## 1. Introduction

The internal tides are inertia-gravity waves that propagate within the ocean and are associated with subsurface currents and vertical isopycnal displacements, the magnitude of which depends on the strength of and proximity to the generation sites (Siedler 1991; Chiswell 2000; Rudnick et al. 2003). In spite of their origin from predictable astronomical forcing, in situ measurements of internal tides frequently exhibit substantial variability (Magaard and McKee 1973; Barnett and Bernstein 1975; Weisberg et al. 1987), which is attributed to refraction, Doppler shifting, and other interactions with the nontidal background (Rainville and Pinkel 2006; Klymak et al. 2008; Chavanne et al. 2010; Zilberman et al. 2011; Park and Farmer 2013).

The dynamics of internal tides may be described, approximately, in terms of horizontally propagating vertical modes that are the solution of a Sturmâ€“Liouville problem, with the modes naturally ordered from fast to slow phase speed (Gill 1982). When the measurement technique permits separation of the first from the higher modes, a substantial fraction of the mode-1 signal is found to be stationary, that is, phase locked with the astronomical forcing. This fact has made it possible to observe internal tides in satellite measurements of sea surface height (SSH), which, because of the nature of the exact-repeat orbits employed, requires multiyear records to extract the stationary tides (Ray and Mitchum 1996; LeProvost 2001). Regional and global maps of the internal tides have been created using both empirical and dynamically constrained harmonic analysis, with higher spatial resolution achieved through combination of multiple missions (Dushaw et al. 2011; Zhao et al. 2011). Other techniques for isolating the low-mode internal tide include inverted echo sounders (Chiswell 2002), moored thermistor chains (Picaut et al. 1995), and acoustic travel time measurements (Dushaw et al. 1995).

There have been several efforts to determine the degree of phase locking and the partitioning of low-mode internal tide signals between the stationary and nonstationary parts. Ray and Zaron (2011) used two techniques to identify both the stationary and nonstationary internal tide from altimetry. The first approach was based on harmonic analyses of subsets of the data and comparison of tides within different epochs or seasons. Some regions of significant seasonal variability were identified; however, the resolution of nonstationarity by this technique is fundamentally constrained by the temporal sampling (once per 10 days for TOPEX/Poseidon and the Jason satellite series). The second approach utilized along-track wavenumber spectra to identify the SSH variance within wavenumber bands associated with internal tide modes. But because the internal tide field is spatially inhomogeneous and anisotropic, this approach may miss internal tide variance at wavenumbers that are unfavorably aligned with the satellite ground tracks. In situ measurements of the low-mode tide generally find nonstationary variance of between 20% and 30% of the stationary variance (Dushaw et al. 1995; Picaut et al. 1995; Chiswell 2002), but these values are representative of specific locations that tend to be located near generation sites.

The present paper takes a new approach in the time domain based on an analysis of dual-satellite crossover data. Data from the *Jason-2* (J2) and *Cryosat-2* (C2) missions are utilized because of the relatively high instrumental precision of the altimeters and the large number and spatial density of the orbit crossovers. A sample statistic, the mean square of J2 minus C2 SSH differences, is computed as a function of time lag and interpreted as a sum of instrumental noise and physical signals. In section 2, the sample statistic is defined and some of its properties are enumerated. In section 3, attributes of the internal tide are inferred and presented. Finally, sections 4 and 5 discuss and summarize the results.

## 2. Background

*I*

_{ij,k}is an indicator function that is equal to 1 if

*i*th and

*j*th observations are within 6 km (at orbit crossovers), and 0 otherwise. The quantity

The quantity *structure functions* in the fluid dynamics literature, and that terminology for

The mesoscale-corrected structure function *Ï„* compared to the expected error of ^{2} at

Figure 2b illustrates the quantity *Ï„* ranging from 1 to 48 h. A smooth curve drawn through *y* axis at approximately 8.6 cm^{2}, which is consistent with the sum of the SSH variance attributed to waveform tracker noise in the two missions, plus radial orbit error. The linear rise from 0- to 5-h lag is consistent with a first-order autoregressive process that is hypothesized to be related to the decorrelation of reflectors (sea swell) within the nominal 6-km-radius patch of the ocean surface contributing to the radar return.

*A*modified by (nondimensional) amplitude modulations

*Î±*and

*Ï•*are uncorrelated and stationary Gaussian random variables, it is a straightforward exercise to derive the following expression for the expected value of the structure function,

## 3. Results

Parameters in the above model are determined in a two-stage process. The first stage consists of neglecting the nonstationary tides and determining the

Table 1 presents the linear regression coefficients obtained by ordinary least squares fitting of ^{2}.

The

A comparison of the values in Table 1 with the ^{2}, which corresponds to 1 cm^{2} of tidal variance. The sum of the model coefficients associated with the semidiurnal band is 0.73 cm^{2} (Table 1, ^{2} variance is associated with the nonstationary semidiurnal tide. Note that this estimate is very sensitive to the amplitude of the semidiurnal oscillation inferred from Fig. 2b; a more precise estimate is made below.

Figure 3 illustrates the *Ï„* values covering three consecutive springâ€“neap cycles. For clarity, both sets of curves have been high-pass filtered for presentation purposes (48-h half-power point). Both

The power spectral density of

The time domain analysis cannot distinguish between barotropic and baroclinic tidal signals in SSH, and it is tacitly assumed that the tidal correction by GOT4.10 removes the majority of barotropic tidal variance from the SSH. The semidiurnal variance estimated from short lags (1 cm^{2}) is definitely larger than the GOT4.10 error variance in deep water (Stammer et al. 2014), and it is almost certainly dominated by the internal tide. It is likely that the majority of the variance associated with the minor tides,

Next, characteristics of the nonstationary tide are identified by examining the semidiurnal band variance in the high-passed residual ^{2}, slightly larger than the estimate obtained above by comparing Fig. 2b and Table 1, which is about 30% of the total (coherent plus incoherent) semidiurnal variance. The demodulate, which is an estimate of ^{âˆ’1} (Chelton and Schlax 1996) would take about 700 h to cross a 150-km-wide internal tide beam. Unfortunately, the signal-to-noise ratio of the

## 4. Discussion

The demodulates shown in Fig. 5 indicate that the nonstationary internal tide rapidly decorrelates as it propagates through the deep ocean. Based on the present analysis in the time domain, approximately 30% of the semidiurnal internal tide SSH variance is nonstationary. This result is somewhat higher than estimated by other methods (Dushaw et al. 1995; Chiswell 2002; Ray and Zaron 2011; Dushaw et al. 2011), although the present work is apparently the first assessment of a near-global average.

One attribute of the internal tide highlighted in previous observational and modeling studies is the spatial inhomogeneity of the associated SSH field (e.g., Simmons et al. 2004). The present analysis has relied exclusively on second-order statistics, that is, an analysis of variance, which, by definition, gives higher weight to those regions with larger SSH amplitude. To examine the possibility of bias, a parallel analysis was conducted using a first-order statistic analogous to

Another aspect of spatial variability is connected with the dynamics of internal tide generation and propagation. For example, the diurnal tides are not expected to propagate poleward of the turning latitude where the waves are subinertial (e.g., approximately 30Â° for _{1} variance is 0.08 cm^{2} poleward of 30Â° and 0.23 cm^{2} equatorward of 30Â° (cf. Table 1). Thus, the statistical approach taken here broadly agrees with our understanding of tidal dynamics. Taking this approach further to develop a spatially resolved picture is difficult, though, as it involves a trade-off between sampling errors, which increase as the number of data points is reduced by subsetting and spatial resolution. This is an area of ongoing efforts.

Model-based studies of the generation and propagation of nonstationary tides have been conducted previously. A result that is directly comparable with the present work is the study of Zaron and Egbert (2014), in which the stationary and nonstationary internal tide variance was mapped near Hawaii. They found that 20% of the

The temporal structure function presented in Fig. 2 is a summary of oceanic variability that complements previous analyses in the (spatial) wavenumber domain (Fu 1983; LeTraon et al. 1990; Stammer 1997; LeTraon et al. 2008; Xu and Fu 2011). It shows that the present altimeter constellation is capable of mapping the ocean mesoscales at time scales greater than 150 h. It also indicates that about 10 cm^{2} of SSH variance remains to be mapped at shorter time scales. This sub-6-day variance may be partitioned into 2.5 cm^{2} of very high-frequency variability (5-h time scale and shorter), 2.5 cm^{2} of tidal variance, and about 5 cm^{2} of other SSH variance, which includes unresolved mesoscale and submesoscale variability, topographic Rossby and Kelvin waves, and other gravity waves (note that these SSH variance estimates correspond to Â½ of the structure function values shown in Fig. 2).

## 5. Conclusions

J2 and C2 crossover data have been used to compute the second-order structure function of SSH for data collocated in space but separated in time. The structure function partitions the SSH variance according to time scale in a manner analogous to the frequency power spectrum, but without the need for evenly spaced time series. For analysis of internal tidal signals, precision was enhanced by subtracting off the known SSH signal (the barotropic tide and the slowly evolving mesoscale) and averaging in space, between Â±55Â° latitude, and time, from 1 January 2010 to 30 May 2014.

The tidal variance present in the structure function was used to compute, for the first time, near-global averages of the amplitudes of the stationary and nonstationary internal tides. Within the semidiurnal band, for which highly accurate prior models are available to remove the barotropic tide, about 1-cm^{2} SSH variance is found, the majority of which may be attributed to the internal tide. For the dominant ^{2} is found in the structure function associated with the

The results obtained also suggest a strategy for predicting the nonstationary low-mode internal tide, as will be needed for detiding data from the future SWOT wide-swath altimeter mission. The temporal filtering scale of the mesoscale signal in the Ssaltoâ€“Duacs (AVISO) gridded SSH maps is approximately 150 h, and this is significantly shorter than the 400 h decorrelation time of the nonstationary semidiurnal tide. Because a significant fraction of the time-variable mesoscale signal is resolved at time scales of shorter than the decorrelation time of the tide, it may be feasible to resolve much of the internal tide variability in a data-assimilative mesoscale-resolving model, provided the model is capable of producing the correct phase-speed modulations associated with the mesoscale SSH. Once demonstrated, such a capability would enhance the efficacy of tidal corrections in the future.

## Acknowledgments

Satellite altimeter data used in this study were extracted from the RADAR Altimetry Database System (RADS; http://rads.tudelft.nl/rads/rads.shtml). The mesoscale SSH fields,

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