Abstract

Though unresolved by Argo floats, internal waves still impart an aliased signal onto their profile measurements. Recent studies have yielded nearly global characterization of several constituents of the stationary internal tides. Using this new information in conjunction with thousands of floats, we quantify the influence of the stationary, mode-1 M2 and S2 internal tides on Argo-observed temperature. We calculate the in situ temperature anomaly observed by Argo floats (usually on the order of 0.1°C) and compare it to the anomaly expected from the stationary internal tides derived from altimetry. Globally, there is a small, positive correlation between the expected and in situ signals. There is a stronger relationship in regions with more intense internal waves, as well as at depths near the nominal mode-1 maximum. However, we are unable to use this relationship to remove significant variance from the in situ observations. This is somewhat surprising, given that the magnitude of the altimetry-derived signal is often on a similar scale to the in situ signal, and points toward a greater importance of the nonstationary internal tides than previously assumed.

1. Introduction

The array of Argo floats, which reached quasi-global coverage in 2004 (Roemmich and Gilson 2009), has been transformative for the in situ study of a wide variety of oceanographic phenomena. Argo floats are well suited to capture processes that occur on time scales from months to years, such as large-scale changes in oceanic heat content (Sutton and Roemmich 2011; Trenberth et al. 2016), salinity properties (Anderson and Riser 2014; Wilson and Riser 2016; von Schuckmann et al. 2009), and biogeochemical tracers (Martz et al. 2008; Hennon et al. 2016; Bushinsky et al. 2017).

Internal waves, on the other hand, are a phenomenon not well resolved by Argo float measurements. The floats nominally profile the upper 2000 dbar of the ocean once every 10 days, which is far below the sampling frequency required to resolve most internal waves. However, their signal is still aliased onto Argo profile measurements, as internal waves vertically heave gradients of temperature, salinity, and tracers. This acts to add variability to Argo observations that are generally considered “noise.” Most baroclinic energy is often within internal tide or near inertial frequency bands. Inertial waves are intermittent and generated by sporadic storm events (D’Asaro 1985; Alford 2001), making them naturally irregular. The forcings for internal tides are astronomical (Munk and Wunsch 1998) and therefore they have a much higher degree of predictability. A moored profiler near the Hawaiian ridge shows that temperature in the thermocline can fluctuate by 1.5°–2.0°C at tidal frequencies (Figs. 1a,c). In this region, at least, it would be valuable to disentangle the strong internal tide signal from nontidal processes measured by Argo floats, as a signal of >1°C could potentially obscure these nontidal processes and, for example, add noise to heat content estimates.

Recent studies have characterized the semidiurnal internal tide from satellite observations (Zhao et al. 2016; Ray and Zaron 2016; Zhao 2017). Using this new information, Zaron and Ray (2017) investigated 1497 Argo float profiles near Hawaii and found that the M2 internal tide only accounted for a small fraction of the overall steric height variability. To build upon this analysis and expand it to the global domain, here we use characterizations of the stationary mode-1 M2 and S2 internal tides (Zhao et al. 2016; Zhao 2017) alongside measurements from 4251 Argo floats to construct a global survey of the impact of internal waves on Argo profile observations, with the purpose of identifying regions where the semidiurnal internal tide may be a significant contributor to Argo-observed variability and investigating the feasibility of a correction.

2. Method examined on moored profiler data

a. Methods

The primary objective of this work is to establish how effectively we can use altimetric observations of the stationary semidiurnal internal tide (M2 and S2) to estimate the influence of internal waves on Argo profile observations. To demonstrate and validate the methods we use when analyzing Argo float data (section 3), we first turn to mooring data. Mooring data provide the temporal resolution to explicitly resolve internal waves, whereas the ~10-day periods of Argo cycles do not. Here, we examine data from a moored profiler (MP) deployed as part of the IWAP experiment (Alford et al. 2007). The MP was fixed to a mooring deployed at 25.49°N, 165.15°W, from 25 April to 4 June 2006, and collected CTD measurements in the upper 1400 m of the water column once every 1.75 h.

While the 40-day record of the MP is much shorter than the typical Argo float lifespan (up to 6 years), the sampling frequency is adequate to resolve semidiurnal tidal signals. The vertical range is also comparable to Argo floats (upper 2000 m). The MP is deployed in a region of strong internal tides and low mesoscale variability (Shum et al. 1990; Chelton et al. 2007), which are ideal conditions for observing clear tidal signatures in temperature. Combined, these factors allow us to develop a rough sense of the “best case” scenario for our methods, which are detailed hereafter.

We focus our analysis on the temperature anomaly observed by the MP TMP, which is defined as

 
TMP(z,t)=TMP(z,t)TMP¯(z),
(1)

where TMP is the temperature recorded by the MP in pressure z and time t coordinates, and TMP¯ is the mean temperature profile (calculated on isobars over the full temporal record). Temperature anomaly TMP is then high-pass filtered along isobars with a fourth-order Butterworth filter using a cutoff period of 30 days. Since the record is only 40 days, this makes very little difference to TMP, but is done to be consistent with the procedure we subsequently use on Argo data, where low-pass filtering is used to reduce signals induced by the lateral migration of floats (section 3b).

Since the semidiurnal tides are well resolved within TMP, it is straightforward to estimate the contributions of the M2 and S2 constituents through, for example, spectral analysis or harmonic fitting. However, the end goal of this work is to demonstrate how well altimetric estimates of the internal tides can be used to predict, and potentially correct, observed variability. Therefore, we use the M2 and S2 sea surface height (SSH) signals to estimate the expected temperature anomaly within the MP record due to the semidiurnal internal tides.

First, we calculate the predicted SSH time series for the record of the MP deployment,

 
SSH(t)=i=M2,S2Ai(x,y)cos[ωit+θi(x,y)],
(2)

where A is the SSH amplitude of the tidal signal, while ω and θ are the corresponding frequency and phase, respectively. Variables A and θ are both obtained from data in Zhao et al. (2016) and Zhao (2017), for the M2 and S2 constituents, respectively. Here, x and y denote longitude and latitude, respectively, and are set to the coordinates of the mooring.

Next, we estimate the internal wave vertical displacement η

 
η(z,t)=SSH(t)R(x,y)Φ(x,y,z).
(3)

Here, SSH is as in Eq. (2), R is the ratio of SSH to the maximum internal wave vertical displacement (which have scales of millimeters and meters, respectively), and Φ is the mode-1 vertical structure of vertical displacement (normalized so the maximum is one). Parameters R and Φ are longitudinally and latitudinally dependent (as with A and θ), and Φ has the additional dimension of depth (z). The R and Φ are derived from the polarization relation for a mode-1 internal tide and are computed from climatological stratification, as elucidated in Zhao et al. (2016). While it would be optimal to use in situ stratification to calculate the mode-1 structure, Argo floats often sample less than half the water column (only down to 2000 m). Moderate deviations between climatological and in situ stratification do not strongly impact the mode-1 vertical structure, so the use of climatology for Φ does not severely alter our results. Finally, we calculate the expected temperature anomaly from the stationary, mode-1 semidiurnal internal tide Ttide by multiplying the vertical displacement by the vertical temperature gradient dT/dz,

 
Ttide(z,t)=η(z,t)dTdz(z,t),
(4)

where dT/dz is calculated from the MP temperature record, which is low-pass filtered (fourth-order Butterworth) in time and pressure (2 days and 100 dbar, respectively). Naturally, Ttide will be large where internal waves are big, and in the thermocline where vertical temperature gradients are strong.

b. MP results

In the main thermocline the observed temperature anomaly TMP is generally similar in structure to the expected temperature anomaly Ttide (Fig. 1). There is good phase agreement, and fortnightly modulation of the spring–neap cycle is present in both TMP and Ttide. However, Ttide significantly underestimates the largest peaks of TMP (up to a factor of ~4), likely because it only includes the first mode of the stationary M2 and S2 signals. Subtracting Ttide from TMP reduces the variance by at least 20% from 250 to 750 m, and by up to 40% from 500 to 750 m. Below 750 m, dT/dz and TMP are very weak, and subtracting Ttide does not appreciably change the overall variance. Likewise, below 750 m the correlation between TMP and Ttide (calculated on each pressure surface) decreases substantially (Fig. 1e). For comparison, on each pressure surface we find the best M2 and S2 fits to TMP using harmonic analysis. The M2 and S2 fits are added together, creating an estimate for the combined stationary semidiurnal signal, Tfit (Fig. 1c, blue line), which serves as a useful comparison to Ttide. The amplitudes of Ttide and Tfit are roughly the same magnitude and very similar in phase, consistent with results found by Zhao et al. (2010).

By subtracting Tfit from TMP, we can slightly improve upon the variance reduction compared to that of Ttide (Fig. 1f). However, the variance reduction when using Ttide only marginally underperforms the variance reduction from Tfit, indicating that the majority of the semidiurnal signal is mode-1 and coherent, and the methods in section 2a work well for this location. Although the stationary, mode-1 M2 and S2 signals do not account for even half of the overall variance, there are many other tidal constituents that are neglected. For example, from 30 April to 4 May, there is a very strong diurnal signal (over 1°C peak to trough), which is not included in our analysis. The diurnal signal adds significant variance near Hawaii (Dushaw et al. 2011), and by using eight total tidal constituents, Dushaw et al. (2011) generally find higher fractions of variance explained than we do here.

3. Argo data and methods

a. Argo float data

We now turn our analysis to Argo floats, where we use very similar methods to examine how the semidiurnal tide influences Argo profile observations in a global context. Estimating the expected temperature anomaly due to the semidiurnal internal tide for Argo float data is not as straightforward as for continuous data from mooring records (section 2). The MP records the precise sample time associated with each CTD measurement collected during profiling, which makes calculating the concurrent SSH signal [Eq. (2)] for any MP measurement trivial. Because Argo floats must transmit their information via satellite, most do not include the precise time of each sample point for profile measurements. Instead, in order to reduce the size of the transmitted data package, frequently only the surface time (when the float breaches the water to transmit data) is included.

Argo floats sample only on the upcast, and with an ascent rate of roughly 10 cm s−1, the duration of the full 2000-m profile spans roughly 6 h. Thus, over the course of a vertical profile the semidiurnal phase changes nearly 180°, so the surface time alone is insufficient to characterize semidiurnal signals over the entire profile. To apply Eq. (2) to Argo float data and calculate the state of the semidiurnal internal tide SSH signal, we must accurately estimate the timing of all profile measurements. Two types of Argo floats allow us to overcome this obstacle (float data are retrieved from http://www.usgodae.org/ftp/outgoing/argo/dac/).

The first is a set of 1312 floats that return coarse time series of pressure during the floats’ vertical profiles, hereafter referred to as group A floats. Using linear interpolation (and extrapolation when necessary), we use this coarse pressure time series to estimate the timing of each CTD measurement. For the floats included in group A, the coarse pressure time series have a minimum of five points during the vertical profiles, though 78% have between 10 and 25 (inclusive).

The second is a set of 2939 floats that use Iridium telemetry and whose only time stamp is at the surface of their profiles, hereafter referred to as group B floats. These floats do not have any temporal information associated with their vertical profile measurements, but, unlike floats that use ARGOS telemetry, the reported surface time is generally accurate to within 5–15 min of the true end (top) of the profile (J. Gilson 2018, personal communication). Therefore, we can estimate the timing of each CTD sample by using the surface time of the profile and extrapolating backward in time, assuming a constant 10 cm s−1 accent rate (roughly consistent with the ascent rate observed in group A floats). While less accurate than the methods for group A floats, modest differences between the assumed and true ascent rate will result in errors of roughly 0.5 h at the bottom of the profile (with less error at shallower depths). This is an acceptably small fraction of the semidiurnal period (~12 h).

Both sets of floats offer coverage over large swaths (Fig. 2). Group A floats have a significant gap in coverage in the North Pacific and eastern Indian Oceans, while group B floats are relatively sparse in the central South Pacific. Together, the two sets of floats offer nearly comprehensive spatial coverage of the world’s oceans.

b. Argo temperature anomaly

We proceed to analyze the Argo data in a similar manner as in section 2. We linearly interpolate profile data to a pressure grid from the surface to 2050 dbar, spaced by 5 dbar. As in Eq. (1), we calculate the Argo temperature anomaly Targo by subtracting the average temperature (from the full float record) along isobars. We high-pass filter Targo with a 30-day cutoff period along isobars. Although this makes little difference to the MP data because it was only deployed about 6 weeks, Argo floats can remain active for up to 6 years, and over that time can be advected across different water masses that cause large signals in temperature. The high-pass filter largely removes these migration-induced changes, so instead we can focus on the profile-to-profile variability that internal waves are most likely to impact. Here it is important to note that if the Argo floats remained at a fixed location like a mooring (and continued to profile every 10 days), the stationary internal tidal signal is assumed constant. High-pass filtering with a 30-day cutoff could remove nearly all of the semidiurnal signal, as it is aliased to frequencies not in the pass band. However, the floats do migrate (often rapidly) through the phase space of the M2 and S2 internal tides, so there is no single stationary signal that is aliased in a manner familiar to simple signal processing exercises. Thus, the 30-day cutoff period should not appreciably diminish the internal wave signal in our observations. The correlations presented here are not very sensitive to the filter cutoff period used. A significantly longer cutoff (100 days) was also tested, with negligible quantitative differences to the correlations between Targo and Ttide.

To compute the predicted temperature anomaly from the internal tides Ttide we again use Eqs. (2)(4), with only two notable changes. First, for the MP data we took A(x, y), θ(x, y), R(x, y), and Φ(x, y, z) to be constant in x and y, reflecting the fixed position of the mooring. Now these variables are allowed to evolve with the lateral migration of the Argo floats (Fig. 3a). Second, dT/dz is only low-pass filtered in depth (again a 100-dbar cutoff), not time, as the cycling frequency of Argo floats (~0.1 days−1) is not high enough to make the same temporal filtering used in section 2 useful.

Using Targo and Ttide (Fig. 3b), we calculate the correlation r to estimate the relationship between in situ temperature anomaly and the anomaly expected from the semidiurnal tide. This correlation is calculated along isobars for the full deployment of each float that met two conditions. First, the float acquired at least 20 profiles of data, and second, at least 75% of those profiles occur where there are valid estimates of SSH from altimetry. This yields one vertical profile of r per float (Fig. 3c). Profiles occurring in regions without valid estimates of the mode-1 M2 (Zhao et al. 2016) and S2 (Zhao 2017) SSH signals, and consequently Ttide, are not included in our calculations of r. While it is possible to break data into smaller blocks (e.g., year-long segments) we opt to keep the longest record possible, as shorter blocks do not change the qualitative results in sections 4 and 5, but add considerable scatter.

4. Argo results

Combining groups A and B, there are initially 4251 floats available for analysis. Due to a combination of spatial gaps in the altimetric SSH data used (Zhao et al. 2016; Zhao 2017) and because some floats have deployments too short for useful analysis, the number of floats with usable data is significantly reduced (section 3b). At 950 m, 2668 floats have sufficient data to use Targo and Ttide to calculate r (the exact number varies slightly by depth because of variable profiling schemes used by the floats).

Among all the floats used to calculate r, the average is weakly positive at all depths (Fig. 4). The mean value of r as a function of depth has a mode-1 resemblance, beginning at zero near the surface, reaching a maximum (~0.08) at 1125 m, and then decaying gradually with increasing depth. Many estimates of r are negative. Of the 772 floats where r < 0 at 950 m, only 37 are significant at the 95% confidence level (4.8% of total), implying that these negative correlations likely arise simply from statistical variation due to nontidal signal. For comparison, of the 1896 floats where r > 0 at 950 m, 438 are significant at the 95% confidence level (23.1% of total), far exceeding the number expected from random processes.

The global patterns of r also illustrate several broad trends (Fig. 5). Most apparently, r is generally positive (red in Fig. 5), but with significant variability. There are regions where the correlation is noticeably stronger, such as the north and southwest Pacific, while other regions are weaker or negligible, such as near the equator. The low correlations at the equator are due to the almost nonexistent stationary tide there, as equatorial jets and changes in stratification decohere the internal tides (Zhao et al. 2016; Buijsman et al. 2017). As with the global aggregate of data (Fig. 4), the maximum correlations appear to be near 1000 m over much of the globe, but the correlation is not strongly depth dependent below the first few hundred meters. The spatial structure of r broadly resembles that of the SSH amplitude of the stationary mode-1 M2 (Zhao et al. 2016) and S2 (Zhao 2017) constituents of the internal tides. Using the temporal average of A [SSH amplitude from Eq. (2)] along each float’s drift path as a proxy, we observe a clear relationship between r and the strength of the semidiurnal tide (Fig. 6).

The use of correlation coefficients is a helpful metric to characterize the phase agreement between Ttide and Targo, but it does not give insight on their comparative magnitudes. To quantify this, we calculate the standard deviation of Ttide (δtide) along isobars. This provides an estimate of the temperature variability expected from the stationary, mode-1 semidiurnal tide. To compare this to the total temperature anomaly observed by Argo floats, we also take the standard deviation of Targo (δargo) along isobars. Naturally, δtide is largest close to the surface (where thermal gradients are stronger), but is also larger near regions of strong internal waves (Fig. 7, left panels). The ratio of δtide and δargo (δtide/δargo) similarly is large around regions where the stationary internal tides are more intense, but does not have the same decay with depth as δtide (Fig. 7, right panels). The highest values of δtide/δargo are generally near the mode-1 maximum (~1000 m).

The methods described in sections 2 and 3 use temperature as the primary variable of interest. When temperature in Eqs. (1) and (4) is replaced with potential density it only very slightly improves the correlations between variability observed by Argo floats and that expected from the stationary semidiurnal tides. Given the marginal difference, we choose to keep analysis in terms of temperature because it is somewhat more intuitive, as well as more applicable to efforts to characterize changes to ocean heat content (Fahrbach et al. 2004; Purkey and Johnson 2010; Lyman and Johnson 2014).

5. Discussion

The magnitude of the temperature variability expected from the stationary mode-1 M2 and S2 internal tides is often a considerable fraction of the total high-passed signal observed by Argo floats, exceeding 0.25 in roughly half the world’s oceans at 950 m (Fig. 7). Because the nonstationary tide is neglected, this fraction (δtide/δargo) represents a lower bound on the total variability contributed by semidiurnal internal tides. The fraction δtide/δargo is generally higher where the correlation r is higher (Fig. 5), which leads to the fairly intuitive conclusion that Argo float measurements are more likely to be affected by internal waves where stationary internal waves are strong.

Ideally, we would subtract Ttide from Targo in order to disentangle the fraction of variance accounted for by the mode-1 semidiurnal tides and the remaining signal (nontidal and other tidal constituents). However, given the low correlations and the noise within both Ttide and Targo, such analysis is generally ineffective for these purposes. Even if we isolate analysis to just the floats with r > 0 significant to the 95% level (438 floats at 950 m, 16% of the total available), Ttide generally only explains ~5% of the total variance in Targo.

This is somewhat surprising, given the relatively strong semidiurnal internal tidal signal present over much of the globe. Yet these results are consistent with the findings of Zaron and Ray (2017), who observed that even near a strong source of internal waves (the Hawaiian Ridge) the M2 internal tide only explained a small percentage of the total variance of steric height. There are many factors that could hinder the ability to observe a stronger relationship between Argo profile variability and the expected internal wave signal.

Some are methodical in nature. Small differences between the in situ phase and the satellite-estimated phase used in Eq. (2) could cause misalignment and reduction in correlation between Targo and Ttide. In section 2 we find good phase agreement between TMP and Ttide. More generally, when internal tide signals are at least moderately strong (SSH amplitude > 5 mm), Zhao et al. (2016) find a fairly tight coupling between the mooring-observed and satellite-estimated phase, so inaccuracies with the satellite-estimated phase are unlikely to be too severe near regions with strong tidal signals. However, when SSH amplitudes are <5 mm Zhao et al. (2016) find much more scatter between mooring-observed and satellite-estimated phase. This is likely an important factor in the negligible correlations associated with floats in regions of weak SSH tidal signals (Fig. 6).

Given multiple time stamps per profile, the timing of measurements in group A floats is accurate to well under 1 h, minimizing any deleterious effects from timing-related phase offsets. However, group B floats only record their surface times, so we are forced to estimate the timing of the CTD measurements during their profiles by using an assumed ascent rate and extrapolating backward. Since a profile spans roughly 6 h, moderate discrepancies between the assumed ascent rate and true ascent rate could contribute to significant phase differences (most pronounced at the bottom of the profile). To address this potential issue, we took all group A floats and used only the surface time to estimate the timing of the CTD samples (as done with group B floats), and recalculated r. There is a tight relationship between the recalculated values of r and the original values of r, as the correlation between the original and recalculated values is 0.96. The original estimates of r have a marginally higher average at 950 m (0.080) than the recalculated values (0.075), likely arising from the slightly better time estimates. However, these differences are minimal, suggesting that the lack of timing information within profiles is not a significant hindrance for group B floats.

Other parameters in Eqs. (2)(4) require brief scrutinization. Parameters R and Φ are computed using climatological datasets. Neither are particularly sensitive to small perturbations, so large errors arising from differences between climatology and in situ conditions are unlikely. The temperature gradient dT/dz is computed directly from Argo profile data, and the step between vertical internal wave displacement η and temperature anomaly Ttide is straightforward [Eq. (4)]. The relationship between the satellite-estimated SSH amplitude A used in Eq. (2) and the in situ amplitude observed by moorings is roughly linear, but with a moderate degree of scatter (Zhao et al. 2016). While the error in altimetric-derived phase and amplitude for weak internal wave signals could be high enough to adversely affect Ttide, estimates of Ttide in regions of strong internal waves should be reasonably accurate. So our question stands, why do we not observe a stronger correlation between Ttide and Targo in regions with prominent internal wave signals?

Although a detailed analysis is beyond the scope of this work, it is important to note several prominent phenomena that can decohere internal tides. Mesoscale eddies are a nearly ubiquitous phenomenon in the ocean everywhere but the equator (Chelton et al. 2011) and can refract internal tidal beams. Rainville and Pinkel (2006) find that the path of the mode-1 internal tide emanating from Hawaii can be significantly altered depending on the state of the mesoscale field. Buijsman et al. (2017) observe that changes in stratification and equatorial jets can erode the stationary tide such that it is nearly nonexistent at the equator. Other fronts, such as the Kuroshio, have strong shear gradients that manifest as potential vorticity barriers capable of reflecting internal tides (Rainville and Pinkel 2004), so variability in front paths can alter internal wave trajectories and weaken the stationary tide. Internal tides are capable of propagating thousands of kilometers (Ray and Mitchum 1997; Alford and Zhao 2007), and any combination of the aforementioned processes can act to decohere internal tides, which in turn can obscure the altimetric signal used to characterize their stationary SSH signature.

Finally, internal tides are clearly not the only signal within Targo. There can be nontidal frequencies that add considerable variance to the internal wave spectrum (Hennon et al. 2014). Additionally, the same processes that refract internal waves can also add variability to Targo, as features such as mesoscale eddies and fronts create uneven gradients in temperature as the floats drift laterally through the ocean (Gould 1985; Chaigneau and Pizarro 2005; Chaigneau et al. 2011). The MP examined (section 2) was at a fixed location in a strongly tidal region, yet the correlation coefficient between Ttide and TMP peaked at ~0.6 (representing a maximum of ~40% variance explained). Since Argo floats undersample semidiurnal signals, can migrate considerable distances between profiles, and are often in regions of relatively weak internal tides, perhaps it is unsurprising that the correlations between Targo and Ttide are mostly small.

6. Summary

We aim to quantify the impact of the stationary, mode-1 M2 and S2 internal tides on the vertical profile measurements of >2500 Argo floats. There is a weak, but positive correlation between the in situ temperature anomaly observed by floats and the temperature anomaly expected from the heaving induced by the stationary semidiurnal internal tide. Although this correlation tends to be stronger in regions where the internal tides are prominent as well as near the mode-1 maximum, it is still somewhat weaker than expected given the of the relative strength of the expected temperature anomaly compared to the observed temperature anomaly (Fig. 7). The inclusion of additional tidal constituents would almost certainly improve the correlations between the observed and expected temperature anomaly, but this does not fully address the cause of the low correlations found.

We scrutinize our methods and find that the errors in methodology are unlikely to account for the low correlations between expected and observed temperature anomalies in regions of strong internal tides. However in regions of weak internal tides perhaps error in the altimetric estimates (particularly phase) contribute to low correlations. A variety of phenomena decohere internal tides as they propagate, such as mesoscale eddies and fronts, weakening the stationary internal tides. Additionally, the lateral migration of the floats presents an additional obstacle in capturing internal wave signatures as signals from other processes add to the temperature variability measured by Argo floats.

Acknowledgments

We thank John Gilson for his expertise and guidance in the use of the Argo data used here. Madeleine Hamann and Arnaud Le Boyer gave helpful feedback that significantly improved the manuscript, as did two anonymous reviewers. This work was funded by NASA under Grant NNX13AD90G and by the Office of Naval Research under Grant N00014-17-1-2112.

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Footnotes

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