Editorial Type:
Article
Article Type:
Research Article

On the Pathways of Wind-Driven Coastal Upwelling: Nonlinear Momentum Flux and Baroclinic Instability

Dou Li aDepartment of Earth and Environment, Boston University, Boston, Massachusetts

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https://orcid.org/0000-0001-7293-2577
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Xiaozhou Ruan aDepartment of Earth and Environment, Boston University, Boston, Massachusetts

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Online Publication:
19 Dec 2023
Print Publication:
01 Jan 2024
DOI:
https://doi.org/10.1175/JPO-D-23-0098.1
Page(s):
63–79
Free access

Abstract

Wind-driven upwelling of cold, nutrient-rich water is a key feature near the eastern boundaries of major ocean basins, with significant implications for the local physical environment and marine ecosystems. Despite the traditional two-dimensional description of upwelling as a passive response to surface offshore Ekman transport, recent observations have revealed spatial variability in the circulation structures across different upwelling locations. Yet, a systematic understanding of the factors governing the spatial patterns of coastal upwelling remains elusive. Here, we demonstrate that coastal upwelling pathways are influenced by two pairs of competing factors. The first competition occurs between wind forcing and eddy momentum flux, which shapes the Eulerian-mean circulation; the second competition arises between the Eulerian-mean and eddy-induced circulation. The importance of nonlinear eddy momentum flux over sloping topography can be described by the local slope Burger number, S = αN/f, where α is the topographic slope angle and N and f are the buoyancy and Coriolis frequencies. When S is small, the classic coastal upwelling structure emerges in the residual circulation, where water upwells along the sloping bottom. However, this comes with the added complexity that mesoscale eddies may drive a subduction route back into the ocean interior. As S increases, the upwelling branch is increasingly suppressed, unable to reach the surface and instead directed offshore by the eddy-induced circulation. The sensitivity of upwelling structures to variable wind stress and surface buoyancy forcing is further explored. The diagnostics may help to improve our understanding of coastal upwelling systems and yield a more physical representation of coastal upwelling in coarse-resolution numerical models.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dou Li, douli@bu.edu

Abstract

Wind-driven upwelling of cold, nutrient-rich water is a key feature near the eastern boundaries of major ocean basins, with significant implications for the local physical environment and marine ecosystems. Despite the traditional two-dimensional description of upwelling as a passive response to surface offshore Ekman transport, recent observations have revealed spatial variability in the circulation structures across different upwelling locations. Yet, a systematic understanding of the factors governing the spatial patterns of coastal upwelling remains elusive. Here, we demonstrate that coastal upwelling pathways are influenced by two pairs of competing factors. The first competition occurs between wind forcing and eddy momentum flux, which shapes the Eulerian-mean circulation; the second competition arises between the Eulerian-mean and eddy-induced circulation. The importance of nonlinear eddy momentum flux over sloping topography can be described by the local slope Burger number, S = αN/f, where α is the topographic slope angle and N and f are the buoyancy and Coriolis frequencies. When S is small, the classic coastal upwelling structure emerges in the residual circulation, where water upwells along the sloping bottom. However, this comes with the added complexity that mesoscale eddies may drive a subduction route back into the ocean interior. As S increases, the upwelling branch is increasingly suppressed, unable to reach the surface and instead directed offshore by the eddy-induced circulation. The sensitivity of upwelling structures to variable wind stress and surface buoyancy forcing is further explored. The diagnostics may help to improve our understanding of coastal upwelling systems and yield a more physical representation of coastal upwelling in coarse-resolution numerical models.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dou Li, douli@bu.edu
Keywords: Baroclinic flows; Coastal flows; Ekman pumping/transport; Nonlinear dynamics; Upwelling/downwelling

1. Introduction

Coastal upwelling occurs when winds blow parallel to the coastline, deflecting the surface water offshore in the ocean’s surface Ekman layer (Ekman 1905; Sverdrup 1937). This process brings up cold, nutrient-rich subsurface water to the sunlit surface layers, enhancing local primary production and fishery yields, making coastal upwelling systems some of the most biologically productive regions in the world’s oceans (e.g., Pauly and Christensen 1995; Falkowski et al. 1998). Despite significant progress made in recent decades, our understanding of coastal upwelling systems, especially of their complex spatial structures, remains incomplete, leading to uncertainties in predicting their future behavior in the face of rapid climate change.

Wind-driven coastal upwelling has been studied extensively in two-dimensional settings where connections have been established between the rate of cross-shore exchange and various environmental factors such as the coastal stratification, local bathymetry, and wind stress and its curl (Huyer 1983; Bakun 1990; Allen et al. 1995; Lentz and Chapman 2004; Jacox and Edwards 2011; Jacox et al. 2018). One of the key and unresolved aspects of coastal upwelling lies in the position of the subsurface return pathway that replenishes the surface offshore Ekman transport and upwells nutrient-rich water toward the surface. Using a compilation of observations at different upwelling sites globally, Lentz and Chapman (2004, hereinafter LC04) noticed that some of the return flows are along the sloping seafloor, while some reside in the ocean interior away from the bottom. The location of the return pathway has important implications for the coastal ecosystems as the flow along the bottom brings significant nutrients into the euphotic zone due to their proximity to bottom sediments (Johnson et al. 1999; Bruland et al. 2001).

LC04 went a step further and proposed a theoretical framework to determine the location of the return pathway. The depth-integrated alongshore momentum balance for wind-driven coastal upwelling can be written as
∂∂t∫−h0udz+∂∂y∫−h0uυdz=−gh∂η∂x+τsx−τbxρ0.
Here, x and y denote the horizontal alongshore and cross-shore directions with y pointing to deeper water (Fig. 1), u and υ are the along- and cross-shore velocities, h marks the depth of sloping topography, η is the sea surface height, τsx and τbx are the surface and bottom stress in the alongshore direction, and ρ0 is a reference density. The term involving the Coriolis frequency disappears as the depth-integrated cross-shelf velocity is negligible. This expression states that at steady state and in the absence of alongshore asymmetries, there is a balance among the surface and bottom stress, as well as the vertically integrated divergence of cross-shore momentum flux [the second term on the left-hand side of Eq. (1)].
Fig. 1.
Fig. 1.

Model setup and snapshots of basic simulation variables. The Rossby number Ro is shown at the surface (z = 0), together with the alongshore velocity (u) at x = 300 km and cross-shore velocity (υ) at y = 320 km. The snapshots are taken on day 300 from run 1 (Table 1). Gray contours indicate isopycnals with an interval of 0.0034 m s−2. A steady, uniform upwelling-favorable wind stress is applied at the surface, driving an upwelling front. The yellow shadow zone denotes the location of the sponge layer.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

Through analytical formulations in two dimensions, LC04 concluded that the relative proportion of the return transport in the ocean interior and in the bottom Ekman layer is dictated by the ratio of the depth-integrated divergence of cross-shore momentum flux to the wind stress (further elaborated in section 3b); this ratio is ultimately determined by the magnitude of the local slope Burger number S = αN/f, where α is the topographic slope angle and N and f are the buoyancy and Coriolis frequencies. For S ≪ 1, the leading-order depth-integrated momentum balance is achieved between the bottom stress and the wind stress, and the onshore return flow primarily takes place in the bottom Ekman layer. As S increases, the bottom stress becomes weaker, and the wind stress is increasingly balanced by the cross-shore momentum flux divergence term, leading to a stronger onshore return flow in the ocean interior. The theoretical prediction agrees qualitatively well with two-dimensional idealized simulations. Unfortunately, two-dimensional simulations suffer from unphysical processes such as the offshore drift of the boundary front/current (e.g., Allen et al. 1995; Austin and Lentz 2002; Jacox and Edwards 2011) and will eventually evolve to a state with homogenized water column due to a lack of restratification mechanism. More importantly, as will be detailed below, the importance of baroclinic instability and its associated circulation is ignored in such settings, leaving the physical picture incomplete.

The contribution of (sub)mesoscale turbulence to the cross-shore exchange has increasingly been accounted for in recent numerical studies of coastal upwelling systems, both in idealized setups (e.g., Lathuilière et al. 2010; Brink 2016; Brink and Seo 2016; He and Mahadevan 2021; Thomsen et al. 2021; Moscoso et al. 2021) and in realistic regional simulations (e.g., Marchesiello et al. 2003; Durski and Allen 2005; Gruber et al. 2011; Colas et al. 2013; Nagai et al. 2015). These studies confirmed the presence of active baroclinic instability over the continental shelf/slope, which has a significant influence on the cross-shore transport. Similar to the “eddy compensation” effect that is widely accepted in the Southern Ocean literature (e.g., Marshall and Radko 2003; Hallberg and Gnanadesikan 2006), it has been found that eddy-induced circulation in the coastal ocean counteracts the wind-driven Eulerian-mean circulation. This counteracting effect limits the nutrient delivery to the euphotic zone since the transport of passive tracers, including nutrients, is determined by the residual circulation, which is the sum of the Eulerian-mean and eddy-induced circulation (Abernathey et al. 2011; Marshall and Speer 2012). For instance, realistic simulations of upwelling within the California Current System show that the eddy-induced circulation subducts and laterally exports nitrate from the coastal euphotic zone, which reduces the local biological productivity (Gruber et al. 2011; Nagai et al. 2015). Using idealized simulations, He and Mahadevan (2021) further demonstrated the control of eddy-induced circulation on the depth of the upwelled water (the “source depth”) in coastal upwelling systems. Thomsen et al. (2021) explored the sensitivity of eddy-mediated transport to surface buoyancy forcing in the presence of an alongshore pressure gradient and geostrophic cross-shore flow. While these studies demonstrated the potentially significant role of eddies in coastal upwelling processes, they are disconnected from the theoretical predictions of coastal upwelling pathways, especially regarding the Eulerian-mean circulation, based on the aforementioned integrated momentum budget, and this is exactly the motivation for our study.

Here, we address this major gap in our understanding of coastal upwelling systems by combining the fundamental constraints of the integrated momentum balance [Eq. (1)] with eddy-induced circulation. To achieve this, we conduct a series of controlled experiments using three-dimensional idealized simulations that go beyond the limitations of unsteady two-dimensional models. By examining the impact of variable slope Burger number S on three-dimensional upwelling structures, we can more accurately account for the influence of baroclinic instability while maintaining a simplified approach. Our findings highlight the significant role of baroclinic instability in shaping the coastal upwelling structure. It exerts a dominant influence not only on the eddy-induced circulation but also on the Eulerian-mean circulation through the divergence of eddy momentum flux. The paper is organized as follows: the model configuration and analysis methods are described in section 2. In section 3, we present the role of nonlinear momentum flux and eddy-induced circulation in determining the residual coastal circulation and their dependence on S. The sensitivity of the residual circulation to wind stress and surface buoyancy flux is also explored in section 3. A summary and discussion are provided in section 4.

2. Model setup and methods

a. Idealized model configuration

An idealized three-dimensional numerical model is designed using the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams 2005). The size of the computational domain is 300 km × 320 km in the alongshore (x) and cross-shore (y) directions with a horizontal resolution of 1 km (Fig. 1). The model utilizes a terrain-following vertical coordinate and contains a total of 30 levels, refined at the surface and bottom to ensure adequate resolution of the boundary layers. Over the midshelf, the vertical resolutions are 0.75 and 2.5 m near the surface and bottom and 10 m in the interior ocean; the vertical resolution becomes coarser moving offshore but generally around 1 and 3 m in the surface and bottom boundary layers. The bottom topography h(y), defined in Eq. (2), consists of a region with constant slope α (0 < y < 0.14/α km), a smooth transition zone (0.14/α ≤ y ≤ 0.29/α km) to avoid abrupt changes in slopes, and a flat bottom in the open ocean with a depth of 200 m (0.29/α < y < 320 km). The bottom topography is uniform in the alongshore direction:
h(y)=1000αy+10,0<y<0.14/αkm,
h(y)=200−50×exp[y−0.29/α0.15/α×log(50α)]−1150α−1,0.14/α≤y≤0.29/αkm, and
h(y)=200,0.29/α<y<320km.
A third-order upstream biased advection scheme is used for horizontal momentum advection and a multidimensional positive definite advection transport algorithm (MPDATA) scheme (Smolarkiewicz and Margolin 1998) is chosen for tracer advection. Vertical mixing is approximated using the Mellor–Yamada level-2.5 turbulence closure. The background diffusivity and viscosity are 3 × 10−5 m2 s−1 in the vertical direction and 3 m2 s−1 in the horizontal. Periodic boundary conditions are used in the alongshore direction. At the inshore (y = 0) and offshore (y = 320 km) boundaries, closed and no-slip boundary conditions are applied. A linear bottom stress is introduced at the seafloor with a friction coefficient r = 5 × 10−4 m s−1.

For simplicity, density is assumed to be a linear function of temperature only and all the model runs are initialized with a constant stratification. A random noise with an amplitude up to 0.001°C is introduced to the background temperature field. Temperature is restored toward the initial values in an offshore sponge layer between 280 and 320 km (marked by yellow in Fig. 1) with a restoring time decreasing from 50 days at y = 280 km to 0.1 day at y = 320 km. All model runs start from rest and are forced by a spatially uniform constant wind stress that is in the upwelling-favorable direction. A constant Coriolis frequency f = 1 × 10−4 s−1 is applied.

Twenty simulations with varying topographic slope angles, stratification, and surface forcing, typical for the upwelling systems, are listed in Table 1. The resulting value of S ranges from 0.5 to 2, which is typical for major upwelling sites globally. For reference, the Northern California shelf typically has an S value of around 0.5, the Oregon shelf around 1, and the Peru shelf around 1.5 (LC04). All the simulations are run for 500 days to ensure the establishment of a statistically steady state, and the results presented are averaged from day 300 to 500 to exclude the initial spinup period. To interpret the simulation results, we use the slope Burger number during the equilibrium state Seq, considering the substantial disparity in stratification between the initial and equilibrium states. The calculation of Seq utilizes a depth-averaged N, typically at a water depth of 150 m, unless otherwise noted.

Table 1.

Summary of the simulation parameters; α is the topographic slope angle, Nini is the initial buoyancy frequency, Sini = αNini/f is the initial slope Burger number, and Ld = NiniH/f is the first baroclinic Rossby radius of deformation (H = 100 m is used in the calculation). The upwelling and subduction percentage are calculated based on Eqs. (17) and (18) with a surface layer thickness δ = 30 m. The calculation of the subduction percentage is excluded in runs where the upwelling percentage is smaller than 5%. The source depth of surface water is calculated using Eq. (19) with δ = 30 m and averaged within a nearshore band that extends 50 km from the coast. The standard deviation is obtained by varying the surface layer thickness δ between 20 and 50 m and the width of the nearshore band between 10 and 60 km.
Table 1.

b. Energetics and streamfunction calculation

The coastal upwelling system features energetic boundary currents and flow instabilities. To illustrate the different energy pathways, we diagnose the kinetic energy and available potential energy, as introduced below. Specifically, we denote the alongshore and time averages as •¯=1/Lx∫0Lx•dx and ⟨•⟩=1/T∫t0t0+T•dt, respectively, where Lx = 300 km, T = 200 days, and t0 is day 300. The mean kinetic energy (MKE) and eddy kinetic energy (EKE) per unit mass are calculated as
MKE=12(u¯2+υ¯2) and
EKE=12(u′2+υ′2),
where the prime terms represent the deviation from the alongshore average. The calculation of available potential energy (APE) relies on a rearranged density field ρ*(z,t), in which all the fluid parcels in the model domain are rearranged adiabatically to a stable, horizontally uniform rest state (Winters et al. 1995; MacCready and Giddings 2016). Accordingly, APE can be calculated as
APE=g∫z*z[ρ−ρ*(z′)]dz′.
The source of EKE can be diagnosed by calculating the dominant terms in the alongshore-averaged EKE budget (Barth 1994; Hetland 2017):
∂∂tEKE=−u′υ′¯ u¯y−u′w′¯ u¯z+w′b′¯,
with the three terms on the right-hand side representing lateral and vertical shear instabilities and baroclinic instability. The sum of the two shear production terms indicates the energy conversion from the MKE to EKE.
To illustrate the transport of buoyancy and nutrients, we turn to the transformed Eulerian-mean framework and calculate the Eulerian-mean, eddy, and residual streamfunctions from the model output. All the streamfunctions are calculated when the statistically steady state is reached. Specifically, the Eulerian-mean streamfunction is calculated as
ψm=∫z0⟨υ¯⟩dz.
According to the alongshore momentum equation, the Eulerian-mean circulation can be influenced not only by surface wind forcing but also by mesoscale eddies, particularly through the effect of eddy momentum fluxes (e.g., Vallis 2017; Manucharyan and Isachsen 2019). Thus, ψm can be further decomposed into three components:
ψm=−τsxρ0f︸ψEk+1f∂∂y∫z0⟨u′υ′¯⟩dz︸ψem+1f∂∂y∫z0⟨u¯ υ¯⟩dz︸ψmm,
where ψEk represents the contribution from wind-driven Ekman pumping and ψem represents the contribution from the cross-shore eddy momentum flux. The third component, ψmm, stems from the mean cross-shore momentum flux and is typically ignored in quasigeostrophic frameworks. In this study, we keep ψmm for completeness, considering the importance of friction-driven boundary layers over a shallow water depth. We will show later that the eddy momentum flux, rather than the mean momentum flux, plays a dominant role in shaping the Eulerian-mean circulation in coastal upwelling systems, alongside the wind forcing.
In the presence of baroclinic instability, the eddy-induced circulation tends to flatten isopycnals and oppose the Eulerian-mean circulation. Its magnitude can be quantified with an eddy streamfunction ψeddy, which is typically defined as ψeddy=υ′b′¯/b¯z in the interior ocean where the vertical buoyancy gradient dominates (Andrews and Mcintyre 1976) and ψeddy=−w′b′¯/b¯y for the boundary layer with surface isopycnal outcropping (Held and Schneider 1999). Here, we adopt a combined form from Cerovečki et al. (2009) where a coordinate stretching factor ϵ is introduced to correct for the small horizontal and vertical buoyancy gradients:
ψeddy=ϵ2⟨υ′b′¯⟩⟨b¯⟩z−⟨w′b′¯⟩⟨b¯⟩y⟨b¯⟩y2+ϵ2⟨b¯⟩z2.
We used ϵ = 10−3 in our study, which is the same as in Mahadevan et al. (2010) and He and Mahadevan (2021). Finally, the residual streamfunction can be obtained by combining both the Eulerian-mean and eddy streamfunctions:
ψres=ψm+ψeddy.

c. Lagrangian float trajectory

In addition to using an Eulerian description to study upwelling dynamics, we also utilize an online Lagrangian tracking module to simulate the movement of neutrally buoyant floats in the coastal ocean. Two different sets of float releases are conducted in our simulations. In the first set of experiments, we release 240 floats along the outer edge of the slope (71 km ≤ y ≤ 74 km) on day 300, by which time the simulations had reached an equilibrium state. These floats are positioned at 10-km intervals in the alongshore direction and within the two bottom-most sigma layers. This serves two purposes: 1) to validate the calculation of residual streamfunction and 2) to provide a more direct visualization of coastal upwelling pathways. In the second set of simulations, we release a total of 3600 floats over a depth-cross-shore section at x = 0. This section covers all 30 vertical levels between y = 0 and y = 240 km. The second set of floats is used to determine the source depth of the upwelled water in the surface boundary layer.

3. Results

a. Spinup and energetics

We begin by examining the spinup adjustments and the associated energetics in two representative model runs, which provide the context for the general circulation patterns in two distinct dynamic regimes. These runs, labeled 1 and 4, share identical model parameters except for the initial stratification (Nini), resulting in two different slope Burger numbers, Sini = 0.5 and Sini = 1.2, as well as the baroclinic Rossby radius of deformation (Ld) (Table 1).

During the spinup period, a frontal current develops near the coast as momentum is transferred from the wind and divergence in the surface Ekman transport pulls isopycnals from depth, enhancing the lateral density gradient (Fig. 1). Consequently, both the MKE and APE increase rapidly with time (Figs. 2a,b). As the isopycnals steepen, the accumulated APE is released predominantly through baroclinic instability in both runs (Fig. 2d). This fuels the growth of finite-amplitude disturbances and local EKE (Fig. 2c). Both simulations reach a statistically steady state after day 200. The run with a stronger initial stratification takes longer to equilibrate and exhibits higher energy levels and energy conversion rates compared to the other run. Consistent with previous studies, our simulations confirm the existence of active baroclinic instability in the coastal upwelling zone (e.g., Marchesiello et al. 2003; Durski and Allen 2005; Gruber et al. 2011; Colas et al. 2013; Nagai et al. 2015). The Rossby number Ro, defined here as the ratio between the vertical component of relative vorticity and the Coriolis frequency (ζ/f), appears O(1) at the surface (Fig. 1), indicating submesoscale turbulence in the surface boundary layer. We have also conducted sensitivity analyses with different spatial resolutions to examine the relative importance of submesoscale and mesoscale processes. The results indicate that the primary circulation structures are mainly sustained by mesoscale processes, although the shallow water depth leads to a deformation radius that can approach or be smaller than 10 km, bringing it closer to the scale of submesoscale processes compared to the open ocean (figure not shown).

Fig. 2.
Fig. 2.

Time series of domain-averaged (per unit mass) evolution of the (a) MKE, (b) APE, (c) EKE, and (d) the energy conversion rate via shear production (SP) and baroclinic instability (BC) [Eq. (6)]. Results are shown for the simulations with Sini = 0.5 (run 1) and Sini = 1.2 (run 4). A moving average with a 10-day window is applied to the daily output in order to highlight the long-term trends.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

b. Eulerian-mean circulation and momentum budget

We now investigate the circulation structures of the wind-driven upwelling system after the simulations have reached statistically steady states. Again, we will begin by focusing on the two representative model runs and later generalize the results to the rest of the runs. For small values of S, the wind-driven upwelling system takes on the classic two-dimensional pattern. The wind stress pushes surface water offshore within the surface Ekman layer, and this redistributed mass generates a sea surface height gradient that sustains a geostrophic current in the interior (Figs. 3a,b). The boundary current interacts directly with the seafloor, causing the bottom stress, which is opposite in direction to the mean flow, to induce a bottom Ekman transport that flows up the topographic slope, thus completing the coastal circulation (Figs. 3b,c).

Fig. 3.
Fig. 3.

Cross-shore sections of time- and alongshore-averaged (a),(d) alongshore velocity ⟨u¯⟩, (b),(e) cross-shore velocity ⟨υ¯⟩, and (c),(f) vertical velocity ⟨w¯⟩ in the simulation with (top) Sini = 0.5 (run 1) and (bottom) Sini = 1.2 (run 4). The black contours in (a) and (d) denote isopycnals. The green dashed lines in (c) represent the offshore locations where the water depth is 50 and 150 m (used in Fig. 5).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

As S increases, the Eulerian-mean circulation pattern undergoes significant changes. In addition to a broader frontal jet and stronger thermal wind shear due to the high initial stratification, the main difference is the location of the return pathway. Instead of concentrating near the bottom, the return flow now follows an interior pathway above the sloping topography, joining the surface Ekman transport close to the coast immediately below the surface boundary layer (Figs. 3e,f). This negligible bottom Ekman transport over the slope results from a weakening of the alongshore velocity (Fig. 3d), which resembles the Ekman arrest adjustment (MacCready and Rhines 1991; Brink and Lentz 2010; Ruan et al. 2019, 2021). Ekman arrest typically occurs when a balanced, along-slope mean flow is present over sloping topography, and the associated cross-slope Ekman transport advects isopycnals up- or downslope depending on the mean flow direction. This advection, along with any turbulent mixing that may occur, produces lateral density gradients and a geostrophic velocity shear that always opposes the mean flow, thus reducing the magnitude of the total along-slope velocity near the bottom and bottom Ekman transport. The presence of isopycnals intersecting the bottom in the large S case and the Ekman arrest adjustment in the bottom Ekman layer may be a necessary condition for the existence of an interior return flow in wind-driven coastal upwelling, which is absent in the small S regime.

A better understanding of the circulation patterns can be gained through the diagnostics of the depth-integrated Eulerian-mean momentum budget, which, at a statistically steady state, can be written as
∂∂y∫−h0⟨u′υ′¯⟩dz+∂∂y∫−h0⟨u¯ υ¯⟩dz=τsx−⟨τbx¯⟩ρ0.
This equation represents a similar formulation as Eq. (1) but with a breakdown of the total cross-shore momentum flux into separate mean and eddy components. In their two-dimensional model, LC04 focused on the mean component that arises from the coupling between the vertically sheared alongshore boundary current (in thermal wind balance) and the surface offshore current and subsurface return flow. In the absence of any eddy contribution, they argued that the divergence of the mean momentum flux determines the structure of the Eulerian-mean circulation over the slope.

Here, we examine and compare the eddy and mean components in our two representative three-dimensional runs. While the mean momentum flux can exhibit comparable or even stronger magnitudes within the surface Ekman layer when eddy activity is weak (Figs. 4a–c), its cross-shore divergence is limited, leading to a smaller contribution to the depth-integrated budget. As S increases, the eddy activity becomes more prominent and dominates the total offshore momentum flux both at the surface and in the interior (Figs. 4d–f). The relative importance of these two components to the total depth-integrated momentum budget (normalized by the surface stress) is summarized in Figs. 5b and 5c for all model runs, illustrating the growing significance of the eddy momentum flux with increasing S. Our analysis contradicts the conclusion of LC04 and highlights the importance of eddy momentum flux in the Eulerian-mean momentum budget.

Fig. 4.
Fig. 4.

Cross-shore sections of (a),(d) ⟨uυ¯⟩, (b),(e) ⟨u¯ υ¯⟩, and (c),(f) ⟨u′υ′¯⟩ in the simulation with (top) Sini = 0.5 (run 1) and (bottom) Sini = 1.2 (run 4). Note the different scales between the upper and lower panels.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

Fig. 5.
Fig. 5.

Diagnostics of the depth-integrated momentum budget in the alongshore direction. (a) The mean isopycnal slope |αρ| and f/N averaged over the slope region between 50- and 150-m water depth (green dashed lines in Fig. 3c) with the surface 5 m excluded for all model runs. The black line indicates a constant slope of 0.25. (b),(c) The ratios of the bottom stress and divergence of cross-shore momentum flux and its decomposition to the wind stress in the depth-integrated alongshore momentum budget. The ratios are calculated at the location where the water depth is 50 m in (b) and 150 m in (c) in all model runs. The solid and dashed curves correspond to the steady theory [Eqs. (12) and (13)] with b = 1.5 and b = 1, respectively. Note that the slope Burger number Seq at equilibrium is used on the x axis.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

Through a series of idealized assumptions about the mean flows, LC04 derived the ratios of the depth-integrated divergence of (mean) momentum flux and the bottom stress to the surface stress:
∂∂y∫−h0uυdz≈τsxρ0bS1+bS/2 and
τbx≈τsx1−bS/21+bS/2.
These two ratios are functions of the slope Burger number S only. According to the derivation, b is a constant that accounts for 1) the constant proportionality a between the local isopycnal slope αρ=−(∂ρ/∂y)/(∂ρ/∂z) and f/N:
αρ=−af/N,
and 2) the uncertainties associated with the idealized assumptions in the theoretical framework. It was determined through a best fit that b = 1 in LC04.

Given the earlier discussion on the comparison between mean and eddy momentum flux, it is necessary to move away from theoretical predictions based solely on mean flows [Eqs. (12) and (13)]. However, due to the absence of established scalings for the eddy momentum flux in wind-driven coastal upwelling systems, we choose to put these formulas to the test. This decision is motivated by the potential for similarities in the profile of the depth-integrated divergence of full momentum flux between the mean-flux-dominated 2D regime and the eddy-flux-dominated 3D regime; the primary difference lies in the partitioning of the two components rather than the sum. As a result, it is important to view the two aforementioned predictions as empirical rather than grounded in a solid physical basis.

We now investigate the magnitude of the different terms in the depth-integrated momentum budget as in Eq. (11) and compare their ratios with the theoretical predictions from Eqs. (12) and (13). Overall, our analysis reveals a robust pointwise agreement between the integrated momentum terms diagnosed from the model and the theoretical predictions, demonstrated by the results across two example depths shown in Figs. 5b and 5c. We report that the best fit for the constant b is 1.5. Based on the theoretical formulas and model diagnostics, it is evident that the magnitude of mass transport in the sloping bottom Ekman layer can be predicted by the slope Burger number S only. When S is small, the bottom stress and transport become comparable to their surface counterparts, resulting in a more classic two-dimensional description of wind-driven upwelling with a return flow concentrated near the bottom. In contrast, a large S leads to a greater contribution from the eddy momentum flux to the integrated momentum budget, causing the solution to evolve toward an interior return pathway. The variation in the spatial structure of the Eulerian-mean circulation with S is more pronounced in its offshore extent than in its vertical depth within the water column. The overall structure exhibits greater sensitivity to changes in S when S is small, and this sensitivity decreases as S becomes larger.

Last, while the constant a in Eq. (14) does not appear directly in the final formulas, it can still serve as a valuable predictor of the overall isopycnal structure in coastal upwelling regions. To determine the constant, we calculate the average isopycnal slope and stratification across the entire shelf region with a depth range between 50 and 150 m. We exclude the inner shelf region shallower than 50 m and a 5-m-thick top layer where the isopycnals can become vertical due to mixing processes. The diagnosed a in our simulations across different cases is approximately 0.25, consistent with the results in LC04 (Fig. 5a); a = 1 would imply that S can be directly interpreted as the ratio between the local topographic slope and the isopycnal slope. The fact that a ≈ 0.25 suggests that this interpretation holds true but with a constant multiplicative factor. This calculation further suggests that the characteristic horizontal length scale associated with coastal upwelling is approximately 4 times the local deformation radius.

c. Eddy-induced and residual circulation

As discussed in previous sections, the wind-driven coastal upwelling system exhibits active baroclinic instability associated with the upwelling front. This can be quantified by examining the distribution of EKE and the energy conversion rate via baroclinic instability (⟨w′b′¯⟩). In both cases, EKE displays strong depth dependence, intensifying toward the surface and decaying with depth (Figs. 6b,e). Unlike the findings of Brink (2016), which showed a concentration of EKE near the coast, our study demonstrates a broad signal throughout the entire domain due to the steady wind forcing applied. In general, weaker EKE is observed near the coast in both runs, although the underlying causes are distinct. At low values of S, where the EKE level is lower (Fig. 2c), the inner shelf region is weakly stratified (Fig. 6a), leading to reduced APE storage and conversion to EKE (Figs. 6b,c). As S increases, both EKE and the energy conversion rate are weak over the lower portion of the slope (Figs. 6e,f), despite the significantly stronger stratification near the coast. This can be attributed to the inhibited baroclinic instability over sloping topography, particularly when the topographic slope is steeper than the isopycnal slope (Fig. 3d) (Blumsack and Gierasch 1972; Hetland 2017; Isachsen 2011).

Fig. 6.
Fig. 6.

Cross-shore sections of time- and alongshore-averaged (a),(d) stratification, (b),(e) EKE, and (c),(f) vertical turbulent buoyancy flux in the simulation with (top) Sini = 0.5 and (bottom) Sini = 1.2. Note the different scales of EKE between (b) and (e).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

The general circulation patterns of the upwelling system with different S, including the Eulerian-mean (ψm), eddy (ψeddy), and residual (ψres) streamfunctions, are shown in Fig. 7. The spatial distribution of the Eulerian-mean streamfunctions is in line with previous findings, indicating that for small S, the return flow is mainly located in the bottom Ekman layer along the sloping bottom, while for large S, the return flow takes an interior pathway (Figs. 7b,f). Recalling Eq. (8), we can decompose the Eulerian-mean streamfunction into three components: ψEk driven by the wind and ψem and ψmm driven by the divergence of eddy and mean momentum flux. When S is not too small, ψmm can be neglected and it is mainly the interplay between ψEk and ψem that determines the Eulerian-mean circulation. In our simulations, the observed shift in the total Eulerian-mean upwelling pathway over the slope can be attributed to the increasing impact of the eddy momentum flux (Figs. 7a,e), which opposes the spatially uniform (positive) wind-induced Eulerian-mean circulation (ψEk, not shown). As the cancellation between the wind-induced and eddy-momentum-flux-driven Eulerian circulation strengthens with higher S, it results in a gradual deviation of the return pathway from the sloping bottom (Figs. 7b,f).

Fig. 7.
Fig. 7.

Cross-shore sections of (a),(e) eddy-momentum-flux-driven Eulerian-mean streamfunction [ψem, Eq. (8)], (b),(f) full Eulerian-mean streamfunction [ψm, Eq. (7)], (c),(g) eddy streamfunction [ψeddy, Eq. (9)], and (d),(h) residual streamfunction [ψres, Eq. (10)] in the simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2. The black dashed line in (d) represents the water depth of 30 m in a nearshore band of 80 km (used in Fig. 8).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

The eddy streamfunction ψeddy is negative, and its magnitude decreases with depth (Figs. 7c,g). This implies that the eddy-induced circulation is counterclockwise and opposes the Eulerian-mean circulation. The distribution of ψeddy resembles that of the energy conversion, with stronger circulation extending to depth with stronger S. Near the surface, ψeddy is the strongest, which is the characteristic of submesoscale baroclinic instability in the surface boundary layer (Fox-Kemper et al. 2008; Colas et al. 2013). The strongest ψeddy is found offshore, consistent with the previously discussed weak EKE signals over the slope.

The residual streamfunction ψres can be obtained by combining ψm and ψeddy. Generally, ψeddy is stronger than ψm in the upper layer, resulting in a counterclockwise circulation near the surface away from the coast. As ψeddy decays with depth and toward the slope, ψm becomes dominant and determines the local distribution of ψres (Figs. 7d,h). Overall, the spatial distribution of ψres exhibits a counterclockwise circulation in the interior region away from the slope, which gradually transitions to a clockwise circulation as it approaches the slope.

In addition to the basic features, the structures of ψres differ significantly between the two cases, indicating distinct pathways for nutrient upwelling. For small S, the residual circulation is dominated by the Eulerian-mean circulation over the slope due to the weak eddy activity, providing a direct upwelling route into the surface boundary layer along the sloping topography (Fig. 7d). Furthermore, the coupling between the clockwise residual circulation over the slope and the counterclockwise circulation away from the coast results in a vigorous residual subduction pathway out of the surface boundary layer into the interior ocean (Fig. 7d). This residual overturning facilitates the transport of nutrient-rich deep water into the euphotic layer along the sloping bottom, thereby supporting primary production. However, it also competes with the subduction process, which has the potential to remove unutilized nutrients from the sunlit surface layers. In contrast, for large S, the Eulerian-mean circulation shifts away from the slope and toward the interior and is increasingly offset by the eddy-induced circulation. Consequently, the residual circulation is confined below 150-m depth, lacking direct access to the surface layers for nutrient upwelling. In this scenario, nutrients are directed away from the coast at depth without being effectively utilized in photosynthesis (Fig. 7h).

To quantify the intensity of residual upwelling in all model runs, we examine the maximum residual streamfunction at a depth of 30 m over the slope, normalized by the Eulerian-mean streamfunction due to Ekman pumping:
Tup=ψresmaxψEk.
This expression provides an estimate of the volume transport along the upwelling branch. Similarly, we calculate the volume transport associated with the subduction by using the difference in streamfunction between the maximum and minimum values, normalized by the same Eulerian-mean streamfunction:
Tsub=ψresmin−ψresmaxψEk.
The smaller the difference in streamfunction (or the larger the magnitude), the stronger the subduction. Overall, we find that the intensity of both upwelling and subduction decreases as Seq increases (Figs. 8a,b). There are significant correlations (p < 0.01) between both upwelling (R = −0.86) and subduction (R = 0.83) and Seq in runs 1–12, where the surface forcing stays the same. We discuss the impact of different surface forcings in the next section. It is worth noting that a potential saturation state is observed when S > 1.5, beyond which the intensity of both upwelling and subduction remains constant and no longer varies with S.
Fig. 8.
Fig. 8.

The diagnosed intensity of (a) upwelling (Tup) and (b) subduction (Tsub) at the depth of 30 m in a nearshore band of 80 km (black dashed line in Fig. 7d). The same symbols are used as in Fig. 5a. The black solid lines in (a) and (b) are linear regressions of the different variables against Seq with the R values being the correlation coefficients for runs 1–12 (excluding the ones with variable surface forcing). The linear regression for runs 17–20 is shown separately by the blue dashed line in (b). The superscript * in the R values indicates statistical significance (p < 0.01). Schematic representation of the competition between ψm (red) and ψeddy (blue) in the different S regimes is shown in (c) and (d).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

The change in the intensity of upwelling and subduction with S can be easily explained by the competition between ψm and ψeddy. With small S, the Eulerian-mean circulation dominates over the slope, while the eddy-induced circulation dominates the offshore region. This offset in space leads to the coexistence of strong upwelling and subduction (Fig. 8c). As S increases, the intensified eddy-induced circulation extends to a greater depth, cancelling the Eulerian-mean circulation that shifts away from the sloping topography in the upper water column (Fig. 8d). These findings indicate that in the small S regime, upwelling is consistently accompanied by subduction, emphasizing the significance of the time scales involved in nutrient uptake during primary production and their retention in the euphotic zone before getting subducted. Conversely, in the large S regime, both the upwelling and subduction decrease concurrently, likely leading to a reduction in productivity in the coastal ocean.

d. Float trajectory and source depth

In addition to the Eulerian description of the coastal upwelling pathway, we also employ a Lagrangian framework to better illustrate the residual upwelling pathway and also to validate the calculation of residual streamfunction. As introduced in section 2c, a total of 240 floats are released near the bottom of the slope and are evolved online for 100 days. In the small S regime, the float distribution confirms our previous observation based on Eulerian calculations, such that strong upwelling and subduction occur at the same time. Over 100 days of float evolution, most floats are upwelled into the surface boundary layer, but many of them are later subducted back into the interior ocean (Figs. 9a,b). In contrast, in the large S regime, most floats recirculate near the bottom of the slope and are directed away from the coast without reaching the surface, consistent with the distribution of residual streamfunction (Figs. 9d,e).

Fig. 9.
Fig. 9.

Spatial distribution of (a),(d) float locations and (b),(e) trajectories during the 100-day evolution after release at the bottom of the slope (71–74 km). The source depths of the floats that reach the surface layer (δ = 30 m) are shown in (c) and (f). Simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2 are shown. The colors in (a) and (d) represent days after release. The gray lines in (b) and (e) denote the trajectories of 120 floats with 4 floats highlighted in different colors. The green dashed lines in (c) and (f) indicate the offshore boundary of the nearshore band that extends 50 km from the coast where the mean source depth (Dmean) is calculated.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

To provide a more quantitative assessment of the intensity of upwelling based on float distribution, we use the following formula to calculate the upwelling percentage:
upwelling percentage=NupNtotal×100%.
Here, Nup represents the number of floats that have reached the surface layer, which is defined with a thickness of δ, and Ntotal = 240 is the total number of floats released. The subduction percentage can be similarly calculated using the ratio between the number of subducted floats (Nsub) and the total upwelled floats (Nup):
subduction percentage=NsubNup×100%.
In the case of small S, 80.4% of 240 floats upwell to the surface layer (δ = 30 m) over the course of 100 days, with 90.7% of them subsequently subducting to the interior away from shore. Conversely, for large S, only 2.9% of the floats reach the surface layer, with most of them being advected offshore near the bottom. The upwelling and subduction percentage across other runs with varying parameters are shown in Figs. 10a and 10b where the trends with S agree well with those of the upwelling and subduction intensity (Fig. 8). Less than 10% of the floats upwell to the surface layer when Seq ≥ 1.
Fig. 10.
Fig. 10.

The diagnosed (a) upwelling and (b) subduction percentage, as well as (c) the mean source depth of the upwelled floats that have reached the surface layer against Seq. Note that the runs with an upwelling percentage smaller than 5% are excluded in (b). The black solid lines are linear regressions of the different variables against Seq with the R values being the correlation coefficients for runs 1–12. The superscript * in the R values indicates statistical significance (p < 0.01).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

Coastal upwelling is characterized by the offshore transport of surface water, which must be replenished by deeper water upwelling from depth. To gain further insights, it is helpful to focus on the surface layers and examine the source depth of the upwelled water in different regimes. In the second set of float experiments, we release 3600 floats along the depth-cross-shore transect at x = 0 at all 30 vertical levels. This enables us to track the movement of water at all depths. The source depth of surface water is calculated by averaging the initial depth of all floats that reach the surface layer in the last 20 days of the 100-day float evolution period:
Dmean=∑i=1NsfcDiniiNsfc.
Here, Nsfc is the number of floats that reach the surface layer over a time period after release and Dinii is the initial depth of the ith float. The floats that start in the surface boundary layer are excluded from the calculation. In general, the source depth of the upwelled water is deeper over the slope than offshore in the two runs with Sini = 0.5 and Sini = 1.2, respectively (Figs. 9c,f). However, in the case of small S, floats originating from ∼150 m could reach the surface, indicating a deeper source depth compared to the large S case, where the source depth is much shallower. The mean source depths are calculated within an offshore boundary of 50 km (the green dashed line in Fig. 9c). The rapid decline of the source depth in the small S scenario, observed in their surface distribution across the offshore boundary (Fig. 9c), suggests the existence of a subduction pathway that disrupts the offshore transport of upwelled water, which would otherwise lead to a more evenly distributed deeper source depth. The mean source depth of surface water near the coast decreases as Seq increases (R = 0.92), across various simulations (Fig. 10c). We speculate that in the large S regimes, eddy-induced diffusion along isopycnals may be responsible for bringing up deeper water toward the surface, in the absence of direct residual upwelling routes.

e. Atmospheric forcing

Our study aims to investigate the spatial characteristics of coastal upwelling by analyzing the interplay between the Eulerian-mean and eddy-induced circulation. Additionally, surface forcing, such as wind stress and surface buoyancy flux, can directly impact coastal upwelling by influencing these two opposing cells. In this section, we focus on the changes in the residual circulation that occur under varying wind and surface buoyancy flux, with the goal of providing a comprehensive understanding of the processes governing coastal upwelling.

We conducted four additional simulations (runs 13–16) with the same parameters as our baseline runs (1–4) but with wind stress reduced by half. In comparison to the original runs with τsx = −0.1 N m−2, the simulations with weaker wind stress retained similar spatial structures for the residual circulation but with substantially weaker intensity (Figs. 11b,e). This is because the weaker wind stress led to a concurrent linear reduction of both the Eulerian-mean and eddy-induced circulation. The reduction in Eulerian-mean flow is most apparent in the intensity of upwelling over the topographic slope in the small S regime, where it dominates the residual circulation (Fig. 11b). The strength of both components of the Eulerian-mean circulation, driven by wind and eddy momentum flux, was observed to be directly proportional to the magnitude of wind stress. The weaker Eulerian-mean circulation results in less outcropping of the isopycnals, leading to stronger stratification and a decrease in APE storage. Consequently, the eddy streamfunction and subduction rate also decrease. Overall, the intensity of upwelling (Tup) and subduction (Tsub) in runs 13–16 (open circles in Figs. 8a,b) still exhibits a negative correlation with Seq, similar to the baseline simulations. However, the magnitudes of both upwelling and subduction are smaller than in the original runs with full wind forcing (solid circles). The reduced upwelling is also reflected in the upwelling percentage, as well as the source depth from float evolution (Table 1).

Fig. 11.
Fig. 11.

Cross-shore sections of residual streamfunction (ψres) in the simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2. (a),(d) The baseline runs (1 and 4) and the runs with (b),(e) half wind stress and (c),(f) added surface heat flux are shown.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

In the context of coastal upwelling, imposed surface buoyancy flux can suppress baroclinic instability and consequently weaken the eddy-induced circulation (Thomsen et al. 2021). To explore this effect in our system, we performed four additional simulations with an added surface heat flux of 200 W m−2 (runs 17–20) compared to the baseline runs. We found that the surface heat flux had a direct impact on the intensity of residual circulation through the modification of eddy-induced circulation; this is particularly evident in the much weaker subduction route in the low S case (Fig. 11c). The impact on the Eulerian-mean circulation was much smaller, mainly through a change in the stratification and S near the surface at equilibrium.

As S increases, the eddy-induced circulation becomes less sensitive to the surface heat flux. This difference in sensitivity may be due to the overall stratification and subsurface vertical buoyancy flux resulting from turbulent mixing. A stronger stratification can reduce the penetration of vertical heat flux, leading to a less affected interior eddy-induced circulation. This trend is illustrated in Fig. 8b, where the difference between the runs with added surface heat flux and the baseline runs (represented by crosses and solid circles, respectively) decreases with increasing S. We anticipate that if the initial stratification decays more rapidly with depth, as opposed to the constant profile used here, the influence of surface heat flux on the eddy-induced circulation could extend to greater depths.

4. Conclusions and discussion

Using idealized three-dimensional numerical simulations, we investigated the pathways of wind-driven coastal upwelling, especially of the residual circulation that transports passive tracers such as nutrients and carbon within the coastal ocean. The residual pathway arises from the large cancellation between the Eulerian-mean and eddy-induced circulation. Our study showed that as the slope Burger number S increases, the Eulerian-mean upwelling shifts from the bottom-concentrated route to an interior pathway above the slope. This shift is determined by whether the wind stress is balanced by the divergence of cross-shore momentum flux or the bottom stress in the depth-integrated alongshore momentum budget [Eq. (1)]. For small S, the surface wind stress is predominantly balanced by the bottom stress, leading to upwelling along the sloping bottom to replenish the surface offshore Ekman transport. However, as S increases, the divergence of cross-shore eddy momentum flux becomes more significant, effectively balancing the surface wind stress and leading to the development of an interior upwelling pathway [Eq. (11)]. These findings contradict the conclusions of LC04 on the Eulerian-mean circulation, where the significance of the cross-shore eddy momentum flux is ignored in the momentum budget.

Our study has also highlighted the key role played by the eddy-induced circulation as APE is stored in the upwelling front and released via baroclinic instability. Observational evidence has begun to reveal distinct surface signatures of upwelled water, such as in the sea surface temperature field, in upwelling regions with comparable wind stress (e.g., He and Mahadevan 2021), indicating the likely presence of eddy compensation in the coastal ocean. Our study further demonstrates that the interaction between the Eulerian-mean and eddy-induced circulation is influenced by the slope Burger number S, which determines the degree of cancellation between the two cells. This is because the eddy-induced circulation mainly occupies the offshore side of the upwelling front and extends to a greater depth with larger S, while its distance to the coast is relatively less sensitive to S. In contrast, the location of the upwelling branch in the Eulerian-mean circulation is primarily determined by S, as discussed previously. When S is small, the upwelling in the Eulerian-mean circulation is close to the sloping bottom and thus cannot be opposed by the eddy-induced circulation due to their offset in space (Fig. 8c). Therefore, the residual circulation takes an upwelling branch near the bottom, followed by subduction into the interior ocean due to the eddy-induced circulation in the opposite direction (Fig. 12a). As S increases, the Eulerian-mean circulation shifts offshore, and it is almost completely opposed by the eddy-induced circulation in the upper water column (Fig. 8d); this constrains the residual cell at depth with no access to the surface layer (Fig. 12b). Generally, the intensity of upwelling and subduction weakens as S increases (Figs. 8 and 10).

Fig. 12.
Fig. 12.

Conceptual diagram of the pathways of wind-driven coastal upwelling in different regimes. The residual circulation is controlled by the competition between the Eulerian-mean and eddy-induced circulation. (a) In the regime with slope Burger number S less than 1, the Eulerian-mean circulation dominates the residual circulation over the slope, leading to an upwelling route along the sloping topography (orange arrow). The eddy-induced circulation is weak near the slope due to weak stratification but dominates in the offshore region, inducing a subduction route back into the interior ocean (blue arrows). (b) For S greater than 1, the Eulerian-mean upwelling shifts from the bottom to an interior pathway and is offset by the opposing eddy-induced circulation. The eddy-induced circulation extends to depth but does not approach the coast due to the suppression of baroclinic instability over the slope. Consequently, the residual cell is directed offshore and cannot reach the surface. Additional surface heat flux tends to weaken mesoscale eddies, which causes the eddy-induced circulation to become weaker and shoals toward the surface. This effect is more apparent in the small S regime, where the subduction intensity becomes weaker. The sensitivity of the response of eddy-induced circulation to surface heat flux is much smaller in the large S regime. The white contours denote isopycnals, and the green and brown dashed lines represent the cancellation depth between the Eulerian-mean and eddy-induced circulation in the runs with and without surface heat flux.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0098.1

Furthermore, the influence of wind stress and surface buoyancy flux on the residual circulation with different S is explored. We find that both the intensity and structures of the residual circulation are sensitive to surface forcing. With weaker wind stress, the residual streamfunction weakens in intensity but keeps the same circulation structure in space because both ψm and ψeddy decrease proportionally to the wind stress. On the other hand, adding surface buoyancy flux leads to a smaller subduction intensity due to the weaker eddy-induced circulation. The impact of surface buoyancy flux on the residual circulation generally becomes less pronounced as S increases.

There are several notable differences between the idealized model setup used in our study and real-world ocean environments that merit discussion. First, both the wind stress and surface buoyancy flux applied in our simulations are spatially uniform and temporally steady. It is important to recognize that the coastal ocean experiences significant temporal and spatial variability in atmospheric forcing, which could introduce additional complexities to the upwelling response. One such complexity arises from the cross-shore variation in wind stress, which has the potential to modify the divergence of mean momentum flux and consequently affect its contribution to the Eulerian-mean momentum budget. Second, we did not consider the intricate bathymetric features commonly found in coastal regions, such as the shelf break and submarine canyons, which typically extend beyond the 200-m depth limit used in our model. These features have the potential to generate complex current systems, altering local circulation patterns and their responses to surface atmospheric forcing, potentially influencing the conclusions drawn from our study. Last, the alongshore symmetry can be disrupted by irregularities in the coastline and variations in alongshore wind. These asymmetries can give rise to alongshore pressure gradients, which, in turn, can support cross-shore geostrophic currents that contribute to the overall cross-shelf exchange and affect the dynamics of coastal upwelling systems (Hickey and Pola 1983; Marchesiello and Estrade 2010; Ndoye et al. 2017; Jacox et al. 2018). For instance, Thomsen et al. (2021) incorporated an alongshore pressure gradient into their numerical model, which led to the inclusion of an undercurrent in the simulated coastal upwelling system. Their surface offshore Ekman transport is entirely compensated by the onshore geostrophic flow resulting from the added pressure gradient, which makes it challenging to directly compare the coastal upwelling circulation with our results. Incorporating these more realistic forcing conditions and accounting for the influence of complex topographic features is the focus of ongoing research.

Despite the idealization, our study offers useful insights into the physical processes that control the circulation dynamics in coastal upwelling systems. Predicting the future behavior of coastal upwelling requires a comprehensive understanding of the complex interactions among various oceanic processes. Even with improved resolution in climate models, the small-scale processes in the coastal ocean will continue to pose challenges in fully capturing the intricate complexities of coastal upwelling systems. Future efforts are needed to address these limitations and improve our ability to forecast and manage coastal upwelling systems under the influence of a changing climate.

Acknowledgments.

We thank Ken Brink and Steve Lentz for their helpful comments that improved this manuscript.

Data availability statement.

The data that support the findings of this study can be obtained at https://zenodo.org/record/8346851. Full output from individual model runs is too large to be hosted in an online repository, but it is available from the corresponding author upon reasonable request.

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    • Export Citation

  • Fox-Kemper, B., R. Ferrari, and R. Hallberg, 2008: Parameterization of mixed layer eddies. Part I: Theory and diagnosis. J. Phys. Oceanogr., 38, 1145–1165, https://doi.org/10.1175/2007JPO3792.1.

    • Search Google Scholar
    • Export Citation

  • Gruber, N., Z. Lachkar, H. Frenzel, P. Marchesiello, M. Münnich, J. C. McWilliams, T. Nagai, and G.-K. Plattner, 2011: Eddy-induced reduction of biological production in eastern boundary upwelling systems. Nat. Geosci., 4, 787–792, https://doi.org/10.1038/ngeo1273.

    • Search Google Scholar
    • Export Citation

  • Hallberg, R., and A. Gnanadesikan, 2006: The role of eddies in determining the structure and response of the wind-driven Southern Hemisphere overturning: Results from the Modeling Eddies in the Southern Ocean (MESO) project. J. Phys. Oceanogr., 36, 2232–2252, https://doi.org/10.1175/JPO2980.1.

    • Search Google Scholar
    • Export Citation

  • He, J., and A. Mahadevan, 2021: How the source depth of coastal upwelling relates to stratification and wind. J. Geophys. Res. Oceans, 126, e2021JC017621, https://doi.org/10.1029/2021JC017621.

    • Search Google Scholar
    • Export Citation

  • Held, I. M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci., 56, 1688–1697, https://doi.org/10.1175/1520-0469(1999)056<1688:TSBOTZ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Hetland, R. D., 2017: Suppression of baroclinic instabilities in buoyancy-driven flow over sloping bathymetry. J. Phys. Oceanogr., 47, 49–68, https://doi.org/10.1175/JPO-D-15-0240.1.

    • Search Google Scholar
    • Export Citation

  • Hickey, B. M., and N. E. Pola, 1983: The seasonal alongshore pressure gradient on the West Coast of the United States. J. Geophys. Res., 88, 7623–7633, https://doi.org/10.1029/JC088iC12p07623.

    • Search Google Scholar
    • Export Citation

  • Huyer, A., 1983: Coastal upwelling in the California Current System. Prog. Oceanogr., 12, 259–284, https://doi.org/10.1016/0079-6611(83)90010-1.

    • Search Google Scholar
    • Export Citation

  • Isachsen, P. E., 2011: Baroclinic instability and eddy tracer transport across sloping bottom topography: How well does a modified Eady model do in primitive equation simulations? Ocean Modell., 39, 183–199, https://doi.org/10.1016/j.ocemod.2010.09.007.

    • Search Google Scholar
    • Export Citation

  • Jacox, M. G., and C. A. Edwards, 2011: Effects of stratification and shelf slope on nutrient supply in coastal upwelling regions. J. Geophys. Res., 116, C03019, https://doi.org/10.1029/2010JC006547.

    • Search Google Scholar
    • Export Citation

  • Jacox, M. G., C. A. Edwards, E. L. Hazen, and S. J. Bograd, 2018: Coastal upwelling revisited: Ekman, Bakun, and improved upwelling indices for the U.S. West Coast. J. Geophys. Res. Oceans, 123, 7332–7350, https://doi.org/10.1029/2018JC014187.

    • Search Google Scholar
    • Export Citation

  • Johnson, K. S., F. P. Chavez, and G. E. Friederich, 1999: Continental-shelf sediment as a primary source of iron for coastal phytoplankton. Nature, 398, 697–700, https://doi.org/10.1038/19511.

    • Search Google Scholar
    • Export Citation

  • Lathuilière, C., V. Echevin, M. Lévy, and G. Madec, 2010: On the role of the mesoscale circulation on an idealized coastal upwelling ecosystem. J. Geophys. Res., 115, C09018, https://doi.org/10.1029/2009JC005827.

    • Search Google Scholar
    • Export Citation

  • Lentz, S. J., and D. C. Chapman, 2004: The importance of nonlinear cross-shelf momentum flux during wind-driven coastal upwelling. J. Phys. Oceanogr., 34, 2444–2457, https://doi.org/10.1175/JPO2644.1.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and P. B. Rhines, 1991: Buoyant inhibition of Ekman transport on a slope and its effect on stratified spin-up. J. Fluid Mech., 223, 631–661, https://doi.org/10.1017/S0022112091001581.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and S. N. Giddings, 2016: The mechanical energy budget of a regional ocean model. J. Phys. Oceanogr., 46, 2719–2733, https://doi.org/10.1175/JPO-D-16-0086.1.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., A. Tandon, and R. Ferrari, 2010: Rapid changes in mixed layer stratification driven by submesoscale instabilities and winds. J. Geophys. Res., 115, C03017, https://doi.org/10.1029/2008JC005203.

    • Search Google Scholar
    • Export Citation

  • Manucharyan, G. E., and P. E. Isachsen, 2019: Critical role of continental slopes in halocline and eddy dynamics of the Ekman-driven Beaufort Gyre. J. Geophys. Res. Oceans, 124, 2679–2696, https://doi.org/10.1029/2018JC014624.

    • Search Google Scholar
    • Export Citation

  • Marchesiello, P., and P. Estrade, 2010: Upwelling limitation by onshore geostrophic flow. J. Mar. Res., 68, 37–62.

  • Marchesiello, P., J. C. McWilliams, and A. Shchepetkin, 2003: Equilibrium structure and dynamics of the California Current System. J. Phys. Oceanogr., 33, 753–783, https://doi.org/10.1175/1520-0485(2003)33<753:ESADOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and T. Radko, 2003: Residual-mean solutions for the Antarctic Circumpolar Current and its associated overturning circulation. J. Phys. Oceanogr., 33, 2341–2354, https://doi.org/10.1175/1520-0485(2003)033<2341:RSFTAC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and K. Speer, 2012: Closure of the meridional overturning circulation through Southern Ocean upwelling. Nat. Geosci., 5, 171–180, https://doi.org/10.1038/ngeo1391.

    • Search Google Scholar
    • Export Citation

  • Moscoso, J. E., A. L. Stewart, D. Bianchi, and J. C. McWilliams, 2021: The Meridionally Averaged Model of Eastern Boundary Upwelling Systems (MAMEBUSv1.0). Geosci. Model Dev., 14, 763–794, https://doi.org/10.5194/gmd-14-763-2021.

    • Search Google Scholar
    • Export Citation

  • Nagai, T., N. Gruber, H. Frenzel, Z. Lachkar, J. C. McWilliams, and G.-K. Plattner, 2015: Dominant role of eddies and filaments in the offshore transport of carbon and nutrients in the California Current System. J. Geophys. Res. Oceans, 120, 5318–5341, https://doi.org/10.1002/2015JC010889.

    • Search Google Scholar
    • Export Citation

  • Ndoye, S., X. Capet, P. Estrade, B. Sow, E. Machu, T. Brochier, J. Döring, and P. Brehmer, 2017: Dynamics of a “low-enrichment high-retention” upwelling center over the southern Senegal shelf. Geophys. Res. Lett., 44, 5034–5043, https://doi.org/10.1002/2017GL072789.

    • Search Google Scholar
    • Export Citation

  • Pauly, D., and V. Christensen, 1995: Primary production required to sustain global fisheries. Nature, 374, 255–257, https://doi.org/10.1038/374255a0.

    • Search Google Scholar
    • Export Citation

  • Ruan, X., A. F. Thompson, and J. R. Taylor, 2019: The evolution and arrest of a turbulent stratified oceanic bottom boundary layer over a slope: Downslope regime. J. Phys. Oceanogr., 49, 469–487, https://doi.org/10.1175/JPO-D-18-0079.1.

    • Search Google Scholar
    • Export Citation

  • Ruan, X., A. F. Thompson, and J. R. Taylor, 2021: The evolution and arrest of a turbulent stratified oceanic bottom boundary layer over a slope: Upslope regime and PV dynamics. J. Phys. Oceanogr., 51, 1077–1089, https://doi.org/10.1175/JPO-D-20-0168.1.

    • Search Google Scholar
    • Export Citation

  • Shchepetkin, A. F., and J. C. McWilliams, 2005: The Regional Oceanic Modeling System (ROMS): A split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean Modell., 9, 347–404, https://doi.org/10.1016/j.ocemod.2004.08.002.

    • Search Google Scholar
    • Export Citation

  • Smolarkiewicz, P. K., and L. G. Margolin, 1998: MPDATA: A finite-difference solver for geophysical flows. J. Comput. Phys., 140, 459–480, https://doi.org/10.1006/jcph.1998.5901.

    • Search Google Scholar
    • Export Citation

  • Sverdrup, H. U., 1937: On the process of upwelling. J. Mar. Res., 1, 155–164.

  • Thomsen, S., X. Capet, and V. Echevin, 2021: Competition between baroclinic instability and Ekman transport under varying buoyancy forcings in upwelling systems: An idealized analog to the Southern Ocean. J. Phys. Oceanogr., 51, 3347–3364, https://doi.org/10.1175/JPO-D-20-0294.1.

    • Search Google Scholar
    • Export Citation

  • Vallis, G. K., 2017: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 964 pp.

  • Winters, K. B., P. N. Lombard, J. J. Riley, and E. A. D’Asaro, 1995: Available potential energy and mixing in density-stratified fluids. J. Fluid Mech., 289, 115–128, https://doi.org/10.1017/S002211209500125X.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation

  • Fox-Kemper, B., R. Ferrari, and R. Hallberg, 2008: Parameterization of mixed layer eddies. Part I: Theory and diagnosis. J. Phys. Oceanogr., 38, 1145–1165, https://doi.org/10.1175/2007JPO3792.1.

    • Search Google Scholar
    • Export Citation

  • Gruber, N., Z. Lachkar, H. Frenzel, P. Marchesiello, M. Münnich, J. C. McWilliams, T. Nagai, and G.-K. Plattner, 2011: Eddy-induced reduction of biological production in eastern boundary upwelling systems. Nat. Geosci., 4, 787–792, https://doi.org/10.1038/ngeo1273.

    • Search Google Scholar
    • Export Citation

  • Hallberg, R., and A. Gnanadesikan, 2006: The role of eddies in determining the structure and response of the wind-driven Southern Hemisphere overturning: Results from the Modeling Eddies in the Southern Ocean (MESO) project. J. Phys. Oceanogr., 36, 2232–2252, https://doi.org/10.1175/JPO2980.1.

    • Search Google Scholar
    • Export Citation

  • He, J., and A. Mahadevan, 2021: How the source depth of coastal upwelling relates to stratification and wind. J. Geophys. Res. Oceans, 126, e2021JC017621, https://doi.org/10.1029/2021JC017621.

    • Search Google Scholar
    • Export Citation

  • Held, I. M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation in the troposphere. J. Atmos. Sci., 56, 1688–1697, https://doi.org/10.1175/1520-0469(1999)056<1688:TSBOTZ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Hetland, R. D., 2017: Suppression of baroclinic instabilities in buoyancy-driven flow over sloping bathymetry. J. Phys. Oceanogr., 47, 49–68, https://doi.org/10.1175/JPO-D-15-0240.1.

    • Search Google Scholar
    • Export Citation

  • Hickey, B. M., and N. E. Pola, 1983: The seasonal alongshore pressure gradient on the West Coast of the United States. J. Geophys. Res., 88, 7623–7633, https://doi.org/10.1029/JC088iC12p07623.

    • Search Google Scholar
    • Export Citation

  • Huyer, A., 1983: Coastal upwelling in the California Current System. Prog. Oceanogr., 12, 259–284, https://doi.org/10.1016/0079-6611(83)90010-1.

    • Search Google Scholar
    • Export Citation

  • Isachsen, P. E., 2011: Baroclinic instability and eddy tracer transport across sloping bottom topography: How well does a modified Eady model do in primitive equation simulations? Ocean Modell., 39, 183–199, https://doi.org/10.1016/j.ocemod.2010.09.007.

    • Search Google Scholar
    • Export Citation

  • Jacox, M. G., and C. A. Edwards, 2011: Effects of stratification and shelf slope on nutrient supply in coastal upwelling regions. J. Geophys. Res., 116, C03019, https://doi.org/10.1029/2010JC006547.

    • Search Google Scholar
    • Export Citation

  • Jacox, M. G., C. A. Edwards, E. L. Hazen, and S. J. Bograd, 2018: Coastal upwelling revisited: Ekman, Bakun, and improved upwelling indices for the U.S. West Coast. J. Geophys. Res. Oceans, 123, 7332–7350, https://doi.org/10.1029/2018JC014187.

    • Search Google Scholar
    • Export Citation

  • Johnson, K. S., F. P. Chavez, and G. E. Friederich, 1999: Continental-shelf sediment as a primary source of iron for coastal phytoplankton. Nature, 398, 697–700, https://doi.org/10.1038/19511.

    • Search Google Scholar
    • Export Citation

  • Lathuilière, C., V. Echevin, M. Lévy, and G. Madec, 2010: On the role of the mesoscale circulation on an idealized coastal upwelling ecosystem. J. Geophys. Res., 115, C09018, https://doi.org/10.1029/2009JC005827.

    • Search Google Scholar
    • Export Citation

  • Lentz, S. J., and D. C. Chapman, 2004: The importance of nonlinear cross-shelf momentum flux during wind-driven coastal upwelling. J. Phys. Oceanogr., 34, 2444–2457, https://doi.org/10.1175/JPO2644.1.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and P. B. Rhines, 1991: Buoyant inhibition of Ekman transport on a slope and its effect on stratified spin-up. J. Fluid Mech., 223, 631–661, https://doi.org/10.1017/S0022112091001581.

    • Search Google Scholar
    • Export Citation

  • MacCready, P., and S. N. Giddings, 2016: The mechanical energy budget of a regional ocean model. J. Phys. Oceanogr., 46, 2719–2733, https://doi.org/10.1175/JPO-D-16-0086.1.

    • Search Google Scholar
    • Export Citation

  • Mahadevan, A., A. Tandon, and R. Ferrari, 2010: Rapid changes in mixed layer stratification driven by submesoscale instabilities and winds. J. Geophys. Res., 115, C03017, https://doi.org/10.1029/2008JC005203.

    • Search Google Scholar
    • Export Citation

  • Manucharyan, G. E., and P. E. Isachsen, 2019: Critical role of continental slopes in halocline and eddy dynamics of the Ekman-driven Beaufort Gyre. J. Geophys. Res. Oceans, 124, 2679–2696, https://doi.org/10.1029/2018JC014624.

    • Search Google Scholar
    • Export Citation

  • Marchesiello, P., and P. Estrade, 2010: Upwelling limitation by onshore geostrophic flow. J. Mar. Res., 68, 37–62.

  • Marchesiello, P., J. C. McWilliams, and A. Shchepetkin, 2003: Equilibrium structure and dynamics of the California Current System. J. Phys. Oceanogr., 33, 753–783, https://doi.org/10.1175/1520-0485(2003)33<753:ESADOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and T. Radko, 2003: Residual-mean solutions for the Antarctic Circumpolar Current and its associated overturning circulation. J. Phys. Oceanogr., 33, 2341–2354, https://doi.org/10.1175/1520-0485(2003)033<2341:RSFTAC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., and K. Speer, 2012: Closure of the meridional overturning circulation through Southern Ocean upwelling. Nat. Geosci., 5, 171–180, https://doi.org/10.1038/ngeo1391.

    • Search Google Scholar
    • Export Citation

  • Moscoso, J. E., A. L. Stewart, D. Bianchi, and J. C. McWilliams, 2021: The Meridionally Averaged Model of Eastern Boundary Upwelling Systems (MAMEBUSv1.0). Geosci. Model Dev., 14, 763–794, https://doi.org/10.5194/gmd-14-763-2021.

    • Search Google Scholar
    • Export Citation

  • Nagai, T., N. Gruber, H. Frenzel, Z. Lachkar, J. C. McWilliams, and G.-K. Plattner, 2015: Dominant role of eddies and filaments in the offshore transport of carbon and nutrients in the California Current System. J. Geophys. Res. Oceans, 120, 5318–5341, https://doi.org/10.1002/2015JC010889.

    • Search Google Scholar
    • Export Citation

  • Ndoye, S., X. Capet, P. Estrade, B. Sow, E. Machu, T. Brochier, J. Döring, and P. Brehmer, 2017: Dynamics of a “low-enrichment high-retention” upwelling center over the southern Senegal shelf. Geophys. Res. Lett., 44, 5034–5043, https://doi.org/10.1002/2017GL072789.

    • Search Google Scholar
    • Export Citation

  • Pauly, D., and V. Christensen, 1995: Primary production required to sustain global fisheries. Nature, 374, 255–257, https://doi.org/10.1038/374255a0.

    • Search Google Scholar
    • Export Citation

  • Ruan, X., A. F. Thompson, and J. R. Taylor, 2019: The evolution and arrest of a turbulent stratified oceanic bottom boundary layer over a slope: Downslope regime. J. Phys. Oceanogr., 49, 469–487, https://doi.org/10.1175/JPO-D-18-0079.1.

    • Search Google Scholar
    • Export Citation

  • Ruan, X., A. F. Thompson, and J. R. Taylor, 2021: The evolution and arrest of a turbulent stratified oceanic bottom boundary layer over a slope: Upslope regime and PV dynamics. J. Phys. Oceanogr., 51, 1077–1089, https://doi.org/10.1175/JPO-D-20-0168.1.

    • Search Google Scholar
    • Export Citation

  • Shchepetkin, A. F., and J. C. McWilliams, 2005: The Regional Oceanic Modeling System (ROMS): A split-explicit, free-surface, topography-following-coordinate oceanic model. Ocean Modell., 9, 347–404, https://doi.org/10.1016/j.ocemod.2004.08.002.

    • Search Google Scholar
    • Export Citation

  • Smolarkiewicz, P. K., and L. G. Margolin, 1998: MPDATA: A finite-difference solver for geophysical flows. J. Comput. Phys., 140, 459–480, https://doi.org/10.1006/jcph.1998.5901.

    • Search Google Scholar
    • Export Citation

  • Sverdrup, H. U., 1937: On the process of upwelling. J. Mar. Res., 1, 155–164.

  • Thomsen, S., X. Capet, and V. Echevin, 2021: Competition between baroclinic instability and Ekman transport under varying buoyancy forcings in upwelling systems: An idealized analog to the Southern Ocean. J. Phys. Oceanogr., 51, 3347–3364, https://doi.org/10.1175/JPO-D-20-0294.1.

    • Search Google Scholar
    • Export Citation

  • Vallis, G. K., 2017: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 964 pp.

  • Winters, K. B., P. N. Lombard, J. J. Riley, and E. A. D’Asaro, 1995: Available potential energy and mixing in density-stratified fluids. J. Fluid Mech., 289, 115–128, https://doi.org/10.1017/S002211209500125X.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Model setup and snapshots of basic simulation variables. The Rossby number Ro is shown at the surface (z = 0), together with the alongshore velocity (u) at x = 300 km and cross-shore velocity (υ) at y = 320 km. The snapshots are taken on day 300 from run 1 (Table 1). Gray contours indicate isopycnals with an interval of 0.0034 m s−2. A steady, uniform upwelling-favorable wind stress is applied at the surface, driving an upwelling front. The yellow shadow zone denotes the location of the sponge layer.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Time series of domain-averaged (per unit mass) evolution of the (a) MKE, (b) APE, (c) EKE, and (d) the energy conversion rate via shear production (SP) and baroclinic instability (BC) [Eq. (6)]. Results are shown for the simulations with Sini = 0.5 (run 1) and Sini = 1.2 (run 4). A moving average with a 10-day window is applied to the daily output in order to highlight the long-term trends.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Cross-shore sections of time- and alongshore-averaged (a),(d) alongshore velocity ⟨u¯⟩, (b),(e) cross-shore velocity ⟨υ¯⟩, and (c),(f) vertical velocity ⟨w¯⟩ in the simulation with (top) Sini = 0.5 (run 1) and (bottom) Sini = 1.2 (run 4). The black contours in (a) and (d) denote isopycnals. The green dashed lines in (c) represent the offshore locations where the water depth is 50 and 150 m (used in Fig. 5).

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Cross-shore sections of (a),(d) ⟨uυ¯⟩, (b),(e) ⟨u¯ υ¯⟩, and (c),(f) ⟨u′υ′¯⟩ in the simulation with (top) Sini = 0.5 (run 1) and (bottom) Sini = 1.2 (run 4). Note the different scales between the upper and lower panels.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Diagnostics of the depth-integrated momentum budget in the alongshore direction. (a) The mean isopycnal slope |αρ| and f/N averaged over the slope region between 50- and 150-m water depth (green dashed lines in Fig. 3c) with the surface 5 m excluded for all model runs. The black line indicates a constant slope of 0.25. (b),(c) The ratios of the bottom stress and divergence of cross-shore momentum flux and its decomposition to the wind stress in the depth-integrated alongshore momentum budget. The ratios are calculated at the location where the water depth is 50 m in (b) and 150 m in (c) in all model runs. The solid and dashed curves correspond to the steady theory [Eqs. (12) and (13)] with b = 1.5 and b = 1, respectively. Note that the slope Burger number Seq at equilibrium is used on the x axis.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Cross-shore sections of time- and alongshore-averaged (a),(d) stratification, (b),(e) EKE, and (c),(f) vertical turbulent buoyancy flux in the simulation with (top) Sini = 0.5 and (bottom) Sini = 1.2. Note the different scales of EKE between (b) and (e).

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Cross-shore sections of (a),(e) eddy-momentum-flux-driven Eulerian-mean streamfunction [ψem, Eq. (8)], (b),(f) full Eulerian-mean streamfunction [ψm, Eq. (7)], (c),(g) eddy streamfunction [ψeddy, Eq. (9)], and (d),(h) residual streamfunction [ψres, Eq. (10)] in the simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2. The black dashed line in (d) represents the water depth of 30 m in a nearshore band of 80 km (used in Fig. 8).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    The diagnosed intensity of (a) upwelling (Tup) and (b) subduction (Tsub) at the depth of 30 m in a nearshore band of 80 km (black dashed line in Fig. 7d). The same symbols are used as in Fig. 5a. The black solid lines in (a) and (b) are linear regressions of the different variables against Seq with the R values being the correlation coefficients for runs 1–12 (excluding the ones with variable surface forcing). The linear regression for runs 17–20 is shown separately by the blue dashed line in (b). The superscript * in the R values indicates statistical significance (p < 0.01). Schematic representation of the competition between ψm (red) and ψeddy (blue) in the different S regimes is shown in (c) and (d).

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Spatial distribution of (a),(d) float locations and (b),(e) trajectories during the 100-day evolution after release at the bottom of the slope (71–74 km). The source depths of the floats that reach the surface layer (δ = 30 m) are shown in (c) and (f). Simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2 are shown. The colors in (a) and (d) represent days after release. The gray lines in (b) and (e) denote the trajectories of 120 floats with 4 floats highlighted in different colors. The green dashed lines in (c) and (f) indicate the offshore boundary of the nearshore band that extends 50 km from the coast where the mean source depth (Dmean) is calculated.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    The diagnosed (a) upwelling and (b) subduction percentage, as well as (c) the mean source depth of the upwelled floats that have reached the surface layer against Seq. Note that the runs with an upwelling percentage smaller than 5% are excluded in (b). The black solid lines are linear regressions of the different variables against Seq with the R values being the correlation coefficients for runs 1–12. The superscript * in the R values indicates statistical significance (p < 0.01).

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Cross-shore sections of residual streamfunction (ψres) in the simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2. (a),(d) The baseline runs (1 and 4) and the runs with (b),(e) half wind stress and (c),(f) added surface heat flux are shown.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Conceptual diagram of the pathways of wind-driven coastal upwelling in different regimes. The residual circulation is controlled by the competition between the Eulerian-mean and eddy-induced circulation. (a) In the regime with slope Burger number S less than 1, the Eulerian-mean circulation dominates the residual circulation over the slope, leading to an upwelling route along the sloping topography (orange arrow). The eddy-induced circulation is weak near the slope due to weak stratification but dominates in the offshore region, inducing a subduction route back into the interior ocean (blue arrows). (b) For S greater than 1, the Eulerian-mean upwelling shifts from the bottom to an interior pathway and is offset by the opposing eddy-induced circulation. The eddy-induced circulation extends to depth but does not approach the coast due to the suppression of baroclinic instability over the slope. Consequently, the residual cell is directed offshore and cannot reach the surface. Additional surface heat flux tends to weaken mesoscale eddies, which causes the eddy-induced circulation to become weaker and shoals toward the surface. This effect is more apparent in the small S regime, where the subduction intensity becomes weaker. The sensitivity of the response of eddy-induced circulation to surface heat flux is much smaller in the large S regime. The white contours denote isopycnals, and the green and brown dashed lines represent the cancellation depth between the Eulerian-mean and eddy-induced circulation in the runs with and without surface heat flux.

Fig. 1.

Model setup and snapshots of basic simulation variables. The Rossby number Ro is shown at the surface (z = 0), together with the alongshore velocity (u) at x = 300 km and cross-shore velocity (υ) at y = 320 km. The snapshots are taken on day 300 from run 1 (Table 1). Gray contours indicate isopycnals with an interval of 0.0034 m s−2. A steady, uniform upwelling-favorable wind stress is applied at the surface, driving an upwelling front. The yellow shadow zone denotes the location of the sponge layer.
Fig. 2.

Time series of domain-averaged (per unit mass) evolution of the (a) MKE, (b) APE, (c) EKE, and (d) the energy conversion rate via shear production (SP) and baroclinic instability (BC) [Eq. (6)]. Results are shown for the simulations with Sini = 0.5 (run 1) and Sini = 1.2 (run 4). A moving average with a 10-day window is applied to the daily output in order to highlight the long-term trends.
Fig. 3.

Cross-shore sections of time- and alongshore-averaged (a),(d) alongshore velocity ⟨u¯⟩, (b),(e) cross-shore velocity ⟨υ¯⟩, and (c),(f) vertical velocity ⟨w¯⟩ in the simulation with (top) Sini = 0.5 (run 1) and (bottom) Sini = 1.2 (run 4). The black contours in (a) and (d) denote isopycnals. The green dashed lines in (c) represent the offshore locations where the water depth is 50 and 150 m (used in Fig. 5).
Fig. 4.

Cross-shore sections of (a),(d) ⟨uυ¯⟩, (b),(e) ⟨u¯ υ¯⟩, and (c),(f) ⟨u′υ′¯⟩ in the simulation with (top) Sini = 0.5 (run 1) and (bottom) Sini = 1.2 (run 4). Note the different scales between the upper and lower panels.
Fig. 5.

Diagnostics of the depth-integrated momentum budget in the alongshore direction. (a) The mean isopycnal slope |αρ| and f/N averaged over the slope region between 50- and 150-m water depth (green dashed lines in Fig. 3c) with the surface 5 m excluded for all model runs. The black line indicates a constant slope of 0.25. (b),(c) The ratios of the bottom stress and divergence of cross-shore momentum flux and its decomposition to the wind stress in the depth-integrated alongshore momentum budget. The ratios are calculated at the location where the water depth is 50 m in (b) and 150 m in (c) in all model runs. The solid and dashed curves correspond to the steady theory [Eqs. (12) and (13)] with b = 1.5 and b = 1, respectively. Note that the slope Burger number Seq at equilibrium is used on the x axis.
Fig. 6.

Cross-shore sections of time- and alongshore-averaged (a),(d) stratification, (b),(e) EKE, and (c),(f) vertical turbulent buoyancy flux in the simulation with (top) Sini = 0.5 and (bottom) Sini = 1.2. Note the different scales of EKE between (b) and (e).
Fig. 7.

Cross-shore sections of (a),(e) eddy-momentum-flux-driven Eulerian-mean streamfunction [ψem, Eq. (8)], (b),(f) full Eulerian-mean streamfunction [ψm, Eq. (7)], (c),(g) eddy streamfunction [ψeddy, Eq. (9)], and (d),(h) residual streamfunction [ψres, Eq. (10)] in the simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2. The black dashed line in (d) represents the water depth of 30 m in a nearshore band of 80 km (used in Fig. 8).
Fig. 8.

The diagnosed intensity of (a) upwelling (Tup) and (b) subduction (Tsub) at the depth of 30 m in a nearshore band of 80 km (black dashed line in Fig. 7d). The same symbols are used as in Fig. 5a. The black solid lines in (a) and (b) are linear regressions of the different variables against Seq with the R values being the correlation coefficients for runs 1–12 (excluding the ones with variable surface forcing). The linear regression for runs 17–20 is shown separately by the blue dashed line in (b). The superscript * in the R values indicates statistical significance (p < 0.01). Schematic representation of the competition between ψm (red) and ψeddy (blue) in the different S regimes is shown in (c) and (d).
Fig. 9.

Spatial distribution of (a),(d) float locations and (b),(e) trajectories during the 100-day evolution after release at the bottom of the slope (71–74 km). The source depths of the floats that reach the surface layer (δ = 30 m) are shown in (c) and (f). Simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2 are shown. The colors in (a) and (d) represent days after release. The gray lines in (b) and (e) denote the trajectories of 120 floats with 4 floats highlighted in different colors. The green dashed lines in (c) and (f) indicate the offshore boundary of the nearshore band that extends 50 km from the coast where the mean source depth (Dmean) is calculated.
Fig. 10.

The diagnosed (a) upwelling and (b) subduction percentage, as well as (c) the mean source depth of the upwelled floats that have reached the surface layer against Seq. Note that the runs with an upwelling percentage smaller than 5% are excluded in (b). The black solid lines are linear regressions of the different variables against Seq with the R values being the correlation coefficients for runs 1–12. The superscript * in the R values indicates statistical significance (p < 0.01).
Fig. 11.

Cross-shore sections of residual streamfunction (ψres) in the simulations with (top) Sini = 0.5 and (bottom) Sini = 1.2. (a),(d) The baseline runs (1 and 4) and the runs with (b),(e) half wind stress and (c),(f) added surface heat flux are shown.
Fig. 12.

Conceptual diagram of the pathways of wind-driven coastal upwelling in different regimes. The residual circulation is controlled by the competition between the Eulerian-mean and eddy-induced circulation. (a) In the regime with slope Burger number S less than 1, the Eulerian-mean circulation dominates the residual circulation over the slope, leading to an upwelling route along the sloping topography (orange arrow). The eddy-induced circulation is weak near the slope due to weak stratification but dominates in the offshore region, inducing a subduction route back into the interior ocean (blue arrows). (b) For S greater than 1, the Eulerian-mean upwelling shifts from the bottom to an interior pathway and is offset by the opposing eddy-induced circulation. The eddy-induced circulation extends to depth but does not approach the coast due to the suppression of baroclinic instability over the slope. Consequently, the residual cell is directed offshore and cannot reach the surface. Additional surface heat flux tends to weaken mesoscale eddies, which causes the eddy-induced circulation to become weaker and shoals toward the surface. This effect is more apparent in the small S regime, where the subduction intensity becomes weaker. The sensitivity of the response of eddy-induced circulation to surface heat flux is much smaller in the large S regime. The white contours denote isopycnals, and the green and brown dashed lines represent the cancellation depth between the Eulerian-mean and eddy-induced circulation in the runs with and without surface heat flux.
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