## 1. Introduction

The abyssal ocean is a large reservoir of heat and carbon for the climate system, and the abyssal overturning circulation has often been hypothesized to be a key player that has modulated climate feedbacks during past climate changes (e.g., Adkins 2013, and references therein). Despite its importance, the mechanisms that govern these changes in the abyssal overturning circulation remain poorly understood.

The classic theoretical picture for the abyssal overturning circulation, still found in textbooks, is built on the early work by Stommel and Munk. Stommel (1961) proposed a two-box model to illustrate the role of temperature and salinity forcing on the overturning circulation, although ocean dynamics and mixing processes that are crucial to return the deep water to the surface are treated implicitly. Munk (1966) and Munk and Wunsch (1998) focused on the buoyancy budget of the abyssal upwelling, assuming it to be uniform throughout the basin and constrained by the balance between vertical advection and turbulent diapycnal diffusion. Stommel and Arons (1959) investigated the horizontal structure of the abyssal circulation, based on the vorticity dynamics in a closed basin, but the rate of deep water formation must be prescribed in their theory.

The modern literature on the abyssal overturning circulation has come to a general agreement that the minimum conceptual model to represent its dynamics should account for not only a basin but also a circumpolar channel representing the Southern Ocean (e.g., Toggweiler and Samuels 1995; Ito and Marshall 2008; Nikurashin and Vallis 2011; Shakespeare and Hogg 2012; Stewart et al. 2014). This view accounts for the fact that a large amount of deep water upwells to the surface in the Southern Ocean. The Southern Ocean circulation is driven by Ekman pumping and suction, which highlights the potential role of wind stress forcing, in addition to the buoyancy forcing and turbulent mixing considered in the classical models (Lumpkin and Speer 2007; Marshall and Speer 2012; Talley 2013, and references therein).

Building upon these findings, recent theories for the dynamics of the abyssal overturning circulation have put the most emphasis on the roles of wind stress and baroclinic eddies in the Southern Ocean (Ito and Marshall 2008; Nikurashin and Vallis 2011; Mashayek et al. 2015). In these theories, it is implicitly assumed that the dynamics in the basin are not essential to constraining the circulation, although the thermodynamics in the basin are relevant. Specifically, Nikurashin and Vallis (2011) developed a scaling theory that matches the wind- and eddy-driven circulation derived from the momentum budget in the Southern Ocean, to a basin that is effectively a one-dimensional advective–diffusive model as envisioned in Munk (1966). Mashayek et al. (2015) suggested improvements by pointing out the necessity of horizontal advection in maintaining the buoyancy budget in the abyssal boundary layer, but the resulting scaling relations do not change fundamentally, as will be discussed below. In both studies, the proposed scaling relations are compared with three-dimensional numerical simulations with an idealized model geometry that consists of a basin and a channel. The simulations confirm the theoretical predictions, indicating that the basin dynamics may indeed be unimportant.

On the other hand, there are scaling theories that focus on the dynamics in the basin, while not explicitly considering the momentum budget in the Southern Ocean (Kamenkovich and Goodman 2000; Jansen and Nadeau 2016, hereinafter JN16). The argument of JN16 is motivated by a need to better understand the circulation impact of a changing surface buoyancy loss around Antarctica (e.g., Shin et al. 2003; Shakespeare and Hogg 2012; Stewart et al. 2014; Ferrari et al. 2014; Watson et al. 2015; Sun et al. 2016). None of the previously discussed channel-based scaling theories (Ito and Marshall 2008; Nikurashin and Vallis 2011; Mashayek et al. 2015) predict the effect on the circulation of changes in the surface buoyancy loss, and JN16 argued that to do so one cannot avoid discussing basin dynamics. While not a closed theory, they accordingly derived a scaling relation that is substantially different from the theories considering the channel dynamics and points toward the important role of the net surface buoyancy loss in the region of bottom water formation. They also confirmed that their scaling is generally supported by their numerical simulations. However, an apparent discrepancy exists between JN16’s theory, which focuses on basin dynamics, and the other existing theories, which focus on channel dynamics.

This study aims to explain this discrepancy by testing the theories and their assumptions with the help of numerical simulations of an idealized model. To ensure that these simulations consist of the essential physics only, we employ a model configuration that is purposely chosen to eliminate complexities that are not taken into account in any of the theories. Taking advantage of this simplicity, we then examine the changes in the abyssal circulation with various parameters and compare the results with the scaling theories. We primarily focus on assessing the theoretical predictions for the overturning strength of the abyssal circulation, as its changes appear to be more robustly defined than other potentially interesting aspects of circulation changes.

The rest of the paper is organized in the following way. The existing theories are formally introduced in section 2, and an extension to close the scaling of JN16 is proposed. Section 3 describes the setup of our model simulations, the parameter range explored, and the details of the diagnostics performed. Section 4 then compares the existing theories with the simulations. In section 5, we investigate the extent to which the results are modified in the presence of bottom topography in the channel. Section 6 summarizes our findings and concludes with some questions for future work.

## 2. Existing theories for overturning strength

### a. Decomposition of the residual streamfunction

*y*,

*z*), which is in turn defined as the isopycnal streamfunction, Ψ

^{†}(

*y*,

*b*), remapped from buoyancy (

*b*) coordinates to depth (

*z*) coordinates. The theories we examine in this study are derived from (and compared with) idealized models that do not resolve mesoscale eddies. The Gent–McWilliams (GM) scheme is adopted in these models to parameterize the meridional transport by mesoscale eddies,

*υ*

^{GM}≡ −∂

_{z}(

*K*∂

_{y}

*z*|

_{b}) with

*K*being the GM diffusivity (Gent and McWilliams 1990). Since resolved transient eddies are generally negligible in coarse-resolution simulations that employ the GM parameterization, the total meridional transport can be approximated as

^{EU}is the Eulerian mean streamfunction, Ψ

^{GM}is the GM streamfunction that represents the parameterized transport by transient mesoscale eddies, and Ψ

^{ST}is the stationary eddy streamfunction due to any zonal asymmetry of the flow. Unlike the residual streamfunction, the Eulerian mean streamfunction, and the GM streamfunction, are defined by zonally integrating the meridional transport at fixed depth:

^{ST}(

*y*,

*z*) = Ψ(

*y*,

*z*) − (Ψ

^{EU}(

*y*,

*z*) + Ψ

^{GM}(

*y*,

*z*)). A summary of the different streamfunction notations and their definitions is provided in Table 1 for easy reference. These different streamfunction notations will be repeatedly brought up in the discussion, since Eq. (3) is the starting point of the existing theories for the abyssal overturning strength, as will be explained in the following.

A summary of the notation used for different streamfunction components and overturning strengths.

### b. Scaling relations based on channel dynamics

We first consider the existing scaling theories that are based on the dynamics in the channel and summarized in Table 2: Ito and Marshall (2008), Nikurashin and Vallis (2011), and Mashayek et al. (2015) (hereinafter collectively referred to as INM). Although the specific derivations in INM differ somewhat, the resulting scaling relations are similar, and will be rederived here. One challenge for the derivation of a scaling theory based on Eq. (3) is that, although the residual circulation is approximately constant along isopycnals below the surface layer in the channel if diabatic transformations are small, the individual terms on the rhs of Eq. (3) generally vary substantially. Averaging Eq. (3) meridionally along isopycnals in the channel alleviates this issue, while allowing us to relate the components of the streamfunction to the surface boundary conditions and the stratification in the basin to the north.

Summary of abyssal-cell scaling theories based on the dynamics in the channel. We note that Mashayek et al. (2015) proposed a second scaling to apply above the abyssal boundary layer, which depends on the *e*-folding depth scale of diapycnal diffusivity. This scaling is not included here because we are not considering the effect of a vertically varying diffusivity. As defined in the main text, *κ* is the diapycnal diffusivity, *K* is the GM diffusivity, *W* is the width of the channel, *τ*_{0} is the strength of wind stress, *ρ* is the reference density, and *f* is the Coriolis parameter. The dimensionless factor *d* encapsulates differing assumptions among these existing scaling theories (see the text).

*b*between the outcrop latitude

*y*

_{O}, where the isopycnal intersects with the bottom of the surface layer (defined here as the minimum buoyancy at the uppermost model level), and the northern end of the channel

*y*

_{C}≡ 49°S; that is,

*l*

_{C}=

*y*

_{C}−

*y*

_{O}. Averaging Eq. (3) then yields

*υ*

_{GM}and assuming

*K*to be constant, we can approximate the GM streamfunction as

*w*(

*y*) being the zonal width of the channel at latitude

*y*. Neglecting variations in

*w*(

*y*), we approximate

*w*(

*y*) ≈

*W*, where

*W*is the characteristic zonal width of the channel.

*S*

_{C}≡

*h*

_{C}/

*l*

_{C}> 0 is the magnitude of the averaged slope of the isopycnal

*b*in the channel, with

*h*

_{C}> 0 the difference between the depths of

*b*at

*y*

_{C}and

*y*

_{O}. If we assume that the Eulerian mean circulation in the channel is solely set by the wind-induced Ekman transport, we can approximate it as

*τ*the wind stress,

*ρ*the reference density, and

*f*the Coriolis parameter. Following INM, we first neglect the effect of zonal asymmetry and ⟨Ψ

^{ST}⟩ in which case Eq. (7) becomes

*τ*and the GM diffusivity

*K*, which characterize the two most important dynamical processes that control the overturning in the channel.

*S*

_{C}, which remains to be constrained. Slope

*S*

_{C}is controlled by the surface boundary conditions and the stratification in the basin to the north, which in turn can be related to the basin buoyancy budget. Specifically, although INM adopted different assumptions for the specifics of the basin buoyancy budget, they all assumed an advective–diffusive balance, and, for scaling purposes, they can all be cast in the general form,

*κ*is the vertical diapycnal diffusivity,

*l*and

*h*are horizontal and vertical length scales, with

*h*characterizing the vertical stratification and

*lW*being the area of diffusive upwelling. Equation (11) suggests that Ψ within the basin is controlled by an aspect ratio,

*h*/

*l*, while the overturning in the channel is controlled by a different aspect ratio

*S*

_{C}=

*h*

_{C}/

*l*

_{C}in Eq. (10). None of the variables,

*h*

_{C},

*l*

_{C},

*h*, and

*l*, are necessarily known a priori and different assumptions are made in the different theories. Nevertheless, we can generally rewrite Eq. (11) as

*d*≡

*S*

_{C}/(

*h*/

*l*) =

*h*

_{C}

*l*/(

*hl*

_{C}) > 0 a dimensionless factor. The Ψ in Eq. (12) is most directly related to the transport into the basin, that is, the streamfunction evaluated at the channel–basin interface (i.e.,

*y*=

*y*

_{C}). Assuming that this quantity is well approximated by ⟨Ψ⟩ evaluated at the same buoyancy (an approximation that holds perfectly if the circulation is adiabatic in the channel), we can simultaneously solve Eqs. (10) and (12) to get

^{1}

*τ*,

*κ*, and

*K*, as well as the unknown parameter

*d*.

*τ*(

*ρf*)

^{−1}⟩ < 0. In the limit

*ε*≪ 1, Eq. (14) implies a scaling:

*d*to recover their specific definition described in Table 2), and this limit case is equivalent to assuming

*ε*≫ 1 in Eq. (14), we obtain an alternative scaling,

*d*described in Table 2). The limit case considered in Eq. (18) is equivalent to assuming that the wind-driven circulation is negligible and thus

It is important to note that for these scaling predictions to be closed, one has to further assume *d* to be relatively insensitive to the external parameters. Only then is Ψ solely a function of *κ*, *K*, and (depending on the considered limit case) *τ*, as suggested by INM, who made different assumptions to constrain *d* ≡ *S*_{C}/(*h*/*l*) = *h*_{C}*l*/(*hl*_{C}). For a thorough comparison between INM, we have listed the implied assumption for *d* for each scaling theory in Table 2. As will be shown below, *d*, as defined here, is found to vary with various external parameters in our simulations, which turns out to be a major limiting factor for the predictive skill of these theories.

### c. Scaling relations based on basin dynamics

^{GM}is of leading-order importance in Eq. (3), due to the sloping isopycnals. For the basin, JN16 instead assumed that the isopycnals can be regarded as mostly horizontal so that Ψ

^{GM}is small relative to Ψ

^{EU}. Retaining the assumption of a negligible contribution by the stationary eddy streamfunction, i.e., Ψ

^{ST}, Eq. (3) in the basin is then reduced to

*h*

_{x}, characterizing the vertical scale over which Ψ

^{EU}increases from zero to its maximum value, and the zonal buoyancy contrast across the basin Δ

*b*

_{x}. The subscript

*x*here is used to denote the scales relating to the thermal wind scaling.

*d*

_{1}≡

*h*

_{x}/

*h*, as the ratio of the two depth scales therein, one derives

*l*, Δ

*b*

_{x}, and

*d*

_{1}are left to be determined. Applying Eq. (22) to the abyssal boundary layer (ABL), JN16 assume

*l*to be the meridional extent of the ABL and to scale with the basin length (

*L*

_{B}) and diagnosed Δ

*b*

_{x}as the meridional buoyancy contrast averaged over the ABL from the simulations. [While it is the zonal buoyancy contrast that is of relevance in the thermal wind balance in Eq. (21), JN16 assumed the zonal and meridional buoyancy contrasts in the ABL to scale with one another]. With these working definitions and assuming

*d*

_{1}to be constant, they found that Eq. (22) explains the sensitivity of the basin streamfunction maximum in their simulations to the surface buoyancy loss integrated over the Southern Ocean deep water formation region,

*B*(cf. their Fig. 9). Since

*B*does not directly appear in Eq. (22), the sensitivity to

*B*is captured implicitly by the change of Δ

*b*

_{x}with

*B*.

*B*, we need to eliminate Δ

*b*

_{x}in Eq. (22) with another scaling relating Δ

*b*

_{x}and

*B*. More readily, the buoyancy flux can be related to the bulk buoyancy contrast across the abyssal cell,

*B*

_{B}is the net meridional buoyancy transport by the abyssal cell at the latitude where Ψ is evaluated. Assuming that the net buoyancy gain by the abyssal cell south of this latitude is relatively small (or amounts to an approximately constant fraction of

*B*), we expect

*B*

_{B}to be proportional to the surface buoyancy loss in the bottom water formation region,

*B*. Introducing another dimensionless factor,

*d*

_{2}≡ Δ

*b*

_{x}/Δ

*b*

_{B}, we can then combine Eqs. (22) and (23) to get

*B*. Equation (25) is a closed expression for the strength of the overturning if the length scale

*l*, as well as the dimensionless factors

*d*

_{1}and

*d*

_{2}, can be assumed constant.

While JN16 did not explicitly test Eq. (25), their simulations are seen to nicely follow Eq. (25). In JN16’s Fig. 9, the simulated streamfunction maximum varies between 2.5 and 4 Sv over a factor-of-6 change in the prescribed buoyancy loss rate, giving a power-law scaling ∝ *B*^{0.26}, which is almost indistinguishable from the prediction of Eq. (25) that Ψ ∝ *B*^{1/4} and therefore numerically supports the scaling argument.

From the above derivations, it becomes clear that the basin-based scaling, Eq. (25), predicts a very different parameter dependence from the channel-based scalings in Eqs. (16) and (18), and indeed is derived from very different dynamics. To better understand how the two sets of theories may be reconciled, we conduct a set of idealized model simulations, which are described in the next section.

## 3. Model and diagnostics

### a. Model configuration

We run our numerical simulations using the hydrostatic Boussinesq version of the Massachusetts Institute of Technology General Circulation Model (Marshall et al. 1997). The model geometry and resolution are as in JN16. That is, we have a basin spanning from 0° to 60°E and from 69°N to 48°S. Attached on its south side is a zonally periodic channel extending to 69°S. The horizontal resolution is one degree. The full depth of the model is *H* = 4 km, modeled with 28 levels with varying vertical resolution.

We use a linear equation of state, with a single buoyancy variable, *b*, which can be interpreted only in a relative sense. Throughout the domain, we apply a spatially uniform vertical diapycnal diffusivity *κ* = 6 × 10^{−5} m^{2} s^{−1}. For the parameterized mesoscale eddy transport, we use the GM parameterization with the transfer coefficient chosen as *K* = 700 m^{2} s^{−1}.

*y*

_{C}= 49°S and

*y*

_{S}= 69°S are the latitudes of the channel–basin interface and the southern end of the channel, respectively. This ensures the wind stress is maximized with its magnitude

*τ*

_{0}= 0.2 N m

^{−2}at the midchannel and vanishes at the channel–basin interface, thus effectively eliminating any direct effect of the continent on the wind-driven circulation in the channel.

*F*

_{b}is applied as

*R*, equivalent to an adjustment time scale of 3 months for the 50-m-deep top model layer, and a constant reference buoyancy,

*b*

_{R}. This results in surface fluxes over the basin being close to horizontally uniform in the simulations. In the channel, we prescribe buoyancy forcing only south of a prescribed latitude, which we interpret as the effective sea ice line,

*y*

_{I}= 60°S. South of the sea ice line, we prescribe an integrated surface buoyancy loss,

*B*, uniformly distributed over the enclosed area,

*A*

_{I}(between

*y*

_{S}and

*y*

_{I}), which amounts to a surface buoyancy loss rate per unit area,

*B*/

*A*

_{I}= 2 × 10

^{−9}m

^{2}s

^{−3}. North of

*y*

_{I}= 60°S in the channel, we impose a no-flux boundary condition, which effectively eliminates the presence of an upper (clockwise) overturning cell, which does not appear in the theories discussed in section 2. The choice to exclude the upper cell therefore aligns with our intention to include only the essential processes that enter in the existing theories, and is consistent with the approach taken by Nikurashin and Vallis (2011) and Mashayek et al. (2015).

To study the parameter dependence predicted by the theories discussed in section 2, we perform a set of sensitivity simulations. We vary one of the following parameters at a time, with their control values as described above: the prescribed surface buoyancy loss *B*, the diapycnal diffusivity *κ*, the latitude extent of the prescribed buoyancy loss (*δ*_{I} ≡ *y*_{I} − *y*_{S}; with fixed *B*), the magnitude of the prescribed wind stress *τ*_{0}, and the GM diffusivity *K*. We integrate each simulation to a statistically steady state, and all diagnostics are computed using the model output from the last 50 years.

### b. Overturning circulation in the control simulation

The meridional overturning circulation in our control simulation, measured via the isopycnal streamfunction, Ψ^{†}(*y*, *b*) [defined in Eq. (1)], and the residual streamfunction, Ψ(*y*, *z*) [defined in Eq. (2)], is shown in Fig. 1. A single well-defined anticlockwise overturning circulation is seen, which is an idealized representation of the abyssal overturning cell. The surface water, exposed to the prescribed buoyancy loss in the southern part of the channel, sinks along the sloping isopycnals in the channel until reaching the bottom of the ocean near the channel–basin interface. This “Antarctic Bottom Water” flows northward into the basin, forming the lower branch of the cell in the basin, and upwells within the basin where it encounters diapycnal diffusion and gains buoyancy as the abyssal stratification decreases toward and vanishes at the bottom, i.e., ∂_{z}(*κ*∂_{z}*b*) > 0. The upwelled water then moves southward in the upper branch of the cell at middepth and is divided into two routes. In one route the water loses buoyancy via diapycnal mixing due to the bottom-intensified stratification near the southern end of the basin, i.e., ∂_{z}(*κ*∂_{z}*b*) < 0, and thus recirculates back into the abyssal basin. In the other route, the water continues to upwell along the isopycnals in the channel back to the surface and closes the overturning loop.

### c. Diagnostic metrics for the overturning strength

Because of the recirculation near the channel–basin interface, the global maximum Ψ (in terms of the absolute value) is seen to locate within the southern part of the basin in our simulations. Such an appearance of the maximum streamfunction within the basin can also be seen in other idealized model simulations (e.g., JN16; Jones and Abernathey 2019) as well as in an ocean state estimate (e.g., Cessi 2019). JN16 apply their scaling relations to explain this streamfunction maximum in the southern part of the basin. On the other hand, as discussed in section 2, the scaling theories of INM are better interpreted as explaining Ψ in the channel or at the channel–basin interface. To account for the difference between JN16 and INM and to compare the model results with both theories, we explicitly consider two diagnostic metrics, with one measuring the overturning strength within the basin and the other measuring the overturning strength within the channel (Fig. 1). The characteristic basin overturning strength, Ψ_{B}, is evaluated as the maximum Ψ at 19°S (i.e., the latitude of the global maximum in the control simulation). The characteristic channel overturning strength, Ψ_{C}, is evaluated as the maximum Ψ at 49°S (i.e., the latitude of the channel–basin interface).

The diagnosed Ψ_{B} and Ψ_{C} are shown as a function of the external parameters in Fig. 2. It is clearly seen that Ψ_{B} and Ψ_{C}, representing the different measures of the overturning strength in the basin and in the channel, respectively, differ substantially in their dependence on the external parameters. This implies that JN16 and INM in fact consider the overturning strengths in different parts of the abyssal cell, which may partially explain the apparent contradiction in the scaling theories they proposed. In the next section, we test the scaling theories for Ψ_{C} and Ψ_{B} separately to illustrate the distinct dynamics controlling them.

The overturning strength in the basin (Ψ_{B}; circles) and in the channel (Ψ_{C}; plus signs) as a function of the external parameters: (left) the rate of surface buoyancy loss *B*, (left center) diapycnal diffusivity *κ*, (center) the latitudinal extent of surface buoyancy loss *δ*_{I}, (right center) the strength of the wind stress *τ*_{0}, and (right) the GM diffusivity *K*.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

The overturning strength in the basin (Ψ_{B}; circles) and in the channel (Ψ_{C}; plus signs) as a function of the external parameters: (left) the rate of surface buoyancy loss *B*, (left center) diapycnal diffusivity *κ*, (center) the latitudinal extent of surface buoyancy loss *δ*_{I}, (right center) the strength of the wind stress *τ*_{0}, and (right) the GM diffusivity *K*.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

The overturning strength in the basin (Ψ_{B}; circles) and in the channel (Ψ_{C}; plus signs) as a function of the external parameters: (left) the rate of surface buoyancy loss *B*, (left center) diapycnal diffusivity *κ*, (center) the latitudinal extent of surface buoyancy loss *δ*_{I}, (right center) the strength of the wind stress *τ*_{0}, and (right) the GM diffusivity *K*.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

## 4. Theory–simulation comparison

### a. Overturning strength in the channel

We first examine the extent to which Ψ_{C}, the maximum Ψ evaluated at the channel–basin interface, can be predicted by the scaling theories of INM that are based on the channel dynamics, i.e., Eqs. (16) and (18). Recall that these two scaling relations are obtained by considering two approximations, Eqs. (17) and (19), respectively. Since they characterize the relative importance of the different components in Eq. (7), we start by comparing these terms to decide which limit case better describes our simulations. We first confirm that Ψ_{C} is indeed qualitatively similar to the isopycnally averaged streamfunction in the channel, i.e., Ψ_{C} ≈ ⟨Ψ⟩ (Fig. 3). The decomposition of ⟨Ψ⟩ into ⟨Ψ^{EU}⟩ and ⟨Ψ^{GM}⟩ according to Eq. (7) is then shown in Fig. 4. In general, we find the dominant balance to be between ⟨Ψ^{EU}⟩ and ⟨Ψ^{GM}⟩. The two substantially cancel each other and result in a relatively small ⟨Ψ⟩. Therefore, we expect that ⟨Ψ⟩ is better predicted by Eq. (16) than Eq. (18).

As in Fig. 2, but for Ψ_{C} (black) and ⟨Ψ⟩ (the isopycnal average computed in the channel; magenta); ⟨Ψ⟩ is evaluated at the same buoyancy as Ψ_{C}, i.e., the buoyancy at which the circulation at the channel–basin interface obtains its maximum. The lines indicate the power-law scaling predicted by Eq. (16). Notice that the *x* axis is linear in the fourth panel, because we are including a case with zero wind stress.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 2, but for Ψ_{C} (black) and ⟨Ψ⟩ (the isopycnal average computed in the channel; magenta); ⟨Ψ⟩ is evaluated at the same buoyancy as Ψ_{C}, i.e., the buoyancy at which the circulation at the channel–basin interface obtains its maximum. The lines indicate the power-law scaling predicted by Eq. (16). Notice that the *x* axis is linear in the fourth panel, because we are including a case with zero wind stress.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 2, but for Ψ_{C} (black) and ⟨Ψ⟩ (the isopycnal average computed in the channel; magenta); ⟨Ψ⟩ is evaluated at the same buoyancy as Ψ_{C}, i.e., the buoyancy at which the circulation at the channel–basin interface obtains its maximum. The lines indicate the power-law scaling predicted by Eq. (16). Notice that the *x* axis is linear in the fourth panel, because we are including a case with zero wind stress.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Decomposition of the channel-averaged streamfunction ⟨Ψ⟩, based on Eq. (7). Notice that the sign of all streamfunction components is reversed, and ⟨Ψ^{ST}⟩ is not shown because it is virtually zero in all cases.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Decomposition of the channel-averaged streamfunction ⟨Ψ⟩, based on Eq. (7). Notice that the sign of all streamfunction components is reversed, and ⟨Ψ^{ST}⟩ is not shown because it is virtually zero in all cases.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Decomposition of the channel-averaged streamfunction ⟨Ψ⟩, based on Eq. (7). Notice that the sign of all streamfunction components is reversed, and ⟨Ψ^{ST}⟩ is not shown because it is virtually zero in all cases.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Equation (16) does qualitatively explain the increase of ⟨Ψ⟩ with increasing *κ*, increasing *K*, and decreasing *τ*_{0}, but it generally fails to predict the correct power-law dependence on these external parameters (Fig. 3). In some cases, such as when *τ*_{0} decreases to zero and ⟨Ψ^{EU}⟩ vanishes, the failure of Eq. (16) can be partially attributed to the breakdown of the approximation in Eq. (17), such that the fit may be improved by considering the generalized solution, Eq. (13).

However, we find that perhaps the most fundamental limitation in the channel scaling theories is that the dimensionless factor *d* ≡ (*S*_{C})/(*h*/*l*) in Eq. (13) also varies with the external parameters. In Fig. 5, we plot *S*_{C} (the numerator of *d*) against *h*/*l* (the denominator of *d*), with all quantities diagnosed from the simulations. It is found that *h*/*l* shows substantial parameter dependence, while *S*_{C} varies comparatively little (presumably due to geometrical constraints). As a result, the ratio, *d*, varies systematically with *κ*, *τ*_{0}, and *K*. Although less strongly, *d* also depends on *B* and *δ*_{I}, which indeed is essential to explain the dependence of ⟨Ψ⟩ on *B* and *δ*_{I}, which do not appear explicitly in Eq. (13). A quantitative improvement on the existing theories therefore requires a theory for how *d* depends on the external parameters, which remains missing.

Evaluating changes in *d* = *S*_{C}/(*h*/*l*): *S*_{C} = *h*_{C}/*l*_{C} is plotted against *h*/*l*. The latter is calculated as *h*/*l* ≡ −*κW*⟨Ψ⟩^{−1} such that Eq. (11) holds perfectly. All values are normalized by the corresponding values in the control simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Evaluating changes in *d* = *S*_{C}/(*h*/*l*): *S*_{C} = *h*_{C}/*l*_{C} is plotted against *h*/*l*. The latter is calculated as *h*/*l* ≡ −*κW*⟨Ψ⟩^{−1} such that Eq. (11) holds perfectly. All values are normalized by the corresponding values in the control simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Evaluating changes in *d* = *S*_{C}/(*h*/*l*): *S*_{C} = *h*_{C}/*l*_{C} is plotted against *h*/*l*. The latter is calculated as *h*/*l* ≡ −*κW*⟨Ψ⟩^{−1} such that Eq. (11) holds perfectly. All values are normalized by the corresponding values in the control simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

### b. Overturning strength in the basin

We now consider the overturning circulation in the basin Ψ_{B}, which we find to be more readily explained by the theoretical prediction than the channel circulation discussed above. The scaling prediction for the residual streamfunction in the basin is described by Eq. (25), which suggests that Ψ_{B} scales as *B*^{1/4}*κ*^{1/2}. This predicted parameter dependence is seen to agree with most of our simulations (Fig. 6). For our varying *B* cases, we see a *B*^{1/4} dependence; for our varying *δ*_{I} but fixed *B* cases, we see no dependence on *δ*_{I}. In addition, we see that Ψ_{B} scales as *κ*^{1/2} but stays approximately constant when *τ*_{0} is varied. If *B* and *κ* are varied together, the predicted *B*^{1/4}*κ*^{1/2} dependence also remains valid (not shown). These results all provide supporting evidence for Eq. (25). An apparent exception appears when *K* is large. In this limit, changes in the GM streamfunction in the basin become nonnegligible, such that Eq. (20) as an underlying assumption behind Eq. (25) no longer holds. This interpretation is supported by the increasing differences between the maximum residual streamfunction, Ψ_{B}, and the maximum Eulerian mean streamfunction

As in Fig. 2, but for Ψ_{B} (the maximum residual streamfunction evaluated at 19°S) and

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 2, but for Ψ_{B} (the maximum residual streamfunction evaluated at 19°S) and

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 2, but for Ψ_{B} (the maximum residual streamfunction evaluated at 19°S) and

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

The above analysis focuses on the maximum value of the residual streamfunction, Ψ_{B}, but the same scaling argument also has skill in predicting the vertical profile in buoyancy space. In Fig. 7, we plot the simulated profiles of Ψ evaluated at 19°S as a function of the buoyancy difference, Δ*b* = *b* − *b*_{B}, where *b*_{B} is the buoyancy where the streamfunction obtains its maximum value (i.e., Ψ_{B}). The streamfunction on the *x* axis is normalized according to the predicted maximum in Eq. (25) and the buoyancy difference on the *y* axis is normalized according to the predicted buoyancy contrast in Eq. (24). Figure 7 shows that, except for the simulations with large *K*, the results collapse reasonably well on each other. While some nonnegligible deviations exist, the scaling argument therefore provides a useful description for not only the maximum value but also the overall shape of Ψ(*b*).

Profiles of Ψ(*b*) evaluated at 19°S and plotted against Δ*b* = *b* − *b*_{B}, with *b*_{B} being the buoyancy where Ψ(*b*) obtains its maximum. Both Ψ and Δ*b* are rescaled to account for the predicted stretching and amplifying effect due to the parameter changes. Specifically, Δ*b* is normalized by (*B*/*B*_{r})^{3/4}(*κ*/*κ*_{r})^{−1/2} according to Eq. (24), and Ψ is normalized by (*B*/*B*_{r})^{1/4}(*κ*/*κ*_{r})^{1/2} according to Eq. (25). The subscript *r* denotes the value obtained in the control simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Profiles of Ψ(*b*) evaluated at 19°S and plotted against Δ*b* = *b* − *b*_{B}, with *b*_{B} being the buoyancy where Ψ(*b*) obtains its maximum. Both Ψ and Δ*b* are rescaled to account for the predicted stretching and amplifying effect due to the parameter changes. Specifically, Δ*b* is normalized by (*B*/*B*_{r})^{3/4}(*κ*/*κ*_{r})^{−1/2} according to Eq. (24), and Ψ is normalized by (*B*/*B*_{r})^{1/4}(*κ*/*κ*_{r})^{1/2} according to Eq. (25). The subscript *r* denotes the value obtained in the control simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Profiles of Ψ(*b*) evaluated at 19°S and plotted against Δ*b* = *b* − *b*_{B}, with *b*_{B} being the buoyancy where Ψ(*b*) obtains its maximum. Both Ψ and Δ*b* are rescaled to account for the predicted stretching and amplifying effect due to the parameter changes. Specifically, Δ*b* is normalized by (*B*/*B*_{r})^{3/4}(*κ*/*κ*_{r})^{−1/2} according to Eq. (24), and Ψ is normalized by (*B*/*B*_{r})^{1/4}(*κ*/*κ*_{r})^{1/2} according to Eq. (25). The subscript *r* denotes the value obtained in the control simulation.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

We caution, however, that there are in fact some subtleties in the derivation of Eq. (25), which may put into question the generality of this scaling relation. Specifically, although Eq. (25) successfully predicts the parameter dependence of the basin overturning strength in our simulations, the dimensionless factors *d*_{1} and *d*_{2}, in Eq. (25) do exhibit nontrivial sensitivity to the external parameters (see the appendix). However,

## 5. Role of bottom topography

The simulations we examined so far have a flat bottom throughout the entire model domain, which eliminates potential effects associated with bottom topography in the Southern Ocean. To address the role of bottom topography, we compare our flat-bottom simulations with a new set of simulations where everything is kept the same except that the model is set up with an idealized bottom topography. That is, a topographical ridge is included, which extends the continent with a maximum height of 2 km throughout the entire channel. The exact form of the ridge is shown in Fig. 8 and is designed to be qualitatively similar to the model setup in Ito and Marshall (2008).

Model setup and control-case results for the simulations with bottom topography: (left) the ocean depth and (right) Ψ(*y*, *z*), with black lines indicating the zonally averaged depth of a given isopycnal in the control simulation. The dashed line indicates the bottom of the surface layer, defined as the mean depth of the isopycnal corresponding to the maximum surface density at a given latitude.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Model setup and control-case results for the simulations with bottom topography: (left) the ocean depth and (right) Ψ(*y*, *z*), with black lines indicating the zonally averaged depth of a given isopycnal in the control simulation. The dashed line indicates the bottom of the surface layer, defined as the mean depth of the isopycnal corresponding to the maximum surface density at a given latitude.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Model setup and control-case results for the simulations with bottom topography: (left) the ocean depth and (right) Ψ(*y*, *z*), with black lines indicating the zonally averaged depth of a given isopycnal in the control simulation. The dashed line indicates the bottom of the surface layer, defined as the mean depth of the isopycnal corresponding to the maximum surface density at a given latitude.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

The overturning circulation in the control simulation is significantly affected by the presence of topography (cf. Fig. 8 to Fig. 1). The global maximum (in terms of absolute value) now appears in the channel rather than the basin. The effect of topography on the overturning strength depends strongly on where the circulation is evaluated. In Fig. 9, we compare the overturning strength between the flat-bottom and the topography simulations, across the full parameter range. As in the previous analysis, we show the two different diagnostic metrics, Ψ_{B} and Ψ_{C}, which measure the overturning strength in the basin and in the channel, respectively. Although the presence of the bottom ridge interferes with the circumpolar zonal flow in the lower portion of the channel, we retain our earlier definition of Ψ_{C} as the maximum streamfunction at 49°S, which again is reasonably similar to the isopycnally averaged streamfunction in the channel, ⟨Ψ⟩ (Fig. 10).

Comparison of the overturning strength between the flat-bottom simulations and the simulations with bottom topography: (left) Ψ_{B} and (right) Ψ_{C}.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Comparison of the overturning strength between the flat-bottom simulations and the simulations with bottom topography: (left) Ψ_{B} and (right) Ψ_{C}.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

Comparison of the overturning strength between the flat-bottom simulations and the simulations with bottom topography: (left) Ψ_{B} and (right) Ψ_{C}.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 3, but for the simulations with bottom topography. The lines indicate the power-law scaling predicted by Eq. (18).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 3, but for the simulations with bottom topography. The lines indicate the power-law scaling predicted by Eq. (18).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 3, but for the simulations with bottom topography. The lines indicate the power-law scaling predicted by Eq. (18).

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

The Ψ_{B} in simulations with topography is only slightly larger than in the flat-bottom simulations; the two scale well with each other and have very similar parameter dependence (left panel in Fig. 9). This suggests that the presence of bottom topography in the channel has little impact on the dynamics controlling the overturning strength within the basin. It also reaffirms that the details of the channel dynamics, which are strongly affected by the bottom topography, may not be as crucial to understand the abyssal circulation in the basin.

On the contrary, the results for Ψ_{C} are substantially modified after the bottom topography is introduced (right panel in Fig. 9). The magnitude of Ψ_{C} in the topography simulations is generally much larger than in the flat-bottom simulations, and the two cases do not seem to scale well with each other.

A decomposition of the channel-averaged streamfunction ⟨Ψ⟩ reveals that the difference between the flat-bottom and topography simulations is in part due to the contribution by the stationary eddy streamfunction ⟨Ψ^{ST}⟩. In the topography simulations, stationary eddies contribute significantly to the total transport, owing to the presence of bottom topography that breaks the zonal symmetry in the channel. As shown in Fig. 11, ⟨Ψ^{ST}⟩ acts to approximately cancel ⟨Ψ^{EU}⟩. This almost perfect standing eddy compensation is especially important in explaining the much weaker *τ*_{0} dependence of ⟨Ψ⟩ in the topography simulations (Fig. 11) relative to the flat-bottom simulations (Fig. 4). We note that a similar behavior is reported by Bishop et al. (2016), although the reasons for and robustness of this compensation remain unclear.

As in Fig. 4, but for the simulations with bottom topography.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 4, but for the simulations with bottom topography.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

As in Fig. 4, but for the simulations with bottom topography.

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

A direct result of the compensation between ⟨Ψ^{ST}⟩ and ⟨Ψ^{EU}⟩ is that ⟨Ψ⟩ is left to mostly follow ⟨Ψ^{GM}⟩, which makes Eq. (19) a decent approximation, albeit not for the reason assumed in the existing theories (i.e., ⟨Ψ^{ST}⟩ ≈ ⟨Ψ^{EU}⟩ ≈ 0). Moreover, we find that the dimensionless factor *d* is less sensitive to the external parameters in the simulations with bottom topography (not shown). Consequently, Eq. (18) more accurately predicts Ψ_{C} in these simulations (Fig. 10).

## 6. Conclusions

This study revisits existing theories for the abyssal overturning strength (Ito and Marshall 2008; Nikurashin and Vallis 2011; Mashayek et al. 2015; JN16), aiming to better interpret their apparent contradiction regarding the importance of channel versus basin dynamics. An ocean general circulation model is set up with idealized geometry and surface forcing conditions that encapsulate only the essential elements in these theories. Using this model, we conducted a series of sensitivity runs to study the parameter dependence of the circulation strength. The external parameters we varied are the integrated surface buoyancy loss rate prescribed in the channel *B*, the diapycnal diffusivity *κ*, the latitudinal extent of the prescribed buoyancy loss *δ*_{I}, the magnitude of the prescribed wind stress *τ*_{0}, and the GM diffusivity *K*. Our key findings are summarized in the following:

The strengths of the residual overturning circulation Ψ measured within the channel and within the basin can vary with the external parameters in distinct ways due to the possibility of a significant recirculation in the basin. Thus, the apparent inconsistency among the theories with different dynamical focuses can at least in part be explained by the fact that they are attempting to explain different measures of the circulation strength.

Within the channel, Ψ is seen to depend on all parameters,

*B*,*κ*,*δ*_{I},*τ*_{0}, and*K*in the simulations with a flat bottom. When a bottom ridge is included in the channel, Ψ is to-the-first-order only sensitive to*κ*and*K*, as predicted by the existing theories that consider channel dynamics [i.e., Eq. (18)]. This difference between the flat-bottom and topography simulations is due to the presence of stationary eddies in the latter, which effectively cancel the mean circulation, allowing the residual circulation to be well approximated by the parameterized eddy-driven circulation. This approximation reduces the complexity of the problem and is a key assumption in the INM theories. Why this almost perfect cancellation occurs, however, remains unclear.Within the basin, Ψ is found to scale with

*B*^{1/4}*κ*^{1/2}, as described in Eq. (25). Building on the basin-based argument proposed by JN16, this scaling is derived without consideration of channel dynamics and is instead based on advective–diffusive balance and thermal wind relation in the basin. Consistent with the importance of basin over channel dynamics, the overturning strength in the basin is similar in simulations with and without channel topography.

A related question is to what extent these theories for the abyssal overturning strength, which have been derived for and validated in single-basin models so far, also apply to multibasin and global models. Nadeau and Jansen (2020) recently showed that the general results for the global overturning circulation in a single-basin setup can be largely carried over to a two-basin setup after accounting for the domain size effects. While these results suggest that single-basin theories remain qualitatively relevant to models with more complicated geometries, in how far the scaling arguments discussed here would hold quantitatively requires more investigation.

In addition, the analyses we performed in this study are limited to the equilibrium response of the circulation changes. All theories we discussed make use of the basin thermodynamic scaling [i.e., Eq. (11)], which assumes the basin buoyancy budget to be in equilibrium. This assumption will need to be modified when the storage term in the budget is nonnegligible—for example, as is expected during anthropogenic climate change in the coming decades.

Despite these remaining questions, we believe our results can provide useful insights that improve our understanding of the relative importance of different mechanisms in controlling the abyssal circulation. Particularly noteworthy is the importance of thermodynamics, specifically buoyancy forcing and diapycnal mixing, and the comparatively small role of wind stress in controlling the abyssal overturning in the basin as well as in the channel when topography is present. A better understanding of the processes that control buoyancy forcing around Antarctica and turbulent mixing in the abyss is thus crucial to better constrain and predict past and future changes in the abyssal overturning circulation.

## Acknowledgments

Computational resources for this project were generously provided by the University of Chicago Research Computing Center. Authors Chang and Jansen acknowledge support from NSF Award OCE-1846821.

## Data availability statement

The MITgcm configuration files used for this study are available online (https://github.com/cyinchang/MITgcm-abyss).

## APPENDIX

### Dimensionless Factors in Eq. (25)

Equation (25) predicts the parameter dependence of the basin overturning strength in terms of external parameters only if the combination of the dimensionless factors, *d*_{1} ≡ *h*_{x}/*h* and *d*_{2} ≡ Δ*b*_{x}/Δ*b*_{B} from our simulations. Specifically, based on Eq. (21) and loosely following JN16, we calculate *h*_{x} as the height of the maximum Eulerian overturning *b*_{x} as the zonal buoyancy difference across the basin and averaged vertically between the bottom and the height of the streamfunction maximum. The definition for Δ*b*_{B} is specified in Eq. (23). As in section 2, we compute *h* from Eq. (11), using the diagnosed Ψ_{B} and assuming *l* = *L*_{B} to get an estimate of *h*.

The obtained results indicate that *d*_{1} and *d*_{2} individually vary with the external parameters. As seen in Fig. A1, the variations in *h*_{x} (numerator of *d*_{1}) and *h* (denominator of *d*_{1}) do not generally scale with each other, so that *d*_{1} is generally not constant. Similarly, Δ*b*_{x} (numerator of *d*_{2}) and Δ*b*_{B} (denominator of *d*_{2}) do not generally scale with each other, leading to variations in *d*_{2}. However,

(top left) The *h*_{x} plotted against *h*, (top right) Δ*b*_{x} plotted against Δ*b*_{B}, (bottom left)

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

(top left) The *h*_{x} plotted against *h*, (top right) Δ*b*_{x} plotted against Δ*b*_{B}, (bottom left)

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

(top left) The *h*_{x} plotted against *h*, (top right) Δ*b*_{x} plotted against Δ*b*_{B}, (bottom left)

Citation: Journal of Physical Oceanography 51, 7; 10.1175/JPO-D-20-0316.1

*h*

_{x}and Δ

*b*

_{x}with

*h*and Δ

*b*

_{B}in Eq. (21). Equation (A1) combined with Eqs. (11), (20), and (23) gives Eq. (25), but with the dependence on

*d*

_{1}and

*d*

_{2}eliminated. Although Eq. (A1) holds in our simulations, it remains unclear to us as to how far this relationship can be expected to be robust.

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^{1}

We have omitted the positive solution because it corresponds to *h* < 0.