Diagnosing Subsurface Vertical Velocities from High-Resolution Sea Surface Fields

Lei Liu; Huijie Xue; Hideharu Sasaki

doi:10.1175/JPO-D-20-0152.1

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Journal of Physical Oceanography

Abstract
1. Introduction
2. Data
- a. The 1/30°-resolution OFES simulation
- b. Data preprocessing
3. Methods
- a. The L19 reconstruction method
- b. The QG omega equation with vertical mixing terms
4. Results
5. Discussion
6. Conclusions
Derivation of Eq. (11)
Decomposition of wOFES

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Fig. 1.

Monthly mean SSH from the OFES simulation in December 2001. The regions to be considered are indicated by the boxes (from Liu et al. 2019). Subsurface w diagnoses are conducted in the larger box, and evaluated in the smaller box to avoid the edge effect.

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Fig. 2.

(a) SSD anomaly and (b) SSH anomaly fields. Solid box delimits the region 144°–150°E, 32°–38°N, as in Fig. 1. (c) The N²(z) profile (dashed line) derived directly from OFES, and the adjusted N²(z) (thick solid line) used by L19. A constant stratification $N_{0}^{2}$ (thin solid line) is also utilized by L19. (d),(f) OFES density and horizontal velocity fields (red vector arrows), and (e),(f) L19 reconstructions (blue arrows), at the 200-m depth. For (g) density and (h) horizontal velocity fields in the upper 1000 m, green lines denote the pattern correlation coefficients between OFES and L19. In (h), correlations are plotted for both u and υ (solid and dashed lines). Plots correspond to the date of 6 Dec 2001.

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Fig. 3.

(a) Prescribed vertical turbulent viscosity profile with a maximum value of A_υ0. (b) Thin and thick lines respectively indicate the time series of A_υ0 and the ML base (H_ML) averaged in the reconstruction region (dashed box in Fig. 1).

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Fig. 4.

Vertical velocity w fields at (top) the 50-m and (middle) 200-m depth: (a),(d) $w_{OFES}^{tg}$ component of the OFES simulation from Eq. (B1), (b),(e) adiabatic w_L19QG constituent of the L19 diagnosis, and (c),(f) adiabatic w_eSQG from the eSQG diagnosis. (g) Spectral correlation between $w_{OFES}^{tg}$ and w_L19QG as a function of horizontal wavenumber and depth. (h) Pattern correlation between $w_{OFES}^{tg}$ and the diagnosed w fields as a function of depth. Solid and dashed lines stand for w_L19QG and w_eSQG, respectively. Gray shading denotes the area-averaged MLD (110 m). Plots correspond to the date of 6 Dec 2001.

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Fig. 5.

The green line denotes the time series of (a) mean correlation between $w_{OFES}^{tg}$ and w_L19QG averaged in the 0–500-m layer, and (c) correlation between SSDA and the density anomaly field at MLD. Black line indicates the time series of MLD. Gray dashed line marks the 100-m depth. (b) Mean spatial correlation between $w_{OFES}^{tg}$ and w_L19QG as a function of depth averaged in the cold (blue line) and warm (red line) season.

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Fig. 6.

Vertical velocity w fields at the 50-m depth: (a) $w_{OFES}^{mix}$ component of the OFES simulation from Eq. (B1) and (b) diabatic w_L19m constituent of the L19 diagnosis. Plus symbols mark the position of the filamentary structures. (c) Fields of MLD (color) and SSDA (contours) derived from OFES data. Subdomains with large MLD horizontal variability are indicated by the boxes. (d) Pattern correlation between $w_{OFES}^{mix}$ and w_L19m as a function of depth. Gray shade denotes the area-averaged MLD (110 m). Plots correspond to the date of 6 Dec 2001.

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Fig. 7.

Vertical velocity w fields at the 20-m depth: (a) $w_{OFES}^{mix}$ component of the OFES simulation from Eq. (B1) and (b) diabatic w_L19m constituent of the L19 diagnosis. Plus symbols mark the position of the filamentary structures. (c) Pattern correlation between $w_{OFES}^{mix}$ and w_L19m as a function of depth. Gray shade denotes the area-averaged MLD (50 m). Plots correspond to the summertime date of 2 May 2002.

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Fig. 8.

Vertical velocity w fields at the 50-m depth: (a) OFES simulated w_OFES and (b) L19 total diagnosis w_L19. (c) Pattern correlation between the simulated and diagnosed w fields as a function of depth. Red and black lines pertain to w_L19 and its adiabatic constituent w_L19QG, respectively. Gray shading denotes the area-averaged MLD (110 m). Plots correspond to the date of 6 Dec 2001.

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Fig. 9.

(a) Pattern correlations between the OFES simulated w_OFES and L19 diagnosis w_L19 as a function of time and depth. (b) As in (a), but for the adiabatic w_L19QG constituent of the L19 diagnosis. (c) The difference taken between (a) and (b) in the upper 300 m. The black line denotes the area-averaged MLD. Notice that the scale is different in (c).

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Fig. 10.

(a)–(c) Mean pattern correlations between the diagnosed and simulated w fields, and (d)–(f) mean rms of different w fields, as a function of depth averaged in (left) 2001/02, (center) the cold season, and (right) the warm season. Gray, red, and black lines stand for w_OFES, w_L19, and w_L19QG, respectively.

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Fig. 11.

(a) Spatial correlation between $w_{OFES}^{tg}$ and w_L19QG as a function of depth. Black and red lines denote the results with L_C being 150 and 100 km, respectively. (b),(c) Spectral correlation between $w_{OFES}^{tg}$ and w_L19QG, with L_C (marked by dashed lines) being 150 km in (b) and 100 km in (c). Plots correspond to the date of 2 May 2002.

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Fig. 12.

Pattern correlations between the diagnosed and simulated w fields as a function of depth on (a) 6 Dec 2001 and (b) 2 May 2002. Black lines stand for w_L19QG. Gray, red, blue, and green lines stand for w_L19 with varying values of A_υ0. The A_υ0 values in (a) and (b) respectively fall in the range of 6 × 10⁻³–6 × 10⁻² m² s⁻¹ and 6 × 10⁻⁴–4.5 × 10⁻³ m² s⁻¹. Gray shading denotes the area-averaged MLD of 110 m in (a) and 50 m in (b).

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Fig. 13.

(a) As in Fig. 9c, but for the turbulent mixing parameterization scheme put forth by Ponte et al. (2013). (b) Red and black lines denote the ML-averaged value of correlations shown in (a) and Fig. 9c, respectively.

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Fig. 14.

(a) As in Fig. 1, but for region 2 (from Liu et al. 2019) in the domain of the North Pacific Current. (b)–(d) As in Figs. 10a–c, but for region 2.

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Fig. 15.

(a) Temporal correlation between w_L19 and w_OFES and (b) autocorrelation functions (solid lines), for the 2-yr simulations averaged in the upper 500 m within the target region delimited by the solid box in Fig. 1. In (b), dashed line marks the integral time scale [ $I = \int_{0}^{T_{0}} c (t) d t$ , with c(t) being the autocorrelation and T₀ its first zero-crossing; Stammer 1997]; gray and red lines represent w_OFES and w_L19, respectively.

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Fig. B1.

Vertical velocity w fields at the 50-m depth on 6 Dec 2001: (a) OFES simulated w_OFES and (b)–(f) its decomposition into five components from Eq. (B1).

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Editorial Type:: Article

Article Type:: Research Article

Diagnosing Subsurface Vertical Velocities from High-Resolution Sea Surface Fields

Lei Liu

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

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Huijie Xue

Huijie Xue State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China
Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou, China
School of Marine Sciences, University of Maine, Orono, Maine

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https://orcid.org/0000-0003-0738-4978

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Hideharu Sasaki

Hideharu Sasaki Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama, Japan

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Online Publication:: 09 Apr 2021

Print Publication:: 01 May 2021

DOI:: https://doi.org/10.1175/JPO-D-20-0152.1

Page(s):: 1353–1373

Received:: 10 Jul 2020

Accepted:: 01 Feb 2021

Published Online:: 09 Apr 2021

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

Using the extended “interior + surface quasigeostrophic” method from the 2019 study by Liu et al. (hereafter L19), subsurface density and horizontal velocities can be reconstructed from sea surface buoyancy and surface height. This study explores the potential of L19 for diagnosing the upper-ocean vertical velocity w field from high-resolution surface information, employing the 1/30° horizontal resolution OFES model output. Specifically, we employ the L19-reconstructed density and horizontal velocity fields in a diabatic version of the omega equation that incorporates a simplified parameterization for turbulent vertical mixing. The w diagnosis is evaluated against OFES output in the Kuroshio Extension region of the North Pacific, and the result indicates that the L19 method constitutes an effective framework. Statistically, the OFES-simulated and L19-diagnosed w fields have a 2-yr-averaged spatial correlation of 0.42–0.51 within the mixed layer and 0.51–0.67 throughout the 1000-m upper ocean below the mixed layer. Including the diabatic turbulent mixing effect has improved the w diagnoses inside the mixed layer, particularly for the cold-season days with the largest correlation improvement reaching 0.31. Our encouraging results suggest that the L19 method can be applied to the high-resolution sea surface height data from the forthcoming Surface Water and Ocean Topography (SWOT) satellite mission for reconstructing 3D hydrodynamic conditions of the upper ocean.

Corresponding author: Huijie Xue, huijiexue@scsio.ac.cn

Abstract

Corresponding author: Huijie Xue, huijiexue@scsio.ac.cn

Keywords: North Pacific Ocean; Mesoscale processes; Mixing; Vertical motion

1. Introduction

Vertical velocity w in the upper ocean plays a fundamental role in the exchanges of heat, salinity, and biogeochemical elements between the surface mixed layer and the ocean interior (Nagai et al. 2008). Despite its importance, direct measurement of w is still very challenging due to its typically short spatiotemporal scales and much smaller magnitude compared to horizontal velocities (Buongiorno Nardelli et al. 2018; Qiu et al. 2020), and therefore w has often been indirectly diagnosed via the omega equation. Among several different versions, the most commonly used one is the quasigeostrophic (QG) omega equation originally put forth for the atmospheric study (Hoskins et al. 1978). This formulation allows diagnosis of adiabatic w from the observations of three-dimensional (3D) density ρ and geostrophic horizontal motions V_g, and has been successfully applied by many investigators (e.g., Pollard and Regier 1992; Rudnick 1996; Buongiorno Nardelli et al. 2001; Rousselet et al. 2019). To study oceanic motions of large Rossby number, the QG omega equation has further been extended to include non-QG processes neglected in the original formulation (Pallàs-Sanz and Viúdez 2005; Giordani et al. 2006; Nagai et al. 2006).

Diagnosis of w through the omega equation requires high-resolution synoptic 3D oceanographic observations that are rarely met and necessarily confined in spatial coverage. To circumvent such a limitation, a number of studies explored the possibility of applying the omega equation to the subsurface 3D ρ and V_g fields that are reconstructed from satellite sea surface measurements (Buongiorno Nardelli et al. 2012; Buongiorno Nardelli 2020). Lapeyre and Klein (2006) modified the surface quasigeostrophic (SQG) theory originally established for the atmosphere (Blumen 1978; Held et al. 1995; Lapeyre 2017), and put forth an effective SQG (eSQG) method to reconstruct the subsurface structures from either the sea surface height (SSH) or sea surface density (SSD). Taking into account the nonnegligible potential vorticity (PV) in the ocean interior (neglected by the original SQG formalism), the eSQG method yielded promising reconstructions of ρ and V_g (Isern-Fontanet et al. 2006, 2008; LaCasce 2012; Smith and Vanneste 2013), based on which w below the mixed layer (ML) was successfully diagnosed through the omega equation (LaCasce and Mahadevan 2006; Klein et al. 2008, 2009; Ponte and Klein 2013; Qiu et al. 2016, 2020). Emphasizing the fact that inside the ML vertical circulation is controlled by the turbulent mixing (Nagai et al. 2006), several recent studies tried to explore simplified parameterizations for turbulent vertical mixing and added the mixing effect (diabatic w) to the eSQG-diagnosed adiabatic w (e.g., Ponte et al. 2013; Chavanne and Klein 2016). Although the eSQG method has been proved to be promising, it is constrained by the underlying assumption that SSD and interior PV are correlated (Wang et al. 2013; González-Haro and Isern-Fontanet 2014; Isern-Fontanet et al. 2014, 2017; Liu et al. 2014, 2019), in which case SSD would have the same phase as the SSH. However, SSH–SSD phase shift is sometimes quite notable in real oceans, indicating that the eSQG method could not capture the vertical phase shift between the SSD and interior PV, which is crucial to the w diagnosis (Qiu et al. 2016, 2020).

Wang et al. (2013) proposed a more elaborate method called “interior + surface QG” (isQG) that employs both SSH and SSD to reconstruct subsurface ρ and V_g. This method first calculates the solution associated with SSD (i.e., surface PV sheet; Bretherton 1966) via the SQG theory; it then solves for subsurface PV using the two gravest vertical modes, whose amplitudes are determined by the residual SSH (the total SSH minus the contribution from the SQG solution). In contrast to the eSQG approach, the isQG method employs both SSH and SSD to constrain the surface and subsurface PV simultaneously, and is not restricted by the SSH–SSD correlation. By evaluating the quality of reconstructed ρ and V_g, several studies (Wang et al. 2013; Liu et al. 2014, 2017; LaCasce and Wang 2015; Yan et al. 2020; Chen et al. 2020) have demonstrated the isQG method to be promising (it is worth mentioning these studies did not include w diagnosis). However, a recent study (Liu et al. 2019) utilizing the 1/30°-resolution Ocean General Circulation Model (OGCM) for the Earth Simulator (OFES) simulation found that, the isQG method employing only the barotropic (BT) and first baroclinic (BC1) modes cannot perform properly (below the ML) for the smaller-scale horizontal motions (<~150 km) that contain significant higher-order-mode signals (Lapeyre 2009; Badin 2014; de La Lama et al. 2016; LaCasce 2017). Therefore, although the isQG method can reasonably capture the abovementioned vertical phase shift between SSD and interior PV (that is crucial for diagnosing w), result of Liu et al. (2019) suggests that the deficiency of only BT and BC1 might adversely impact the w diagnosis beneath the ML, especially in the deeper layers.

In preparation for the upcoming Surface Water and Ocean Topography (SWOT) satellite mission (Durand et al. 2010; Fu and Ubelmann 2014; Su et al. 2018; Torres et al. 2018; Wang et al. 2018, 2019; Wang and Fu 2019; Li et al. 2019) that aims at capturing high-resolution SSH, Liu et al. (2019) devised an extended isQG method (denoted L19 hereafter): at larger scales (>~150 km) where BT and BC1 signals are prominent, the isQG scheme is retained; while at smaller scales (≤~150 km) where other higher-order BC modes become important, the eSQG framework (that can reasonably represent these BC modes) is adopted. Evaluation of the isQG, eSQG and L19 methods demonstrated that, in terms of both the ρ and V_g fields in the upper 1000 m, the L19 method can achieve the most satisfactory reconstructions. Although encouraging results validated the potential applicability of the L19 method to the SWOT-like high-resolution sea surface data, the study (Liu et al. 2019) only investigated the reconstructability of ρ and V_g, whereas the L19 method was not explored for the upper-ocean w diagnosis.

The goal of this work is to extend the recent study of Liu et al. (2019) and explore the applicability of the L19 method in diagnosing subsurface w from high-resolution sea surface variables. We utilize a diabatic version of the QG omega equation that takes into account the effects of turbulent mixing. The rest of this article is structured as follows. The 1/30°-resolution OFES simulation serving as the basis of our investigations is described in section 2. Section 3 briefly reviews the L19 method and describes the QG omega equation including vertical mixing terms. In section 4 we evaluate the L19 performance for diagnosing subsurface w. Limitations of the L19 method and the mixing parameterization scheme are discussed in section 5. Finally, section 6 provides the conclusions.

2. Data

a. The 1/30°-resolution OFES simulation

To evaluate the L19 performance for the subsurface w diagnosis, the OFES (Masumoto et al. 2004; Komori et al. 2005) simulation output with a 1/30° horizontal resolution and 100 vertical levels is used as “true” field in this study. Atmospheric forcing of this high-resolution simulation, an extended version of the global 1/10° OFES simulation (Sasaki et al. 2008; Nonaka et al. 2016), is from the 6-hourly Japanese 25-yr reanalysis (Onogi et al. 2007). The model applies a biharmonic operator for horizontal mixing of momentum and tracers to reduce numerical noises and employs the scheme of Noh and Kim (1999) for vertical mixing. Notice that the length scales of eddy variability captured by the 1/30° OFES simulation are similar to those expected to be detected by the SWOT satellite mission (Qiu et al. 2016).

The 1/30° OFES simulation was run for the North Pacific Ocean of 20°S–66°N and 100°E–70°W, and integrated for 3 years from 2000 to 2002 (Qiu et al. 2014; Sasaki and Klein 2012; Sasaki et al. 2014, 2017). In this study, the daily mean temperature, salinity, horizontal and vertical velocities, and SSH in the last two years (2001 and 2002) are used. As pointed out by Sasaki et al. (2017), filtering out a large part of near-inertial motions, the daily average retains only the balanced part of the flow. We conduct the subsurface w diagnosis using surface variables from the OFES model as input, in the Kuroshio Extension region of 30°–40°N and 142°–152°E (10° × 10° dashed box in Fig. 1). To avoid the edge effect, diagnosed w is evaluated in a smaller domain (32°–38°N, 144°–150°E; 6° × 6° solid box in Fig. 1) against the OFES simulation itself.

b. Data preprocessing

The input of the L19 method includes three variables: SSD anomaly field (SSDA), SSH anomaly field (SSHA), and stratification vertical profile N²(z).

In the OFES simulation, potential density is estimated from potential temperature, salinity and pressure following the method of Jackett and McDougall (1995) based on UNESCO1983. For consistency, according to the UNESCO 1983 equation of state (Fofonoff and Millard 1983), we utilize the temperature and salinity data from OFES to derive potential density fields, which are then used to calculate an area-averaged N²(z) by

N^{2} (z) = - \frac{g}{ρ_{υ}} \frac{\partial \bar{ρ}}{\partial z} .

(1)

Here g is the gravity constant, and ρ_υ and

\bar{ρ}

are the volumetric and horizontal mean of potential density, respectively, within the dashed box in Fig. 1. Such N²(z) approaches zero near the surface, and could cause unrealistic overshoots when it is directly utilized in the isQG framework (Wang et al. 2013). Therefore, according to Wang et al. (2013), we adjust the N²(z) profile in the ML: replace the surface value by a ML-averaged one; then a linear interpolation, between this new surface value and the one at the base of ML (defined as where the stratification reaches its maximum), is used as the adjusted stratification profile in the ML. The L19 method (see section 3a) also requires a constant stratification

N_{0}^{2}

that is set to be an averaged value of the unadjusted N²(z) in the top 1000 m (Isern-Fontanet et al. 2008, 2017; Liu et al. 2019).

To get the anomaly fields of SSD and SSH (i.e., SSDA and SSHA), we respectively subtract their low-pass-filtered fields, defined as a least squares fit of a field to the quadratic surface S(x, y) (Wang et al. 2013),

S (x, y) = [C_{0}, C_{1}, C_{2}, C_{3}, C_{4}, C_{5}] {[1, x, y, x^{2}, y^{2}, x y]}^{T} .

(2)

Here x and y denote the zonal and meridional coordinates, respectively. The superscript T represents matrix transpose. For the purpose of Fourier transform (Qiu et al. 2016), we apply a two-dimensional trapezoid window to the SSDA and SSHA fields, in a 1° band along the box edges.

As an example, Figs. 2a–c present the input fields required by the L19 method as applied to the reconstruction region (dashed box in Fig. 1) on 6 December 2001. In Fig. 2c, the dashed line shows the N²(z) derived directly from the OFES data, the thick solid line represents the adjusted N²(z), and the thin solid line denotes the constant $N_{0}^{2}$ .

3. Methods

a. The L19 reconstruction method

In the QG theory, the geostrophic streamfunction Ψ is related to PV q as

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial}{\partial z} \frac{f_{0}^{2}}{N^{2}} \frac{\partial}{\partial z}) Ψ = q, - H < z < 0,

(3)

with N and f₀ being the Brunt–Väisälä frequency and Coriolis parameter, respectively (Pedlosky 1987). The linear Eq. (3) permits a decomposition of Ψ into a homogeneous (SQG) solution Ψ^sur and a particular (interior) solution Ψ^int (Charney 1971; Hoskins 1975; Ferrari and Wunsch 2010) such that Ψ = Ψ^sur + Ψ^int. The SQG solution Ψ^sur can be derived from buoyancy anomalies at the boundaries, while Ψ^int from the interior PV, q. Specifically,

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial}{\partial z} \frac{f_{0}^{2}}{N^{2}} \frac{\partial}{\partial z}) Ψ^{sur} = 0,

{\frac{\partial}{\partial z} Ψ^{sur} |}_{z = 0} = f_{0}^{- 1} b (x, y, 0),

{\frac{\partial}{\partial z} Ψ^{sur} |}_{z = - H} = f_{0}^{- 1} b (x, y, - H) = 0,

(4)

(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial}{\partial z} \frac{f_{0}^{2}}{N^{2}} \frac{\partial}{\partial z}) Ψ^{int} = q,

{\frac{\partial}{\partial z} Ψ^{int} |}_{z = 0, - H} = 0.

(5)

At the bottom boundary (in this study H = 4000 m), the value of buoyancy anomaly b is set to zero. The lateral boundary conditions are taken as doubly periodic. Equation (4) can be numerically resolved for Ψ^sur, given SSDA and N²(z). However, Ψ^int cannot be directly solved from Eq. (5), because we do not know the subsurface q.

Emphasizing that both Ψ^sur and Ψ^int contribute to SSH, as in Wang et al. (2013), the L19 method projects downward the residual SSH (the total SSH minus the SQG estimate of SSH) to determine Ψ^int. The novel point of L19 is the recognition that the downward projection (into BT and BC structures) should be a function of horizontal scale (Wortham and Wunsch 2014): the L19 method introduces a cutoff scale L_C (prescribed as 150 km here) and implements a scale-dependent vertical projection.

For larger horizontal scales (>L_C), the L19 method retains the isQG two-gravest-mode scheme. Specifically, Ψ^int is expanded in terms of vertical normal modes (that form a complete basis), and then truncated at the two gravest ones (i.e., BT and BC1):

{\hat{Ψ}}^{int} (k, l, z) = \sum_{n} A_{n} (k, l) F_{n} (z) \approx A_{0} (k, l) F_{0} (z) + A_{1} (k, l) F_{1} (z) .

(6)

Here the hat represents the Fourier transform with (k, l) being the horizontal wavenumbers. The variable F_n is the (flat bottom) vertical normal mode with A_n being the modal coefficient (Pedlosky 1987). Both Ψ^sur and Ψ^int contribute to the surface and bottom pressure such that

{\hat{Ψ}}^{sur} (k, l, 0) + {\hat{Ψ}}^{int} (k, l, 0) = \frac{g}{f_{0}} \hat{η} (k, l),

(7)

{\hat{Ψ}}^{sur} (k, l, - H) + {\hat{Ψ}}^{int} (k, l, - H) = 0.

(8)

Here η is the SSHA that determines the surface pressure anomaly

(g / f_{0}) \hat{η}

, and the bottom pressure anomaly is neglected. With these two boundary conditions, Eqs. (7) and (8), Eq. (6) can be solved for Ψ^int. Note that, in Eq. (7), the residual SSHA is implicitly defined, i.e.,

(g / f_{0}) \hat{η} (k, l) - {\hat{Ψ}}^{sur} (k, l, 0)

For smaller scales (≤L_C), at which higher-order-mode contributions are dominant, the L19 method exploits the eSQG framework (Lapeyre and Klein 2006; Isern-Fontanet et al. 2008) for the vertical structure of Ψ^int (that decays away from the surface exponentially), and determines its amplitude by matching at the surface to the residual SSHA:

{\hat{Ψ}}^{int} (k, l, z) = [\frac{g}{f_{0}} \hat{η} (k, l) - {\hat{Ψ}}^{sur} (k, l, 0)] \exp (\frac{N_{0}}{f_{0}} κ z), for κ \geq \frac{2 π}{L_{C}} .

(9)

Here N₀ is a constant stratification, and κ = (k² + l²)^1/2 is the modulus of the wavenumber vector. Now Ψ^int can be determined, at all scales, and thus the total solution Ψ.

As an example for 6 December 2001, Figs. 2d–h illustrate the L19 reconstructability for subsurface ρ and V_g: horizontal structures of the OFES fields can be well captured by the L19 solution (cf. Figs. 2d and 2e; also see red and blue arrows in Fig. 2f), with strong spatial correlation exceeding 0.9 in the upper 1000 m (green lines in Figs. 2g,h).

b. The QG omega equation with vertical mixing terms

The QG version of the omega equation is adopted to diagnose subsurface w from the L19-reconstructed ρ and V_g. Furthermore, for diagnosis inside the ML where the turbulent mixing controls w (Giordani et al. 2006; Nagai et al. 2006; Ponte et al. 2013; Qiu et al. 2016, 2020), we add the vertical mixing effects to the QG omega equation. Due to the lack of the mixing coefficient from the OFES output, we follow the simplified vertical mixing parameterization scheme proposed by Chavanne and Klein (2016): for the whole study region (dashed box in Fig. 1), the turbulent viscosity A_υ(z) is prescribed as horizontally invariant and a parabolic profile in the vertical (with the values being zero at the surface and ML base, and below the ML; Fig. 3a):

A_{υ} (z) = - 4 A_{υ 0} \frac{z}{H_{ML}} (1 + \frac{z}{H_{ML}}), - H_{ML} \leq z \leq 0.

(10)

Here H_ML is the area-averaged ML depth (MLD) estimated as where the stratification (used by the L19 reconstruction in section 3a) reaches its maximum, A_υ0 the maximum viscosity value. It is important to emphasize that A_υ0 should have a seasonal variability (i.e., smaller values in summertime; Schudlich and Price 1998; Shrira and Almelah 2020) in agreement with the seasonal cycle of H_ML (see thin and thick lines in Fig. 3b). According to the vertical diffusivity K_υ estimated from the 10-yr mooring measurements at the Kuroshio Extension Observatory site (Cronin et al. 2015, see their Fig. 10), and meanwhile taking the viscosity A_υ be equal to K_υ (Xie et al. 2017), we set A_υ0 to the typical ML values as shown by the thin line in Fig. 3b.

Provided that the Rossby (Ro) and Ekman (Ek) numbers are small, the diabatic QG omega equation including the vertical mixing terms can be written as (a detailed derivation is given in the appendix A and also can be found in Chavanne and Klein 2016)

(f_{0}^{2} \frac{\partial^{2}}{\partial z^{2}} + N^{2} \nabla^{2}) w_{L 19} = 2 \nabla \cdot Q + \frac{d^{2} A_{υ}}{d z^{2}} \nabla^{2} b .

(11)

Equation (11) employs the same area-averaged N²(z) as used by the L19 reconstruction. The adiabatic forcing Q = (g/ρ₀)(∇V_g)^T ⋅ ∇ρ (with ρ₀ and ∇ being the reference density and horizontal gradient operator, respectively), known as the QG Q vector (also named the frontogenesis vector), represents the kinematic deformation of the horizontal density field induced by the geostrophic horizontal motions (Hoskins et al. 1978). The second forcing on the right-hand side represents the diabatic dynamics (viz., the vertical mixing). Both forcing terms can be calculated from the L19-reconstructed V_g and ρ. Hereafter, the adiabatic constituent (forced by the geostrophic kinematic deformation) of the L19-diagnosed total solution w_L19 is denoted as w_L19QG; while the diabatic constituent (generated by the mixing process) is represented by w_L19m, i.e., w_L19 = w_L19QG + w_L19m.

Equation (11) is numerically solved using finite differences with zero Dirichlet boundary conditions at the surface and bottom [i.e., w_L19(z = 0) = w_L19(z = −H) = 0] first in the spectral space (with lateral boundary conditions being doubly periodic), then in the physical space through inverse Fourier transform (also see appendix B). Note that, due to the discontinuity of dA_υ/dz at the ML base H_ML (see Fig. 3a), w_L19m therein must be treated differently. The same way as Chavanne and Klein (2016) did, we vertically integrate Eq. (11) across H_ML over an infinitesimally thin layer to yield

f_{0}^{2} ({\frac{\partial w_{L19m}}{\partial z} |}_{z = - H_{ML}^{+}} - {\frac{\partial w_{L19m}}{\partial z} |}_{z = - H_{ML}^{-}}) = {\nabla^{2} b |}_{z = - H_{ML}^{+}} {\frac{d A_{υ}}{d z} |}_{z = - H_{ML}^{+}},

(12)

which is discretized at z = −H_ML when numerically solving Eq. (11).

It is important to point out that, due to the assumption of small Ek (a parameter physically related to the ratio of the Ekman depth to the MLD), this parameterization scheme does not resolve the surface Ekman layer. To include the effects of Ekman dynamics, the surface boundary condition for w would have to be modified, as suggested by Nagai et al. (2006). Meanwhile, the scheme actually has not considered the atmospheric forcing (with A_υ and K_υ being zero at the surface).

4. Results

In this section, we first focus on typical case studies to analyze individually each constituent of the diagnosed w_L19, i.e., w_L19QG and w_L19m. Specifically, w_L19QG and w_L19m are evaluated against their OFES counterparts: $w_{OFES}^{tg}$ forced by the geostrophic kinematic deformation and $w_{OFES}^{mix}$ induced by the turbulent mixing, which are derived through decomposing the OFES simulated w_OFES (see appendix B for details) the same way as Giordani et al. (2006) and Qiu et al. (2020) did. Then, to assess the practical applicability of our L19 framework in the real oceans, we assess the diagnosed total solution w_L19 against the undecomposed w_OFES throughout the 2-yr simulation period (i.e., 2001 and 2002).

a. Performance of w_L19QG

The wintertime example on 6 December 2001 is first presented to elucidate the effectiveness of our adiabatic solution w_L19QG in capturing the geostrophic kinematic deformation process (i.e., kinematic deformation of density anomalies by horizontal geostrophic motions; Giordani et al. 2006). As shown in Figs. 4a and 4b comparing $w_{OFES}^{tg}$ and w_L19QG at the 50-m depth inside the ML (with the MLD averaged in the target region being 110 m; see Fig. 2c), the diagnosed w_L19QG matches the simulated $w_{OFES}^{tg}$ field quite well with pattern correlation reaching as high as 0.91. It is worth pointing out that w_L19QG captures primarily the mesoscale features. This can be more clearly illustrated by Fig. 4g in the spectral domain: the 50-m spectral correlation between $w_{OFES}^{tg}$ and w_L19QG generally exceeds (falls below) 0.7 for signals with wavelengths longer (shorter) than about 100 km. At the 200-m depth below the ML (Figs. 4d,e), the $w_{OFES}^{tg}$ and w_L19QG patterns also show a good match with spatial correlation being 0.86. Notice this value is slightly smaller than its 50-m counterpart, because the L19 performance (actually the eSQG framework), at smaller scales, deteriorates gradually with an increasing depth (Liu et al. 2019): the 200-m spectral correlation in general falls below 0.4 at wavelengths shorter than about 100 km (see Fig. 4g). However, as the magnitude of the smaller-scale signals tends to diminish with depth (Sasaki et al. 2014; Qiu et al. 2016; also compare Figs. 4a,d), Fig. 4h shows that w_L19QG matches $w_{OFES}^{tg}$ well with pattern correlations (solid line) exceeding 0.8 throughout the upper 1000-m layer. In this case study, we also present the comparison between w_L19QG and the adiabatic w_eSQG (diagnosed from the eSQG method using SSHA; Qiu et al. 2016). Although w_eSQG is similar to $w_{OFES}^{tg}$ (cf. Figs. 4c and 4a; Figs. 4f and 4d), correlations for w_eSQG fall by and large in between 0.7 and 0.8 in the upper 1000 m (dashed line in Fig. 4h), which are smaller than those for w_L19QG (solid line).

Figure 5a shows the mean correlation between w_L19QG and $w_{OFES}^{tg}$ averaged in the 0–500-m layer throughout the 2-yr simulations. A closer look indicates that the w_L19QG performance (green line) undergoes a seasonal cycle (i.e., higher correlation in winter) in accordance with that of the MLD (black line), although the seasonal variability appears less prominent for w_L19QG (green line). We then divide the two simulation years into two seasons: the cold and warm seasons with the MLD deeper and shallower than 100 m (see the gray dashed line in Fig. 5c), respectively. The mean correlation in the upper 1000 m averaged separately in these two seasons (Fig. 5b) also demonstrates a better performance of w_L19QG in the cold season (blue line) than in the warm season (red line). This result agrees with the conclusion of Isern-Fontanet et al. (2008) that the SQG reconstruction can be more successful when SSDA resembles the density anomaly at the MLD (ρ_MLD), as seen from the green line in Fig. 5c showing that SSDA better resembles ρ_MLD with higher spatial correlations (>~0.5) in the cold season.

b. Performance of w_L19m

As a winter example case, here fields on 6 December 2001 (results are typical for all other wintertime days) are used again to analyze the performance of our diabatic solution w_L19m. We prescribe A_υ0 to be 2 × 10⁻² m² s⁻¹ (see Fig. 3b), which is similar to that used by previous studies of the eSQG-based w diagnosis [4 × 10⁻² m² s⁻¹ by Ponte et al. (2013); 1.5 × 10⁻² m² s⁻¹ by Chavanne and Klein (2016)]. At 50 m inside the ML (upper 110 m), plane views of the $w_{OFES}^{mix}$ field (Fig. 6a) reveal a complex submesoscale patterns of elongated filaments or cells of upwelling and downwelling. Although noticeable departures of w_L19m (Fig. 6b) from $w_{OFES}^{mix}$ can be found (with the spatial correlation being 0.37), our diabatic diagnosis does reasonably reproduce much of the simulated positive and negative stripes of the vertical motion that alternately appear in Fig. 6a, such as the noticeable south–north filamentary and dipolar structure around 147°E and 32°–34°N, the west–east one around 34°–35°N, and those denoted by the plus symbols. Notice that in the OFES simulation with realistic external forcings, the abovementioned structures are blurred by patchy structures (not found in w_L19m) probably caused by air–sea fluxes such as the wind stress, which actually are not included in our simplified mixing parameterization (see section 3b). It is also important to emphasize that our parameterization utilizes a horizontally invariant H_ML, which is not beneficial for recovering the w signals associated with the combined effects of the H_ML topology (i.e., horizontal variability) and the surface mechanical energy (Giordani et al. 2006). Therefore, in the subdomains with noticeable MLD topology (such as the ones delimited by the boxes in Fig. 6c; see the color for MLD), large discrepancies are found between Fig. 6a ( $w_{OFES}^{mix}$ ) and Fig. 6b (w_L19m). Meanwhile, these large discrepancies may also be partly caused by that, the assumption of small Ro behind the parameterization becomes less reasonable at strong density fronts (contour in Fig. 6c) correlated with the notable MLD topology. As shown by Fig. 6d, our mixing parameterization tends to be more effective in the near-surface shallower layers, with the spatial correlations (between $w_{OFES}^{mix}$ and w_L19m) reaching 0.5 around the 20-m depth, below which the w_L19m performance deteriorates gradually with an increasing depth.

To contrast the wintertime performance of the mixing parameterization, we present in Fig. 7 a typical summertime example on 2 May 2002 with the area-averaged MLD being 50 m. In this case A_υ0 is prescribed to be smaller (1.5 × 10⁻³ m² s⁻¹; see section 3b and Fig. 3b), because the vertical mixing is weakened as the MLD shoals toward summer (Mensa et al. 2013; Callies et al. 2015). Visually unlike the winter example shown above, here the 20-m diagnosed w_L19m (Fig. 7b) deviates notably from the simulated $w_{OFES}^{mix}$ field (Fig. 7a) with the pattern correlation being 0.31. However, it is worth noticing that our parameterization does reasonably recover some of the elongated filaments in $w_{OFES}^{mix}$ , such as those around 32.5°N and 144°–146°E; 36°–37°N and 146°–148°E (see the plus symbols). Figure 7c displays the spatial correlations between $w_{OFES}^{mix}$ and w_L19m reaching the maximum of 0.38 in the shallower part of the ML, which are weaker than those presented in Fig. 6d, further demonstrating that the w_L19m performance is less effective in the summertime ML.

c. Evaluation of w_L19 against w_OFES

To explore the potential applicability of L19 in the real oceans, hereafter we evaluate the diagnosed total solution w_L19 (i.e., w_L19QG + w_L19m) against the undecomposed w_OFES. For the wintertime example on 6 December 2001 inside the ML (110-m depth), the 50-m visual comparison between Figs. 8a and 8b shows a favorable correspondence between the simulated w_OFES and diagnosed w_L19, particularly for those alternating filaments of upwelling and downwelling triggered by the diabatic vertical mixing (recall $w_{OFES}^{mix}$ in Fig. 6a). The spatial correlation between Figs. 8a and 8b is 0.43. Compared to the eSQG-based results of Qiu et al. (2016) emphasizing that diagnosing w from surface information is a real challenge, although this correlation for w_L19 is lower than those for the reconstructed ρ and V_g (green lines in Figs. 2g and 2h; see also the results of Liu et al. 2019), we believe the 50-m w_L19 diagnosis presented here is satisfactory. It is important to emphasize that, although the w_L19QG constituent has been demonstrated to be effective in section 4a, this adiabatic solution alone cannot approximate w_OFES in the ML: pattern correlation between the 50-m w_OFES (Fig. 8a) and w_L19QG (recall Fig. 4b) fields is very low with the value being only 0.16. This is because that the non-QG processes (such as the turbulent vertical mixing; Ponte et al. 2013) significantly contribute to the near-surface w_OFES signal, which can be seen in appendix B (Fig. B1) that analyzes the composition of w_OFES field. Figure 8c shows the spatial correlations in the upper 1000 m. Within the ML (gray shading) where the vertical mixing effect is prominent, w_L19QG alone almost fails to recover the w_OFES with weak correlations especially near the surface (with values approaching 0.1; black line). After including the turbulent mixing contribution (i.e., w_L19m), correlations for w_L19 (red line) notably improve with the maximum increase up to 0.31 in the near-surface layer (notice the difference between the red and black lines). In the 110–1000-m layer below the ML, the diagnosed w_L19 has correlations with w_OFES falling in between 0.59 and 0.79 (red line). Note that the diabatic mixing contribution becomes insignificant beneath the ML, and w_L19 is largely determined by its adiabatic w_L19QG constituent: correlations for w_L19 (red line) and w_L19QG (black line) are similar.

Furthermore, w diagnoses throughout the 2-yr OFES simulations are examined. Figure 9 displays the spatial correlations between the simulated and diagnosed w fields as a function of time and depth. We first analyze the w_L19QG results shown in Fig. 9b. Inside the ML (see black line marking the MLD), correlations for w_L19QG are generally low with the values being smaller than about 0.5, particularly in the near-surface layers. As indicated in sections 4a and 4b, the reason for these low correlations may be threefold. First and most important, the vertical motions within the ML are considerably dictated by the ageostrophic contributions such as the vertical mixing process, which are not represented in the adiabatic w_L19QG. Second, at submesoscales (<~50 km), w_L19QG (actually the eSQG framework employed by the L19 method) cannot reasonably represent w signals that are forced by the geostrophic kinematic deformation. Third, at wavelengths shorter than L_C (150 km), the L19 (actually the eSQG framework) performance in the shallower layers could be degraded by the SSH–SSD phase shift (Liu et al. 2019). Below the ML, primarily due to the weakened mixing process, w_L19QG has on average good correlations (exceeding 0.6) with w_OFES. Figure 9a shows the result for w_L19 diagnosis that takes into account the vertical mixing process. A visual comparison between Figs. 9a and 9b shows that the w_L19 correlations have notably improved within the ML. Beneath the ML, correlations remain almost unchanged as expected, due to the weakening of the mixing process therein. To illustrate the improvement more clearly, we present in Fig. 9c the difference between Figs. 9a and 9b. It is worth noticing that the improvement occurs primarily in the shallower part of the ML, and tends to undergo a regular seasonal cycle consistent with that of the ML: larger increase of correlation (reaching up to 0.31) can be found during wintertime when the MLD (black line) deepens. Consistent with the case studies analyzed in section 4b, Fig. 9c shows that when the turbulent mixing process becomes weakened in summer, the employed mixing parameterization scheme is less effective.

Figure 10a shows the mean correlation averaged in 2001/02. For the w_L19QG field (black line) below the 100-m depth, correlations fall by and large in between 0.51 and 0.67, and similar results have been reported in the previous study of the eSQG-based w diagnosis by Qiu et al. (2016, see their Fig. 3c). Within the ML shallower than 100 m, although the w_L19QG correlations are relatively weaker in the range of 0.31–0.51, they appear to be a little better than the eSQG counterparts in Qiu et al. (2016, see also their Fig. 3c). After including the vertical mixing effect, results for w_L19 (red line) improve inside the ML especially in the near-surface layers, with the w_L19 correlations in the range of 0.42–0.51 (see also Table 1). Figure 10b shows the statistical results in the cold season: within and below the ML (upper 180 m) the ranges for w_L19QG correlations are 0.27–0.51 and 0.51–0.67 (black line), respectively; by adding the diabatic mixing effect, w_L19 correlations (red line) notably improve inside the ML with values falling in the 0.42–0.51 range. In the summer (see Fig. 10c), compared to the w_L19QG correlations (falling in between 0.35 and 0.51 in the upper 50 m; black line), although the w_L19 performance (red line) does improve in the ML statistically, the improvement is small with the largest increase being only 0.07 (see Table 1). As mentioned above (in Fig. 9c), our parameterization for vertical mixing tends to be less effective when the turbulent mixing process is weakened in the shallow warm-season ML. Notice that, taking into account all components of w_OFES (see appendix B), here the correlations (between w_L19QG and w_OFES) are weaker in the cold season (cf. black lines in Figs. 10b,c), contrary to those (between w_L19QG and $w_{OFES}^{tg}$ ) presented in Fig. 5b.

Table 1.

Mean pattern correlations between the diagnosed and simulated w fields averaged in 2001/02, the cold season, and the warm season.

Although the correlation analyses are focused on in this study as did by Qiu et al. (2016, 2020) and Liu et al. (2019), we did examine the evaluation in terms of amplitude in Figs. 10d–f. The 2-yr-mean statistics (Fig. 10d) show that: the root-mean-square (rms) of w_L19QG (black line) is generally consistent with that of w_OFES (gray line), but the w_L19QG amplitude is slightly smaller above the 400-m depth; after adding the diabatic mixing effect, the mean rms of w_L19 (red line) improves and becomes closer to the OFES simulation (gray line) in the upper 400 m. Statistics in the cold (Fig. 10e) and warm (Fig. 10f) seasons further indicate that, in terms of amplitude, our parameterization for vertical mixing is less effective in the warm season. Notice that, differences between w_OFES and the diagnosed w are relatively larger in the 200–400-m layer, which may be partly caused by the use of area-averaged N²(z). Employing the localized N²(x, y, z) might improve the diagnoses therein.

5. Discussion

a. The w_L19QG diagnosis

Considering that L_C is empirically chosen as 150 km in this study, we have tested the sensitivity by prescribing a smaller value of 100 km. As an example, Fig. 11 gives the result on 2 May 2002. In the upper 400 m, as displayed in Fig. 11a showing the spatial correlation between w_L19QG and $w_{OFES}^{tg}$ , w_L19QG using L_C = 100 km (red line) mildly outperforms that using L_C = 150 km (black line). As L_C changes from 150 to 100 km, the W13 framework takes effect for a wider scale range (i.e., from >150 to >100 km), meaning that the SSD–SSH phase relationship (and thus the w field) can be more reasonably captured (diagnosed). As also seen in the spectral domain: compared to the result using L_C = 150 km (Fig. 11b), spectral correlations between $w_{OFES}^{tg}$ and w_L19QG using L_C = 100 km (Fig. 11c) generally improve at the 100–150-km band in the upper 400 m. However, in the deeper layer, due to the W13 limitation (Liu et al. 2019), using a smaller L_C (100 km) is not beneficial: smaller scales within the 100–150-km band may be falsely overprojected into deeper layer as spurious signals. This could cause the slightly degraded correlations for w_L19QG using L_C = 100 km, in the 500–1000-m layer (red line in Fig. 11a). Of course, as pointed out by Liu et al. (2019), future effort is required for a better choice of L_C. Notice that the QG Q vector is the nonlinear product of V_g and ρ gradients, and thus we have cross product of smaller-scale V_g by larger-scale ρ and larger-scale V_g by smaller-scale ρ. Impacts of these terms are not analyzed in this study.

Here it is worth mentioning the two important limitations of our L19 method. At scales smaller than L_C: 1) the L19 method (actually the eSQG framework) still is subject to the restriction that SSD and SSH have to be aligned, and 2) it cannot reasonably capture the submesoscale signals. As emphasized by Liu et al. (2019), new theoretical frameworks should be pursued in future works. A prospective way may be using the semi-geostrophic approximation (Badin 2013; Ragone and Badin 2016; Lapeyre 2017).

b. The w_L19m diagnosis

We have also examined the sensitivity to the value of A_υ0. Figure 12 shows the results for the wintertime (6 December 2001) and summertime (2 May 2002) examples, which are typical of the results obtained throughout the 2-yr simulation period. With the values of A_υ0 varying in the typical range (6 × 10⁻³–6 × 10⁻² m² s⁻¹ for winter and 6 × 10⁻⁴–4.5 × 10⁻³ m² s⁻¹ for summer; according to Price and Schudlich 1987; Schudlich and Price 1998; Cronin et al. 2015), inside the ML the parameterization always proves to be functioning and beneficial (cf. black line with others), and tends to be more effective when A_υ0 becomes larger. It is worth mentioning that changes of A_υ0 affect actually the magnitude of the w_L19m field, rather than its spatial pattern that is correlated with the Laplacian of the density [see Eq. (11)]. Notice that our choice of A_υ0 in section 4 (red lines in Fig. 12) is arbitrary and preliminary, and a more realistic A_υ0 is desired for the application of this parameterization scheme.

It is worth noticing that (see Fig. 9c), near and beneath MLD, degradation rather than improvement is found for some dates after including the mixing parameterization (degradation is also discernible in the statistical results shown in Fig. 10c). In this study, we have also tried another simplified turbulent mixing parameterization scheme (denoted P13) put forth by Ponte et al. (2013):

w_{L 19 m} = (g / f_{0}^{2} ρ_{0}) A_{υ 0} \nabla^{2} ρ,

(13)

and results are shown in Fig. 13. Figure 13a shows the correlation improvement brought about by the P13 scheme. Compared to Fig. 9c, the P13-induced improvement (Fig. 13a) exhibits similar characteristics: it undergoes a seasonal cycle with larger value in the cold season, and occurs mainly within the shallower layers. This is as expected (Chavanne and Klein 2016), because both schemes [see Eqs. (11) and (13)] are consistent with the scaling analysis in Garrett and Loder (1981): the derived w field is correlated with the Laplacian of the density. Notice that the analytical solution of P13 exerts influences only in the ML. Therefore, in Fig. 13a below the ML, employing the P13 mixing parameterization does not change the correlations. Comparison between the two mixing parameterizations is further illustrated in Fig. 13b that shows the ML-averaged correlation time series. Although the P13 scheme (see red line) does alleviate the abovementioned degradations (see also Figs. 13a and 9c) for some dates, the P13-induced improvement is generally smaller (cf. red and black lines).

For the idealized mixing parameterization adopted in this study, three points are worth noting.:

The use of horizontally invariant N²(z), H_ML, and A_υ(z) leads to a rather simplified formulation of the omega equation Eq. (11) [also see Eqs. (A8) and (A9) in the appendix A], and this may be insufficient. Particularly, w signals associated with the H_ML topology (Giordani et al. 2006) cannot be reasonably recovered, for which one may then consider employing localized N²(x, y, z) (also beneficial for w_L19QG; Liu et al. 2019), H_ML(x, y), and A_υ(x, y, z).
Due to the assumption of small Ek, this scheme does not include the effects of Ekman dynamics. A physically more realistic scheme may be to add a thin Ekman layer at the top boundary, as indicated by Nagai et al. (2006). Also, the assumption of small Ro may be not suitable at strong fronts.
The parameterization has not considered the atmospheric forcing (e.g., momentum flux at the surface; see Thompson 2000; Thomas and Lee 2005), which should be explored in future works. One may consider taking the wind stress as a surface boundary condition to include the wind effect (Estrada-Allis et al. 2019).

Clearly, as pointed out by Qiu et al. (2020) and Yang et al. (2021), a better parameterization and understanding of the turbulent mixing process in the ML is called for.

c. Diagnoses in another region

Besides the energetic Kuroshio Extension subdomain shown in Fig. 1, we have also conducted the diagnoses in another subdomain of the North Pacific Current (dashed box in Fig. 14a). Results are consistent with those obtained in the Kuroshio Extension subdomain: after adding the mixing effect, the mean correlations averaged in 2001/02 for w_L19 (red line in Fig. 14b) improve inside the ML, and the parameterization for vertical mixing is more effective in the cold season (cf. Figs. 14c,d). Notably in this second subdomain that is less energetic, the L19 w diagnoses are less satisfactory (with correlations falling generally in between 0.4 and 0.5 below the 100-m depth), which is consistent with the result reported by Qiu et al. (2016, see their Figs. 10a,g).

6. Conclusions

In preparation for the future SWOT satellite mission scheduled for 2021, the present study employs the 1/30°-resolution OFES simulation to evaluate the applicability of the L19 method for diagnosing subsurface w from the surface information. Specifically, we utilize the L19-reconstructed subsurface ρ and V_g fields (from our earlier study of Liu et al. 2019) for the diagnosis, through a diabatic version of the QG omega equation that includes a simplified parameterization for turbulent vertical mixing. Compared to the eSQG-based results of Qiu et al. (2016) emphasizing that the w diagnosis is a real challenge, our encouraging results suggest the great potential of L19 for the 3D w estimation from high-resolution surface data.

The adiabatic w_L19QG constituent of the total diagnosis is computed from the L19 reconstructions of ρ and V_g. It can well reproduce the signals generated by the geostrophic kinematic deformation forcing, which is the dominant contributor to the w_OFES field below the MLD. Averaged over the 2-yr OFES simulation period, below the ML to the 1000-m depth, the spatial correlations between w_L19QG and w_OFES fall in the 0.51–0.67 range, similar to the results obtained by Qiu et al. (2016). Within the ML, the 2-yr-averaged correlations for w_L19QG fall in between 0.31 and 0.51. Although these correlations are not strong, primarily due to the vertical mixing process (considerably driving w inside the ML) not considered by the adiabatic solution w_L19QG, they are generally larger than their counterparts in Qiu et al. (2016) of the eSQG-based w diagnosis. This is because that, at scales larger than L_C (150 km), the L19 method (actually the isQG framework) can better represent the phase relationship between SSD and SSH, and thus can better capture the vertical phase shift between the SSD and interior PV, which is crucial to the w diagnosis.

The diabatic constituent w_L19m (associated with the vertical turbulent mixing process most active inside the ML) is estimated from the L19-reconstructed ρ, according to the idealized mixing parameterization proposed by Chavanne and Klein (2016): the vertical turbulent viscosity A_υ is prescribed as a parabolic profile with a maximum value of A_υ0. Of course, such a strong oversimplification of vertical mixing may be questionable [see the scheme of Noh and Kim (1999) employed by OFES or the K-profile parameterization of Large et al. (1994)]. However, as a simplified possible scheme within the QG framework, it does allow an analysis of the influence of turbulent mixing within the ML. After adding the diabatic mixing induced w_L19m to the adiabatic w_L19QG, the mean correlations over two years between w_L19 (i.e., w_L19QG + w_L19m) and w_OFES improve to 0.42–0.51 inside the ML, with the maximum increase being 0.11 (compared to the 2-yr-averaged w_L19QG correlations of 0.31–0.51; see Table 1). However, our parameterization tends to be less effective in the warm-season shallow ML where the mixing signals become weakened: the wintertime mean w_L19 correlations are 0.42–0.51 with the maximum increase of 0.15 (compared to 0.27–0.51 for w_L19QG; Table 1); while the maximum increase is only 0.07 for the summertime mean w_L19 correlations (being 0.42–0.51; recall 0.35–0.51 for w_L19QG; Table 1). Below the ML, due to the weakening of vertical mixing, w_L19 is dominated by its adiabatic constituent w_L19QG. Therefore, the w_L19 correlations (with w_OFES) are similar to those for w_L19QG. In addition to spatial correlation analyses, we have also calculated the temporal correlation between w_L19 and w_OFES for the 2-yr simulations, averaged in the upper 500 m within the 6° × 6° target region (solid box in Fig. 1). As shown in Fig. 15a, the correlation decays to below 0.3 after 1 day, indicating that the w_L19 diagnosis could be no longer reasonable after one day (notice that the daily mean fields are analyzed in this study).

There are two points worth remarking on in concluding this work.

As suggested by Morrow et al. (2019), internal gravity waves would probably dominate the SWOT SSH at scales smaller than about 100 km. However, the OFES simulation forming the basis of this study does not include tidal forcing. Therefore, the extent to which the internal gravity wave signal can impact our w diagnoses cannot be examined. Future studies similar to Qiu et al. (2020) that can quantify the effects of internal tide and wave are needed.
Short temporal decorrelation scales for w_OFES (~1.5 days; see gray lines in Fig. 15b) and w_L19 (~2.1 days; red lines) indicate that, the diagnosed w_L19 would be sensitive to the temporal smoothing of the SSH field (Qiu et al. 2016). To assess the impact of sampling and measurement errors (intrinsic to the SWOT SSH) on our L19 diagnoses, future works employing the SWOT simulator (Gaultier et al. 2016; Qiu et al. 2016, 2020) are desired.

Acknowledgments

We thank Shiqiu Peng, Rui Xin Huang, Bo Qiu, Bruno Buongiorno Nardelli, and Yu-Kun Qian for valuable discussions. We also thank the anonymous reviewers for their insightful and constructive comments. This work was supported by the National Natural Science Foundation of China (Grant 41806036), the Strategic Priority Research Program of Chinese Academy of Sciences (Project XDB42000000), and Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (GML2019ZD0303). The 1/30° OFES simulation was conducted by using the Earth Simulator under support of JAMSTEC. We gratefully acknowledge the use of high performance computing clusters at the South China Sea Institute of Oceanology, Chinese Academy of Sciences.

APPENDIX A

Derivation of Eq. (11)

Consider flow regimes in QG balance and for which the Rossby (Ro) and Ekman (Ek) numbers are small. Including vertical mixing but neglecting horizontal mixing terms, the horizontal momentum equations and the buoyancy equation can be written as follows (Nagai et al. 2006):

\frac{D_{q} u_{g}}{D t} - f_{0} υ_{a} = \frac{\partial}{\partial z} (A_{υ} \frac{\partial u_{g}}{\partial z}),

(A1)

\frac{D_{q} υ_{g}}{D t} + f_{0} u_{a} = \frac{\partial}{\partial z} (A_{υ} \frac{\partial υ_{g}}{\partial z}),

(A2)

\frac{D_{q} b}{D t} + w \frac{\partial b}{\partial z} = \frac{\partial}{\partial z} (K_{υ} \frac{\partial b}{\partial z}) .

(A3)

Here D_q/Dt = ∂/∂t + u_g(∂/∂x) + υ_g(∂/∂y) is the QG derivative operator. The terms u and υ are the east–west and north–south component of velocity, with the subscripts g and a denoting the geostrophic and ageostrophic parts, and the vertical velocity w is ageostrophic. The terms A_υ and K_υ are the vertical viscosity and diffusivity, respectively.

Taking the vertical derivative of (A1) and (A2) on one hand, with the thermal wind balance [∂b/∂y = −f₀(∂u_g/∂z); ∂b/∂x = f₀(∂υ_g/∂z)] we obtain

f_{0} \frac{D_{q}}{D t} (\frac{\partial u_{g}}{\partial z}) - \frac{\partial u_{g}}{\partial x} \frac{\partial b}{\partial y} + \frac{\partial u_{g}}{\partial y} \frac{\partial b}{\partial x} - f_{0}^{2} \frac{\partial υ_{a}}{\partial z} = f_{0} \frac{\partial^{2}}{\partial z^{2}} (A_{υ} \frac{\partial u_{g}}{\partial z}),

(A4)

f_{0} \frac{D_{q}}{D t} (\frac{\partial υ_{g}}{\partial z}) - \frac{\partial υ_{g}}{\partial x} \frac{\partial b}{\partial y} + \frac{\partial υ_{g}}{\partial y} \frac{\partial b}{\partial x} + f_{0}^{2} \frac{\partial u_{a}}{\partial z} = f_{0} \frac{\partial^{2}}{\partial z^{2}} (A_{υ} \frac{\partial υ_{g}}{\partial z}) .

(A5)

On the other hand, taking separately the x and y derivatives of (A3) (note that ∂b/∂z = N²) and also using the thermal wind balance, we obtain

f_{0} \frac{D_{q}}{D t} (\frac{\partial υ_{g}}{\partial z}) + \frac{\partial u_{g}}{\partial x} \frac{\partial b}{\partial x} + \frac{\partial υ_{g}}{\partial x} \frac{\partial b}{\partial y} + \frac{\partial}{\partial x} (N^{2} w) = \frac{\partial}{\partial x} \frac{\partial}{\partial z} (K_{υ} \frac{\partial b}{\partial z}),

(A6)

- f_{0} \frac{D_{q}}{D t} (\frac{\partial u_{g}}{\partial z}) + \frac{\partial u_{g}}{\partial y} \frac{\partial b}{\partial x} + \frac{\partial υ_{g}}{\partial y} \frac{\partial b}{\partial y} + \frac{\partial}{\partial y} (N^{2} w) = \frac{\partial}{\partial y} \frac{\partial}{\partial z} (K_{υ} \frac{\partial b}{\partial z}) .

(A7)

By combining (A4)–(A7), the time derivatives can be eliminated and using the continuity equation (∂u_a/∂x + ∂υ_a/∂y + ∂w/∂z = 0), we obtain the diabatic QG omega equation considering turbulent vertical mixing effects, as

(f_{0}^{2} \frac{\partial^{2}}{\partial z^{2}} + N^{2} \nabla^{2}) w = 2 \nabla \cdot Q+ \nabla^{2} \frac{\partial}{\partial z} (K_{υ} \frac{\partial b}{\partial z}) - f_{0} \frac{\partial}{\partial z} [\frac{\partial}{\partial x} \frac{\partial}{\partial z} (A_{υ} \frac{\partial υ_{g}}{\partial z}) - \frac{\partial}{\partial y} \frac{\partial}{\partial z} (A_{υ} \frac{\partial u_{g}}{\partial z})] .

(A8)

Notice that, in this study, we use A_υ(z), K_υ(z), and N²(z) that are all independent of x and y. Therefore, the second term on the right-hand side vanishes (note that N² = ∂b/∂z). After a little algebra (A8) can be further expressed as

(f_{0}^{2} \frac{\partial^{2}}{\partial z^{2}} + N^{2} \nabla^{2}) w = 2 \nabla \cdot Q + \frac{d^{2} A_{υ}}{d z^{2}} \nabla^{2} b .

(A9)

APPENDIX B

Decomposition of w_OFES

To identify different physical sources of w_OFES, we follow the approach of Giordani et al. (2006) to decompose w_OFES using the omega equation within the primitive equation (PE) framework (see also Qiu et al. 2020). Under the Boussinesq, hydrostatic, and f-plane approximations, the PE omega equation in the generalized Q-vector formulation is expressed as

f_{0}^{2} \frac{\partial^{2} w}{\partial z^{2}} + \nabla (N^{2} \cdot \nabla w) = \nabla \cdot Q .

(B1)

It is important to notice that Eq. (B1) uses a localized N²(x, y, z) different from the area-averaged N²(z) employed in the L19 diagnosis (see section 3). Components of the Q vector are expressed as follows:

Q = Q_{th} + Q_{dm} + Q_{tg} + Q_{tag} + Q_{dag} + Q_{dr} .

(B2)

On the right-hand side of Eq. (B2), the first two terms Q_th and Q_dm denote the forcing by turbulent (buoyancy and momentum) fluxes. The remaining terms represent the geostrophic kinematic deformation (Q_tg), ageostrophic kinematic deformation (Q_tag), thermal wind imbalance deformation (Q_dag), and material derivative of the thermal wind imbalance (Q_dr), respectively, with

Q_{tg} = \frac{2 g}{ρ_{0}} {(\nabla V_{g})}^{T} \cdot \nabla ρ,

(B3)

Q_{tag} = \frac{2 g}{ρ_{0}} {(\nabla V_{a})}^{T} \cdot \nabla ρ,

(B4)

Q_{dag} = - f_{0} {[\nabla (k \times V)]}^{T} \cdot \frac{\partial V_{a}}{\partial z},

(B5)

Q_{dr} = - f_{0} k \times \frac{D}{D t} (\frac{\partial V_{a}}{\partial z}) .

(B6)

Given the OFES output, we first calculate the geostrophic velocity V_g from the density field via the thermal wind balance (using 1500 m as the reference depth of no motion), and thus the ageostrophic part V_a. Then, Q_tg, Q_tag, Q_dag, and Q_dr can be evaluated through Eqs. (B3)–(B6) in the target 10° × 10° region (dashed box in Fig. 1). After calculating the divergence of each Q vector, we subtract their large-scale signals [defined as a least squares fit of a field to the quadratic surface S(x, y), Eq. (2)]; then the 2D trapezoid window is applied in a 1° band along the box edges for the purpose of Fourier transform (with lateral boundary conditions being doubly periodic). Using w = 0 at the surface and bottom boundaries, contributions to w_OFES from Q_tg, Q_tag, Q_dag, and Q_dr (denoted $w_{OFES}^{tg}$ , $w_{OFES}^{tag}$ , $w_{OFES}^{dag}$ and $w_{OFES}^{dr}$ ) are separately derived via Eq. (B1) (due to its linearity) first in the spectral space, then in the physical space through inverse Fourier transform.

Due to the lack of the OFES information about the turbulent mixing, we cannot directly evaluate $w_{OFES}^{th}$ and $w_{OFES}^{dm}$ forced by Q_th and Q_dm. Following Qiu et al. (2020), we estimate $w_{OFES}^{th} + w_{OFES}^{dm}$ (denoted $w_{OFES}^{mix}$ ) as the residual between the simulated w_OFES and the sum of $w_{OFES}^{tg} + w_{OFES}^{tag} + w_{OFES}^{dag} + w_{OFES}^{dr}$ . As an example, Fig. B1 shows the w_OFES field and its decomposition at 50-m depth on 6 December 2001.

REFERENCES

Badin, G., 2013: Surface semi-geostrophic dynamics in the ocean. Geophys. Astrophys. Fluid Dyn., 107, 526–540, https://doi.org/10.1080/03091929.2012.740479.
- Crossref
- Search Google Scholar
- Export Citation
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.
- Crossref
- Search Google Scholar
- Export Citation
Blumen, W., 1978: Uniform potential vorticity flow. Part I: Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci., 35, 774–783, https://doi.org/10.1175/1520-0469(1978)035<0774:UPVFPI>2.0.CO;2.
- Crossref
- Search Google Scholar
- Export Citation
Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325–334, https://doi.org/10.1002/qj.49709239302.
- Crossref
- Search Google Scholar
- Export Citation
Buongiorno Nardelli, B., 2020: A multi-year timeseries of observation-based 3D horizontal and vertical quasi-geostrophic global ocean currents. Earth Syst. Sci. Data, 12, 1711–1723, https://doi.org/10.5194/essd-12-1711-2020.
- Crossref
- Search Google Scholar
- Export Citation
Buongiorno Nardelli, B., S. Sparnocchia, and R. Santoleri, 2001: Small mesoscale features at a meandering upper-ocean front in the Western Ionian Sea (Mediterranean Sea): Vertical motion and potential vorticity analysis. J. Phys. Oceanogr., 31, 2227–2250, https://doi.org/10.1175/1520-0485(2001)031<2227:SMFAAM>2.0.CO;2.
- Crossref
- Search Google Scholar
- Export Citation
Buongiorno Nardelli, B., S. Guinehut, A. Pascual, Y. Drillet, S. Ruiz, and S. Mulet, 2012: Towards high resolution mapping of 3-D mesoscale dynamics from observations. Ocean Sci., 8, 885–901, https://doi.org/10.5194/os-8-885-2012.
- Crossref
- Search Google Scholar
- Export Citation
Buongiorno Nardelli, B., S. Mulet, and D. Iudicone, 2018: Three-dimensional ageostrophic motion and water mass subduction in the Southern Ocean. J. Geophys. Res. Oceans, 123, 1533–1562, https://doi.org/10.1002/2017JC013316.
- Crossref
- Search Google Scholar
- Export Citation
Callies, J., R. Ferrari, J. Klymak, and J. Gula, 2015: Seasonality in submesoscale turbulence. Nat. Commun., 6, 6862, https://doi.org/10.1038/ncomms7862.
- Crossref
- Search Google Scholar
- Export Citation
Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, 1087–1095, https://doi.org/10.1175/1520-0469(1971)028<1087:GT>2.0.CO;2.
- Crossref
- Search Google Scholar
- Export Citation
Chavanne, C. P., and P. Klein, 2016: Quasigeostrophic diagnosis of mixed layer dynamics embedded in a mesoscale turbulent field. J. Phys. Oceanogr., 46, 275–287, https://doi.org/10.1175/JPO-D-14-0178.1.
- Crossref
- Search Google Scholar
- Export Citation
Chen, Z., X. Wang, and L. Liu, 2020: Reconstruction of three-dimensional ocean structure from sea surface data: An application of isQG method in the Southwest Indian Ocean. J. Geophys. Res. Oceans, 125, e2020JC016351, https://doi.org/10.1029/2020JC016351.
- Crossref
- Search Google Scholar
- Export Citation
Cronin, M. F., N. A. Pelland, S. R. Emerson, and W. R. Crawford, 2015: Estimating diffusivity from the mixed layer heat and salt balances in the North Pacific. J. Geophys. Res. Oceans, 120, 7346–7362, https://doi.org/10.1002/2015JC011010.
- Crossref
- Search Google Scholar
- Export Citation
de La Lama, M. S., J. H. LaCasce, and H. Fuhr, 2016: The vertical structure of ocean eddies. Dyn. Stat. Climate Syst., 1, dzw001, https://doi.org/10.1093/climsys/dzw001.
- Crossref
- Search Google Scholar
- Export Citation
Durand, M., L.-L. Fu, D. P. Lettenmaier, D. E. Alsdorf, E. Rodríguez, and D. Esteban-Fernandez, 2010: The surface water and ocean topography mission: Observing terrestrial surface water and oceanic submesoscale eddies. Proc. IEEE, 98, 766–779, https://doi.org/10.1109/JPROC.2010.2043031.
- Crossref
- Search Google Scholar
- Export Citation
Estrada-Allis, S., B. Barceló-Llull, E. Pallàs-Sanz, A. Rodríguez-Santana, J. Souza, E. Mason, J. McWilliams, and P. Sangrà, 2019: Vertical velocity dynamics and mixing in an anticyclone near the Canary Islands. J. Phys. Oceanogr., 49, 431–451, https://doi.org/10.1175/JPO-D-17-0156.1.
- Crossref
- Search Google Scholar
- Export Citation
Ferrari, R., and C. Wunsch, 2010: The distribution of eddy kinetic and potential energies in the global ocean. Tellus, 62, 92–108, https://doi.org/10.3402/tellusa.v62i2.15680.
- Crossref
- Search Google Scholar
- Export Citation
Fofonoff, N. P., and R. C. Millard Jr., 1983: UNESCO technical papers in marine science: Algorithms for computation of fundamental properties of seawater. Unesco Technical Papers in Marine Science Rep. 44, 58 pp.
Fu, L.-L., and C. Ubelmann, 2014: On the transition from profile altimeter to swath altimeter for observing global ocean surface topography. J. Atmos. Oceanic Technol., 31, 560–568, https://doi.org/10.1175/JTECH-D-13-00109.1.
- Crossref
- Search Google Scholar
- Export Citation
Garrett, C., and J. Loder, 1981: Dynamical aspects of shallow sea fronts. Philos. Trans. Roy. Soc. London, 302A, 563–581, https://doi.org/10.1098/rsta.1981.0183.
- Search Google Scholar
- Export Citation
Gaultier, L., C. Ubelmann, and L.-L. Fu, 2016: The challenge of using future SWOT data for oceanic field reconstruction. J. Atmos. Oceanic Technol., 33, 119–126, https://doi.org/10.1175/JTECH-D-15-0160.1.
- Crossref
- Search Google Scholar
- Export Citation
Giordani, H., L. Prieur, and G. Caniaux, 2006: Advanced insights into sources of vertical velocity in the ocean. Ocean Dyn., 56, 513–524, https://doi.org/10.1007/s10236-005-0050-1.
- Crossref
- Search Google Scholar
- Export Citation
González-Haro, C., and J. Isern-Fontanet, 2014: Global ocean current reconstruction from altimetric and microwave SST measurements. J. Geophys. Res. Oceans, 119, 3378–3391, https://doi.org/10.1002/2013JC009728.
- Crossref
- Search Google Scholar
- Export Citation
Held, I. M., R. T. Pierrehumbert, S. T. Garner, and K. L. Swanson, 1995: Surface quasigeostrophic dynamics. J. Fluid Mech., 282, 1–20, https://doi.org/10.1017/S0022112095000012.
- Crossref
- Search Google Scholar
- Export Citation
Hoskins, B. J., 1975: The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci., 32, 233–242, https://doi.org/10.1175/1520-0469(1975)032<0233:TGMAAT>2.0.CO;2.
- Crossref
- Search Google Scholar
- Export Citation
Hoskins, B. J., I. Draghici, and H. C. Davies, 1978: A new look at the ω-equation. Quart. J. Roy. Meteor. Soc., 104, 31–38, https://doi.org/10.1002/qj.49710443903.
- Crossref
- Search Google Scholar
- Export Citation
Isern-Fontanet, J., B. Chapron, G. Lapeyre, and P. Klein, 2006: Potential use of microwave sea surface temperatures for the estimation of ocean currents. Geophys. Res. Lett., 33, L24608, https://doi.org/10.1029/2006GL027801.
- Crossref
- Search Google Scholar
- Export Citation
Isern-Fontanet, J., G. Lapeyre, P. Klein, B. Chapron, and M. W. Hecht, 2008: Three dimensional reconstruction of oceanic mesoscale currents from surface information. J. Geophys. Res., 113, C09005, https://doi.org/10.1029/2007JC004692.
- Crossref
- Search Google Scholar
- Export Citation
Isern-Fontanet, J., M. Shinde, and C. González-Haro, 2014: On the transfer function between surface fields and the geostrophic stream function in the Mediterranean Sea. J. Phys. Oceanogr., 44, 1406–1423, https://doi.org/10.1175/JPO-D-13-0186.1.
- Crossref
- Search Google Scholar
- Export Citation
Isern-Fontanet, J., J. Ballabrera-Poy, A. Turiel, and E. García-Ladona, 2017: Remote sensing of ocean surface currents: A review of what is being observed and what is being assimilated. Nonlinear Processes Geophys., 24, 613–643, https://doi.org/10.5194/npg-24-613-2017.
- Crossref
- Search Google Scholar
- Export Citation
Jackett, D. R., and T. J. McDougall, 1995: Minimal adjustment of hydrostatic profiles to achieve static stability. J. Atmos. Oceanic Technol., 12, 381–389, https://doi.org/10.1175/1520-0426(1995)012<0381:MAOHPT>2.0.CO;2.
- Crossref
- Search Google Scholar
- Export Citation
Klein, P., B. L. Hua, G. Lapeyre X. Capet, S. L. Gentil, and H. Sasaki, 2008: Upper ocean turbulence from high-resolution 3D simulations. J. Phys. Oceanogr., 38, 1748–1763, https://doi.org/10.1175/2007JPO3773.1.
- Crossref
- Search Google Scholar
- Export Citation
Klein, P., J. Isern-Fontanet, G. Lapeyre, G. Roullet, E. Danioux, B. Chapron, S. Le Gentil, and H. Sasaki, 2009: Diagnosis of vertical velocities in the upper ocean from high resolution sea surface height. Geophys. Res. Lett., 36, L12603, https://doi.org/10.1029/2009GL038359.
- Crossref
- Search Google Scholar
- Export Citation
Komori, N., K. Takahashi, K. Komine, T. Motoi, X. Zhang, and G. Sagawa, 2005: Description of sea-ice component of coupled Ocean Sea-Ice Model for the Earth Simulator (OIFES). J. Earth Simul., 4, 31–45.
- Search Google Scholar
- Export Citation
LaCasce, J. H., 2012: Surface quasigeostrophic solutions and baroclinic modes with exponential stratification. J. Phys. Oceanogr., 42, 569–580, https://doi.org/10.1175/JPO-D-11-0111.1.
- Crossref
- Search Google Scholar
- Export Citation
LaCasce, J. H., 2017: The prevalence of oceanic surface modes. Geophys. Res. Lett., 44, 11 097–11 105, https://doi.org/10.1002/2017GL075430.
- Crossref
- Search Google Scholar
- Export Citation
LaCasce, J. H., and A. Mahadevan, 2006: Estimating subsurface horizontal and vertical velocities from sea surface temperature. J. Mar. Res., 64, 695–721, https://doi.org/10.1357/002224006779367267.
- Crossref
- Search Google Scholar
- Export Citation
LaCasce, J. H., and J. Wang, 2015: Estimating subsurface velocities from surface fields with idealized stratification. J. Phys. Oceanogr., 45, 2424–2435, https://doi.org/10.1175/JPO-D-14-0206.1.
- Crossref
- Search Google Scholar
- Export Citation
Lapeyre, G., 2009: What vertical mode does the altimeter reflect? On the decomposition in baroclinic modes and on a surface-trapped mode. J. Phys. Oceanogr., 39, 2857–2874, https://doi.org/10.1175/2009JPO3968.1.
- Crossref
- Search Google Scholar
- Export Citation
Lapeyre, G., 2017: Surface quasi-geostrophy. Fluids, 2, 7, https://doi.org/10.3390/fluids2010007.
- Crossref
- Search Google Scholar
- Export Citation
Lapeyre, G., and P. Klein, 2006: Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory. J. Phys. Oceanogr., 36, 165–176, https://doi.org/10.1175/JPO2840.1.
- Crossref
- Search Google Scholar
- Export Citation
Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32, 363–403, https://doi.org/10.1029/94RG01872.
- Crossref
- Search Google Scholar
- Export Citation
Li, Z., J. Wang, and L.-L. Fu, 2019: An observing system simulation experiment for ocean state estimation to assess the performance of the SWOT mission: Part I-A twin experiment. J. Geophys. Res. Oceans, 124, 4838–4855, https://doi.org/10.1029/2018JC014869.
- Crossref
- Search Google Scholar
- Export Citation
Liu, L., S. Peng, J. Wang, and R. X. Huang, 2014: Retrieving density and velocity fields of the ocean’s interior from surface data. J. Geophys. Res. Oceans, 119, 8512–8529, https://doi.org/10.1002/2014JC010221.
- Crossref
- Search Google Scholar
- Export Citation
Liu, L., S. Peng, and R. X. Huang, 2017: Reconstruction of ocean’s interior from observed sea surface information. J. Geophys. Res. Oceans, 122, 1042–1056, https://doi.org/10.1002/2016JC011927.
- Crossref
- Search Google Scholar
- Export Citation
Liu, L., H. Xue, and H. Sasaki, 2019: Reconstructing the ocean interior from high-resolution sea surface information. J. Phys. Oceanogr., 49, 3245–3262, https://doi.org/10.1175/JPO-D-19-0118.1.
- Crossref
- Search Google Scholar
- Export Citation
Masumoto, Y., and Coauthors, 2004: A fifty-year eddy-resolving simulation of the world ocean: Preliminary outcomes of OFES (OGCM for the Earth Simulator). J. Earth Simul., 1, 35–56.
- Search Google Scholar
- Export Citation
Mensa, J. A., Z. Garraffo, A. Griffa, T. M. Ozgokmen, A. Haza, and M. Veneziani, 2013: Seasonality of the submesoscale dynamics in the Gulf Stream region. Ocean Dyn., 63, 923–941, https://doi.org/10.1007/s10236-013-0633-1.
- Crossref
- Search Google Scholar
- Export Citation
Morrow, R., and Coauthors, 2019: Global observations of fine-scale ocean surface topography with the Surface Water and Ocean Topography (SWOT) mission. Front. Mar. Sci., 6, 232, https://doi.org/10.3389/fmars.2019.00232.
- Crossref
- Search Google Scholar
- Export Citation
Nagai, T., A. Tandon, and D. L. Rudnick, 2006: Two-dimensional ageostrophic secondary circulation at ocean fronts due to vertical mixing and large-scale deformation. J. Geophys. Res., 111, C09038, https://doi.org/10.1029/2005JC002964.
- Crossref
- Search Google Scholar
- Export Citation
Nagai, T., A. Tandon, N. Gruber, and J. C. McWilliams, 2008: Biological and physical impacts of ageostrophic frontal circulations driven by confluent flow and vertical mixing. Dyn. Atmos. Oceans, 45, 229–251, https://doi.org/10.1016/j.dynatmoce.2007.12.001.
- Crossref
- Search Google Scholar
- Export Citation
Noh, Y., and H. J. Kim, 1999: Simulations of temperature and turbulence structure of the oceanic boundary layer with the improved near-surface process. J. Geophys. Res., 104, 15 621–15 634, https://doi.org/10.1029/1999JC900068.
- Crossref
- Search Google Scholar
- Export Citation

Sections

References

Figures

Cited By

Metrics

Related Content

GENERAL CONDITIONS

No. III

BAROMETRIC PRESSURE

VERIFICATIONS

MISCELLANEOUS PHENOMENA

Diagnosing Subsurface Vertical Velocities from High-Resolution Sea Surface Fields

Abstract

Abstract

1. Introduction

2. Data

a. The 1/30°-resolution OFES simulation

b. Data preprocessing

3. Methods

a. The L19 reconstruction method

b. The QG omega equation with vertical mixing terms

4. Results

a. Performance of wL19QG

b. Performance of wL19m

c. Evaluation of wL19 against wOFES

5. Discussion

a. The wL19QG diagnosis

b. The wL19m diagnosis

c. Diagnoses in another region

6. Conclusions

Acknowledgments

APPENDIX A

Derivation of Eq. (11)

APPENDIX B

Decomposition of wOFES

REFERENCES

a. Performance of w_L19QG

b. Performance of w_L19m

c. Evaluation of w_L19 against w_OFES

a. The w_L19QG diagnosis

b. The w_L19m diagnosis

Decomposition of w_OFES