a. Inferring dynamics from one-dimensional horizontal wavenumber spectra

The spectral slopes and power ratios between the components of horizontal KE wavenumber spectra, together with their depth dependence, are useful diagnostics that may indicate the presence of certain dynamical regimes (e.g., Callies and Ferrari 2013). For isotropic flows, either purely rotational or divergent, the power-ratio between the cross- and along-track components is directly proportional to the KE spectral slope. Dominantly rotational motions, such as in a quasi-geostrophy (QG) turbulence regime (e.g., interior QG; Charney 1971), project primarily onto the cross-track component. Predominantly along-track energy would be indicative of divergent, ageostrophic motions (Callies and Ferrari 2013), which may include a variety of phenomena with differing dynamics (IGWs, mixed layer instabilities, fronts and filaments, etc.). For flows that contain both nondivergent and divergent components, there are no set expectations for the power ratio between the cross- and along-track components of KE, nor for its connection to the KE spectral slope.

Besides interpretation of the above two spectral properties and their relationship, we use the two-step decomposition method of Bühler et al. (2014) to obtain both the rotational and divergent spectral components (aka Helmholtz decomposition), as well as the vortex (balanced) and wave (IGW) components of the KE spectra. This method allows a better dynamical diagnostic. A similar approach has been used for atmospheric flight-track data (Lindborg 2007, 2015).

The Bühler et al. (2014) decomposition has strict assumptions, namely, that the rotational and divergent fields are stationary, homogeneous, isotropic random functions. While the isotropy assumption can be relaxed under specific circumstances (see Bühler et al. 2017), a one-dimensional (1D) Helmholtz decomposition can be made without assuming anything about the relationship between the rotational and divergent fields (see supplement text and Lindborg 2015). The vortex–wave decomposition of KE further assumes that IGWs obey the Garrett and Munk (1972, GM) spectrum or a similar model (e.g., Levine 2002, hereafter L02). This last step also assumes that all divergence is solely provided by linear IGWs that propagate in a horizontally uniform background stratification that varies slowly in the vertical, consistent with the WKB approximation. We implement the method according to the steps outlined in appendix C of Rocha et al. (2016a), relying on the exact solution of the coupled ordinary differential equation (ODE) pair proposed by Bühler et al. (2014).

This decomposition is a statistical tool: it requires a sufficiently large number of ensemble members in order to yield robust along-track and cross-track velocity spectra. Even when confidence limits suggest that derived quantities such as the transition scale are valid, they must be interpreted cautiously, particularly at the extremes of the spectrum, e.g., near the Nyquist wavenumber or at the lowest resolvable wavenumber. To apply this approach to ship tracks with wide spatial coverage (Fig. 1), we assume that the statistics of the velocity spectra are homogeneous within an area larger than the transect lengths over which spectra are calculated.

b. ADCP data

We use ADCP data from two repositories: 1) the Joint Archive of Shipboard ADCP data (JASADCP) and 2) the Rolling Deck to Repository (R2R) shipboard ADCP collection housed at the National Centers for Environmental Information (NCEI). While JASADCP contains only ready-to-use ADCP cruise data (processed largely with University of Hawaii software and methods, see Firing and Hummon 2010), the NCEI repository contains a large amount of unprocessed ADCP data, including many uninterrupted science-free transits. Processed ADCP cruise data consist of vertical profiles of ocean velocity relative to the sea floor, averaged over some time interval, typically 2–5 min, and organized in discrete depth bins. The two repositories were searched for underway ship transects within the regions of interest that were collected with minimal interruptions (e.g., for hydrographic stations) and sufficiently fast, meaning with an average ship speed above ∼3.5 m s⁻¹, an upper bound of speed of the first baroclinic mode. As part of our study, previously unprocessed cruises were processed with UHDAS (University of Hawaii Data Acquisition System) software, permission was sought as needed to release proprietary data, and data were archived at JASADCP. The resulting dataset is more temporally and geographically diverse than data used in previous studies (see ship tracks in Fig. 1). The set constitutes a blend of sonar types under various profiling configurations (e.g., pulse length, ping frequency, blanking interval, averaging interval) that affect depth range and resolution as well as accuracy and horizontal resolution. The list of cruises used in this study is in Table S1 in the online supplemental material.

This analysis focuses on the near surface, using the bin centered nearest to 30-m depth. To minimize potential near-field artifacts, we discard data from the top two bins. We also preferentially select high-frequency sonars and/or sonars operating in broadband mode when available, yielding typical bin widths of 2–8 m, narrow enough to afford a detailed view of the upper ocean. The 30-m depth bin emerges as a good compromise between data quantity, quality, and surface proximity.

We analyze data collected from 1994 to 2018. These vary in quality mostly because, prior to the early 2000s, data acquisition systems and processing techniques limited ping-data screening and editing capabilities. During the 1990s, not only were GPS fixes sparser, but signals for civilian use were dithered [known as selective availability (SA)] to lower accuracy. However, we found older data, in general, to be adequate for our purposes, with statistics that differ little from the more modern data, possibly reflecting the fact that most ships were equipped with P (Precise) code GPS, which bypassed SA. We also have not found evidence that sonar type, frequency, or configurations affect the spectral characteristics of the data in a systematic manner other than shifting noise levels at the highest wavenumbers, corresponding to scales shorter than 10 km.

The data treatment is as follows. We analyze 256 mostly uninterrupted and gap-free transects (no gap larger than 20 km, no more than 10% small gap coverage) that are at least 500 km long, drawn from our blended dataset. Small gaps are filled through linear interpolation. Each transect is visually inspected for possible issues that might have escaped quality control procedures or that were not resolvable by prior processing, and any transects deemed contaminated are discarded; for this study we discarded three transects because of suspect data. Transects that pass screening are then block-averaged to a 2-km regularly spaced grid.

To calculate KE spectra, we first subdivide transects into 500-km-long segments, with 50% overlap between segments (when transects exceed 750 km in length). This length yields sufficient ensemble members to produce sensible averages and allows us to resolve wavelengths between ∼167 km (corresponding to 3 cycles per record length, the lowest wavenumber free of windowing effects) and 4 km, our Nyquist wavelength. Segments that deviate from a geodesic by more than 2% are discarded. Prior to forming spectral estimates, the velocities are rotated into an along- and cross-track coordinate system based on the segment’s median ship velocity vector. All velocity components are then detrended and windowed with a Blackman–Harris window (setting effective degrees of freedom to 1.98 times the total number of segments, determined according to Percival and Walden 1993). As in Rocha et al. (2016a), results are not sensitive to the choice of window. We obtain the KE wavenumber spectrum of each segment by Fourier transforming these velocities, multiplying the Fourier coefficients by their complex conjugates, and then correcting the spectral variance for the windowing. Finally, robust spectral estimates are formed by averaging specific segments together, with confidence limits estimated assuming that the Fourier coefficients are normally distributed and that their magnitudes squared follow a χ² distribution (Bendat and Piersol 2010). Uncertainties for the Helmholtz decomposition are estimated by propagating the upper and lower limits of the KE spectral confidence limits.

As in previous studies, we use the fast-tow assumption, treating the data as if the ships sampled the 500-km segments instantaneously. In reality 18–30 h are required, so wavenumber estimates are Doppler shifted. This issue is discussed in section 6b; other less common sampling and measurement issues are left for a future article.

c. Ancillary data

a. Regional annual-mean spectra

The KE spectra and their cross- and along-track components are aggregated by month and averaged within each of the three domains shown in Fig. 1. To avoid biasing the means, given the likely existence of seasonality (as discussed below), we use the following strategy: we retain the same number of segments for each season (where the year is split into two seasons, defined as consecutive 6-month periods), discarding excess data from oversampled months while at the same time attempting to maintain consistent geographical coverage during the two periods (thus also discarding samples from isolated areas that were not sampled in other months). The annual mean along- and cross-track components of the KE spectra are displayed in Figs. 2a–c.

Annual-mean spectra of kinetic energy (KE, thin black lines), and the cross- (${\widehat{C}}^{\upsilon }$, red lines) and along-track (${\widehat{C}}^u$, blue lines) components, for ADCP data collected in (a) the northeast tropical Pacific, (b) the deep tropical northeast Pacific, and (c) the southeast tropical Pacific. (d)–(f) Corresponding Helmholtz decomposition of the KE spectra into rotational (${\widehat{K}}^{\psi }$, green lines) and divergent (${\widehat{K}}^{\varphi }$, yellow lines) parts. (g)–(i) The partition of KE into wave (KE_W, orange lines) and vortex (KE_V, purple lines) components, using the Garrett–Munk (GM) spectrum (thin dark lines), Levine (L02) spectrum (thick light lines), and a constant $f^2/{\omega }_{°}^2$ (values for each region shown in Fig. 3). Shading denotes the 95% confidence interval. Negative spectral energy levels are masked. Dashed lines show the noise floor (black, see the appendix) and the reference k⁻² and k⁻³ curves (gray). In-panel labels are applicable column wise.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Annual-mean spectra of kinetic energy (KE, thin black lines), and the cross- (${\widehat{C}}^{\upsilon }$, red lines) and along-track (${\widehat{C}}^u$, blue lines) components, for ADCP data collected in (a) the northeast tropical Pacific, (b) the deep tropical northeast Pacific, and (c) the southeast tropical Pacific. (d)–(f) Corresponding Helmholtz decomposition of the KE spectra into rotational (${\widehat{K}}^{\psi }$, green lines) and divergent (${\widehat{K}}^{\varphi }$, yellow lines) parts. (g)–(i) The partition of KE into wave (KE_W, orange lines) and vortex (KE_V, purple lines) components, using the Garrett–Munk (GM) spectrum (thin dark lines), Levine (L02) spectrum (thick light lines), and a constant $f^2/{\omega }_{°}^2$ (values for each region shown in Fig. 3). Shading denotes the 95% confidence interval. Negative spectral energy levels are masked. Dashed lines show the noise floor (black, see the appendix) and the reference k⁻² and k⁻³ curves (gray). In-panel labels are applicable column wise.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Annual-mean spectra of kinetic energy (KE, thin black lines), and the cross- (${\widehat{C}}^{\upsilon }$, red lines) and along-track (${\widehat{C}}^u$, blue lines) components, for ADCP data collected in (a) the northeast tropical Pacific, (b) the deep tropical northeast Pacific, and (c) the southeast tropical Pacific. (d)–(f) Corresponding Helmholtz decomposition of the KE spectra into rotational (${\widehat{K}}^{\psi }$, green lines) and divergent (${\widehat{K}}^{\varphi }$, yellow lines) parts. (g)–(i) The partition of KE into wave (KE_W, orange lines) and vortex (KE_V, purple lines) components, using the Garrett–Munk (GM) spectrum (thin dark lines), Levine (L02) spectrum (thick light lines), and a constant $f^2/{\omega }_{°}^2$ (values for each region shown in Fig. 3). Shading denotes the 95% confidence interval. Negative spectral energy levels are masked. Dashed lines show the noise floor (black, see the appendix) and the reference k⁻² and k⁻³ curves (gray). In-panel labels are applicable column wise.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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In the NETP the KE spectra roll off with a least-squares-fit slope of −2.15 ± 0.1 from 15 to 200 km (Fig. 2a). The power ratio between the cross- (${\widehat{C}}^{\upsilon }$) and along- (${\widehat{C}}^u$) track components is only consistent with the slope for low wavenumbers, where it has a value near 2, and thus indicates that geostrophic dynamics dominate at scales larger than 100 km. The power ratio quickly decreases to one at around 30-km wavelength and then to about 0.9 for the remainder of the high wavenumber band, indicative of a loss of geostrophic balance. In the DTNEP, the KE spectra appear to be slightly flatter (−1.95 ± 0.1 slope), and in the SETP they are flatter still (−1.8 ± 0.12) for the 15–200-km wavelength range (Figs. 2b,c). The power ratio at the lowest resolvable wavenumber (∼170 km) is 1.8 for the DTNEP and 1.4 for the SETP. Hence, consistency with geostrophy appears only at these very large scales as well. The power ratio then sharply decreases to one at about 60–70-km wavelength and then decreases again relatively more slowly (relative to what was observed in the NETP) to about 0.8 for both these regions. The slope of the cross-track component is slightly steeper than the along-track component in all regions, hence driving most of the ratio changes.

Overall, the KE spectra differ significantly between the three regions mainly for scales ≳ 50 km, with higher levels of energy in the NETP and DTNEP relative to the SETP. In general, the regional differences are larger for the cross-track component of KE (red lines in Figs. 2a–c), with more energy at most wavenumbers in the NETP, followed by the DTNEP and then by the SETP. The along-track component is also less energetic at low wavenumbers in the SETP (Fig. 2c) when compared with the two other regions, contributing to the lower levels of KE in this region. Below scales of 40–50 km, the along-track component spectra in all three regions are nearly identical. At scales ≲ 15 km, the cross-track spectra in all three regions are nearly identical, and we note further flattening at these very high wavenumbers (scales ≲ 15–5 km) for this component, which we attribute to noise. This noise effect leads to an artificial increase in the power ratio at very high wavenumbers.

The Helmholtz-decomposed spectra are shown in Figs. 2d–f. The rotational component (K^ψ) clearly dominates the divergent component (K^ϕ) only for scales ≳ 70 km in the NETP (Fig. 2d), and ≳200 km for the DTNEP and SETP (Fig. 2e,f). For the SETP, this dominance is less reliable (note the overlapping shading in Fig. 2f). In the NETP, K^ψ and K^ϕ contribute equally around 30-km wavelength (Fig. 2d). The divergent component slightly dominates the rotational component for smaller scales. In the DTNEP and SETP, the divergent component starts to dominate at scales of about 80 km, with a larger ratio of divergence to rotation seen in the SETP than in the DTNEP, and much larger still than seen in the NETP. This dominance means that K^ϕ typically represents 60%–70% of the KE at these scales, and this fraction is much larger than what has been reported elsewhere, except by Qiu et al. (2017) in a small area of the western tropical Pacific. In the NETP and DTNEP, the spectra of K^ψ appear to roll off as k^−2.5 between 20- and 200-km scales.

In Figs. 2d–f, the small rise in rotational energy for scales smaller than ∼10 km is ascribable to noise. This illustrates that even modest levels of (mostly white) noise can impact the accuracy of the Helmholtz decomposition at high wavenumbers. Because the Helmholtz decomposition depends on the differences in power between the cross- and along-track components, the rotational and divergent components are more uncertain (see shading in Figs. 2d–f) than the cross-track and along-track components (Figs. 2a–c).

The results in Figs. 2d–f show that in the submesoscale-transition range, ageostrophic and divergent motions are important for the NETP and are likely dominant in the DTNEP and SETP regions of the eastern tropical Pacific. With spectral slopes flatter than −3, interior QG does not appear to be the most relevant framework to interpret the total KE. The spectra do not display the signature characteristics of surface quasigeostrophy (SQG; Blumen 1978) described by Callies and Ferrari (2013) such as flattening at scales longer than the injection scale (estimated here to be at 10–30 km), or abrupt steepening with depth (Fig. S1), indicating that SQG is unlikely to explain the observed KE. Other frameworks for balanced dynamics, e.g., those originating from mixed layer baroclinic instabilities (Callies et al. 2016; Cao et al. 2021), may still be relevant to interpret the spectra, particularly in the NETP, where rotational energy is comparable to divergent energy over many smaller scales. This evaluation, however, will require isolating the balanced component, since many ageostrophic flows also have a rotational component. The easiest case to analyze with 1D data is a flow composed of only linear IGWs and nondivergent eddies.

b. The wave–vortex decomposition and the transition scale

To separate wave and vortex contributions to the KE with the Bühler et al. (2014) method, we require knowledge of the frequency content of the IGW field. This knowledge is encapsulated by the empirical parameter $f^2/{\omega }_{*}^2(k)$, which is related to the ratio ${\widehat{C}}_{W,\psi }^u/{\widehat{C}}_{W,\varphi }^{\upsilon }$, where ${\widehat{C}}_{W,\varphi }^{\upsilon }$ is the spectrum of the divergent cross-track velocity of the wave component and ${\widehat{C}}_{W,\psi }^u$ its rotational along-track velocity component. Following Bühler et al. (2014), an external model of IGW energy is used to determine $f^2/{\omega }_{*}^2(k)$, the squared ratio of the local inertial frequency f to a wavenumber-dependent frequency function ${\omega }_{*}$. This yields the unknown ${\widehat{C}}_{W,\psi }^u$, under the assumption that all of the divergent energy is due to IGWs, i.e., that ${\widehat{C}}_{W,\varphi }^{\upsilon }={\widehat{C}}_{\varphi }^{\upsilon }$, where ${\widehat{C}}_{\varphi }^{\upsilon }$ is the spectrum of the divergent cross-track velocity from the Helmholtz decomposition. Isotropy determines ${\widehat{C}}_{W,\psi }^{\upsilon }$ and thus the wave KE, and the vortex KE is found by subtracting the wave from the total KE. We use the GM model and the L02 version of GM to estimate $f^2/{\omega }_{*}^2(k)$ as part of an evaluation of the sensitivity of the wave–vortex decomposition. This is possible because of the energy equipartition property of IGWs (Bühler et al. 2014). The L02 formulation shows a better fit to frequency spectra in the tropics (L02; Polzin and Lvov 2011).

We first explore the sensitivity of $f^2/{\omega }_{*}^2$ to specific formulations and parameter choices of the IGW models. Parameters for the GM model were obtained from exponential fits to the N profiles shown in Figs. 3a–c and are tabulated in Table 2. The L02 model uses the observed N profiles in Figs. 3a–c. The calculations are carried out between 4000 and 30 m, the depth at which most of our ADCP data were measured. We show $f^2/{\omega }_{*}^2$ predictions for the GM and L02 models in Figs. 3d–f (gray lines), for annual mean, winter/spring, and summer/fall stratification conditions. In all three domains the L02 model predicts smaller ratios at all wavenumbers. The differences result almost entirely from the shape of the L02 frequency spectrum. Stratification has little impact on model ratios; sensitivity tests show that these models require much larger variations of N with depth than are observed either seasonally or even regionally, in order to impact the IGW behavior near the surface (as indicated by the thin, often superimposed, gray lines in Figs. 3d–f).

(a)–(c) Buoyancy frequency (N) observed in the vicinity of the cruise transects used for each averaging domain; annual mean (indigo lines), winter/spring (blue) and summer/fall means (red) are shown. (d)–(f) The ratio of ${\widehat{C}}_{\psi }^u$ to ${\widehat{C}}_{\varphi }^{\upsilon }$, which is equivalent to $f^2/{\omega }_{*}^2$, derived from the Helmholtz decomposition of ADCP data in each domain, color-coded by season; also shown are $f^2/{\omega }_{*}^2$ predictions by the Garrett–Munk (GM, dark gray) and Levine (L02, light gray) spectral models for the various N profiles, along with the values for the M₂ internal tide and for the frequency ${\omega }_{\mathrm{°}}$ that best matches the observed ratio.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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(a)–(c) Buoyancy frequency (N) observed in the vicinity of the cruise transects used for each averaging domain; annual mean (indigo lines), winter/spring (blue) and summer/fall means (red) are shown. (d)–(f) The ratio of ${\widehat{C}}_{\psi }^u$ to ${\widehat{C}}_{\varphi }^{\upsilon }$, which is equivalent to $f^2/{\omega }_{*}^2$, derived from the Helmholtz decomposition of ADCP data in each domain, color-coded by season; also shown are $f^2/{\omega }_{*}^2$ predictions by the Garrett–Munk (GM, dark gray) and Levine (L02, light gray) spectral models for the various N profiles, along with the values for the M₂ internal tide and for the frequency ${\omega }_{\mathrm{°}}$ that best matches the observed ratio.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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(a)–(c) Buoyancy frequency (N) observed in the vicinity of the cruise transects used for each averaging domain; annual mean (indigo lines), winter/spring (blue) and summer/fall means (red) are shown. (d)–(f) The ratio of ${\widehat{C}}_{\psi }^u$ to ${\widehat{C}}_{\varphi }^{\upsilon }$, which is equivalent to $f^2/{\omega }_{*}^2$, derived from the Helmholtz decomposition of ADCP data in each domain, color-coded by season; also shown are $f^2/{\omega }_{*}^2$ predictions by the Garrett–Munk (GM, dark gray) and Levine (L02, light gray) spectral models for the various N profiles, along with the values for the M₂ internal tide and for the frequency ${\omega }_{\mathrm{°}}$ that best matches the observed ratio.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Table 1

Transition scale estimates from wave–vortex decomposition (km). For ADCP-derived scales, the range shows estimates using the Garrett–Munk (GM) model (low values) and narrowband model (high values). The LLC4320-derived scales in this study use the 1D method with the GM model only for zonal transects and ship transects (in parentheses). The LLC4320 scales obtained by Qiu et al. (2018) (Fig. 1) were averaged in the three domains, where values in parenthesis only average areas visited by ship tracks. The asterisk denotes that the scales are underestimated in the DTNEP and SETP because undefined values (white pixels in Fig. 1) are set to 500 km.

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Table 2

Parameters used in the Garrett–Munk model of IGW energy, and domain-averaged values for mixed layer depth (MLD) from Argo data and background mesoscale eddy kinetic energy (EKE) from gridded altimetry data. The latitude used is the mean latitude of the selected ship tracks; see section 3b for details on the calculation of the other parameters.

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There is also an alternative approach. In a regime where wave energy dominates, ${\widehat{C}}_{\psi }^u/{\widehat{C}}_{\varphi }^{\upsilon }$ from the Helmholtz decomposition is directly comparable to $f^2/{\omega }_{*}^2$, and is thus a useful test for an IGW model. For ADCP spectra, this would occur for scales less than 40 km for the NETP (Fig. 2d) and for scales less than 100 km for the DTNEP (Fig. 2e) and SETP (Fig. 2f). In this IGW regime, regardless of season, ADCP-derived ${\widehat{C}}_{\psi }^u/{\widehat{C}}_{\varphi }^{\upsilon }$ (colored lines in Figs. 3d–f) plateaus between 0.5 and 0.8 depending on the region, before increasing again at high wavenumbers presumably due to a combination of noise and the boundary condition of the method (Rocha et al. 2016a, or the supplemental text).

The fact that ${\widehat{C}}_{\psi }^u/{\widehat{C}}_{\varphi }^{\upsilon }$ ratios do not converge to the GM or L02 predictions implies that the IGW models may have shortcomings. Additionally, the high ratio specifically implies that the observations are dominated by near-inertial-like IGWs. This result argues for the use of a narrowband wave–vortex decomposition prescribed by a constant $f^2/{\omega }_{°}^2$ (dash–dotted lines in Figs. 2g–i for annual-mean KE wave–vortex decomposition). In all regions, the use of $f^2/{\omega }_{\mathrm{°}}^2$ in lieu of a wave model leads to a larger wave to vortex KE ratio and results in steeper spectra for KE_V (Figs. 2g–i). This contrasts with the lower sensitivity of the wave–vortex partition when choosing between the GM or L02 models (solid lines in Figs. 2g–i). Differences between the narrowband ${\omega }_{\mathrm{̥}}$ and broadband ${\omega }_{\ast }$ estimates are as large as, and at times exceed, the uncertainty range obtained by propagating the statistical uncertainty through the wave–vortex step (the shading is omitted in Figs. 2g–i and 5c,d for clarity).

Overall, our analysis indicates that in the eastern tropical Pacific, IGWs account for ∼70% of the annual-mean total KE (and hence most or even all of the shape of the spectra) below a certain scale (which is domain dependent) (Figs. 2g–i). This generally holds whether using $f^2/{\omega }_{°}^2$ or the GM model, with the former method implying larger fractions of wave KE and weaker, steeper vortex spectra. Following previous work (Qiu et al. 2017; Chereskin et al. 2019), we define the transition scale from balanced to unbalanced motions as the KE_W and KE_V crossover wavelength in the wave–vortex decomposition (e.g., the scale at which the purple line is equal to the orange line in Figs. 2g–i). The transition scale, as summarized in Table 1, is shorter in the NETP, where it is 50–100 km, is around 200 km in the DTNEP, and is longest in the SETP. The narrowband IGW model yields longer transition scales than broadband IGW models such as GM, but the choice of GM versus L02 does not impact the transition scale.

c. Seasonality in spectra and transition scale

The KE spectra and their wave–vortex partitioning often exhibit time variability in the form of seasonality (Callies et al. 2015; Rocha et al. 2016b; Qiu et al. 2017, 2018). Whereas the area-averaged mixed layer depth undergoes a well-defined seasonal cycle in all regions, with deeper layers in winter/spring than in summer/fall (Table 2), background EKE does not display much seasonality (Table 2). The background EKE has larger interannual than seasonal variability; however, we lack sufficient ADCP data to resolve the former. To explore seasonality in the ADCP spectra and transition scales, we define seasons using the mixed layer depth and contrast the spectra, Helmholtz decompositions, and wave–vortex decompositions for July–October with those for January–April (December–March for the NETP, Figs. 4 and 5).

Seasonal kinetic energy (KE) spectra for ADCP data in the northeast tropical Pacific NETP. Light colors denote winter/spring means, while darker colors denote the summer/fall means. (a) Total KE, (b) cross-track component, and (c) along-track component. The hashed area in (a)–(c) is the interval between the noise floor estimated in winter/spring and the noise floor in summer/fall. Noise is typically higher in winter.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Seasonal kinetic energy (KE) spectra for ADCP data in the northeast tropical Pacific NETP. Light colors denote winter/spring means, while darker colors denote the summer/fall means. (a) Total KE, (b) cross-track component, and (c) along-track component. The hashed area in (a)–(c) is the interval between the noise floor estimated in winter/spring and the noise floor in summer/fall. Noise is typically higher in winter.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Seasonal kinetic energy (KE) spectra for ADCP data in the northeast tropical Pacific NETP. Light colors denote winter/spring means, while darker colors denote the summer/fall means. (a) Total KE, (b) cross-track component, and (c) along-track component. The hashed area in (a)–(c) is the interval between the noise floor estimated in winter/spring and the noise floor in summer/fall. Noise is typically higher in winter.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Helmholtz and wave–vortex decompositions of the seasonal kinetic energy (KE) spectra for ADCP data in the northeast tropical Pacific NETP. Light colors denote winter/spring means, while darker colors denote the summer/fall means. (a) The rotational component, (b) the divergent component, (c) the wave–vortex decomposition using the Garrett–Munk spectrum, and (d) the decomposition using the constant $f^2/{\omega }_{*}^2$.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Helmholtz and wave–vortex decompositions of the seasonal kinetic energy (KE) spectra for ADCP data in the northeast tropical Pacific NETP. Light colors denote winter/spring means, while darker colors denote the summer/fall means. (a) The rotational component, (b) the divergent component, (c) the wave–vortex decomposition using the Garrett–Munk spectrum, and (d) the decomposition using the constant $f^2/{\omega }_{*}^2$.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Helmholtz and wave–vortex decompositions of the seasonal kinetic energy (KE) spectra for ADCP data in the northeast tropical Pacific NETP. Light colors denote winter/spring means, while darker colors denote the summer/fall means. (a) The rotational component, (b) the divergent component, (c) the wave–vortex decomposition using the Garrett–Munk spectrum, and (d) the decomposition using the constant $f^2/{\omega }_{*}^2$.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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In all three regions, ADCP energy levels in winter/spring spectra differ slightly from summer/fall spectra, but only in the NETP are the differences significant at the 95% level (Fig. 4a and Fig. S2). We find no significant differences in slope between seasons in all regions. In the NETP, there is a substantial increase in KE at scales less than 100 km (Fig. 4a) in winter/spring relative to summer/fall, but this contrast is absent elsewhere. Hence, we focus solely on the NETP here. At the longest scales, the along-track component of KE (Fig. 4c) accounts for most of this energy increase, yielding a power ratio near one at these scales. For scales less than about 100 km the cross-track component (Fig. 4b) contributes more for the seasonal increase than the along-track.

Divergence energy in winter/spring is similar to summer/fall for most wavenumbers (but weaker at larger scales, Fig. 5b). The seasonal increase in KE for scales less than 100 km is driven by the rotational component (Fig. 5a). In winter/spring, ${\widehat{K}}^{\psi }$ is flatter than in summer/fall, and the scale at which ${\widehat{K}}^{\psi }$ equals ${\widehat{K}}^{\varphi }$ decreases, from 40 km in summer/fall to 20 km in winter/spring. Unlike the directly observed KE components, the seasonal changes in the energy density of the Helmholtz components are highly uncertain. Further decomposition into wave and vortex KE (Figs. 5c,d) suggests that IGWs undergo little seasonality, regardless of the IGW model used, with substantial changes only at the largest scales, which we discuss in section 6.

Statistical reliability of the decompositions aside, we find a pronounced seasonality in the vortex KE component, with seasonal vortex KE differences found for both IGW models for scales below ∼150 km. The winter/spring vortex KE spectrum appears to be slightly flatter than the summer/fall spectrum, at least for scales longer than ∼50 km (cf. dark and light purple lines in Figs. 5c,d). The use of the narrowband near-inertial model to perform the decomposition results in a noticeably steeper summer/fall vortex spectrum (Fig. 5d).

The estimates for the transition scale for each region and season are shown in Table 1. The scale shortens only slightly during winter/spring in the NETP, but lengthens considerably during summer/fall; the precise scale and its seasonal amplitude depends again on the type of IGW model. The transition scale in the NETP, in any case, is expected to be no more than 100 km. The seasonality in this transition is due to the increase in vortical KE at scales of 50–100 km. In determining the NETP winter/spring transition scale for the narrowband method, we ignore the crossover of the vortex and wave KE spectra at the largest scales (Fig. 5d); in section 6 we discuss why we do not consider the large-scale elevated along-track variance to be exactly IGW related. In the SETP and DTNEP, the transition scale in both seasons remains much longer than in the NETP, at or above 200 km (increasing in summer/fall but unreliably), and depends little on the choice of IGW model. This results from the year-round dominance of IGWs, which undergo weak or nonexistent seasonality (e.g., Fig. S2).

In the three regions considered here, seasonal changes in stratification are confined to the upper 150 m, with the amplitude of the seasonal signal quickly decreasing with depth (Figs. 3a–c). The cycle is relatively small in the NETP and is more pronounced in the DTNEP and SETP, where summer/fall N increases by nearly an order of magnitude. This provides an explanation for the low seasonality in IGW KE near the surface for the NETP. It seems inconsistent with the weak IGW seasonality in the DTNEP and SETP, where we see sufficiently strong restratification to support increased IGW amplitude in summer/fall.

In terms of the mixed layer, in the DTNEP the mixed layer depth cycle is weaker than in the other regions, and summer/fall mixed layers in this area are expected to be shallower than 30 m, while elsewhere the mixed layer is always deeper than 30 m (Table 2). The NETP and SETP see a large cycle in mixed layer depth (Table 2). The deep (>50 m) winter/spring mixed layers in the two regions would favor the onset of mixed layer baroclinic instabilities (Boccaletti et al. 2007), which potentially energize the balanced spectra (Callies et al. 2016) and small-scale divergence (e.g., Cao et al. 2021). This is consistent with the vortex results for the NETP but not for the SETP. For instance, recalculation of NETP spectra using ADCP data from April or May, when averaged mixed layers are shallower (instead of November or December), suppresses the seasonality somewhat through a decrease in the vortex KE.

In contrast, we do not expect background EKE to drive seasonality in the vortex spectra. Seasonal differences in EKE are small, and regional differences are more noteworthy. Background EKE is largest in the DTNEP, where it is roughly twice that of the NETP and 5 times stronger than in the SETP (Table 2). Background EKE in the NETP and SETP peaks around July and is weakest in December–January; in the DTNEP, the EKE seasonal cycle is less defined. The cycles’ amplitudes are close to the uncertainty (Table 2).

a. Are assumptions satisfied?

To assess the fidelity of the results, we evaluated the assumptions underlying the Helmholtz decomposition for 1D data. Bühler et al. (2017) listed statistical metrics for 1D current observations to diagnose the presence of anisotropy and serial correlation between rotational and divergent fields. One metric for anisotropy is the expected value of the difference between variance in the along-track component and variance in the cross-track component (i.e., $\mathbb{E}[u^2-{\upsilon }^2]$). A second metric is significant correlations between u and υ, i.e., $\left|\mathbb{E}[u\upsilon ]\right|\ne 0$, or more usefully the cross-spectrum between u and υ (see also supplement text). Isotropic rotational and divergent fields will lead to zero cospectra and quadrature spectra. A significant quadrature spectrum indicates that the two fields are correlated (Bühler et al. 2017).

These quantities, computed for example subsets of the ADCP data in Fig. 6 (also in Fig. S4 for the other regions), indicate that the assumptions of isotropy and uncorrelated rotational and divergent components are sufficiently satisfied. This is demonstrated by the fact 1) that we either find small values for both $\mathbb{E}[u^2-{\upsilon }^2]$ and $\mathbb{E}[u\upsilon ]$ [O(1%) of the variance] or we find $\mathbb{E}[u\upsilon ]\ll \mathbb{E}[u^2-{\upsilon }^2]$ when $\mathbb{E}[u^2-{\upsilon }^2]$ is O(10%) of the variance (Figs. 6c,d), and 2) that there is a lack of a significant coherent cross-spectra between the along- and cross-track velocities (in either the imaginary or real components; Figs. 6a,b). For context, in the anisotropic Gulf Stream region, Bühler et al. (2017) found $\mathbb{E}[u\upsilon ]$ to be 20% of the variance and the ratio of $\mathbb{E}[u\upsilon ]$ to $\mathbb{E}[u^2-{\upsilon }^2]$ to be close to 1. In contrast, we find ratios closer to 0.1. These results are insensitive to the amount of data (cf. the blue colors and filled lines for collocated data to the orange and dashed lines for all data in Fig. 6) or to any particular choice of data aggregation (Figs. S3d–f).

The cross-spectra (real and imaginary components) and spectral coherence squared (γ², right y axis) for collocated segments (i.e., consistent geographical coverage; solid lines, circles) and for all available segments (dashed lines, stars) in the northeast tropical Pacific (NETP) domain during (a) summer/fall and (b) winter/spring; the magenta lines are the mean cross- to along-track power ratios. Shading gives the 95% confidence interval, and we only show coherences that are significant at the 95% level. (c),(d) $\mathbb{E}[u^2-{\upsilon }^2]$, $\mathbb{E}[u\upsilon ]$, and $\mathbb{E}[u^2+{\upsilon }^2]$ (variance) for summer/fall and winter/spring in the NETP, respectively; the blue bars consider only transect segments that are collocated, orange all segments available within the region/season; the colored bars show twice the standard error of the mean and the black bars show the 5%–95% range.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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The cross-spectra (real and imaginary components) and spectral coherence squared (γ², right y axis) for collocated segments (i.e., consistent geographical coverage; solid lines, circles) and for all available segments (dashed lines, stars) in the northeast tropical Pacific (NETP) domain during (a) summer/fall and (b) winter/spring; the magenta lines are the mean cross- to along-track power ratios. Shading gives the 95% confidence interval, and we only show coherences that are significant at the 95% level. (c),(d) $\mathbb{E}[u^2-{\upsilon }^2]$, $\mathbb{E}[u\upsilon ]$, and $\mathbb{E}[u^2+{\upsilon }^2]$ (variance) for summer/fall and winter/spring in the NETP, respectively; the blue bars consider only transect segments that are collocated, orange all segments available within the region/season; the colored bars show twice the standard error of the mean and the black bars show the 5%–95% range.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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The cross-spectra (real and imaginary components) and spectral coherence squared (γ², right y axis) for collocated segments (i.e., consistent geographical coverage; solid lines, circles) and for all available segments (dashed lines, stars) in the northeast tropical Pacific (NETP) domain during (a) summer/fall and (b) winter/spring; the magenta lines are the mean cross- to along-track power ratios. Shading gives the 95% confidence interval, and we only show coherences that are significant at the 95% level. (c),(d) $\mathbb{E}[u^2-{\upsilon }^2]$, $\mathbb{E}[u\upsilon ]$, and $\mathbb{E}[u^2+{\upsilon }^2]$ (variance) for summer/fall and winter/spring in the NETP, respectively; the blue bars consider only transect segments that are collocated, orange all segments available within the region/season; the colored bars show twice the standard error of the mean and the black bars show the 5%–95% range.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Small, nonzero values of $\mathbb{E}[u^2-{\upsilon }^2]$ and $\mathbb{E}[u\upsilon ]$ can arise in at least two other ways: biased sampling and nonwhite measurement noise (see the appendix). The latter may be relevant to understand the presence of weak cross-spectral energy (e.g., Figs. 6a,b) and small . The former provides an explanation for the larger $\mathbb{E}[u^2-{\upsilon }^2]$ relative to $\mathbb{E}[u\upsilon ]$ (e.g., Figs. 6c,d; i.e., the low ratio mentioned above): a negative $\mathbb{E}[u^2-{\upsilon }^2]$ can arise in a rotation-dominated isotropic field or in an anisotropic field that is preferentially sampled parallel to anisotropy in the rotational or sampled perpendicular to anisotropy in the divergent field. While conceivable, the anisotropic case is arguably less likely than the isotropic case; furthermore, we found no sensitivity to sampling direction (Fig. S12).

b. Significance tests through randomized resampling experiments

We detect seasonality in the NETP KE spectra and power ratio (Figs. 4a–c): seasonal differences, though not particularly strong, exceed 95% uncertainty levels. We use randomized resampling experiments to more formally test whether these seasonal differences in power ratio, slope, and anisotropy metrics are likely to arise by chance sampling. For the NETP seasonality experiment, 60 samples were randomly drawn without repetition from the combined summer/fall and winter/spring data pool to create two groups. We perform 10 000 realizations and record the differences in the above parameters between the two groups. We plot the differences in power ratio that were frequently recorded (the 33rd and 66th percentiles, dotted lines in Fig. 7a), and those rarely recorded (the 2.5th and 97.5th percentiles, dashed lines in Fig. 7a). The observed power ratio (blue line in Fig. 7a) lies mostly within the less-likely range, especially for the largest scales and for the ∼25–50-km band, showing that the seasonal difference is unlikely to be a statistical fluke.

(a) The difference in power ratio between winter/spring and summer/fall averages (blue; data have been averaged by decade) in the northeast tropical Pacific vs the 33rd and 66th percentiles (dotted black lines) and 5th and 95th percentiles (dashed black lines) of the differences between two random subsets from the resampling experiment, using data from both seasons. (b) Cumulative distribution functions for the differences in $\mathbb{E}[u^2-{\upsilon }^2]$ (blue) and $\mathbb{E}[u\upsilon ]$ (orange) and slope fit (green); the color-matched dashed lines show the observed value of the difference in the parameters.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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(a) The difference in power ratio between winter/spring and summer/fall averages (blue; data have been averaged by decade) in the northeast tropical Pacific vs the 33rd and 66th percentiles (dotted black lines) and 5th and 95th percentiles (dashed black lines) of the differences between two random subsets from the resampling experiment, using data from both seasons. (b) Cumulative distribution functions for the differences in $\mathbb{E}[u^2-{\upsilon }^2]$ (blue) and $\mathbb{E}[u\upsilon ]$ (orange) and slope fit (green); the color-matched dashed lines show the observed value of the difference in the parameters.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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(a) The difference in power ratio between winter/spring and summer/fall averages (blue; data have been averaged by decade) in the northeast tropical Pacific vs the 33rd and 66th percentiles (dotted black lines) and 5th and 95th percentiles (dashed black lines) of the differences between two random subsets from the resampling experiment, using data from both seasons. (b) Cumulative distribution functions for the differences in $\mathbb{E}[u^2-{\upsilon }^2]$ (blue) and $\mathbb{E}[u\upsilon ]$ (orange) and slope fit (green); the color-matched dashed lines show the observed value of the difference in the parameters.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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The differences between seasons in slope, $\mathbb{E}[u^2-{\upsilon }^2],\mathbb{E}[u\upsilon ]$ (cf. Fig. 6c versus Fig. 6d and dashed lines in Fig. 7b) could be a chance outcome (they do not exceed the 95% likelihood, Fig. 7b). The same resampling experiment applied to the DTNEP and SETP indicates that the small seasonal differences could arise from chance sampling. The resampling experiments also show that the uncertainty due to sampling in area-mean spectral properties, in all cases, is significantly smaller than the χ² based uncertainty (Figs. S3a–c), and we do not find significant differences between any two random subsets, including for the diagnostic metrics (Figs. S3d–f). This result suggests fairly homogeneous statistics within the three domains, as further implied by the statistically insignificant differences between choice subregions within the domains.

a. Comparison with previous studies: How different is the SETP/NETP from other regions?

b. Caveats

2) Assumptions for the 1D Helmholtz and wave–vortex decomposition

The decompositions presented here rely on the assumptions that the velocity statistics are stationary, homogeneous, and isotropic. While we were able to evaluate the last two, we do not know how well stationarity holds. The overall sparseness of the cruise data prevents adequate binning to unambiguously assess the role of interannual variability, which is seen in the two key environmental drivers of the KE spectra: mixed layer depth and background EKE.

Another important assumption is that all divergence is due to IGWs. Further model analysis by Torres et al. (2018) supports use of this assumption for the low EKE regions that we examine here, yet during winter, the model shows nonnegligible (∼30%) contributions from nonwave motions at high wavenumbers in some areas (Figs. 11a,b in Torres et al. 2018, or Fig. S9). It is not obvious that these are also the case for the observations; since the spectra of divergence have no seasonality, attribution of the divergent energy to a specific process is ambiguous. To circumvent this issue, we take the opposite approach: we assume that all ADCP rotational energy is due to eddying motions and estimate the Rossby number as $R_o=\sqrt{k^3{\hat{K}}^{\psi }}/f$ (Callies et al. 2015). The R_o at submesoscales (Fig. 11) are at most O(0.1), so unlikely to support significant divergence, implying the original assumption holds. Divergent corrections to the vortical KE, however small they may be, would still be useful to explore, as this assumption impacts the accuracy of the result that KE within the submesoscale transition range (Figs. 3d–f) appears to be composed of NIWs.

Rossby number R_o estimated from the rotational component of ADCP data in the three domains in the east tropical Pacific. The dashed black line marks R_o = 0.5.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Rossby number R_o estimated from the rotational component of ADCP data in the three domains in the east tropical Pacific. The dashed black line marks R_o = 0.5.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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Rossby number R_o estimated from the rotational component of ADCP data in the three domains in the east tropical Pacific. The dashed black line marks R_o = 0.5.

Citation: Journal of Physical Oceanography 52, 8; 10.1175/JPO-D-21-0139.1

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The appropriateness of an IGW spectral model is difficult to evaluate independently. Our results agree with those of Chereskin et al. (2019) in showing rather limited agreement with the GM model. This could be a near-surface effect; GM behavior could occur deeper in the water column, but ADCP signal-to-noise issues prevent more in-depth analysis at present. Analysis of deeper layers are thus left for a future study.

This surface behavior is unlikely to be caused by wave refraction processes and could reflect NIW sources. The L02 framework indicated little impact of the mixed layer in near-surface values of the wave rotation to divergence ratio. While the Bühler et al. (2014) IGW energy equipartition statement is not formally valid in the abrupt transition between the mixed layer and the thermocline, we estimate, using the L02 framework, that the resulting error of the WKB approximation breakdown is smaller than the uncertainties reported so far. Details of these analyses are in the supplement. Finally, as the mixed layer depth varies along transects, the ADCP 30-m bin averages together mixed layer and thermocline velocities. We cannot therefore rule out that a mixture of IGW and nonwave or mixed layer-only processes contributes to the high rotation to divergence ratio.

c. Implications for high resolution and swath satellite altimetry

In the eastern tropical Pacific, the fact that a large portion of the variability of near-surface currents is divergent, potentially IGW driven, reduces the range of usefulness of altimetric derived current data. Our estimates of the transition scale suggest that we should expect dominant geostrophic currents for scales longer than about 200 km over most of this area, potentially for scales as short as 50 km during certain seasons and limited areas (e.g., NETP). The usefulness of the transition scale to interpret any given or a few submesoscale-permitting altimeter passes/swaths is limited by our inability to constrain the stationarity assumption. Nonetheless, for those interested in the absolute transition scales, we note that there are no obvious reasons to doubt their estimated ranges. Accessing the balanced KE at wavelengths near or immediately below the transition scale will depend not only on the altimeter’s true signal-to-noise ratio, but also on the frequency content of the IGWs. Neither is currently well known. If NIWs indeed dominate the near-surface unbalanced wavenumber spectrum, their weak projection onto SSH (Gill 1982) would not hinder detection of the balanced component. Since an analysis by Callies and Wu (2019) further suggests that the IGW continuum lies below the SWOT noise floor, this part of the IGW spectrum would also not hinder the balanced component detection in the tropical eastern Pacific. In contrast, extracting the balanced component from a spectrum with strong incoherent internal tides, as in the LLC4320 output (Arbic et al. 2022), would be more challenging.