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  • View in gallery

    (a) Schematic maps of the Kuroshio, location of the PN section, climatological mean wind stress vectors (red arrows), and bottom velocity at the PN section (blue bars) in winter and (b) geostrophic velocity at the PN section in winter. In (a), black dots indicate positions of survey stations along the PN section, and color shading and numbered contours denote isobaths (m).

  • View in gallery

    (a) Squared buoyancy frequency N2, (b) squared vertical shear of cross-sectional velocity , and (c) vertical eddy viscosity coefficient derived using the Munk–Anderson scheme.

  • View in gallery

    (a) Streamfunction (colored shading) and cross-shelf velocity field (arrows) at the PN section, as induced by the Kuroshio vertical shear stress, assuming that no flow penetrates through the bottom Ekman layer. (b) Vertical velocity at the top of the bottom Ekman layer, as induced by the bottom along-shelf geostrophic current.

  • View in gallery

    Cross-shelf velocity derived by calculating (a) , (b) , and (c) ; and that velocity derived inversely from solving the momentum equations with the effects of (d) vertical eddy viscosity, (e) vertical eddy viscosity and wind stress, and (f) vertical and horizontal eddy viscosity and wind stress.

  • View in gallery

    Cross-shelf velocity contributed by (a) the vertical viscosity term in the σ-coordinate system and the four components of horizontal viscosity term (b) , (c) , (d) , and (e) expressed in the σ-coordinate system, and (f) their sum.

  • View in gallery

    (a) Along-shelf, (b) cross-shelf, and (c) vertical velocities estimated in the σ-coordinate system by taking into account both horizontal and vertical eddy viscosities. (d) Along-shelf and (e) cross-shelf velocity difference [relative to that derived from the linear momentum equations in (a) and (b)] solved from nonlinear momentum equations. (f) Cross-shelf velocity derived from nonlinear equations. Green arrows in (b) denote two offshore transport pathways.

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Cross-Shelf Circulation Induced by the Kuroshio Shear Stress in the East China Sea

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  • 1 State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou, China
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Abstract

This study solves two-dimensional (cross-shelf and depth directions) steady-state nonlinear primitive equations to infer the cross-shelf circulation induced by the Kuroshio shear stress in the East China Sea (ECS). The Kuroshio velocity data are estimated from hydrographic observations at the PN section in the ECS. Nonlinear momentum equations are solved using an iterative approach in a terrain-following coordinate system, which helps to adequately take into account the boundary conditions over complex topography. The vertical shear stress of the Kuroshio is shown to induce two offshore transport pathways over the continental shelf, which are related to the structure of the interior geostrophic current and bottom Ekman transport, respectively. As a result of the vertical shear stress, an upwelling is induced above the bottom Ekman layer on the continental slope. The horizontal shear stress of the Kuroshio has the effect of inducing onshore transport at the flow core. The advection terms in the primitive equations are found to amplify the cross-shelf velocity solved from the linear equations. This study reveals that the Kuroshio has a substantial effect on the cross-shelf circulation and that it might drive multiple transport pathways.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yanzhou Wei, weiyanzhou@sio.org.cn

Abstract

This study solves two-dimensional (cross-shelf and depth directions) steady-state nonlinear primitive equations to infer the cross-shelf circulation induced by the Kuroshio shear stress in the East China Sea (ECS). The Kuroshio velocity data are estimated from hydrographic observations at the PN section in the ECS. Nonlinear momentum equations are solved using an iterative approach in a terrain-following coordinate system, which helps to adequately take into account the boundary conditions over complex topography. The vertical shear stress of the Kuroshio is shown to induce two offshore transport pathways over the continental shelf, which are related to the structure of the interior geostrophic current and bottom Ekman transport, respectively. As a result of the vertical shear stress, an upwelling is induced above the bottom Ekman layer on the continental slope. The horizontal shear stress of the Kuroshio has the effect of inducing onshore transport at the flow core. The advection terms in the primitive equations are found to amplify the cross-shelf velocity solved from the linear equations. This study reveals that the Kuroshio has a substantial effect on the cross-shelf circulation and that it might drive multiple transport pathways.

Denotes content that is immediately available upon publication as open access.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yanzhou Wei, weiyanzhou@sio.org.cn

1. Introduction

Cross-shelf transport in the East China Sea (ECS) is important both to biogeochemical processes and to material exchange between local seas and the open ocean, and has therefore been studied extensively by oceanographers (Chen and Wang 1999; Isobe et al. 2004; Guo et al. 2006; Matsuno et al. 2009; Zhao and Guo 2011). Cross-shelf circulation is a complex process influenced by many factors, such as wind stress and along-shelf currents (Huthnance 1995; Lentz and Chapman 2004; Brink 2016). Under the influence of wind stress, cross-shelf circulation is characterized by a surface Ekman flow and a bottom compensation flow (Yanagi et al. 1996; Estrade et al. 2008). However, it is also affected by along-shelf currents (Hsuesh and O’Brien 1971; Liu and Su 1990; Lozier and Gawarkiewicz 2001; Zhou et al. 2015; Zhang et al. 2017) and regulated by frontal eddies (Sugimoto et al. 1988; Pollard and Regier 1992; Isobe et al. 2004; Isobe and Beardsley 2006; Matsuno et al. 2009), such that it displays complex variability over a wide range of spatial and temporal scales.

The horizontal pattern of cross-shelf transport in the ECS has been well documented. The Kuroshio intrudes onshore in areas northeast of Taiwan and southwest of Kyushu (Japan) (Isobe 2000; Guo et al. 2006; Liu and Gan 2012; Zhou et al. 2015; Ding et al. 2016; Zhang et al. 2017). However, ECS shelf water can also occasionally penetrate into the Kuroshio intermediate layer and then spread into the western Pacific Ocean with the Kuroshio, as suggested by chemical tracers and turbidity observed in the middle shelf of the ECS (approximately at the PN section; Fig. 1) (Isobe et al. 2004; Matsuno et al. 2009).

Fig. 1.
Fig. 1.

(a) Schematic maps of the Kuroshio, location of the PN section, climatological mean wind stress vectors (red arrows), and bottom velocity at the PN section (blue bars) in winter and (b) geostrophic velocity at the PN section in winter. In (a), black dots indicate positions of survey stations along the PN section, and color shading and numbered contours denote isobaths (m).

Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0204.1

The understanding of the vertical structure of the cross-shelf circulation within the ECS is still subject to large uncertainties, and associated published research is scarce. Ito et al. (1995) reported a snapshot of cross-shelf velocity based on direct acoustic Doppler current profiler (ADCP) observations, and found an intensive upwelling over the continental slope. Yanagi et al. (1996) diagnosed cross-shelf velocity at the PN section using primitive equations with the density nudged to the observations, and found a surface Ekman flow and a bottom compensation flow. However, no prior study has observed a more complex structure in the cross-shelf flow field in the ECS, which we present here.

Previous studies have demonstrated that western boundary currents can influence the cross-shelf circulation, creating more-complex structures. For instance, Schaeffer et al. (2013) and Archer et al. (2017) observed an offshore intermediate flow, and onshore surface and near-bottom flows, which they attributed to both the East Australian Current and surface wind stress forcing. The along-shelf current may influence cross-shelf circulation in two ways. First, the along-shelf current can drive upwelling of cold waters at the shelf break through bottom friction effects (e.g., Hsuesh and O’Brien 1971; Schaeffer et al. 2013; Archer et al. 2017). Second, the vertical gradient of along-shelf geostrophic current (which is equivalent to horizontal gradient of seawater density) can induce cross-shelf transport as formulated by Garrett and Horne (1978) and Garrett and Loder (1981) (see also section 3). Therefore, accounting for the Kuroshio shear stress, cross-shelf circulation in the ECS would likely display a more complex structure than that previously diagnosed (e.g., Yanagi et al. 1996), as demonstrated in this study.

A system of 2D differential equations (cross-shelf and depth directions) is often used to diagnose cross-shelf circulation by ignoring the along-shelf gradients (Garrett and Loder 1981; Wang 1984; Chen and Wang 1990). Following the equations utilized by Garrett and Loder (1981), this study solved the 2D steady-state nonlinear primitive equations to infer the effect of the Kuroshio shear stress on cross-shelf circulation. The Kuroshio shear stress is an important factor on cross-shelf circulation, especially in the deep ocean layer beyond the influence of wind stress. The equations of Garrett and Loder (1981) were originally written in a z-coordinate system; however, in this study, they were transformed to the σ-coordinate system for its convenience in dealing with boundary conditions over complex topography and solving the 3D velocities from the nonlinear momentum equations.

2. Data and methodology

a. Data

The PN section (Fig. 1) is a well-known array of hydrographic observation that transects the Kuroshio in the ECS. It is surveyed routinely by the Japan Meteorological Agency. This section has been surveyed since 1955 with only sparse stations initially; since April 1987, the survey stations become dense with 16 regular stations being fixed for quarterly observation during October 1996–January 2010 (Fig. 1; Wei et al. 2013). Therefore, this study only uses temperature and salinity data acquired between April 1987 and January 2010. The 16 survey stations are not distributed uniformly along the PN line from PN-6 (126°E, 29°N) to PN-1 (128.25°E, 27.5°N); instead, they are concentrated on the continental slope where the topography is steep and the current is strong. One or two of these 16 stations were occasionally omitted during April 1987–May 1991; in such cases, the missing data were estimated from those at neighboring stations using a regression method (Wei et al. 2013). From the temperature and salinity data acquired along the PN section, the along-shelf geostrophic current can be obtained using an inverse technique (Wunsch 1978) based on property conservation, as illustrated and discussed in Wei et al. (2013, 2017). The resultant velocity and volume transport were found to agree well with observations (Wei et al. 2013, 2015). In particular, the negative bottom velocity over the continental slope (negative as in southwestward; Fig. 1) is consistent with previous observations (James et al. 1999; Nakamura et al. 2008).

The surface wind stress data were derived from QuikSCAT (version 4). The QuikSCAT wind speed vector was obtained from the SeaWinds scatterometer launched on 19 June 1999 on board the QuikSCAT satellite (Hoffman and Leidner 2005). Monthly wind speed data are available from July 1999 to November 2009 at a 0.25° × 0.25° spatial resolution. The wind stress is calculated from the 10-m wind speed using the bulk formula (τ1, τ2) = ρa × cd × |U| × (U1, U2), where ρa is the air density (ρa = 1.2 kg m−3), cd is the drag coefficient (cd = 0.0039/|U| for |U| < 3.0 m s−1, and cd = 0.0013 for |U| > 3.0 m s−1), U = (U1, U2), |U| is the 10-m wind speed, and (U1, U2) are its zonal and meridional components, respectively (Edson et al. 2013).

The objective of this study is to illustrate the effect of Kuroshio shear stress on cross-shelf circulation. The seasonal variability of the Kuroshio is weaker relative to its climatological mean structure (Wei et al. 2015). Therefore, this study diagnoses the cross-shelf circulation in winter only. The winter season is chosen here because the wind stress is strongest during this season, which can help to examine the relative effect of the Kuroshio shear stress in comparison to the strongest wind stress forcing. To remove signals from small-scale perturbations, the climatological mean data during boreal winter (December–February) was determined and used for the diagnostic analyses. The climatological density and geostrophic current at the PN section were derived from data acquired between April 1987 and January 2010. Geostrophic velocity data were determined in between each pair of adjacent stations (n = 16) along the PN section at a vertical resolution of 1 m. The climatological wind stress data at the PN line were derived using data acquired between July 1999 and November 2009.

b. 2D primitive equations

A 2D model is employed to diagnose cross-shelf velocity (Garrett and Loder 1981; Wang 1984). Assuming the fields are uniform along the direction of the Kuroshio y, the steady-state primitive equations can be written as follows:

  • The continuity equation is given by
    e1
  • the momentum equations are given by
    e2
    e3
  • and the hydrostatic equation is given by
    e4
where x denotes the cross-shelf direction (seaward positive), z denotes the vertical direction (upward positive), and u, υ, and w denote the cross-shelf, along-shelf, and vertical velocities, respectively. Here, P represents the seawater pressure, f is the Coriolis parameter, ρ denotes the seawater density, and g denotes gravitational acceleration. The AV and AH represent the vertical and horizontal eddy viscosity coefficients, respectively. Finally, υg denotes the geostrophic component of the along-shelf velocity that is equal to , where ρ0 denotes the mean value of seawater density. In this study, υg takes the climatological mean geostrophic current calculated using the inverse method by Wei et al. (2013) during boreal winter.

The boundary conditions are as follows:

  • at the surface,
    e5
  • at the bottom,
    e6
  • and at the lateral boundaries,
    e7
where τx and τy denote the wind stresses in the cross-shelf and along-shelf directions, respectively. The wind stresses are taken at each mooring location as interpolated between nearest grid points. The cross-shelf velocity can be solved from the momentum equations and then used to solve the vertical velocity from the continuity equation.

Although the z-coordinate system is simple, it is inconvenient when dealing with complex bathymetry and solving nonlinear momentum equations. The σ-coordinate system, which incorporates a free surface and irregular bottom topography, can elegantly avoid these problems, and hence it is utilized here. We denote , is the surface height elevation, H is the bottom topography, and D is the depth of the water column. Let , and then and . Then, to any scalar field S, it has
eq1
eq2
Insert this relationship into the right-hand side of the momentum equations, and drop the asterisk; denote , , , and x = (1/L)x, where L denotes the length of the PN line. Similarly, Vg(x, σ) = υg(x, z). Then the primitive equations can be written as follows:
e8
e9
e10
Now, let , and insert them into Eqs. (8)(10) as unknown variables. Then the above equations are changed to the following:
e11
e12
e13

The boundary conditions now become the following:

  • at the surface,
    e14
  • at the bottom,
    e15
  • and at the lateral boundaries,
    e16

The vertical velocity field in the z-coordinate system is .

Note that the above derivations assume that the eddy viscosity coefficients are constant; if they display spatial variations, the equations must be written in different forms (see the appendix). The terrain-following coordinate model can adequately simulate the bottom boundary layer, which is of great benefit to oceanographers; however, it has a problem of pressure gradient error (Mellor et al. 1994, 1998). This can produce detectable velocity errors in the case of an initially horizontal homogeneous density field, which should produce zero velocities in theory (Mellor et al. 1994, 1998). In the present study, that problem is avoided because the pressure-gradient-related velocity field υg is initially derived in the z-coordinate system and then transformed into the σ-coordinate system.

c. Eddy viscosity

Eddy viscosity characterizes the diffusion of velocities; hence, it is also known as momentum eddy diffusivity. It varies with background ocean conditions such as stratification and current shear, and has large uncertainties in the ocean. Few studies have attempted to estimate the eddy viscosity in the ECS, and available estimations have been limited to vertical eddy viscosity through analysis of velocity spirals in the bottom Ekman layer (Yoshikawa and Endoh 2015). Previous studies have often adopted different values for eddy viscosity. Garrett and Horne (1978) took AH = 10 m2 s−1 and AV = 0.001 m2 s−1; Chen and Wang (1990) took AH = 50 m2 s−1 and estimated AV from a mixed layer model; Yanagi et al. (1996) took AH = 10 m2 s−1 and AV = 0.0001–0.001 m2 s−1. In textbooks, values for horizontal eddy viscosity range from 10 to 1000 m2 s−1, while those for vertical eddy viscosity vary from 0.0001 to 0.1 m2 s−1 (Pedlosky 1987; Cushman-Roisin and Beckers 2011). Therefore, AH might vary from 10 to 100 m2 s−1, and AV might vary from 0.0001 to 0.01 m2 s−1. The following analysis adopts AH = 10 m2 s−1 and AV = 0.001 m2 s−1 in accordance with Yanagi et al. (1996), who studied the same area as the present study.

As eddy viscosity in the ocean is difficult to measure directly, it is often approximated using different schemes (e.g., James 1984; Guo and Yanagi 1998), such as the Munk–Anderson scheme (e.g., Munk and Anderson 1948) or the turbulence closure model (Mellor and Yamada 1982). These schemes account for the influence of background stratification and current shear. This study employs the Munk–Anderson scheme, which expresses the vertical eddy viscosity coefficient in the following form:
e17
e18
where A0 = 0.001 m2 s−1 and denotes the background viscosity, Ri represents the Richardson number, N is the buoyancy frequency, and ∂υ/∂z denotes along-shelf current shear. As can be seen from Fig. 2, N2 is on the order of approximately 10−4 s−2, and is on the order of approximately 10−5 s−2. To let N2 and both enter into effect, ε is set to 0.1. The vertical eddy viscosity coefficient derived from this scheme lies within the range 10−4 to 10−3 m2 s−1 (Fig. 2c), consistent with empirical knowledge (Yanagi et al. 1996).
Fig. 2.
Fig. 2.

(a) Squared buoyancy frequency N2, (b) squared vertical shear of cross-sectional velocity , and (c) vertical eddy viscosity coefficient derived using the Munk–Anderson scheme.

Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0204.1

3. Results

a. Preliminary results derived in z-coordinate system

Some interesting relationships can be obtained if assuming small values of the Rossby number and the Ekman number (Garrett and Horne 1978; Garrett and Loder 1981). In this way, the momentum equations can be written as follows:
e19
e20
Denote
e21
then u = ∂φ/∂z and w = −∂φ/∂x, where φ(x, z) is the streamfunction of the cross-shelf velocity (u, w), and ϕ(x) can be any function of x. This relationship is adopted from Garrett and Horne (1978) and Garrett and Loder (1981) to depict the effect of vertical eddy viscosity and density gradients on cross-shelf velocity. It should be noted that this equation is incorrect in the surface and bottom Ekman layers, as pointed out by Garrett and Loder (1981).
The term in Eq. (21) illustrates the effect of the density structure on the interior cross-shelf circulation. It is simple to taper the to zero at the bottom by incorporating a depth-independent function . Their sum is denoted by . Let ; then Eq. (21) can be written as
e22
where ϕ0(x, z) reflects the depth-dependent term and ϕ0(x, z)|D = 0, and the term ∂D denotes the top of the bottom Ekman layer. That means there is no flow perpendicular to ∂D in the velocity field determined from ϕ0(x, z). The term ϕE(x) is a function independent of depth, which can be obtained from
e23
where wE is the vertical velocity at ∂D (e.g., Cushman-Roisin and Beckers 2011). For simplicity, the eddy viscosity coefficients are set to AV = 0.001 m2 s−1 and AH = 10 m2 s−1 in this subsection.

Figure 3 shows ϕ0(x, z) and the associated velocity fields (u0, w0) (Fig. 3a), as well as ϕE(x) and the associated vertical velocity wE (Fig. 3b). Multiple cells are found in the cross-shelf streamfunction (Fig. 3a), and an upwelling is evident at the bottom (Fig. 3b). These results suggest that density structure, bottom Ekman suction, and topography all play important roles in cross-shelf circulation.

Fig. 3.
Fig. 3.

(a) Streamfunction (colored shading) and cross-shelf velocity field (arrows) at the PN section, as induced by the Kuroshio vertical shear stress, assuming that no flow penetrates through the bottom Ekman layer. (b) Vertical velocity at the top of the bottom Ekman layer, as induced by the bottom along-shelf geostrophic current.

Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0204.1

Note that the streamfunction φ(x, z) derived in Eq. (22) can be used only to explain the cross-shelf circulation near the bottom. Near the surface, the vertical velocity derived from this equation cannot satisfy the boundary condition w = 0. The following subsection discusses how to modify the vertical velocity w to make it satisfy the boundary conditions. The streamfunction helps illustrate the cross-shelf circulation because a local maximum corresponds to an anticlockwise circulation, while a local minimum corresponds to a clockwise circulation. As evident, the cross-shelf circulation consists of multiple cells. It has an anticlockwise circulation centered at 127.2°E and 300-m depth and a clockwise circulation over the shelf break at station PN-4. This illustrates how complex the cross-shelf circulation can be induced by the Kuroshio shear stress.

If the advection terms in the momentum equations [Eqs. (2) and (3)] are ignored, the equations are linear, and so the cross-shelf velocity can be obtained easily using a numerical method. There are 15 horizontal grid points in the middle of two adjacent stations, the same as υg, and the vertical interval is 5 m. Here, the viscosity terms are discretized using the second-order central difference. The boundary conditions at the surface and the two lateral boundaries enter into effect as explicit terms when discretizing these second-derivative terms. As there are two corresponding difference equations at each grid point, the number of unknowns is the same as that of the equations. The discretized form of Eqs. (2) and (3) can be written as
e24
where matrix depends on the discretization scheme and is invertible. Here, v, u are two column vectors that denote the unknown velocities at each grid point, and f(τ) is a column vector that represents the effect of wind stress, which only has nonzero values that equal τ/ρ0fδz (where δz is the vertical interval distance, i.e., 5 m) at the surface grid point. Thus, the along-shelf and cross-shelf velocities can be solved by
e25

Figures 4a–c show the cross-shelf velocities induced by the vertical shear stress [i.e., ], horizontal shear stress [i.e., ], and their sum [i.e., ], respectively. As can be seen from Fig. 4a, the vertical shear stress of the along-shelf current can induce an offshore transport over the continental shelf, which can extend to the Kuroshio intermediate layer at a depth of approximately 200 m. Figure 4b suggests that the horizontal shear stress can induce an onshore transport over PN (d′–c) and an offshore transport over PN (4–d′). Figure 4c suggests that the cross-shelf velocity can display more complicated structure under their net effect.

Fig. 4.
Fig. 4.

Cross-shelf velocity derived by calculating (a) , (b) , and (c) ; and that velocity derived inversely from solving the momentum equations with the effects of (d) vertical eddy viscosity, (e) vertical eddy viscosity and wind stress, and (f) vertical and horizontal eddy viscosity and wind stress.

Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0204.1

Figures 4d–f show the cross-shelf velocities derived by solving the momentum equations inversely [Eq. (25)] with consideration of 1) vertical eddy viscosity, 2) wind stress and vertical eddy viscosity, and 3) wind stress and both horizontal and vertical eddy viscosities, respectively. The solutions derived inversely (Figs. 4d–f) are smooth and clean relative to those derived from direct calculation (Fig. 4c) and have two distinct Ekman layers at the surface and bottom (Fig. 4f). The results shown in Figs. 4a and 4d suggest that the vertical shear stress of the Kuroshio has the effect of inducing an onshore transport over the left side of the Kuroshio [PN (6–c)] and an offshore transport at the right side [PN (c–1)] at the surface, if there is no wind stress effect. This is because υg is a concave (convex) function with respect to depth z (upward positive) on the left (right) side of the Kuroshio at the surface (Fig. 1b). When the wind stress is incorporated, an Ekman layer is seen at the surface (Fig. 4e).

b. Solutions derived in the σ-coordinate system without advection terms

The 3D velocities derived in the σ-coordinate system in this subsection ignored the advection terms. There are 30 grid points in horizontal direction and 60 grid points in the vertical direction in the σ-coordinate system. The values of υg, initially recorded at 1-m interval in the vertical direction for each pair of adjacent stations in the z-coordinate system, were interpolated to these grid points in the σ-coordinate system.

The importance of full transformation of the horizontal viscosity term [Eqs. (9) and (10)] is addressed first. From Eq. (10), the cross-shelf velocity can be approximated as follows:
e26
The contributions of the five components in Eq. (26) are calculated and shown in Fig. 5 (AV = 0.001 m2 s−1 and AH = 10 m2 s−1 in this calculation). It is evident that the vertical viscosity term has the effect of inducing an offshore transport over the continental shelf [at PN (5–4)], similar to that derived in the z-coordinate system (Fig. 4a). Furthermore, the first term in the horizontal viscosity term [see Eq. (26)] has the effect of inducing an onshore transport at PN (d′–c), similar to that derived in the z-coordinate system (Fig. 4b). The other three terms also have substantial effects on cross-shelf velocity (Figs. 5c–e), especially in areas of steep topography and large along-slope velocity shear (Fig. 1b). The second term can induce positive cross-shelf velocity over the continental slope at the depth of approximately 200–300 m. The third term can induce negative cross-shelf velocity over the continental slope (Fig. 5d), while the fourth term can induce positive cross-shelf velocity at the continental slope (Fig. 5e). Therefore, full transformation of the horizontal diffusion term needs to be used here. The magnitude of cross-shelf velocity induced by these three terms is on the order of approximately (10−3 m s−1) and becomes small at the surface as γ approaches zero. The sum of the vertical viscosity term (Fig. 5a) and horizontal viscosity term (Figs. 5b–e) is shown in Fig. 5f. It is evident that the cross-shelf velocity induced by vertical and horizontal stresses derived in the σ-coordinate system coincides well with that derived in the z-coordinate system (Fig. 4c).
Fig. 5.
Fig. 5.

Cross-shelf velocity contributed by (a) the vertical viscosity term in the σ-coordinate system and the four components of horizontal viscosity term (b) , (c) , (d) , and (e) expressed in the σ-coordinate system, and (f) their sum.

Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0204.1

In the following, the 3D velocities are solved. The vertical eddy viscosity coefficient AV takes the result estimated from the Munk–Anderson scheme to consider its dependence on background ocean conditions, while AH remains constant (10 m2 s−1). Similar to the procedure used in the z-coordinate system, the nonlinear advection terms are ignored temporarily. Then, the momentum equations [Eqs. (12) and (13)] can be discretized and expressed in the form
eq3
where matrix depends on the discretization scheme and is invertible, and represents the effect of wind stress. Then,
eq4
After the cross-shelf velocity is solved, the vertical velocity can be calculated using the continuity equation [Eq. (11)]:
e27
Note that the vertical velocity calculated using this equation is incompatible with both the surface and the bottom boundary conditions simultaneously, because is not satisfied. This is an inherent drawback of the steady-state 2D model. Therefore, some corrections are necessary on the calculated in Eq. (27). The full form of the continuity equation [Eq. (11)] is , which means Eq. (11) has larger uncertainty in the upper layer because the magnitude of is larger. Therefore, less credit is given to the vertical velocity calculated in the upper layer using Eq. (27), by adopting the following simple approach:
e28
Thus, the obtained vertical velocity can satisfy the surface and bottom boundary conditions simultaneously and clearly illustrates the structure in the deep layer at the expense of loss of precision in the upper layer.

The 3D velocities calculated in the σ coordinate are presented in Figs. 6a–c. The wind stress induces an Ekman transport that is onshore at the surface. The negative velocity in the initial field Vg in the bottom layer (Fig. 1) induces an offshore current to balance the strong velocity shear (Fig. 6b). The amplitude of the cross-shelf velocity in the bottom Ekman layer depends on both the strength of the bottom along-shelf geostrophic current and the vertical eddy viscosity coefficient. Along with this offshore transport pathway, there is another offshore transport pathway located in the intermediate layer over the continental shelf (as indicated by the two arrows in Fig. 6b), which is related to the interior structure of density {because from Eq. (21)}. Onshore transport appears mainly in the position of the flow core and is contributed by the horizontal shear stress of the Kuroshio. The vertical velocity is shown in Fig. 6c. A clear upwelling appears on top of the bottom Ekman layer at the continental slope, which is consistent with the result shown in Fig. 3b. Both the bottom offshore transport and the upwelling are induced by the negative bottom along-shelf velocity over the slope (James et al. 1999; Nakamura et al. 2008; Fig. 1). This upwelling is essential in moving the cold, nutrient-rich water upward and in sustaining the uplift isopycnals.

Fig. 6.
Fig. 6.

(a) Along-shelf, (b) cross-shelf, and (c) vertical velocities estimated in the σ-coordinate system by taking into account both horizontal and vertical eddy viscosities. (d) Along-shelf and (e) cross-shelf velocity difference [relative to that derived from the linear momentum equations in (a) and (b)] solved from nonlinear momentum equations. (f) Cross-shelf velocity derived from nonlinear equations. Green arrows in (b) denote two offshore transport pathways.

Citation: Journal of Physical Oceanography 48, 7; 10.1175/JPO-D-17-0204.1

c. Effects of advection terms

To solve the nonlinear equations, the advection terms are approximated by linear terms. For example, the multiple of two terms uυ/∂x can be approximated by . Initial fields (u0, υ0, w0) are obtained by solving the linear equations, as illustrated above. From the initial velocity field (u0, υ0), the new velocity (u1, υ1) can be solved using the linearized momentum equations. Note that in this process the vertical velocity retains its initial value. The derived (u1, υ1) are then modified to (u1, υ1) = (u1, υ1)/2 + (u0, υ0)/2 and used to update the linearized momentum equations to solve the new velocities (u2, υ2), which are similarly modified to (u2, υ2) = (u2, υ2)/2 + (u1, υ1)/2 and used to update the momentum equations for the next step. This operation is run iteratively until the velocity field reaches a stable state.

Note the importance of updating the solution slowly [i.e., ]; otherwise the derived solution would “blow up” after a few steps. This method is similar to that used in solving a simple equation [i.e., x2 = a (a > 0)], which can be approximated to x0x = a. Starting from an initial value x0 (≠0), the solution at the next step should be x1 = (1/2)(x0 + a/x0) instead of x1 = a/x0; otherwise the ultimate solution cannot be obtained.

After 20 iterations, the solution can reach a stable state; that is, there is less than 10−5 m s−1 difference between the results of step 20 and the previous step. The effects of advection terms on the along-shelf and cross-shelf velocities are evident in the surface Ekman layer and in the region over the continental slope (Figs. 6d and 6e). This can be explained as follows: From the equation , the amplitude of the cross-shelf current decreases (relative to that derived without the advection terms) in areas where ∂υ/∂x and wυ/∂z are positive. However, the cross-shelf transport pathways are almost unchanged by the effects of the advection terms (Fig. 6f), compared with those derived using the linear equations (Fig. 6b).

In summary, the cross-shelf velocity is affected by the vertical eddy viscosity, horizontal eddy viscosity, and advection terms. It is found that the vertical shear stress of the Kuroshio can induce two offshore transport pathways (Fig. 4e). One can extend from the shallow sea to the Kuroshio intermediate layer (Fig. 4e); this path is related to the interior structure of density {remember, }. The second transport pathway can extend from the shallow sea to the Okinawa Trough, following a path confined to a thin layer over the topography; this path is related to the bottom Ekman transport induced by the negative geostrophic current near the bottom. The first pathway can be diagnosed directly from in situ hydrographic observations, while the second one requires an adequate geostrophic current near bottom. The horizontal shear stress of the Kuroshio can induce an onshore transport at the flow core (Fig. 4b). The advection terms tend to amplify the cross-shelf velocity induced by the Kuroshio shear stress (Figs. 6b and 6f). This study has shown that the effects of the vertical and horizontal shear stresses of the Kuroshio and of the advection terms are comparable, when eddy viscosity coefficient values are taken as in the present study.

4. Discussion

Previous studies of cross-shelf exchange in the ECS have often concentrated on the transport across the 200-m isobath (Guo et al. 2006; Liu and Gan 2012; Zhou et al. 2015; Ding et al. 2016; Zhang et al. 2017), with little attention paid to the cross-shelf circulation transecting the isobaths (Ito et al. 1995; Yanagi et al. 1996). However, the latter plays a fundamental role in material exchange between the shelf sea and the deep ocean. Cross-shelf circulation might have been less studied because of the following factors. First, transport across the 200-m isobath is often considered sufficiently representative for most studies, because the transport that affects the shelf sea must pass the 200-m isobath. Second, the cross-shelf circulation transecting the isobaths is much more difficult to extract because it is weak and often masked by noise. Actually, cross-shelf circulation is influenced by many factors, and is often studied with some simplifications (Brink 2016). This study examined the cross-shelf circulation induced by the Kuroshio shear stress. Despite inevitable uncertainties, the results provide new insights on the cross-shelf circulation transecting the Kuroshio.

a. Uncertainties

The derived cross-shelf circulation might be biased because of the neglect of along-shelf pressure gradients, which determine the cross-shelf geostrophic velocity. Certainly, the Kuroshio shear stress can combine with the along-shelf pressure gradients to affect cross-shelf exchange. However, except for regions northeast of Taiwan and southwest of Kyushu, there is no evident cross-shelf geostrophic velocity, as inferred from the absolute dynamic topography data (Liu and Gan 2012). Therefore, it is difficult to resolve the cross-shelf geostrophic velocity at the PN section adequately. Using satellite altimetry data and climatological hydrographic data to estimate cross-shelf geostrophic velocity at the PN section cannot provide favorable results. First, the usefulness of altimetry data in the continental area must be considered carefully, while the use of climatological hydrographic dataset can underestimate the Kuroshio geostrophic velocity (with zero reference velocity at bottom) considerably, as highlighted by Ding et al. (2016). Second, there is the problem of angle error; that is, a small error in defining the true along-shelf direction can lead to substantial changes in cross-shelf velocity (Brink 2016). Therefore, the effects of along-shelf pressure gradients cannot yet be adequately evaluated.

The cross-shelf circulation induced by the Kuroshio shear stress is also sensitive to the eddy viscosity coefficients. These coefficients are the least known variables in the ocean, because of the difficulty of sustaining turbulence observations. Estimates of the vertical eddy viscosity are limited to the bottom Ekman layer through analysis of observed velocity spirals (Yoshikawa and Endoh 2015). Although a scheme for estimating turbulent eddy viscosity is employed to account for its dependence on the background conditions, the extent to which this can resolve the eddy viscosity value and structure at the PN section remain unclear. Bryden (1982) found that the horizontal eddy viscosity at the Gulf Stream was −900 m2 s−1, where the negative eddy viscosity indicated that energy was transferred from small scales to large scales. However, the validity of this value has not been confirmed in the other regions. In most previous studies (e.g., Garrett and Horne 1978; Chen and Wang 1990; Yanagi et al. 1996), researchers often chose the values of AH and AV within the ranges 10 to 50 and 0.0001 to 0.01 m2 s−1, respectively. This study adopted values of AH =10 m2 s−1 and AV = 0.001 m2 s−1 in accordance with Yanagi et al. (1996). Henceforth, the derived cross-shelf velocity might have an uncertainty of one order of magnitude.

The cross-shelf velocity derived in this study is on the order of 0.1 cm s−1 given the eddy viscosity AV (AH) = 0.001 (10) m2 s−1. It might increase to the order of 1.0 cm s−1 if the eddy viscosity AV (AH) = 0.01 (100) m2 s−1, with the velocity structure kept almost unchanged. In general, the cross-shelf velocity diagnosed in this study is mainly determined by the linear combination of vertical and horizontal viscosity terms and is amplified by incorporation of the advection term. Only when the eddy viscosity coefficients are especially small (less than the values used in this study), could the advection term dominate the velocity structure, but such small viscosity values are rarely used in the present study area.

b. Differences from previous studies

Few previous studies have addressed the cross-shelf circulation at the PN section because it is difficult to measure and often masked by noise. Ito et al. (1995) reported a snapshot of the cross-shelf velocity and vertical velocity at the PN section, as observed by a towed ADCP. They found an intensive upwelling that could be >2.0 cm s−1, which they attributed to Kuroshio frontal waves. The vertical velocity derived in this study is 0.01 cm s−1, that is, considerably less than that reported by Ito et al. (1995). This is because the upwelling was induced by Kuroshio frontal waves in Ito et al. (1995), whereas it was induced by bottom Ekman transport in this study. In addition to frontal waves, onshore and offshore migration of the Kuroshio also has the potential to influence cross-shelf circulation, as does the case in relation to the East Australian Current reported by Schaeffer et al. (2013) and Archer et al. (2017).

The present results differ from those of Yanagi et al. (1996), who examined cross-shelf circulation at the PN section. Their results showed that cross-shelf circulation was dominated by a wind-driven circulation that had a surface Ekman transport and a bottom compensation flow. Their results likely underestimated the effect of the Kuroshio shear stress. In contrast to Yanagi et al. (1996), we isolated the effect of the Kuroshio shear stress on cross-shelf circulation, establishing that it might drive multiple offshore transport pathways.

Note that the analyses employed here were based on the climatological mean geostrophic current in winter, which differs from using a single cruise study that might be influenced by the Kuroshio migration and frontal waves (Sugimoto et al. 1988; Pollard and Regier 1992; Ito et al. 1995; James et al. 1999; Isobe and Beardsley 2006). The extent to which frontal waves can contribute to cross-shelf circulation from the perspective of the climatological mean remains unclear, and it should be investigated in the future using a high-resolution prognostic 3D model.

5. Conclusions

This study solved the steady-state nonlinear 2D primitive equations to infer the cross-shelf circulation induced by the shear stresses of the Kuroshio. The cross-shelf transport induced by the Kuroshio exists in the deep ocean, such that it has a substantial effect on the cross-shelf transport of nutrients, which are rich in the deep layer. The cross-shelf circulation can display multiple cells (Fig. 3a) under the influence of the Kuroshio, different from the single cell structure (a surface Ekman flow and a bottom compensation flow) induced purely by wind stress. These results highlight the importance of the Kuroshio to cross-shelf circulation. The results derived in this study could also be used to test the cross-shelf transport simulated by ocean models.

Acknowledgments

The author is grateful to two anonymous reviewers for their valuable suggestions and comments. The hydrographic data at the PN section were downloaded from the Japan Oceanographic Data Center (JODC; http://www.jodc.go.jp). QuikSCAT data were produced by Remote Sensing Systems and sponsored by the NASA Ocean Vector Winds Science Team (and are available at www.remss.com). This study is partially supported by Scientific Research Fund of the Second Institute of Oceanography, SOA (Grant JB1808), and the National Natural Science Foundation of China (Grants 41730535, 41621064, and 91528304).

APPENDIX

Derivation of the σ-Coordinate Equations

When eddy viscosity coefficients display spatial variation, the momentum equations should be written in a different form. Denote ψ = σ(∂D/∂x); then to any scalar field S, there is
eq5
Then, Eqs. (8)(10) are revised to the following:
ea1
ea2
ea3
Now, denoting and inserting them into the above equations as unknown variables, Eqs. (A1)(A3) are changed to the following:
ea4
ea5
ea6

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