1. Introduction
The abyssal meridional overturning circulation is fed by waters that become dense enough to sink into the ocean abyss at high latitudes and return to the surface through convoluted three-dimensional pathways. While it is well established that most of the sinking is confined to the Labrador and Greenland Seas in the North Atlantic and the Weddell and Ross Seas around Antarctica, the return pathways are less understood (Talley et al. 2011). The consensus is that widespread small-scale mixing generated by breaking internal waves provides most of the energy to lift the North Atlantic Deep Waters (NADWs) and the Antarctic Bottom Waters (AABWs) up to a depth of approximately 2000 m; internal waves break within a few hundred meters of abyssal ridges, mountains, and rises, which are most ubiquitous below 2000 m (Munk 1966; Ferrari 2014). Once the waters rise above 2000 m, they flow southward along density surfaces to the Southern Ocean, where the Roaring Forties lift them to the surface to be transformed into intermediate waters that flow back toward Antarctica and the North Atlantic, thereby closing the overturning loop [see the review by Marshall and Speer (2012)]. There would appear to be a major inconsistency in this scenario: strong internal wave–driven mixing is typically bottom intensified and thus drives sinking, not rising, of waters through density surfaces (e.g., St. Laurent et al. 2001; Klocker and McDougall 2010). In this paper, the conundrum is resolved by arguing that abyssal waters come to the surface along turbulent bottom boundary layers, rather than in the stratified ocean interior.
We begin by establishing a few well-documented observations about the ocean. First, the ocean is stably stratified on time scales longer than a few hours (e.g., Talley et al. 2011). Second, estimates of the zonally averaged global overturning circulation based on inverse calculations from ocean observations (Ganachaud and Wunsch 2000; Lumpkin and Speer 2007) and numerical models constrained to observations (Wunsch and Heimbach 2009) show that waters sink to the ocean bottom at high latitudes and then slowly rise throughout the rest of the oceans crossing density surfaces at least up to 2000 m (Marshall and Speer 2012; Talley 2013). Third, there is growing evidence from in situ measurements that the turbulent kinetic energy generated by breaking internal waves is large within a few hundred meters of rough-bottom topography and decays to weaker values farther up in the water column. The bottom enhancement of turbulence reflects the generation of energetic waves by tides and geostrophic flows impinging over topography and breaking locally (e.g., Wunsch and Ferrari 2004; Garrett and Kunze 2007; Nikurashin and Ferrari 2013). During a breaking event most of the energy is lost to dissipation, but a fraction is expended in mixing the stratified fluid (Osborn 1980). Mixing rates have been estimated to be two to three orders of magnitude above interior levels within a few hundred meters of rough ocean topography (Polzin et al. 1997; St. Laurent et al. 2012). The term mixing is used here to refer to the turbulent density flux generated by breaking waves and not the turbulent diffusivity, which is equal to minus the turbulent flux divided by the density stratification. A recent comprehensive compilation of available mixing profiles confirms that ocean mixing rates are stronger within a kilometer of rough ocean bottom topography and weaker above, while mixing rates are weak and nearly uniform above smooth topography (Waterhouse et al. 2014).
The emerging evidence that mixing increases toward the ocean bottom, at least above rough topography where mixing is strong and contributes significantly to water mass transformations, represents the key departure from the textbook explanation of how mixing contributes to waters rising from the abyss (Munk 1966; Munk and Wunsch 1998). Waters that sink to the abyss along Antarctica flood the ocean below 2000 m. Munk (1966) argued that these abyssal waters are transformed into lighter waters in the ocean interior as a result of internal wave–driven mixing. However, waters sink (they do not rise) in regions where the rate of mixing increases toward the ocean bottom (Gargett 1984; Polzin et al. 1997; St. Laurent et al. 2001; Simmons et al. 2004; Klocker and McDougall 2010; Kunze et al. 2012). The goal of this paper is to resolve the conundrum of how bottom waters are transformed into lighter waters. We will argue that abyssal waters rise toward the surface as lighter waters in narrow turbulent boundary layers that develop along the continental margins and abyssal ridges.
This paper focuses on the diabatic transformation of abyssal water masses, that is, the part of the overturning circulation that crosses density surfaces. The global transport of tracers like heat and freshwater is affected by additional flows along density surfaces, like isopycnal stirring and mixing by geostrophic eddies, which will not be discussed. Our approach is similar to the one taken by Nikurashin and Ferrari (2013) and by de Lavergne et al. (2016), who studied the role of mixing in driving water mass transformations in the deep ocean using available estimates of internal wave–driven mixing. But our focus is on understanding and quantifying the crucial role played by the turbulent bottom boundary layers.
The paper is organized as follows. In section 2, we use heuristic and formal arguments to explain why a bottom enhancement of abyssal mixing results in diapycnal sinking of waters in the stratified ocean interior and diapycnal rising of waters along bottom boundary layers. Section 3 introduces the idealized numerical model used to test our theory. In section 4, we use the numerical model to illustrate how ocean mixing and the shape of bathymetry affect the abyssal ocean overturning circulation and stratification. We then show in section 5 that our claim that abyssal waters rise along deep sloping boundaries is supported by estimates of mixing profiles in the global ocean. Finally, in section 6, we conclude.
2. Theory
There is a rich literature on the role of mixing on water mass transformations in the ocean interior [see references in Wunsch and Ferrari (2004)]. Much work has also been done on mixing-driven flows along sloping boundary layers (e.g., Armi 1978; Garrett et al. 1993). However, the two lines of research have proceeded largely independently of each other. In this section we use Walin’s (1982) approach to emphasize the connection between upwelling along sloping boundaries and interior sinking in response to bottom-intensified mixing.
We begin our discussion with an illustrative example. Consider the schematic in Fig. 1a. A water parcel in the middle of the ocean water column, sketched as a gray square, experiences stronger mixing through its lower face than its upper face, consistent with the observational evidence that vertical mixing is stronger at depth. Mixing will make this water parcel denser by diffusing more heavy waters from below than light waters from above. A mixing profile that increases toward the ocean bottom acts to increase the density of water parcels. (We consider only mixing in the vertical because in the ocean interior the diapycnal direction is very close to vertical.)
(a) A longitude–depth section of the ocean above a sloping rough-bottom topography. A water parcel way above the ocean bottom boundary layer (left gray square) experiences more diapycnal mixing of density through its lower face than through its upper one as a result of the bottom enhancement of mixing. Mixing is represented as black curly arrows, whose length is proportional to the strength of mixing. This water parcel becomes denser and sinks in response to mixing. A water parcel sitting right on the ocean bottom topography (right gray square) experiences mixing only through its upper surface. This water parcel becomes lighter and rises in response to mixing. (b) A cut through a conical ocean basin with area increasing toward the surface. The two light gray horizontal stripes along the ocean boundary represent two density surfaces. Only the portion of the density surfaces experiencing strong mixing in the near-boundary region is shown. As the upper surface is larger than the lower one, the mixing integrated over the upper surface may exceed that integrated on the lower surface, even though mixing increases toward the ocean bottom along every vertical profile (before going to zero right at the ocean bottom).
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(a) A longitude–depth section of the ocean above a sloping rough-bottom topography. A water parcel way above the ocean bottom boundary layer (left gray square) experiences more diapycnal mixing of density through its lower face than through its upper one as a result of the bottom enhancement of mixing. Mixing is represented as black curly arrows, whose length is proportional to the strength of mixing. This water parcel becomes denser and sinks in response to mixing. A water parcel sitting right on the ocean bottom topography (right gray square) experiences mixing only through its upper surface. This water parcel becomes lighter and rises in response to mixing. (b) A cut through a conical ocean basin with area increasing toward the surface. The two light gray horizontal stripes along the ocean boundary represent two density surfaces. Only the portion of the density surfaces experiencing strong mixing in the near-boundary region is shown. As the upper surface is larger than the lower one, the mixing integrated over the upper surface may exceed that integrated on the lower surface, even though mixing increases toward the ocean bottom along every vertical profile (before going to zero right at the ocean bottom).
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(a) A longitude–depth section of the ocean above a sloping rough-bottom topography. A water parcel way above the ocean bottom boundary layer (left gray square) experiences more diapycnal mixing of density through its lower face than through its upper one as a result of the bottom enhancement of mixing. Mixing is represented as black curly arrows, whose length is proportional to the strength of mixing. This water parcel becomes denser and sinks in response to mixing. A water parcel sitting right on the ocean bottom topography (right gray square) experiences mixing only through its upper surface. This water parcel becomes lighter and rises in response to mixing. (b) A cut through a conical ocean basin with area increasing toward the surface. The two light gray horizontal stripes along the ocean boundary represent two density surfaces. Only the portion of the density surfaces experiencing strong mixing in the near-boundary region is shown. As the upper surface is larger than the lower one, the mixing integrated over the upper surface may exceed that integrated on the lower surface, even though mixing increases toward the ocean bottom along every vertical profile (before going to zero right at the ocean bottom).
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
To understand how water parcels rise to the surface as lighter waters, one must consider what happens close to the ocean solid boundaries. Mixing vanishes right at the seafloor because ocean turbulence does not mix density across the solid boundary. Thus, the rate of mixing drops to zero within a thin boundary layer close to the ocean bottom. The parcel in the bottom boundary layer (BBL) sketched in Fig. 1a experiences mixing only on its upper face and becomes lighter. The implication of this simple thought experiment is that water parcels are transformed into heavier density classes in the stratified ocean interior, while they are transformed into lighter density classes along the oceanic boundaries.
Figure 1b depicts a hypothetical conical ocean basin, where the area of the basin decreases with depth. The two horizontal strips along the ocean boundary represent two constant density surfaces, taken to be flat for simplicity. (The density surfaces should tilt and intersect the boundary at right angles within the narrow BBL, but this correction is irrelevant in this example.) The horizontal strips represent the portions of the density surfaces that experience the strong near-boundary mixing. If the width of the strips is constant with depth, then the area of the upper strip is larger than that of the lower strip because of the shape of the ocean topography. Thus, the mixing integrated over the upper density surface can be larger than the mixing integrated over the lower density surface, even though mixing increases toward the bottom in every vertical profile. Whether the whole volume between the two density surfaces is mixed more with the waters above or below will depend on the relative rate at which the ocean area experiencing mixing decreases with depth versus the rate at which mixing increases with depth. It is important to remark that what matters is the rate of change of the near-boundary area experiencing mixing and not the rate of change of the whole ocean area (the hypsometry).
This heuristic argument can be cast in more rigorous mathematical terms. The equation for the mass transport across a density surface in the ocean interior is derived in McDougall (1984) for seawater, which has an equation of state that depends nonlinearly on temperature, salinity, and pressure. Here, we will consider the simpler case of a linear equation of state, which is sufficient to study the qualitative aspects of mixing-driven flows. The reader is referred to the recent paper by de Lavergne et al. (2016) for a discussion of the mass transport generated by mixing in the abyssal ocean with a nonlinear equation of state. Our goal is to illustrate how interior mixing drives both downwelling across density surfaces in the ocean interior and upwelling along the boundaries, rather than providing an accurate estimate of the observed ocean overturning circulation.
Equation (3) confirms that a diapycnal velocity 
Munk’s (1966) picture came into question when the first measurements of dissipation in the abyssal ocean were collected as part of the Brazil Basin Experiment (Polzin et al. 1997). The dissipation profiles of ϵ were found to increase by one to two orders of magnitude as the turbulent probes approached the ocean bottom above rough topography but were still in the stratified ocean above the BBL. Numerical simulations and laboratory experiments found that Γ varies within a factor of 2 of the canonical value of 0.2 (Smyth et al. 2001; Peltier and Caulfield 2003; Mashayek and Peltier 2013a). Mashayek and Peltier (2013b) further showed with numerical simulations that the neglect of nonlocal redistribution of energy in the Osborn (1980) model adds to the uncertainty in the estimate of 
The density flux does not increase all the way to the ocean bottom, because it must vanish at the solid boundary (or match the weak negative geothermal flux in the real ocean, thus reinforcing our result). The transition between the strong interior density flux and the zero value at the ocean boundaries occurs within a turbulent BBL, where the isopycnals become normal to the boundary so as to obey the no flux boundary condition. In the BBLs the dissipation of kinetic energy ϵ is large, whereas the density flux 
We will refer to the mixing-driven flows in the stratified interior and along the BBLs as diapycnal upwelling and downwelling. The diapycnal velocities are by definition normal to density surfaces. In the stratified interior, density surfaces are approximately horizontal and the diapycnal direction is vertical. In the BBL, density surfaces intersect the bottom boundary at normal angles and the diapycnal direction is along the solid boundary. The transformation of light waters into denser ones in the stratified interior is associated with a vertical downward velocity or diapycnal downwelling. The transformation of dense waters into lighter ones along the BBL is instead associated with an along-boundary flow as per Eq. (3). To the extent that the abyssal BBLs are sloped, this flow will have a vertical upward component, and we refer to it as a diapycnal upwelling. However, the “diapycnal upwelling” is horizontal if the BBL is flat.
A longitude–depth section of the near-boundary region of the ocean above rough topography is characterized by enhanced turbulence generated by tidal and geostrophic flows impinging over topography. The region closest to the boundary is characterized by a weakly stratified BBL a few tens of meters thick. Above the BBL, there is a SML, a region where topographically generated turbulence is still strong but not so strong as to substantially reduce the stratification. The combination of strong turbulence and large stratification results in a strong density flux 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
A longitude–depth section of the near-boundary region of the ocean above rough topography is characterized by enhanced turbulence generated by tidal and geostrophic flows impinging over topography. The region closest to the boundary is characterized by a weakly stratified BBL a few tens of meters thick. Above the BBL, there is a SML, a region where topographically generated turbulence is still strong but not so strong as to substantially reduce the stratification. The combination of strong turbulence and large stratification results in a strong density flux
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
A longitude–depth section of the near-boundary region of the ocean above rough topography is characterized by enhanced turbulence generated by tidal and geostrophic flows impinging over topography. The region closest to the boundary is characterized by a weakly stratified BBL a few tens of meters thick. Above the BBL, there is a SML, a region where topographically generated turbulence is still strong but not so strong as to substantially reduce the stratification. The combination of strong turbulence and large stratification results in a strong density flux
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Equation (9) differs in one very important aspect from that derived in Klocker and McDougall (2010). Klocker and McDougall (2010) took the average over the whole isopycnal layer and thus their equation had the full isopycnal area A(ρ) on the right-hand side. Observing that isopycnals are nearly flat in the stratified ocean, they concluded that it is the increase in the area of the ocean basins with depth (the hypsometry) that makes 
The implication of applying Eq. (9) to Amix rather than A is well illustrated by McDougall (1989), who studied the impact of bottom-enhanced mixing around an isolated seamount. The increase with density/depth of the circumference of a seamount implies that seamounts act as a net sink of seawater from the surrounding ocean because ∂Amix/∂ρ > 0, even though the area of the ocean decreases downward around the seamount and ∂A/∂ρ < 0. We thus expect that net diabatic upwelling is more likely near continental boundaries and ridges, where the perimeter of incrop lines increases with height above the bottom, rather than in the vicinity of seamounts.
The flow induced by mixing at a sloping boundary in a stratified fluid has a long history dating back to Wunsch (1970), Thorpe (1982), and Garrett (1990). These studies concentrated on mixing induced by a source of energy at the boundary, and the major focus was whether the energy was spent “mixing mixed fluid” or whether some of the boundary-generated energy was spent mixing stratified fluid above the BBL, where it is more effective in causing along-boundary diffusion and advection (Garrett 1990). By contrast, our present study assumes knowledge of the strength of mixing in the stratified fluid above the BBL. The density budget and the advection of fluid in the BBL is a slave to the flux of density it receives at its upper boundary. The difference in approach is motivated by the evidence, accumulated in the last 20 years, that topographically generated internal waves drive a mixing profile that increases toward the ocean bottom in the stratified interior above the BBL. The circulation driven by mixing is not confined to the BBL.
While the tendency for mixing to drive along-boundary flows has long been recognized, the implications for the overall ocean circulation have not. Our claim is that deep waters are transformed into denser waters by the bottom-enhanced mixing in the stratified ocean interior above rough topography and sink toward the BBLs. Once in the BBLs, the waters are transformed back to lighter waters and rise toward the surface along the ocean boundaries. In this picture, it is crucial that the boundaries are sloping because otherwise the diabatic boundary flow cannot bring waters back toward the surface. This view is quite different from the traditional explanation that ocean mixing away from boundaries drives the rise of abyssal waters through the water column. We are proposing that abyssal waters rise in the net despite the action of interior ocean mixing that, by itself, causes diapycnal downwelling.
3. Numerical model setup
Given the dearth of deep mixing measurements in the global ocean, we turn to idealized numerical simulations to test our claim that BBLs are key pathways for returning abyssal waters to the surface. We use the Massachusetts Institute of Technology General Circulation Model (MITgcm; Marshall et al. 1997) in two different configurations sketched in the top panels of Fig. 3. Both configurations include a single basin bounded by walls to the north, east, and west and connected to a reentrant channel to the south. The Box configuration has a flat bottom 4000 m deep, as shown in Fig. 3 (top left), while the Bowl configuration consists of a bowl-shaped basin and a flat-bottom, 4000-m-deep channel, as shown in Fig. 3 (top right) (the deepest point in the center of the basin reaches 4000 m). The model vertical resolution is uniform and equal to 80 m. The horizontal resolution is 2° × 2°. The domain at the surface extends over 130° of longitude and from 60°S to 60°N for a total basin surface area approximately equal to that of the Pacific Ocean. There are no lateral boundaries between 60° and 40°S, and the flow in this range is zonally periodic. This setup is best thought of as an idealization of the circulation in a closed basin, like the Pacific Ocean, connected to a laterally unbounded channel to the south, like the Southern Ocean.
(top) Model topography for the Box and Bowl simulations. (bottom) Turbulent density flux profiles 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(top) Model topography for the Box and Bowl simulations. (bottom) Turbulent density flux profiles
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(top) Model topography for the Box and Bowl simulations. (bottom) Turbulent density flux profiles
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
The model is forced at the surface by restoring temperature to a prescribed profile on a time scale of a month over the 80-m-thick topmost grid cell. The restoring temperature profile, shown in the left panel of Fig. 4, increases monotonically with latitude from the southern boundary to the equator and then remains constant; the profile is constant in the zonal direction. We use a linear equation of state and set salinity to a constant; thus, density is only a function of temperature and equal to ρ = ρ0[1 − α(T − T0)], where ρ0 = 1035 kg m−3, T0 = 6.5°C, and α = 2 × 10−4 °C−1. The restoring temperature profile is chosen to drive convection and formation of dense waters at the southern edge of the channel but not in the Northern Hemisphere, like in today’s North Pacific Ocean. A zonally uniform wind stress, shown in the right panel of Fig. 4, blows only over the channel; the stress peaks at 0.2 N m−2 in the center of the channel and decays toward its northern and southern edges. We do not impose a wind stress over the basin to avoid spinning wind-driven gyres of no relevance to the deep stratification and circulation investigated here. Similar model setups have been employed in the literature to study the ocean overturning circulation (e.g., Nikurashin and Vallis 2011; Wolfe and Cessi 2011; Munday et al. 2013; Mashayek et al. 2015).
(left) The air temperature profile to which the ocean surface temperature is restored and (right) the wind stress profile applied at the ocean surface for all the simulations described in the paper.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(left) The air temperature profile to which the ocean surface temperature is restored and (right) the wind stress profile applied at the ocean surface for all the simulations described in the paper.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(left) The air temperature profile to which the ocean surface temperature is restored and (right) the wind stress profile applied at the ocean surface for all the simulations described in the paper.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
A number of processes need to be parameterized because they arise at scales not resolved by the model grid. The slumping of density surfaces by baroclinic instability is parameterized with the Gent and McWilliams (1990) scheme with a lateral diffusivity of 1000 m2 s−1. We do not prescribe any background along-isopycnal diffusivity because our only tracer is temperature and does not have any gradients along temperature/density surfaces. Turbulent momentum fluxes are parameterized with horizontal and vertical viscosities of νH = 2 × 105 m2 s−1 and νV = 1.2 × 10−4 m2 s−1, respectively, and a linear bottom drag with a drag coefficient of 5 × 10−4 m s−1.
Representation of vertical mixing of density
Our choice of imposing the turbulent density flux should not be read as implying that the profile of mixing is independent of stratification; for example, mixing ought to vanish in the absence of stratification. We impose the turbulent density flux because our goal is to diagnose what circulation and stratification are consistent with a particular mixing profile rather than to simulate the interactions between the large-scale ocean structure and the small-scale turbulent mixing. To the extent that the solution is characterized by a stratified abyss and a strong overturning circulation that crosses density surfaces, we will conclude that the solution is qualitatively consistent with the observed present-day ocean. Otherwise, we will conclude that the turbulent density flux profile is not consistent with what we know about the abyssal ocean. Anticipating our results, we will find that bottom-intensified mixing is consistent with the observed abyssal ocean only if the ocean basin has sloping lateral walls, but not if it has vertical walls.
We will consider four simulations with different mixing and topography profiles. Two Box simulations have vertical sidewalls and mixing profiles that either decrease or increase above the BBL (and go to zero within the BBL), as shown in the two left panels at the bottom of Fig. 3. Two Bowl simulations have sloping sidewalls and mixing profiles that decrease with height above the BBL (before going to zero in the BBL) but such that the area-integrated mixing profile increases with height z. One Bowl simulation uses the same mixing profile everywhere, while the other has a larger mixing on the eastern half of the domain, so as to mimic a situation with nonhomogeneous mixing. The topographic shapes and the corresponding mixing profiles are shown in Fig. 3.
4. Numerical model results
In all simulations the zonally averaged circulation in the southern channel is the result of two opposing forces. The eastward winds drive a thermally indirect overturning circulation, that is, a circulation with dense waters rising at the southern edge of the channel and light waters sinking at its northern edge. The parameterized baroclinic eddies drive an opposing thermally direct circulation with dense waters sinking at the southern edge of the channel and light waters rising at its northern edge (Marshall and Radko 2003). In our setup, the thermally direct circulation dominates because the surface cooling at the southern edge of the channel weakens the vertical stratification by driving upright convection to the ocean bottom (the analog of dense water formation in the Southern Ocean.). In weakly stratified waters, the slumping of isopycnals by baroclinic instability is very rapid and generates a very strong eddy-driven overturning circulation. In all our simulations the overturning circulation is thermally direct with abyssal waters sinking in the channel and rising toward the surface in the closed basin, regardless of the mixing profile. Nikurashin and Vallis (2011) pointed out that without mixing in the ocean basin, this circulation would be confined to the Southern Ocean. Mixing in the basin is crucial in connecting the overturning circulation and stratification between the channel and the basin.
In this section, we illustrate how the overall solution changes for different profiles of vertical mixing. In sections 4a and 4b, we show that in an ocean with a flat-bottom topography, a deep stratification can be maintained only if the turbulent density flux increases toward the surface, contrary to what is observed in the abyssal ocean. Section 4c shows that, if the boundaries are sloped, a deep stratification can be maintained with a turbulent density flux that decreases with height above the BBL. This stable stratification can exist in steady state because waters upwell toward the surface along the sloping boundaries, where the turbulent density flux divergence is positive. Section 4d explores the effect of variations in the profile of mixing along the sloping topography.
a. Box simulation with a turbulent density flux profile increasing above the BBL
In this first Box-MixTop experiment, we consider an ocean with a flat bottom, vertical sidewalls, and a turbulent density flux profile that starts from zero at the ocean bottom and increases toward the surface before dropping back to zero over the topmost grid cell (see Fig. 3). This is a scenario where the turbulent density flux divergence generated by mixing in the ocean interior is positive, unlike in the present ocean, and thus waters are expected to rise from the abyss toward the surface everywhere in the basin.
(left) Results from the Box-MixTop simulation with vertical sidewalls and a turbulent density flux profile increasing toward the ocean surface and shown in the MixTop panel of Fig. 3. The top panel shows the residual overturning streamfunction in Sv calculated using Eq. (16) (gray shading) and sample isopycnals spaced by 0.025 kg m−3 at depths where there is substantial overturning (black lines). The middle panel shows the diapycnal velocity, along four longitude–depth sections in the basin, computed using Eq. (4). The diapycnal velocities in the basin are everywhere positive and reach 4 × 10−7 m s−1. The black lines are isopycnal contours spaced by 0.1 kg m−3. The bottom panel shows the water mass transformations (Sv) computed with Eq. (6) for the portion of the isopycnal surface in the ocean interior above the BBL (blue bars) in the BBL (red bars) and for the sum of the two contributions (black bars). (right) Results from the Box-MixBottom simulation with vertical sidewalls and a turbulent density flux profile decreasing toward the ocean surface and shown in the Mix-Bottom panel of Fig. 3. The panels in the right column show the same diagnostics as in the left column. In this simulation the diapycnal velocities are positive in the BBL with values reaching 3 × 10−5 m s−1 and negative in the ocean interior with values reaching only −1 × 10−6 m s−1.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(left) Results from the Box-MixTop simulation with vertical sidewalls and a turbulent density flux profile increasing toward the ocean surface and shown in the MixTop panel of Fig. 3. The top panel shows the residual overturning streamfunction in Sv calculated using Eq. (16) (gray shading) and sample isopycnals spaced by 0.025 kg m−3 at depths where there is substantial overturning (black lines). The middle panel shows the diapycnal velocity, along four longitude–depth sections in the basin, computed using Eq. (4). The diapycnal velocities in the basin are everywhere positive and reach 4 × 10−7 m s−1. The black lines are isopycnal contours spaced by 0.1 kg m−3. The bottom panel shows the water mass transformations (Sv) computed with Eq. (6) for the portion of the isopycnal surface in the ocean interior above the BBL (blue bars) in the BBL (red bars) and for the sum of the two contributions (black bars). (right) Results from the Box-MixBottom simulation with vertical sidewalls and a turbulent density flux profile decreasing toward the ocean surface and shown in the Mix-Bottom panel of Fig. 3. The panels in the right column show the same diagnostics as in the left column. In this simulation the diapycnal velocities are positive in the BBL with values reaching 3 × 10−5 m s−1 and negative in the ocean interior with values reaching only −1 × 10−6 m s−1.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(left) Results from the Box-MixTop simulation with vertical sidewalls and a turbulent density flux profile increasing toward the ocean surface and shown in the MixTop panel of Fig. 3. The top panel shows the residual overturning streamfunction in Sv calculated using Eq. (16) (gray shading) and sample isopycnals spaced by 0.025 kg m−3 at depths where there is substantial overturning (black lines). The middle panel shows the diapycnal velocity, along four longitude–depth sections in the basin, computed using Eq. (4). The diapycnal velocities in the basin are everywhere positive and reach 4 × 10−7 m s−1. The black lines are isopycnal contours spaced by 0.1 kg m−3. The bottom panel shows the water mass transformations (Sv) computed with Eq. (6) for the portion of the isopycnal surface in the ocean interior above the BBL (blue bars) in the BBL (red bars) and for the sum of the two contributions (black bars). (right) Results from the Box-MixBottom simulation with vertical sidewalls and a turbulent density flux profile decreasing toward the ocean surface and shown in the Mix-Bottom panel of Fig. 3. The panels in the right column show the same diagnostics as in the left column. In this simulation the diapycnal velocities are positive in the BBL with values reaching 3 × 10−5 m s−1 and negative in the ocean interior with values reaching only −1 × 10−6 m s−1.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
The rising of waters from the abyss in the closed ocean basin is confirmed in the middle-left panel of Fig. 5, where we show representative longitude–depth sections in the basin of the diapycnal velocity 
The top-left panel of Fig. 5 shows that the stratification consists of steeply sloped isopycnals (black contours) in the channel and nearly horizontal isopycnals in the basin. The isopycnal slope in the channel is set through a balance between the wind stress and the eddy-driven circulation (e.g., Marshall and Radko 2003; Nikurashin and Vallis 2011). The stratification in the basin is instead maintained through a balance between the advection of density by the residual overturning and vertical diffusion of density driven by mixing, consistent with Munk’s (1966) “abyssal recipes” hypothesis. In the upper ocean, the stratification becomes very large because the turbulent density flux is balanced by a weak advection (not shown). In the real ocean, this does not happen because shallow waters are ventilated by wind forcing and Northern Hemisphere convection, while turbulent mixing becomes second order.
The turbulent diffusivity implied by the stratification and turbulent density flux obtained from the simulation at equilibrium is calculated using Eq. (14). Its horizontal average over the basin is shown in Fig. 6a. The turbulent diffusivity increases from ~1 × 10−4 m2 s−1 at the ocean bottom to ~2 × 10−3 m2 s−1 at 2000-m depth. (The sudden decrease of κ in the BBL reflects the vanishing of the turbulent density flux there.) A turbulent diffusivity that increases with height above the ocean bottom is not consistent with observations.
Diapycnal turbulent diffusivities computed with Eq. (14) based on the turbulent density fluxes and the stratification obtained in the various simulations (solid lines). The diapycnal diffusivity based on the turbulent density flux and stratification for the Pacific Ocean based on the calculation described in Nikurashin and Ferrari (2013) (dashed lines). (a) Box simulation with a turbulent density flux increasing above the BBL. (b) Box simulation with a turbulent density flux profile decreasing above the BBL. (c) Bowl simulation with a turbulent density flux profile decreasing above the BBL with the same bottom value everywhere. (d) Bowl simulation with a density flux profile decreasing everywhere above the BBL, but twice as large in the eastern half of the domain than in the western half.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Diapycnal turbulent diffusivities computed with Eq. (14) based on the turbulent density fluxes and the stratification obtained in the various simulations (solid lines). The diapycnal diffusivity based on the turbulent density flux and stratification for the Pacific Ocean based on the calculation described in Nikurashin and Ferrari (2013) (dashed lines). (a) Box simulation with a turbulent density flux increasing above the BBL. (b) Box simulation with a turbulent density flux profile decreasing above the BBL. (c) Bowl simulation with a turbulent density flux profile decreasing above the BBL with the same bottom value everywhere. (d) Bowl simulation with a density flux profile decreasing everywhere above the BBL, but twice as large in the eastern half of the domain than in the western half.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Diapycnal turbulent diffusivities computed with Eq. (14) based on the turbulent density fluxes and the stratification obtained in the various simulations (solid lines). The diapycnal diffusivity based on the turbulent density flux and stratification for the Pacific Ocean based on the calculation described in Nikurashin and Ferrari (2013) (dashed lines). (a) Box simulation with a turbulent density flux increasing above the BBL. (b) Box simulation with a turbulent density flux profile decreasing above the BBL. (c) Bowl simulation with a turbulent density flux profile decreasing above the BBL with the same bottom value everywhere. (d) Bowl simulation with a density flux profile decreasing everywhere above the BBL, but twice as large in the eastern half of the domain than in the western half.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
The take home message is that this simulation with vertical sidewalls resulted in an overturning circulation and stratification broadly consistent with those observed in the abyssal Pacific Ocean, but it required a turbulent density flux divergence and a vertical profile of turbulent diffusivity opposite to the ones observed in the abyssal ocean. Can one obtain a similar circulation and stratification for a realistic turbulent density flux profile?
b. Box simulation with a turbulent density flux profile decreasing above the BBL
Naively one might guess that this mixing profile ought to drive downwelling in the basin, resulting in a thermally indirect overturning circulation with cold waters upwelling along the southern edge of the channel and then flowing toward the basin to sink back into the abyss. However, a thermally indirect overturning circulation is inconsistent with the air–sea cooling applied over the channel. Indeed the overturning circulation diagnosed from the simulation and shown in the top-right panel of Fig. 5 is thermally direct with cold waters sinking along the southern edge of the channel and rising nearly adiabatically along the steep isopycnals in the basin. Contrary to the naive inference, the overturning is thermally direct, but the deep ocean is very weakly stratified, unlike in the Box-MixTop simulation.
To gain insight in this simulation, it is useful to draw an analogy with the tropical atmosphere. The prescribed mixing profile drives an increase in temperature (and hence a decrease in density) over the bottom grid cell and cooling (an increase in density) in the rest of the water column, much like the radiative budget of the atmosphere, which is characterized by warming at the sea–land surface and radiative cooling above. As observed in the tropical atmosphere, the bottom heating triggers upright convection and pretty much erases any vertical stratification in the basin (ocean eddies in the basin are too weak to maintain a stratification, like eddies in the tropical atmosphere and unlike eddies in the extratropical atmosphere). The model output confirms that convective adjustment is active below 1500 m, where the stratification is weak. The thickness of this unstratified abyssal layer depends on the particular profile of vertical mixing and the value of the air–sea density flux, which is weak and negative along most of the basin.
While the vertical stratification in the bottom convective layer is weak, there is a substantial horizontal stratification. The waters flow northward along the BBL, where the turbulent density flux divergence is positive and allows a flow toward lighter density classes. The middle-right panel of Fig. 5 confirms that the diapycnal velocity is positive along the BBL as waters flow horizontally and cross the isopycnals that intersect the ocean bottom; the diapycnal upwelling in the BBL is horizontal in the absence of sloping boundaries. The waters then flow back toward the channel approximately along the curved isopycnals, but at a somewhat smaller slope, as they gain density through the imposed negative turbulent density flux divergence. The middle-right panel of Fig. 5 confirms that the diapycnal velocity is negative in the interior and much weaker in magnitude than in the BBL.
The bottom-right panel of Fig. 5 shows that waters are transformed to lighter density classes in the net, but this transformation is the result of a large cancellation between the transformation into lighter density classes within the BBL, where 
The solution with hardly any abyssal stratification depicted in the right column of Fig. 5 is very different from that observed in the deep oceans. An ocean with vertical sidewalls and a realistic turbulent density flux profile results in a stratification not consistent with observations. This solution is also somewhat physically inconsistent. Mixing is generated by breaking internal waves. In the limit of weak stratification, as obtained in the simulation, the internal wave field would vanish together with the associated mixing.
Figure 6b shows the turbulent diffusivity calculated from Eq. (14) and averaged horizontally over the whole basin. A very large diffusivity κ, increasing beyond 10−2 m2 s−1 at the ocean bottom, is required to drive a bottom-intensified turbulent density flux with a traditional diffusive closure: 
c. Bowl simulations with a turbulent density flux profile decreasing above the BBL
The shape of the topography and the value of turbulent density flux divergence are chosen such that the turbulent density flux profile integrated over the full basin area at fixed depth increases toward the surface, even though each individual flux profile decreases with height above the BBL (both profiles are shown in the bottom panel for the Mix Symmetric simulation in Fig. 3). The value of Ft is larger than in the Box simulations to reduce by 30% the turbulent density flux divergence. With a smaller divergence, it is easier to pick a topographic shape such that the second term on the right-hand side of Eq. (9) dominates over the first, resulting in net upwelling throughout the basin. The parabolic topography has very weak slopes toward the bottom of the basin so that the whole ocean area experiences strong mixing and Amix = A in the deepest 1 km of the bowl.2 The area of strong mixing Amix hardly increases above the bottom kilometer, unlike the full area A, because strong mixing is confined to narrow ribbons along the boundaries whose area is nearly constant with height. According to Eq. (6), it is the area experiencing mixing along an isopycnal, and not at constant depth, that sets the sign of the water mass transformation in the basins. However, isopycnals are mostly flat in the Bowl simulations, except in the thin BBLs where they bend to intersect topography at right angles. Thus, the sign of the divergence of the turbulent density flux is not very sensitive to whether the integration is taken along isobaths or isopycnals. This is also true for the estimate of the turbulent density flux for the global ocean presented in the next section.
The left column of Fig. 7 shows the results for the first Bowl simulation. Starting from the top-left panel, we see that the ocean is fully stratified at depth, despite a turbulent density flux profile that decreases toward the surface. In contrast to the Box-MixBottom simulation, no convection develops at depth. The zonally averaged overturning circulation is thermally direct with waters sinking at the southern edge of the channel and rising back toward the surface in the basin. However, the middle-left panel of Fig. 7 shows that the zonally averaged picture is somewhat misleading because the overturning is the result of waters sinking toward higher densities in the basin interior (blue diapycnal velocities) and waters rising toward lighter density classes at much faster speeds along the thin BBL (red diapycnal velocities). The bottom-left panel of the figure confirms that the interior water mass transformation into heavier density classes is overwhelmed by a larger transformation into lighter density classes along the BBLs: 
(left) Results from a Bowl simulation with a turbulent density flux profile decreasing with height above the parabolic ocean bottom as shown in the Mix Symmetric panel of Fig. 3. The three panels show the overturning streamfunction, the diapycnal velocity, and the water mass transformation, respectively, as described in the caption of Fig. 5. The diapycnal velocities are positive along the BBL with values reaching 1 × 10−5 m s−1 and negative in the ocean interior with values reaching only −1 × 10−6 m s−1. (right) Results from a Bowl simulation with a turbulent density flux profile decreasing above the bottom, but twice as large in the eastern as in the western half of the domain as shown in the Mix Asymmetric panel of Fig. 3. The three panels in the right column show the same diagnostics as in the left column. In this simulation the diapycnal velocities in the eastern half of the basin are twice as large as those in the western half.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(left) Results from a Bowl simulation with a turbulent density flux profile decreasing with height above the parabolic ocean bottom as shown in the Mix Symmetric panel of Fig. 3. The three panels show the overturning streamfunction, the diapycnal velocity, and the water mass transformation, respectively, as described in the caption of Fig. 5. The diapycnal velocities are positive along the BBL with values reaching 1 × 10−5 m s−1 and negative in the ocean interior with values reaching only −1 × 10−6 m s−1. (right) Results from a Bowl simulation with a turbulent density flux profile decreasing above the bottom, but twice as large in the eastern as in the western half of the domain as shown in the Mix Asymmetric panel of Fig. 3. The three panels in the right column show the same diagnostics as in the left column. In this simulation the diapycnal velocities in the eastern half of the basin are twice as large as those in the western half.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(left) Results from a Bowl simulation with a turbulent density flux profile decreasing with height above the parabolic ocean bottom as shown in the Mix Symmetric panel of Fig. 3. The three panels show the overturning streamfunction, the diapycnal velocity, and the water mass transformation, respectively, as described in the caption of Fig. 5. The diapycnal velocities are positive along the BBL with values reaching 1 × 10−5 m s−1 and negative in the ocean interior with values reaching only −1 × 10−6 m s−1. (right) Results from a Bowl simulation with a turbulent density flux profile decreasing above the bottom, but twice as large in the eastern as in the western half of the domain as shown in the Mix Asymmetric panel of Fig. 3. The three panels in the right column show the same diagnostics as in the left column. In this simulation the diapycnal velocities in the eastern half of the basin are twice as large as those in the western half.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
This circulation pattern confirms our theoretical prediction that topography exerts a strong control on the ocean circulation. The rapid increase in the length/perimeter of isobaths with height is crucial to allow waters to upwell in the net, despite a mixing profile that decays with height above the bottom. Unexpectedly, the net overturning circulation is the residual of two much larger and opposite diapycnal transports: downwelling of waters in the ocean interior (blue bars) and upwelling of waters along the boundaries (red bars). In the next section, we will provide tantalizing evidence that this is not only the case in our idealized simulations but possibly in the abyssal ocean as well.
The importance of topography is best illustrated by comparing the Box-MixBottom and Bowl simulations. In both simulations the air–sea fluxes in the channel convert light waters into heavy ones that sink to the bottom and flow toward the basin. In the Box-MixBottom simulation with a turbulent density flux profile decreasing toward the surface, the isopycnals in the basin bend downward to intersect the BBL at the ocean flat bottom, where waters can be converted back into lighter waters to close the overturning circulation loop. In the Bowl configuration, isopycnals in the basin are nearly flat and intersect the BBL at the lateral boundaries. The topography is crucial in allowing the ocean basin to remain stratified with a thermally direct circulation.
Figure 6c shows the basin-averaged turbulent diffusivity as a function of height above the bottom implied by the turbulent density flux and stratification in the equilibrated solution. The diffusivity peaks at 2 × 10−3 m2 s−1 above the BBL and decays by an order of magnitude over 1000 m. These values are consistent with the diffusivity profiles implied by the turbulent density fluxes for the abyssal ocean estimated by Nikurashin and Ferrari (2013) for the Pacific Ocean, the basin with a circulation and stratification closest to our simulations (see dashed line in Fig. 6c and next section). We also show a longitude–depth map of the turbulent diffusivity averaged over the latitudes corresponding to the closed basin in Fig. 8 (left panel). The map highlights that the largest values of κ are confined to the deepest part of the basin, where stratification is weakest, and they decrease upward.
Diapycnal turbulent diffusivities computed with Eq. (14) based on the turbulent density fluxes and the stratification obtained in the (left) Bowl-Symmetric and (right) Bowl-Asymmetric simulations. The diffusivities are averaged meridionally over the latitudes corresponding to the closed basin and plotted as a function of depth and longitude.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Diapycnal turbulent diffusivities computed with Eq. (14) based on the turbulent density fluxes and the stratification obtained in the (left) Bowl-Symmetric and (right) Bowl-Asymmetric simulations. The diffusivities are averaged meridionally over the latitudes corresponding to the closed basin and plotted as a function of depth and longitude.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Diapycnal turbulent diffusivities computed with Eq. (14) based on the turbulent density fluxes and the stratification obtained in the (left) Bowl-Symmetric and (right) Bowl-Asymmetric simulations. The diffusivities are averaged meridionally over the latitudes corresponding to the closed basin and plotted as a function of depth and longitude.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
The last simulation we consider is equivalent to the Bowl-MixSymmetric simulation we just described, except that the turbulent density flux profile is different on the eastern and western halves of the domain, as shown in the Mix Asymmetric panel of Fig. 3. We use the same vertical profile given in Eq. (19), but we multiply the western profile by 0.7 and the eastern profile by 1.3, so that the horizontally averaged density profile remains the same as in the Bowl-MixSymmetric simulation, but there is an east–west asymmetry. This choice is a crude attempt to mimic the variability in mixing profiles observed in the real ocean. In particular, we were inspired by the iconic mixing profiles reported by Polzin et al. (1997) from the Brazil Basin, where the turbulent density flux was shown to be nearly an order of magnitude larger on the eastern side of the basin than on the western side.
Results from this simulation are quite similar to those with the same mixing profile on the two sides of the basin. The only difference is that the diapycnal velocities (Fig. 7) and turbulent diffusivities (Fig. 6d and Fig. 8, right panel) along the eastern boundary are about twice those along the western boundary, both in the interior and in the BBL, consistent with our choice of imposing a turbulent density flux divergence twice as large in the eastern side than in the western side. We infer that, in an ocean with variable mixing profiles, both the interior sinking and the BBL upwelling are going to be most intense above rough boundaries where the turbulent density flux divergence is largest.
5. Mixing and water mass transformations in the global ocean
De Lavergne et al. (2016) have recently estimated the water mass transformation induced by mixing in the stratified ocean interior. Here, we revisit their result by focusing on the crucial role played by the BBLs. De Lavergne and collaborators considered the contribution of geothermal heat flux and mixing generated by topographic waves breaking locally and nonlocally, that is, close and far from where they were generated. We start our analysis by considering local mixing generated by topographic waves following Nikurashin and Ferrari (2013). We neglect mixing induced by wind-generated waves, which dominate mixing in the upper ocean, because our focus is on the abyssal water mass transformations close to the bottom topography. Changes in bathymetry are most pronounced below 2000 m, as we show below, and this is where we expect upwelling along BBLs. Extending Nikurashin and Ferrari (2013), we explicitly compute the impact of the interior transformation within the BBLs and show that, consistent with our theoretical analysis, the BBL transformation in the abyssal ocean has the opposite sign and is substantially larger than the overall interior transformation.
Abyssal mixing in the stratified ocean above the BBL is driven by breaking internal waves. These waves are primarily generated by tidal and geostrophic currents flowing over irregular bottom topography (Polzin et al. 1997; St. Laurent et al. 2012). Nikurashin and Ferrari (2013) used linear wave theory to estimate the energy radiated into topographic internal waves and then assumed that ⅓ of that energy was locally dissipated through wave breaking. By further assuming that only ⅙ of the total energy lost in a breaking wave ends up mixing the fluid3 (Osborn 1980) and that it generates a turbulent density flux that decays exponentially with height above the bottom topography (Jayne and St. Laurent 2001), they generated a map of the turbulent density flux at the top of the BBL, Fb(x, y). The full column turbulent density flux was then estimated by substituting Fb(x, y) in Eq. (18) with an e-folding scale of d = 500 m, Ft = 0 kg m−2 s−1.
Nikurashin and Ferrari (2013) computed the water mass transformation using their turbulent density flux map with Eq. (7) but replaced density with neutral density (Jackett and McDougall 1997), a variable that conveniently eliminates irrelevant compressive effects from the full density (an important correction when dealing with the seawater nonlinear equation of state). Their map had a vertical resolution of 250 m below 1500 m, the depth range of interest here. The result of their calculation is shown as blue bars in Fig. 9 and suggests that stratified waters above the BBL are transformed to denser classes by breaking waves, as expected for a density profile decreasing upward.4 Notice that our focus is on density classes greater than approximately 28.0 kg m−3, which sit below 2000 m, where topographic waves are believed to dominate mixing rates.
Water mass transformation for the global ocean based on the estimate of the mixing induced by breaking topographic waves published in Nikurashin and Ferrari (2013). The blue bars indicate the water mass transformation induced by waves breaking in the stratified ocean interior above the BBL. The red bars represent the water mass transformation within the BBL. The solid black bars are the net transformation given by the sum of the interior and BBL transformations. The water mass transformations are plotted as a function of neutral density. The dashed line indicates the 28.1 kg m−3 neutral density surface that separates the two major water masses within the density range considered: AABW and NADW.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Water mass transformation for the global ocean based on the estimate of the mixing induced by breaking topographic waves published in Nikurashin and Ferrari (2013). The blue bars indicate the water mass transformation induced by waves breaking in the stratified ocean interior above the BBL. The red bars represent the water mass transformation within the BBL. The solid black bars are the net transformation given by the sum of the interior and BBL transformations. The water mass transformations are plotted as a function of neutral density. The dashed line indicates the 28.1 kg m−3 neutral density surface that separates the two major water masses within the density range considered: AABW and NADW.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Water mass transformation for the global ocean based on the estimate of the mixing induced by breaking topographic waves published in Nikurashin and Ferrari (2013). The blue bars indicate the water mass transformation induced by waves breaking in the stratified ocean interior above the BBL. The red bars represent the water mass transformation within the BBL. The solid black bars are the net transformation given by the sum of the interior and BBL transformations. The water mass transformations are plotted as a function of neutral density. The dashed line indicates the 28.1 kg m−3 neutral density surface that separates the two major water masses within the density range considered: AABW and NADW.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
We extend the Nikurashin and Ferrari (2013) map by imposing that the turbulent density flux goes linearly to zero over the last grid point. This is tantamount to assuming that the turbulent density flux goes from zero to its interior value across a 250-m-deep BBL. The water mass transformation over the BBL is given as red bars in Fig. 9 and confirms that waters become lighter along the BBL. The sum of the negative interior and positive BBL transformations represent the net overturning circulation shown as black bars in Fig. 9. Much like in our idealized simulations, the maximum net deep overturning of about 10 Sv across the neutral density γ = 28.1 kg m−3 is a small residual between nearly 40 Sv of waters rising along the boundaries and nearly 30 Sv of waters sinking in the stratified interior.
Figure 10b shows the profile of the area-integrated turbulent density flux across neutral density surfaces based on the estimate of Nikurashin and Ferrari (2013), which was used to generate Fig. 9. The dashed line represents the integral taken over the whole area of each neutral density surface. The continuous line is the integral taken only over Amix, defined as the isopycnal area where the turbulent density flux divergence is larger than 1.1 × 10−11 kg m−3 s−1, the value of the globally averaged turbulent density flux 1000 m above the bottom. The two lines are very close, confirming that the topographic mixing is confined to Amix. The integral of the turbulent density flux increases toward lighter waters up to 28.0 kg m−3, consistent with upwelling of those waters as per Eq. (8) and Fig. 9. The turbulent density flux changes are smaller for lighter waters and thus we do not expect topographic wave breaking to drive strong flows there, as confirmed in Fig. 9.
(a) The total area A of ocean isopycnals and the Amix portion of that area near the ocean boundaries where isopycnals experience the bottom-enhanced mixing; Amix is defined as the portion of the isopycnal area where the turbulent density flux divergence is larger than 1.1 × 10−8 kg m−3 s−1. The turbulent density flux is based on the flux estimates of Nikurashin and Ferrari (2013). Only neutral densities higher than 27.0 kg m−3 are considered. (b) The turbulent density flux integrated over Amix and A. (c) The turbulent density flux averaged over Amix and A.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(a) The total area A of ocean isopycnals and the Amix portion of that area near the ocean boundaries where isopycnals experience the bottom-enhanced mixing; Amix is defined as the portion of the isopycnal area where the turbulent density flux divergence is larger than 1.1 × 10−8 kg m−3 s−1. The turbulent density flux is based on the flux estimates of Nikurashin and Ferrari (2013). Only neutral densities higher than 27.0 kg m−3 are considered. (b) The turbulent density flux integrated over Amix and A. (c) The turbulent density flux averaged over Amix and A.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
(a) The total area A of ocean isopycnals and the Amix portion of that area near the ocean boundaries where isopycnals experience the bottom-enhanced mixing; Amix is defined as the portion of the isopycnal area where the turbulent density flux divergence is larger than 1.1 × 10−8 kg m−3 s−1. The turbulent density flux is based on the flux estimates of Nikurashin and Ferrari (2013). Only neutral densities higher than 27.0 kg m−3 are considered. (b) The turbulent density flux integrated over Amix and A. (c) The turbulent density flux averaged over Amix and A.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Figure 10a plots the vertical profile of the area of near-boundary mixing Amix (solid line). This area represents the near-boundary strip of an isopycnal surface that experiences bottom-enhanced mixing. The along-boundary strip extends from a few tens to a few hundreds of kilometers from the solid boundary to the interior.5 Topographic roughness on scales less than O(10) km contributes some fuzziness at the edges of Amix but does not affect the overall area. The calculation of Amix is based on the ETOPO5 (National Geophysical Data Center 1993) global 5-min topography, which is reasonably accurate at scales larger than 10 km. The major uncertainty in the calculation is likely to arise from unknown variations in the e-folding scale of mixing.
The area Amix increases by two orders of magnitude between the ocean bottom and the neutral density 28.0 kg m−3, and it is a full order of magnitude smaller than the full ocean area (dashed line). Densities larger than 28.0 kg m−3 correspond to Antarctic Bottom Water and the densest North Atlantic Deep Water. We thus expect that topographic effects and BBL upwelling will be most important for these deep waters and less above. The rapid increase in Amix is primarily associated with the increase of the arc length of the bathymetry as one moves upward in the water column up to approximately 2000 m. At shallower depths, Amix remains approximately constant because the majority of ridges and seamounts in the ocean do not reach above 2000 m.
Finally, in Fig. 10c, we show the turbulent density flux averaged (not integrated) over Amix (solid line) and averaged over the whole isopycnal area (dashed line). The area-averaged turbulent density flux decreases by at most a factor of 5 between the bottom and the 28.0 kg m−3 isopycnal, while the area Amix drops by two orders of magnitude over the same depth range. According to Eq. (9), this implies that the net upwelling of waters denser than 28.0 kg m−3 is primarily a result of the increase in the area of near-topography mixing.
A full study of water mass transformations in the deep ocean should include the effect of mixing associated with internal waves generated locally and nonlocally. The turbulent density flux map produced by Nikurashin and Ferrari (2013) considered only mixing associated with topographic waves that break locally near their generation sites. De Lavergne et al. (2016) attempted to estimate the mixing associated with nonlocal waves, but their results were quite uncertain. To get a sense of the overall importance of these nonlocal waves in driving diapycnal flows in the abyss, we appeal to the observational evidence that, away from regions where strong topographic waves are generated, ocean mixing is weak and characterized by a background turbulent diffusivity of about 10−5 m2 s−1 (Waterhouse et al. 2014). Figure 11 shows the water mass transformation calculated with Eq. (7) and a turbulent density flux given by 
Water mass transformation for the global ocean induced by background mixing represented as a turbulent density flux 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Water mass transformation for the global ocean induced by background mixing represented as a turbulent density flux
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
Water mass transformation for the global ocean induced by background mixing represented as a turbulent density flux
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0244.1
It should be clear that our global estimate of the water mass transformation is very uncertain, as it relies on a poorly constrained estimate of the turbulent density fluxes induced by mixing. But the pattern of sinking in the interior and upwelling along the abyssal BBLs is robust because it depends only on having a turbulent density flux decreasing upward above the BBL, as suggested by many independent observations (Polzin et al. 1997; St. Laurent et al. 2012; Waterman et al. 2013; Waterhouse et al. 2014), and a turbulent density flux going to zero in the BBL, as demanded by the no flux boundary condition through the ocean bottom.
De Lavergne et al. (2016) studied the effect of the geothermal heat flux in their analysis, which we ignored, and showed that it lightens waters along the ocean boundaries and drives a few additional Sverdrups to the total BBL transport. Based on the work presented in this paper, this implies that the geothermal heat flux further reinforces the diapycnal upwelling of bottom waters along the BBLs. De Lavergne et al. (2016) also considered the effects of nonlinearities in the equation of state of seawater, which we ignored, and found that those effects introduce minor corrections.
6. Conclusions
The global overturning circulation is thought to consist of sinking of dense waters at high latitudes, a concentrated upwelling in the latitude band of the Southern Ocean and a broader upwelling spread across all ocean basins. The upwelling in the Southern Ocean is along isopycnals and driven by the surface westerlies. The upwelling in the basins crosses density surfaces (i.e., it is a diapycnal upwelling) and has been thought to be associated with widespread mixing as first proposed by Munk (1966) in his seminal paper on abyssal recipes. In this manuscript, we have shown that the recent observational evidence that mixing is most intense close to the ocean bottom is inconsistent with this picture. The observed vertical profile of mixing drives diapycnal sinking of deep waters toward the abyss in the ocean interior. We have argued that the diapycnal upwelling of waters is instead confined to thin, turbulent boundary layers along the sloping ocean boundaries. The boundary layer diapycnal upwelling must be so intense as to overcome the interior diapycnal sinking and allow waters to rise toward the surface in the net. This hypothesis has been illustrated with theory, idealized simulations, and a global map of ocean mixing.
Our result applies to the abyssal density layers that outcrop only in the Southern Ocean and across which waters upwell diapycnally in the ocean interior. Density surfaces that outcrop both in the North Atlantic and in the Southern Ocean have flow mostly along density surfaces and are believed to respond more to surface wind patterns and air–sea fluxes than to interior mixing. Also, mixing profiles are observed to decay rapidly above the bottom in regions with rough topography and intense wave breaking, while the mixing profiles are more uniform over abyssal plains. We thus expect boundary diapycnal upwelling to be concentrated along rough boundaries rather than along smooth ones.
Our work differs in a fundamental way from previous investigations that emphasized the role of enhanced mixing along the ocean boundaries on the ocean overturning circulation (Scott and Marotzke 2002; Spall 2001). In our study, the focus is not on the enhancement of mixing along the boundaries; rather, we are emphasizing the decrease in mixing within the bottom boundary layers that drive the upwelling of abyssal waters. Our work differs also from the literature on diffusion-driven flows along sloping boundaries (Garrett et al. 1993). That literature investigates the flows that arise within a boundary in response to local mixing, while we illustrate how the boundaries accommodate the rising of waters that sink in response to bottom-intensified mixing in the stratified interior.
St. Laurent et al. (2001) and Thurnherr et al. (2005) diagnosed the overturning circulation for the Brazil Basin using an inverse model together with vertical profiles of temperature, salinity, and turbulence collected as part of the Brazil Basin Tracer Release Experiment. They inferred that waters sunk away from boundaries in response to the interior mixing, while they upwelled along the rough boundary of the Mid-Atlantic Ridge. Polzin (2009) confirmed this overall picture with a theoretical study of the depth profile of turbulent dissipation in the region. Here, we have generalized these results to the global ocean, emphasizing the importance of the ocean’s bathymetry or more precisely the length of density incrop lines as a function of density in determining whether the along-boundary upwelling exceeds the interior sinking.
Our results have important implications for the horizontal abyssal circulation in addition to the overturning. Stommel, Arons, and Faller in a series of papers from 1958 to 1960 laid the foundation for our present understanding of the horizontal abyssal circulation (Stommel 1958; Stommel et al. 1958; Stommel and Arons 1960). They argued that mixing drives a pervasive diapycnal upwelling of waters from the abyss, resulting in a stretching of water columns within all ocean basins. This stretching in turn would force waters to flow poleward to conserve their potential vorticity and to return equatorward along deep western boundary currents. McDougall (1989) and Rhines (1993) pointed out that sloping boundaries modify this picture. Our argument that ocean waters sink across density surfaces in response to mixing in the stratified ocean interior and rise along lateral boundaries results in an even more complex geometry of squeezing and stretching of water columns, further modifying the overall deep gyre circulation. There is a lot more to understand about this new emerging view of the ocean abyssal circulation.
An interesting puzzle emerging from our work is whether large-scale numerical models capture the dichotomy of diapycnal sinking of ocean waters in the stratified interior and diapycnal rising along the boundaries. Numerical models used for climate studies typically represent mixing with a prescribed vertical diffusivity profile either constant or small in the upper ocean and large below about 2000 m (Bryan and Lewis 1979). With both choices, the resulting turbulent density flux profile 
To conclude, we believe that our results call for detailed observational programs of the flows and mixing in the turbulent bottom boundary layers. There are hardly any observations of the turbulent fluxes in the bottom boundary layers of the ocean abyss, while there are more observations of deep-ocean mixing in the stratified interior. A full picture of the global ocean overturning requires sampling of turbulence and the vertical motions in both regions.
Acknowledgments
The work benefitted from discussions with Carl Wunsch and Joern Callies and email exchanges with Chris Garrett, Andrew Hogg, and Gurvan Madec. Matthew Alford generously shared an unpublished manuscript on observational estimates of the sinking induced by mixing in the abyssal ocean. Louise Bell is thanked for preparing the first two illustrations. We gratefully acknowledge National Science Foundation support through Award OCE-1233832 (R.F.) and Australian Research Council support though Grant DE150100937.
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The divergence of the flux component normal to the boundary scales with 
The bathymetry has small slopes in the bottom 1 km and mixing extends a distance d/tanθ ~ 10 latitude degrees away from the lateral boundaries. Thus, the area experiencing mixing increases with the basin area in the bottom kilometer.
Osborn (1980) argued that 
Nikurashin and Ferrari (2013) misinterpreted their calculation of the water mass transformation in the stratified interior and incorrectly stated that their calculation implied diabatic rising, rather than sinking, of abyssal waters.
Equation (10) shows that the width of Amix is given by the ratio of the e-folding scale of the turbulent density flux over the tangent of the topographic slope, measured on scales larger than 10 km, that is, scales larger than those of the waves that are radiated from the topography and generate the mixing. For a typical e-folding scale of 500 m and a mild topographic slope of ⅝, we infer a width of 200 km [setting α = 1 in Eq. (10)]. Steeper slopes would result in a narrower Amix.
