, etc.) or through wave-wave interactions, energy may escape as nonlinear internal waves (as in the South China Sea). However, a large fraction likely escapes as linear internal tides—waves at the tidal frequencies, more or less sinusoidal over several periods, that propagate as free waves in a stratified rotating fluid (either as modes or vertically propagating waves). This paper focuses on the linear internal tide.
 Satellite altimetry [Ray and Mitchum, 1996, 1997; Zhao and Alford, 2009; Dushaw et al., 2011], numerical models [Niwa and Hibiya, 2001; Merrifield and Holloway, 2002; Niwa and Hibiya, 2004], and, to a lesser extent, moored observations [e.g., Alford and Zhao, 2007] have been used to identify signatures of internal tides propagating great distances from their generation regions.
 The strong tidal currents and three-dimensional topography of Luzon Strait are expected to generate a complex internal wave field, where several waves superimpose upon each other to create an interference pattern [Rainville et al., 2010]. In addition, the Kuroshio Current and its associated mesoscale field [Farris and Wimbush, 1996; Liang et al., 2003; Caruso et al., 2006] can modulate internal wave formation by altering the background stratification, and affect wave propagation through refraction or trapping. The fast time scales (hours), large spatial scales (wavelengths hundreds of kilometers), and variability (spring-neap cycles of the forcing and interactions with the background stratification and ocean currents) of internal tides present an observational challenge.
 This paper characterizes the complex internal tides near Luzon Strait by exploiting the persistence and extensive spatial coverage provided by an array of autonomous ocean gliders. Section 'Measurements and Methods' presents the instruments and methods used to estimate the amplitude and phase of the internal tides. Linear fits to time series of isopycnal depths provide estimates of semidiurnal and diurnal internal tide amplitude and phase. Estimates are obtained as a function of location and time by fitting over moving windows, with independent estimates obtained for each isopycnal. The spatial and temporal distributions of internal tide displacements at approximately 500 m depth are discussed in section 'Spatial and Temporal Structure of the Internal Tides'. The lowest dynamical (normal) modes dominate the vertical structure and can be used to extrapolate glider observations confined to the upper 1000 m to the full water column. With propagation direction inferred from spatial variations of the phase, estimates of depth-integrated energy flux vectors are obtained (section 'Mode-1 Displacement and Energy Flux'). A summary follows.
2. Measurements and Methods
2.1. Glider Measurements in and Around Luzon Strait
 Gliders are 50 kg, 2 m long autonomous underwater vehicles designed to glide from the ocean surface to as deep as 1000 m in a sawtooth pattern while collecting profiles of water properties, including temperature, salinity, pressure, dissolved oxygen concentration, and optical properties [Eriksen et al., 2001; Sherman et al., 2001; Rudnick et al., 2004]. They steer through the water by controlling attitude (pitch and roll) while moving at about 0.25 m s−1, or roughly 20 km d−1. Gliders can navigate between waypoints in a variety of oceanographic conditions and, with skillful piloting, even in environments with currents that exceed their own forward speed.
 In 2007 and 2008, a collaborative program of glider-based observations was launched by groups at the Scripps Institution of Oceanography and the Applied Physics Laboratory, funded by the Office of Naval Research (ONR), to study the Kuroshio Current and its associated mesoscale variability [Rudnick et al., 2011]. During the 15 month program, 13 Seagliders and Spray gliders conducted missions in the Kuroshio and Luzon Strait. Gliders occupied a series of sections across the Kuroshio, zigzagging downstream from Luzon Island (Philippines) to Taiwan, demonstrating that gliders can navigate and occupy sampling patterns through the strong Kuroshio Current.
 Data from 16 Seaglider missions, obtained as part of the ongoing Origins of the Kuroshio-Mindanao Currents program (OKMC, funded by ONR), are also included in this study. For this program, three Seagliders have been concurrently sampling two regions in the Pacific sector immediately east of Taiwan and of Luzon Island (northeast and southeast of Luzon Strait, respectively), and a section on the western side of Luzon Strait. The combined Kuroshio and OKMC gliders collectively provide the 22,914 profiles used here (13,533 dives), covering a total horizontal distance of nearly 51,400 km and spanning 2806 glider days (Figure 1).
 The temperature depth-time map for one mission (Seaglider 121, July to October 2007; Figure 2) illustrates spatial and temporal variability typical of the region around Luzon Strait. The glider samples primarily the Pacific side of Luzon Strait, approaching the eastern ridge on several occasions. The time series of temperature, here from 402 dives, shows motions on many scales, including heaving of isopycnals due to the large-scale Kuroshio (shallower isopycnal on the western side) and mesoscale eddies. At shorter time scales, the glider also captures the relatively rapid vertical displacements associated internal waves, primarily associated with the diurnal and semidiurnal tidal frequencies.
2.2. Barotropic Tide
 Both the semidiurnal (principal lunar, M2, and principal solar, S2) and diurnal (lunisolar diurnal, K1, and principal lunar diurnal, O1) barotropic tidal components are important in Luzon Strait. Barotropic zonal currents at 20.5°N, 121.5°E, near the middle of Luzon Strait, illustrate the relative strengths of the tidal constituents (Figure 3). Tidal predictions are obtained from the TPXO7.2 regional tidal model in the seas around China [Egbert and Erofeeva, 2002; Zu et al., 2008] between the two ridges. In each band, the minor barotropic tidal components have a significant magnitude relative to M2 and K1 (for example, the magnitude of the S2 and O1 components are 35% and 79% that of M2 and K1, respectively) and therefore both the amplitude and phase of the combined semidiurnal and diurnal barotropic tides vary in time. Amplitudes and phases are obtained using a complex demodulation with a four-point phase-preserving Butterworth filter with a bandwidth of 1/3 days−1, identical to the width of the time window used in the baroclinic displacement fits below (section 'Observations of Internal Waves from Glider Measurements').
 For each frequency component, phases are expressed relative to Greenwich mean time (GMT), the convention used for barotropic tides [e.g., Egbert and Erofeeva, 2002]. The frequencies and time convention used for the calculation of phase are the same as in the TPXO tidal model, such that phases calculated from the least square fit to a year-long time series of M2 (or K1) barotropic tide generated by the tidal model exactly match phases of the M2 and K1 tidal constituents listed in the model. Baroclinic phases follow the same convention.
 Just as there is a fortnightly cycle in the amplitudes of the semidiurnal and diurnal barotropic tides (Figure 3b), phases of the semidiurnal and diurnal barotropic tides also exhibit temporal variability (Figure 3c). The phase of the semidiurnal barotropic tide varies by as much as 80° (∼2.5 h), with a standard deviation of 19° (0.6 h) over the fortnightly cycle. For the diurnal tide, the K1 and O1 constituents have comparable magnitudes and, therefore, larger interference: the phases vary by as much as 140° (∼9 h), with a standard deviation of 33° (2.2 h). These temporal variations are much larger than the spatial variations of phase across the region. For example, from 119° to 124°E (over 500 km in the general propagation direction of the M2 and K1 barotropic tides), phases of the barotropic tidal currents for both constituents vary by less than 10°. As a consequence of the wandering phase of the barotropic tides, even if internal tides were phase locked with their forcing, their timing at a given location would vary by several hours during the fortnightly cycle.
2.3. Observations of Internal Waves from Glider Measurements
 Gliders move slowly, typically completing dives to 1000 m every 5–7 h for the missions used in this analysis. This sampling adequately resolves displacements at the diurnal frequency. This paper aims to demonstrate that this sampling also captures processes occurring at semidiurnal frequency, although a careful evaluation of the errors associated with these glider-based measurements is necessary.
 To estimate the baroclinic displacements associated with internal tides, the times and depths at which the glider crosses selected isopycnals are identified for each profile (Figure 4). The vertical displacement of an isopycnal is calculated relative to its 3 day running-mean depth, which filters spatial and temporal variations associated with the mesoscale and large-scale circulation. This calculation is repeated for isopycnals separated on average by 10 m.
 Assuming that the internal waves responsible for the signal do not vary significantly over a period of 3 days (corresponding to a distance of roughly 60 km), fits of sinusoidal waves to the measured displacements, η(z), provide estimates of internal tide amplitude and phase (Figure 4):
where A and Φ are the amplitude and phase of the sine wave at frequency ω, for both the semidiurnal (SD) and diurnal (D) constituents, chosen to the M2 (1/12.4206 h−1) and K1 (1/23.9345 h−1) frequencies. The phase speed of both the semidiurnal and diurnal internal tides is over an order of magnitude faster than the horizontal speed of the glider, so the waves effectively propagate past the glider. In round numbers, a 3 day window admits periods between 18 and 36 h for the diurnal band and 10.3–14.4 h for the semidiurnal fit. Smaller horizontal wavelengths, nonsinusoidal, or higher-frequency waves can be estimated through vertical velocity fluctuations in the glider's path [Rudnick et al., 2013]. The method presented here exploits the persistence and broad spatial coverage provided by the gliders to produce a novel characterization of internal waves propagating away from Luzon Strait.
 Several conditions, listed in Appendix A, are imposed to ensure that both phase and amplitude are captured accurately. Ultimately, amplitude uncertainties are 3 m or less. Overall, 88% of the depth-space region covered by the gliders provides good estimates in both diurnal and semidiurnal bands. Averaged across all the missions near Luzon Strait and all tracked isopycnals, the sum of semidiurnal and diurnal waves explains 47% of the total estimated displacement variance for periods shorter than 3 days.
 Time variability in forcing amplitude is reflected in the amplitudes of the internal tide in each band (Figure 5). Generally speaking, amplitude grows as a function of depth—recall that gliders sample only the top 1000 m of the water column—and is greater when the barotropic forcing in Luzon Strait is large. The pattern also includes significant spatial variability. For example, no large semidiurnal isopycnal displacements are observed during the spring tide around 12 October 2007, presumably because the glider is far from the generation site at this time. These variations in space and time are explored in the next section.
3. Spatial and Temporal Structure of the Internal Tides
3.1. Vertical Displacements at 500 m
 The complexity of the internal wave field near Luzon Strait is even more evident when considering the ensemble of all 29 missions. For both frequency bands, isopycnal displacement amplitudes at 500 m (Figures 6a and 6b) are generally larger near the middle of the strait (∼121–122°E) and smaller further afield. Larger diurnal amplitudes are observed on the southwestern side of Luzon Strait, suggesting asymmetric generation and propagation. Maps show internal tide phase generally increasing from the middle of the strait (Figures 6e and 6f), indicating propagation both to the east and to the west, away from generation sites on the ridges. Phases are plotted relative to the phase of the barotropic M2 (177° GMT) and K1 (296° GMT) barotropic zonal currents in the middle of Luzon Strait (20.5°N, 121.5°E) for the semidiurnal and diurnal displacements, respectively.
 Zonal sections depicting the ensemble of 500 m semidiurnal and diurnal internal tide amplitude and phase estimates between 19.5°N and 21°N illustrate phase propagation away from the ridge (Figures 6c, 6d, 6g, and 6h). Amplitude plots (Figures 6c and 6d) are color-coded by forcing magnitude, while phase plots are coded by forcing phase (Figures 6g and 6h). Amplitudes are further binned to show the mean and standard deviations along this zonal section. Binned amplitudes are calculated using the middle 50% of the distribution of tidal current magnitude, excluding times where the forcing is either very large or very weak. Thin lines represent one standard deviation, smoothed by a 100 km running mean.
 Semidiurnal amplitudes (Figure 6c) generally decrease away from the middle of the ridge. Some variation as a function of range is expected due to the vertically propagating nature of “beams” generated at a steep ridge. Diurnal internal tides (Figure 6d) exhibit a similar behavior, although no decrease in amplitude is observed in the South China Sea. Note that the diurnal mode-1 wavelength (or, alternatively, the distance between the surface bounces of a vertically propagating diurnal free wave) is about 350 km (over 3° of longitude)—implying that all the observations presented here for diurnal waves lie within one wavelength of the ridges of Luzon Strait. Complex bathymetry and interactions between beams generated at a few specific sites likely leads to the observed complexity—all these measurements are very much in the “near field,” where one could expect more beam-like structures and nonuniform distribution in depth.
 While there is a suggestion that displacement amplitudes generally scale with the magnitude of barotropic forcing (Figures 6c and 6d) as it is expected by theory [Baines, 1982; Llewellyn Smith and Young, 2002; St. Laurent et al., 2003], the confused relationship, particularly for the semidiurnal band, points to the importance of interactions with the ocean stratification and background currents.
 Across Luzon Strait, the phases of semidiurnal and diurnal internal tide displacements (Figures 6g and 6h) show an increase from the middle of the ridge, at a rate close to that predicted by the theoretical mode-1 wavelength (150 km for M2, 340 km for K1, gray line). The phase obtained for each frequency component of the fit (equation (1)) provides information about spatial propagation:
where is the horizontal wave number vector of the wave of magnitude kH. ϕ0 is a constant, with neither space nor time dependence for a single propagating wave generated by a steady forcing. As the waves from Luzon Strait travel more or less zonally, their phases increase away from the ridges at a rate that is dominantly prescribed by the mode-1 wavelengths . The phase progressions shown in Figures 6g and 6h are direct observations, from continuous measurements over the upper 1000 m of the water column that span a large spatial domain, of patterns observed in remotely sensed sea surface height that have been interpreted as internal tides [e.g., Ray and Mitchum, 1997; Zhao and Alford, 2009].
 There is significant scatter in the phase estimate, which is associated with both phase variability of the forcing (Figures 6g and 6h) and factors affecting the propagation speed of the waves through the ocean. Binned average of the phase estimates as a function of longitude shows propagation away from the middle of Luzon Strait, at a speed that matches the mode-1 phase speed. Direct observation of the overall phase progression by thousands of independent estimates obtained within the water column is unique to this work.
3.2. Vertical Structure
 All estimates falling between 19.5°N and 21°N are used to produce composite depth-longitude sections of diurnal and semidiurnal internal tide amplitude and phase (Figure 7) across Luzon Strait. Amplitudes and phases are averaged separately. Because of uncertainties in the amplitude relationship, no scaling relative to the magnitude of the barotropic tide is applied. To minimize potential biases in each band, estimates corresponding to times when the barotropic forcing is in the top or bottom 10% of its respective speed distribution are excluded from the composite. Phase estimates are adjusted to be relative to the phase of the barotropic tide at the reference point in the same time window (Figure 3) and are averaged using a complex representation on the unit circle. Depth and amplitude are scaled to account for the nonuniform stratification: the depth coordinate is stretched to emphasize the region of strong stratification near the surface, and the amplitudes are scaled following Wentzel-Kramers-Brillouin (WKB) normalization [e.g., Althaus et al., 2003], using a constant reference buoyancy frequency of N0 = 6.7 × 10−3, the average stratification of the upper 1000 m on the Pacific side. Meridional median of the bottom depth between 20° and 21°N is shown as the gray shading. In this stretched coordinate system, the bottom is effectively shallower than it is in reality. Note that although the gliders sample only the top 1000 m, they effectively sample more than half of the density range when one takes into account that the stratification is stronger in the upper ocean than in the deep.
 Beam-like features of the internal tide are starting to be visible (Figure 7). Phases show a little more structure near the ridges, indicative of the presence of higher modes and vertical propagation, but are generally more uniform in depth further from topography. Regions of elevated amplitude displacements are likely due to interactions between the ridges and superposition of beams, as suggested in several modeling studies [Chao et al., 2007; Jan et al., 2008; Simmons et al., 2011]. There appears to be a build up of semidiurnal energy between the ridges, likely due to the fact that the semidiurnal internal tide is close to resonance (the ridges are spaced by almost exactly one mode-1 wavelength [Echeverri et al., 2011]).
4. Mode-1 Displacement and Energy Flux
4.1. Mode-1 Displacements
 Even in the very energetic internal wave environment of Luzon Strait and in complex regions close to the generation sites, glider-based estimates of isopycnal displacements capture most of the displacement variance, with phases that remain relatively constant with depth (Figure 7). Given this, the fitting technique can be adapted to focus on the low modes (long vertical wavelengths). By definition, a mode-1 internal wave is a sinusoidal wave with long horizontal wavelength (150 km for M2, 340 km for K1) and depth dependence prescribed by the background stratification [e.g., Rainville and Pinkel, 2006]. Glider-based sampling is particularly appropriate for resolving these temporal and spatial scales. For a given frequency, the oceanic normal modes (vertical dependence) and eigenvalues (ce) are set by stratification and ocean depth. The magnitude of the wave number vector for each mode is determined by the dispersion relation, . Most of the internal tide energy and energy flux are generally found in the lowest mode, which makes it a particularly useful descriptor.
 Mode-1 solutions are obtained by a least square fit, with vertical structure prescribed by , derived from the stratification. The one amplitude and phase that best explain the displacements of all the isopycnals tracked within a 3 day window are found (Figure 8):
where Am1 and Φm1 are the amplitude and phase of the mode-1 fit, constant for each time window. Only one frequency is listed in equation (4), but generally both the semidiurnal and diurnal bands are used, as in equation (1). As before, the phase obtained from the fit is primarily a representation of the propagation of the waves in space. In the next section, this information is used to infer the direction of propagation. Over all the missions near Luzon Strait, the sum of semidiurnal and diurnal mode-1 waves explains on average 29% of the total estimated high-frequency displacement variance (for periods shorter than 3 days).
 Comparisons between estimated mode-1 internal tide amplitudes and phases and those associated with tidal forcing relate the observed variability to barotropic forcing over the ridges (Figure 9). Estimates are taken in narrow bands away from the ridges, spanning 25 km centered around 120°E and 123°E (∼150 km on either side of the reference site for the barotropic tidal currents: 20.5°N, 121.5°E) and between 20 and 21°N. Particularly for the diurnal component, there is a general increase in the internal wave amplitude as a function of the barotropic forcing. Similarly, the phases of the internal tide are also generally related to the phase of the forcing, although the relationship for the semidiurnal tide is less clear. The large scatter, particularly for the semidiurnal tide west of Luzon Strait, can presumably be attributed to part of the energy going into high-frequency nonlinear internal waves, interactions with the mesoscale eddies and the Kuroshio, or interference patterns. Internal tides in Luzon Strait are neither static nor phase locked. Further studies are required to relate this variability to specific processes. Glider-based observations such as those presented here can provide the persistent presence and extensive spatial coverage needed for such investigations.
4.2. Energy Flux
 Glider-based estimates of the spatial variability in mode-1 amplitude and phase can be used to produce maps of internal tide energy flux. Although gliders sample only the top 1000 m (about half of the “effective” water column, in a WKB-stretched sense, Figure 7), the shape of the dynamic modes is known for all depths, effectively allowing us to extrapolate observed displacements over the entire water column. Total baroclinic mode-1 energy and energy flux magnitude are related directly to the amplitude of the mode-1 displacements by the internal wave equation [Gill, 1982].
 At any given point, normal mode and “beam” (free wave) representations are mathematically equivalent only when considering the entire water column. Particularly near generation regions, a large fraction of the internal tide field might be below the profiling range of the gliders. Here we mitigate this effect by averaging the mode-1 amplitudes and phases in 100 × 100 km boxes for the semidiurnal tide and 200 × 200 km boxes for the diurnal tide, corresponding to roughly 2/3 of one wavelength. Only boxes with a mean depth larger than 3000 m are considered. The resulting mode-1 energy and energy fluxes are therefore valid only on those scales. Much of the “high-wave number”/“high mode” variance should quickly dissipate, generally near the first surface bounce of a beam [Martin et al., 2006].
 The depth-integrated mode-1 energy flux can be calculated as:
where pm1(t,z) and um1(t,z) are mode-1 profiles, counterparts of equation (4) for baroclinic pressure and velocity along the wave propagation direction. and are the normalized mode-1 depth profiles of baroclinic pressure and velocity in the direction of the wave, fully described by the internal wave polarization relations, stratification, and total water depth. ρ is the water density. Note that velocity is not measured—the magnitude of the energy flux is estimated from the mode-1 fit to isopycnal displacements (baroclinic pressure) alone.
 The spatial distribution of phase (its variation as a function of latitudinal and longitudinal coordinates) is a signature of the propagating nature of internal tides. Propagation direction of a mode-1 wave is calculated by minimizing the misfit between observed and modeled mode-1 phase as a function of horizontal coordinates (Figure 10). This calculation effectively works only when one mode-1 wave dominates the signal in a 100 km box. No propagation direction (nor energy flux) is assigned when the depth varies too much over the area, or when a nontilted plane (not propagating wave) fits the data better than the plane with the expected phase speed. For example, it is likely that there is an interference east of 123°E between the semidiurnal mode-1 waves propagating southward from the Ryukyu Ridge (∼24.5°N) and those from Luzon Strait.
 Direct, glider-based observations provide a map of baroclinic energy flux (Figure 11) over a nearly 600 × 800 km region. Observed patterns closely resemble those produced by numerical simulations [Niwa and Hibiya, 2004; Chao et al., 2007; Jan et al., 2008; Simmons et al., 2011]. This is striking because maps of energy flux derived from direct observations rarely resolve this degree of spatial detail over such expansive regions. The total depth-integrated and meridionally integrated energy flux going into the South China Sea (along 120°E) is 3.7 GW and 5.3 GW for the semidiurnal and diurnal components, respectively. On the Pacific side (123°E), the values are 2.7 GW and 3.3 GW, respectively. These values are consistent with estimates provided by models for the semidiurnal internal tide (e.g., Niwa and Hibiya  report M2 fluxes of 4.2 and 3.2 GW to the west and east of Luzon Strait, respectively).
5. Summary and Discussion
 Strong internal tides generated in Luzon Strait and propagating away into the South China Sea and the Pacific are observed directly and quantified by autonomous gliders. The results presented here are complementary to the recent efforts directed at obtaining estimates of the internal wave energy through vertical velocity inferred from the motion of autonomous gliders [Merckelbach et al., 2010; Frajka-Williams et al., 2011; Rudnick et al., 2013].
 Gliders are directed to collect profiles over defined sections and sampling patterns. Multiple, long missions provided broad coverage spanning the region around Luzon Strait and captured the vertical displacements associated with the low-mode internal tides, which allows us to characterize the mean internal wave field and quantify its variability over a roughly 600 km × 800 km region. Sampling extended to the Pacific sector just east of the ridges—an area that has not been well sampled by previous observational efforts. In situ glider-based measurements in the upper 1000 m of the water column capture phase progression associated with propagation of waves away from their generation sites within Luzon Strait.
 The gliders reveal a rich internal wave field in and around Luzon Strait. If the internal tides were generated by the regular barotropic tidal currents always acting on the same stratification and bathymetry, and if they propagated though an ocean without mesoscale or large-scale structure (or a time invariant structure), the amplitude and phase of the internal tide displacements at a given position and depth would remain proportional to and phased locked with the barotropic tidal forcing at the ridge. However, observations show significant variations, even for estimates at the same location. The relationship between amplitudes of the internal tide and magnitudes of barotropic forcing at suspected generation sites is unclear at best.
 Shear and vorticity associated with the Kuroshio and associated mesoscale eddies likely modulate both the generation and the propagation of internal tides near Luzon Strait. Changes in stratification in Luzon Strait might also impact where waves are generated along the ridges. Complex topography creates potential resonances and interferences that can introduce signals with spatial scales barely resolved by the method described here. It is also possible, perhaps even likely, that using model tides at a single location in the strait as a proxy for barotropic forcing is inadequate. Relating observed variations in the internal wave field to the conditions near where the waves are presumably generated, and to the conditions through which the waves have propagated, is an obvious next step. Realistic high-resolution numerical models, particularly models resolving eddies and a time-varying Kuroshio, could be used in conjunction with this data set to interpret the variability of the low-mode internal wave field near Luzon Strait.
 The linear internal tide fits capture roughly half of the isopycnal displacement variance for periods shorter than 3 days (Figure 12). The spatial pattern of the residual is similar to the distribution of signal and the fit, suggesting that in the near field (recall that all these observations are within one to two mode-1 wavelengths of the generation area) the amplitude and phase of the internal tide might vary on scales smaller than the ones used for the fits. Furthermore, the distribution of the residual is also similar to the standard deviation of the glider vertical velocity derived by Rudnick et al.  as a measure of the energy of nontidal processes such as the inertial wave continuum, inertial waves, sharp fronts, and eddies.
 Direct mooring and ship-based measurements were collected in Luzon Strait itself as part of the IWISE (Internal Waves In Straits Experiment) program in summers 2010 [Alford et al., 2011] and 2011. These measurements show that the isopycnal displacements are not pure sine waves, and not varying only at slow temporal and long spatial scales. The technique described here, resolving variations on time scales of 3 days and horizontal scales of 30–50 km, arguably misses part of the signal. On the other hand, by providing persistence and broad spatial coverage, gliders capture the mean and variations in the internal wave field in the Luzon Strait region.
Appendix A: Error Estimates on Fits
 Because gliders complete only 3–4 profiles per day, they only marginally resolve the displacement associated with the semidiurnal internal waves. The cumulative effects of instrument measurement errors, the sampling frequency, and interval between displacement estimates are taken into account to compute the error associated with the internal tide fit for each particular window. Details of the error analysis are provided here.
A1. Error in Isopycnal Depths Due to Slow Sampling
 The ability to track an isopycnal is in part limited by the sampling, and by the quality of the measurement of pressure and potential density. During a typical mission, gliders sample temperature, conductivity, and pressure every several seconds (5–30 s for Seagliders, every 8 s for the Spray gliders). The slow sampling frequency, when multiplied by the vertical velocity of the glider, translates into a vertical spacing of 1–2 m between estimates, varying across the water column depending on the oceanic and flight conditions. Given the high accuracy of the sensors, the slow sampling is ultimately what dominates the uncertainty in determining the depth of a given isopycnal. Note that Seagliders use an unpumped Sea-Bird Electronics (SBE) temperature sensor and conductivity cell and sample both on the dive and the climb, and Spray gliders use a pumped SBE conductivity-temperature-depth (CTD) sampling only on the way up. Data from both platforms are quality controlled and processed such that there are no biases, spikes, or unphysical trends for temperature, salinity, and other measured variables (e.g., no systematic difference between up and down profiles for Seagliders). For each estimate of isopycnal depth (which determines η(t,z)), the uncertainty due to vertical spacing is tracked and translated into uncertainty in the amplitude and phase using test waves.
A2. Test Waves: Impact of Profiling Slowly and Irregularly for the Fits
 In order to get an estimate of the accuracy of internal tide amplitudes and phases, a series of test waves at the M2 and K1 frequencies are generated with known amplitudes and phases and are sampled as a glider would observe them. A random noise (normal distribution with a standard deviation equal to the displacement uncertainty ϵ) is added to obtain the “displacement estimates.” The amplitude and phase of the test wave is then reworked from the displacement estimates. In a Monte Carlo-like approach, this is repeated for 40 different realizations of the noise and 20 different phases of the test wave (covering 0–360°), and the fit error is calculated as the standard deviation of the differences between the estimates and the “true” test waves. For example, for , a more regular sampling in the midwater column results to a 0.70 m fit error (Figure 13a), and twice that amount (1.50 m) for the more uneven sampling near the bottom of the range, where the estimates from the down and up profiles are nearby in time (Figure 13b). Note that the fit error is independent of the test wave amplitude—while it is useful for illustration purposes to have finite test wave amplitude, the results are the same if it is set to zero. In other words, this method determines what sine waves can be fitted to random noise.
 In order to represent a 95% confidence interval, the amplitude fit error is taken as twice the standard deviation of the test wave calculation. The amplitude fit error is linearly proportional to the magnitude of the displacement random noise, ϵ (Figure 13c). The absolute error in the estimate of internal tide amplitude is obtained using the actual glider sampling and uncertainty in isopycnal depth (shown in Figure 14 for a representative period). For the quantitative analysis presented in the paper, estimates are ignored if the error is larger than 3 m (only about 4% of the fits).
 To further decrease the likelihood of unphysical results, isopycnals have to be sampled at least 10 times over the course of the 3 day window (eliminates 3% of the fits). In addition, the variance of the sum of the inferred semidiurnal and diurnal internal tides has to be smaller than twice the variance of the data used to obtain the fit (i.e., if there was only one wave, its amplitude would be less than twice the standard deviation of the data). This last condition eliminates 1% of the fits. Overall, about 87% of the fits are considered good with absolute error less than 3 m.
A3. Internal Tides in a Fully Dynamical Ocean
 The above discussion considers the random error (instrument noise) combined with aliasing due to the slow profiling. Another important factor to consider is how the estimates are affected by motions in the ocean that are not linear internal tides (i.e., pure sine waves), including phenomena at higher frequencies like solitary waves or nonlinear internal waves, and the continuum spectrum of internal waves. The method described in this paper only estimates the amplitude and phase of the sine wave that are most consistent with the observations over a 3 day period. A reasonable question is how much signal is missed by the glider (and the method). This is a difficult task, because by definition the glider doesn't resolve the full spectrum. However, we can validate this method by using a high-resolution time series recorded in Luzon Strait and asking how much a glider-type sampling would capture.
 A Sea-Bird Electronics moored CTD sensor (SBE37) was deployed at 1600 m on mooring MP-N (20.6°N 121.33°E) for a 3 week period in late summer 2010 as part of the pilot IWISE program [Alford et al., 2011]. The instrument sampled every 20 s. Assuming that the measured density fluctuations are due to a vertical heaving of the water column, they can be expressed as vertical displacement using the mean density gradient at that depth (Figure 15a). Although 1600 m is below the profiling range of gliders, in this thought experiment we sample these displacements the way sg124 would sample an isopycnal located around 500 m during its spring 2008 mission (as in Figure 14). Using 6–8 estimates of displacements per day, the amplitude and phase of the semidiurnal and diurnal internal tides are obtained using the method described in this paper (shown in Figures 15a and 15b with the full time series). The spectra (Figure 15c) show that glider estimates capture the peaks, but miss a small fraction of the variance, and miss the higher harmonics completely, as does the spectrum of the band-passed signal obtained from two fourth-order Butterworth filters with a bandwidth of 1/3 day−1 centered at M2 and K1. The “glider” fits are comparable to band-pass filters typically used on mooring data.
 We are grateful to the Office of Naval Research, whose support has been essential to the development of the gliders and to the observational program described here. Gliders mission in the Kuroshio were supported by ONR grants N00014-06-1-0751, N00014-06-1-0776, and N00014-10-1-038. The success of the glider operations is largely due to the dedication and commitment from the Integrative Observational Platforms group at APL/UW, and the Instrument Development Group at SIO/UCSD.