Editorial Type:
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Article Type:
Research Article

Seasonal Mode-1 M2 Internal Tides from Satellite Altimetry

Zhongxiang Zhao aApplied Physics Laboratory, University of Washington, Seattle, Washington

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https://orcid.org/0000-0002-5897-089X
Online Publication:
07 Sep 2021
Print Publication:
01 Sep 2021
DOI:
https://doi.org/10.1175/JPO-D-21-0001.1
Page(s):
3015–3035
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Abstract

The seasonal variability of mode-1 M2 internal tides is investigated using 25 years of multisatellite altimeter data from 1992 to 2017. Four seasonal internal tide models are constructed using seasonally subsetted altimeter data and World Ocean Atlas seasonal climatologies. This work is made possible by a newly developed mapping procedure that can significantly suppress model errors. Seasonal-mean and seasonally variable internal tide models are derived from the four seasonal models. All of the models are intercompared and evaluated using independent CryoSat-2 data. The seasonal-mean model is overall the best model because averaging the four seasonal models further reduces model errors. The seasonally variable models are better in the tropical zone, where large seasonal signals may overcome model errors. Each seasonal model works best in its own season and worst in its opposite season. These internal tide models reveal that mode-1 M2 internal tides are subject to significant seasonal variability and that their seasonal variations are a function of location. Large seasonal variations dominantly occur in the tropical zone, where the World Ocean Atlas climatology shows strong seasonal variations in ocean stratification. Seasonal phase variations are obtained from the directionally decomposed internal tide components. They are dominantly ±60° at the equator and up to ±120° in the central Arabian Sea. Incoherence caused by seasonal phase variations is usually less than 10% but may be up to 40%–50% in the tropical zone.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Zhongxiang Zhao, zzhao@apl.uw.edu

Abstract

The seasonal variability of mode-1 M2 internal tides is investigated using 25 years of multisatellite altimeter data from 1992 to 2017. Four seasonal internal tide models are constructed using seasonally subsetted altimeter data and World Ocean Atlas seasonal climatologies. This work is made possible by a newly developed mapping procedure that can significantly suppress model errors. Seasonal-mean and seasonally variable internal tide models are derived from the four seasonal models. All of the models are intercompared and evaluated using independent CryoSat-2 data. The seasonal-mean model is overall the best model because averaging the four seasonal models further reduces model errors. The seasonally variable models are better in the tropical zone, where large seasonal signals may overcome model errors. Each seasonal model works best in its own season and worst in its opposite season. These internal tide models reveal that mode-1 M2 internal tides are subject to significant seasonal variability and that their seasonal variations are a function of location. Large seasonal variations dominantly occur in the tropical zone, where the World Ocean Atlas climatology shows strong seasonal variations in ocean stratification. Seasonal phase variations are obtained from the directionally decomposed internal tide components. They are dominantly ±60° at the equator and up to ±120° in the central Arabian Sea. Incoherence caused by seasonal phase variations is usually less than 10% but may be up to 40%–50% in the tropical zone.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Zhongxiang Zhao, zzhao@apl.uw.edu
Keywords: Ocean; Internal waves; Tides; Altimetry; Seasonal cycle; Satellite observations; Sea level; Topographic effects

1. Introduction

Internal tides are generated by barotropic tidal currents flowing over varying topography in the stratified ocean, and propagate over long distances from their generation sites (Wunsch 1975; Ray and Mitchum 1996; Egbert and Ray 2000; Garrett and Kunze 2007; Zhao et al. 2016). Leaving their generation sites, internal tides dephase with astronomical potential forcing in propagation. They are widespread in the global ocean and provide about half of the mechanical power required for ocean interior mixing (Munk 1966; Munk and Wunsch 1998; Jayne and St. Laurent 2001; MacKinnon et al. 2017; Vic et al. 2019; de Lavergne et al. 2020; Whalen et al. 2020). Internal tides owe their existence to ocean stratification; therefore, their generation and propagation are controlled by time-varying ocean stratification. Satellite and field measurements show that internal tides are subject to multiscale temporal variations ranging from days to decades (Mitchum and Chiswell 2000; Chiswell 2002; Alford 2003; Colosi and Munk 2006; Ray and Zaron 2011; Nash et al. 2012; Kelly et al. 2015; Zaron 2015; Zhao 2016a,b; Löb et al. 2020). A fraction of internal tides remains phase-locked, making them detectable by satellite altimetry (Ray and Mitchum 1996, 1997). Satellite altimetry is one unique technique for observing the global internal tide field from space. Thanks to satellite altimeter data accumulated over the past more than two decades, our knowledge of internal tides has substantially progressed, evidenced by a few reliable global internal tide models constructed using the altimeter data (Carrere et al. 2021). The altimeter-derived models only account for the phase-locked coherent (stationary) component (Dushaw 2015; Ray and Zaron 2016; Zhao et al. 2016; Zaron 2019; Zhao 2019). Our understanding of the multiscale temporal modulations on internal tides is still poor; nevertheless, this knowledge is important not only to quantify their incoherent (nonstationary) component, but also to study their dissipation mechanisms and interactions with other ocean motions. As the first step of addressing this issue, this paper focuses on the dominant seasonal cycle of mode-1 M2 internal tides.

The seasonal variations of internal tides have long been shown by internal tide theory and simulated by numerical models. Since their generation and propagation are largely determined by ocean stratification, internal tides are inevitably modulated by the seasonally variable ocean stratification (Munk 1981; Garrett and Kunze 2007; Wunsch 2013). The seasonal variations have been confirmed by contrasting numerical simulations using different stratification profiles. For example, Gerkema et al. (2004) simulate the effect of seasonal thermocline on the generation and propagation of internal tides in the Bay of Biscay. Jan et al. (2008) simulate seasonal internal tides around the Luzon Strait using two stratification profiles taken from summer and winter climatologies, respectively. Recent years have seen more and more studies following this technical strategy (e.g., Osborne et al. 2011; Hall et al. 2014; Jeon et al. 2014; Liu et al. 2019; Song and Chen 2020; Yan et al. 2020). On the other hand, high-resolution global numerical models can simulate large-scale ocean circulation, mesoscale eddies, and internal tides simultaneously (Arbic et al. 2004, 2010; Müller et al. 2012; Li and von Storch 2020). These models are driven by astronomical tidal potential and atmospheric forcing, so they can generate time-varying ocean stratification. Based on global numerical simulations, the seasonal variations of internal tides have been successfully achieved (Müller et al. 2012; Buijsman et al. 2017). Yet, these regional and global numerical models cannot fully reproduce the seasonally variable ocean stratification, as evidenced by significant discrepancies in the propagation speed of internal tides when compared with satellite observations. In this regard, the satellite results presented in this paper may offer remote sensing observations for calibrating numerical models and improving parameterization schemes.

Field mooring measurements have shown significant seasonal variations of internal tides (Nash et al. 2012; Zhao et al. 2012; Kelly et al. 2015; Liu et al. 2015; Shang et al. 2015; Zhou et al. 2015; Cao et al. 2017). Nash et al. (2012) investigate a number of moorings on continental slopes and report strong seasonal variability of internal tides. Zhou et al. (2015) analyze 15 years of mooring measured upper-ocean hydrographic profiles to the north of the Solomon Strait and find remarkable seasonal variations. On the contrary, Vic et al. (2021) analyze an array of seven moorings over the Reykjanes Ridge and find no obvious seasonal variations in the kinetic energy of the semidiurnal internal tides. Altogether, the mooring measurements reveal that the seasonal variations of internal tides are a function of location, which makes it even challenging to quantify their global variations by a limited number of field moorings. This knowledge gap can be partially filled by satellite altimetric observations.

The seasonal variations of internal tides have been previously reported by satellite altimetry (Ray and Zaron 2011; Zaron 2019). Ray and Zaron (2011) divide 17 years of TOPEX/Poseidon–Jason altimeter data into winter and summer subsets and extract along-track M2 internal tide harmonic constants. They find outstanding phase variations in the South China Sea and the Amazon River plume. Zaron (2019) explicitly determines two annual constituents MA2 and MB2, whose combination represents the annual modulation on M2, from 25 years of satellite altimeter data. Zaron (2019) confirms that the South China Sea and the Amazon River plume have strong seasonal variations, and further identifies more notable regions including the western Pacific and the Arabian Sea. However, neither of these two studies reports seasonal variations in regions of weak internal tides. One likely reason is that the seasonal variations of internal tides in most of the global ocean are very weak so that the seasonal signals are overwhelmed by model errors in the satellite-derived internal tides. Note that all the above recognized regions have strong internal tides, whose seasonal variations are assumed to be greater than model errors. Therefore, constructing low-noise-level internal tide models is one key step for exploring their seasonal variability.

This paper explores the seasonality of global mode-1 M2 internal tides from satellite altimetry using a method different from those in Ray and Zaron (2011) and Zaron (2019). To suppress errors, this paper employs a new mapping technique comprising multiple spatiotemporal filtering (Zhao 2019). The resultant internal tide models have smaller errors, making it possible to detect weak seasonal signals of internal tides in most of the global ocean. The new mapping technique has the following advantages. First, it explicitly constructs four seasonal internal tide models using seasonally subsetted altimeter data and seasonally varying dynamic parameters (e.g., wavelength). Second, it decomposes each internal tide model into multiple components by propagation direction. The new internal tide models are useful to detect the spatiotemporal variations in internal tide’s amplitude and propagation speed. Among them, the latter conveys important information on stratification structure and ocean heat content (Zhao 2016a,b).

The rest of this paper is arranged as follows. Section 2 describes the seasonal variations in ocean stratification and internal tide dynamic parameters. Section 3 describes the satellite altimeter data and mapping techniques used in this study. Section 4 presents the satellite-derived M2 internal tide models and conducts model comparisons. Section 5 evaluates the resultant internal tide models using independent satellite altimeter data. Section 6 examines the seasonal variations of M2 internal tides in the global ocean. Section 7 contains a summary. Limitations of this work and future improvements are discussed in section 8.

2. Seasonal variations of ocean stratification

The seasonal variations of ocean stratification and internal tide wavelength are demonstrated in this section. The ocean stratification data are from hydrographic profiles in the World Ocean Atlas 2013 (WOA2013) (Locarnini et al. 2013; Zweng et al. 2013). WOA2013 provides seasonal and seasonal-mean hydrographic profiles at a grid of 0.25° by 0.25° in the global ocean. The seasons are divided as follows: winter consists of January–March, spring consists of April–June, summer consists of July–September, and fall consists of October–December. Ocean depth is extracted from the 1-arc-min topography database developed using in situ and satellite measurements (Smith and Sandwell 1997). Internal tides in the stratified ocean can be described by a set of orthogonal modes, each of which has its characteristic vertical modal structure and freely propagates in the horizontal direction like a wave in a homogeneous fluid. The modal structures Φn(z) and eigenvalue speeds Cne are determined by the Sturm–Liouville equation (Munk 1981; Gill 1982)
d2Φn⁡(z)dz2+N2⁡(z)Cne2=0,
subject to rigid boundary conditions Φn(0) = Φn(−H) = 0, where N(z) is stratification profile, H is ocean depth, and n is mode number. Assuming a free-surface condition yields similar modal structure and a little faster speed Cne (Kelly 2016). With Earth’s rotation, phase speeds Cnp differ from eigenvalue speeds and can be derived utilizing the dispersion relation following
Cnp=ωω2−f2Cne,
where ω and f are the tidal and inertial frequencies, respectively. The differences between phase speed and group speed are explained in Zhao (2017b). From phase speeds and the tidal frequency, it is straightforward to calculate wavelengths following
λ=Cnpω=Cneω2−f2.
The wavelengths of M2 internal tides are a function of ocean stratification, ocean depth, latitude, and mode number. The wavelength (or wavenumber) is one prerequisite parameter for mapping mode-1 M2 internal tides by plane wave analysis (section 3).

The global maps of the seasonal and seasonal-mean wavelengths are calculated (Fig. 1). The results are consistent with those obtained by R. Ray and E. Zaron using the same method and shown in one presentation entitled “Variability of internal tides: A view from altimetry” at the 2017 Surface Water Ocean Topography (SWOT) meeting. Figures 1a and 1b show that the seasonal-mean and winter wavelengths are very close. The spring, summer and fall maps have similar patterns (not shown). To highlight the seasonal variations, their wavelength anomalies are calculated and shown in Figs. 1c–1f. The seasonal anomalies here are defined to be the differences between the seasonal wavelengths and the seasonal-mean wavelengths. The results reveal that strong seasonal variations mainly occur in the tropical zone. At middle and high latitudes, the seasonal variations are relatively weak but robust over large spatial scales. A notable feature is the opposite patterns between the southern and northern hemispheres. Internal tide wavelengths from WOA2013 and WOA2018 are almost the same (Fig. S1 in the online supplemental material); therefore, the global internal tides would not be affected if the WOA2018 climatology is employed in this paper.

Fig. 1.
Fig. 1.

Seasonal variations of wavelengths of mode-1 M2 internal tides. Wavelengths are calculated from WOA2013 seasonal and seasonal-mean hydrographic profiles. Seasonal anomalies denote the differences between the seasonal and seasonal-mean wavelengths. Shown are the (a) seasonal-mean wavelength, (b) winter wavelength, (c) winter anomaly, (d) spring anomaly, (e) summer anomaly, and (f) fall anomaly.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

3. Data and methods

a. Satellite altimeter data

The sea surface height (SSH) data used in this study are from seven exact-repeat-track satellite altimeter missions (Fig. S2 in the online supplemental material). They are TOPEX/Poseidon (TP), Jason-1, Jason-2, Jason-3, European Remote Sensing Satellite-2 (ERS-2), Environmental Satellite (Envisat), and Geosat Follow-On (GFO). Their SSH measurements are merged into four datasets according to their orbital configurations, and are labeled as TPJ (TP-Jason), TPT (TP-Jason tandem), ERS, and GFO, respectively. There are 1002, 488, 254, and 254 tracks for ERS, GFO, TPJ, and TPT, respectively. Each dataset has wide-spaced ground tracks, making it difficult to extract internal tides, because wavelengths of mode-1 M2 internal tides range from 100 to 200 km. The merged data have denser ground tracks and higher spatial resolution, which are important for building internal tide models. These four datasets have different spatial and temporal sampling patterns. All SSH measurements have been processed by applying standard corrections for atmospheric effects, surface wave biases, and geophysical effects. The barotropic tide and loading tide are corrected using the global ocean tide model GOT4.8 (Ray 2013). In previous studies, the same data have been used to study global mode-1 M2 internal tides (Zhao 2019) and mode-2 M2 internal tides (Zhao 2018).

b. Seasonal data subsetting

The 25 years of satellite altimeter data are divided into four seasonal subsets to construct seasonal internal tide models. They are divided following the convention employed in the World Ocean Atlas (WOA) products (section 2). It is thus convenient to match the satellite results with the WOA climatological parameters without causing confusion. Each seasonal subset is 3-month-long, 25% of the original all-year all-month data.

To compensate for the short seasonal data records, this paper employs fitting windows of 250 km by 250 km, in contrast to 160 km by 160 km in Zhao (2019). The larger fitting windows may slightly underestimate the amplitude of internal tides but do not affect the phase of internal tides or their seasonal variability. The local wavenumber of mode-1 M2 internal tides is a prerequisite parameter in constructing internal tide models. In this study, the seasonal wavenumbers are from ocean stratification profiles in the WOA2013 seasonal climatologies (section 2). The seasonal wavenumbers are slightly different from the annual-mean values (Fig. 1). The four seasonal internal tide models are constructed using their respective seasonal wavenumbers. The seasonal models are also constructed using the seasonal-mean wavenumbers, and the results are almost the same as those using the seasonal wavenumbers (Fig. S3 in the online supplemental material).

c. Two key techniques

In this study, mode-1 M2 internal tides are extracted using the same mapping procedure presented in Zhao (2019). This procedure is a combination of along-track spatial filtering, harmonic analysis, plane wave analysis, and 2D spatial filtering. The principles underlying these techniques are (i) mode-1 M2 internal tides are quasi-periodic motions driven by astronomical potential in the sun–Earth–moon system, so they can be extracted using their known tidal frequency (Doodson 1921), and (ii) mode-1 M2 internal tides are overall stationary, so they can be extracted using theoretical wavenumbers determined by ocean stratification. These two principles imply that only the stationary internal tides in one given period can be extracted, missing the incoherent component. Two key techniques, plane wave analysis and 2D spatial filtering, are described below.

1) Plane wave analysis

Plane wave analysis is developed for mapping internal tides from satellite altimetry (Zhao 2016b, 2017a, 2018). It extracts internal tides from satellite altimeter data by fitting plane waves using all measurements in one given fitting window, in contrast to pointwise harmonic analysis. The amplitude a, phase ϕ, and propagation direction θ of target internal tidal waves are determined by least squares fits following
η⁡(x,y,t)=∑m=1Mam cos⁡(kx cosθm+ky sinθm−ωt−ϕm),
where ω and k are the frequency and wavenumber of the target waves, x and y are the local Cartesian coordinates, and t is time. The seasonal wavenumbers are used for the four seasons, respectively (section 2). An iterative algorithm has been developed to extract an arbitrary number of waves. In this study, five waves are extracted in each fitting window. Plane wave analysis has the following advantages: First, it can map internal tidal waves using satellite data along widely spaced tracks; Second, nontidal noise can be significantly reduced by utilizing a large number of independent data; Third, internal tidal waves of different propagation directions can be separately resolved.

2) 2D spatial filtering

The M2 internal tides are mapped at spatially regular grids by plane wave analysis from satellite data along discrete ground tracks. The 2D wavenumber spectrum of the internal tide field reveals that the variance is mainly around the theoretical wavenumber (Zhao 2019). The variance falling outside the theoretical wavenumber range is due to errors and higher baroclinic modes. Taking advantage of the spatially regular field, M2 internal tides can be further filtered by removing components that do not meet the wavenumber requirement. First, the internal tide field is converted to the 2D wavenumber spectrum S(k) by Fourier transform. Second, S(k) is truncated using the local theoretical wavenumber K0 following
S˜⁡(k)={0,k<0.8K0S⁡(k),1.25K0>k≥0.8K0.0,k≥1.25K0
Third, S˜⁡(k) is converted to the internal tide field by inverse Fourier transform. The width [0.8 1.25] of the bandpass filter is empirically determined after testing several different values. It is chosen to be as narrow as possible without filtering out real internal tide signals.

d. Mapping procedure

This study employs a five-step mapping procedure presented in Zhao (2019). The five steps, their goals, and key parameters are summarized in Table 2.

  • Step 1 is along-track spatial filtering, in which the filter is used to remove large-scale errors such as geodetic residual and barotropic tide residual. This study uses a fourth-order Butterworth high-pass filter with cutoff wavelength of 2000 km (Zhao 2019).

  • Step 2 is pointwise harmonic analysis, in which M2 internal tides are obtained by harmonic analysis of along-track filtered data. A fraction of mesoscale variance is leaked into the internal tide field as a result of tidal aliasing; therefore, the harmonically fitted M2 internal tides are noisy. The harmonic constants are used to reconstruct time series at every along-track point. The reconstructed data are used for mapping internal tides in the following steps.

  • Step 3 involves mapping internal tides by plane wave analysis, in which M2 internal tides are determined by fitting plane waves in numerous overlapping windows of 250 km by 250 km using the reconstructed data in step 2. In this step, internal tidal waves are mapped from discrete satellite tracks to regular spatial fields. The five largest mode-1 M2 internal tidal waves are determined at each grid point, and their sum gives the M2 internal tide solution at the point. In this study, the resultant internal tides are on a regular grid of 0.2° longitude by 0.2° latitude.

  • Step 4 is 2D spatial filtering, in which a horizontal 2D bandpass filter is employed to further remove nontidal noise and higher baroclinic modes. It constrains internal tides by requiring them to match theoretical wavelengths. Over the global ocean, the filter is applied to a number of overlapping windows of 850 km by 850 km. Seasonal wavenumbers are used in this step again (Table 2; Fig. 1).

  • Step 5 is multidirectional decomposition by plane wave analysis; here, the goal is to separately resolve internal tidal waves in different propagation directions. Plane wave analysis is called again to decompose the internal tide field. In this step, plane wave analysis is applied to the 2D filtered internal tide field obtained in step 4. The decomposition is conducted using the same parameters as in step 3 except that the fitting window is 160 km by 160 km. The resultant five waves are saved separately with their amplitudes, phases, and propagation directions.

e. Demonstration of intermediate processes

The mapping procedure of the winter internal tide model is given in Fig. 2. This example shows how the SSH variance decreases with multiple spatial and temporal filters. Other seasons show similar intermediate mapping results (not shown). Figures 2a–2d show the global maps of SSH variances in steps 1–4, respectively. Their global mean variances are calculated and given in the figure.

Fig. 2.
Fig. 2.

Mapping mode-1 M2 internal tides from satellite altimetry (the winter model is shown as an example): variance of the (a) raw satellite altimeter data. (b) preprocessed satellite altimeter data by along-track high-pass filtering (step 1) and harmonic analysis (step 2), (c) M2 internal tides obtained by plane wave analysis (step 3), and (d) M2 internal tides after 2D spatial filtering (step 4). In (a)–(d), SSH variances are spatially averaged in 250-km by 250-km windows and shown in logarithmic scale. Also shown are snapshot M2 internal tide fields (e) obtained by plane wave analysis (step 3) and (f) after 2D spatial filtering (step 4). In (a)–(f), the black contours indicate regions of strong currents. The numbers in (a)–(d) are the global mean variances (excluding regions of strong currents).

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

Figure 2a shows the variance map of the raw altimeter data. Its mean is 1611 mm2, because it contains variance of many ocean dynamic processes. Figure 2b shows the variance map of the preprocessed altimeter data. The raw altimeter data are bandpass filtered (step 1) and harmonic analyzed (step 2). Its mean becomes 162 mm2. It means that steps 1 and 2 combined remove about 90% of the raw variance, because most of the variance is irrelevant to internal tides and can be removed by high-pass filtering and harmonic analysis. In step 3, mode-1 M2 internal tides are extracted by plane wave analysis. Figure 2c shows the variance map of M2 internal tides. The mean variance is reduced to 33 mm2. In step 4, the M2 internal tide field is cleaned by 2D spatial filtering. Figure 2d shows the variance map. The mean variance now is 19 mm2, about 1.2% of the raw variance. This example clearly shows that the variance constantly decreases by multiple rounds of spatial and temporal filtering (Figs. 2a–d). Meanwhile, the weak internal tide signals become clean and clear. Figures 2e and 2f show the internal tide models obtained in steps 3 and 4, respectively. The latter is much cleaner than the former, consistent with the variance decrease. The decomposed multidirectional internal tide field in step 5 is not shown here, because this step only decomposes the internal tide field, but barely reduces noise.

This example indicates that mode-1 M2 internal tides account for about 1.2% of the raw SSH variance. This fact explains why it is difficult to extract internal tides from satellite altimeter data. Internal tides are usually overwhelmed by nontidal components (model errors), which dominate the raw SSH variance. Mode-1 M2 internal tides can be extracted and cleaned in multiple rounds of filtering by taking advantage of their known wavenumber and frequency. The present short altimeter data are still the main limiting factor, which will be ameliorated with the accumulation of SSH measurements from multiple altimeter missions. The new mapping procedure presented in this paper works well in most of the global ocean, yet it cannot extract internal tides in regions of strong boundary currents such as the Gulf Stream, the Kuroshio Extension region, the Leeuwin Current, the East Australia Current, the Antarctic Circumpolar Current, and the Malvinas Current. Dedicated techniques and/or carefully chosen parameters are needed to address the extremely challenging regions. These regions are indicated by black contours in Fig. 2 and are excluded from the statistical analyses in this paper.

4. Global M2 internal tide models

a. Four seasonal models

Following the mapping procedure described in section 3, four seasonal internal tide models are constructed using the seasonally subsetted altimeter data. The four models are labeled as SN1, SN2, SN3, and SN4 (Table 1). For convenience, they are also called the winter, spring, summer and fall models, respectively. Figures 3a–3d give the global maps of their SSH amplitudes. Here the model amplitudes are plotted in logarithmic scale to better show some low values. Although the SSH amplitudes are as low as a few mm, they are pretty robust as a result of multiple filters in time and space. Individual internal tidal beams can be clearly seen throughout the ocean. In some cases, these beams propagate across one another. The most remarkable feature is that the four seasonal models have almost identical spatial patterns. The four seasonal models have large errors due to short seasonally subsetted satellite data. In some regions, the model errors may be larger than the seasonal signals; in other regions, the seasonal signals are larger. This feature will be further explored in the following sections.

Table 1.

Eight M2 internal tide models from satellite altimetry. The first five models are constructed directly from different altimeter data subsets by the same mapping technique. The last three models are derived from the four seasonal models.

Table 1.
Fig. 3.
Fig. 3.

The M2 internal tide models from satellite altimetry (the SSH amplitudes are shown in logarithmic scale) for (a) winter, (b) spring, (c) summer, (d) fall, (e) the seasonal-mean model (vector mean of four seasonal models), and (f) the 25-yr-coherent model. The black contours indicate regions of strong boundary currents, where all models are not reliable.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

b. Two time-mean models

The seasonal-mean internal tide model is derived from the four seasonal models. At each grid point, its amplitude and phase are from the vector mean of internal tides of the four seasonal models following
SN0⁡(a0,ϕ0)=a0eiϕ0=14∑n=14aneiϕn,
where an and ϕn are the amplitude and phase of the nth seasonal model. The seasonal-mean internal tide model is labeled as SN0 (Table 1) and shown in Fig. 3e.

The 25-yr-coherent internal tide model is directly constructed using all the satellite altimeter data following the same mapping procedure (Table 1). This model is labeled as Zhao21. Note that Zhao21 in this study and Zhao19b in Zhao (2019) are developed using the same data and the same mapping parameters except that Zhao21 is determined in 250-km by 250-km windows and Zhao19b is determined in 160-km by 160-km windows (Table 2). Zhao21 has slightly lower amplitudes. The dependence of the amplitude of internal tides on the size of fitting windows will be reported elsewhere. The 25-yr-coherent model is shown in Fig. 3f. Comparisons show that SN0 and Zhao21 are almost the same, suggesting that these two methods are equivalent (section 4f).

Table 2.

The five-step mapping procedure. The operation, goal, and parameters in each step are listed. A prerequisite parameter, wavenumber K0, is needed in steps 3–5. Seasonal wavenumbers are computed from the WOA2013 seasonal hydrographic climatologies.

Table 2.

c. Two seasonally variable models

To describe the seasonal variations of M2 internal tides, two seasonally variable internal tide models are derived from the four seasonal models. They are labeled SNa and SNb, respectively (Table 1). SNa is a step function of the four seasonal models:
SNa⁡(t)={SN1⁡(a1,ϕ1),t∈ winterSN2⁡(a2,ϕ2),t∈ springSN3⁡(a3,ϕ3),t∈ summerSN4⁡(a4,ϕ4),t∈ fall,
where an and ϕn represent the amplitude and phase in the nth seasonal model. SNa means that the internal tide field in each season is described using the respective seasonal model. For comparison, SNb is similar to SNa but has seasonally invariable amplitudes:
SNb⁡(t)={SN1⁡(a0,ϕ1),t∈ winterSN2⁡(a0,ϕ2),t∈ springSN3⁡(a0,ϕ3),t∈ summerSN4⁡(a0,ϕ4),t∈ fall,
where ϕn represents the phase in the nth seasonal model, and a0 is taken from the seasonal-mean model SN0. SNa is a phase-variable, amplitude-variable model; while SNb is a phase-variable, amplitude-invariable model. The seasonal-mean model SN0 can be taken as a phase-invariable, amplitude-invariable model. Because both the amplitude and phase of M2 internal tides vary seasonally, their seasonal variations and respective contributions can be examined by comparing these models. These models will be evaluated in section 5, and the seasonal variations will be studied in section 6.

d. Basic features

These internal tide models, four seasonal models and two time-mean models, have similar spatial patterns (Fig. 3). They all show that strong internal tides are associated with major topographic features in the open ocean such as the Hawaiian Ridge, the French Polynesian Ridge, and the Mariana Ridge. Internal tides are also generated on steep continental slopes such as India’s southwestern continental slope and the Amazon River plume off Brazil, and in straits such as the Luzon Strait and the Lombok Strait. Internal tides are observed to travel hundreds to thousands of kilometers. These features have been reported in previous studies by satellite altimetry (Ray and Zaron 2016; Zhao et al. 2016; Zaron 2019; Zhao 2019). This paper further reveals that the seasonal models have similar spatial features, suggesting that the global internal tide field is overall stationary around the year. Comparisons of the seasonal models further discover weak but significant seasonal cycles in the global ocean.

e. Model errors

Model errors are indicated by the 95% confidence levels calculated in MATLAB’s built-in function regress using the standard formula for Gaussian distributed data. Figure 4a shows the global map of the amplitude errors of the 25-yr-coherent model. It shows that the amplitude errors are a function of location, which mimics the spatial pattern of variance of the raw altimeter data (Fig. 2). Overall, the amplitude errors are lower than 1 mm (Fig. S4 in the online supplemental material). Because there are O(104) SSH measurements in each fitting window, the resultant small-amplitude errors make it possible to detect weak internal tides and their seasonal variations (Fig. 3).

Fig. 4.
Fig. 4.

Model errors: (a),(b) amplitude errors and (c),(d) phase errors in the (left) 25-yr-coherent model and (right) winter model. Different color map ranges are used for better comparisons. For amplitude, they differ by a factor of 2. For phase, they differ by a shift of 3°.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

The four seasonal models have the same error levels, because they are from evenly subsetted seasonal data records. The amplitude errors generally decrease with the increasing number of measurements involved in plane wave analysis following σ = σm/N1/2, where σm is the measurement errors and N is the number of independent measurements. Because the seasonal subsets are about one quarter of the original whole data, it is derived that the errors in seasonal models are about 2 times those in the 25-yr-coherent model. Figure 4b shows the amplitude errors of the winter model. Figures 4a and 4b are plotted using the same color map, but the ranges are different by a factor of 2. It shows that Fig. 4b has the same spatial pattern as Fig. 4a, but the errors are about 2 times as large (Fig. S4).

The M2 internal tides determined by pointwise harmonic analysis have even larger errors, because the time series at one given point has O(102) SSH measurements. Thus, M2 internal tides determined by harmonic analysis have errors that are about 10 times those in the 25-yr-coherent model. The large errors mask weak seasonal signals of internal tides, except for regions of strong internal tides such as the Luzon Strait and the Amazon River plume (Ray and Zaron 2011; Zaron 2019).

The phase errors in the internal tide models are affected by amplitude errors and wave amplitudes (roughly the ratios of amplitude errors to amplitudes). For the same amplitude error, a larger internal tide has a smaller phase error. Figures 4c and 4d give the phase errors in the 25-yr-coherent and winter models, respectively. Shown here are the phase errors associated with the largest (first) waves at each grid point. They have similar spatial patterns that mimic their spatial patterns of model amplitudes (Fig. 3). Small phase errors are associated with large-amplitude internal tides, as explained above. Statistical analysis shows that their phase errors are different by about three degrees. To highlight this feature, Figs. 4c and 4d are shown using the same color map, but their ranges being shifted by three degrees (Fig. S4). One might wonder why their phase errors are different by three degrees, instead of by a factor of 2 (as their amplitude errors). That is because the seasonal models have amplitudes about 27% greater than the 25-yr-coherent model (Fig. S5 in the online supplemental material).

f. Model comparisons

These internal tide models are directly compared. For each pair of models, two-dimensional histograms of amplitudes and phases are shown in logarithmic scale (Fig. S6 in the online supplemental material). Comparisons of the seasonal-mean and 25-yr-coherent models reveal that their root-mean-square (RMS) differences in amplitude and phase are 0.7 mm and 14°, respectively. The good agreement means that the two time-mean models are equivalent, which will be confirmed by model evaluation using independent data (section 5).

Comparisons of the four seasonal models are made for six model pairs. They yield almost the same results. For simplicity, only the comparisons of the winter and summer models are given (Fig. S6). The RMS differences are 2.8 mm for amplitude and 63° for phase. The winter and 25-yr-coherent models are compared (Fig. S6). Their RMS differences (2.2 mm and 42°) are between the above two cases, as expected. Further analysis reveals that the amplitudes in the four seasonal models are about 27% larger than those in the 25-yr-coherent model (Fig. S5). The larger amplitudes should have two sources: (i) large model errors due to short seasonal data records (Fig. 4) and (ii) real seasonal coherent signals instead of 25-yr-long coherent. The latter is the research goal of this paper, which may be smeared by the former.

5. Model evaluations using independent data

a. Evaluation method

The internal tide models are evaluated using independent CryoSat-2 altimeter data, which are not used in the above model construction. The CryoSat-2 data used in this study are exactly nine years long from January 2010 to January 2019 (Fig. S2 in the online supplemental material). The evaluation method has been employed in previous studies (Ray and Zaron 2016; Zaron 2019; Zhao 2016b, 2019; Carrere et al. 2021). For each CryoSat-2 raw SSH measurement of known time and location ηr(λ, ϕ, t), where λ, ϕ, and t are longitude, latitude, and time, respectively, the M2 internal tide signal ηM2(λ, ϕ, t) is predicted using the model under evaluation. The corrected SSH measurement ηc(λ, ϕ, t) is obtained following
ηc⁡(λ,ϕ,t)=ηr⁡(λ,ϕ,t)−ηM2⁡(λ,ϕ,t).
Their variances can be written as ηc2, ηr2 and ηM22, respectively. By making internal tide correction, the variance reduction δ(λ, ϕ, t) is the difference between the raw and corrected variances
δ⁡(λ,ϕ,t)=ηr2⁡(λ,ϕ,t)−ηc2⁡(λ,ϕ,t).
All CryoSat-2 SSH measurements are processed in the same way. In the end, the four quantities (ηc2, ηr2, ηM22, and δ) are binned into half-overlapping 2° by 2° windows on a regular grid of 1° longitude by 1° latitude. The performance of one internal tide model in question is indicated by variance reduction. All internal tide models are evaluated by this method (Figs. S7–S11 in the online supplemental material).

b. Using all CryoSat-2 data

The internal tide models are evaluated using nine years of CryoSat-2 data following the above method (Figs. S7–S11). Figure 5 shows the global maps of their variance reductions. All models cause variance reduction in most of the ocean, indicating the models are overall robust. In particular, large variance reductions are associated with source regions of internal tides such as the Hawaiian Ridge, the Polynesian Ridge, and the northwestern Pacific Ocean. Positive variance reductions can be seen throughout the low-latitude Atlantic Ocean. If there are no model errors, one internal tide model will always reduce variance. However, the satellite-derived internal tide models contain large model errors, which lead to variance increase. When internal tide signals are larger, the model leads to positive variance reduction. When model errors are larger, the model leads to negative variance reduction. Note that the model errors are not uniform, but a function of location (Fig. 4). Regions of strong boundary currents have large model errors and thus negative variance reductions (black contours).

Fig. 5.
Fig. 5.

Variance reductions (binned into windows of 2° longitude by 2° latitude for six internal tide models) obtained by making internal tide correction to 9 years of CryoSat-2 data, shown for (a) winter, (b) spring, (c) summer, (d) fall, (e) the seasonal-mean model, and (f) the 25-yr-coherent model. The global mean variance reductions are given in the inset boxes. The black contours indicate regions of strong boundary currents (excluded from global integrations).

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

To quantify their performance, the global mean variance reduction is calculated. The calculation excludes regions of boundary current, where neither model works well (Fig. 5, black contours). The four seasonal models have mean variance reductions of 13.9, 13.0, 13.5, and 13.3 mm2, respectively. The four-season mean is 13.4 mm2. The result suggests that the four seasonal models have similar performances. The winter model is a little better than the others, suggesting that the winter internal tide field overall is stronger in terms of SSH variance. The two time-mean (seasonal mean and 25-yr coherent) models yield almost the same variance reduction (17.4 mm2). The good agreement, combined with their small RMS differences in pointwise comparisons, suggests that these two models are almost the same.

The seasonal-mean model can reduce 4 mm2 (17.4 − 13.4) or 30% (the ratio of 4 to 13.4) more variance than the four seasonal models. It suggests that the seasonal-mean model is better than the four seasonal models. To examine why they are different, the seasonal-mean model and the four seasonal models are compared in terms of model variance and variance reduction. The mean model variance and variance reduction of the four seasonal models are used (Fig. S12 in the online supplemental material). Their model variance difference and variance reduction difference are of exactly the same magnitude but with opposite signs. The seasonal-mean model has smaller errors as a result of seasonal vector mean, so it has smaller model variance but larger variance reduction.

c. Using CryoSat-2 seasonal subsets

The seasonal variability is studied using seasonal internal tide models and seasonal independent data. The nine years of CryoSat-2 data are divided into four seasonal subsets. Each subset is used to evaluate internal tide models using the same method. Each internal tide model is applied to the four seasonal subsets, respectively, and yields four global variance reduction maps. The resultant global maps are not shown, because they are similar to those in Fig. 5. The global mean variance reductions are calculated and shown in Fig. 6a.

Fig. 6.
Fig. 6.

Global mean variance reductions obtained by making internal tide correction to CryoSat-2 seasonal subsets, shown for (a) four seasonal models [each model reduces variance the most in its own season (filled symbols) and the least in its opposite season] and (b) three seasonally variable models. SNa is by definition composed of four seasonal peaks in (a).

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

Figure 6a has two outstanding features. First, the four seasonal models all have seasonal cycles in variance reduction. Each seasonal model reduces most variance in its own season (filled symbols) and least variance in its opposite season. For example, the winter model reduces most variance for the winter subset (filled blue circle). On the contrary, the summer model reduces most variance for the summer subset (filled pink triangle). This feature confirms the seasonal variations of mode-1 M2 internal tides. Second, for each CryoSat-2 seasonal subset, the best model is its corresponding seasonal model. For example, for the spring subset, the spring model reduces the most variance. Because the spring model is constructed using multisatellite spring data, it is reasonable that it is the best model for the CryoSat-2 spring subset.

Following the same procedure, internal tide models SNa, SNb and SN0 are also evaluated using the CryoSat-2 seasonal subsets (Fig. 6b). By definition, SNa is composed of the four seasonal peaks shown in Fig. 6a (filled symbols). Figure 6b reveals that the three models rank in the order of SN0 > SNb > SNa in terms of variance reduction. This feature can be explained using model errors. The four seasonal models have large errors, which can be suppressed by seasonal vector mean. The seasonal-mean amplitudes in SNb have smaller errors than SNa amplitudes; therefore, SNb reduces more variance than SNa. Similarly, the seasonal-mean phases in SN0 have smaller errors than SNb phases; therefore, SN0 reduces most variance. This feature underscores the importance of suppressing model errors in the study of internal tides and their seasonal variations.

d. Spatial features of model performance

Both seasonal signals and model errors are a function of location; therefore, the performance of internal tide models should also be a function of location. To examine the spatial features of the model performance, the global maps of variance reduction differences are studied. Because the four seasonal models are similar, the four-season mean values are calculated and used in the comparison. Figures 7a and 7b show the differences of SNa and SNb with respect to SN1–SN4. Figures 7c and 7d show the differences of SNa and SNb with respect to SN0. The global mean values are given for all panels.

Fig. 7.
Fig. 7.

Global maps of variance reduction differences. The 4-season average denotes the mean variance reduction of the four seasonal models SN1–SN4. The numbers in the insets are the global mean variance reduction differences (regions of strong boundary currents are excluded). Shown are (a) SNa minus 4-seasonal average, (b) SNb minus 4-seasonal average, (c) SNa minus SN0, (d) and SNb minus SN0.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

Figures 7a and 7b show that both SNa and SNb are better than SN1–SN4. They reduce 1.7 and 3.3 mm2 more variance, respectively, which are about 13% and 25% of the four-season mean of 13.4 mm2. As explained earlier, the seasonal-mean amplitudes in SNb have smaller errors; therefore, SNb reduces 1.6 (3.3 − 1.7) mm2 more variance than SNa. Figure 7b shows that the differences (SNb − SN1–SN4) are uniformly positive (with few exceptional regions), suggesting that SNb is better than the four seasonal models on a global scale.

Figures 7c and 7d show that both SNa and SNb reduce less variance than SN0. The variance reduction differences are −2.2 and −0.7 mm2, respectively. As explained above, SN0 has smaller errors in both amplitude and phase, because it is the vector mean of the four seasonal models. From the numbers in Fig. 7, one can deduce that the amplitude mean increases variance reduction by 1.5 mm2, and the phase mean increases variance reduction by 0.7 mm2.

The variance reduction differences are not uniform but a function of location. Figures 7c and 7d show that both SNa and SNb can reduce more variance than SN0 in the tropical zone. This feature can be better seen in the longitudinally averaged results (Fig. S13 in the online supplemental material). SNa and SNb are better in the tropical zone. In fact, SNa is the best in the tropical zone. It is because the tropical zone has the largest seasonal signals (amplitude and phase). Outside the tropical zone, SN0 works better, because it has smaller model errors. The results suggest that, in the tropical zone, the seasonal signals are stronger than model errors; therefore, the seasonally variable models can reduce more variance. Outside the tropical zone, the seasonal signals are smaller than model errors, so the seasonally variable models reduce less variance.

6. Seasonal variations of M2 internal tides

In this section, the seasonal variations of internal tides are examined by directly comparing the four seasonal models. The amplitude and phase of M2 internal tides are separately studied. In particular, the decomposed multidirectional components allow us to track the seasonal variations along individual internal tidal beams.

a. Seasonal contrasts

From the satellite-derived seasonal models, it is straightforward to examine seasonal differences. Here the four models are divided into two pairs of opposite seasons: winter versus summer, and spring versus fall. For each pair, their amplitudes and phases are compared, respectively. The differences between two models are calculated point by point and smoothed by seven-point running mean in both zonal and meridional directions. The spatial smoothing averages out model errors and small-scale features, highlighting their large-scale spatial patterns. Regions of weak internal tides (amplitude lower than 2 mm) and strong boundary currents are discarded. The resultant global maps of amplitude and phase differences are shown in Fig. 8.

Fig. 8.
Fig. 8.

Seasonal differences of M2 internal tides: (a),(c) winter minus summer and (b),(d) spring minus fall for (top) amplitude and (bottom) phase. The differences are calculated pointwise and are smoothed by 7-point running mean in both the zonal and meridional directions. Internal tides with amplitudes of <2 mm are discarded. Regions of strong boundary currents are excluded.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

Figure 8 reveals significant differences in both amplitude and phase for the two pairs of models. The complex spatial patterns may partly be caused by model errors. For example, large differences in the central Indian Ocean are likely due to model errors, because the internal tides in this region are weak and model errors are larger. Despite model errors, the large-scale features are obvious and robust. For example, large-amplitude differences are observed around the Luzon Strait, in the western Pacific Ocean, the Amazon River plume, and the Arabian Sea. The regions have been recognized in previous satellite investigations by Ray and Zaron (2011) and Zaron (2019). The good agreement suggests that the new methods and results presented in this study are robust. In addition, large-amplitude differences are observed throughout the tropical Atlantic Ocean, in the Bay of Bengal, off northwest Australia, and in the Madagascar–Mascarene region.

The seasonal phase differences can be studied as well. Figures 8c and 8d show that large phase differences predominantly occur in the tropical zone, consistent with strong seasonal variations in propagation speed (Fig. 1). Both Figs. 1 and 8 show opposite seasonal patterns in the southern and northern hemispheres. The largest phase differences are in the central Arabian Sea (more than 90 degrees). This feature explains why the seasonally variable models (SNa and SNb) are better than the seasonal-mean model (SN0) in the tropical zone. Additionally, significant phase differences are observed in the Northeast Pacific Ocean. The results suggest that the seasonal phase variations should be taken into account for future internal tide models, in contrast to present internal tide models of constant harmonics (Carrere et al. 2021). Note that the complex spatial patterns of seasonal differences are partly caused by the multiwave internal tide field.

b. Model decomposition

To decipher the seasonal variations, the seasonal models should be decomposed into multiple components. One advantage of the new mapping technique is that the multiwave internal tide field has been resolved by five waves of different propagation directions. Using the five-wave solutions, each seasonal internal tide model can be decomposed into four components by direction: southward (225°–315°), northward (45°–135°), westward (135°–225°), and eastward (315°–45°). The four seasonal internal tide models are decomposed following the same method (Figs. S14–S18 in the online supplemental material). The decomposed fields show that mode-1 M2 internal tides mainly originate from outstanding topographic features such as the Hawaiian Ridge, the Mendocino Ridge, the Great Meteor, the French Polynesian Ridge, and so on. Besides, internal tides are also from the Lombok Strait and the Luzon Strait. The southward and northward components have relatively small errors, because satellite ground tracks are generally in the south–north direction. On the contrary, the eastward and westward components have large errors, due to the wide intertrack gaps (Zhao 2019). The separately resolved components make it possible to study the phase variation with propagation.

c. Seasonal amplitude variations

In this section, the seasonal amplitude variations are studied using the northward and southward components. The comparisons are made for two model pairs: winter minus summer and spring minus fall. The differences are calculated point by point and smoothed by seven-point running mean. Regions of weak internal tides (amplitude lower than 1 mm) are discarded. The resultant amplitude differences are given in Fig. 9. Cophase charts are superimposed to show the long-range propagation of internal tides. One can see that the amplitude differences of the decomposed components are different from those of the summed fields (Figs. 8a,b). The decomposed maps show more details and convey useful information after dropping the westward and eastward components. Figure 9 shows that the amplitude differences are generally smaller than 3 mm. Considering that the internal tide’s amplitudes range from 5 to 15 mm, however, the seasonal variations are estimated to be up to 20% of the internal tide amplitudes.

Fig. 9.
Fig. 9.

Seasonal amplitude variations of M2 internal tides: (top) winter minus summer and (bottom) spring minus fall for the (a),(c) northward component (45°–135°) and (b),(d) southward component (225°–315°). The differences are calculated pointwise and are smoothed by 7-point running mean in both the zonal and meridional directions. Internal tides with amplitudes of <1 mm are discarded. Regions of strong boundary currents are excluded. Cophase charts are shown as gray lines.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

Figure 9 shows that large-amplitude differences are mainly in the tropical Indian, Pacific and Atlantic Oceans. For example, in the western tropical Pacific Ocean, both the southward and northward internal tides have large-amplitude differences (negative or positive). It agrees with the strong seasonal variation of ocean stratification in the tropical zone (Fig. 1). There are some large differences at high latitudes, for example, the northward internal tides from Meteor Rise and Cape Rise in the South Atlantic Ocean (Fig. 9a). Figure 9 also shows that the seasonal amplitude variations are a function of location. In some regions, positive and negative values neighbor to each other in space. For example, in the Madagascar–Mascarene region, the northward internal tides have both negative and positive values (Figs. 9a,c). In the Arabian Sea, the southward internal tides have negative and positive values (Figs. 9b,d). Off the U.S. West Coast, the southward internal tides from the Mendocino Ridge have negative and positive values (Figs. 9b,d).

The seasonal amplitude variations (Fig. 9) may contain rich information on the energetics of internal tides. To the south of the Lombok Strait, the southward internal tides have greater amplitudes in spring (Fig. 9d). The seasonal amplitude differences should be caused by the seasonal variations in the barotropic-to-baroclinic tidal conversion. In the Amazon River plume, the northward internal tides have smaller amplitudes in winter and spring (Figs. 9a,c). However, in the surrounding region of the plume, both the southward and northward components have greater amplitudes in winter and spring. This contrast feature of seasonal variations is likely due to the seasonal variation of Amazon’s freshwater discharge. The satellite-derived results give the SSH amplitude, not internal tide energy flux. The transfer relation from SSH amplitude to internal tide energy varies with the seasonal variation in ocean stratification (Wunsch 2013; Lahaye et al. 2019) [see Fig. A2 in Zhao et al. (2016)]. The seasonal ocean stratification may also modulate the energy distribution among baroclinic modes (Nycander 2005; Falahat et al. 2014; Vic et al. 2019).

d. Seasonal phase variations

Phase anomalies are obtained from the phase differences between the four seasonal models and the seasonal-mean model. They are calculated point by point and smoothed by seven-point running mean. Internal tides with amplitudes < 1 mm are discarded. The four decomposed components are analyzed separately (Figs. S14–S18 in the online supplemental material). Figures 10 and 11 give the seasonal phase anomalies of the northward and southward components, respectively. As explained earlier, the westward and eastward components are noisy and are not discussed in this paper.

Fig. 10.
Fig. 10.

Seasonal phase anomalies of the northward component (45°–135°) for (a) winter, (b) spring, (c) summer, and (d) fall. Seasonal phase anomalies are defined to be the differences between the four seasonal models and the seasonal-mean model. Pointwise differences are smoothed by two-dimensional 7-point running mean. Internal tides with amplitudes of <1 mm are discarded. Regions of strong boundary currents are excluded. Cophase charts are shown as gray lines.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

Fig. 11.
Fig. 11.

As in Fig. 10 but for the southward M2 internal tide component (225°–315°). Note that the largest phase anomalies of ~±120° occur in the Arabian Sea, where colors are saturated.

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

The global maps reveal three pronounced features (Figs. 10 and 11). First, the phase anomalies of opposite seasons have opposite signs. For example, the northward internal tides have opposite signs in winter (Figs. 10a) versus summer (Figs. 10c), and opposite signs in spring (Fig. 10b) versus fall (Fig. 10d). The same features can be observed in the southward component (Fig. 11). Second, the phase anomalies have opposite signs in the southern and northern hemispheres. For example, Fig. 10a shows that the phase anomalies in the Pacific Ocean are negative in the Southern Hemisphere and positive in the Northern Hemisphere. Third, large phase differences mainly occur in the tropical zone, similar to amplitude differences. For example, significant phase anomalies are observed in the Bay of Bengal, to the south of the Lombok Strait, in the South China Sea, and in the Amazon River plume. For both southward and northward components, the seasonal phase anomalies at the equator vary over ±60°. The largest anomalies are in the Arabian Sea, varying from −120° in spring (Fig. 11b) to 120° in fall (Fig. 11d). All these features confirm the seasonal phase variations of internal tides in the global ocean. They are consistent with the seasonal variations of internal tide speed estimated from WOA2013 hydrographic climatologies (Fig. 1). The good agreement suggests that the seasonal phase anomalies are mainly caused by different propagation speeds in the four seasons. Though, large-scale ocean circulations and mesoscale eddies may also modulate the speed of internal tides at seasonal time scales.

Figures 10 and 11 show that the phase anomalies increase with propagation distances, as a result of seasonally varying propagation speeds. An example is discussed using the northward internal tides from the Hawaiian Ridge (Fig. 10). The phase anomalies are almost zero at the Hawaiian Ridge, and accumulate in their 3500-km-long propagation. In the far field, the phase anomalies are up to ±30°. From the propagation time of 20 tidal periods, the change rate of propagation speed is estimated to be 0.4%. For comparison, the northward internal tides from the Polynesian Ridge are in the tropical zone. The seasonal phase anomalies accumulate to ±60° over 14 tidal periods, from which the change rate of propagation speed is estimated to about 1.2%. The seasonal change rate of propagation speed contains important information on ocean stratification and vertical distribution of ocean heat content (OHC). Previous studies have reported the interannual change rate of propagation speed and demonstrated a method to estimate the OHC change (Zhao 2016a,b).

e. Incoherence due to seasonal variations

The above analyses reveal that internal tides are subject to significant seasonal variations. The incoherence (nonstationarity) caused by seasonal phase variations is estimated following
incoherence%=1−⁡(14∑n=14eiϕn)2,
where ϕn is the phase of the nth seasonal model. The square in this equation means that the incoherence is in terms of internal tide energy (instead of amplitude). The incoherence indicates the percentage of the incoherent component in the total internal tide (coherent plus incoherent). The incoherence is calculated pointwise and spatially smoothed by five-point running mean along both longitude and latitude. Because low-amplitude internal tides have large phase errors, internal tides with amplitudes of < 1 mm are discarded. Figure 12 shows the resultant incoherence for the southward and northward components.
Fig. 12.
Fig. 12.

Incoherence of M2 internal tides induced by seasonal phase variations. It is calculated pointwise and is smoothed by two-dimensional 5-point running mean. Shown are the (a) northward component (45°–135°) and (b) southward component (225°–315°).

Citation: Journal of Physical Oceanography 51, 9; 10.1175/JPO-D-21-0001.1

One remarkable feature is that the incoherence of strong internal tides is largely within 10% (Fig. 12, blue patches). It suggests that strong internal tides are seasonally stationary. As a result of seasonally varying propagation speed, the seasonal phase anomalies increase with propagation distances; therefore, high incoherence can be seen in the far field. For example, the northward internal tides from the Hawaiian Ridge have incoherence of 10%–20% near the Alaskan coast (Fig. 12a, cyan patches). Another remarkable feature is that the incoherence in the tropical zone is as high as 40%–50% (Figs. 12a,b; red patches). This feature agrees with the large phase anomalies in the tropical zone (Figs. 10 and 11).

Zaron (2017) estimates the incoherence of M2 internal tides using 23 years of satellite altimeter data. He estimates the nonstationary (i.e., incoherent) internal tides from original and harmonic variances integrated along wavenumber spectra. Zaron (2017) reports that on the global scale the incoherence accounts for about 44% of the total energy and that the incoherence is spatially inhomogeneous (see his Fig. 9). These two independent studies are consistent in the global patterns of incoherence—both show high incoherence in the tropical zone. Additionally, Zhao (2016a) studies the interannual phase variations along the northward internal tides from the Polynesian Ridge and finds high incoherence in the tropical zone due to the influence of El Niño. The incoherence in Zaron (2017), however, is caused by the combined effect of all ocean processes including seasonal variations, interannual variations and mesoscale eddies. The differences preclude us from making a quantitative comparison of these results. However, the combination of these studies may offer an approach to distinguish the respective contributions of these ocean processes.

7. Summary

This study has investigated the seasonal variability of mode-1 M2 internal tides on a global scale using 25 years of exact-repeat-track satellite altimeter data from 1992 to 2017. It is a challenging task to detect their seasonal cycle, because 1) mode-1 M2 internal tides account for only about 1.2% of the satellite-measured SSH variance, 2) the seasonally subsetted data records are shorter, and 3) the multiwave internal tide field is complex. These issues are addressed by a new mapping technique that combines along-track filtering, harmonic analysis, plane wave analysis and 2D bandpass filtering (Fig. 2). Theoretical seasonal wavelengths of internal tides from WOA2013 climatologies are used (Fig. 1). In addition, larger fitting windows of 250 km by 250 km are used to compensate for the data loss due to seasonal subsetting. All of these measures contribute to construct reliable internal tide models. Four seasonal internal tide models are constructed using seasonally subsetted multisatellite altimeter data (Figs. 3a–d). Two seasonally variable internal tide models are derived to account for the seasonal variations. The seasonal-mean internal tide model is from the vector mean of the four seasonal models (Fig. 3e). The 25-yr-coherent model is constructed using the whole data following the same procedure (Fig. 3f). All these models are evaluated using 9 years of independent CryoSat-2 data from 2010 to 2019 (Fig. 5).

The four seasonal internal tide models have larger model errors than the seasonal-mean model. Their amplitude errors are twice those of the seasonal-mean model (Fig. 4). Their large errors are due to the short seasonally subsetted data records. The seasonal models are evaluated using CryoSat-2 seasonal subsets. Each seasonal internal tide model reduces most variance in its own season and least variance in its opposite season (Fig. 6). Based on integrated variance reductions, the four seasonal internal tide models have similar performances, with the winter model being a little better than the others (Fig. 6).

The seasonal-mean model and the 25-yr-coherent model are almost the same. It suggests that the two mapping methods are equivalent: the former is from the vector mean of four models obtained using four subsetted datasets, and the latter is directly constructed using the whole data. The seasonal-mean model works better than the four seasonal models, because it can reduce 30% more variance. The improvement of the seasonal-mean model is because of smaller model errors achieved in the vector mean. It is found that the model variance difference and variance reduction difference are of the same magnitude but opposite sign (Fig. S12 in the online supplemental material).

The two seasonally variable internal tide models are derived from the four seasonal models. The first one is a phase-variable, amplitude-variable model. The second one is a phase-variable, amplitude-invariable model. Evaluations using CryoSat-2 data reveal that both models are better than the four seasonal models by reducing 13% and 25% more variance, respectively (Figs. 7a,b). It is found that the performance of the internal tide models is determined by seasonal signals and model errors, both of which are a function of location. Wherever the seasonal signals are larger than model errors, one model will increase variance reduction. Based on global integrations, the two seasonally variable models are worse than the seasonal-mean model, because the latter has small model errors (Fig. 6b). However, the two seasonally variable models are better than the seasonal-mean model inside the tropical zone and worse than the seasonal-mean model outside the tropical zone (Figs. 7c,d). It is because the seasonal variations of internal tides are strongest in the tropical zone and decrease with increasing latitudes. The main limiting factor is the large model errors caused by the short satellite altimeter data currently available.

The seasonal variations of mode-1 M2 internal tides are investigated by directly comparing the four seasonal internal tide models. Significant seasonal differences are observed throughout the global ocean (Fig. 8). This paper agrees with previous studies (Ray and Zaron 2011; Zaron 2019), but can show seasonal variations not only in amplitude but also in phase (Fig. 8). In particular, the seasonal variations are investigated along long-range internal tidal beams in the decomposed components. Internal tides in the tropical zone have the strongest seasonal signals, in both amplitude and phase (Figs. 9–11). The seasonal amplitude variations may be ±20% of the internal tide amplitudes. The seasonal phase variations in the tropical zone may be up to ±60° (Figs. 10 and 11). The largest seasonal phase variations of ±120° are in the Arabian Sea (Fig. 11). The along-beam phase anomalies increase with propagation, because of the seasonal variations in propagation speed. The change rates of propagation speed as low as 0.4% are detectable along long-range internal tidal beams. Incoherence caused by seasonal phase variations is estimated (Fig. 12). Strong internal tides have incoherence lower than 10%, suggesting that internal tides are seasonally stationary. In the tropical zone, the incoherence may be up to 40%–50%, due to large seasonal variations in ocean stratification and phase speed.

8. Discussion

This paper investigates the seasonality of mode-1 M2 internal tides using 25 years of satellite altimeter data. One achievement of this paper is that model errors are significantly suppressed by a new mapping technique. Although mode-1 M2 internal tides account for only about 1.2% of the raw satellite data in terms of SSH variance, low-noise-level seasonal internal tides are constructed and used for examining their seasonality. However, the major limitation of this work is still the large model errors stemming from the short satellite data currently available. It is expected that the model errors can be further reduced with the accumulation of SSH measurements from future altimeter missions. In addition, the eastward and westward internal tides are not studied in this paper, because they have even larger errors. This issue may be addressed using the two-dimensional SSH measurements made by the upcoming SWOT mission (Fu and Ubelmann 2014; Li et al. 2019; Qiu et al. 2018; Wang et al. 2019).

This paper investigates the seasonal variations of internal tides in the open ocean. In contrast, most of the field measurements are on continental slopes or in marginal seas (Nash et al. 2012; Kelly et al. 2015; Shang et al. 2015; Cao et al. 2017). It is found that these regions are subject to even stronger seasonal variations due to a variety of ocean processes with seasonal cycles (Nash et al. 2012; Kelly et al. 2015). In addition, internal tide dynamics in shallow waters are more complicated including nonlinearity, scattering and reflection. Dedicated field experiments, high-resolution numerical models and two-dimensional SWOT measurements combined may address the challenge.

This paper investigates the seasonal cycles of mode-1 M2 internal tides, which are the largest tidal constituent in the ocean. It has been reported that the same data and methods can be applied to O1 and K1 constituents (Zhao 2014; Zaron 2019) and mode-2 M2 internal modes (Johnston et al. 2003; Zhao 2018). To extract even weaker seasonal signals associated with these constituents, one needs to choose optimal mapping parameters as discussed in this paper.

Acknowledgments

This work was supported by the National Aeronautics and Space Administration (NASA) via projects NNX17AH14G and NNX17AH57G and the National Science Foundation (NSF) via project OCE1634041. The author thanks Ed Zaron, Loren Carrere, and Ernst Schrama for their constructive comments on an earlier draft presented at the OSTST 2020 meeting. The author thanks two anonymous reviewers for their insightful suggestions and comments that greatly improved this paper.

Data availability statement

The satellite altimeter products were produced by Ssalto/Duacs and distributed by Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO) with support from CNES (http://www.aviso.altimetry.fr) and the Copernicus Marine Environment Monitoring Service (http://marine.copernicus.eu). The World Ocean Atlas 2013 is produced and made available by NOAA National Oceanographic Data Center (https://www.nodc.noaa.gov/OC5/woa13/). The seasonal internal tide models constructed in this study are available online (https://doi.org/10.6084/m9.figshare.14759094). Contact the author for policy and instructions (zzhao@apl.uw.edu).

REFERENCES

  • Alford, M. H., 2003: Redistribution of energy available for ocean mixing by long-range propagation of internal waves. Nature, 423, 159–162, https://doi.org/10.1038/nature01628.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Arbic, B. K., S. T. Garner, R. W. Hallberg, and H. L. Simmons, 2004: The accuracy of surface elevations in forward global barotropic and baroclinic tide models. Deep-Sea Res. II, 51, 3069–3101, https://doi.org/10.1016/j.dsr2.2004.09.014.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Arbic, B. K., A. J. Wallcraft, and E. J. Metzger, 2010: Concurrent simulation of the eddying general circulation and tides in a global ocean model. Ocean Modell., 32, 175–187, https://doi.org/10.1016/j.ocemod.2010.01.007.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buijsman, M. C., B. K. Arbic, J. G. Richman, J. F. Shriver, A. J. Wallcraft, and L. Zamudio, 2017: Semidiurnal internal tide incoherence in the equatorial Pacific. J. Geophys. Res. Oceans, 122, 5286–5305, https://doi.org/10.1002/2016JC012590.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Cao, A., Z. Guo, X. Lv, J. Song, and J. Zhang, 2017: Coherent and incoherent features, seasonal behaviors and spatial variations of internal tides in the northern South China Sea. J. Mar. Syst., 172, 75–83, https://doi.org/10.1016/j.jmarsys.2017.03.005.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Carrere, L., and Coauthors, 2021: Accuracy assessment of global internal tide models using satellite altimetry. Ocean Sci., 17, 147–180, https://doi.org/10.5194/os-17-147-2021.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chiswell, S. M., 2002: Energy levels, phase, and amplitude modulation of the baroclinic tide off Hawaii. J. Phys. Oceanogr., 32, 2640–2651, https://doi.org/10.1175/1520-0485-32.9.2640.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Colosi, J. A., and W. Munk, 2006: Tales of the venerable Honolulu tide gauge. J. Phys. Oceanogr., 36, 967–996, https://doi.org/10.1175/JPO2876.1.

  • de Lavergne, C., and Coauthors, 2020: A parameterization of local and remote tidal mixing. J. Adv. Model. Earth Syst., 12, e2020MS002065, https://doi.org/10.1029/2020MS002065.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Doodson, A. T., 1921: The harmonic development of the tide-generating potential. Proc. Roy. Soc. Lond., 100A, 305–329, https://doi.org/10.1098/rspa.1921.0088.

    • Search Google Scholar
    • Export Citation

  • Dushaw, B. D., 2015: An empirical model for mode-1 internal tides derived from satellite altimetry: Computing accurate tidal predictions at arbitrary points over the world oceans. University of Washington Applied Physics Laboratory Tech. Rep., 114 pp.

  • Egbert, G. D., and R. D. Ray, 2000: Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature, 405, 775–778, https://doi.org/10.1038/35015531.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Falahat, S., J. Nycander, F. Roquet, and M. Zarroug, 2014: Global calculation of tidal energy conversion into vertical normal modes. J. Phys. Oceanogr., 44, 3225–3244, https://doi.org/10.1175/JPO-D-14-0002.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Fu, L.-L., and C. Ubelmann, 2014: On the transition from profile altimeter to swath altimeter for observing global ocean surface topography. J. Atmos. Oceanic Technol., 31, 560–568, https://doi.org/10.1175/JTECH-D-13-00109.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Garrett, C., and E. Kunze, 2007: Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech., 39, 57–87, https://doi.org/10.1146/annurev.fluid.39.050905.110227.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gerkema, T., F.-P. A. Lam, and L. R. Maas, 2004: Internal tides in the Bay of Biscay: Conversion rates and seasonal effects. Deep-Sea Res. II, 51, 2995–3008, https://doi.org/10.1016/j.dsr2.2004.09.012.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Hall, R. A., M. H. Alford, G. S. Carter, M. C. Gregg, R.-C. Lien, D. J. Wain, and Z. Zhao, 2014: Transition from partly standing to progressive internal tides in Monterey Submarine Canyon. Deep-Sea Res. II, 104, 164–173, https://doi.org/10.1016/j.dsr2.2013.05.039.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jan, S., R.-C. Lien, and C.-H. Ting, 2008: Numerical study of baroclinic tides in Luzon Strait. J. Oceanogr., 64, 789–802, https://doi.org/10.1007/s10872-008-0066-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jayne, S. R., and L. C. St. Laurent, 2001: Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett., 28, 811–814, https://doi.org/10.1029/2000GL012044.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jeon, C., and Coauthors, 2014: Seasonal variation of semidiurnal internal tides in the East/Japan Sea. J. Geophys. Res. Oceans, 119, 2843–2859, https://doi.org/10.1002/2014JC009864.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Johnston, T. M. S., M. A. Merrifield, and P. E. Holloway, 2003: Internal tide scattering at the Line Islands Ridge. J. Geophys. Res., 108, 3365, https://doi.org/10.1029/2003JC001844.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kelly, S. M., 2016: The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography. J. Phys. Oceanogr., 46, 3777–3788, https://doi.org/10.1175/JPO-D-16-0131.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kelly, S. M., N. L. Jones, G. N. Ivey, and R. J. Lowe, 2015: Internal-tide spectroscopy and prediction in the Timor Sea. J. Phys. Oceanogr., 45, 64–83, https://doi.org/10.1175/JPO-D-14-0007.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lahaye, N., J. Gula, and G. Roullet, 2019: Sea surface signature of internal tides. Geophys. Res. Lett., 46, 3880–3890, https://doi.org/10.1029/2018GL081848.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Li, Z., and J.-S. von Storch, 2020: M2 internal-tide generation in STORMTIDE2. J. Geophys. Res. Oceans, 125, e2019JC015453, https://doi.org/10.1029/2019JC015453.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Li, Z., J. Wang, and L.-L. Fu, 2019: An observing system simulation experiment for ocean state estimation to assess the performance of the SWOT mission: Part 1—A twin experiment. J. Geophys. Res. Oceans, 124, 4838–4855, https://doi.org/10.1029/2018JC014869.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Liu, J., Y. He, D. Wang, T. Liu, and S. Cai, 2015: Observed enhanced internal tides in winter near the Luzon Strait. J. Geophys. Res. Oceans, 120, 6637–6652, https://doi.org/10.1002/2015JC011131.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Liu, K., J. Sun, C. Guo, Y. Yang, W. Yu, and Z. Wei, 2019: Seasonal and spatial variations of the M2. J. Geophys. Res. Oceans, 124, 1115–1138, https://doi.org/10.1029/2018JC014819.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Löb, J., J. Köhler, C. Mertens, M. Walter, Z. Li, J.-S. von Storch, Z. Zhao, and M. Rhein, 2020: Observations of the low-mode internal tide and its interaction with mesoscale flow south of the Azores. J. Geophys. Res. Oceans, 125, e2019JC015879, https://doi.org/10.1029/2019JC015879.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Locarnini, R. A., and Coauthors, 2013: World Ocean Atlas 2013, Volume 1: Temperature. National Oceanographic Data Center, 40 pp., http://data.nodc.noaa.gov/woa/WOA13/DOC/woa13_vol1.pdf.

  • MacKinnon, J. A., and Coauthors, 2017: Climate process team on internal wave-driven ocean mixing. Bull. Amer. Meteor. Soc., 98, 2429–2454, https://doi.org/10.1175/BAMS-D-16-0030.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mitchum, G. T., and S. M. Chiswell, 2000: Coherence of internal tide modulations along the Hawaiian Ridge. J. Geophys. Res., 105, 28 653–28 661, https://doi.org/10.1029/2000JC900140.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Müller, M., J. Y. Cherniawsky, M. G. G. Foreman, and J.-S. von Storch, 2012: Global M2 internal tide and its seasonal variability from high resolution ocean circulation and tide modeling. Geophys. Res. Lett., 39, L19607, https://doi.org/10.1029/2012GL053320.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Munk, W. H., 1966: Abyssal recipes. Deep-Sea Res., 13, 707–730, https://doi.org/10.1016/0011-7471(66)90602-4.

  • Munk, W. H., 1981: Internal waves and small-scale processes. Evolution of Physical Oceanography, B. A. Warren and C. Wunsch, Eds., MIT Press, 264–291.

  • Munk, W. H., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, https://doi.org/10.1016/S0967-0637(98)00070-3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nash, J., E. Shroyer, S. Kelly, M. Inall, T. Duda, M. Levine, N. Jones, and R. Musgrave, 2012: Are any coastal internal tides predictable? Oceanography, 25, 80–95, https://doi.org/10.5670/oceanog.2012.44.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nycander, J., 2005: Generation of internal waves in the deep ocean by tides. J. Geophys. Res., 110, C10028, https://doi.org/10.1029/2004JC002487.

  • Osborne, J. J., A. L. Kurapov, G. D. Egbert, and P. M. Kosro, 2011: Spatial and temporal variability of the M2. J. Phys. Oceanogr., 41, 2037–2062, https://doi.org/10.1175/JPO-D-11-02.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Qiu, B., S. Chen, P. Klein, J. Wang, H. Torres, L.-L. Fu, and D. Menemenlis, 2018: Seasonality in transition scale from balanced to unbalanced motions in the world ocean. J. Phys. Oceanogr., 48, 591–605, https://doi.org/10.1175/JPO-D-17-0169.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., 2013: Precise comparisons of bottom-pressure and altimetric ocean tides. J. Geophys. Res. Oceans, 118, 4570–4584, https://doi.org/10.1002/jgrc.20336.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and G. T. Mitchum, 1996: Surface manifestation of internal tides generated near Hawaii. Geophys. Res. Lett., 23, 2101–2104, https://doi.org/10.1029/96GL02050.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and G. T. Mitchum, 1997: Surface manifestation of internal tides in the deep ocean: Observations from altimetry and island gauges. Prog. Oceanogr., 40, 135–162, https://doi.org/10.1016/S0079-6611(97)00025-6.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and E. D. Zaron, 2011: Non-stationary internal tides observed with satellite altimetry. Geophys. Res. Lett., 38, L17609, https://doi.org/10.1029/2011GL048617.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and E. Zaron, 2016: M2 internal tides and their observed wavenumber spectra from satellite altimetry. J. Phys. Oceanogr., 46, 3–22, https://doi.org/10.1175/JPO-D-15-0065.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shang, X., Q. Liu, X. Xie, G. Chen, and R. Chen, 2015: Characteristics and seasonal variability of internal tides in the southern South China Sea. Deep-Sea Res. I, 98, 43–52, https://doi.org/10.1016/j.dsr.2014.12.005.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Smith, W. H. F., and D. T. Sandwell, 1997: Global sea floor topography from satellite altimetry and ship depth soundings. Science, 277, 1956–1962, https://doi.org/10.1126/science.277.5334.1956.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Song, P., and X. Chen, 2020: Investigation of the internal tides in the Northwest Pacific Ocean considering the background circulation and stratification. J. Phys. Oceanogr., 50, 3165–3188, https://doi.org/10.1175/JPO-D-19-0177.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vic, C., and Coauthors, 2019: Deep-ocean mixing driven by small-scale internal tides. Nat. Commun., 10, 2099, https://doi.org/10.1038/s41467-019-10149-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vic, C., B. Ferron, V. Thierry, H. Mercier, and P. Lherminier, 2021: Tidal and near-inertial internal waves over the Reykjanes Ridge. J. Phys. Oceanogr., 51, 419–437, https://doi.org/10.1175/JPO-D-20-0097.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, J., L.-L. Fu, H. S. Torres, S. Chen, B. Qiu, and D. Menemenlis, 2019: On the spatial scales to be resolved by the surface water and ocean topography Ka-band radar interferometer. J. Atmos. Oceanic Technol., 36, 87–99, https://doi.org/10.1175/JTECH-D-18-0119.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Whalen, C. B., C. de Lavergne, A. C. Naveira Garabato, J. M. Klymak, J. A. MacKinnon, and K. L. Sheen, 2020: Internal wave-driven mixing: Governing processes and consequences for climate. Nat. Rev. Earth Environ., 1, 606–621, https://doi.org/10.1038/s43017-020-0097-z.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wunsch, C., 1975: Internal tides in the ocean. Rev. Geophys. Space Phys., 13, 167–182, https://doi.org/10.1029/RG013i001p00167.

  • Wunsch, C., 2013: Baroclinic motions and energetics as measured by altimeters. J. Atmos. Oceanic Technol., 30, 140–150, https://doi.org/10.1175/JTECH-D-12-00035.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yan, T., Y. Qi, Z. Jing, and S. Cai, 2020: Seasonal and spatial features of barotropic and baroclinic tides in the northwestern South China Sea. J. Geophys. Res. Oceans, 125, e2018JC014860, https://doi.org/10.1029/2018JC014860.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zaron, E. D., 2015: Nonstationary internal tides observed using dual-satellite altimetry. J. Phys. Oceanogr., 45, 2239–2246, https://doi.org/10.1175/JPO-D-15-0020.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zaron, E. D., 2017: Mapping the nonstationary internal tide with satellite altimetry. J. Geophys. Res. Oceans, 122, 539–554, https://doi.org/10.1002/2016JC012487.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zaron, E. D., 2019: Baroclinic tidal sea level from exact-repeating mission altimetry. J. Phys. Oceanogr., 49, 193–210, https://doi.org/10.1175/JPO-D-18-0127.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., 2014: Internal tide radiation from the Luzon Strait. J. Geophys. Res. Oceans, 119, 5434–5448, https://doi.org/10.1002/2014JC010014.

  • Zhao, Z., 2016a: Internal tide oceanic tomography. Geophys. Res. Lett., 43, 9157–9164, https://doi.org/10.1002/2016GL070567.

  • Zhao, Z., 2016b: Using CryoSat-2 altimeter data to evaluate M2 internal tides observed from multisatellite altimetry. J. Geophys. Res. Oceans, 121, 5164–5180, https://doi.org/10.1002/2016JC011805.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., 2017a: The global mode-1 S2 internal tide. J. Geophys. Res. Oceans, 122, 8794–8812, https://doi.org/10.1002/2017JC013112.

  • Zhao, Z., 2017b: Propagation of the semidiurnal internal tide: Phase velocity versus group velocity. Geophys. Res. Lett., 44, 11 942–11 950, https://doi.org/10.1002/2017GL076008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., 2018: The global mode-2 M2 internal tide. J. Geophys. Res. Oceans, 123, 7725–7746, https://doi.org/10.1029/2018JC014475.

  • Zhao, Z., 2019: Mapping internal tides from satellite altimetry without blind directions. J. Geophys. Res. Oceans, 124, 8605–8625, https://doi.org/10.1029/2019JC015507.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., M. H. Alford, R.-C. Lien, M. C. Gregg, and G. S. Carter, 2012: Internal tides and mixing in a submarine canyon with time-varying stratification. J. Phys. Oceanogr., 42, 2121–2142, https://doi.org/10.1175/JPO-D-12-045.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., M. H. Alford, J. B. Girton, L. Rainville, and H. L. Simmons, 2016: Global observations of open-ocean mode-1 M2 internal tide. J. Phys. Oceanogr., 46, 1657–1684, https://doi.org/10.1175/JPO-D-15-0105.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhou, X.-H., D.-P. Wang, and D. Chen, 2015: Validating satellite altimeter measurements of internal tides with long-term TAO/TRITON buoy observations at 2°S–156°E. Geophys. Res. Lett., 42, 4040–4046, https://doi.org/10.1002/2015GL063669.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zweng, M., and Coauthors, 2013: Salinity. Vol. 2, World Ocean Atlas 2013, NOAA Atlas NESDIS 74, 39 pp., https://www.ncei.noaa.gov/data/oceans/woa/WOA13/DOC/woa13_vol2.pdf.

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Journal of Physical Oceanography

  • Alford, M. H., 2003: Redistribution of energy available for ocean mixing by long-range propagation of internal waves. Nature, 423, 159–162, https://doi.org/10.1038/nature01628.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Arbic, B. K., S. T. Garner, R. W. Hallberg, and H. L. Simmons, 2004: The accuracy of surface elevations in forward global barotropic and baroclinic tide models. Deep-Sea Res. II, 51, 3069–3101, https://doi.org/10.1016/j.dsr2.2004.09.014.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Arbic, B. K., A. J. Wallcraft, and E. J. Metzger, 2010: Concurrent simulation of the eddying general circulation and tides in a global ocean model. Ocean Modell., 32, 175–187, https://doi.org/10.1016/j.ocemod.2010.01.007.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buijsman, M. C., B. K. Arbic, J. G. Richman, J. F. Shriver, A. J. Wallcraft, and L. Zamudio, 2017: Semidiurnal internal tide incoherence in the equatorial Pacific. J. Geophys. Res. Oceans, 122, 5286–5305, https://doi.org/10.1002/2016JC012590.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Cao, A., Z. Guo, X. Lv, J. Song, and J. Zhang, 2017: Coherent and incoherent features, seasonal behaviors and spatial variations of internal tides in the northern South China Sea. J. Mar. Syst., 172, 75–83, https://doi.org/10.1016/j.jmarsys.2017.03.005.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Carrere, L., and Coauthors, 2021: Accuracy assessment of global internal tide models using satellite altimetry. Ocean Sci., 17, 147–180, https://doi.org/10.5194/os-17-147-2021.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Chiswell, S. M., 2002: Energy levels, phase, and amplitude modulation of the baroclinic tide off Hawaii. J. Phys. Oceanogr., 32, 2640–2651, https://doi.org/10.1175/1520-0485-32.9.2640.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Colosi, J. A., and W. Munk, 2006: Tales of the venerable Honolulu tide gauge. J. Phys. Oceanogr., 36, 967–996, https://doi.org/10.1175/JPO2876.1.

  • de Lavergne, C., and Coauthors, 2020: A parameterization of local and remote tidal mixing. J. Adv. Model. Earth Syst., 12, e2020MS002065, https://doi.org/10.1029/2020MS002065.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Doodson, A. T., 1921: The harmonic development of the tide-generating potential. Proc. Roy. Soc. Lond., 100A, 305–329, https://doi.org/10.1098/rspa.1921.0088.

    • Search Google Scholar
    • Export Citation

  • Dushaw, B. D., 2015: An empirical model for mode-1 internal tides derived from satellite altimetry: Computing accurate tidal predictions at arbitrary points over the world oceans. University of Washington Applied Physics Laboratory Tech. Rep., 114 pp.

  • Egbert, G. D., and R. D. Ray, 2000: Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature, 405, 775–778, https://doi.org/10.1038/35015531.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Falahat, S., J. Nycander, F. Roquet, and M. Zarroug, 2014: Global calculation of tidal energy conversion into vertical normal modes. J. Phys. Oceanogr., 44, 3225–3244, https://doi.org/10.1175/JPO-D-14-0002.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Fu, L.-L., and C. Ubelmann, 2014: On the transition from profile altimeter to swath altimeter for observing global ocean surface topography. J. Atmos. Oceanic Technol., 31, 560–568, https://doi.org/10.1175/JTECH-D-13-00109.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Garrett, C., and E. Kunze, 2007: Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech., 39, 57–87, https://doi.org/10.1146/annurev.fluid.39.050905.110227.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gerkema, T., F.-P. A. Lam, and L. R. Maas, 2004: Internal tides in the Bay of Biscay: Conversion rates and seasonal effects. Deep-Sea Res. II, 51, 2995–3008, https://doi.org/10.1016/j.dsr2.2004.09.012.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gill, A. E., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Hall, R. A., M. H. Alford, G. S. Carter, M. C. Gregg, R.-C. Lien, D. J. Wain, and Z. Zhao, 2014: Transition from partly standing to progressive internal tides in Monterey Submarine Canyon. Deep-Sea Res. II, 104, 164–173, https://doi.org/10.1016/j.dsr2.2013.05.039.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jan, S., R.-C. Lien, and C.-H. Ting, 2008: Numerical study of baroclinic tides in Luzon Strait. J. Oceanogr., 64, 789–802, https://doi.org/10.1007/s10872-008-0066-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jayne, S. R., and L. C. St. Laurent, 2001: Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett., 28, 811–814, https://doi.org/10.1029/2000GL012044.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jeon, C., and Coauthors, 2014: Seasonal variation of semidiurnal internal tides in the East/Japan Sea. J. Geophys. Res. Oceans, 119, 2843–2859, https://doi.org/10.1002/2014JC009864.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Johnston, T. M. S., M. A. Merrifield, and P. E. Holloway, 2003: Internal tide scattering at the Line Islands Ridge. J. Geophys. Res., 108, 3365, https://doi.org/10.1029/2003JC001844.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kelly, S. M., 2016: The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography. J. Phys. Oceanogr., 46, 3777–3788, https://doi.org/10.1175/JPO-D-16-0131.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kelly, S. M., N. L. Jones, G. N. Ivey, and R. J. Lowe, 2015: Internal-tide spectroscopy and prediction in the Timor Sea. J. Phys. Oceanogr., 45, 64–83, https://doi.org/10.1175/JPO-D-14-0007.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lahaye, N., J. Gula, and G. Roullet, 2019: Sea surface signature of internal tides. Geophys. Res. Lett., 46, 3880–3890, https://doi.org/10.1029/2018GL081848.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Li, Z., and J.-S. von Storch, 2020: M2 internal-tide generation in STORMTIDE2. J. Geophys. Res. Oceans, 125, e2019JC015453, https://doi.org/10.1029/2019JC015453.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Li, Z., J. Wang, and L.-L. Fu, 2019: An observing system simulation experiment for ocean state estimation to assess the performance of the SWOT mission: Part 1—A twin experiment. J. Geophys. Res. Oceans, 124, 4838–4855, https://doi.org/10.1029/2018JC014869.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Liu, J., Y. He, D. Wang, T. Liu, and S. Cai, 2015: Observed enhanced internal tides in winter near the Luzon Strait. J. Geophys. Res. Oceans, 120, 6637–6652, https://doi.org/10.1002/2015JC011131.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Liu, K., J. Sun, C. Guo, Y. Yang, W. Yu, and Z. Wei, 2019: Seasonal and spatial variations of the M2. J. Geophys. Res. Oceans, 124, 1115–1138, https://doi.org/10.1029/2018JC014819.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Löb, J., J. Köhler, C. Mertens, M. Walter, Z. Li, J.-S. von Storch, Z. Zhao, and M. Rhein, 2020: Observations of the low-mode internal tide and its interaction with mesoscale flow south of the Azores. J. Geophys. Res. Oceans, 125, e2019JC015879, https://doi.org/10.1029/2019JC015879.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Locarnini, R. A., and Coauthors, 2013: World Ocean Atlas 2013, Volume 1: Temperature. National Oceanographic Data Center, 40 pp., http://data.nodc.noaa.gov/woa/WOA13/DOC/woa13_vol1.pdf.

  • MacKinnon, J. A., and Coauthors, 2017: Climate process team on internal wave-driven ocean mixing. Bull. Amer. Meteor. Soc., 98, 2429–2454, https://doi.org/10.1175/BAMS-D-16-0030.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mitchum, G. T., and S. M. Chiswell, 2000: Coherence of internal tide modulations along the Hawaiian Ridge. J. Geophys. Res., 105, 28 653–28 661, https://doi.org/10.1029/2000JC900140.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Müller, M., J. Y. Cherniawsky, M. G. G. Foreman, and J.-S. von Storch, 2012: Global M2 internal tide and its seasonal variability from high resolution ocean circulation and tide modeling. Geophys. Res. Lett., 39, L19607, https://doi.org/10.1029/2012GL053320.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Munk, W. H., 1966: Abyssal recipes. Deep-Sea Res., 13, 707–730, https://doi.org/10.1016/0011-7471(66)90602-4.

  • Munk, W. H., 1981: Internal waves and small-scale processes. Evolution of Physical Oceanography, B. A. Warren and C. Wunsch, Eds., MIT Press, 264–291.

  • Munk, W. H., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res. I, 45, 1977–2010, https://doi.org/10.1016/S0967-0637(98)00070-3.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nash, J., E. Shroyer, S. Kelly, M. Inall, T. Duda, M. Levine, N. Jones, and R. Musgrave, 2012: Are any coastal internal tides predictable? Oceanography, 25, 80–95, https://doi.org/10.5670/oceanog.2012.44.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nycander, J., 2005: Generation of internal waves in the deep ocean by tides. J. Geophys. Res., 110, C10028, https://doi.org/10.1029/2004JC002487.

  • Osborne, J. J., A. L. Kurapov, G. D. Egbert, and P. M. Kosro, 2011: Spatial and temporal variability of the M2. J. Phys. Oceanogr., 41, 2037–2062, https://doi.org/10.1175/JPO-D-11-02.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Qiu, B., S. Chen, P. Klein, J. Wang, H. Torres, L.-L. Fu, and D. Menemenlis, 2018: Seasonality in transition scale from balanced to unbalanced motions in the world ocean. J. Phys. Oceanogr., 48, 591–605, https://doi.org/10.1175/JPO-D-17-0169.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., 2013: Precise comparisons of bottom-pressure and altimetric ocean tides. J. Geophys. Res. Oceans, 118, 4570–4584, https://doi.org/10.1002/jgrc.20336.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and G. T. Mitchum, 1996: Surface manifestation of internal tides generated near Hawaii. Geophys. Res. Lett., 23, 2101–2104, https://doi.org/10.1029/96GL02050.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and G. T. Mitchum, 1997: Surface manifestation of internal tides in the deep ocean: Observations from altimetry and island gauges. Prog. Oceanogr., 40, 135–162, https://doi.org/10.1016/S0079-6611(97)00025-6.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and E. D. Zaron, 2011: Non-stationary internal tides observed with satellite altimetry. Geophys. Res. Lett., 38, L17609, https://doi.org/10.1029/2011GL048617.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Ray, R. D., and E. Zaron, 2016: M2 internal tides and their observed wavenumber spectra from satellite altimetry. J. Phys. Oceanogr., 46, 3–22, https://doi.org/10.1175/JPO-D-15-0065.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Shang, X., Q. Liu, X. Xie, G. Chen, and R. Chen, 2015: Characteristics and seasonal variability of internal tides in the southern South China Sea. Deep-Sea Res. I, 98, 43–52, https://doi.org/10.1016/j.dsr.2014.12.005.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Smith, W. H. F., and D. T. Sandwell, 1997: Global sea floor topography from satellite altimetry and ship depth soundings. Science, 277, 1956–1962, https://doi.org/10.1126/science.277.5334.1956.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Song, P., and X. Chen, 2020: Investigation of the internal tides in the Northwest Pacific Ocean considering the background circulation and stratification. J. Phys. Oceanogr., 50, 3165–3188, https://doi.org/10.1175/JPO-D-19-0177.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vic, C., and Coauthors, 2019: Deep-ocean mixing driven by small-scale internal tides. Nat. Commun., 10, 2099, https://doi.org/10.1038/s41467-019-10149-5.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vic, C., B. Ferron, V. Thierry, H. Mercier, and P. Lherminier, 2021: Tidal and near-inertial internal waves over the Reykjanes Ridge. J. Phys. Oceanogr., 51, 419–437, https://doi.org/10.1175/JPO-D-20-0097.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, J., L.-L. Fu, H. S. Torres, S. Chen, B. Qiu, and D. Menemenlis, 2019: On the spatial scales to be resolved by the surface water and ocean topography Ka-band radar interferometer. J. Atmos. Oceanic Technol., 36, 87–99, https://doi.org/10.1175/JTECH-D-18-0119.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Whalen, C. B., C. de Lavergne, A. C. Naveira Garabato, J. M. Klymak, J. A. MacKinnon, and K. L. Sheen, 2020: Internal wave-driven mixing: Governing processes and consequences for climate. Nat. Rev. Earth Environ., 1, 606–621, https://doi.org/10.1038/s43017-020-0097-z.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wunsch, C., 1975: Internal tides in the ocean. Rev. Geophys. Space Phys., 13, 167–182, https://doi.org/10.1029/RG013i001p00167.

  • Wunsch, C., 2013: Baroclinic motions and energetics as measured by altimeters. J. Atmos. Oceanic Technol., 30, 140–150, https://doi.org/10.1175/JTECH-D-12-00035.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yan, T., Y. Qi, Z. Jing, and S. Cai, 2020: Seasonal and spatial features of barotropic and baroclinic tides in the northwestern South China Sea. J. Geophys. Res. Oceans, 125, e2018JC014860, https://doi.org/10.1029/2018JC014860.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zaron, E. D., 2015: Nonstationary internal tides observed using dual-satellite altimetry. J. Phys. Oceanogr., 45, 2239–2246, https://doi.org/10.1175/JPO-D-15-0020.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zaron, E. D., 2017: Mapping the nonstationary internal tide with satellite altimetry. J. Geophys. Res. Oceans, 122, 539–554, https://doi.org/10.1002/2016JC012487.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zaron, E. D., 2019: Baroclinic tidal sea level from exact-repeating mission altimetry. J. Phys. Oceanogr., 49, 193–210, https://doi.org/10.1175/JPO-D-18-0127.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., 2014: Internal tide radiation from the Luzon Strait. J. Geophys. Res. Oceans, 119, 5434–5448, https://doi.org/10.1002/2014JC010014.

  • Zhao, Z., 2016a: Internal tide oceanic tomography. Geophys. Res. Lett., 43, 9157–9164, https://doi.org/10.1002/2016GL070567.

  • Zhao, Z., 2016b: Using CryoSat-2 altimeter data to evaluate M2 internal tides observed from multisatellite altimetry. J. Geophys. Res. Oceans, 121, 5164–5180, https://doi.org/10.1002/2016JC011805.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., 2017a: The global mode-1 S2 internal tide. J. Geophys. Res. Oceans, 122, 8794–8812, https://doi.org/10.1002/2017JC013112.

  • Zhao, Z., 2017b: Propagation of the semidiurnal internal tide: Phase velocity versus group velocity. Geophys. Res. Lett., 44, 11 942–11 950, https://doi.org/10.1002/2017GL076008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., 2018: The global mode-2 M2 internal tide. J. Geophys. Res. Oceans, 123, 7725–7746, https://doi.org/10.1029/2018JC014475.

  • Zhao, Z., 2019: Mapping internal tides from satellite altimetry without blind directions. J. Geophys. Res. Oceans, 124, 8605–8625, https://doi.org/10.1029/2019JC015507.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., M. H. Alford, R.-C. Lien, M. C. Gregg, and G. S. Carter, 2012: Internal tides and mixing in a submarine canyon with time-varying stratification. J. Phys. Oceanogr., 42, 2121–2142, https://doi.org/10.1175/JPO-D-12-045.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhao, Z., M. H. Alford, J. B. Girton, L. Rainville, and H. L. Simmons, 2016: Global observations of open-ocean mode-1 M2 internal tide. J. Phys. Oceanogr., 46, 1657–1684, https://doi.org/10.1175/JPO-D-15-0105.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhou, X.-H., D.-P. Wang, and D. Chen, 2015: Validating satellite altimeter measurements of internal tides with long-term TAO/TRITON buoy observations at 2°S–156°E. Geophys. Res. Lett., 42, 4040–4046, https://doi.org/10.1002/2015GL063669.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zweng, M., and Coauthors, 2013: Salinity. Vol. 2, World Ocean Atlas 2013, NOAA Atlas NESDIS 74, 39 pp., https://www.ncei.noaa.gov/data/oceans/woa/WOA13/DOC/woa13_vol2.pdf.

  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Seasonal variations of wavelengths of mode-1 M2 internal tides. Wavelengths are calculated from WOA2013 seasonal and seasonal-mean hydrographic profiles. Seasonal anomalies denote the differences between the seasonal and seasonal-mean wavelengths. Shown are the (a) seasonal-mean wavelength, (b) winter wavelength, (c) winter anomaly, (d) spring anomaly, (e) summer anomaly, and (f) fall anomaly.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Mapping mode-1 M2 internal tides from satellite altimetry (the winter model is shown as an example): variance of the (a) raw satellite altimeter data. (b) preprocessed satellite altimeter data by along-track high-pass filtering (step 1) and harmonic analysis (step 2), (c) M2 internal tides obtained by plane wave analysis (step 3), and (d) M2 internal tides after 2D spatial filtering (step 4). In (a)–(d), SSH variances are spatially averaged in 250-km by 250-km windows and shown in logarithmic scale. Also shown are snapshot M2 internal tide fields (e) obtained by plane wave analysis (step 3) and (f) after 2D spatial filtering (step 4). In (a)–(f), the black contours indicate regions of strong currents. The numbers in (a)–(d) are the global mean variances (excluding regions of strong currents).

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    The M2 internal tide models from satellite altimetry (the SSH amplitudes are shown in logarithmic scale) for (a) winter, (b) spring, (c) summer, (d) fall, (e) the seasonal-mean model (vector mean of four seasonal models), and (f) the 25-yr-coherent model. The black contours indicate regions of strong boundary currents, where all models are not reliable.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Model errors: (a),(b) amplitude errors and (c),(d) phase errors in the (left) 25-yr-coherent model and (right) winter model. Different color map ranges are used for better comparisons. For amplitude, they differ by a factor of 2. For phase, they differ by a shift of 3°.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Variance reductions (binned into windows of 2° longitude by 2° latitude for six internal tide models) obtained by making internal tide correction to 9 years of CryoSat-2 data, shown for (a) winter, (b) spring, (c) summer, (d) fall, (e) the seasonal-mean model, and (f) the 25-yr-coherent model. The global mean variance reductions are given in the inset boxes. The black contours indicate regions of strong boundary currents (excluded from global integrations).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Global mean variance reductions obtained by making internal tide correction to CryoSat-2 seasonal subsets, shown for (a) four seasonal models [each model reduces variance the most in its own season (filled symbols) and the least in its opposite season] and (b) three seasonally variable models. SNa is by definition composed of four seasonal peaks in (a).

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Global maps of variance reduction differences. The 4-season average denotes the mean variance reduction of the four seasonal models SN1–SN4. The numbers in the insets are the global mean variance reduction differences (regions of strong boundary currents are excluded). Shown are (a) SNa minus 4-seasonal average, (b) SNb minus 4-seasonal average, (c) SNa minus SN0, (d) and SNb minus SN0.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Seasonal differences of M2 internal tides: (a),(c) winter minus summer and (b),(d) spring minus fall for (top) amplitude and (bottom) phase. The differences are calculated pointwise and are smoothed by 7-point running mean in both the zonal and meridional directions. Internal tides with amplitudes of <2 mm are discarded. Regions of strong boundary currents are excluded.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Seasonal amplitude variations of M2 internal tides: (top) winter minus summer and (bottom) spring minus fall for the (a),(c) northward component (45°–135°) and (b),(d) southward component (225°–315°). The differences are calculated pointwise and are smoothed by 7-point running mean in both the zonal and meridional directions. Internal tides with amplitudes of <1 mm are discarded. Regions of strong boundary currents are excluded. Cophase charts are shown as gray lines.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Seasonal phase anomalies of the northward component (45°–135°) for (a) winter, (b) spring, (c) summer, and (d) fall. Seasonal phase anomalies are defined to be the differences between the four seasonal models and the seasonal-mean model. Pointwise differences are smoothed by two-dimensional 7-point running mean. Internal tides with amplitudes of <1 mm are discarded. Regions of strong boundary currents are excluded. Cophase charts are shown as gray lines.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    As in Fig. 10 but for the southward M2 internal tide component (225°–315°). Note that the largest phase anomalies of ~±120° occur in the Arabian Sea, where colors are saturated.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Incoherence of M2 internal tides induced by seasonal phase variations. It is calculated pointwise and is smoothed by two-dimensional 5-point running mean. Shown are the (a) northward component (45°–135°) and (b) southward component (225°–315°).

Fig. 1.

Seasonal variations of wavelengths of mode-1 M2 internal tides. Wavelengths are calculated from WOA2013 seasonal and seasonal-mean hydrographic profiles. Seasonal anomalies denote the differences between the seasonal and seasonal-mean wavelengths. Shown are the (a) seasonal-mean wavelength, (b) winter wavelength, (c) winter anomaly, (d) spring anomaly, (e) summer anomaly, and (f) fall anomaly.
Fig. 2.

Mapping mode-1 M2 internal tides from satellite altimetry (the winter model is shown as an example): variance of the (a) raw satellite altimeter data. (b) preprocessed satellite altimeter data by along-track high-pass filtering (step 1) and harmonic analysis (step 2), (c) M2 internal tides obtained by plane wave analysis (step 3), and (d) M2 internal tides after 2D spatial filtering (step 4). In (a)–(d), SSH variances are spatially averaged in 250-km by 250-km windows and shown in logarithmic scale. Also shown are snapshot M2 internal tide fields (e) obtained by plane wave analysis (step 3) and (f) after 2D spatial filtering (step 4). In (a)–(f), the black contours indicate regions of strong currents. The numbers in (a)–(d) are the global mean variances (excluding regions of strong currents).
Fig. 3.

The M2 internal tide models from satellite altimetry (the SSH amplitudes are shown in logarithmic scale) for (a) winter, (b) spring, (c) summer, (d) fall, (e) the seasonal-mean model (vector mean of four seasonal models), and (f) the 25-yr-coherent model. The black contours indicate regions of strong boundary currents, where all models are not reliable.
Fig. 4.

Model errors: (a),(b) amplitude errors and (c),(d) phase errors in the (left) 25-yr-coherent model and (right) winter model. Different color map ranges are used for better comparisons. For amplitude, they differ by a factor of 2. For phase, they differ by a shift of 3°.
Fig. 5.

Variance reductions (binned into windows of 2° longitude by 2° latitude for six internal tide models) obtained by making internal tide correction to 9 years of CryoSat-2 data, shown for (a) winter, (b) spring, (c) summer, (d) fall, (e) the seasonal-mean model, and (f) the 25-yr-coherent model. The global mean variance reductions are given in the inset boxes. The black contours indicate regions of strong boundary currents (excluded from global integrations).
Fig. 6.

Global mean variance reductions obtained by making internal tide correction to CryoSat-2 seasonal subsets, shown for (a) four seasonal models [each model reduces variance the most in its own season (filled symbols) and the least in its opposite season] and (b) three seasonally variable models. SNa is by definition composed of four seasonal peaks in (a).
Fig. 7.

Global maps of variance reduction differences. The 4-season average denotes the mean variance reduction of the four seasonal models SN1–SN4. The numbers in the insets are the global mean variance reduction differences (regions of strong boundary currents are excluded). Shown are (a) SNa minus 4-seasonal average, (b) SNb minus 4-seasonal average, (c) SNa minus SN0, (d) and SNb minus SN0.
Fig. 8.

Seasonal differences of M2 internal tides: (a),(c) winter minus summer and (b),(d) spring minus fall for (top) amplitude and (bottom) phase. The differences are calculated pointwise and are smoothed by 7-point running mean in both the zonal and meridional directions. Internal tides with amplitudes of <2 mm are discarded. Regions of strong boundary currents are excluded.
Fig. 9.

Seasonal amplitude variations of M2 internal tides: (top) winter minus summer and (bottom) spring minus fall for the (a),(c) northward component (45°–135°) and (b),(d) southward component (225°–315°). The differences are calculated pointwise and are smoothed by 7-point running mean in both the zonal and meridional directions. Internal tides with amplitudes of <1 mm are discarded. Regions of strong boundary currents are excluded. Cophase charts are shown as gray lines.
Fig. 10.

Seasonal phase anomalies of the northward component (45°–135°) for (a) winter, (b) spring, (c) summer, and (d) fall. Seasonal phase anomalies are defined to be the differences between the four seasonal models and the seasonal-mean model. Pointwise differences are smoothed by two-dimensional 7-point running mean. Internal tides with amplitudes of <1 mm are discarded. Regions of strong boundary currents are excluded. Cophase charts are shown as gray lines.
Fig. 11.

As in Fig. 10 but for the southward M2 internal tide component (225°–315°). Note that the largest phase anomalies of ~±120° occur in the Arabian Sea, where colors are saturated.
Fig. 12.

Incoherence of M2 internal tides induced by seasonal phase variations. It is calculated pointwise and is smoothed by two-dimensional 5-point running mean. Shown are the (a) northward component (45°–135°) and (b) southward component (225°–315°).
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