Estimating the Ocean Interior from Satellite Observations in the Kerguelen Area (Southern Ocean): A Combined Investigation Using High-Resolution CTD Data from Animal-Borne Instruments

Lei Liu aState Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China
bSouthern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou, China

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Huijie Xue cState Key Laboratory of Marine Environmental Science, Xiamen University, Xiamen, China
dDepartment of Physical Oceanography, College of Ocean and Earth Sciences, Xiamen University, Xiamen, China

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Abstract

Observational surface data are utilized to reconstruct the subsurface density and geostrophic velocity fields via the “interior + surface quasigeostrophic” (isQG) method in a subdomain of the Antarctic Circumpolar Current (ACC). The input variables include the satellite-derived sea surface height (SSH), satellite-derived sea surface temperature (SST), satellite-derived or Argo-based sea surface salinity (SSS), and a monthly estimate of the stratification. The density reconstruction is assessed against a newly released high-resolution in situ dataset that is collected by a southern elephant seal. The results show that the observed mesoscale structures are reasonably reconstructed. In the Argo-SSS-based experiment, pattern correlations between the reconstructed and observed density mostly exceed 0.8 in the upper 300 m. Uncertainties in the SSS products notably influence the isQG performance, and the Argo-SSS-based experiment yields better density reconstruction than the satellite-SSS-based one. Through the two-dimensional (2D) omega equation, we further employ the isQG reconstructions to diagnose the upper-ocean vertical velocities (denoted wisQG2D), which are then compared against the seal-data-based 2D diagnosis of wseal. Notable discrepancies are found between wisQG2D and wseal, primarily because the density reconstruction does not capture the seal-observed smaller-scale signals. Within several subtransects, the Argo-SSS-based wisQG2D reasonably reproduce the spatial structures of wseal, but present smaller magnitude. We also apply the isQG reconstructions to the 3D omega equation, and the 3D diagnosis of wisQG3D is very different from wisQG2D, indicating the limitations of the 2D diagnostic equation. With reduced uncertainties in satellite-derived products in the future, we expect the isQG framework to achieve better subsurface estimations.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Huijie Xue, hjxue@xmu.edu.cn

Abstract

Observational surface data are utilized to reconstruct the subsurface density and geostrophic velocity fields via the “interior + surface quasigeostrophic” (isQG) method in a subdomain of the Antarctic Circumpolar Current (ACC). The input variables include the satellite-derived sea surface height (SSH), satellite-derived sea surface temperature (SST), satellite-derived or Argo-based sea surface salinity (SSS), and a monthly estimate of the stratification. The density reconstruction is assessed against a newly released high-resolution in situ dataset that is collected by a southern elephant seal. The results show that the observed mesoscale structures are reasonably reconstructed. In the Argo-SSS-based experiment, pattern correlations between the reconstructed and observed density mostly exceed 0.8 in the upper 300 m. Uncertainties in the SSS products notably influence the isQG performance, and the Argo-SSS-based experiment yields better density reconstruction than the satellite-SSS-based one. Through the two-dimensional (2D) omega equation, we further employ the isQG reconstructions to diagnose the upper-ocean vertical velocities (denoted wisQG2D), which are then compared against the seal-data-based 2D diagnosis of wseal. Notable discrepancies are found between wisQG2D and wseal, primarily because the density reconstruction does not capture the seal-observed smaller-scale signals. Within several subtransects, the Argo-SSS-based wisQG2D reasonably reproduce the spatial structures of wseal, but present smaller magnitude. We also apply the isQG reconstructions to the 3D omega equation, and the 3D diagnosis of wisQG3D is very different from wisQG2D, indicating the limitations of the 2D diagnostic equation. With reduced uncertainties in satellite-derived products in the future, we expect the isQG framework to achieve better subsurface estimations.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Huijie Xue, hjxue@xmu.edu.cn

1. Introduction

Over the past decades, satellite technology has profoundly revolutionized the field of oceanography by providing spatiotemporally well-sampled measurements of surface variables on a global scale, such as sea surface height (SSH), temperature (SST), and salinity (SSS). However, satellites cannot directly observe subsurface fields. Because subsurface in situ observations are still sparse in space and time, three-dimensional (3D) estimations of the ocean’s interior from satellite measurements become a key issue in oceanography.

While studies relying on statistical methods to estimate the upper-ocean structures have reported satisfactory results (Buongiorno Nardelli et al. 2012; Wang et al. 2012; Chu et al. 2014; Fresnay et al. 2018; Su et al. 2021), dynamical methods (e.g., Zhang et al. 2014) can obviate the requirement of well-sampled historical statistics and provide dynamically consistent fields. Over the past years, the surface quasigeostrophic (SQG) theory has been explored fruitfully within the oceanographic context (LaCasce 2012; Lapeyre 2017; Miracca-Lage et al. 2022). LaCasce and Mahadevan (2006) relaxed the zero interior potential vorticity (PV) assumption in the standard SQG theory, and they derived subsurface PV from the surface density through an empirical relation derived from in situ data. Using SST and a climatological stratification, their modified model yielded better subsurface estimations compared to the standard SQG model. Lapeyre and Klein (2006) suggested an alternate effective SQG (eSQG) method. The method adjusts the vertical scale by a modified stratification, which is assumed to be constant. The eSQG method can be used to reconstruct the subsurface structures from either the SSH or sea surface density (SSD) and has been successfully employed by many investigators (e.g., Isern-Fontanet et al. 2006; Klein et al. 2008, 2009; Ponte et al. 2013; Ponte and Klein 2013; Chavanne and Klein 2016; Qiu et al. 2016, 2020). However, the eSQG formalism assumes good correlation between SSD and interior PV, in which case SSD and SSH would have the same phase (Isern-Fontanet et al. 2014; González-Haro and Isern-Fontanet 2014). This assumption indicates that, when there exists a notable SSD–SSH phase shift (as sometimes found in real oceans), the eSQG reconstruction could be degraded (Wang et al. 2013; Liu et al. 2019).

Wang et al. (2013) put forth a more elaborate dynamical method called “interior + surface QG” (isQG). This method first links SSD (i.e., surface PV sheet; Bretherton 1966) to the subsurface field via the SQG formalism; it then augments the subsurface field by the addition of the two gravest vertical [barotropic (BT) and first baroclinic (BC1)] modes, whose amplitudes are determined by matching to the residual SSH (the total SSH minus the SQG contribution). In contrast to the eSQG approach using a single SSH (or SSD) image, the isQG method employs both SSD and SSH to constrain the surface and subsurface PV simultaneously, and does not require good SSH–SSD correlation. Effectiveness of the isQG method for estimating subsurface density and geostrophic velocity fields has been demonstrated by several studies utilizing fields with moderate resolution (Liu et al. 2014, 2017; Chen et al. 2020; Yan et al. 2020), all of which yielded promising results. LaCasce and Wang (2015) extended the isQG method by employing an alternate BC mode (the first surface mode; see de La Lama et al. 2016; LaCasce 2017). Employing idealized stratification profile, their method (denoted LW15) uses analytical solutions for the vertical projection and greatly simplifies the isQG method. Liu et al. (2019) pointed out the isQG limitation (arising from the deficiency of only two gravest modes) that emerges in the high-resolution context and proposed an extended isQG method (denoted L19) geared toward the upcoming high-resolution SSH from the SWOT (Surface Water and Ocean Topography) satellite mission (Durand et al. 2010; Fu and Ubelmann 2014; Torres et al. 2018; Wang et al. 2018, 2019; Wang and Fu 2019; Klein et al. 2019; Morrow et al. 2019; Du et al. 2021). Results of Liu et al. (2019, 2021) highlight the potential applicability of L19 to the future SWOT SSH. All of these isQG-related works were carried out in the context of numerical model outputs, except for the study of Liu et al. (2017) that was implemented in the real oceans. However, the scarcity of subsurface observations in Liu et al. (2017) prevented the investigation of vertical velocity.

High-resolution in situ observations collected by an elephant seal, east of the Kerguelen Islands in the ACC from 20 October 2014 to 16 January 2015, have been newly released by the International MEOP (Marine Mammals Exploring the Oceans Pole to Pole) Consortium. This dataset provides an invaluable testbed for the study of subsurface reconstruction in the real oceans. In this article, considering the moderate resolution (1/4°) of the available satellite SSH, we focus on the isQG method by Wang et al. (2013). The observational surface data employed in isQG include satellite-derived and Argo-based SSS and satellite-derived SST and SSH. Subsurface isQG estimations are compared against the seal-derived data. Comparisons concern two variables: density anomaly ρ and vertical velocity w, the latter of which is diagnosed through the adiabatic 2D QG omega equation.

In the following section 2, we give a description of the employed in situ data, and then the isQG experiment setup in which the satellite data are introduced, followed by a brief review of the 2D QG omega equation. In section 3, we evaluate the isQG solutions (denoted by the subscript “isQG”) of density anomalies (ρisQG) based on two different SSS products, against the seal-observed density anomalies ρobs. In section 4, through the 2D omega equation, we first diagnose the seal-based vertical velocities wseal, then the ones (denoted by the subscript “isQG2D”; wisQG2D) based on the isQG reconstruction of ρisQG and geostrophic velocities (VisQG). These isQG-based wisQG2D are compared against wseal serving as the testbed. Results from the present study and perspectives for the future work are discussed in section 5. Finally, in section 6, we provide a conclusion of the results and caveats of the isQG method.

2. Data and method

a. Seal-borne observations of seawater density

The high-resolution dataset was collected by a female southern elephant seal, over a period of almost 3 months (20 October 2014 to 16 January 2015; see triangles in Fig. 1) and a distance exceeding 5000 km (black/red dots in Fig. 1). The seal was equipped with a CTD-SRDL (conductivity–temperature–depth satellite relay data logger) tag (Siegelman et al. 2019a) and continuously recorded conductivity, temperature, and pressure during its trip. The postprocessed gridded fields of temperature and salinity (Siegelman et al. 2019b), with horizontal and vertical resolutions being 1 km and 1 m, respectively, extends from ∼15 m down to ∼400-m depth. We refer readers to Siegelman et al. (2020a) for more details on the dataset and correction procedure. Considering the isQG performance is more effective in energetic regions (Wang et al. 2013; Liu et al. 2014), here we analyze the transect spanning from 1000 to 2300 km (2–24 November 2014; red dots in Fig. 1) that lies within the eddy-rich area of the energetic ACC (Wang et al. 2016), as shown by the SSH contours in Fig. 1. Numerous missing values below 300-m depth (see Fig. S1 in the online supplemental material) would hinder the diagnosis of w, and we fill them employing the ISAS-15 (In Situ Analysis System; Gaillard et al. 2016; Kolodziejczyk et al. 2017) monthly temperature–salinity dataset (1/2°) in November 2014. Specifically, we first fill the missing values in the 1000–2000-m layer using the linearly collocated ISAS data along the seal’s path, then in the upper 1000 m with the 2D linear interpolation.

Fig. 1.
Fig. 1.

Geographical location of the study region (dashed box; 80°–90°E, 57.5°–47.5°S) and the 3-month seal trajectory (black/red dots) from 20 Oct 2014 (0 km; green triangle) to 16 Jan 2015 (5270 km; black triangle), with arrows indicating the direction of the seal. Subsurface estimations are conducted in the dashed box, and evaluated within the solid smaller box (82°–88°E, 55.5°–49.5°S) against the seal’s 1000–2300-km transect (from 2 to 24 Nov 2014; red dots). A snapshot of SST (color) and SSH (white contours) on 18 Nov 2014 is superimposed.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

Using the processed temperature–salinity and the TEOS-10 equation (IOC et al. 2010), we calculate the potential density. For the stratification profile N2(z) required by the w diagnosis, we derive a trajectory-averaged Nobs2(z) (hereafter subscript “obs” indicates the value derived from in situ seal observations; see Fig. 2a) by
Nobs2(z)=gρυρ¯z.
Here ρυ represents the mean density within the entire 1300-km transect, ρ¯ the horizontal means at different depths, and g the gravity constant. Since our study focuses on the anomaly field, we obtain the potential density anomalies (denoted ρobs; Fig. 3a) by subtracting the mean value along the 1300-km trajectory. For reference, we present the unfilled seal-observed density anomalies in the supplemental material (Fig. S1a).
Fig. 2.
Fig. 2.

(a) Trajectory-averaged stratification profile computed from the seal observations. (b) Solid line denotes the trajectory-averaged stratification profile computed from the monthly fields and used by isQG (also L19), with gray shading indicating the surface layer (upper 130 m), the dashed line the exponential stratification used by LW15, and the dotted line the constant stratification required by L19.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

Fig. 3.
Fig. 3.

(a) The 1000–2300-km (red dots in Fig. 1) vertical section of seal-measured density anomalies ρobs with the missing values filled. (b) The SMOS-SSS-based isQG solution of density anomalies ρisQGSMOS that are linearly collocated with the seal’s 1000–2300-km transect. (c) As in (b), but for the ISAS-SSS-based isQG solution of ρisQGISAS. Characters “C” and “A” mark the positions of the seal-observed cyclonic and anticyclonic eddies, respectively.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

b. The isQG experiment setup

We give a brief review of the isQG method in appendix A, and a more detailed one can be found in Wang et al. (2013). We conduct the isQG explorations using satellite-derived (and also Argo-based) surface variables as input, in the Kerguelen region (Indian sector of the Southern Ocean) of 57.5°–47.5°S and 80°–90°E (see the 10° × 10° dashed box in Fig. 1), from 2 to 24 November 2014. To avoid the edge effect, subsurface estimations are compared in a smaller domain (55.5°–49.5°S, 82°–88°E; 6° × 6° solid box in Fig. 1) against the in situ seal-derived data. Three satellite-derived daily gridded products are employed: the Ssalto/Duacs L4 1/4°SSH provided by CMEMS (Copernicus Marine and Environment Monitoring Service); the 1/100° MUR (Multiscale Ultrahigh Resolution) SST by JPL (Jet Propulsion Laboratory); and the ESA (European Space Agency) SMOS (Soil Moisture and Ocean Salinity) product generated at the BEC (Barcelona Expert Center), the L4 1/20° SSS [see Olmedo et al. (2021) for details on this newly released product].

It is necessary to first assess the employed MUR SST and BEC-L4 SSS. Assessments against the seal observations, three other satellite products (Good et al. 2020; Huang et al. 2020), and the ISAS SSS are detailed in the supplemental material (see Table S1 and Fig. S2 and the related text). Although the assessments are not thorough, and meanwhile the satellite and in situ measurements are inherently different (Boutin et al. 2016; Vinogradova et al. 2019), results partly support our choice of MUR and BEC-L4. [Here it is worth emphasizing that satellite measures temperature/salinity at the first centimeter (or skin layer) of the sea surface, while the topmost level of seal observation is at the depth of 15–20 m below the surface.] Noticeably, the ISAS SSS (based on in situ observations) statistically appears to be more realistic than BEC-L4, although the former has a much coarser (1/2° monthly) resolution. In this work, each of these two SSS products, combined with satellite (MUR) SST and (CMEMS) SSH, are used as input of isQG to check the influence of SSS uncertainties.

Before obtaining the anomalies of SSD (SSDA) and SSH (SSHA) required by isQG, we fill the missing values in BEC-L4 SSS that hinder the isQG calculation, using the ARMOR3D reanalysis SSS (Buongiorno Nardelli 2012) that was employed by Olmedo et al. (2021) for assessing BEC SSS; details are presented in the supplemental material (Fig. S3). To be consistent with the spatial resolution of SSH, we low-pass filter the higher-resolution fields (of 1/100° MUR SST and 1/20° BEC SSS) onto the 1/4°SSH grid: specifically, these fields are first low-pass filtered [using the fast Fourier transform (FFT) algorithm and a cutoff wavelength of 120 km] to avoid aliasing, and then linearly interpolated onto the 1/4° grid. Here it is worth mentioning that we have examined three cutoff-wavelength values (90, 120, and 150 km), and very similar results are obtained. The 1/2° ISAS SSS is directly and linearly interpolated onto the 1/4° grid. Then we combine the interpolated SST and SSS to calculate SSD using the TEOS-10 equation. To get SSDA and SSHA, we remove the large-scale background information from SSD and SSH. Here the background information is defined as the mean value of the Lagrangian time series (of SSD or SSH) extracted along the seal’s 1300-km path (red dots in Fig. 1), consistent with that defined for the seal observation. As in Qiu et al. (2016) for the purpose of Fourier transform, a two-dimensional trapezoid window is then applied to SSHA and SSDA in a 1° band along the 10° × 10° box edges (dashed rectangle in Fig. 1). Vertical stratification profile is also required by isQG. For practical application, our isQG calculations focus on the stratification readily derived from the monthly dataset. Specifically, using the ISAS monthly data and TEOS-10 equation, we first obtain the collocated potential density (along the seal’s trajectory). Then we compute the trajectory-averaged Nmon2(z) (with subscript “mon” indicating the value estimated from ISAS monthly fields; black solid line in Fig. 2b) according to Eq. (1).

c. 2D QG omega equation

We use the 2D version of the QG omega equation (Siegelman et al. 2020a) written as
(f022z2+N22s2)w=2gρ0s(Vsρs).
Here s denotes the cross-front direction, V the cross-front horizontal geostrophic velocity, ρ0 the reference density, and f0 the trajectory-averaged Coriolis parameter. The term on the right-hand side represents the kinematic deformation of the density anomaly induced by the geostrophic velocity. We numerically solve Eq. (2) using finite differences for the vertical derivatives with zero Dirichlet surface and bottom boundary conditions [w(z=0)=w(z=H)=0, with H set to 2000 m in this study]; computations for the horizontal coordinate are conducted in the spectral space using periodic lateral boundary conditions, then reverted to the physical space via the inverse Fourier transform.

3. The isQG reconstruction of density

For comparison with the in situ observation, we collocate the isQG-reconstructed 3D ρisQG fields (with spatiotemporal resolutions of 1/4° and 1 day; recall section 2b) with the seal’s 2D transect (with spatiotemporal resolutions of 1 km and ∼0.3 h). Specifically, for each individual date from 2 to 24 November 2014, the isQG solutions are linearly interpolated to the locations of seal measurements acquired within the corresponding 1-day time window. Figures 3b and 3c show the collocated isQG reconstructions of ρisQGSMOS and ρisQGISAS. Hereafter superscripts “SMOS” and “ISAS” denote the solutions based on SMOS-BEC-L4 SSS and ISAS SSS, respectively.

Observation (Fig. 3a) shows that the seal crossed several interacting mesoscale eddies extending down to depths of at least 400 m, with the anomalies being positive in cyclonic eddies (denoted by C1–C5) and negative in anticyclonic eddies (A1A4). Both the reconstructed transects of ρisQGSMOS (Fig. 3b) and ρisQGISAS (Fig. 3c) are visually similar to the observed one (Fig. 3a): by and large, structures of the observed mesoscale eddies are reasonably captured by the reconstructions. The transect of ρisQGISAS appears to bear a more satisfactory resemblance to ρobs, as further corroborated by Fig. 4a presenting the pattern correlations between ρisQGSMOS/ρisQGISAS and ρobs, calculated for the entire 1300-km transect. [Results are not displayed in the upper 20 m and beneath 300-m depth where the in situ data are poorly sampled (recall Figs. S2e and S1a).] Correlations between ρisQGSMOS and ρobs fall within the 0.4–0.6 range in the 300-m upper ocean (blue line); in contrast, correlations for ρisQGISAS (black line) are notably stronger and exceed 0.8. Lower spatial correlations for ρisQGSMOS could be caused by 1) the larger biases in SMOS-BEC-L4 SSS (compared with ISAS SSS; recall Table S1 and Fig. S2) and 2) the artifacts introduced during our procedure of filling missing values in SMOS SSS (see Fig. S3). Influences of these two factors are also revealed in Fig. 4b showing the root-mean-square (rms) of density: in the upper 100 m, the magnitude of SMOS-derived density (blue line) is notably larger than that of seal-based one (red line), while the ISAS-derived value (black line) appears to be moderately smaller.

Fig. 4.
Fig. 4.

(a) Pattern correlations between the seal-observed and isQG-reconstructed density anomalies in the 1300-km transect as a function of depth. (b) The rms of observed (red line) and isQG-reconstructed density anomalies in the 1300-km transect as a function of depth. Blue and black lines denote the isQG solutions based on SMOS and ISAS SSS, respectively.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

Now we focus on the more realistic solution of ρisQGISAS (Fig. 3c) to analyze the isQG performance. Spatial uncertainties in the gridded surface fields (particularly the coarse SSH and SSS) that arise from the optimal interpolation for producing the 2D maps can inevitably cause a spatial shift in the isQG reconstruction near the periphery of eddies. This issue has been reported by Liu et al. (2017) and is also discernible here [e.g., the region between reconstructed A1 and C2 is shifted earlier toward the start (1000 km) location]. Biases within the gridded surface inputs can lead to subsurface discrepancies shown in Fig. 4. Meanwhile, magnitude discrepancies (Fig. 4b) could also be caused by the trajectory-averaged estimate of the stratification profile employed by the isQG method. Here it is important to mention that the isQG density solutions do not reproduce the fine-scale signals present in ρobs, as also revealed by the along-trajectory density gradients (see Figs. S4b, S4d, and S4f in the supplemental material). This could be caused because 1) the spatiotemporal resolution of input gridded fields (of isQG) is relatively coarse, particularly the daily 1/4°SSH and monthly 1/2° (ISAS) SSS or 2) the isQG formalism only depicts the balanced motions and thus cannot capture the signature of internal gravity waves experienced by the seal (Siegelman et al. 2020b).

To illustrate the isQG density reconstruction more explicitly in three dimensions, we show in Fig. 5 an example case on 21 November 2014, during which the seal was on its trip of 2159–2206 km (marked by the dots in Fig. 5) and was approaching the center of the cyclonic eddy C5 (around 54.5°S, 87°E). For clarity, here we only present the ISAS-SSS-based results, and the SMOS-SSS-based ones are given in the supplemental material (Fig. S5). Figures 5a, 5b, and 2b (solid line) present the input fields required by the isQG method as applied to the 10° × 10° reconstruction region (i.e., dashed box in Fig. 1). In Fig. 2b, the trajectory-averaged stratification reaches its maximum at 130-m depth and we mark the surface layer (denoted SL, upper 130 m) with the gray shading. Figures 5c and 5d give the isQG reconstructions of density and geostrophic velocity anomaly fields in the 6° × 6° region of our interest (solid box in Figs. 1, 5a, and 5b) at two different depths. Within the SL, the 50-m density reconstruction (Fig. 5c) is mainly contributed by its surface portion (Wang et al. 2013) and is similar to SSDA (Fig. 5a). Below the SL, the 400-m isQG solution is determined by its interior portion, and thus the density solution (Fig. 5d) bears a good resemblance to SSHA albeit the opposite signs (Fig. 5b).

Fig. 5.
Fig. 5.

(a) ISAS-SSS-based SSD anomaly and (b) SSH anomaly fields. As in Fig. 1, solid box delimits the region 82°–88°E, 55.5°–49.5°S. (c),(d) ISAS-SSS-based isQG reconstructions of density anomaly field ρisQGISAS (color) and geostrophic velocity anomaly field VisQG (black vector arrows) at 50- and 400-m depth, respectively. Plots correspond to the date of 21 Nov 2014, when the seal (with its 2159–2206-km track marked by dots) was within the cyclonic eddy C5.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

4. 2D diagnosis of vertical velocity

In this section, we investigate the 2D diagnosis of w, associated with submesoscale fronts that are generated by the ambient mesoscale strain field (i.e., by the geostrophic kinematic deformation forcing; Giordani et al. 2006; Liu et al. 2021), through the 2D version of the omega equation (2). The seal-based diagnosis (denoted wseal) serving as the testbed is first calculated. Then we derive the 2D diagnoses of wisQG2DSMOS and wisQG2DISAS that are based on the isQG solutions of ρisQGSMOS (Fig. 3b) and ρisQGISAS (Fig. 3c), respectively, and compare them against wseal. The 2D Eq. (2) assumes the trajectory to be normal to the density front. However, the seal’s track is often oblique (rather than perpendicular) to the fronts (Siegelman et al. 2020a,b). To correct for the sampling orientation, assuming fronts to be aligned with the stretching FSLEs (finite size Lyapunov exponents; d’Ovidio et al. 2004), we perform the data normalization following Siegelman et al. (2020a,b). Details of the normalization scheme are given in the supplemental material (Fig. S4 and the related text).

a. 2D diagnosis based on seal-borne observations

We diagnose wseal using the seal-observed cross-front (normalized) density gradients ∂sρobs (Fig. S4a) and Nobs2(z) derived from the seal data (Fig. 2a). Note that subsurface geostrophic velocities are not sampled by the seal. Siegelman et al. (2020a) assumed them to be depth invariant and used the altimetry-derived surface values to represent those in the interior ocean. In this study we employ the isQG-reconstructed geostrophic velocities (the ISAS-SSS-based ones VisQGISAS) decaying with depth, to derive the cross-front geostrophic velocity gradients (sVisQGISAS; see Fig. S4g). Actually, diagnoses from both schemes (i.e., depth-invariant/variant geostrophic velocities) are similar in the upper 400 m of our interest (see Fig. S6).

As shown in Fig. 6a, vertical section of wseal along the 1300-km path reveals positive and negative stripes of vertical velocity that alternately appear and have a width of 5–20 km, with the magnitude reaching up to 80 m day−1. Vertical velocities are intensified in the ocean interior (and do not necessarily penetrate into the SL; Siegelman 2020), below the SL down to at least 400-m depth. Since direct measurement of w is known to be extremely difficult in the open ocean because of its relatively small amplitude (Buongiorno Nardelli 2020), we use wseal presented in Fig. 6a as the reference velocities in this study.

Fig. 6.
Fig. 6.

The 1000–2300-km vertical sections of (a) seal-based 2D diagnosis of wseal, (b) SMOS-isQG-based 2D diagnosis of wisQG2DSMOS, and (c) ISAS-isQG-based 2D diagnosis of wisQG2DISAS. Dashed boxes mark the two subtransects of 1130–1360 km and 1880–2020 km.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

It is worth mentioning that our results of wseal are very different from those reported by Siegelman et al. (2020a, see their Fig. 5b). We utilize an altimetry product different from the one (currently not available online) employed by Siegelman et al. (2020a). Different SSH could directly result in different altimetry-based geostrophic velocities. On the other hand, the FSLEs are computed using the altimetry-based geostrophic velocities, and thus different FSLEs could lead to differences in the normalization procedure for the cross-front gradients (of both density and velocity; recall Fig. S4 and the related text). Considering the fact that our wseal and those in Siegelman et al. (2020a) are all not directly observed, in this study we do not investigate their discrepancies. We focus on the comparison between wseal and the isQG-based 2D diagnosis of wisQG2D (see the following subsection).

b. The isQG 2D diagnosis

We use the ρisQGSMOS and ρisQGISAS 2D transects (i.e., Figs. 3b and 3c) to derive the along-track density gradients (Figs. S4d and S4f), and then the cross-front gradients (Figs. S4c and S4e) via our normalization scheme. For explicit comparison with wseal, here we use the same cross-front velocity gradient sVisQGISAS (Fig. S4g) [for the diagnosis of wisQG2DSMOS, usages of sVisQGISAS and sVisQGSMOS yield similar results (not shown)]. The monthly Nmon2(z) (solid line in Fig. 2b) is employed [usage of Nobs2(z) (Fig. 2a) derived from seal observations will be discussed in section 5a].

The isQG 2D diagnoses are shown in Fig. 6. In the entire 1300-km transect, a visual comparison between Figs. 6a and 6b/6c indicates notable structure discrepancies between wseal and wisQG2DSMOS/wisQG2DISAS especially for the structures around 1000–1100 km. Meanwhile, the magnitude for wisQG2DSMOS/wisQG2DISAS is quite smaller than that for wseal. As further demonstrated in Figs. 7a and 7d for 1000–2300 km within the upper 400 m, rms of wisQG2DSMOS/wisQG2DISAS (blue/black line in Fig. 7a) is smaller; pattern correlations between wseal and wisQG2DSMOS/wisQG2DISAS are weak and fall in the range from −0.2 to 0.4 (blue/black line in Fig. 7d). As analyzed in section 3, ρisQGISAS is more realistic than ρisQGSMOS. Therefore, correlations for wisQG2DISAS (black line in Fig. 7d) appear to be stronger than those for wisQG2DSMOS (blue line in Fig. 7d), although the magnitude of wisQG2DSMOS (blue line in Fig. 7a) is closer to that of wseal (red line in Fig. 7a).

Fig. 7.
Fig. 7.

(top) The rms of vertical velocities and (bottom) pattern correlations between the seal-based wseal and isQG-based 2D diagnoses of wisQG2D computed within (a),(d) the entire 1000–2300 km, (b),(e) 1130–1360 km, and (c),(f) 1880–2020 km. Red, blue, and black lines represent the results for wseal, SMOS-isQG-based wisQG2DSMOS, and ISAS-isQG-based wisQG2DISAS, respectively.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

Within several subtransects, a close scrutiny reveals that wisQG2DISAS can reasonably capture wseal in terms of spatial structure. As the most noticeable example, within the 1130–1360-km subtransect (left dashed boxes in Fig. 6), wisQG2DISAS (Fig. 6c) bears a good resemblance to wseal (Fig. 6a), with pattern correlations between the two generally exceeding 0.6 in the upper 400 m (black line in Fig. 7e). Figure 7f further presents another example for the 1880–2020-km subtransect (right dashed boxes in Fig. 6): here pattern correlations between wseal and wisQG2DISAS (black line) fall in between 0.6 and 0.9 below 150 m. Consistent with the results for the entire 1000–2300-km trajectory, in these two subtransects wisQG2DSMOS also appears to be less satisfactory than wisQG2DISAS. Correlations for wisQG2DSMOS are still quite poor (blue lines in Figs. 7e and 7f), although the magnitude (blue lines in Figs. 7b and 7c) is closer to that of the reference velocities (red lines in Figs. 7b and 7c).

5. Discussion

In this section, we discuss the reasons for the magnitude discrepancies between wseal and wisQG2DISAS mentioned in Fig. 6, using additional experiment. Furthermore, considering that the isQG calculation is actually performed in three dimensions, we directly implement the isQG diagnosis via the 3D version of the QG omega equation and examine the difference between the 2D and 3D diagnostic equations. Finally, subsurface estimations from two other methods, LW15 and L19, are discussed.

a. The isQG experiment using Nobs2(z)

For practical application, in this study the isQG calculations focus on the usage of Nmon2(z) that can be readily derived from the monthly dataset (solid line in Fig. 2b). As a test, we redo the ISAS-SSS-based isQG experiment using Nobs2(z) measured by seal-borne instruments (Fig. 2a). Different from the results seen in Fig. 3c, here the density reconstruction (Fig. 8a) shows several subsurface local maxima in the vertical, which is caused by the strong vertical variations in Nobs2(z). However, the vertical velocity diagnosis (Fig. 8b) is very similar to wisQG2DISAS (Fig. 6c) using Nmon2(z). Such a similarity indicates that the significant magnitude discrepancy between wseal and wisQG2DISAS has little association with the difference between Nobs2(z) and Nmon2(z). Since both wseal and wisQG2DISAS diagnoses use the same velocity gradients sVisQGISAS (Fig. S4g), it is the differences between ∂sρobs (Fig. S4a) and sρisQGISAS (Fig. S4e) [i.e., ρobs (Fig. 3a) and ρisQGISAS (Fig. 3c)] that lead to the magnitude discrepancy. Such differences could arise from 1) limited resolution of the surface input fields (of isQG) and 2) unbalanced motions (experienced by the seal) that cannot be depicted by the isQG framework.

Fig. 8.
Fig. 8.

(a),(b) As in Figs. 3c and 6c, but for solutions based on Nobs2(z) that is computed from the seal observations.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

b. The isQG 3D diagnosis

Normalization procedure in the 2D case is no longer needed here. The isQG 3D reconstructions of ρisQG and VisQG, along with the monthly Nmon2(z), are directly applied to the 3D version of the QG omega equation (Hoskins et al. 1978) written as
(f022z2+Nmon22)wisQG3D=2Q.
Here Q = (g/ρ0)(∇VisQG)T ⋅ ∇ρisQG is the QG Q vector (also known as the frontogenesis vector). Superscript T represents matrix transpose and ∇ the horizontal gradient operator. Equation (3) is solved numerically in the same way as Eq. (2) (recall section 2c). As an example on 21 November 2014, Fig. S7 shows the diagnoses at 200 m (with wisQG3DSMOS and wisQG3DISAS denoting the isQG 3D diagnoses based on ρisQGSMOS and ρisQGISAS, respectively). Then we linearly collocate wisQG3DSMOS and wisQG3DISAS with the seal’s 1000–2300-km track, and the obtained transects are shown in Fig. 9.
Fig. 9.
Fig. 9.

(a) The SMOS-isQG-based 3D diagnosis of wisQG3DSMOS and (b) ISAS-isQG-based 3D diagnosis of wisQG3DISAS, that are linearly collocated with the seal’s 1000–2300-km transect.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

Most noticeably, the collocated 3D diagnoses [wisQG3DSMOS (Fig. 9a) and wisQG3DISAS (Fig. 9b)] are very different from their 2D counterparts [i.e., wisQG2DSMOS (Fig. 6b) and wisQG2DISAS (Fig. 6c)]. As corroborated by the blue/black line in Fig. 10a for the entire 1300-km transect, weak spatial correlations between wisQG3DSMOS/wisQG3DISAS and wisQG2DSMOS/wisQG2DISAS fall around 0/−0.1. Figures 10b and 10c also examine the two subtransects. Correlations are generally very weak; for the 1880–2020-km transect, correlations between wisQG3DISAS and wisQG2DISAS are stronger but negative (black line in Fig. 10c). This result indicates the difference between the 3D and 2D diagnoses through different versions of the omega equation. As pointed out by Nagai et al. (2006), localized mesoscale and submesoscale 3D deformation fields cannot be fully resolved in the 2D diagnosis. The 3D diagnoses are also very different from wseal, even in those two subtransects, as demonstrated by the blue/black lines in Figs. 10d–f presenting poor correlations between wisQG3DSMOS/wisQG3DISAS and wseal.

Fig. 10.
Fig. 10.

(top) Pattern correlations between the isQG-based 3D diagnosis of wisQG3D and 2D diagnosis of wisQG2D. (bottom) Pattern correlations between wisQG3D and the seal-based wseal. Correlations are computed for (a),(d) the entire 1000–2300 km, (b),(e) 1130–1360 km, and (c),(f) 1880–2020 km. Blue and black lines represent the results for wisQG3DSMOS and wisQG3DISAS, respectively.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

c. The LW15 and L19 estimations

The LW15 method is briefly reviewed in appendix B, and we refer readers to LaCasce and Wang (2015) for details. Here we examine the LW15 solutions (denoted by the subscript “LW15”) employing pure exponential stratification (dashed line in Fig. 2b), and implement the ISAS-SSS-based experiment. In terms of density anomaly structure, the LW15 ρLW15ISAS (Fig. 11a) reasonably reproduces ρobs (Fig. 3a), but the spatial correlations for ρLW15ISAS (falling generally in between 0.7 and 0.8; green line in Fig. 11c) are slightly weaker than those for the isQG counterpart ρisQGISAS (black line in Fig. 11c). In terms of w structure, the LW15 2D diagnosis (wLW152DISAS; Fig. 11b) is also outperformed by its isQG counterpart wisQG2DISAS: correlations for wLW152DISAS in the two subtransects are weaker (cf. green and black lines in Figs. 11e and 11f), especially in 1880–2020 km with values falling around 0.1 (green line in Fig. 11f). Note that, as in section 4b, here we also use the velocity gradient sVisQGISAS (Fig. S4g) [usages of sVisQGISAS and sVLW15ISAS yield similar results (not shown)]. In terms of the magnitude, the LW15 solutions are even smaller than their isQG counterparts (cf. Fig. 11a and Fig. 3c; Fig. 11b and Fig. 6c). Surface mixed layer (SML) is deep in our target high-latitude region. We expect that adding a SML could improve the abovementioned simple exponential solutions, as indicated by LaCasce and Wang (2015). An alternate prospective approach was suggested by the SQG-based work of Chavanne and Klein (2016) taking into account the presence of the SML. These approaches will be pursued in our future studies.

Fig. 11.
Fig. 11.

(a),(b) As in Figs. 3c and 6c, but for results using the LW15 method. (c) Green line denotes spatial correlations between the ISAS-based LW15 density solution ρLW15ISAS and the seal-observed ρobs, within the entire 1000–2300 km. (d)–(f) Green lines denote spatial correlations between the ISAS-LW15-based 2D diagnosis wLW152DISAS and the seal-based wseal, for the entire 1000–2300 km, 1130–1360 km, and 1880–2020 km, respectively. In (c)–(f), black lines denote correlations for the isQG solution and are the same as those in Figs. 4a, 7d, 7e, and 7f.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

The L19 method, which is geared toward the SWOT-like high-resolution surface data, is briefly reviewed in appendix C [see Liu et al. (2019) for details]. Because of the resolution constraint imposed by the input data (particularly the 1/4° SSH and 1/2° SSS), the advantage of L19 over isQG does not manifest itself (see Liu et al. 2019), and the L19 ISAS-SSS-based subsurface estimations (denoted by the subscript “L19”; Fig. 12) are very similar to their isQG counterparts. Here the L19 2D diagnosis also uses sVisQGISAS (Fig. S4g) [usages of sVisQGISAS and sVL19ISAS yield similar results (not shown)].

Fig. 12.
Fig. 12.

As in Fig. 11, but for results using the L19 method.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0183.1

6. Conclusions

In this study, we apply the isQG method to estimate the ocean subsurface density and vertical velocity fields using observations. The observational data serving as input of isQG include the satellite SSH and SST, satellite/Argo-based SSS, and an estimate of vertical stratification from Argo-based monthly fields. Usage of two (satellite and Argo-based) SSS products is to investigate the impacts of SSS uncertainties on the isQG performance. The isQG-reconstructed density anomalies are directly evaluated against in situ measurements collected by an elephant seal in a subdomain of the ACC (east of the Kerguelen Islands), within the Indian sector of the Southern Ocean. The isQG-based vertical velocities (that are diagnosed using reconstructed density and geostrophic velocity fields, via the 2D QG omega equation) are compared against the seal-based ones (that are derived using in situ seal observations, through the 2D QG omega equation).

For the density anomaly field, the structures of the mesoscale eddies represented in the seal-observed transect are reasonably captured by the isQG solutions. Quality of the surface input variables inevitably influences the isQG performance. Due to smaller uncertainties in ISAS SSS, the ISAS-SSS-based ρisQGISAS outperforms the SMOS-SSS-based ρisQGSMOS. In the upper 300 m, pattern correlations between ρisQGISAS and observation are quite strong with values mostly exceeding 0.8, while they are weaker for ρisQGSMOS (0.4–0.6). Meanwhile, discrepancies between ρisQGISAS and ρobs are discernible: 1) the magnitude of reconstructed ρisQGISAS is weaker than the observation; 2) the pattern of ρisQGISAS is slightly shifted in space near the periphery of eddies; and 3) lateral gradients of ρisQGISAS are significantly smaller than the ones computed from high-resolution seal observations. Such discrepancies could partly arise from the coarse spatiotemporal resolution of the gridded inputs (especially the 1/4° daily SSH and 1/2° monthly SSS) and the measurement uncertainties therein.

The above-concluded density discrepancies can lead to differences in the diagnosed vertical velocity fields. In terms of spatial structure, differences between wseal and the isQG 2D diagnoses are quite noticeable within the entire 1300-km transect, especially for wisQG2DSMOS based on the less reasonable ρisQGSMOS (compared to ρisQGISAS); pattern correlations between wseal and wisQG2DSMOS(wisQG2DISAS) fall in between −0.2 and 0 (0.2 and 0.4) in the upper 400 m. However, within specific subtransects, wisQG2DISAS can reasonably reproduce the structures of wseal; for the 1130–1360-km (1880–2020 km) trajectory, pattern correlations exceed 0.6 in the upper 400 m (fall in between 0.6 and 0.9 below 150 m). In terms of magnitude, both wisQG2DISAS and wisQG2DSMOS (especially the former) are smaller than wseal.

Two points are worth remarking on in concluding our study. First, the isQG method is limited by the uncertainties within the employed moderate-resolution satellite products. We expect that advanced products in the future can improve the isQG performance. Meanwhile, since the SQG framework cannot reasonably capture scales smaller than about 30 km (Klein et al. 2009), new methods should be pursued in future works.

Second, the vertical velocities serving as our testbed (i.e., wseal) are not directly observed, but are diagnosed through the 2D omega equation. They could differ from the true vertical motion in the real oceans: 1) diagnoses from the 2D and 3D omega equations could be quite different, even in the context of the same density and geostrophic velocity fields; 2) wseal only represents the signal that is forced by the geostrophic kinematic deformation, and signals generated by other forcings (such as turbulent mixing) are not included (Ponte et al. 2013; Chavanne and Klein 2016; Xie et al. 2017; Estrada-Allis et al. 2019; Liu et al. 2021; Yang et al. 2021). Noticeably, using a different SSH product, our wseal are very different from those reported in Siegelman et al. (2020a). Future works employing observed or more realistic w (e.g., Yu et al. 2019) are needed.

Acknowledgments.

We thank Lia Siegelman, Rui Xin Huang, Shiqiu Peng, Bo Qiu, Bruno Buongiorno Nardelli, and Yu-Kun Qian for helpful discussions. We also thank the editor Joseph LaCasce, and both reviewers for their valuable and constructive comments. This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Project XDB42000000), Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (GML2019ZD0303), the Strategic Priority Program on Space Science, the Chinese Academy of Sciences (XDA15020901), the National Natural Science Foundation of China (Grants 41806036), and the Independent Research Project Program of State Key Laboratory of Tropical Oceanography (LTOZZ2203). We are grateful to the freely available data for this study: the marine mammal data were collected by the International MEOP Consortium and the national programs that contribute to it (http://www.meop.net); SSH and OSTIA SST are provided by CMEMS (http://marine.copernicus.eu); OISST is provided by NOAA (https://www.ncei.noaa.gov/products/optimum-interpolation-sst); MUR SST is obtained from JPL (https://coastwatch.pfeg.noaa.gov/erddap/griddap/jplMURSST41.html); SMOS SSS is provided by BEC (http://bec.icm.csic.es/); FSLE is provided by AVISO (https://www.aviso.altimetry.fr); the monthly temperature and salinity gridded fields are from ISAS15 (https://www.seanoe.org/data/00412/52367). We gratefully acknowledge the use of high performance computing clusters at the South China Sea Institute of Oceanology, Chinese Academy of Sciences.

APPENDIX A

The isQG Method

Considering a flow in QG balance (Pedlosky 1987), the geostrophic streamfunction Ψ and PV q are related by
(2x2+2y2+zf02Nmon2z)Ψ=q,H<z<0,
where x and y are the zonal and meridional coordinates. The averaged Brunt–Väisälä frequency Nmon and Coriolis parameter f0 are the same as in section 2. Linearity of this equation permits a decomposition of Ψ: Ψ = Ψsur + Ψint (Charney 1971; Hoskins 1975; Ferrari and Wunsch 2010). The homogeneous (SQG) solution Ψsur can be calculated from buoyancy anomalies at the boundaries:
(2x2+2y2+zf02Nmon2z)Ψsur=0,
zΨsur|z=0=f01b(x,y,0),
zΨsur|z=H=f01b(x,y,H)=0,
while the particular (interior) solution Ψint from the interior PV:
(2x2+2y2+zf02Nmon2z)Ψint=q,
zΨint|z=0,H=0.
At the bottom boundary (H = 2000 m in this study), buoyancy anomaly b = −/ρ0 (with ρ being the density anomaly) is set to zero. We take the lateral boundary conditions as doubly periodic, and numerically resolve Eq. (A2) for Ψsur, given SSDA and Nmon2(z). However, since q is mostly unknown, Ψint cannot be directly solved from Eq. (A3).
Emphasizing that both solutions (Ψsur and Ψint) contribute to SSH, the isQG method projects downward the residual SSH (i.e., the total SSH minus the SQG contribution) to determine Ψint. Specifically, the method employs a two-gravest-mode scheme: Ψint is expanded in terms of BT–BC modes, and then is truncated at the two gravest ones (BT and BC1),
Ψ^int(k,l,z)=nAn(k,l)Fn(z)A0(k,l)F0(z)+A1(k,l)F1(z).
Here the caret denotes the horizontal Fourier transform with (k, l) being the wavenumbers. The variable Fn is the traditional (flat bottom) normal mode (Pedlosky 1987), with An being the modal coefficient. At the surface and bottom boundaries, both Ψsur and Ψint contribute to the pressure anomaly field such that
Ψ^sur(k,l,0)+Ψ^int(k,l,0)=gf0η^(k,l),
Ψ^sur(k,l,H)+Ψ^int(k,l,H)=0.
Here (g/f0)η^ is the surface pressure anomaly with η being the SSHA. We assume that the bottom pressure anomaly vanishes. Based on Eqs. (A4)(A6), Ψint can be solved, and thus the total solution Ψ. Furthermore, subsurface density and geostrophic velocity anomaly fields are deduced according to ρisQG = −(ρ0f0/g)∂Ψ/∂z and VisQG = k × ∇Ψ, with k being the unit vector in the direction of the z axis.

APPENDIX B

The LW15 Method (Pure Exponential Solution)

The LW15 method also employs the two-component decomposition of Eq. (A1), but alternately uses a different bottom boundary condition of
limzΨ^sur=Ψ^int=0.
Assuming an idealized exponential N2 profile (N = N0ez/h), the SQG solution is analytically derived as
Ψ^sur(k,l,z)=b^(k,l,0)N0κez/hI1(Leκez/h)I0(Leκ).
Here In are the modified Bessel functions, and κ=(k2+l2)1/2 the modulus of the wavenumber vector. The term Le = N0h/f0 (associated with the exponential stratification) is a deformation-like scale.
With zero flow at depth, the BT mode is absent (LaCasce 2012). Therefore, the amplitude (γ1) of an alternate BC mode (the first surface mode; de La Lama et al. 2016; LaCasce 2017) is the single unknown and is determined from the residual SSHA. The interior solution is then
Ψ^int(k,l,z)=γ1ez/hJ1(2.4048ez/h).
Here J1 is the Bessel function of the first kind. The term γ1 is determined as
γ1=1J1(2.4048)[gη^(k,l)f0b^(k,l,0)N0κI1(Leκ)I0(Leκ)].

The exponential N (see the dashed line in Fig. 2b) is obtained by fitting the deeper portion (bottom to −200 m) of monthly Nmon. This exponential is also used in the LW15 2D diagnosis of w.

APPENDIX C

The L19 Method

Recognizing that SSH at smaller scales significantly reflects higher-order BC modes (Lapeyre 2009; Smith and Vanneste 2013; Badin 2014; Wortham and Wunsch 2014), the L19 method employs a scale-dependent vertical projection of SSH.

  1. At larger scales (>LC; LC = 150 km in this study) where BT and BC1 signals are dominant, the L19 method retains the isQG two-gravest-mode scheme, i.e., Eqs. (A4)(A6).

  2. At smaller scales (≤LC) where other higher-order-mode contributions are prominent, the L19 method exploits the eSQG framework for the vertical structure of Ψint (Isern-Fontanet et al. 2008), with its amplitude determined by matching to the residual SSHA [i.e., (g/f0)η^(k,l)Ψ^sur(k,l,0) implicitly defined in Eq. (A5)] at the surface:
    Ψ^int(k,l,z)=[gf0η^(k,l)Ψ^sur(k,l,0)]exp(NL19f0κz),forκ2πLC.

    Here the constant NL19 is an averaged value of monthly Nmon in the top 1000 m (see the dotted line in Fig. 2b). Now Ψint can be determined at all scales, and thus Ψ.

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  • Badin, G., 2014: On the role of non-uniform stratification and short-wave instabilities in three-layer quasi-geostrophic turbulence. Phys. Fluids, 26, 096603, https://doi.org/10.1063/1.4895590.

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