Abstract

Using the generalized omega equation and cruise observations in July 2012, this study analyzes the 3D vertical circulation in the upwelling region and frontal zone east of Hainan Island, China. The results show that there is a strong frontal zone in subsurface layer along the 100-m isobath, which is characterized by density gradient of O(10−4) kg m−4 and vertical eddy diffusivity of O(10−5–10−4) m2 s−1. The kinematic deformation term SDEF, ageostrophic advection term SADV, and vertical mixing forcing term SMIX are calculated from the observations. Their distribution patterns are featured by banded structure, that is, alternating positive–negative alongshore bands distributed in the cross-shelf direction. Correspondingly, alternating upwelling and downwelling bands appear from the coast to the deep waters. The maximum downward velocity reaches −5 × 10−5 m s−1 within the frontal zone, accompanied by the maximum upward velocity of 7 × 10−5 m s−1 on two sides. The dynamic diagnosis indicates that SADV contributes most to the coastal upwelling, while term SDEF, which is dominated by the ageostrophic component SDEFa, plays a dominant role in the frontal zone. The vertical mixing forcing term SMIX, which includes the momentum and buoyancy flux terms SMOM and SBUO, is comparable to SDEF and SADV in the upper ocean, but negligible below the thermocline. The effect of the vertical mixing on the vertical velocity is mainly concentrated at depths with relatively large eddy diffusivity and eddy diffusivity gradient in the frontal zone.

1. Introduction

The vertical circulation is one of the essential components of the ocean circulation system. In particular, in the coastal upwelling region and frontal zones, where high bioproductivity and active air–sea interactions occur, the vertical circulation serves as a carrier of vertical transport of mass and energy (Allen et al. 2005; Thomas et al. 2010; Ferrari 2011). Thus, determining the vertical circulation and its dominant mechanisms is an important work for understanding the physical and biogeochemical processes in the study ocean area (Pollard and Regier 1992; Barth et al. 2007; Nagai et al. 2008).

However, it is hard to measure the accurate vertical velocity directly using the state-of-the-art cruise observation technologies. As a consequence, indirect methods have been developed to calculate the vertical velocity from the cruise data. The quasigeostrophic (QG) omega equation derived by Hoskins et al. (1978) has been broadly adopted for diagnosing the vertical circulation from the three-dimensional (3D) density fields in the frontal zones (Allen and Smeed 1996; Rudnick 1996; Shearman et al. 1999; Naveira Garabato et al. 2001; Hu et al. 2011). For motions with larger Rossby number, the generalized omega equation, a modified QG equation, has been developed by taking into account the ageostrophic advection and other non-QG terms (Viúdez et al. 1996; Giordani et al. 2006). Pallàs-Sanz and Viúdez (2005) used this equation for diagnosing the mesoscale vertical motion at the Atlantic jet and the western Alboran Gyre using conductivity–temperature–depth (CTD) and acoustic Doppler current profiler (ADCP) observations. They found that the non-QG terms are important for calculation of the vertical circulation in the case of the large Rossby number. Pallàs-Sanz et al. (2010b) further modified the generalized omega equation by including the effects of the vertical mixing and found that the vertical velocity is quite sensitive to the vertical mixing parameterization.

This study is interested in the vertical circulation in a coastal upwelling region accompanied with strong ocean fronts on the continental shelf east of Hainan Island of China, as shown in Fig. 1a (Xie et al. 2012; Jing et al. 2015). Previous investigators have estimated the upwelling velocity based on conservation of the salinity flux and fluctuation of the isopycnals from hydrographic observations (Han et al. 1990; Wu 1990; Deng et al. 1995) and by numerical models driven by wind (Guo and Shi 1998; Su and Pohlmann 2009). Their estimated vertical velocities vary within a range of two orders of magnitude from 5.6 × 10−6 to 2 × 10−4 m s−1, implying that there is a big uncertainty in the estimated values of this key parameter for coastal ocean dynamics. Meanwhile, there is little knowledge about the 3D structure of the vertical circulation in the coastal upwelling and associated frontal zones. The effects of the vertical mixing on the vertical circulation also need to be clarified with the directly measured microstructure data. These challenges motivate us to apply the generalized omega equation to diagnose the vertical circulation in the study area, using in situ data of CTD, ADCP, and the microstructure mixing cruise observed in July 2012. This paper is organized as follows: The next section describes the cruise observation program, the data processing, and the generalized omega equation. Section 3 examines the observed 3D density, horizontal velocity, and mixing fields. Section 4 diagnoses the dynamic terms in the generalized omega equation. Section 5 analyzes the 3D structure of the vertical circulation and contributions of each dynamic term. Sections 6 and 7 give discussion and conclusions.

Fig. 1.

(a) Map of the study area in the northern South China Sea. Background shows the climatological monthly SST (color, °C) and wind (white arrows) in July. (b) Map of cruise observation sections and stations. Red dots represent the stations with CTD, ADCP, and TurboMAP measurements. Blue dots are stations without TurboMAP measurements. Black dots are interpolation grids. Gray lines are isobaths (m). Qinglan and Boao are two landmarks at the coast.

Fig. 1.

(a) Map of the study area in the northern South China Sea. Background shows the climatological monthly SST (color, °C) and wind (white arrows) in July. (b) Map of cruise observation sections and stations. Red dots represent the stations with CTD, ADCP, and TurboMAP measurements. Blue dots are stations without TurboMAP measurements. Black dots are interpolation grids. Gray lines are isobaths (m). Qinglan and Boao are two landmarks at the coast.

2. Data and methodology

a. Cruise observation program

A joint physical and biogeochemical measurement program was carried out by R/V Tianlong of Guangdong Ocean University on the northwestern continental shelf of the South China Sea (SCS) in July 2012. As shown in Fig. 1b, cruise observations were conducted along four cross-shelf sections G, H, F, and K in the continental shelf water east of Hainan Island from 13 to 18 July 2012. The cross-shelf sections are spaced by about 25 km, and the stations are spaced by about 5 km. The observed depths are down to 150 m. The dataset includes CTD profiles with SBE 19 plus manufactured by Seabird Inc., horizontal velocity measurements with a 300k Hz ADCP manufactured by RD Instruments Inc., and microstructure mixing measured by TurboMAP-L of JFE Advantech. With the pre/postcruise calibration, the accuracy of the CTD sensors is about 0.0005 S m−1 for the conductivity, 0.005°C for the temperature, and 1 dbar for the pressure. The bin size of ADCP measurements was set to 4 m, and the sampling time interval was set to 30 s. The ADCP was hung at about 6 m below the ocean surface on the windward side of the R/V when it stopped and thus allowed continuous observations for 5 min at the shallowest station and 50 min at the deepest station. The accuracy of the velocity is 2.0 cm s−1. The maximum range of TurboMAP-L equipped with two shear sensors (airfoil) is 500 m with a sampling rate of 512 Hz. At each station, TurboMAP-L was deployed 2–3 times to obtain repeated profiles. The observed high-frequency shear fluctuation data are used to estimate the dissipation rate of the turbulent kinetic energy. Noise level of the turbulent dissipation rate is less than 1 × 10−10 W kg−1.

b. Data processing and interpolation

The raw CTD data are quality controlled and averaged into 1-m bins. The profiles are then smoothed by a 3-m boxcar moving average. Then density profiles and squared Brunt–Väisälä frequency N2 with a 1-m resolution are calculated. ADCP measurements provide east u and north υ current components with a vertical resolution of 4 m. The u/υ profiles at each station are averaged over time of about 5–50 min depending on the depth. Thus, the high-frequency variation variations are eliminated from the time-averaged velocity profile.

From microstructure shear measurements by airfoil probes, the 2-m turbulent kinetic energy dissipation rate ε can be estimated (Zhang et al. 2012). Most ε profiles have reliable values at depths below 7 m, where the profiler is in a required quasi-freefall state and the shear spectra from observations for estimating ε compare well with the Nasmyth universal spectrum. The averaged ε profiles from repeated observations at each station are used to calculate the turbulent eddy diffusivity Kυ using the formula Kυ = 0.2ε/N2 (Osborn 1980).

To obtain a regular gridded 3D matrix for vertical velocity calculation, the density, ADCP horizontal velocity components, and the eddy diffusivity are horizontally interpolated with linear interpolation. The grid sizes in the x and y directions are both set to be 0.05°. The interpolation ranges are 18.6°–19.6°N and 110.6°–111.6°E, as shown in Fig. 1b. The gridded data are then smoothed with a 2D Gaussian filter with scales of Lx = 35 km and Ly = 25 km. These values are chosen to minimize the singularity generated in differential calculation and filter out contamination of small-scale processes but not to damage the essential spatial patterns. The vertical grid is set to be 5 m, and the depth range is 0–160 m. The density, velocity, and eddy diffusivity profiles from observations are averaged in the depth bins of 0, 5, 10, …, 160 m. For bins without observations near the surface and bins close to but above the ocean bottom, values are extrapolated using the nearest observed data. Specifically, the density profiles are extended to the surface with constant values as that in the observed first bin in the mixed layer and to the bottom with linear extrapolation to keep the stratification. The velocities near the surface are filled with the topmost observations at about 10 m, while the velocities below the last observed bins are assumed to decrease linearly to zero at the ocean bottom (no-slip boundary condition). The eddy diffusivities near the surface are reasonably filled with the topmost values from observations at stations where the mixed layer is deeper than the first observation depth and assumed to be 10 times larger if the mixed layer is shallower than the first observation depth. Thus, we obtain the 3D gridded dataset, which has (nx, ny, nz) = (21, 21, 33) nodes with grid sizes (Δx, Δy, Δz) = (5 km, 5 km, 5 m).

c. Calculation of geostrophic and ageostrophic velocities

Using the gridded 3D density data, the geostrophic velocities referred to a no motion depth are derived from the thermal wind balance:

 
formula

and

 
formula

Here, ug and υg are the east and north components of geostrophic velocity at depth bin zk and station grid xi, yj, uref and υref are the velocity components at the reference depth, f = 2Ω sinφ0 is the Coriolis parameter at the mean latitude φ0, and is the negative buoyancy b with ρ0 as the mean density and g as the gravitational acceleration. We use 50 m as the reference depth of no motion (see discussion in section 7b). At stations with bottom depths shallower than 50 m, the geostrophic velocities at the ocean bottom are assumed to be zero. This means that the bottom pressure (dynamic height) at the shallower station is assumed to be equal to the pressure at the same depth (not the bottom pressure) of the nearby seaward station. The ageostrophic velocities u′ are then derived by subtracting the calculated geostrophic velocities ug, υg from the total velocity u, υ observed by ADCP, that is, u′ = (u′, υ′) = (uug, υυg).

d. Generalized omega equation

In the QG theories, the vertical velocity wg on the f plane can be derived from the Q-vector QG omega equation (Hoskins et al. 1978):

 
formula

where is the squared Brunt–Väisälä frequency, f is the Coriolis parameter, and is the horizontal geostrophic frontogenetical Q vector, in which ug = (ug, υg) is the horizontal geostrophic velocity vector. The right-hand side of Eq. (3) is dynamically defined as geostrophic kinematic deformation term SDEFg.

For a flow with a high Rossby number, a higher-order generalized omega equation for an inviscid, isentropic, and Boussinesq flow on the f plane is in the form of (Viúdez et al. 1996; Pallàs-Sanz and Viúdez 2005)

 
formula

where W is the vertical velocity, is the total Q vector, is the horizontal ageostrophic pseudovorticity, is the ageostrophic horizontal velocities, and is the differential vertical vorticity of horizontal velocity minus the differential geostrophic vorticity . Neglecting (i) the material rate of the change in , (ii) terms and , and (iii) the horizontal change in , the generalized omega equation [Eq. (4)] is simplified to

 
formula

The first term on the right-hand side is total kinetic deformation SDEF, which includes the geostrophic component SDEFg in Eq. (3) and ageostrophic deformation SDEFa . The second term is the ageostrophic advection term SADV.

Further, taking the momentum and buoyancy turbulent flux induced by the vertical mixing into account, the generalized omega equation [Eq. (5)] has two more terms (Pallàs-Sanz et al. 2010b):

 
formula

where is the horizontal pseudovorticity vector, SMOM and SBUO are the vertical mixing fluxes of momentum and buoyancy, and Kυ and Aυ are the vertical eddy diffusivity and the vertical eddy viscosity, respectively.

3. 3D density and velocity fields

a. Sea surface wind and sea surface temperature fields

Figure 2 shows the sea surface temperature (SST) and the surface wind fields on 13 and 18 July 2012 in the northern SCS during the cruise observation. The SST data are 6 × 6 km2 daily products downloaded from website of the Group for High Resolution Sea Surface Temperature (GHRSST) project (http://data.nodc.noaa.gov/ghrsst/L4/GLOB/UKMO/OSTIA/). They are merged products with in situ, microwave, and infrared satellite SSTs produced by the U.K. Meteorological Office. The wind data are downloaded from the NOAA National Climatic Data Center (NCDC) website (ftp://eclipse.ncdc.noaa.gov/pub/seawinds/), which are derived from blended multiple satellite measurements of the Special Sensor Microwave Imager (SSM/I), the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI), and the Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E) at a 0.25° resolution.

Fig. 2.

Sea surface wind (white arrows, m s−1) and SST (color, °C) fields on 13 and 18 Jul 2012 during the cruise observations. Landmarks QL and BO are labeled on the map for reference.

Fig. 2.

Sea surface wind (white arrows, m s−1) and SST (color, °C) fields on 13 and 18 Jul 2012 during the cruise observations. Landmarks QL and BO are labeled on the map for reference.

As shown in Fig. 2, southerly winds prevailed during the observation period. This wind condition favors the coastal upwelling along the eastern coast of Hainan Island, China, which is characterized by cold water with SST lower than 28°C. The upwelling hugging the coast between the Qinglan (QL) Bay and Boao (BO) Bay is robust during the whole period, while the background temperature far offshore in the deeper SCS increases from about 29°C on 13 July to above 30°C on 18 July 2012.

b. Temperature, salinity, and density fields

Figure 3 shows the horizontal distribution maps of the water temperature and salinity at depths 10, 20, 30, and 50 m. One can see that the water of low temperature (<24°C) and high salinity (>34 psu) exists in the subsurface layer along the coast from BO to QL. An upwelling center with the temperature 5°C lower and the salinity 1 psu higher than that in the deep water appears near BO at depths above 30 m. This low-temperature and high-salinity upwelling center extends eastward and northward with the depth increased. The coastal waters north of QL are 1°C warmer and 0.2 psu fresher than that near BO.

Fig. 3.

Horizontal (x–y) distribution maps of the (a)–(d) observed temperature T (°C) and (e)–(h) salinity S (psu) at 10, 20, 30, and 50 m. Landmarks QL and BO are labeled on the map for reference. Black dots represent the observation stations. Gray dashed lines represent the 100-m isobath.

Fig. 3.

Horizontal (x–y) distribution maps of the (a)–(d) observed temperature T (°C) and (e)–(h) salinity S (psu) at 10, 20, 30, and 50 m. Landmarks QL and BO are labeled on the map for reference. Black dots represent the observation stations. Gray dashed lines represent the 100-m isobath.

From Fig. 4, one can see that the potential density increases 2.0–4.0 kg m−3 shoreward from the ocean area deeper than 100 m to near the coast. At 10 m, a strong density front with a maximum gradient of 1 × 10−4 kg m−4 appears along the 100-m isobath, which separates the coastal upwelling dense water from the deep-ocean light water. Another weaker front with a gradient of 0.6 × 10−4 kg m−4 appears in the coastal water near QL, where the density is about 1 kg m−3 lower than the maximum potential density of 23.5 kg m−3 near BO. In the deeper layer, the upwelling dense water extends northward. The frontal zone is divided into more than two bands with the maximum gradient of 0.7 × 10−4 kg m−4 appearing in the northeastern observation area. Below 50 m, the density gradient becomes smaller than 0.2 × 10−4 kg m−4 with relatively large value in the deep-ocean area.

Fig. 4.

As in Fig. 3, but for the (a)–(d) potential density σθ (kg m−3) and the (e)–(h) horizontal gradient of potential density ∇σθ (10−4 kg m−4). Black lines in (a)–(d) represent the isopycnals after the Gaussian filter (contour interval is 0.2 kg m−3; σθmin = 20.6, 21.0, 21.8, and 23.8 kg m−3 at 10, 20, 30, and 50 m, respectively). Arrows in (e)–(h) represent the vector ∇σθ. Gray dashed lines represent the 100-m isobath. Black boxes represent the frontal zone.

Fig. 4.

As in Fig. 3, but for the (a)–(d) potential density σθ (kg m−3) and the (e)–(h) horizontal gradient of potential density ∇σθ (10−4 kg m−4). Black lines in (a)–(d) represent the isopycnals after the Gaussian filter (contour interval is 0.2 kg m−3; σθmin = 20.6, 21.0, 21.8, and 23.8 kg m−3 at 10, 20, 30, and 50 m, respectively). Arrows in (e)–(h) represent the vector ∇σθ. Gray dashed lines represent the 100-m isobath. Black boxes represent the frontal zone.

c. Horizontal velocity field and vorticity

Figure 5a shows the total horizontal velocities Vtotal observed by ADCP. One can see that the most velocity vectors at the upper 20 m are directing northeastward and generally parallel to the coastal line. The larger velocities appear near the coast with a maximum value of about 0.4 m s−1. The velocity direction turns anticlockwise and the magnitudes decrease with the depth. Near the bottom, the flows generally turn shoreward in the ocean area deeper than 100 m, transporting deep-ocean waters shoreward. Figure 5b shows the geostrophic velocities Vgeo referred to 50 m derived from the gridded densities and the thermal wind balance. One can see that the geostrophic velocities in the water deeper than 100 m are mostly northeastward with a maximum value of 0.4 m s−1 at the upper layer. Their amplitudes decrease to zero at the reference depth of 50 m, and the directions turn to southwestward in the deeper layer. The maximum southwestward geostrophic current is about 0.1 m s−1 at 80 m. In the shallow waters near the coast, the geostrophic velocities are smaller than 0.1 m s−1 and directing shoreward near QL. Figure 5c shows the ageostrophic velocities calculated by subtracting the geostrophic velocities from the total velocities, that is, Vageo = VtotalVgeo. One can see that the ageostrophic velocities near the coast are larger than 0.3 m s−1 and directing northeastward along the coast like the total velocities, indicating that the ageostrophic component has larger contribution to the total velocity in the shallow coastal water. In the deep water outside the front zone, the ageostrophic velocities are smaller than that in the coastal water and mainly directing shoreward below 50 m, implying the onshore transport supporting the coastal upwelling.

Fig. 5.

Distribution maps of (a) the total horizontal velocity Vtotal observed by the ADCP, the (b) geostrophic velocity referred to 50 m Vgeo, and the (c) ageostrophic velocity Vageo = VtotalVgeo at 10, 20, 30, 50, and 80 m. Gray dashed lines represent the 100-m isobath. Black boxes represent the frontal zone.

Fig. 5.

Distribution maps of (a) the total horizontal velocity Vtotal observed by the ADCP, the (b) geostrophic velocity referred to 50 m Vgeo, and the (c) ageostrophic velocity Vageo = VtotalVgeo at 10, 20, 30, 50, and 80 m. Gray dashed lines represent the 100-m isobath. Black boxes represent the frontal zone.

Figure 6 shows the scaled relative vorticity ζ/f (also named the Rossby number) derived from the total velocity field. One can see that it has larger values with a maximum/minimum of ±0.6 in the coastal water shallower than 100 m, indicating that ageostrophic advection should be considered in the dynamic analysis of the coastal water. In the deep ocean outside the front zone, the scaled relative vorticity is generally smaller than 0.2, so that QG effects may become important. These results coincide with that from the ageostrophic velocities shown in Fig. 5c. Furthermore, the positive and negative vorticities appear alternatingly in the cross-shelf direction, similar to patterns of alternating large and small density gradients shown in Fig. 4.

Fig. 6.

As in Fig. 3, but for the scaled relative vorticity ζ/f.

Fig. 6.

As in Fig. 3, but for the scaled relative vorticity ζ/f.

d. Vertical mixing coefficient

The turbulent dissipation rate ε and the vertical eddy diffusivity Kυ derived from the cruise microstructure observations are shown in Fig. 7. One can see that ε has a maximum value of 4 × 10−7 W kg−1 at 10 m, which appears on the seaward side of the frontal zone. In the frontal zone along the 100-m isobath, ε is generally one order of magnitude smaller than that on its two sides. In the cross-shelf direction, alternating smaller and larger ε bands appear as banded structure. The term ε dramatically decreases with the depth to smaller than 2.5 × 10−9 W kg−1 at 50 m. The banded structure persists in the deep layers with smaller ε on the shoreward side of the frontal zone along the 100-m isobath. The distribution of Kυ mainly follows that of ε due to relatively uniform Brunt–Väisälä frequency (not shown). At 10 m, Kυ is smaller than 1 × 10−5 m2 s−1 along the 100-m isobath and jumps to 1 × 10−4 m2 s−1 on the seaward side of the frontal zone. The maximum value of Kυ = 3 × 10−4 m2 s−1 appears in the deep water outside the front zone, where the mixed layer is deeper than 10 m. The vertical eddy diffusivity Kυ is generally smaller than 1 × 10−5 m2 s−1 at water depths below 20 m, as shown in Figs. 7f–h. The mixed layer depths are shallower than 20 m at all stations in the study area. The weak mixing in layers below 20 m results from strong stratification with the Brunt–Väisälä frequency larger than 5 × 10−1 s−1. In the cross-shelf direction, alternating large and small alongshore bands of Kυ appear at all layers.

Fig. 7.

As in Fig. 3, but for the (a)–(d) turbulent dissipation rate log10(ε) (W kg−1) and the (e)–(h) eddy diffusivity log10(Kυ) (m2 s−1). Color scale is logarithmic.

Fig. 7.

As in Fig. 3, but for the (a)–(d) turbulent dissipation rate log10(ε) (W kg−1) and the (e)–(h) eddy diffusivity log10(Kυ) (m2 s−1). Color scale is logarithmic.

4. Dynamic terms of the generalized omega equation

a. Kinematic deformation term

Figures 8a–d show the horizontal distribution maps of the kinematic deformation term SDEF , which is derived from the observed total horizontal velocity and density at depths 10, 20, 30, and 50 m. One can see that SDEF appears as alternating positive and negative alongshore bands with the maximum absolute values of 30 × 10−16 m−1 s−3 at 10 m near the 100-m isobath. At 20 m, the distribution pattern of SDEF is similar to that at 10 m, but the maximum absolute value decreases to 20 × 10−16 m−1 s−3 near the 100-m isobath and smaller than 10 × 10−16 m−1 s−3 on the shoreward side. Though the values of SDEF decrease with the depth, the dominant banded structures are the same, that is, alternating positive and negative alongshore bands with a horizontal length scale of about 20 km.

Fig. 8.

Distribution maps of kinematic deformation term SDEF (10−16 m−1 s−3) at (a) 10, (b) 20, (c) 30, and (d) 50 m. Gray dashed lines represent the 100-m isobath. Black dots represent the observation stations. Black boxes represent the frontal zone.

Fig. 8.

Distribution maps of kinematic deformation term SDEF (10−16 m−1 s−3) at (a) 10, (b) 20, (c) 30, and (d) 50 m. Gray dashed lines represent the 100-m isobath. Black dots represent the observation stations. Black boxes represent the frontal zone.

We further calculate the geostrophic and ageostrophic components of the deformation term SDEFg and SDEFa from the calculated geostrophic and ageostrophic velocities. The results indicate that SDEFa is generally larger than SDEFg at all layers and has larger values in the shallow waters near the coast and the frontal zone. The banded distribution of SDEF is dominated by SDEFa, indicating the significant contribution of the ageostrophic deformation.

b. Ageostrophic advection term

The ageostrophic advection term SADV is related to the ageostrophic pseudovorticity vector and the Laplacian of uh . We do not calculate directly from the ageostrophic velocity but derive it from the total pseudovorticity vector and the horizontal buoyancy gradient, that is,

Figure 9 shows the distribution maps of ageostrophic advection term SADV at 10, 20, 30, and 50 m. One can see that SADV has alternating positive and negative bands with values above 30 × 10−16 m−1 s−3 at 10 m. The values of SADV decrease with the depth to about 10 × 10−16 m−1 s−3 below 20 m. The positive bands of SADV are always accompanied by the negative bands, supporting the banded structure. Comparing to kinematic deformation term SDEF shown in Fig. 8, ageostrophic advection term SADV has larger values in the shallow waters and smaller values near the 100-m isobath. They counteract each other with similar values but opposite signs.

Fig. 9.

As in Fig. 8, but for the ageostrophic advection term SADV.

Fig. 9.

As in Fig. 8, but for the ageostrophic advection term SADV.

c. Vertical mixing forcing term

Taking the Prandtl number to be 1 and the vertical eddy coefficient Aυ equal to the diffusivity Kυ shown in Figs. 7e–h, the vertical mixing (SMIX) of momentum flux term SMOM and buoyancy flux term SBUO can be derived from the observations. As shown in Fig. 10, SMOM has the largest value of 10 × 10−16 m−1 s−3 at 10 m in the frontal zone, where the upper mixed layer depths are about 5–9 m and the eddy diffusivity varies significantly from the mixed layer to the thermocline. The maximum value of SMOM decreases rapidly to 2 × 10−16 m−1 s−3 at 20 m and 0.05 × 10−16 m−1 s−3 below 30 m, which is one to two orders of magnitude smaller than that of SDEF and SADV. The buoyancy flux term SBUO is larger than SMOM at 10 m and has the maximum value of 20 × 10−16 m−1 s−3 in the frontal zone. It also rapidly decreases with the depth and has a maximum value of about 0.5 × 10−16 m−1 s−3 at 20 m, smaller than that of SMOM. At depths below 30 m, the values of SBUO are smaller than 0.05 × 10−16 m−1 s−3. The distribution patterns of SMOM and SBUO are also featured by the banded structure, following that of the eddy diffusivity Kυ. The signs of SMOM and SBUO bands are mostly opposite.

Fig. 10.

As in Fig. 8, but for terms (a)–(d) SMOM and (e)–(h) SBUO.

Fig. 10.

As in Fig. 8, but for terms (a)–(d) SMOM and (e)–(h) SBUO.

Figure 11 shows the total vertical mixing forcing term SMIX (=SMOM + SBUO) at 10, 20, 30, and 50 m. One can see that SMIX is also featured by the banded structure similar to that of SMOM and SBUO. At 10 m, SMIX has similar distribution patterns and values to that of SBUO. Its largest value 20 × 10−16 m−1 s−3, comparable to that of SDEF and SADV, appears in the frontal zone near the 100-m isobath. At 20 m, the largest value of SMIX decreases by an order of magnitude to −2 × 10−16 m−1 s−3 at the deepest stations, which is dominated by SMOM. In the deeper layers, SMIX decreases to smaller than 0.1 × 10−16 m−1 s−3. The values of the total mixing forcing term SMIX are one to two orders of magnitude smaller than that of SDEF and SADV at depths below 20 m. This implies the limited effects of mixing on the vertical circulation in the upper ocean.

Fig. 11.

As in Fig. 8, but for the total vertical mixing forcing term SMIX.

Fig. 11.

As in Fig. 8, but for the total vertical mixing forcing term SMIX.

5. 3D vertical circulation

Assuming vertical velocity W = 0 at the surface, the ocean bottom, and the lateral boundaries, we derive the 3D structure of the vertical circulation from the cruise observations using the generalized omega equation [Eq. (6)]. In calculation, the SMIX at grids in the northern study area without microstructure observations are extrapolated with the nearest values, so that the values of SDEF and SADV at these grids can be used.

Figure 12 shows the calculation results of vertical velocities W under the effects of SDEF, SADV, and SMIX. One can see that a strong downwelling band with a maximum velocity of −5 × 10−5 m s−1 appears at 10 m in the frontal zone with concentrated isopycnals. An upwelling band with a maximum velocity of 5 × 10−5 m s−1 appears in the lighter water on the seaward side of the frontal zone. In the shallow coastal water, stronger upwelling with a maximum velocity core of 7 × 10−5 m s−1 dominates near QL, where the concentrated isopycnals are perpendicular to the coastline. The distribution patterns of alternatingly appearing upwelling and downwelling bands at 20, 30, and 50 m are similar to that at 10 m, while the maximum velocities decrease with the depth. Generally, the distribution of the vertical circulation also shows the banded structure as alternating upwelling and downwelling from the coast to the deep ocean.

Fig. 12.

Distribution maps of vertical velocity W (10−5 m s−1) at (a) 10, (b) 20, (c) 30, and (d) 50 m. White lines represent the zero contour. Black lines represent the isopycnals in Figs. 4a–d. The gray line represents the 100-m isobath. Pink boxes represent the frontal zone.

Fig. 12.

Distribution maps of vertical velocity W (10−5 m s−1) at (a) 10, (b) 20, (c) 30, and (d) 50 m. White lines represent the zero contour. Black lines represent the isopycnals in Figs. 4a–d. The gray line represents the 100-m isobath. Pink boxes represent the frontal zone.

Figure 13 shows the distribution of W along two cross-shelf sections H and F. One can see the positive and negative stripes of the vertical velocity alternatingly appear along the sections. At section H, upwelling near the coast has the vertical velocity of about 2 × 10−5 m s−1 at depths of 10–20 m. Downwelling with a maximum of −3 × 10−5 m s−1 appears at 10–20 m at stations in the frontal zone 40–60 km offshore, and upwelling with a maximum of about 4 × 10−5 m s−1 appears at 10–40 m at stations outside the frontal zone 60–80 km offshore. At section F, the vertical velocity near the coast reaches 5 × 10−5 m s−1, which is greater than that at section H. Another strong upwelling center with the vertical velocity larger than 4 × 10−5 m s−1 appears at 5–30 m at stations outside the frontal zone 70 km offshore. Between the two strong upwelling centers, downwelling with maximum velocity of −4 × 10−5 m s−1 appears at 10–70 m in the frontal zone. In general, two sections of the vertical circulation show similar patterns. However, there are remarkable differences: 1) the vertical circulation at section F is stronger than that at section H, and 2) the downwelling band at section F is wider than that at section H. This reflects a northward-intensifying trend of upwelling and frontal zone.

Fig. 13.

Vertical distribution of vertical velocity W (10−5 m s−1) along sections (a) H and (b) F. Black lines represent the sea bottom. The isopycnals are shown as black dashed lines, σθmin = 20.4 kg m−3; the contour interval is 0.2 kg m−3. Pink boxes represent the frontal zone.

Fig. 13.

Vertical distribution of vertical velocity W (10−5 m s−1) along sections (a) H and (b) F. Black lines represent the sea bottom. The isopycnals are shown as black dashed lines, σθmin = 20.4 kg m−3; the contour interval is 0.2 kg m−3. Pink boxes represent the frontal zone.

We further calculate alongshore-averaged W and compare its cross-shelf variation with that of dynamic terms SDEF, SADV, and SMIX. The results at depths 10, 20, 30, and 50 m are shown in Figs. 14a–d. One can see that the alongshore-averaged W and all dynamic terms undulate along the cross-shelf direction with a horizontal length scale of O(20–40) km. Above 30 m, SDEF, which is dominated by its ageostrophic component SDEFa, is the dominant dynamic term in deep waters, especially in the frontal zone and the nearby deep-ocean water at about 50–70 km from the coast. In the shallow waters within 40 km offshore, SADV becomes the dominant term. At 50 m, the frontal zone weakens and SADV dominates most sections. The term SMIX is dominated by SBUO and is comparable to SDEF and SADV at 10 m. It is dominated by SBUO and is one order of magnitude smaller than SDEF and SADV at 20 m. This term is two orders of magnitude smaller than SDEF and SADV and negligible below 30 m. The variation of W follows that of SDEFa and SADV. It always has two positive extrema: one is located in the shallow coastal water at 20 km offshore, and another is located in the deep ocean outside the frontal zone 70 km offshore. The maximum W is 2 × 10−5 m s−1 at 10–20 m in the coastal upwelling region and 3 × 10−5 m s−1 outside the frontal zone, respectively.

Fig. 14.

Alongshore-averaged dynamic terms SDEF, SADV, SMIX, SMOM, SBUO, SDEFa, and W in the cross-shelf direction at (a) 10, (b) 20, (c) 30, and (d) 50 m. Pink boxes represent the frontal zone.

Fig. 14.

Alongshore-averaged dynamic terms SDEF, SADV, SMIX, SMOM, SBUO, SDEFa, and W in the cross-shelf direction at (a) 10, (b) 20, (c) 30, and (d) 50 m. Pink boxes represent the frontal zone.

6. Discussion

a. Mechanism of banded structure of the vertical velocity

Previous investigators have found banded distribution of the forcing term and vertical velocity in the mesoscale and submesoscale fronts, where the vertical velocity is upward on the light (warm) side and downward on the heavy (cold) side (Capet et al. 2008; McWilliams 2016). In this study, the vertical circulation shows multiple bands in the cross-shelf direction, which has downwelling in the frontal zone, and upwelling on two sides of the frontal zone. This is a bit different from typical one upwelling–downwelling cell in the fronts but similar to the results by Thomas and Lee (2005) and Nagai et al. (2012), which show two or multiple cells of cross-frontal circulation forced by downfront wind stress. We suggest that the banded structure obtained by this study results from the coexistence of the frontal zone and coastal upwelling induced by wind-driven Ekman suctions.

According to Stern (1965) and Pallàs-Sanz et al. (2010a), the nonlinear Ekman transport Me caused by the wind stress τh interacting with a background flow is

 
formula

and the total Ekman suctions We is

 
formula

where ζ0 is the geostrophic vertical vorticity at the surface. The first term on the right-hand side of Eq. (8) is the classic linear Ekman pumping due to the wind curl. The second term represents the interaction of the wind stress and perpendicular gradient of the vorticity. This indicates that the gradient of ζ0 generates the vertical velocity even when the wind stress curl is zero. We use the averaged wind stress over 13–18 July 2012 for calculation. The results of Me and We are shown in Fig. 15a. One can see that Ekman transport is mostly eastward with a magnitude of O(1) m2 s−1. The total Ekman suction, which is mainly induced by the interaction of the wind and the gradient of ζ0, shows banded structure in the cross-shelf direction. Upwelling bands appear near the coast and outside the 100-m isobath with the maximum vertical velocity of 3 × 10−5 and 5 × 10−5 m s−1, respectively. Between the upwelling bands, downwelling dominates with a maximum velocity of −5 × 10−5 m s−1.

Fig. 15.

(a) Nonlinear Ekman transport Me (arrows) and Ekman suction We (color, 10−5 m s−1) and (b) vertical velocity WSADV (10−5 m s−1) forced by SADV at 10 m. The potential density (black lines, σθmin = 20.6 kg m−3; contour interval is 0.2 kg m−3) at 10 m is shown. The gray dashed line represents the 100-m isobath. White lines represent the zero contours. Pink boxes represent the location of the frontal zone.

Fig. 15.

(a) Nonlinear Ekman transport Me (arrows) and Ekman suction We (color, 10−5 m s−1) and (b) vertical velocity WSADV (10−5 m s−1) forced by SADV at 10 m. The potential density (black lines, σθmin = 20.6 kg m−3; contour interval is 0.2 kg m−3) at 10 m is shown. The gray dashed line represents the 100-m isobath. White lines represent the zero contours. Pink boxes represent the location of the frontal zone.

The results of We are similar to the vertical velocity WSADV only forced by the dynamic term SADV as shown in Fig. 15b. Wind influence is included implicitly in SADV through the ageostrophic vertical shear [Eq. (5)]. In this case, the ageostrophic currents are mainly related to the Ekman currents that are stronger near the coast. Thus, WSADV dominates the total W near the coast, where the upwelling band appears as shown in Fig. 13.

b. Effects of mixing on the vertical circulation

Nagai et al. (2006) and Pallàs-Sanz et al. (2010b) investigated the effect of vertical mixing on the vertical circulation in ocean fronts using parameterized eddy viscosity due to the lack of observed mixing data. In this study, vertical mixing parameters are derived from direct measurements with the Turbulence Ocean Microstructure Acquisition Profiler (TurboMAP) and are included in the generalized omega equation for the first time. Our analysis shows that the mixing forcing term SMIX is comparable to deformation and advection terms only at 10 m in the frontal zone but one to two orders of magnitude smaller at depths below 30 m. To test the effects of the mixing on the vertical circulation, we calculate the vertical velocity without SMIX, that is, Wnmix. The results show that Wnmix has similar distribution patterns and values as that of W at most stations and depths (not shown). Figures 16a–d show the difference between the vertical velocities with and without mixing forcing, that is, ΔWmix = WWnmix, at 10, 20, 30, and 50 m. One can see that the maximum of ΔWmix reaches 2 × 10−5 m s−1 at 10 m in the frontal zone, which is about 30%–40% of the total W there. The maximum of ΔWmix decreases rapidly to one order of magnitude smaller than W at 20 m and negligible below 30 m. As shown in Fig. 16e, the horizontal mean value of |ΔWmix|, that is, |ΔWmix|xy, only has values larger than 0.5 × 10−6 m s−1, that is, 10%–15% of |W|xy (at 5–30 m), with a maximum value at 10 m. Below 30 m, |ΔWmix|xy becomes two orders of magnitude smaller than |W|xy. Compared to the distribution of the vertical eddy diffusivity shown in Fig. 16f, one can see that the larger |ΔWmix|xy in the upper 30-m layer corresponds to strong mixing with |Kυ|xy larger than 3 × 10−6 m2 s−1. This is similar to that suggested by Nagai et al. (2006, 2008), that is, the influence of the vertical mixing on vertical velocity may be important when the eddy diffusivity and the associated scaled Ekman number with deformation rate are large. Furthermore, the largest |ΔWmix|xy appears at the depth where the vertical gradient of the eddy diffusivity |gradz(Kυ)|xy is the largest, implying the vertical variation of the eddy diffusivity is also important to the vertical velocity.

Fig. 16.

Distribution maps of difference ΔWmix between vertical velocities with and without mixing forcing at (a) 10, (b) 20, (c) 30, and (d) 50 m. (e) Vertical distribution of horizontally averaged |ΔWmix|xy (green) and |W|xy (blue). (f) Vertical distribution of horizontally averaged eddy diffusivity |log10(Kυ)|xy (blue) and its vertical gradient log10|gradz(Kυ)|xy (green). Gray dashed lines in (a)–(d) represent the 100-m isobath. White lines represent the zero contours. Pink boxes represent the frontal zone.

Fig. 16.

Distribution maps of difference ΔWmix between vertical velocities with and without mixing forcing at (a) 10, (b) 20, (c) 30, and (d) 50 m. (e) Vertical distribution of horizontally averaged |ΔWmix|xy (green) and |W|xy (blue). (f) Vertical distribution of horizontally averaged eddy diffusivity |log10(Kυ)|xy (blue) and its vertical gradient log10|gradz(Kυ)|xy (green). Gray dashed lines in (a)–(d) represent the 100-m isobath. White lines represent the zero contours. Pink boxes represent the frontal zone.

The confined effects of mixing on the vertical velocity in the upper ocean derived from our observations, in general, are similar to the results estimated by Nagai et al. (2006) and Pallàs-Sanz et al. (2010b). However, contribution of vertical mixing to intensification of vertical velocity is not as great as predicted by previous studies. The reason is that cruise-observed eddy diffusivities in our case are smaller than the values estimated by the previous investigators (Nagai et al. 2006; Pallàs-Sanz et al. 2010b). In fact, as pointed out by Pallàs-Sanz et al. (2010b), the effects of vertical mixing on the vertical velocity are quite sensitive to the vertical mixing parameterization, and the intensification factor of the vertical velocity by mixing significantly decreases from 2 to 0.5 as the parameterized eddy viscosity decreases from 1 × 10−2 to 6 × 10−3 m2 s−1. In this study, the eddy diffusivities from observations have values smaller than 3 × 10−4 m2 s−1, which is one to two orders of magnitude smaller than the parameterized mixing coefficients used by the previous investigators. If the missed eddy diffusivities in the mixed layer are filled with values two orders of magnitude larger, the maximum ΔWmix could reach ±20 × 10−5 m s−1 at 10 m in the frontal zone, which is about 300%–400% of the vertical velocity without mixing.

c. Sensitivity of the vertical velocity to the geostrophic reference level

For calculation of geostrophic velocities, we use 50 m as the reference level of no motion. As shown in Fig. 5b, the directions of the resulted geostrophic velocities turn to southward at 80 m. The north component of the geostrophic velocities at section H shown in Fig. 17a also has negative values below 50 m, which is the opposite of the northward flow in the upper layer. This is consistent with the downward tilt of isopycnals toward the coast in the deep layer, indicating the choice of no motion at 50 m is feasible. We further calculate the geostrophic velocities and geostrophic and ageostrophic deformation terms with levels of no motion at 70 m and the ocean bottom for comparison. As shown in Figs. 17b and 17c, the patterns and values of the geostrophic velocities in the upper 40 m change little with reference levels changing from 50 m to the ocean bottom. Meanwhile, dynamic terms SDEFg and SDEFa in the upper layers are insensitive to the reference level, as in the example shown in Fig. 17d. One can see that below 70 m, the variation of SDEFg and SDEFa counteract with each other, making their sum unchanged. In fact, SDEF is derived from the total velocity observed by ADCP in our calculation and is independent of the reference level.

Fig. 17.

Northward component of geostrophic velocities (m s−1) referred to (a) 50 m, (b) 70 m, and the (c) ocean bottom. Alongshore-averaged SDEFg and SDEFa at (d) 10 and (e) 80 m. White lines in (a)–(c) represent the zero contours.

Fig. 17.

Northward component of geostrophic velocities (m s−1) referred to (a) 50 m, (b) 70 m, and the (c) ocean bottom. Alongshore-averaged SDEFg and SDEFa at (d) 10 and (e) 80 m. White lines in (a)–(c) represent the zero contours.

For the vertical velocity derived from the omega equation, the forcing term is geostrophic deformation SDEFg in QG assumption, which is determined by two key horizontal derivatives of the density and the geostrophic velocity (Hoskins et al. 1978). Thus, careful attention should be paid to the calculation of geostrophic velocities (Rudnick 1996). Different from the QG equation used by the previous investigators, in this study we use the generalized omega equation to calculate the vertical circulation. Forcing terms SDEF, SADV, and SMIX are derived from the in situ density measured by CTD, the eddy diffusivity measured by TurboMAP, and the total horizontal velocity measured by ADCP, instead of the calculated geostrophic velocity. The ageostrophic component is explicit in term SADV, but it is independent of the choice of geostrophic reference level because is derived from the in situ density and the observed total velocity (i.e., ). Thus, in our case the selection of the geostrophic reference level has no effect on the final results of the vertical velocity.

d. Effects of bottom topography

In this study, we assume the vertical velocities at the ocean bottom as zero. To examine the dynamic effect of the bottom topography, we do numerical experiments using the kinetic boundary condition:

 
formula

Where ubot and υbot are the horizontal components of the total velocity at the ocean bottom and h is the bottom depth. At stations where the ADCP observations do not reach the bottom, ubot and υbot are assumed to be the same as that at the deepest observed depth bin, which are mostly about 0.05–0.15 m s−1.

Figures 18a–d show the vertical velocity with the kinetic boundary, that is, Wkb, at 10, 20, 30, and 50 m. One can see that Wkb at 10 m has values and banded distribution patterns of alternating upwelling and downwelling bands quite close to that of W shown in Fig. 12. Below 10 m, Wkb still has similar banded structure in the cross-shelf direction, but the maximum values of the upwelling and downwelling become larger than that of W, especially in the shallow waters near the coast. As shown in Figs. 18e–h, the difference between Wkb and W, that is, ΔWbot = WkbW, in the upwelling bands near the coast increases significantly with the depth from about 0.5 × 10−5 m s−1, 10% of Wkb, at 10 m to larger than 3 × 10−5 m s−1, 50% of Wkb, at 50 m. In the frontal zone and the seaward deep water, the maximum ΔWbot is only about 1 × 10−5 m s−1 appearing at 20 and 30 m, about 15%–20% of Wkb. The signs of ΔWbot are coherent with that of Wkb. The maximum ΔWbot appears in the deeper layer on the edge of the upwelling bands near the coast, where W is close to zero under no-slip boundary condition. This implies that the effect of the bottom topography on the vertical circulation is evident in the deep layer when the kinetic boundary condition with bottom horizontal velocities larger than 0.05 m s−1 is used in the study area.

Fig. 18.

As in Fig. 12, but for (a)–(d) vertical velocity Wkb (kinetic bottom boundary condition) and (e)–(h) the difference ΔWkb between Wkb and W (zero bottom boundary condition).

Fig. 18.

As in Fig. 12, but for (a)–(d) vertical velocity Wkb (kinetic bottom boundary condition) and (e)–(h) the difference ΔWkb between Wkb and W (zero bottom boundary condition).

e. Of low-frequency contamination

In this study, the total horizontal velocities used for vertical velocity calculation have been averaged from 5–50-min ADCP observations to eliminate high-frequency variations. However, random, low-frequency contamination might still be contained. In coastal oceans, tidal currents are usually the strongest components to contaminate the total velocity. Previous long-time observations have shown that the tidal currents are smaller than 0.1 m s−1 in the shelf water near our study area (Xu et al. 2011; He et al. 2012). We observed the maximum velocity amplitudes of K1 and M2 tides in the study area during other cruise projects. The results are 0.15 m s−1 in the upper layer and smaller than 0.05 m s−1 in the deep water. The horizontal scale of the tidal ellipse is of O(10) km. Thus, in order to remove the tidal contamination, we preprocess the cruise-observed data with a 2D Gaussian filter of 35 × 25 km2 prior to calculating the vertical velocities. The near-inertial oscillations (NIOs) are another potential contamination factor. However, the NIOs occur as an episodic event due to the sudden variation of strong wind. The strong signals of NIOs only occur spatially within an ocean area near the ground track and temporally within a week after the passage of a hurricane (or typhoon or storm; Zheng et al. 2006; Zhang et al. 2014; Sun et al. 2015). Our study area is about 500 km west of the ground track of the last tropical storm Doksuri, which passed over the northern SCS from 26 to 30 June 2012. Our cruise observations were carried out during a period from 13 to 18 July 2012, which was two weeks after the typhoon passage. As shown in Fig. 2, the southerly winds were steady during the whole observation period. The currents of NIOs in the study area should be quite small. The 2D spatial smoothing could also further filter out their influence. Thus, we believe that the NIO contamination to the velocity is negligible.

7. Conclusions

This study deals with the calculation of vertical circulation in the upwelling region and frontal zone east of the Hainan Island, China, using the generalized omega equation and the CTD, ADCP, and TurboMAP observations in July 2012. The effects of the kinematic deformation, the ageostrophic advection, and the vertical mixing forcing, which is derived directly from microstructure observations for the first time, on the vertical circulation are diagnosed. The major findings are summarized as follows:

  1. In summer southwesterly wind season, dense water upwells near the coast east of the Hainan Island, China, where the temperature and the salinity near the surface are 5°C lower and 1 psu higher than that in the ocean area deeper than 100 m. The horizontal velocities are northeastward along the coast in the surface layer and turns anticlockwise with the depth to shoreward near the bottom, supporting the cold, salty deep waters to upwell in the coastal water.

  2. A strong, frontal zone with the density gradient of O(10−4) kg m−4 appears along the 100-m isobath in the subsurface layer. In the frontal zone, the kinetic energy dissipation rate ε and the vertical eddy diffusivity Kυ on the shoreward (dense water) side are one order of magnitude smaller than that on the seaward (light water) side. Alongshore bands of high and low mixing rate alternatingly appear in the cross-shelf direction.

  3. The diagnosis dynamic terms of kinematic deformation SDEF, ageostrophic advection SADV, and vertical mixing effect SMIX all show banded structure with a horizontal spatial scale of about 20 km distributed in the cross-shelf direction. Term SDEF is dominated by the ageostrophic component SDEFa and dominant in the frontal zone. The term SADV is the largest one in the shallow coastal waters and at the bottom layer. The mixing effect SMIX on the vertical circulation should be considered in the upper ocean, especially at depths with relatively large vertical eddy diffusivity and eddy diffusivity gradient in the frontal zone but is negligible in deep layers below thermocline.

  4. Consistently with the banded structure of the dynamic terms, alternating alongshore upwelling and downwelling bands appear from the coast to the deep water in the observation area. The maximum downwelling velocity reaches −5 × 10−5 m s−1 in the frontal zone near the 100-m isobath, while upwelling on two sides of the frontal zone has comparable maximum vertical velocities of 5 ~ 7 × 10−5 m s−1. The stronger upwelling band near the coast can be attributed to the nonlinear Ekman suctions due to the interaction of wind and vorticity gradient.

Acknowledgments

We thank the crews of R/V Tianlong for help in the observation program. The sea surface temperature and the sea surface wind data are downloaded from websites (http://data.nodc.noaa.gov/ghrsst/L4/GLOB/UKMO/OSTIA/ and ftp://eclipse.ncdc.noaa.gov/pub/seawinds/). The ETOPO2 data are also available online (downloaded from https://www.ngdc.noaa.gov/mgg/global/global.html). This work is supported by the National Nature Science Foundation of China (Grants 41476009, U1405233, 41676008, 41106012, and 41506018) and the Open Foundation of State Laboratory of Tropic Ocean of China (Grant LTO1404). We appreciate the anonymous reviewers for the comments and suggestions that were significant for improving the paper.

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Footnotes

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