1. Introduction
The wind- and buoyancy-driven circulation in the Southern Ocean plays a primary role in setting the properties in the deep ocean, with important implications for global circulation and climate. For example, this region is responsible for about 30%–40% of oceanic anthropogenic carbon uptake, an oversized contribution given its relative size (Gruber et al. 2009; Frölicher et al. 2015). How that carbon budget has varied since the Industrial Revolution, and how the budget will change with global warming, are open questions (Le Quéré et al. 2017). Since there is a strong vertical gradient in dissolved inorganic carbon in the near-surface Southern Ocean (Gruber et al. 2009), the vertical transport of water parcels (i.e., overturning) is a major physical process affecting the air–sea carbon fluxes and has been shown to be a leading component in controlling anthropogenic carbon (Ito et al. 2010). Thus, an understanding of upwelling and downwelling into the deep ocean and how it responds to changes in wind and buoyancy forcing is necessary for analyzing the carbon budget.
Traditionally, we call the circulation responsible for the vertical and meridional transport of tracers the residual overturning circulation, because it is regarded as a small remainder of two main competing processes, wind-driven overturning that steepens isopycnals and eddy-driven overturning that acts to relax those isopycnals (Marshall and Radko 2003; Marshall and Speer 2012). This understanding was developed for an idealized ocean that lacks bottom topography or other zonal asymmetries. The underlying theory and related streamfunction calculations permit zonal asymmetries, and the “standing-eddy” overturning streamfunction has been calculated in an attempt to reconcile the zonally averaged circulation with the zonally asymmetric nature of the Southern Ocean (Bishop et al. 2016; Dufour et al. 2012). While Marshall and Radko (2003) discuss the fact that the residual flow need not be restricted to zonally symmetric flows, they revert to zonally symmetric forcing without topography for building a conceptual picture of the overturning. This approach carries over to global theory of circulation, such as Nikurashin and Vallis (2011), who construct a global circulation scheme using a symmetric annulus for the Southern Ocean. Yet, many aspects of this zonally averaged theory have been shown to be overly simple due to the existence of topography (e.g., Youngs et al. 2017, 2019; Abernathey et al. 2011; Bishop et al. 2016; Kong and Jansen 2021; Dufour et al. 2012, 2013; Farneti et al. 2015). This leaves the question of how the topography modifies the overturning circulation, and how it changes in response to variations in the winds or buoyancy forcing, not only in the zonally averaged sense, but also in the local sense.
There is a rich history of studies investigating how the Southern Ocean overturning varies with changes in wind in both idealized and realistic configuration (e.g., Hallberg and Gnanadesikan 2006; Abernathey et al. 2011; Meredith et al. 2012; Bishop et al. 2016; Kong and Jansen 2021). It is hypothesized that as the wind strengthens, the wind-driven overturning strengthens and the isopycnal slopes steepen, leading to an increase in the eddy driven circulation, at least partially compensating for the change in the wind driven circulation, a process therefore called eddy compensation (Marshall and Speer 2012; Abernathey et al. 2011). Abernathey et al. (2011) show that in an idealized flat-bottomed Southern Ocean, the overturning scales linearly with the wind, with stronger winds leading to a stronger upper cell and a weaker lower cell due to the strengthening of the wind-driven circulation, and partial eddy compensation. In addition, there is a related concept of eddy saturation that describes the sensitivity of the isopycnal slopes to changes in the wind; this has been studied in many ways including in a two-layer quasigeostrophic channel (Youngs et al. 2019; Zhang et al. 2023). Youngs et al. (2019) find that the presence of the overturning circulation and an undersea ridge modifies the sensitivity of the isopycnal slopes to changes in wind forcing. In particular, at weak winds the isopycnal slopes fan out before converging at strong winds due to the interaction between the barotropic flow and the overturning forcing. However, there is a gap in understanding the dynamical processes of eddy compensation and how they differ between a flat-bottomed ocean and one with undersea topography. These leave room for understanding the response in an intermediate model, one that is eddy-resolving but with idealized geometry, paired with a three-dimensional analysis.
It is challenging to connect the zonally integrated residual-mean theory of overturning to a three-dimensional circulation dominated by ridges that are prominent features in the Southern Ocean. We know that local vertical velocities are elevated near topographic features. The Southern Ocean State Estimate shows that, in the multiyear mean, regions of strong vertical velocity are isolated near topographic features, like Drake Passage, Macquarie Ridge, and Kerguelen Plateau (Fig. 1), suggesting the importance of the topography for driving upwelling (Mazloff et al. 2010). This has been confirmed in models which show particle upwelling localized to regions downstream of topography using a Lagrangian viewpoint of the circulation (Tamsitt et al. 2018; Viglione and Thompson 2016). The localized upwelling is thought to be caused by the enhancement of baroclinic eddies downstream of topography (Tamsitt et al. 2018; Barthel et al. 2022). Additionally, downwelling has been shown to be driven locally by the formation of Antarctic Intermediate Water and Antarctic Bottom Water (Talley 2013). Radko and Marshall (2006) connect the residual overturning circulation to the three-dimensional nature of the overturning using a perturbation expansion on the zonally averaged circulation, but it is difficult to generalize. To quantify the net overturning transport, we need to adapt residual-mean overturning theory in a three-dimensional sense to discover where the transport is occurring, how strong it is, and what is driving the transport.
In this study, we use an MITgcm channel model to represent a section of the Southern Ocean with idealized geometry, topography, and forcings, to investigate how the topography localizes the overturning circulation. We describe the numerical model and methods used to analyze the flow in section 2. In section 3 we present our results, and in section 4 we discuss the key dynamical processes and the implications of our analysis. We conclude our study in section 5.
2. Methods
We investigate the full three-dimensional circulation around topography in the Southern Ocean using the simplest model that encapsulates the necessary complexity: a channel with an idealized meridional ridge (north–south) Gaussian ridge, representing a sector about one-quarter of the zonal extent of the Southern Ocean. Boundary conditions are chosen to create an appropriate magnitude overturning circulation with realistic mean wind stress values. In this configuration we test ideas of the overturning circulation using both the residual-mean formulation in two dimensions and thickness-weighted circulation in three dimensions. Particles are advected in the thickness-weighted circulation to visualize this flow that represents the meridional overturning circulation.
a. Ocean model
We use an MITgcm (Marshall et al. 1997) channel configuration that is 4000 km long × 2000 km wide, reentrant in longitude (Fig. 2). The model is run primarily at 10 km horizontal resolution with select simulations at 5 km. The deformation radius in this configuration is about 15 km, but features tend to be larger than the deformation radius (Pedlosky 1987), making this configuration adequately eddy-resolving. The depth is 4000 m, with 32 points in vertical, from 10 m grid spacing at the surface to 280 m at the bottom, as used in previous studies (Abernathey et al. 2011; Youngs et al. 2017). Here, the model has a constant salinity where the buoyancy/density is entirely controlled by temperature. For topography, a meridional Gaussian ridge of half-width 200 km, 2000 m tall, is located 800 km downstream of the channel entrance (Fig. 2), representing a characteristic undersea ridge in a sector of the Antarctic Circumpolar Current. We run simulations with a flat bottom to quantify the impact of the ridge. We employ a 600 s time step and a diffusivity varying from 0.01 m2 s−1 in the 40-m-thick mixed layer to 1 × 10−5 m2 s−1 below, transitioning with a tanh profile: κ = 10−5 + (10−2 − 10−5) × [tanh(z + 40) − 1]. There is a linear bottom drag with a drag coefficient of 1.1 × 10−3m s−1 and free-slip sides.
b. Boundary conditions
There are two main options for setting up surface heat flux conditions: a fixed flux condition where a prescribed amount of heat is put in or pulled out of the surface ocean, or a relaxation to a set meridional temperature profile at the surface, where the amount of heat fluxed in or out of the surface is proportional to the difference of the surface temperatures from the selected profile (Haney 1971; Abernathey et al. 2011). We use the latter condition as it provides better representation of the air–sea exchange, and as a result, the overturning becomes considerably more sensitive to the wind changes than in the case of a fixed flux (Abernathey et al. 2011). A fixed heat flux effectively constrains the overturning, making it insensitive to the wind.
We need to be careful, however, when setting up the relaxation condition. A fixed surface relaxation temperature profile for both the flat-bottom and ridge cases leads to wildly different overturning circulations when using the same relaxation profile for both. One configuration or the other lacks one of the overturning cells. To make a fair comparison, we have adopted a method which allows both the use of a relaxation boundary condition and a comparable overturning streamfunction in the flat-bottom and ridge simulations following Abernathey et al. (2011).
In summary, we ran one flat and one ridge simulation with fixed flux to determine the correct temperature relaxation profiles, then fixed the profiles for the wind perturbation experiments. This creates comparable overturning circulations in our control integrations, as seen in the overturning streamfunctions in depth and temperature space in Fig. 4, even though we are using different relaxation temperature profiles with and without a ridge (Fig. 3c). The zonally averaged heat fluxes are the same between the two cases, but the overturning is different because the surface buoyancy gradient is different [Eq. (3)]. In particular the magnitude of the lower cell is weaker in the ridge simulations, but is as close as possible with the same heat fluxes.
c. Thickness-weighted circulation
We examine the spatial distribution of upwelling and downwelling, as well as determine the relative importance of the different dynamical components, by tracking particles in the thickness-weighted mean velocity
We use Young’s (2012) formulation of thickness-weighted horizontal and vertical velocities because the overturning circulation is the net Lagrangian transport of water parcels advected by both the Eulerian and time-varying, eddying, flow. We need the thickness-weighted velocities because the mean velocities are not always parallel to the mean buoyancy surfaces
We use the “layers” package of the MITgcm to calculate
d. Particle tracking
To visualize the three-dimensional overturning, we track particles in a single release in the thickness-weighted velocities. We release particles at x = 100 km, upstream of the topography, at every y and z location on the native grid, and then advect the particles with the time-mean thickness-weighted velocities using a Runge–Kutta second-order time-stepping scheme. The particles are catalogued based on their initial temperature. They are advected from the western edge (entrance of the channel) until they either exit the domain to the east or run into the diabatic region in the north. We discard particles that enter the top 250 m, and those that enter the northern temperature relaxation region.
The computation takes advantage of the fact that the relevant tracer advection is by the thickness-weighted velocities
3. Results
In this section we explore the meridional overturning circulation of our Southern Ocean–like channel model. First, we calculate the overturning streamfunction using the residual-mean theory in two dimensions and examine the sensitivity of the overturning to changes in wind, building on Abernathey et al. (2011). Second, we use particle tracking in the thickness-weighted velocities (Young 2012) to provide a three-dimensional estimate of the meridional overturning circulation. From this we expand on the relative contribution of different drivers of the overturning and show that the flat-bottom configuration reflects the residual-mean theory.
a. Residual-mean overturning streamfunction and partition of flow into cells
The overturning streamfunctions calculated for these configurations are reasonable representations of the Southern Ocean meridional overturning circulation. Water at middepths is transformed by the northern relaxation condition, where it then exits the northern forcing region from the north and upwells along isopycnals (as expressed by the residual overturning streamfunction). Once it reaches the surface, some of this splits to the south to form the downwelling branch of the lower cell, while the rest travels north and downwells in the upper cell. We can identify these two branches based on the temperatures of the overturning in temperature space, which corresponds to the northernmost section of the domain in Fig. 4c where the streamfunction contours are flat. The sign of the streamfunction in Fig. 4 indicates whether it is the upper cell (red) or lower cell (blue). By definition the flow is parallel to streamfunction contours, and the gradient (based on the color) dictates the direction of the flow. So, the maximum of the streamfunction delineates where the upwelling and downwelling branches split and the flow changes direction; see arrows drawn on Figs. 4b and 4d.
We split up the overturning into different cells and branches based on the overturning in the northern section of the domain, differentiating them based on regions of water mass import and export at the northern boundary. Based on these conventions, we describe the overturning branches as shown in Figs. 4a and 4c. The lower-cell downwelling branch is located in the coldest temperature range from 0° to 0.5°C in the flat-bottom and from 1° to 1.8°C for the ridge configuration. The lower-cell upwelling occupies the temperature range of about 0.5°–2°C for the flat-bottom configuration and 1.8°–2.5°C for the ridge configuration. The upper-cell upwelling is characterized by the temperature range of 2°–6°C for the flat-bottom configuration and 2.5°–5°C for the ridge configuration.
One potential pitfall of this zonally integrated analysis is that mapping a zonally varying flow with a zonal mean can be misleading. When we consider the overturning in temperature space for the ridge simulations (Fig. 4c), we see a blue “lobe” of overturning attached to the lower-cell overturning. We show that this feature is entirely within the envelope of surface temperatures observed at a given latitude across all longitudes and times (Fig. 4c), which combined with a thought experiment of the thermal pathway of water in the standing meander indicate that this is due to the water in the meander being warmed as it is brought north and cooled as it moves south. However, when this feature is projected into depth space using the zonally averaged isopycnals the surface meander ends up partially below the “surface” of the ocean, appearing to indicate diabatic processes at depth. This is entirely a result of the use of the zonally averaged isopycnals for the depth projection; the flow is (almost) entirely adiabatic below the surface. Due to the lack of diabatic mixing, cross-isotherm transport has been shown to be minimal in configurations like this (Marshall and Speer 2012; Marshall and Radko 2003; Gille et al. 2022). Thus, we regard the “true” overturning as that seen in the northern third of the domain; this is the overturning that is in the ocean interior and not affected by surface transformation.
Isopycnal slopes are much steeper for the flat-bottomed simulations, Figs. 4b and 4d, causing the lower cell to be much deeper in the flat-bottomed simulations than simulations with the ridge (Abernathey and Cessi 2014). This has implications for which water masses reach the surface and where they are exchanged with the rest of the ocean (Ferrari et al. 2014). For example, if isopycnal slopes are steeper then denser water masses exchange with the surface further north. In addition, the baroclinicity of the flow is on average higher in the zonally symmetric flow, but sometimes significantly higher locally in the ridge configuration (not shown).
b. Sensitivity of the overturning strength to wind
The addition of a ridge modifies the equilibrium sensitivity of the overturning to changes in wind stress despite the fact that we carefully calibrated the surface forcing such that the overturning streamfunctions are comparable in the control integrations. Simulations with varying wind stress were run until the transport and the overturning stabilized, about 40 years. Figure 5 shows how the equilibrated overturning strength changes with the wind stress for both flat-bottomed and ridged simulations. The flat-bottomed simulations here reproduce the findings of Abernathey et al. (2011). The sensitivity to the wind stress is qualitatively similar for the upper cell for both cases, but the lower cell responds very differently. The lower-cell strength weakens significantly in response to increased wind forcing in the flat-bottom configuration, but with a ridge, the overturning barely changes. To ensure this was not an artifact of resolution, we also ran the simulations with a 5 km resolution and observed the same behavior: with a ridge, the lower-cell overturning hardly responds to wind changes, in contrast to a marked sensitivity with a flat bottom. We will discuss the dynamical process at play in section 4.
How the overturning responds to changes in winds depends critically on the response of the isopycnal slopes. As the isopycnals move, the surface outcrop positions change, modifying the heat fluxes, which in turn changes the strength of the overturning, as seen through Eq. (3). For our analysis, the isopycnal slopes are computed locally and then averaged over each overturning cell, as defined by temperature and overturning streamfunction in the previous section. The slopes in the upper cell increase dramatically as the wind stress increases in the flat-bottom configuration (Fig. 5). When there is a ridge, however, the upper-cell slope approaches a constant value (Fig. 5). This mirrors the behavior seen in Youngs et al. (2019): as the winds increase beyond a critical threshold, all of the wind changes are absorbed by the standing meander. For the lower cell, isopycnal slopes increase with the wind in the flat-bottom configuration, but with a ridge, they decrease as the winds increase. The reason for this is laid out in Youngs et al. (2019): the key is a strengthening of the meander as the wind stress increases and its interaction with the overturning circulation. As the wind stress goes to zero with a negative overturning (lower cell), the buoyancy forcing itself supports the isopycnal slopes and the baroclinic transport, with eddies uniform throughout the domain (and therefore less efficient than with localized eddies), leading to a steeper isopycnal slopes with a weak wind. As the wind increases, the flow barotropizes and the flow “feels” the bottom topography, generating a standing meander and making the eddies more efficient at extracting available potential energy, making the isopycnal slopes flatter. For a positive overturning (upper cell), at weak winds, the overturning by itself does not support the isopycnal slopes, leading the slopes to be flatter with less wind forcing. This is also seen in Fig. 5. The isopycnal slope changes then dictate how the overturning changes with the wind.
c. Quantifying the overturning with particle displacements
Now we turn our analysis to focus on the three-dimensional structure of the overturning by visualizing water parcel trajectories via the thickness-weighted circulation. We first examine particles advected on three different temperature surfaces. In Fig. 6 we show the position of particles released at the entrance of the domain in the ridge simulation. The vertical displacements are shown in colors. First, we observe that particles on all temperature surfaces shift north ahead of the ridge, and then back southward downstream of the ridge, generally fanning out downstream. In terms of the vertical displacement, the particles rise up ahead of the topography and then travel downward almost to their initial height by their northernmost point specifically in the colder temperature limits, Figs. 6a and 6b. For these colder particles, just downstream of the topography, the displacements are downward in the south and upward in the north, a signature of the isopycnal flattening due to eddy activity in the 1000 km downstream of the ridge. In the far downstream region, there is a general downwelling trend for the coldest temperature, 1°C, Fig. 6a; consistent with the fact that this temperature is associated with the branch of the overturning circulation (see Fig. 4c). For the middle temperature, 2°C, there is a general upwelling trend; this temperature is part of the upwelling branch of the overturning (see Fig. 4c). For the warmest temperature surface, there is a general upwelling in the downstream of the ridge, with a slight downwelling far downstream, this part of the upper-cell upwelling branch, Fig. 6c.
We want to expand this analysis to understand the overturning transport on all temperatures. Figure 6 only provided a partial picture because we could only look at particles on a few specific temperatures. To achieve this goal, we advect particles in all depth (therefore all temperatures) and latitude positions on the native grid, starting near the western edge (the entrance) of the channel, calculating the displacements for each particle. From this, we collect the particles into 0.1°C bins, and then we average across all latitudes (Y) in each temperature bin to quantify the cumulative upwelling over the channel as a function of temperature, allowing a visualization of the upwelling (Fig. 7). We examine the cumulative displacements because the local upwelling is difficult to visualize since the horizontal velocities (and thus residence times) vary significantly across the domain.
The vertical displacement is also normalized by the total time it takes to transit the domain, as the total displacement in one transit of the channel is much larger for the colder temperature bins due to the fact that the transit time is much slower. The transit time is about 2 years for the warmer bins, but closer to 9 years for the colder bins. We normalize by dividing by the mean total time of each temperature layer, but multiplying by the mean of all temperature layers to preserve the units. This allows a clearer comparison between the different temperature layers. Accounting for the difference in transit time, however, makes the total displacement appear to be much smaller for the flat-bottomed configuration. This is because the flow is significantly faster in the flat-bottomed configuration due to significantly larger zonal transport; particles have less time to move vertically per transit through the channel. The net overturning is similar in both configurations by construction, and particles simply need to circumnavigate the flat-bottomed channel multiple times to experience the same displacement as the ridge configuration.
We want to move away from using the term residual transport for the net transport because of the lack of cancellation between the mean and the eddy transports. Instead, we call the full vertical displacement experienced by the particles in the thickness-weighted velocity the total displacement because it includes all of the components defined below. To calculate the Eulerian-mean displacement, we tabulate the vertical displacement from just the Eulerian-mean velocity in depth space. The particles are still advected in x, y, and z using the full thickness-weighted flow, but the vertical displacement is calculated by multiplying the local time-averaged Eulerian-mean w by the time step and adding it to the previous time step’s running mean to calculate the mean displacement. This is the cumulative integral of the Eulerian-mean vertical velocity experienced by the particle. Then, we define the transient eddy component as the difference between the Eulerian mean and the total thickness-weighted component. Finally, we further split up the Eulerian-mean displacement into two components, an Ekman displacement and a standing eddy displacement. The Ekman displacement is the displacement calculated using the Ekman velocity [
1) Eulerian mean versus transient eddies
Now we investigate the three-dimensional circulation using our new method to show the complex nature of the overturning. The mean cumulative vertical displacement of particles transiting the domain west to east in temperature bins is shown in Fig. 7. In Figs. 7a–c we show the cumulative displacement in the ridge simulations, and Figs. 7d–f show the same for the flat-bottomed simulations for comparison. The total vertical displacement is shown in Figs. 7a and 7d, the Eulerian mean in Figs. 7b and 7e, and the transient eddies in Figs. 7c and 7f. First, we examine the total displacement with a flat bottom, Fig. 7d, which reveals a purely linear increase (or decrease) in particle displacement, consistent with the zonally uniform flow of the flat-bottomed simulations. In the upwelling branches of the upper and lower cells, characterized by temperature of 0.5°–5°C, particles are displaced upward, except for warmer temperatures than 4°C when the numbers of particles used for the calculation is too limited. The lower-cell downwelling, appearing only for the coldest temperatures, shows downward displacement. As is well known, this net transport is the residual between two much larger terms, the Eulerian mean (Fig. 7e) and the transient eddies (Fig. 7f), which exhibit opposing signs. In the flat-bottom simulations, the mean component is nearly balanced by a transient eddy component, leaving a small residual, well described by the residual-mean theory (Abernathey et al. 2011).
The ridge case exhibits more complex displacements. Most of the structure comes from the upstream and within 1000 km downstream of the ridge, Fig. 7b. The total upwelling is very localized in longitude with peaks located to the west of the ridge at 600 km and in the downstream region at 1200 km, Fig. 7a. By the end of the domain, we see the net upward and downward displacement we expect, with net upwelling in the temperature range of 1.8°–4.5°C (upper- and lower-cell upwelling) as seen in Fig. 4c, and a similar correspondence for the lower-cell downwelling (0.5°–1.8°C). Thus, the total trajectory-based calculations reflect the net overturning as quantified in the “usual” way. The Eulerian-mean component, Fig. 7b, exhibits a complex pattern that we discuss in the next section. The transient eddies (Fig. 7c) are localized in the downstream region within 1000 km of the ridge, where they consistently drive upward or downward displacement on a given temperature surface, with little variation in longitude. In the upper cell, there is remarkably little cancellation between the Eulerian mean and eddy circulation, the latter dominates the total. In the lower cell, downwelling is driven by both transient eddies and the Eulerian mean, again with a notable lack of cancellation. The lower-cell and upper-cell upwelling are driven by distinct processes, which has implications for how the overturning responds to changes in the wind stress, as shown in Fig. 5. This technique really shows the difference between the residual-mean understanding of the overturning and the localized, thickness-weighted understanding.
2) Decomposing the Eulerian-mean overturning
In this section, we describe the Eulerian-mean vertical displacement decomposed into the Ekman overturning and the standing eddy overturning. As seen in Fig. 7b, the pattern of the Eulerian-mean vertical transport in the ridge simulation exhibits two main components, a first region localized near the topography and a second region where displacement grows gradually, linearly increasing or decreasing throughout the rest of the domain. To better understand this structure, we split the Eulerian velocity into an Ekman component (computed directly from the surface stress) and the remainder, which is associated with the standing eddies. This allows us to compute the displacement from each component separately. In Figs. 8a–c, we show the cumulative displacement in the ridge simulations, and in Figs. 8d–f we show the comparison to the flat-bottomed simulations. The Eulerian displacement is shown in Figs. 8a and 8d, the Ekman displacement in Figs. 8b and 8e, and the remainder, the standing eddy component, in Figs. 8c and 8f. In the flat-bottomed case, Eulerian displacements are associated with Ekman flow (Fig. 8), a consistency check as there should be no standing eddy component in this configuration. The small standing eddy displacement here reflects the finite time mean, and that the distribution of particles is not necessarily representative of the total volume of water.
For the ridge case, much of the longitudinal structure is associated with the standing eddy component, which dominates in the upstream and the near-downstream regions. The Ekman component, however, also exhibits significant longitudinal structure. This is because the wind-driven upwelling, that is, the mass flux in isopycnal layers driven by the Ekman velocity is not entirely zonally symmetric in the ridge case. The isopycnal slopes vary considerably throughout the domain, leading to northward and southward excursions of the temperature surfaces, and in turn, variations in the wind forcing in longitude along an isopycnal. Thus, while the Ekman velocities are the same for the flat and the ridge cases, the upwelling within isopycnal layers differ because of the thermal structures. In the ridge simulation, all of the components of the overturning displacements are zonally asymmetric.
4. Discussion
We have examined the role of topography in the overturning circulation, quantifying the three-dimensional nature of the overturning with a new technique based on the thickness-weighted mean velocities. This technique has shown that in the presence of bottom topography, the meridional overturning circulation is transport driven by the wind, standing and transient eddies, in similar proportions in a highly asymmetric and localized way. In this section, we discuss the implications of this more detailed understanding of the overturning.
a. The standing eddy role in eddy compensation
The standing eddy component provides a leading-order contribution to the overturning circulation in the ridge simulations. We have defined the standing eddy circulation as the difference between the total Eulerian mean and the Ekman component. This differs from previous studies where it is characterized as the deviation from the zonal average (Bishop et al. 2016; Dufour et al. 2012). We chose this definition because the wind-driven (Ekman) component is not zonally symmetric, due to the variation of the outcrop positions, and we wanted to exclude this seemingly standing eddy-driven component that is really driven by the Ekman (wind) flow. The standing eddy component drives enhanced upward displacement at all levels upstream of the topography (Fig. 8c). Downstream of the ridge, the standing meander drives significant downward transport in the lower cell, and upward transport in the upper cell. This supports the conclusions of several studies with more realistic geometries, which also highlight the dominant role of standing eddies in transport (e.g., Bishop et al. 2016; Mazloff et al. 2010; Kong and Jansen 2021).
We showed how the isopycnal slopes respond to increases or decreases in wind speed and how this alters the surface buoyancy fluxes (section 3b). Our understanding of the sensitivity is incomplete, particularly concerning which dynamical components of the overturning respond to changes in the wind. In eddy-permitting models, several studies found that the standing eddies absorb the changes in the wind (Dufour et al. 2012; Bishop et al. 2016). In addition, standing eddies have been shown to sharpen in response to increases in wind stress (Thompson and Naveira Garabato 2014).
The standing eddy component is especially important in the lower cell. If the standing eddy effectively absorbs changes in the wind (Bishop et al. 2016), we would expect that the overturning would not be very sensitive to wind strength. On the other hand, the processes driving the upper cell are more similar between the flat-bottom and ridge integrations, with a wind stress balanced by transient eddies (albeit localized in the ridge case), so the upper cell’s response to the wind is qualitatively similar in both configurations. As the processes driving in the upper and lower cell exhibit different sensitivities to the wind, all components of the overturning need to be modeled correctly in order to produce an appropriate overall sensitivity, which is relevant for modeling changes in future climate.
b. Metrics for evaluating models
1) Localized enhanced diffusivities
The localization of eddies just downstream of the ridge is very important for how the system responds to winds, controlling the impact of varying wind strength on the isopycnal slopes. In addition, the enhanced localized eddy activity causes eddy-driven upwelling in the vicinity of the topography (Fig. 7). Regions of enhanced eddies are evident in observations and more realistic models (Hogg et al. 2015; Bishop et al. 2016). This increased eddy activity leads to elevated eddy diffusivities near the topography (Abernathey and Cessi 2014; Sallée et al. 2011), with implications for the response to changes in wind forcing.
In a comprehensive review of modeling studies examining the overturning response to wind forcing, Gent (2016) found that model configurations where the Gent–McWilliams diffusivity κ was allowed to vary in latitude and longitude, based on the local stratification, were less sensitive to changes in wind (see also Farneti et al. 2015). We suspect this is because the localization of the eddy activity becomes part of the parameterized Gent–McWilliams diffusivity. Allowing a spatially variable diffusivity suggests a way forward to ensure that climate models appropriately respond to changes in wind stress (Kong and Jansen 2021).
2) Three-dimensional signatures in zonally averaged streamfunction diagnostics
Now that we have examined the three-dimensional structure to the overturning in detail, we can ask if it is possible to see its signature in the standard zonally integrated streamfunction diagnostics. Figure 9 shows a conventional, zonally integrated streamfunction of the overturning calculation split into Eulerian-mean and eddy streamfunctions. Figures 9a and 9d is the same as Figs. 4b and 4d replotted for convenience and a new color scale. We calculate the Eulerian-mean streamfunction as the integral of the Eulerian-mean velocities in constant depth. Calculating the Eulerian-mean streamfunction on isopycnals instead of depth, would enable a different split into standing and transient eddies (e.g., Bishop et al. 2016), but here we want to emphasize the implications of the more naive calculation.
For the ridge case, the total transport in this zonally integrated streamfunction framework appears to be a small residual of the Eulerian-mean and eddy components, in contrast to the particle tracking diagnostics where the total, Eulerian-mean, and eddy components are all the same magnitude (Fig. 7). In this sense, the zonally integrated streamfunction framework reflects a different breakdown of the driving factors of the overturning transport of water parcels. In this context, the total overturning is set by the surface and northern boundary conditions. The Eulerian mean is dominated by the wind stress, which is larger than the thermal boundaries can support. The eddies must offset the wind-driven transport, but they do so differently between flat and ridge configurations. In the flat-bottomed case, the eddies act uniformly along the channel, while in the ridge case the eddies are all concentrated near the ridge. All that remains of this difference between the ridge and flat cases in the zonal mean diagnostics are the vertically banded features in the Eulerian mean and the eddy overturning for the ridge case due to the zonal asymmetry (Figs. 9b,c).
Using the particle tracking (Fig. 7), however, suggests that by the end of the domain, the Eulerian-mean component is small relative to the total overturning. This is in contrast to the traditional residual-mean overturning streamfunction diagnostics, where the Eulerian-mean component (dominated by the winds) is larger than the total. Why is the Eulerian-mean component so much smaller, relative to the other components, for the particle tracking diagnostics? We argue that this is due to the fact that particles travel north and south throughout the domain, so they experience the Ekman velocity to varying degrees depending on their position in Y. In addition, we average over Y in temperature capturing a partial cancellation between flow in the north and south. The effect is larger for the ridge case because of the flatter isopycnals. This highlights the need for caution when analyzing the zonal-mean overturning: it hides the zonally asymmetric nature of the transport. The manner of averaging is critical for capturing the effective transport.
We have computed all of the above zonally integrated diagnostics by averaging along latitude bands (or in constant Y). Another alternative is to calculate the zonal average along streamlines as done by Viebahn and Eden (2012). This effectively transfers the contribution of the standing eddy component to the transient eddy component. There are many different ways to choose the streamlines, however, complicating this approach. For this reason, we average along constant latitudes, and consider both the standing and transient eddy flow.
5. Conclusions
We have diagnosed the three-dimensional structure of the meridional overturning circulation using the thickness-weighted circulation in an idealized Southern Ocean–like channel. This reveals that the zonally integrated residual-mean theory provides an insufficient picture due to the localized nature of the overturning, which we have illustrated schematically in Fig. 10. This is highlighted using a technique to interpret the three-dimensional thickness-weighted circulation with particle tracking. The zonally integrated thickness-weighted circulation reduces to the residual-mean limit (Marshall and Radko 2003; Young 2012), providing a connection between our work and the established theory. We also build on recent work investigating the connection between upwelling and eddies (Tamsitt et al. 2018; Barthel et al. 2022) by definitively showing a causal relationship between enhanced eddy activity near topography to enhanced overturning.
Zonally averaged diagnostics hide the three-dimensional nature of the flow in the channel with topography. The zonally averaged and net overturning looks broadly similar between the ridge (zonally asymmetric) and flat-bottomed (zonally symmetric) case by construction. However, the topography dramatically reduces the sensitivity of the lower-cell overturning to changes in wind. This is consistent with eddy compensation arguments, which posit that standing eddies absorb much of the change of the wind (Dufour et al. 2012; Bishop et al. 2016). Thus, including the topography is essential for predicting the sensitivity. In addition, in the downstream region of the ridge configuration, where the flow is essential zonally symmetric, the compensation between eddy and mean is not like the zonally symmetric understanding but is instead something different because the ridge effectively eliminates eddy activity anywhere but the region within 1000 km downstream of the ridge (see also Youngs et al. 2019).
The idealized geometry provides a clear picture of the overturning, but leaves out many other factors likely to be important for Southern Ocean overturning, such as the Drake Passage restriction, other topography, ice, and feedback from coupling to the north. While we expect the importance of the standing meanders to the overturning and its response to wind changes will still apply in a more realistic geometry, more investigation is necessary to tie our results to the real-world circulation. We suggest using the time-mean thickness-weighted velocities to track the overturning in a more complex model with realistic geometry.
Other limitations of our study concern missing processes. In the Southern Ocean, Antarctic Bottom Water forms primarily at two very localized places, the Weddell and Ross Seas (Talley 2013; Rintoul 2018). We did not replicate that localization, or the specific overflow processes that generate the water mass. In addition, the lack of atmospheric coupling is another limitation, due to the importance of local air–sea heat fluxes which are crudely represented by our surface relaxation condition.
Nonetheless, our results suggest the value of a better theoretical understanding of how eddies develop in baroclinic flow around and across the ridge. Topography can destabilize a baroclinic flow and/or create meanders, which are subject to potential vorticity wave instabilities: barotropic and baroclinic instability (Youngs et al. 2017). In addition, the upstream flow is, in reality, time dependent, so that transient growth and waves can grow downstream rather than just in place. We believe that experiments like ours can provide insight into the spatial and temporal development of the transient eddies and their impact on the time-mean flows. In addition, the technique developed here could be of use to the community by allowing a Lagrangian analysis of water parcels that is less computationally intensive.
Our study emphasizes that a full three-dimensional analysis of the zonally asymmetric transport is fundamental for understanding the global overturning, and that the localization of the flow is also important in other ways. In our experiments, the heat flux was tuned to be similar between the flat-bottom and ridge configurations, but it is distributed in space nonuniformly (Fig. 3). We expect this inhomogeneity to alter the feedback on the atmosphere as well. In addition, biological processes, which are very sensitive to vertical velocity, will be altered by topographic effects which are often not well resolved in climate models. As highlighted in Youngs (2020), the flow near the ridge leads to very specific patterns of carbon outgassing, suggesting the necessity of highly localized, in situ measurements near topography to appropriately sample the carbon budget and vertical transport in the Southern Ocean.
Acknowledgments.
We acknowledge helpful discussions with Mara Freilich, Andrew Stewart, David Marshall, Nicole Lovenduski, Michael Spall, and Edwin Gerber. MKY acknowledges funding from the National Defense Science and Engineering Graduate Fellowship, a NOAA Climate and Global Change Postdoctoral Fellowship, NSF OCE-1536515, and an allocation at NCAR CISL UMIT0025. This research received support by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program. GRF acknowledges funding from NSF OCE-1459702. Computational resources for the SOSE were provided by NSF XSEDE Resource Grant OCE130007 and SOCCOM NSF Award PLR-1425989.
Data availability statement.
The code used to run the model is available at https://doi.org/10.5281/zenodo.4730689. The SOSE output is available at sose.ucsd.edu.
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