Rossby and Yanai Modes of Tropical Instability Waves in the Equatorial Pacific Ocean and a Diagnostic Model for Surface Currents

Minyang Wang State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China
University of Chinese Academy of Sciences, Beijing, China
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
Department of Mathematics and Statistics, San Diego State University, San Diego, California

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Shang-Ping Xie Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Samuel S. P. Shen Department of Mathematics and Statistics, San Diego State University, San Diego, California
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California

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Yan Du State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China
University of Chinese Academy of Sciences, Beijing, China
Southern Marine Science and Engineering Guangdong Laboratory, Guangzhou, China

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Abstract

Mesoscale activities over the equatorial Pacific Ocean are dominated by the Rossby and Yanai modes of tropical instability waves (TIWs). The TIW-induced surface velocity has not been accurately estimated in previous diagnostic models, especially for the meridional component across the equator. This study develops a diagnostic model that retains the acceleration terms to estimate the TIW surface velocity from the satellite-observed sea surface height. Validated against moored observations, the velocity across the equator is accurately estimated for the first time, much improved from existing products. The results identify the Rossby- and Yanai-mode TIWs as the northwest–southeastward (NW–SE) velocity oscillations north of the equator and the northeast–southwestward (NE–SW) velocity oscillations on the equator, respectively. Barotropic instability is the dominant energy source of the two TIW modes. The NE–SW velocity oscillation of the Yanai mode is associated with the counterclockwise shear of the South Equatorial Current on the equator. The two TIW modes induce different sea surface temperature patterns and vertical motions. Accurate estimates of TIW velocity are important for studying equatorial ocean dynamics and climate variability in the tropical Pacific Ocean.

Corresponding author: Yan Du, duyan@scsio.ac.cn

Abstract

Mesoscale activities over the equatorial Pacific Ocean are dominated by the Rossby and Yanai modes of tropical instability waves (TIWs). The TIW-induced surface velocity has not been accurately estimated in previous diagnostic models, especially for the meridional component across the equator. This study develops a diagnostic model that retains the acceleration terms to estimate the TIW surface velocity from the satellite-observed sea surface height. Validated against moored observations, the velocity across the equator is accurately estimated for the first time, much improved from existing products. The results identify the Rossby- and Yanai-mode TIWs as the northwest–southeastward (NW–SE) velocity oscillations north of the equator and the northeast–southwestward (NE–SW) velocity oscillations on the equator, respectively. Barotropic instability is the dominant energy source of the two TIW modes. The NE–SW velocity oscillation of the Yanai mode is associated with the counterclockwise shear of the South Equatorial Current on the equator. The two TIW modes induce different sea surface temperature patterns and vertical motions. Accurate estimates of TIW velocity are important for studying equatorial ocean dynamics and climate variability in the tropical Pacific Ocean.

Corresponding author: Yan Du, duyan@scsio.ac.cn

1. Introduction

Tropical instability waves (TIWs) are the dominant mesoscale (700–2000 km) variability in the eastern equatorial Pacific Ocean, internally generated by the meridional shears of background zonal currents (Philander 1978; Yu et al. 1995; Lyman et al. 2007; Zhou and Boyd 2009). Velocity observations revealed two distinct modes of TIWs in the region with periods of ~28–35 days (Wyrtki 1978; Miller et al. 1985; McPhaden 1996; Kennan and Flament 2000) and ~15–23 days (Düing et al. 1975; Halpern et al. 1988; Qiao and Weisberg 1995), referred to as the 33- and 17-day TIWs by Lyman et al. (2007). The two modes are characterized by the ~33-day tropical instability vortices (TIVs; Flament et al. 1996) near 5°N and the ~17-day meridional velocity oscillations on the equator, respectively. Both TIW modes contribute to the sea surface temperature (SST) disturbances along the north equatorial front (Legeckis 1977; Pullen et al. 1987; Chelton et al. 2000), which alter the surface winds (Xie et al. 1998; Chelton et al. 2001) and may impact El Niño–Southern Oscillation (ENSO) via TIW-induced ocean heat transport (Jochum et al. 2007; An 2008; Imada and Kimoto 2012; Holmes et al. 2019; Xue et al. 2020). While the 33-day TIW velocity is easily estimated from satellite-observed sea surface height (SSH) via geostrophic approximation, the 17-day TIW velocity has not been well produced from satellite observations (Johnson et al. 2007).

Nonlinear numerical models and linear stability analyses have suggested that the 33- and 17-day TIW velocity perturbations are pressure-driven and vertically amplified near the surface (McCreary and Yu 1992; Donohue and Wimbush 1998; Masina and Philander 1999; Lyman et al. 2005, 2007; Zhou and Boyd 2009). SSH variability associated with the 33-day TIWs is much stronger north of the equator centered at ~5°N. The 17-day TIWs have an antisymmetric structure of SSH or thermocline depth centered around 2°N and 2°S. The two TIW modes resemble the first meridional mode of Rossby waves and the Yanai waves (Matsuno 1966; Yanai and Maruyama 1966), respectively, although their phases are tilted (Liu et al. 2019) and amplitudes are asymmetric about the equator (Lyman et al. 2007). Lyman et al. (2007) interpreted the 33- and 17-day TIWs as the unstable Rossby and Yanai waves, respectively. The altimeter-observed SSH have confirmed that the two TIW modes have dispersion characteristics similar to the westward-propagating waves with a Kelvin wave phase speed of ~2.8 m s−1 (Farrar 2008; Shinoda et al. 2009; Shinoda 2010). For their distinct patterns, we call the 33- and 17-day TIWs as the Rossby and Yanai modes hereafter, respectively.

Yanai waves are characterized by the ageostrophic meridional velocity on the equator (Yanai and Hayashi 1969). In the classic theory of Matsuno (1966), the equatorial meridional velocity of Yanai waves lags the northward pressure gradient by 90° in phase, achieving ageostrophic balance through the local acceleration term ∂υ/∂t. The quasi-linear and steady diagnostic model of Bonjean and Lagerloef (2002) successfully calculated the low-frequency equatorial zonal currents, but not the equatorial meridional currents (Johnson et al. 2007), probably due to the lack of acceleration terms in their model. Using a nonlinear one-and-a-half-layer shallow water model, Zhou and Boyd (2009) reproduced Yanai-mode TIWs with a pattern similar to the classic Yanai waves. In addition to the pressure-driven component, the TIW-SST-induced surface wind anomalies were suggested to feedback on TIW currents by high-resolution coupled model simulations (Seo et al. 2007; Small et al. 2009). While the Rossby-mode TIWs are widely known, the spatial structure and variability of the Yanai-mode TIWs remain to be derived from observations.

This study investigates the spatial patterns and estimates the surface velocity of the TIW modes using satellite observations. We identify the TIW modes via a joint empirical orthogonal function (EOF) analysis of SSH and SST and compare against velocity observations from equatorial moorings. The new diagnostic model with the critically important acceleration terms is able to accurately produce the TIW velocity on the equator from satellite observed SSH, as highly correlated with the moored velocity. Such improvements in estimating the Pacific TIW currents may be useful for future investigations of TIW-induced effects on the ocean heat (Jochum et al. 2007), freshwater (Lee et al. 2012) and nutrient (Strutton et al. 2001) transports, and the mesoscale air–sea interaction (Small et al. 2008).

The next section introduces the satellite and in situ observations. Section 3 presents the dispersion characteristics and spatial patterns of the Yanai- and Rossby-mode TIWs. Section 4 develops a diagnostic model to calculate the equatorial currents associated with TIWs. Section 5 is a summary with discussions.

2. Data and their preprocessing

a. Satellite and model data

We use the following satellite datasets to identify surface patterns of TIWs in the equatorial Pacific:

  1. daily SSH from the Archiving, Validation, and Interpretation of Satellite Oceanographic Data (AVISO), merged from multiple satellite altimeters (Le Traon et al. 1998), available since 1993 on a 1/4° × 1/4° grid;

  2. daily SST from the NOAA 1/4° Optimum Interpolation SST (OISST), blended with the Advanced Very High Resolution Radiometer (AVHRR) data (Reynolds et al. 2002), available since 1982; and

  3. surface current from the 5-day Ocean Surface Current Analysis Real-Time (OSCAR) satellite field-derived ocean surface currents on 1/3° × 1/3° grids (Bonjean and Lagerloef 2002), available since 1992.

The Cube92 model output of Estimating the Circulation and Climate of the Ocean, phase II (ECCO2), is used to develop a diagnostic model to estimate the equatorial currents associated with TIWs. The data include 2D SSH; SST; wind stress; 3D temperature; salinity; and zonal, meridional, and vertical components of velocity on a 1/4° × 1/4° daily grid since 1992. The ECCO2 data assimilation analyses are obtained by least squares fit of the Massachusetts Institute of Technology General Circulation Model (MITgcm; Marshall et al. 1997) configuration to the available satellite and in situ data (Menemenlis et al. 2008), of which the Cube92 model output is the optimized model solution.

The websites for the data access are listed in the acknowledgments. The SSH, SST, and OSCAR currents used here are from 1 January 1993 to 31 December 2018, and from 1992 to 2018 for ECCO2 output. Because TIWs have zonal wavelength larger than 600 km (wider than 300 km in latitude) and period longer than 14 days, all the datasets are linearly interpolated to a 3-day interval and on a 1° latitude–longitude grid prior to our analysis. The TIW signals, denoted with a prime hereafter, are isolated by temporal and spatial filtering at periods of 14–40 days and wavelengths of 600–3000 km, with only westward wavenumbers retained, via 2D discrete fast Fourier transform (2DFFT) (Farrar 2008). The Hovmöller diagram of SSH′ and SST′ along 140°W from April 2010 to April 2011 shows high-frequency, meridional oscillations within 4° of the equator (Fig. 1a).

Fig. 1.
Fig. 1.

April 2010–April 2011 segments of (a) SSH′ (shading) from AVISO and SST′ (contour; °C) from OISST along 140°W, (b) the reconstructed u′ (shading), and (c) the reconstructed υ′ (shading). The 20°C isotherms (black curve) in (b) and (c) are regarded as the thermocline at 140°W equator. The prime means TIW-band anomalies (section 2).

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

b. In situ observations

We use the velocity data from the 140° and 110°W equator moorings (referred to as “140°W equator” and “110°W equator”) in the Tropical Atmosphere Ocean (TAO) array (McPhaden et al. 1998) to obtain the surface velocity (defined as the mean at the 10–30-m-depth layer) and validate our diagnostic model. Zonal and meridional components of velocity are from the moored current meters (CrM) and acoustic Doppler current profilers (ADCP), and averaged daily. At 140°W equator—a hot spot of TIW currents—the CrM and ADCP measurements are generally available at depths of 10, 25, 45, 80, 120, 160, 200, and 250 m since 1983, and from 35 to 250 m at 5-m intervals since 1990, respectively. As TIW currents are amplified in the upper ocean (Lyman et al. 2007; Wang et al. 2019; Liu et al. 2019), only the measurements above 160 m are used.

Here the CrM and ADCP measurements are merged into a 5-m gridded dataset over 10–160-m depth during 1983–2018, by filling the gaps in ADCP data with the available CrM measurements, mostly at 10 and 25 m. Gaps within 2 days and vertically 50 m in the merged data are linearly interpolated. The TIW-associated zonal and meridional velocities, represented as u′ and υ′, are isolated by removing the seasonal cycle and bandpass filtering at periods of 14–40 days via 1DFFT. Note that the 14–40-day mooring velocities related to the large-scale processes (not TIWs), e.g., the intraseasonal Kelvin waves (peaking near the 60-day period) (Rydbeck et al. 2019), are neglected here. As a reference, the OSCAR 14–40-day, zonally ≥3000 km u′ is weak (standard deviation, STD ≤ 0.04 m s−1) in the eastern equatorial Pacific. In filtering, gaps in each profile are temporarily replaced by the nearest available values to guarantee the vertical consistency of the filtered data. The same approach is applied to the observations at 110°W equator. This produces 8334 (63%) and 7956 (55%) “good” profiles without gaps, out of 13 158 (year 1983–2018) and 14 278 (1980–2018) total profiles at 140° and 110°W equator, respectively. However, there are still a lot of “bad” profiles with gaps above 35 m, accounting for ~40% of the available profiles at two moorings.

To keep these profiles with gaps above 35 m, we introduce a spectral optimal gridding (SOG) method based on EOF analyses (Shen et al. 2017) to fill the missing data. The vertical patterns of u′ and υ′ at each mooring (dashed curves in Figs. 2c,f) are derived from the EOF analysis of u′ and υ′ on the good profiles over 10–160 m. The three leading EOFs, cumulatively accounting for more than 90% of the variance, are used as the vertical patterns to be linearly regressed against the bad profiles, which need to have at least five samples above 100 m to guarantee a stable solution and the ability to describe the TIW variability. The gaps in the bad profiles are then replaced by the regressed EOFs, increasing the number of good profiles to more than 90% of the total observations. The new EOF1s from the new good profiles (solid curves in Figs. 2c,f) are much similar to the vertical patterns. The velocities u′ and υ′ are finally reconstructed as the projections onto their first three new EOFs (more than 90% of the total variance) to ensure the vertical and temporal consistency of the data and are averaged at a 3-day interval. The reconstructed TIW velocities have almost the same amplitude and variability as the original observations. Figures 1b and 1c show the time–depth sections of the reconstructed u′ and υ′ at 140°W equator in one particular year. The in situ surface TIW velocities used hereafter are the upper 10–30-m averages of the reconstructed velocities.

Fig. 2.
Fig. 2.

Power spectral densities (PSDs) [log10 (m2 s−2 cpd−1); cpd is cycles per day] and EOF1s of the moored velocity at (top) 140°W equator and (bottom) 110°W equator. (a),(d) PSDs of the reconstructed u′; (b),(e) PSDs of the reconstructed υ′; and (c),(f) EOF1s of u′ (red curve) and υ′ (blue curve). The values in the brackets of the legends (e) and (f) are the variance explained in percentage. The solid (dashed) curves in (c) and (f) are based on the reconstructed (original) data from TAO (see the details in section 2b). The dashed black line indicates the mean thermocline depth.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

3. Characteristics of Rossby and Yanai modes

Figure 2 shows the power spectra (appendix A) of u′ and υ′ at 140° and 110°W equator. Compared with the typical 33-day u′ (Figs. 2a,d), the meridional velocity υ′ is of a relatively broadband and higher frequency, peaking around 17.5 days at 140°W (Fig. 2b) and 22 days at 110°W equator (Fig. 2e). The spectra suggest the existence of two TIW modes in the region: the 33-day variability and the higher-frequency meridional oscillations. However, the limited observations from moorings are not enough to identify their dispersion characteristics and spatial patterns. As will be clear, SSH′ and SST′ from satellite observations allow us to identify the two TIW modes.

a. Dispersion characteristics

Given that the two TIW modes have different SSH structures across the equator (Shinoda et al. 2009; Shinoda 2010), a cross-spectral analysis of AVISO SSH at 2°S and 2°N is used to distinguish them. The cross-power spectral density (C-PSD) between two normalized time series is calculated as the real part of the product of one Fourier transform and another transform’s complex conjugate. The positive (negative) C-PSD value implies the positive (negative) covariance between two time series with the given spectra. Figure 3a shows the mixed spatial–temporal cross spectra of SSH at 2°S and 2°N over 180°–90°W during the years 1993–2018. Figure 3b shows the 1D cross spectra of SSH′ between 2°S and 2°N computed at every longitude from 180° to 90°W. Figure 3c is the 1D cross spectra of surface υ′ at 0°N, 140°W with SSH′ along 140°W at different latitudes. Figures 3d–f are the same as Figs. 3a–c, respectively, but for SST from OISST.

Fig. 3.
Fig. 3.

Cross spectra (see the details in section 3a): (a) the mixed spatial–temporal cross spectra of SSH at 2°S and 2°N over 180°–90°W, “cp1000km” means cycles per 1000 km; (b) the cross spectra of SSH′ between 2°S and 2°N computed at every longitude from 180° to 90°W; and (c) the cross spectra of υ′ at 140°W equator with SSH′ along 140°W at different latitudes. (d)–(f) As in (a)–(c), but for SST. The unit of the shading in (a) and (d) is var cpd−1 cp1000km−1, and in (b), (c), (e), and (f) it is var cpd−1. The five black dashed lines in (a) and (d) are the dispersion relations of linear equatorial waves (Matsuno 1966) of the first baroclinic mode, with the phase velocity of long gravity wave at c = 2.8 m s−1: “R0,” “R1–3,” and “K” are abbreviations of Yanai, first–third meridional-mode Rossby, and Kelvin waves, respectively. The rectangular box in (a) and (d) is the TIW spectral domain: 14–40 days, 600–3000 km westward.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

The spatial–temporal cross spectra (Fig. 3a) and the longitude-varying cross spectra (Fig. 3b) of SSH highlight the antisymmetric SSH signals (negative values) in the high-frequency bands, where the PSD of SSH is generally weak (Farrar 2008). The antisymmetric signals exist between 160° and 100°W with periods of ~15–26 days and wavelengths of ~1000–3000 km. Specifically, the peaks of the periods are around 17.5 days at 140°W and 22 days at 110°W (Fig. 3b), corresponding to the periods of the equatorial υ′ from the two moorings (Figs. 2b,e). The cross spectra between the equatorial υ′ and SSH′ at 140°W demonstrate that the υ′ is related with the antisymmetric structures of SSH′ across the equator (Fig. 3c). The same is evident at 110°W. The antisymmetric signals are superimposed on the much stronger symmetric ones (positive values in Fig. 3b) in SSH, rendering it difficult to isolate the higher-frequency variability from SSH alone. While the antisymmetric SST signals exist in the broad bands (Figs. 3d,e), the cross spectra between the equatorial υ′ and SST′ at 140°W favor the higher-frequency variability (Fig. 3f), suggesting the possibility to identify the Yanai mode from joint SSH′ and SST′ analysis.

b. Spatial patterns

We perform a joint EOF analysis on SSH′ and SST′ during the 26-yr period of 1993–2018 in the domain centered at 140°W equator (4°S–4°N, 145°–135°W). SSH′ and SST′ are divided by their domain-averaged standard deviations before conducting the EOF analysis. The “140°W equator” region is selected because of the marked SSH structures across the equator (Figs. 3b,c) and the long record of moored observations at 0°N, 140°W. The EOFs display the spatial patterns associated with the two TIW modes and their principal components (PCs) are highly correlated with the equatorial surface currents from the mooring.

As TIWs are westward propagating waves, EOFs come in pairs to represent the propagation. The first four EOFs account for ~28%, 24%, 8%, and 7% of the total variance. To show their patterns against the moored velocity at 140°W equator, SSH′, SST′, u′, and υ′ are lead–lag regressed onto the corresponding PCs. The regressions on PC1 and PC3 are shown in Figs. 4a and 4b to present the leading two pairs of EOFs. The regression results on PC2 and PC4 are almost the same as those on PC1 and PC3, except with 90° phase shift in the zonal direction, indicating wave propagation. The surface velocity data on the equator are from the mooring, while for the region at 3° poleward of the equator, the velocity is calculated from SSH′, based on geostrophic balance. By their distinct SSH patterns, we identify the Rossby and Yanai modes as the first (Fig. 4a) and second (Fig. 4b) pairs of EOFs, respectively. For the Rossby mode, SSH signals are symmetric about the equator, but with much stronger amplitude (~4 cm) to the north. For the Yanai mode, SSH signals are nearly antisymmetric off the equator between 3°S and 3°N, with slightly stronger amplitude (~1 cm) to the south. The asymmetry in SSH amplitude for two modes is consistent with the meridional structure of the temperature amplitude at the thermocline (Lyman et al. 2007). An additional SSH signal (~1.5 cm) exists between 4° and 8°N in the Yanai-mode EOFs (Fig. 4b), which is also evident in the cross spectra of SSH′ on υ′ (Fig. 3c). The generation mechanism of that signal remains unclear, although the signal might be related to the dynamic interaction between the two modes (Zhou and Boyd 2009).

Fig. 4.
Fig. 4.

(a),(b) Lead–lag regressions of SSH′ (shading), SST′ (contour; °C), and surface velocities (arrow) along 140°W onto PC1 and PC3, respectively. (c) PSDs of PC1 (red) and PC3 (blue); the dashed curves represent the 95% confidence level based on the F test. (d) Maxima of the absolute lag-correlation coefficients between the first 10 PCs and u′ (red), υ′ (blue) at 140°W equator; the dashed curves represent the corresponding 95% confidence level of the correlations based on the Student’s t test. As the EOFs come in pairs, the regressions on PC1 and PC3 are shown to represent the EOF1/2 and EOF3/4. The surface velocities are from the mooring at 140°W equator and from SSH′ at 3° poleward of the equator based on geostrophic balance. All the plots are based on the joint EOF analysis of SSH′ and SST′ in the domain of 4°S–4°N, 145°–135°W during years 1993–2018.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

The PSD of PC1 peaks around 33 days, while PC3 is broadband, with two peaks at ~17.5 and 33 days (Fig. 4c), consistent with those of u′ and υ′ at 140°W equator (Figs. 2a,b), respectively. The high lag correlations between PCs and (u′, υ′) suggest that the Rossby- and Yanai-mode PCs explain much of u′ and υ′ on the equator (Fig. 4d), respectively.

The Yanai-mode TIWs show the northeast–southwestward (NE–SW) velocity oscillations (Liu et al. 2019; Fig. 4b). The Rossby mode is found to induce the NE–SW velocity oscillations on the equator also, but is dominated by the northwest–southeastward (NW–SE) velocity oscillations north of the equator (Fig. 4a). The equatorial υ′ of the Rossby mode has an amplitude of 0.08 m s−1 on average, slightly weaker than u′ (0.11 m s−1). The cross-equatorial velocity of the Rossby mode is ageostrophic and possibly related to the meridional shears of zonal currents on the equator. The Yanai mode is dominated by υ′ across the equator, with an amplitude of 0.18 m s−1 compared to 0.06 m s−1 for u′. The relationship between υ′ and SSH′ in the Yanai mode (Fig. 4b) resemble the theoretical Yanai waves, where υ′ lags the pressure gradient in phase (Matsuno 1966). However, whereas the theoretical neutral Yanai υ′ lags the pressure gradient by 90°, we find a lag of ~45° (Fig. 4b), indicating a westward (forward) advection effect on the mesoscale currents by the westward background zonal currents.

4. Diagnostic model for TIW currents

Mesoscale surface currents can be estimated from satellite observations with diagnostic models (Bonjean and Lagerloef 2002). The OSCAR dataset, one of the most widely used surface current products, represents the low-frequency equatorial zonal currents well, but not the equatorial meridional currents across frequencies (Johnson et al. 2007, their Fig. 3) and the high-frequency [>(33 day)−1] equatorial zonal currents. The equatorial meridional currents are dominated by the ageostrophic high-frequency oscillations associated with the Yanai-mode TIWs (Lyman et al. 2007) and cannot be captured by the steady diagnostic model of OSCAR. In this section, we develop an equatorial time-varying diagnostic model based on the ECCO2 output and estimate the TIW currents from satellite observations.

a. Model and validation based on ECCO2

The ECCO2 model is physically consistent and can well produce two TIW modes, whose spatial patterns are similar to the observational results (Figs. 4a,b). The spatial-temporal spectra of TIW-associated surface u′ and υ′ from ECCO2 are elevated at periods of 14–40 days and wavelengths of 600–3000 km (figure not shown), consistent with the satellite observations (Fig. 2a), although the mean amplitudes of the ECCO2 TIW velocities (~0.18 m s−1; Figs. B2c,d) are only ~70% of the observations (~0.25 m s−1; Figs. 2c,f). We use the ECCO2 output to develop a diagnostic model for the TIW currents, by simplifying the momentum equations (see the details in appendix B). The momentum balance based on the ECCO2 output shows that the equatorial TIW currents are dominantly pressure-driven and influenced by the background zonal advection (Fig. B1). After neglecting wind stress, vertical viscosity and other nonlinear terms, our diagnostic model is as follows:

(t+δx)u+Uux(fUy)υ=ghx,
(t+δy)υ+Uυx+fu=ghy,

where velocity is vertically averaged in the upper 30-m layer. The term U(x, y, t) denotes the basic state of zonal current, obtained by zonally 3000-km and 40-day low-pass filtering. The subscripts (x and y) mean the zonal and meridional partial differentiations, respectively. The notation h′ denotes SSH′. The Coriolis parameter is f = βy, with β = 2.3 × 10−11 s−1 m−1, and y is the distance northward from the equator. The gravity acceleration is g = 9.8 m s−2. The Rayleigh damping rates are δx = (7 day)−1 and δy = (4 day)−1, based on the ECCO2 output (see appendix B). The damping rates may be adjusted with the temporal and spatial resolutions of the diagnostic model to achieve realistic amplitudes of u′ and υ′: A higher resolution corresponds to a larger damping rate. The difference between δx and δy indicates the different time of the adjustment of geostrophic balance for the Rossby and Yanai modes, respectively. The diagnostic model is able to estimate the high-frequency, ageostrophic pressure-driven currents on the equator through the acceleration terms. It transitions to geostrophic balance at ~3° poleward of the equator where the Coriolis force becomes dominant.

Fig. 5.
Fig. 5.

Standard deviations (STDs) of: (a) u′ and (b) υ′ estimated from AVISO SSH′ and OSCAR U with the new diagnostic model [Eq. (1)]; (c) u′ and (d) υ′ from OSCAR. The scales of the color bars in (a),(b) and (c),(d) are different. Correlation coefficients between the new diagnostic model results and OSCAR: (e) for u′ and (f) for υ′. The dashed contours in (e) and (f) indicate the 0.70 level of correlation coefficient. The 95% confidence level of the correlation is at most 0.20 in the whole region. The two purple squares in (a) and (b) indicate the equatorial mooring sites at 140° and 110°W for the in situ validations.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

Fig. 6.
Fig. 6.

In situ validations of the new diagnostic υ′ (red) with TAO observations (blue), and comparison with OSCAR (green) at (top) 140°W equator and (bottom) 110°W equator: (a),(d) PSDs; (b),(e) coherence spectra with TAO; and (c),(f) lag correlation with TAO. The error bars in (a), (b), (d), and (e) are the corresponding half STDs (appendix A), indicating the 95% confidence interval. The 95% confidence level of the correlations in (c) and (f) is ~0.20 at two locations.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

The ECCO2 output is used to validate the diagnostic model in estimating the surface TIW currents. The input of h′ and U is from ECCO2. The model [Eq. (1)] is numerically solved on the C-grid and a semi-implicit time scheme to calculate u′ and υ′. The boundary and initial conditions are simply set to zero, which only impact the very-near-boundary areas because of the Rayleigh damping. The estimated u′ and υ′ show high correlations with ECCO2 in most TIW-active regions (Figs. B2a–d), with correlation coefficients larger than 0.70 (Figs. B2e,f). The poor performance east of ~120°W, especially in u′, is probably due to the rough parameterization of the vertical viscosity terms, which is large in the eastern equatorial Pacific (the cyan lines in Fig. B1c). The diagnostic model is also independently tested by using another model output: the QuikSCAT-run eddy-resolving OGCM for the Earth Simulator (OFES) during 1999–2009 (Masumoto 2010; Wang et al. 2017). The cross correlations between the diagnostic results and the OFES TIW velocities (figure not shown) have the correlation coefficients and spatial patterns similar to those of ECCO2 (Figs. B2e,f).

b. Estimating TIW currents with the new diagnostic model

TIW currents are estimated from the satellite observed SSH using the new diagnostic model [Eq. (1)] and validated with TAO observations. The input, low-frequency flow U is from OSCAR, and h′ is from AVISO. Here we do not consider the mapping errors (Ducet et al. 2000) in AVISO. The mapping errors may lead to systematic errors on hx near the equator due to the cross-track gaps in altimeter observations, which would influence the estimation of u′ on the equator. Using the diagnostic model with input of h′ and U, we compute u′ and υ′ over the domain of (170°E–85°W and 10°S–10°N) during 1993–2018, on a 3-day and 1° × 1° grid. The variance of the estimated u′ is high north of the equator (Fig. 5a), and that of υ′ is elevated in two regions: both north and slightly south (~0.5°S) of the equator (Fig. 5b). The amplitudes of our estimated u′ (~0.25 m s−1) and υ′ (~0.25 m s−1) agree with the mooring observations, and twice of those of OSCAR u′ (~0.15 m s−1) and υ′ (~0.12 m s−1) in the high variance regions (Figs. 5c,d). Our diagnostic velocities are highly correlated with OSCAR at ~3° poleward of the equator (Figs. 5e,f), where the geostrophic velocities dominate.

Fig. 7.
Fig. 7.

As in Fig. 6, but for u′.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

The equatorial υ′ (Fig. 6) and u′ (Fig. 7) are validated with TAO-mooring-observed velocity at 140° and 110°W equator. Our estimations are highly correlated with TAO in υ′, with correlation coefficients of 0.76 at 140°W equator and 0.70 at 110°W equator, much improved from the corresponding OSCAR–TAO correlations (0.49 and 0.47 at 140° and 110°W equator, respectively) (Figs. 6c,f). The PSDs of υ′ show that our diagnostic model captures the strong meridional oscillations comparable to the observations; but OSCAR cannot (Figs. 6a,d). The coherences of υ′ with the observations increase much in our model, especially near the peaking bands of the Yanai mode (Figs. 6b,e). Particularly, our model rectifies the time lag of 3 days in the OSCAR velocity (Figs. 6 and 7c,f), which is probably due to the lack of acceleration and zonal advection terms in OSCAR model. The correlation coefficients decrease to ~0.5 without the advection term in our model. It implies that the local time acceleration, and also the advection effect from the background zonal currents are important for the estimation of the high-frequency, ageostrophic meridional currents in this equatorial region.

Fig. 8.
Fig. 8.

(a) The meridional profile of the mean U from OSCAR at 140°W; (b),(c) linear regressions of SSH′ (shading), SST′ (contour), and the estimated velocity (arrow) onto PC1 and PC3, respectively; and (d),(e) the regressed w′ (shading) onto PC1 and PC3, respectively, with SST′ (contour) overlaid. The w′ is estimated from the divergence of the horizontal velocity, averaged in the upper 30 m layer. Panels (b)–(e) are based on the joint EOF analysis of the estimated u′ and υ′ in the domain of 8°S–8°N, 145°–135°W during 1993–2018.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

Fig. 9.
Fig. 9.

Meridional profiles of energetics for (left) the Rossby-mode TIWs and (right) the Yanai-mode TIWs along 140°W: EKE (black curves), barotropic conversion rate (BTR) (red curves), and baroclinic conversion rate (BCR) (blue curves). The error bars are the corresponding half STDs, indicating the 95% confidence interval. The results are based on the joint EOF analysis of the estimated u′ and υ′ in the domain of 8°S–8°N, 145°–135°W during 1993–2018.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

For the estimated u′, the correlation coefficients with TAO are 0.47 at 140°W equator and 0.49 at 110°W equator (0.47 and 0.24 for OSCAR at 140° and 110°W equator, respectively) (Figs. 7c,f). The 95% confidence level of correlation is ~0.20 at the two locations based on the Student’s t test with near 700 effective samples during 1993–2018. The lower correlations in u′ than υ′ are consistent with the ECCO2 validation results (Figs. B2e,f), and are also possibly due to the impact of mapping errors from AVISO on hx. The coherences (Figs. 7b,e) of u′ with TAO suggest that our model performs better than OSCAR in the Yanai-mode bands: ~(17.5 day)−1 at 140°W equator and ~(22 day)−1 at 110°W equator, but functions slightly less well in the low-frequency band: <(33 day)−1. The reason is that our model is designed to estimate the high-frequency ageostrophic pressure-driven currents, but not the low-frequency geostrophic zonal currents on the equator. The u′ at 110°W equator has strong ageostrophic component of the Yanai-mode TIWs (Fig. 2d), for which our model has a higher skill than OSCAR (Fig. 7f). While the u′ at 140°W equator is dominated by the low-frequency geostrophic component of the Rossby-mode TIWs (Fig. 2a), for which OSCAR can yield a better estimate (Fig. 7c).

The diagnostic model is evaluated additionally against the three moorings on the 125°W meridional line at latitudes 2°S, equator, and 2°N from the TAO array. The three moorings have available velocity observations from the current meters at 10 m only during 2006–07, although not as long as the 140° and 110°W equator observations. Validated with the bandpass filtered mooring velocities during 2006–07, our diagnostic υ′ has correlation coefficients of 0.69, 0.80, and 0.81 (0.76, 0.39, and 0.87 for OSCAR) at 2°N, 125°W; equator; and 2°S, respectively, with a better match to the TAO velocity amplitudes than OSCAR at the three locations (figures not shown). Our model performs better on the estimation of equatorial υ′ than OSCAR, consistent with the results at 140° and 110°W. In terms of u′, our model has correlation coefficients of 0.17, 0.43, and 0.30 (0.69, 0.16, and 0.26 for OSCAR) at 2°N, equator, and 2°S, respectively, with a better strength than OSCAR. The 95% confidence level of correlation is ~0.40 at the three locations given the short time series length. The results indicate that our diagnostic u′ and υ′ during 2006–07 is less accurate at 2° off the equator along 125°W than on the equator. The cross-equatorial pattern of the model performance is consistent with the ECCO2 validation results (Figs. B2e,f). The reason might be the neglected time difference of the adjustment of geostrophic balance across the equator in our model (the damping rates).

In general, our diagnostic model produces accurate, equatorial ageostrophic u′ (r ≈ 0.50) and υ′ (r ≈ 0.70) with an amplitude comparable to the observations, which is an improvement from the existing product—OSCAR u′ (r ≈ 0.20–0.50) and υ′ (r ≈ 0.50).

c. TIW energetics

The joint EOF analysis of the estimated u′ and υ′ identifies the Rossby and Yanai modes of TIWs (Figs. 8b,c), which are consistent with the results from SSH′, SST′, and moored velocity (Fig. 4). The first four leading EOFs come in pairs, accounting for 17.6%, 16.7%, 8.8%, and 8.1% of the total variance. The regressions on PC1 (Fig. 8b) and PC3 (Fig. 8c) represent the first two pairs of EOFs. The NE–SW velocity oscillations (u'υ'¯>0) of two modes on the equator are opposite to the counterclockwise shear (Uy < 0) on the southern flank of the South Equatorial Current (SEC) (Fig. 8a). And, the NW–SE velocity oscillations (u'υ'¯<0) of the Rossby mode north of the equator are opposite to the clockwise shear (Uy > 0) between SEC and the North Equatorial Countercurrent (NECC). These imply barotropic instability (Uyu'υ'¯>0) at different latitudes as energy sources of TIW modes, as suggested by Donohue and Wimbush (1998) using a two-and-a-half-layer model. The associated vertical velocity w′ is calculated as the divergence of horizontal velocity: w=D/2(ux+υy), assuming that u′ and υ′ are uniform in the upper D = 30 m layer (Figs. 1b,c). TIW modes induce strong vertical motions near the equator (Figs. 8d,e). Specifically, the Yanai mode has stronger vertical motions, reaching amplitudes of 2 m day−1 in average, consistent with the estimates from observations and modeling TIW studies (Kennan and Flament 2000; Weisberg and Qiao 2000; Menkes et al. 2006). This indicates that baroclinic instability plays a role in the TIW evolution (Yu et al. 1995).

Fig. 10.
Fig. 10.

Annual means for (a) EKE, (b) BTR (shading) and zonal current (contour; m s−1), and (c) BCR (shading) and SST (contour; °C) during 1993–2018. EKE, BTR, and BCR are calculated with the estimated u′ and υ′ from the time-varying diagnostic model [Eq. (1)]. Mean zonal current in (b) and SST in (c) are from OSCAR and OISST.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

To clarify the realistic energy conversion processes responsible for the TIW eddy kinetic energy (EKE), we evaluate the barotropic and baroclinic conversion rate (BTR and BCR, respectively), based on the estimated u′ and υ′. Following Qiao and Weisberg (1998), EKE, BTR, and BCR are given by

EKE=12(uu¯+υυ¯),
BTR=Uyuυ¯,
BCR=gϱw¯,

where u′ and υ′ are from our diagnostic model (section 4b), U from OSCAR, and w=D/2(ux+υy), all vertically averaged in the upper layer D = 30 m. The bars denote Reynolds averaging, obtained by zonally 3000-km and 40-day low-pass filtering. In Eq. (3), we neglect Uxuu¯, Vxuυ¯, and Vyυυ¯, which are weak or negative in BTR (Wang et al. 2017). Additionally, ϱ=ρ/ρ0αSST, where the thermal expansion coefficient α = 3 × 10−4 °C−1, we neglect salinity’s contribution (Lee et al. 2012) and assume that temperature is mixed well in the upper 30 m. Then, Eq. (4) becomes

BCR=gαSSTw¯.

Positive BTR (BCR) indicates barotropic (baroclinic) instability, transferring energy from the mean kinetic energy (eddy potential energy) to EKE (Lorenz 1955). Energetics for the two TIW modes (Fig. 9) are investigated by using the velocities and SST perturbations projected onto PC1 (Rossby) and PC3 (Yanai) from the joint EOF analysis of u′ and υ′ (Fig. 8). The 140°W profiles of energetics show the averaging of the results over year 1993–2018 and 145°–135°W. EKE of the Rossby mode is elevated north of the equator, while the Yanai mode peaks on the equator (Figs. 9a,b). The positive BTRs in different latitudes are the dominant energy sources for two TIW modes (red curves in Figs. 9c,d). For the Rossby mode, the barotropic instability is caused by the clockwise shear (Uy > 0) between SEC and NECC over 2°–7°N (Fig. 8a). Meanwhile, the barotropic instability for the Yanai mode is attributed to the counterclockwise shear (Uy < 0) of SEC over 2°S–2°N (Fig. 8a). The additional barotropic instability near 5°N in the Yanai mode is possibly responsible for the SSH signals centered at ~6°N in the second pairs of EOFs (Figs. 4b and 8c). The baroclinic instability exists along the north equatorial front in the two TIW modes, and plays an important role in the energetics of the Yanai mode (blue curves in Figs. 9c,d).

Fig. B1.
Fig. B1.

STDs of (left) zonal and (right) meridional momentum terms in Eq. (B1): (a),(b) averaged between 170° and 110°W, 1°S and 1°N and (c),(d) averaged between 1°S and 1°N, based on ECCO2 output.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

The annual mean of EKE associated with total TIWs is elevated over the domain of (160°–100°W and 2°S–6°N), with the maxima close to the equator (Fig. 10a). The high EKE on the equator is dominated by the Yanai-mode TIWs (Fig. 9b), often underestimated by ocean general circulation models (Holmes and Thomas 2016; Wang et al. 2017). The distribution of BTR illuminates that the equatorial barotropic instability is responsible for the elevated EKE on the equator (Fig. 10b). The instability is related to the counterclockwise shear (Uy < 0) along the equatorward edge of SEC on the equator (contour in Fig. 10b). BCR is elevated along the north equatorial front and plays a secondary role in the TIW energetics (Fig. 10c).

Fig. B2.
Fig. B2.

Validation of the diagnostic model [Eq. (B4)]: STDs of (a) u′ and (b) υ′ estimated from ECCO2 SSH and U with the diagnostic model; (c) u′ and (d) υ′ from ECCO2 output. Cross-correlation coefficients between the diagnostic results and ECCO2 output: (e) for u′ and (f) for υ′. The dashed contours indicate the 0.70 level of correlation coefficient. The 95% confidence level of the correlation is at most 0.20 in the whole region.

Citation: Journal of Physical Oceanography 50, 10; 10.1175/JPO-D-20-0063.1

5. Summary and discussion

This study identifies the Yanai- and Rossby-mode TIWs from the joint EOF analysis of SSH′ and SST′ data in the eastern equatorial Pacific. To estimate the TIW surface currents, we develop a time-varying diagnostic model with acceleration terms. Based on the diagnostic results, we calculate TIW energetics associated with the two TIW modes, including EKE, BTR, and BCR.

The Yanai-mode TIWs are prominent in the meridional velocity across the equator, but weak in the off-equatorial SSH. On the other hand, the Rossby-mode TIWs feature strong SSH variability north of the equator. Our time-varying diagnostic model estimates the TIW surface currents from SSH′. Validated with the ECCO2 output, the diagnostic velocity based on ECCO2 SSH′ is highly correlated with ECCO2 velocity, with correlation coefficients larger than 0.70 in most regions. Validated against the equatorial moored observations, the estimated TIW velocity based on the satellite-observed SSH shows a large improvement from the existing product in estimating the equatorial high-frequency currents, especially the meridional component υ′. Based on the estimated velocity, we find that the Yanai-mode TIWs induce strong vertical motions near the equator. TIW energetics illustrate that the equatorial barotropic instability is the dominant energy source of the Yanai-mode TIWs, and baroclinic instability along the north equatorial front plays a secondary role in the energy gaining for EKE. The counterclockwise shear (Uy < 0) of SEC on the equator explains the equatorial NE–SW velocity oscillations in two TIW modes via the barotropic instability. And the north-equatorial NW–SE velocity oscillations of the Rossby mode are explained by the clockwise shear (Uy > 0) between SEC and NECC also via the barotropic instability.

We neglect the vertical viscosity in the diagnostic model and ignore the mapping errors in the AVISO SSH, both of which might make the estimation of u′ less accurate than υ′. Nevertheless, our estimations of υ′ are still very accurate and help describe the TIW dynamics, because the meridional component of TIW velocity is known to dominantly induce lateral advection of eddy momentum, vorticity (Kug et al. 2010; Holmes et al. 2014), temperature (Chelton et al. 2000), salinity (Lee et al. 2012), and nutrients (Strutton et al. 2001) in the region. The temperature advection by TIWs was suggested to be an important contributor to the equatorial mixed layer heat budget (Jochum et al. 2007). The TIW-induced SST oscillations lead to robust changes in the atmospheric boundary layer, e.g., surface wind speed (Xie et al. 1998) and cloud cover (Deser et al. 1993), which can feedback on oceanic TIWs and may impact El Niño–Southern Oscillation (ENSO) (Imada and Kimoto 2012; Holmes et al. 2019; Xue et al. 2020). Considering the Rossby- and Yanai-mode TIWs have different velocity patterns, more research is needed to explore their effects on the regional hydrography and atmospheric boundary layer, and the feedbacks on the largescale oceanic and atmospheric variability.

Acknowledgments

We thank two reviewers, Yu Zhang, Xudong Wang, J. Thomas Farrar, Shineng Hu, and Thomas Kilpatrick for their helpful comments and discussions on this work, and gratefully acknowledge the financial support from China Scholarship Council during Minyang's visiting in San Diego, California. Minyang Wang and Yan Du at SCSIO/CAS are supported by the Chinese Academy of Sciences (XDA15020901, 133244 KYSB20190031, ZDRW-XH-2019-2), the National Natural Science Foundation of China (41830538, 41525019), the State Oceanic Administration of China (GASI-IPOVAI-02), and the Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (GML2019ZD0303, 2019BT2H594). Shang-Ping Xie is supported by the U.S. NSF (AGS 1637450). Samuel S.P. Shen is partially supported by a U.S. NSF grant (AGS-1419526). We thank CNES for providing the AVISO data (http://www.aviso.altimetry.fr/en/data/data-access.html), GTMBA Project Office of NOAA/PMEL for the TAO observations (https://www.pmel.noaa.gov/tao/drupal/disdel/), NASA JPL for the ECCO2 Cube92 model output (https://ecco.jpl.nasa.gov/products/all/), and NOAA NCDC for the OISST data (https://www.ncdc.noaa.gov/oisst/data-access). The OSCAR data were obtained from the NASA EOSDIS Physical Oceanography Distributed Active Archive Center (PO.DAAC) at the JPL, Pasadena, CA (http://dx.doi.org/10.5067/OSCAR-03D01). We thank Dennis L. Hartmann for sharing the notes of “Cross Spectrum Analysis” online (https://atmos.washington.edu/~dennis/552_Notes_6c.pdf).

APPENDIX A

Calculation of Spectra

All the spectra, including the power, cross, and coherence spectra in the paper, are computed as the averages of spectra over 600-day running segments of data during 1993–2018. The temporal and mixed temporal–spatial spectra are estimated by using the 1DFFT and 2DFFT, respectively. Most of the spectra pass the 95% confidence level based on the F test (dashed curves in Fig. 4c), as the 14–40-day signals have large number of realizations over the 600-day segments. The 95% confidence interval of spectra (Figs. 6 and 7) is calculated as the half STD of spectra over different 600-day running segments of data during 1993–2018.

APPENDIX B

Derivation of Diagnostic Model

The basic equation governing the surface TIW currents is

utAC+(uux)ADVx+(υuy)ADVy+(wuz)ADVz+ifuCF=ghPGF+BBF+[τρ0DAuz(D)D]EVFz+RES,

where velocity is expressed in the complex variable notation u(x, y, z, t) ≡ u + , and the primes denote TIW-band anomalies (section 2). In addition, w is the vertical component of velocity, the subscripted ut and uz are the time and vertical partial differentiations, and ∇ ≡ ∂/∂x + i∂/∂y is the gradient operator. SSH′ is denoted by h′(x, y, t), the buoyancy force is B(x,y,t)=(g/D)D0ρ/ρ0dz, and the surface wind stress τ′(x, y, t) ≡ τx + y. The Coriolis parameter f = βy, β = 2.3 × 10−11 s−1 m−1, and y is the distance northward from the equator. The gravity acceleration g = 9.8 m s−2, the seawater density constant ρ0 = 1024 kg m−3, and the turbulent viscosity is estimated here as Aυ ≅ 3 × 10−3 m2 s−1, which is related to the water stratification and the vertical current shear (Li et al. 2001). AC, ADV, CF, PGF, BF, and EVF are short for acceleration, advection, Coriolis force, pressure gradient force (surface), buoyancy force, and eddy viscous force (vertical), respectively, all vertically averaged in the upper layer D = 30 m. The residual RES(x, y, z, t) ≡ RESx + iRESy is from subgrid processes, horizontal dissipations, and the errors in the estimated turbulent viscosity.

In the ECCO2 output, AC, ADVx, and PGF dominate in the momentum balance on the equator (Figs. B1a,b). Additionally, ADVy and EVFz are important for the zonal momentum balance east of 130°W (Fig. B1c). Meanwhile, the estimations of some momentum terms based on the satellite SSH and moored velocity show that the amplitude and zonal variability of AC, PGF, and EVFz are similar to the ECCO2 results (figure not shown). The two dominant nonlinear terms: (uux)′ and (υuy)′ are linearized by decompositions of

(uux)=Uux+uUx+Res,
(υuy)=Vuy+υUy+Res,

where the capital letters denote the basic state, obtained by zonally 3000-km and 40-day low-pass filtering. The residuals contain nonlinear terms from the temporal and vertical decompositions.

It turns out that Uux and υUy are dominant on the right hands of Eqs. (B2) and (B3) on the equator. Neglecting the weak terms, we finally get a time-varying diagnostic model with AC, linearized ADVx (additionally υUy in the zonal equation), PGF, and CF retained

(t+δx)u+Uux(fUy)υ=ghx,
(t+δy)υ+Uυx+fu=ghy,

where the Rayleigh damping terms δxu′ and δyυ′ are used to guarantee the stability of the numerical computation and roughly parameterize EVFz. The damping rates δx and δy are generally (2–10 day)−1, which are adjusted with the temporal and spatial resolutions of the diagnostic model: A higher resolution corresponds to a larger damping rate. Here they are artificially determined to be δx = (7 day)−1 and δy = (4 day)−1 to achieve realistic amplitudes of ECCO2 u′ and υ′. The slight changes of δx and δy only influence the amplitudes of the estimated u′ and υ′. The diagnostic results from ECCO2 SSH with Eq. (B4) are shown and validated in Fig. B2.

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Shen, S. S. P., G. P. Behm, Y. T. Song, and T. Qu, 2017: A dynamically consistent reconstruction of ocean temperature. J. Atmos. Oceanic Technol., 34, 10611082, https://doi.org/10.1175/JTECH-D-16-0133.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Shinoda, T., 2010: Observed dispersion relation of Yanai waves and 17-day tropical instability waves in the Pacific Ocean. SOLA, 6, 1720, https://doi.org/10.2151/SOLA.2010-005.

    • Crossref
    • Search Google Scholar
    • Export Citation
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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Small, R. J., and Coauthors, 2008: Air–sea interaction over ocean fronts and eddies. Dyn. Atmos. Oceans, 45, 274319, https://doi.org/10.1016/j.dynatmoce.2008.01.001.