## 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 *N*^{2}(*z*).

*N*

^{2}(

*z*) by

*g*is the gravity constant, and

*ρ*

_{υ}and

*N*

^{2}(

*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

*N*

^{2}(

*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*

^{2}(

*z*) in the top 1000 m (Isern-Fontanet et al. 2008, 2017; Liu et al. 2019).

*S*(

*x*,

*y*) (Wang et al. 2013),

*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 *N*^{2}(*z*) derived directly from the OFES data, the thick solid line represents the adjusted *N*^{2}(*z*), and the thin solid line denotes the constant

(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 *N*^{2}(*z*) profile (dashed line) derived directly from OFES, and the adjusted *N*^{2}(*z*) (thick solid line) used by L19. A constant stratification *u* and *υ* (solid and dashed lines). Plots correspond to the date of 6 Dec 2001.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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 *N*^{2}(*z*) profile (dashed line) derived directly from OFES, and the adjusted *N*^{2}(*z*) (thick solid line) used by L19. A constant stratification *u* and *υ* (solid and dashed lines). Plots correspond to the date of 6 Dec 2001.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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 *N*^{2}(*z*) profile (dashed line) derived directly from OFES, and the adjusted *N*^{2}(*z*) (thick solid line) used by L19. A constant stratification *u* and *υ* (solid and dashed lines). Plots correspond to the date of 6 Dec 2001.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

## 3. Methods

### a. The L19 reconstruction method

*q*as

*N*and

*f*

_{0}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,

*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

*N*

^{2}(

*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.

*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):

*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

*η*is the SSHA that determines the surface pressure anomaly

^{int}. Note that, in Eq. (7), the residual SSHA is implicitly defined, i.e.,

*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:

*N*

_{0}is a constant stratification, and

*κ*= (

*k*

^{2}+

*l*

^{2})

^{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

*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):

*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.

(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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

*N*

^{2}(

*z*) as used by the L19 reconstruction. The adiabatic forcing

**Q**= (

*g*/

*ρ*

_{0})(∇

**V**

_{g})

^{T}⋅ ∇

*ρ*(with

*ρ*

_{0}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}.

*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

*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} (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*_{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*_{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*_{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*_{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*_{L19QG} and the adiabatic *w*_{eSQG} (diagnosed from the eSQG method using SSHA; Qiu et al. 2016). Although *w*_{eSQG} is similar to *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).

Vertical velocity *w* fields at (top) the 50-m and (middle) 200-m depth: (a),(d) *w*_{L19QG} constituent of the L19 diagnosis, and (c),(f) adiabatic *w*_{eSQG} from the eSQG diagnosis. (g) Spectral correlation between *w*_{L19QG} as a function of horizontal wavenumber and depth. (h) Pattern correlation between *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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

Vertical velocity *w* fields at (top) the 50-m and (middle) 200-m depth: (a),(d) *w*_{L19QG} constituent of the L19 diagnosis, and (c),(f) adiabatic *w*_{eSQG} from the eSQG diagnosis. (g) Spectral correlation between *w*_{L19QG} as a function of horizontal wavenumber and depth. (h) Pattern correlation between *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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

Vertical velocity *w* fields at (top) the 50-m and (middle) 200-m depth: (a),(d) *w*_{L19QG} constituent of the L19 diagnosis, and (c),(f) adiabatic *w*_{eSQG} from the eSQG diagnosis. (g) Spectral correlation between *w*_{L19QG} as a function of horizontal wavenumber and depth. (h) Pattern correlation between *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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

Figure 5a shows the mean correlation between *w*_{L19QG} and *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.

The green line denotes the time series of (a) mean correlation between *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*_{L19QG} as a function of depth averaged in the cold (blue line) and warm (red line) season.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

The green line denotes the time series of (a) mean correlation between *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*_{L19QG} as a function of depth averaged in the cold (blue line) and warm (red line) season.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

The green line denotes the time series of (a) mean correlation between *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*_{L19QG} as a function of depth averaged in the cold (blue line) and warm (red line) season.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

### 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} m^{2} s^{−1} (see Fig. 3b), which is similar to that used by previous studies of the eSQG-based *w* diagnosis [4 × 10^{−2} m^{2} s^{−1} by Ponte et al. (2013); 1.5 × 10^{−2} m^{2} s^{−1} by Chavanne and Klein (2016)]. At 50 m inside the ML (upper 110 m), plane views of the *w*_{L19m} (Fig. 6b) from *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*_{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*_{L19m}) reaching 0.5 around the 20-m depth, below which the *w*_{L19m} performance deteriorates gradually with an increasing depth.

Vertical velocity *w* fields at the 50-m depth: (a) *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*_{L19m} as a function of depth. Gray shade denotes the area-averaged MLD (110 m). Plots correspond to the date of 6 Dec 2001.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

Vertical velocity *w* fields at the 50-m depth: (a) *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*_{L19m} as a function of depth. Gray shade denotes the area-averaged MLD (110 m). Plots correspond to the date of 6 Dec 2001.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

Vertical velocity *w* fields at the 50-m depth: (a) *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*_{L19m} as a function of depth. Gray shade denotes the area-averaged MLD (110 m). Plots correspond to the date of 6 Dec 2001.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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} m^{2} 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*_{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.

Vertical velocity *w* fields at the 20-m depth: (a) *w*_{L19m} constituent of the L19 diagnosis. Plus symbols mark the position of the filamentary structures. (c) Pattern correlation between *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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

Vertical velocity *w* fields at the 20-m depth: (a) *w*_{L19m} constituent of the L19 diagnosis. Plus symbols mark the position of the filamentary structures. (c) Pattern correlation between *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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

Vertical velocity *w* fields at the 20-m depth: (a) *w*_{L19m} constituent of the L19 diagnosis. Plus symbols mark the position of the filamentary structures. (c) Pattern correlation between *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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

### 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* 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.

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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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.

(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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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 *N*^{2}(*z*). Employing the localized *N*^{2}(*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*_{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*_{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.

(a) Spatial correlation between *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*_{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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(a) Spatial correlation between *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*_{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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(a) Spatial correlation between *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*_{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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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} m^{2} s^{−1} for winter and 6 × 10^{−4}–4.5 × 10^{−3} m^{2} 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.

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} m^{2} s^{−1} and 6 × 10^{−4}–4.5 × 10^{−3} m^{2} s^{−1}. Gray shading denotes the area-averaged MLD of 110 m in (a) and 50 m in (b).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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} m^{2} s^{−1} and 6 × 10^{−4}–4.5 × 10^{−3} m^{2} s^{−1}. Gray shading denotes the area-averaged MLD of 110 m in (a) and 50 m in (b).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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} m^{2} s^{−1} and 6 × 10^{−4}–4.5 × 10^{−3} m^{2} s^{−1}. Gray shading denotes the area-averaged MLD of 110 m in (a) and 50 m in (b).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

*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).

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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

The use of horizontally invariant

*N*^{2}(*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*N*^{2}(*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).

### 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).

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

## 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).

(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 [*c*(*t*) being the autocorrelation and *T*_{0} its first zero-crossing; Stammer 1997]; gray and red lines represent *w*_{OFES} and *w*_{L19}, respectively.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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 [*c*(*t*) being the autocorrelation and *T*_{0} its first zero-crossing; Stammer 1997]; gray and red lines represent *w*_{OFES} and *w*_{L19}, respectively.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

(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 [*c*(*t*) being the autocorrelation and *T*_{0} its first zero-crossing; Stammer 1997]; gray and red lines represent *w*_{OFES} and *w*_{L19}, respectively.

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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)

*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.

*b*/∂

*y*= −

*f*

_{0}(∂

*u*

_{g}/∂

*z*); ∂

*b*/∂

*x*=

*f*

_{0}(∂

*υ*

_{g}/∂

*z*)] we obtain

*x*and

*y*derivatives of (A3) (note that ∂

*b*/∂

*z*=

*N*

^{2}) and also using the thermal wind balance, we obtain

*u*

_{a}/∂

*x*+ ∂

*υ*

_{a}/∂

*y*+ ∂

*w*/∂

*z*= 0), we obtain the diabatic QG omega equation considering turbulent vertical mixing effects, as

*A*

_{υ}(

*z*),

*K*

_{υ}(

*z*), and

*N*

^{2}(

*z*) that are all independent of

*x*and

*y*. Therefore, the second term on the right-hand side vanishes (note that

*N*

^{2}= ∂

*b*/∂

*z*). After a little algebra (A8) can be further expressed as

## APPENDIX B

### Decomposition of *w*_{OFES}

*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

*N*

^{2}(

*x*,

*y*,

*z*) different from the area-averaged

*N*

^{2}(

*z*) employed in the L19 diagnosis (see section 3). Components of the

**Q**vector are expressed as follows:

**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

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

Due to the lack of the OFES information about the turbulent mixing, we cannot directly evaluate **Q**_{th} and **Q**_{dm}. Following Qiu et al. (2020), we estimate *w*_{OFES} and the sum of *w*_{OFES} field and its decomposition at 50-m depth on 6 December 2001.

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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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).

Citation: Journal of Physical Oceanography 51, 5; 10.1175/JPO-D-20-0152.1

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