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.

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(a) The full record of temperature anomaly from the moored profiler $T_{\text{MP}}^{\prime }$. (b) The expected temperature anomaly based on altimetric estimates of the semidiurnal tide $T_{\text{tide}}^{\prime }$. (c) The time series of $T_{\text{MP}}^{\prime }$ (gray) and $T_{\text{tide}}^{\prime }$ (red) at 300 m [indicated by the dashed line in (a) and (b)]. A harmonic fit of the stationary M₂ and S₂ signal $T_{\text{fit}}^{\prime }$ is also included (blue). The cutout shows the period from 30 Apr to 4 May in more detail. The dashed line in the cutout alternates between black/gray every 24 h in order to highlight the diurnal peaks. (d) A map of the amplitude of the stationary, mode-1 M₂ internal tide (mm), with the white “×” showing the MP location. (e) The correlation r profile between $T_{\text{MP}}^{\prime }$ and $T_{\text{tide}}^{\prime }$. (f) A profile of the RMS of $T_{\text{MP}}^{\prime }$ (gray), RMS when $T_{\text{tide}}^{\prime }$ is subtracted from $T_{\text{MP}}^{\prime }$ (red), and the RMS when $T_{\text{fit}}^{\prime }$ is subtracted from $T_{\text{MP}}^{\prime }$ (blue).

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-19-0121.1

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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

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.

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A map of the global coverage of floats in group A and B (see section 3a). The color in the 2° × 2° bins denotes the presence of profile data from group A (light blue), group B (teal), or both A and B (dark blue). Bins with no profile data are white.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-19-0121.1

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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 $T_{\text{argo}}^{\prime }$ by subtracting the average temperature (from the full float record) along isobars. We high-pass filter $T_{\text{argo}}^{\prime }$ 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 $T_{\text{argo}}^{\prime }$ and $T_{\text{tide}}^{\prime }$.

To compute the predicted temperature anomaly from the internal tides $T_{\text{tide}}^{\prime }$ 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.

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(a) The trajectory of float 5904055 (black line) from its first (green “×”) to last profile (red “×”), with the color indicating the magnitude of the stationary, mode-1 M₂ SSH amplitude (mm). (b) The time series of $T_{\text{argo}}^{\prime }$ (gray) and $T_{\text{tide}}^{\prime }$ (red) at 950 m over the multiyear deployment of float 5904055. Note: the temperature anomaly range is significantly lower than in Fig. 1c, primarily due to the different depths highlighted. (c) The correlation profile r (computed from $T_{\text{argo}}^{\prime }$ and $T_{\text{tide}}^{\prime }$ on pressure surfaces), where the thick line is the correlation, thin lines show the range of the 95% confidence interval, and the dashed line delineates between positive and negative correlations.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-19-0121.1

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Using $T_{\text{argo}}^{\prime }$ and $T_{\text{tide}}^{\prime }$ (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 $T_{\text{tide}}^{\prime }$, 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 $T_{\text{argo}}^{\prime }$ and $T_{\text{tide}}^{\prime }$ 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.

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The distribution of correlation coefficients for the entirety of all float data. The color indicates the number of estimates N that fall in each bin (in log₁₀ scale). Bins are separated by 5 dbar in pressure and 0.01 in r. The dashed line delineates between positive and negative correlations. The solid black line shows the average r as a function of pressure.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-19-0121.1

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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).

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A global plot of the correlation coefficient between $T_{\text{argo}}^{\prime }$ and $T_{\text{tide}}^{\prime }$ at 950 m. (top) The trajectory of all floats (both groups A and B), where the color represents the value of r calculated over the full record of data. (bottom) As in the top panel, but averaged in 4° × 4° bins so that regional trends are more apparent. Red hues represent positive correlations, while blues are negative correlations. The coverage of data here is less than in Fig. 2 because altimetric observations used for Eq. (2) are absent in regions of high mesoscale activity (e.g., the Antarctic Circumpolar Current and western boundary currents) and poleward of 60°.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-19-0121.1

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The correlation coefficient r at 950 m plotted against the average amplitude A of SSH estimated from altimetry along each float path. Each gray circle is representative of one full Argo deployment. The dashed line delineates between positive and negative correlations. The thick line is the average correlation coefficient calculated over bins spaced by 2 mm (minimum 20 values per bin), and the vertical ranges show the standard deviation within each bin.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-19-0121.1

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The use of correlation coefficients is a helpful metric to characterize the phase agreement between $T_{\text{tide}}^{\prime }$ and $T_{\text{argo}}^{\prime }$, but it does not give insight on their comparative magnitudes. To quantify this, we calculate the standard deviation of $T_{\text{tide}}^{\prime }$ (δ_tide) 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 $T_{\text{argo}}^{\prime }$ (δ_argo) along isobars. Naturally, δ_tide 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 δ_tide and δ_argo (δ_tide/δ_argo) similarly is large around regions where the stationary internal tides are more intense, but does not have the same decay with depth as δ_tide (Fig. 7, right panels). The highest values of δ_tide/δ_argo are generally near the mode-1 maximum (~1000 m).

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(left) Logarithmic scale of the standard deviation of $T_{\text{tide}}^{\prime }$ (δ_tide) averaged into 4° × 4° bins. (right) The ratio of the standard deviation of $T_{\text{tide}}^{\prime }$ (δ_tide) and the standard deviation of $T_{\text{argo}}^{\prime }$ (δ_argo) averaged into 4° × 4° bins. Gray contours are at 0.25.

Citation: Journal of Physical Oceanography 49, 10; 10.1175/JPO-D-19-0121.1

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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 (δ_tide/δ_argo) represents a lower bound on the total variability contributed by semidiurnal internal tides. The fraction δ_tide/δ_argo 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 $T_{\text{tide}}^{\prime }$ from $T_{\text{argo}}^{\prime }$ 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 $T_{\text{tide}}^{\prime }$ and $T_{\text{argo}}^{\prime }$, 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), $T_{\text{tide}}^{\prime }$ generally only explains ~5% of the total variance in $T_{\text{argo}}^{\prime }$.

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 $T_{\text{argo}}^{\prime }$ and $T_{\text{tide}}^{\prime }$. In section 2 we find good phase agreement between $T_{\text{MP}}^{\prime }$ and $T_{\text{tide}}^{\prime }$. 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 $T_{\text{tide}}^{\prime }$ 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 $T_{\text{tide}}^{\prime }$, estimates of $T_{\text{tide}}^{\prime }$ in regions of strong internal waves should be reasonably accurate. So our question stands, why do we not observe a stronger correlation between $T_{\text{tide}}^{\prime }$ and $T_{\text{argo}}^{\prime }$ 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 $T_{\text{argo}}^{\prime }$. 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 $T_{\text{argo}}^{\prime }$, 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 $T_{\text{tide}}^{\prime }$ and $T_{\text{MP}}^{\prime }$ 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 $T_{\text{argo}}^{\prime }$ and $T_{\text{tide}}^{\prime }$ 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.