The upwelling diapycnal limb of the ocean’s meridional overturning circulation is driven by divergence of diabatic turbulent buoyancy fluxes 〈w′b′〉 across density surfaces. A global assessment of zonally averaged internal-wave-driven turbulent diapycnal buoyancy fluxes from a strain-based finescale parameterization is used to infer mean diapycnal transports in the interior and near the bottom boundary. Bulk interior diabatic transports dominate above 2500-m depth (buoyancies |B| = gγn/ρ0 < 0.267 m s−2, neutral densities γn < 27.9 kg m−3), upwelling at 10–11 Sv (1 Sv = 106 m3 s−1); 2, 5, and 3–4 Sv in the Indian, Pacific, and Atlantic, respectively, but are weak in the abyss. Boundary water-mass transformations peak at 18–25 Sv (4–6, 10–14, and 4–5 Sv in the Indian, Pacific, and Atlantic) near buoyancy |B| ~ 0.268 m s−2 (γn ~ 28.1 kg m−3, 4500-m depth) between bottom and lower deep waters, consistent with published 20–30-Sv global Antarctic Bottom Water (AABW) transport estimates. Interior transports above 2500-m depth fall below inverse estimates, consistent with a more adiabatic ocean interior where diapycnal mixing occurs at Southern Hemisphere high-latitude surface density outcrops.
The smallest O(1) cm scales of motion in the ocean associated with turbulent mixing are linked to the largest O(10 000) km of the meridional overturning circulation (aka the global conveyer belt) through intermediate scales O(1–10) km associated with internal gravity waves that break to generate most of the turbulence in the stratified ocean interior. The meridional overturning circulation connects most of the World Ocean, its abyssal limb storing heat and solutes on climatological time scales ~O(1000) years. Dense bottom waters form in Antarctic shelf seas and spill down continental slopes (Price and Baringer 1994) in narrow outflow plumes to the abyss where they are carried around deep basins by diverse pathways controlled by eddy and mean-flow interactions with topography (Bower et al. 2009; Lozier 2012), cascading through narrow passages and mixing with overlying waters en route. Lighter deep and intermediate waters created in marginal seas bordering the North Atlantic follow a similar path, though some of these are convectively formed at the surface so do not have overflow plumes on boundaries. The return pathway of these dense waters to the surface, as well as the stratification of the deep and abyssal ocean, are controlled in part by turbulent diapycnal (across density) mixing (Munk 1966; Munk and Wunsch 1998), although adiabatic interior processes coupled with surface mixing at Southern Hemisphere high-latitude surface outcrops have been implicated for lower-latitude water masses shallower than ~2000-m depth (Toggweiler and Samuels 1998; Samelson 2004; Haertel and Fedorov 2012). Without diapycnal mixing, the overturning circulation is confined to the Southern Ocean and driven by wind (Ferrari et al. 2016).
Half a century ago, primarily for lack of measurements, turbulent mixing in the stratified ocean interior was usually taken to be uniform (e.g., Stommel and Arons 1960; Munk 1966), though it was appreciated that nonuniform mixing would impact deep circulation patterns (Stommel et al. 1958). More recent work has established that ocean mixing is heterogeneous in both the vertical and horizontal. Diapycnal diffusivities K are weak ~O(0.1 × 10−4) m2 s−1 over smooth abyssal plains and slopes (Toole et al. 1994; Kunze and Sanford 1996; Polzin et al. 1997), while orders of magnitude higher over abrupt or rough topography (Polzin et al. 1997; Kunze and Toole 1997; St. Laurent et al. 2001; Kunze et al. 2006; Thurnherr et al. 2005; Kunze 2017) associated with internal tide generation (Ray and Mitchum 1997; Simmons et al. 2004; Nycander 2005; Klymak et al. 2006), critical internal-wave reflection (Müller and Liu 2000a,b; Nash et al. 2004, 2007; Kelly et al. 2012; Martini et al. 2013), scattering from rough topography (Müller and Xu 1992; Johnston and Merrifield 2003), and hydraulic flow through narrow passages between abyssal basins (Hogg et al. 1982; Roemmich et al. 1996; Ferron et al. 1998; Bryden and Nurser 2003; Alford et al. 2013). Spontaneous generation (Nagai et al. 2015) and lee-wave generation (Scott et al. 2011; Nikurashin and Ferrari 2011; St. Laurent et al. 2012; Waterman et al. 2014; Wright et al. 2014) appear to be less important. Climate simulations suggest that the distribution of mixing, not just its average magnitude, impact important climatological features of the ocean (Jochum 2009; Friedrich et al. 2011; Melet et al. 2016) such that mixing is more important where stratification is high, for example, the pycnocline, and near deep-water sources.
Turbulent mixing in the ocean’s stratified interior, that is, away from boundaries, is driven by the breaking of finescale internal waves. Low-mode internal waves forced by tides and wind on boundaries at 2.0 ± 0.6 TW redistribute energy over the full water column and horizontally over distances of thousands of kilometers (Alford 2001; Zhao et al. 2016). They lose energy through a cascade from larger energy-containing vertical scales ~O(1000) m to the finescale ~O(10) m where wave breaking produces turbulent dissipation and mixing of 2.1 ± 0.6 TW globally (Kunze 2017).
In this paper, zonally averaged turbulent diapycnal buoyancy fluxes 〈w′b′〉 = −γε = −KN2, where ε is the turbulent kinetic energy dissipation rate, γ the mixing coefficient (Osborn 1980; Gregg et al. 2017), and N the buoyancy frequency, are utilized to quantify the internal-wave-driven component of the diabatic upwelling limb of the ocean interior’s deep and abyssal meridional overturning circulation. The buoyancy fluxes were inferred globally (Kunze 2017) from a well-established finescale parameterization based on the internal wave–wave interaction theory (McComas and Müller 1981; Henyey et al. 1986) that utilized internal-wave strain (Gregg and Kunze 1991; Wijesekera et al. 1993; Mauritzen et al. 2002; Kunze et al. 2006) from ~30 000 hydrographic casts. As well, the role of sloping boundaries in driving integrated buoyancy flux divergences and diapycnal transports is quantified. As in St. Laurent et al.’s (2001) Brazil Basin study, boundary upwelling is found to be critical for the abyssal ocean.
Section 2 lays out the underlying geometry and equations for interior and boundary diapycnal transports driven by area-integrated buoyancy flux divergences, and section 3 describes the internal-wave-driven diapycnal mixing database. Section 4 presents zonally averaged transects of turbulent dissipation rates ε and diapycnal diffusivities K as functions of depth (or buoyancy) and latitude in the Indian, Pacific, and Atlantic; section 5 interior and boundary diapycnal velocity and transport transects; and section 6 the meridional overturning streamfunction. Section 7 describes bulk ocean diapycnal velocity and transport profiles. Section 8 compares transport results with those from inverse methods, section 9 discusses bottom-generated turbulence, and section 10 provides a summary and discussion.
2. Control volumes for interior and boundary diapycnal transports
Before estimating zonally averaged meridional overturning circulations by ocean basin, some preliminary considerations are needed to clarify the control volumes and their implications in the interior and near boundaries. It is convenient to work in buoyancy (density) space following Walin (1982), McDougall and Ferrari (2017), de Lavergne et al. (2016), and Ferrari et al. (2016). Control volumes (Fig. 1) are bound (i) in the “vertical” by iso-B surfaces separated by constant δB where B can be any conservative variable but is taken here to be buoyancy B (=−gγn/ρ0, where γn is neutral density; www.teos-10.org), (ii) in the zonal (east and west) by either solid boundaries where isopycnals ground or surfaces perpendicular to isopycnal surfaces, and (iii) in the meridional (north and south) by the bottom or surfaces perpendicular to isopycnal surfaces. Using lateral boundaries perpendicular to isopycnal surfaces ensures that isopycnal advective (bolus) and turbulent transports do not impact buoyancy conservation (Walin 1982), obviating the need for basinwide integrals as in McDougall (1989) and Klocker and McDougall (2010).
For such a steady control volume in an incompressible fluid with a linear equation of state, conservation of volume is
and conservation of buoyancy B
where variables have been split into a microscale turbulent component, for example, w′ and b′, associated with diapycnal turbulent buoyancy fluxes 〈w′b′〉 = −KN2, and a background u, υ, w, and B encompassing the climatological mean, eddies, and internal waves; if B was taken to be the climatological mean, then eddy and internal waves would become part of the perturbation and so formally contribute to the diapycnal buoyancy flux (Davis 1994), but since neither of these contribute net diapycnal buoyancy fluxes, it makes more dynamical sense to partition eddies and internal waves into the background; the buoyancy binning in section 4 takes this into account. Diapycnal velocities w are normal to buoyancy surfaces, while u and υ are parallel to isopycnals. The terms Au and Ad (Az) denote upper and lower B surfaces, aE and aW (ax) east and west areas, and aN and aS (ay) north and south areas bounding the volume. Letting δB → 0, then, by the virtue of the design of the control volume, BE = BW = BS = BN = B = constant so can be taken out of the ax and ay area integrals in (2), while Bd/u can be expressed as B ± δB/2 with δB also a constant that can be taken out of the isopycnal Az area integrals, such that (2) can be rewritten
The term in parentheses on the left-hand side vanishes by virtue of (1), leaving the powerful result for the diapycnal transport W,
that is, diapycnal transports W are controlled entirely by divergence ∂/∂B of the area-integrated buoyancy fluxes 〈w′b′〉 = −γε across buoyancy surfaces (with the above restrictions in effect), eliminating bolus fluxes from the diapycnal balance because they are adiabatic. This is a special case of Walin (1982). Thus, to maintain steady state in B, divergence of the turbulent diapycnal buoyancy flux transport 〈w′b′〉δA across an iso-B surface requires a diapycnal transport δW. Expression (4) has also been derived in the appendixes of Klocker and McDougall (2010) and McDougall and Ferrari (2017), who show that the assumption of steady state is not necessary, allowing eddies and internal waves to be part of the background as assumed here. Cabbeling, thermobaricity, and double diffusion all act in the opposite sense to turbulent mixing, lowering background potential energy rather than raising it. De Lavergne et al. (2016) found that cabbeling and thermobaricity associated with nonlinearity of the equation of state contributed little to density transformation by diapycnal mixing below the pycnocline.
From (4), diapycnal transports W can be driven by diapycnal divergences ∂/∂B of either (i) the turbulent buoyancy flux 〈w′b′〉 = −γε or (ii) isopycnal area δA as recognized by McDougall (1989) and further developed by Klocker and McDougall (2010), who pointed out the role of hypsometry in basinwide integrals of (4). But (4) applies to any area bounded by normals to isopycnals, allowing us to isolate interior and near-boundary contributions.
a. Interior case
In the stratified interior (Fig. 1c), turbulent diapycnal buoyancy fluxes 〈w′b′〉 = −γε = −KN2. Isopycnal areas in the interior do not vary with depth (by construction in Fig. 1), so that ∂A/∂B = 0, and diapycnal transports Wi are driven by buoyancy flux divergences ∂〈w′b′〉/∂B alone
using the chain rule and ∂B/∂z = Bz = N2 in the last equality. This is the classical abyssal recipes result of Munk (1966).
b. Bottom boundary layer case
Gradients of isopycnal area ∂A/∂B (4) are associated with basin areas shrinking with depth in proximity of near-bottom slopes s = |∇hH|, where H is the bottom depth. If isopycnals were naively assumed to be flat (upper panel of Fig. 1) then transports in region b can be expressed as
since the boundary perimeter varies only weakly with buoyancy B or depth z, using ∂/∂B = N−2∂/∂z and assuming that the average boundary diapycnal diffusivity K varies more weakly with depth z than the area integral in the latitude bins (section 4). In this approximation, hypsometrically driven buoyancy flux divergences depend only on the perimeter-averaged diapycnal diffusivity K and perimeter integral of the inverse bottom slope .
However, neglecting geothermal fluxes, the turbulent no-flux condition at the bottom requires that isopycnals be perpendicular to the bottom (lower panel of Fig. 1) with Bx = sN2 and Bz = s2N2, where N2 is the vertical stratification in the interior outside the boundary layer. This implies that the buoyancy gradient ∂/∂B will be parallel to the bottom in the bottom boundary layer (BBL). Likewise any diapycnal turbulent buoyancy fluxes 〈w′b′〉 = –γε will be parallel to the bottom in the bottom boundary layer. Assuming that the bottom boundary layer γε integrated around the basin perimeter dℓ varies no more with buoyancy B than in the interior, the primary source of divergence in (4) is associated with the changes in isopycnal area with B (Fig. 1b) because of the bottom slope s. The bottom boundary layer (Fig. 1a) contributes little compared to the region of large ∂A/∂B (Fig. 1b).
Assuming that the transition from normal to flat isopycnals occurs as illustrated in Fig. 2 in coordinates (sx, z) so that dA = r(cosθ + sinθ/s)dθdℓ, where r is defined such that x = (r/s)cosθ and z = rsinθ (Fig. 2) and dℓ is the increment perimeter around the basin (into the page of Fig. 2), (4) can be written as
assuming that the buoyancy flux γε does not systematically vary along the π/2 arc in the θ integral; for gentle slopes s ≪ 1, the segment of the integral closest to the bottom, where mixing γε might go to zero, proves not to contribute significantly to the integral so this assumption might be moot. The buoyancy gradient Br = δB/δr = N2 is constant from the definition of r and δB constant, so that
Equation (8) is independent of boundary layer thickness as shown by McDougall and Ferrari (2017). Because of the integral’s weighting, it depends primarily on the average diapycnal diffusivity Kb in the stratified waters outside the bottom boundary layer as well as the perimeter-averaged bottom slope s, which sets the divergence in surface area ∂A/∂z = as previously. In this formulation, the integrated buoyancy flux divergence does not depend on 〈w′b′〉=−KN2 → 0 at the bottom (de Lavergne et al. 2016; Ferrari et al. 2016; McDougall and Ferrari 2017) since the near-bottom region contributes little to the integral. It does not matter if the area-integrated buoyancy flux diverges because of the no-flux bottom boundary condition or because of hypsometry; the divergence will be the same. The difference between curved [(8); lower panel of Fig. 1] and flat isopycnals [(6); upper panel of Fig. 1] is only a factor of π/2 ~ O(1), so either is a reasonable measure of near-boundary flux divergence associated with changes of isopycnal surface area. Equation (8) hinges on changes in near-boundary isopycnal area with buoyancy B being much larger than changes in basin perimeter-integrated buoyancy flux , which cannot be tested at this time.
In section 5, we will use zonally averaged turbulent kinetic energy dissipation rates ε = −〈w′b′〉/γ in the three major oceans (section 4) to infer the meridional overturning circulation caused by internal-wave-driven mixing in the interior. Boundary transport estimates will assume a constant near-boundary diffusivity Kb coupled with Smith and Sandwell (1997) bathymetry.
3. Diapycnal mixing database
Dissipation rates ε and diapycnal diffusivities K were inferred from an internal-wave strain finescale parameterization applied to ~30 000 hydrographic profiles. Details of the processing, methodology, and limitations can be found in Kunze (2017). Data are absent from marginal seas around the North Atlantic and sparse in the Southern Ocean poleward of 60°S but otherwise well-distributed with latitude and longitude. The parameterization provides dissipation rates ε with 128-m vertical resolution from which one can estimate the buoyancy flux 〈w′b′〉 = −γε and diapycnal diffusivity K = γε/N2, where a mixing efficiency γ = 0.2 is used throughout (Osborn 1980; Oakey 1982; St. Laurent and Schmitt 1999; Gregg et al. 2017). Since the parameterization cannot distinguish internal-wave strain from sharp changes in background stratification associated with pycnoclines (Mauritzen et al. 2002; Kunze et al. 2006; Whalen et al. 2012), estimates above 380-m depth were rejected. Estimates were also rejected if the buoyancy frequency N fell below either (i) a noise threshold of 3 × 10−4 rad s−1, below which strain estimates are unreliable because of digitization noise, or (ii) twice the Coriolis frequency, below which internal wave–wave interaction cascades are not expected because of insufficient bandwidth. Contamination of the strain field is possible because of double-diffusive thermohaline staircases and interleaving as well as the finescale vortical mode (Kunze 2017).
The finescale parameterization is only valid for interior internal-wave-driven mixing caused by weakly nonlinear wave–wave interactions. Thus, it does not accurately represent turbulence from direct boundary forcing (Klymak et al. 2006) such as hydraulically controlled turbulence production (Roemmich et al. 1996; Ferron et al. 1998; Alford et al. 2013), in abyssal and shelf-slope canyons (Thurnherr et al. 2005; Kunze et al. 2012) or associated with solitons. It has been found to overestimate turbulence in the Southern Ocean where lee-wave generation is predicted (Waterman et al. 2014). It is not reliable along the equator because it is based on midlatitude rotating internal gravity wave dynamics that does not include equatorially trapped waves.
The globally integrated dissipation rates from the proxy mixing dataset used here are 2.0 ± 0.6 TW (Kunze 2017), consistent with 2.1 ± 0.7 TW internal-wave power inputs from tide and wind and so are energetically plausible; lee-wave generation (Scott et al. 2011; Nikurashin and Ferrari 2011; Wright et al. 2014) and spontaneous imbalance (Nagai et al. 2015) appear to be smaller and within the uncertainties. Regional variability of one to two orders of magnitude is mostly related to underlying topography. For the most part, K is independent of depth, including over rough topography, which implies that 80%–90% of the dissipation ε = KN2/γ and buoyancy flux −KN2 are in the lower-latitude, well-stratified pycnocline, with only 0.08 TW below 2000-m depth. This contrasts with other recent diapycnal transport investigations (Ferrari et al. 2016; McDougall and Ferrari 2017) that assumed depth-independent background interior buoyancy fluxes, augmented by bottom-enhanced tidal mixing parameterizations (e.g., St. Laurent et al. 2002). Average diffusivity profiles are consistent with Lumpkin and Speer’s (2007) inverse estimates to within their factors of 2–3 uncertainty, though consistently higher at lower densities. The inferred diffusivities K are consistent with average microstructure profiles at the 17 sites reported by Waterhouse et al. (2014), including the Brazil Basin measurements, and indicate that microstructure sampling has been biased toward regions of elevated internal tide generation (Kunze 2017). The probability distribution function for 128-m diapycnal diffusivity K is approximately lognormal with modes at K ~ 0.06 × 10−4 m2 s−1 in the Indian and Pacific and 0.1 × 10−4 m2 s−1 in the Atlantic. Arithmetic-averaged diffusivities are 0.4, 0.3, and 0.44 × 10−4 m2 s−1 in the Indian, Pacific, and Atlantic, respectively, an order of magnitude larger than that expected for a canonical Garrett–Munk internal-wave level (Gregg 1989). These diffusivities are significantly smaller than early inverse estimates (e.g., Ganachaud 2003; Macdonald 1998) but roughly consistent with more recent local inverses (e.g., Lumpkin and Speer 2007), finding, for example, generally low values ~O(0.05 × 10−4) m2 s−1 in much of the deep and abyssal northeast Pacific [see Hautala (2017, manuscript submitted to Prog. Oceanogr.), who also reports flows largely along isolines of potential vorticity and salinity as would be expected for weak mixing].
4. Zonally averaged turbulence sections
Zonally averaged diapycnal diffusivities K display similar weak dependence with depth (200-m resolution) and latitude (5° resolution) in all three oceans (Fig. 3d). Weaker diffusivities ~O(10−5 m2 s−1) are found along the equator and stronger diffusivities ~O(10−4 m2 s−1) poleward of 40° latitude in the Atlantic and Indian Oceans. In the Pacific, weakly elevated diffusivities extend throughout the water column near ±30° latitude. In the Southern Ocean, the diffusivity pattern is mirrored by wind forcing, stratification, and interactions of Antarctic Circumpolar Currents with Kerguelen Plateau and Drake Passage topography (Kunze 2017), making its source uncertain. Because dissipation rates ε = KN2/γ and buoyancy fluxes 〈w′b′〉 = −KN2 scale with N2, they display the same mid- and low-latitude high (Fig. 3c) as the bowl of pycnocline stratification N2 (Fig. 3b), largely decreasing with depth or density in an upwelling-favorable fashion [(4)]. Exceptions can be found, notably between 0° and 30°S near 4000-m depth in the South Hemisphere (Fig. 3c) at the transition between the northward-flowing Antarctic Bottom Water and the southward return flow.
Not surprisingly, zonally averaged sections with buoyancy B (=−gγn/ρ0) (with bin size δB ~ 10−4 m s−2, δγn ~ 10−2 kg m−3) and latitude (Fig. 4) resemble deformed versions of these patterns. Elevated dissipation rates ε associated with the subtropical pycnoclines stand out most strongly in the Pacific. Weakly elevated diffusivities K extend over the whole water column in bands about ±30° latitude in the Pacific only. The different coordinate system squeezes low stratification N and dissipation rates ε into a sharply delineated layer at the highest |B| > 0.266 m s−2.
5. Interior and boundary diapycnal velocity and transport sections
In this section, interior diapycnal velocities wi and interior diapycnal transports Wi of the meridional overturning circulation (MOC) will be deduced from zonally averaged turbulent dissipation rates ε in the Indian, Pacific, and Atlantic. Interior diapycnal transports Wi are based on (5) and boundary transports Wb on (8). Isopycnal areas are based on Smith and Sandwell (1997) bathymetry at average depths of the bins. To the extent possible, the analysis will be presented in both depth z and buoyancy B (=−gγn/ρ0, where γn denotes neutral density) coordinates.
Boundary transports (8) assume Kb = 10−4 m2 s−1, based on averages in the bottommost bin of the diffusivities (Fig. 5), and Smith and Sandwell (1997) bathymetry for bottom slopes s = |∇hH| and boundary perimeters as a function of depth z. Average near-bottom diffusivities Kb exhibit no systematic dependence on near-bottom buoyancy frequency Nb (Fig. 5a) or bottom slope s (Fig. 5b), while McDougall and Ferrari (2017) assumed invariant buoyancy flux −KbN2 with implications for the depth distribution of Kb since N2 is a function of depth; McDougall and Ferrari’s results require an average near-bottom diffusivity Kb = 50 × 10−4 m2 s−1. Unlike McDougall and Ferrari (2017) and Ferrari et al. (2016), no distinction is made here between the stratified interior proper and the near-bottom stratified interior since near-bottom average diffusivities of (0.8–1.5) × 10−4 m2 s−1 (Fig. 5) are only ~4 times larger than interior averages (0.3–0.44) × 10−4 m2 (Kunze 2017). Thus, dissipation rates ε = <w′b′>/γ increase with buoyancy B on average globally. Dissipation rates decrease but diffusivities K increase because of decreased N2 at the largest buoyancy magnitudes |B| = 0.268 (neutral density γn = 28.1). There is order-of-magnitude scatter in Kb (Fig. 5) but no systematic dependence on available environmental parameters could be identified. Thus, choosing a constant Kb may not be appropriate for (8) but is the best that can be done for the moment. The issue of overestimated boundary transports associated with very small bottom slope s arose in only a small percentage of cases for which (4) was used with the area of the Smith-and-Sandwell resolution rather than (8). The perimeter in the integral of (8) is only a weakly increasing function of depth over 1000–4500 m because of rough topography corrugating the boundary. The circumference ~ 105 km is also almost independent of buoyancy B over the most of the range of interest, only exhibiting a peak near |B| = 0.268 (γn = 28.1), near the transition from bottom to lower deep water as emphasized by de Lavergne et al. (2016). Boundary transports were converted from depth z to buoyancy B coordinates using the average buoyancy in each latitude–depth bin.
Interior diapycnal velocities wi in depth–latitude coordinates (Fig. 6a) are dominated by ~O(0.2) cm day−1 upwelling though there is a layer of downwelling near 4000-m depth in the Southern Hemisphere of all three oceans associated with the top of the high stratification layer capping Antarctic Bottom Water (Fig. 3b) that deepens toward the north, grounding at the bottom near the equator in the Indian Ocean and near 30°N in the Pacific and Atlantic. There is also spatially incoherent downwelling near the bottom and at high latitudes in the North and South Atlantic.
The largest interior transports Wi (Fig. 6b) are in the subtropical pycnoclines of the North and South Pacific, principally because of their strong stratifications inducing large buoyancy flux divergences and large areas. Contrary to concerns about buoyancy fluxes increasing toward the bottom driving diapynal downwelling as found in the Brazil Basin (St. Laurent et al. 2001), negative abyssal Wi is weak and not widespread in the zonal averages. This suggests that different locales may be dominated by interior up- and downwelling with corresponding squeezing and stretching of isopycnals. Upwelling boundary transports Wb are only strong below 2500-m depth where they exceed abyssal interior transports by an order of magnitude.
In buoyancy–latitude coordinates (Fig. 7), the role of the midlatitude North and South Pacific pycnocline is further emphasized; strong up- and downwelling in neighboring bins may be an artifact of the buoyancy bin size. The 4000-m depth downwelling layer (Fig. 6) is only coherent in the Indian Ocean at |B| ~ 0.257 m s−2 (γn ~ 0.269 kg m−3), suggesting that downwelling and the Antarctic Bottom Water (AABW) cap are not consistently associated with a single buoyancy surface in the Pacific and Atlantic. Boundary transports Wb are only significant in a narrow layer of the highest buoyancy magnitudes |B| as will be clearer in section 7.
6. The meridional overturning streamfunction Ψ
The MOC streamfunction Ψ is only calculated in buoyancy–latitude coordinates so as not to introduce meridional flows crossing buoyancy surfaces:
where Ψ is summed from the northern boundary under the assumption that it vanishes there; this is a good approximation in the Indian Ocean but less so in the Atlantic with its deep- and intermediate-water sources in the Labrador and Nordic Seas north of 50° latitude or even in the Pacific where many lighter isopycnals outcrop to the north of 35°N. Two broad cells stand out (Fig. 7d), one in the upper deep water (UDW) layer (or its base in the Atlantic) and one in the intermediate water of the pycnocline. Meridional transports V = ΔBΨi + ΔBΨb (Fig. 7e) are finely striated in the vertical in contrast to the smooth two cells usually depicted. While this might be attributed to noise introduced by the second derivative of the buoyancy flux required for this quantity, it may also be due to heterogeneity in turbulent mixing tied to topographic features driving local overturning circulations of small horizontal and vertical scale (McDougall 1989; Reid 1997), which are not entirely smoothed out by the zonal averaging. Hautala (2017, manuscript submitted to Prog. Oceanogr.) finds vertical overturning scales of only 300 m in the deep and abyssal northeast North Pacific. Southward transports appear to be more narrowly confined than northward transports, at least in the Pacific and Atlantic. Diabatically driven meridional velocities υ (not shown) are O(1) mm s−1, consistent with an Argo float–based Sverdrup analysis Gray and Riser (2014).
7. Bulk ocean profiles
Basin-averages indicate interior diapycnal velocities wi ~ 0.25–0.35 cm day−1 above 2500 m and 0.15–0.2 cm day−1 at greater depths in all three oceans, comparable to but smaller than the canonical abyssal recipes 1 cm day−1 (Munk 1966) and more than an order of magnitude smaller than the median of 5 cm day−1 in Macdonald’s (1998) inverse estimates; Macdonald et al.’s (2009) later estimates in the Pacific are more comparable though reported to only be significant near the bottom.
Ocean interior diapycnal transports Wi are largest above 2500-m depth (UDW and shallower) at roughly 2, 5, and 3–4 Sv (1 Sv = 106 m3 s−1) in the Indian, Pacific, and Atlantic (Table 1) and are less than 1 Sv in abyssal waters. The global interior transport Wi = 10–11 Sv contrasts with the 2.5 Sv inferred by McDougall and Ferrari (2017) assuming a constant diffusivity K = 0.1 × 10−4 m2 s−1 in the interior, motivating them to neglect interior diapycnal transports. It is comparable to upper-ocean interior transports inferred by Ferrari et al. (2016)
Upwelling boundary transports Wb are the largest below 3000-m depth (|B| > 0.268 m s−2, γn > 28.1 kg m−3) at 4–6, 10–14, and 4–5 Sv in the Indian, Pacific, and Atlantic (Fig. 8; Table 1), exceeding interior abyssal transports by an order of magnitude. The abyssal boundary transport peak is deeper and denser in the Atlantic than in the Indian and Pacific. Global boundary transports exceed 20 Sv at 4500-m depth (|B| ~ 0.268 m s−2, γn ~ 28.1 kg m−3 between bottom and lower deep waters) in the 20–30-Sv range of global AABW transport estimates (Johnson 2008; Ganachaud and Wunsch 2000; Lumpkin and Speer 2007; Talley 2013). In the Indian Ocean, MacKinnon et al. (2008) inferred 3 Sv through southwest Indian Passage in the west and Sloyan (2006) ~4–6 Sv into Perth Basin in the east. In the Pacific, Roemmich et al. (1996) inferred 11 Sv through Samoan Passage, in the range of our numbers. In the Atlantic, Speer and Zenk (1993) and Hogg and Owens (1999) estimated 4–6 Sv of AABW entering Brazil Basin. Pacific and Atlantic transports are generally consistent with those inferred here but Indian Ocean transports appear to be a bit higher. This buoyancy level corresponds to the maximum isopycnal bottom incrop area (de Lavergne et al. 2016), consistent with the bowl shape of ocean basins and the peak in ocean basin perimeter length . McDougall and Ferrari (2017) inferred similar maximum net boundary transports of ~18 Sv but at 3000-m depth. Restricting integrals to north of 30°S, Atlantic interior transports are unchanged while other interior and boundary transports are reduced by ~20%.
The difference between abyssal (boundary) and shallower (interior) upwelling transports implies a 10–15-Sv return flow to the Southern Ocean in lower deep water (LDW). This becomes ~20 Sv when geothermal fluxes are included.
8. Comparison with inverse-method estimates
Inverse-method estimates represent least squares fits of O(1000) equations with O(3000) unknowns (e.g., Ganachaud 2003). Diapycnal transports and diffusivities require very accurate estimates of horizontal transport divergences that are calculated from differences between large, and largely adiabatic, horizontal or isopycnal transports based on quasigeostrophic and wind-driven balances and levels of no motion. They assume a steady ocean though they use multiple hydrographic sections that individually can take 1–2 months to complete, comparable to eddy evolution time scales, and are often collected years apart (Macdonald 1998). These limitations make estimation of diapycnal diffusivities and transports using inverse budgets extremely challenging. For example, Wijffels (1993) could not find solutions where all diffusivities were nonnegative and distinguishable from zero. High early inverse diapycnal numbers have generally become smaller and more consistent with microstructure and dye-release diffusivities over time (e.g., Lumpkin and Speer 2007; Hautala 2017, manuscript submitted to Prog. Oceanogr.).
On the other hand, the finescale parameterization only quantifies internal-wave-driven mixing in the stratified ocean interior, so does not include direct boundary-forced turbulence that will impact both boundary and interior transports or other mixing sources such as double diffusion. It is not reliable near the equator with its plethora of equatorially trapped waves. Inferred turbulent dissipation rates and diffusivities from Kunze (2017) may be over- or underestimated depending on variability in the shear–strain variance ratio Rω and physics that is poorly represented by the parameterization (Polzin et al. 1995; Polzin et al. 2014; Ijichi and Hibiya 2015). Estimated factor of 3 uncertainties in individual estimates (Polzin et al. 1995; Whalen et al. 2015) are certainly reduced in averages, but systematic biases may remain that can impact the divergences and cannot be reliably assessed here.
Interior diapycnal transports Wi here are 2, 5, and 3–4 Sv in the Indian, Pacific, and Atlantic, mostly contributed above 2500-m depth, while boundary transports Wb are 4–6, 10–14, and 4–5 Sv from below 2500 m (Fig. 8; Table 1). Because interior and boundary transports contribute at different depths and water masses, they should not be summed together. Globally, boundary transports exceed 20 Sv at 4500-m depth (|B| ~ 0.268 m s−2, γn ~ 28.1 kg m−3, though deeper and denser in the Atlantic) between bottom and lower deep waters and are consistent with published 20–30-Sv AABW transport estimates, even more so when account is made for the additional 5-Sv abyssal diapycnal transport caused by geothermal heating (Emile-Geay and Madec 2009; de Lavergne et al. 2016). Shallower interior diapycnal transports are smaller than inverse-method estimates, which do not distinguish between interior and boundary transports and can differ among themselves by as much as an order of magnitude depending on the constraints applied (depth of no motion, choice of stations and sections, tracer conservation, surface forcing, etc.). For example, in the Indian Ocean, inverse estimates collated by Huussen et al. (2012) range from 3 (Fu 1986) to ~20 Sv (Macdonald 1998; Sloyan and Rintoul 2001), with a small majority of ~10 Sv (Robbins and Toole 1997; Ganachaud et al. 2000; Bryden and Beal 2001; McDonagh et al. 2008), a factor of 5 larger than those inferred here. Ganachaud (2003) inferred 11 ± 4, 7 ± 2, and 6 ± 1.3 Sv, Lumpkin and Speer (2007) inferred 12 ± 3, 15 ± 3, and 17 ± 4 Sv, and Talley (2013) summarized 16, 9, and 26 Sv for the Indian, Pacific, and Atlantic, respectively. A more recent inverse in the Pacific finds MOC transports of only 1–3 Sv (Macdonald et al. 2009), which is a factor of 2 smaller than our estimate. Pacific and Atlantic values found here are consistent within uncertainties with those of Ganachaud, while those of Lumpkin and Speer and Talley are higher by a factor of 2 or more.
Above 2000-m depth, weak interior mixing is consistent with the notion that much diapycnal mixing occurs at high-latitude density outcrops (Toggweiler and Samuels 1998; Samelson 2004; Haertel and Fedorov 2012) with lower-latitude stratification controlled by lateral advection and potential vorticity conservation. Interior internal-wave-driven diapycnal transports (Fig. 8; Table 1) cannot simply be increased by increasing the mixing 〈w′b′〉 = −γε = −KN2 because the transports are driven by buoyancy flux divergences ∂〈w′b′〉/∂B, not by the buoyancy fluxes themselves. Stronger vertical gradients in the dissipation rate ε would be required. These might arise seasonally (Whalen et al. 2012). St. Laurent et al. (2001) demonstrated the value of using microstructure measurements as additional constraints in inverses in the Brazil Basin. It would be illuminating to use the mixing rates inferred here as an additional constraint for future inversions to determine if they are truly inconsistent or whether weaker inverse-method solutions are possible.
9. Bottom-generated turbulence
While our boundary water-mass transformations (Fig. 8) appear to be consistent with expectations from AABW transport estimates in the literature, some discussion of direct bottom-generated turbulent mixing is warranted as it is not well captured by the finescale parameterization (section 3) and may have consequences for both boundary transport Wb and the near-bottom contribution to interior transports Wi. While non-internal-wave mechanisms are largely confined to the bottom, the bottom is not always at the same depth or density so may influence deep as well as abyssal waters. At bottom turbulent hot spots, turbulence can be 100–1000 times background levels, requiring them to occupy 1%–10% of the ocean bottom if they are to produce the highest inverse diffusivities of ~O(10−3) m2 s−1. Carter and Gregg (2002) and Kunze et al. (2012) argued that there may be as much mixing in shelf-slope canyons as in the deep-ocean interior, which is not enough to explain an average diffusivity of 10−3 m2 s−1. Bryden and Nurser (2003) argued that mixing in hydraulically controlled flow in topographically constrained passages between abyssal basins is large enough to overwhelm any mixing in basin interiors. Their estimates, based on density changes between basins and transport estimates through hydraulically controlled passages, yield bulk diffusivities of 10−1 m2 s−1, comparable to the highest diffusivities found by direct microstructure measurements and an order of magnitude above microstructure averages of <K> ~ 10−2 m2 s−1 (Ferron et al. 1998; Alford et al. 2013). Taking the deep-ocean area below 2000-m (4000 m) depth to be 2.8 × 1014 m2 (1.8 × 1014 m2) and regions of K ~ 10−1 m2 s−1 to occupy 1010 m2 in area, 20 (15) such hot spots are needed globally to produce an average Kb ~ 10−3 m2 s−1, while only 5–6 have been documented in detail (Vema Channel, Romanche Fracture Zone, Discovery Gap, Samoan Passage, Luzon Strait, etc.), and these appear to have average <K> ~ 10−2 m2 s−1. A more complete global census of abyssal hot spots is needed to assess their role. However, as noted by McDougall and Ferrari (2017), while increasing near-bottom turbulent mixing will amplify boundary upwelling, it will also enhance compensating near-bottom interior downwelling (McDougall and Ferrari 2017; Ferrari et al. 2016) for little net change in the net diapycnal transport. The exclusion of abyssal regions of very low stratification here is unlikely to be important because of the lack of density contrast.
10. Summary and discussion
Using diapycnal buoyancy fluxes 〈w′b′〉 = −γε = −KN2 (Figs. 3–5) inferred from application of a finescale parameterization for turbulence production based on internal-wave strain to the global hydrographic database (Kunze 2017), zonally averaged interior [(4)] and boundary [(8)] diapycnal transports W of the meridional overturning circulation were diagnosed in the Indian, Pacific, and Atlantic (Figs. 6–8). Interior transports were 2, 5, and 3–4 Sv in the upper deep water and pycnocline above 2500-m depth, while boundary transports were 4–6, 10–14, and 4–5 Sv between the bottom and lower deep waters at 4000–4500-m depth in the Indian, Pacific, and Atlantic, respectively. When 5-Sv upwelling transport from geothermal heating near 4000-m depth (γn = 28.11) is added (Emile-Geay and Madec 2009; de Lavergne et al. 2016), the maximum total boundary water-mass transformations exceeding 20 Sv from bottom to lower deep waters found here (Fig. 8) are consistent with 20–30-Sv AABW production (Ganachaud and Wunsch 2000; Lumpkin and Speer 2007; Johnson 2008; Talley 2013) despite the finescale parameterization not accurately accounting for direct boundary-generated turbulence and mixing in hydraulically controlled passages between deep basins (section 9); this may indicate that direct bottom-forced turbulence is not a major contributor globally or the finescale parameterization approximately captures its contribution, but uncertainties are large enough that direct forcing cannot be discounted. The 10–15-Sv difference between abyssal and upper deep-water transformation is presumably transported in a return flow to the Southern Ocean. While zonal averages were used here, there may be sufficient coverage in the assembled dataset to examine more regional diapycnal transport balances and test alternative parameterizations.
It is difficult if not misleading to assign uncertainties to these estimates. Like inverse methods, internal error bars do not reflect unknown biases in sampling and methodology. Interior transports Wi are probably more robust since the finescale parameterization was developed for these conditions and the sampling is better (Figs. 3a, 4a). However, with the exception of Ganachaud (2003) in the Pacific and Atlantic, inverse estimates are typically higher by more than a factor of 2. Bottom transports Wb rely on a less well-constrained Kb that does not reliably capture direct boundary-forced turbulence, but these agree well with AABW transports, except possibly in the Indian Ocean, perhaps because they are primarily controlled by well-known bathymetry through bottom slopes s and basin perimeters .
While interior transports fall below inverse estimates (section 8), inverse values show up to order-of-magnitude scatter among themselves depending on assumed constraints, so it would be premature to dismiss any of these estimates until greater consistency can be achieved. Above 2000-m depth, weak diapycnal transports might be explained by most of the diapycnal mixing occurring at high-latitude surface outcrops (Toggweiler and Samuels 1998; Samelson 2004; Haertel and Fedorov 2012; Marshall and Speer 2012), but this does not explain middepth discrepancies (z = 1300–2800 m, γn = 26.07–27.65 kg m−3) between inverse-method and fine/microstructure estimates in the Indian Ocean (Huussen et al. 2012).
Boundary transports Wb making use of the Walin (1982) framework in idealized models and ocean climatology have been the subject of several recent papers on abyssal water-mass transformation (de Lavergne et al. 2016; McDougall and Ferrari 2017; Ferrari et al. 2016). McDougall and Ferrari (2017) and Ferrari et al. (2016) assumed weak interior diffusivities K ~ 0.1 × 10−4 m2 s−1, or constant interior buoyancy fluxes, with enhanced buoyancy fluxes in the bottom 500 m motivated by local tidal and lee-wave mixing parameterizations (e.g., St. Laurent et al. 2002) based on microstructure measurements at a limited number of sites. De Lavergne et al. (2016) tested sensitivity to the bottom-enhancement decay scale and different vertical and horizontal distributions of remote mixing, finding little sensitivity for decay scales ≤1000 m. In contrast, buoyancy fluxes used here are from a finescale parameterization applied to ~30 000 oceanwide CTD casts (Kunze 2017) that show elevated diffusivities over rough topography extending almost uniformly into the pycnocline with decay scales of ~2000 m on average. The inferred distributions in Kunze (2017) would be most consistent with an even larger local decay scale ~2000 m, dissipation rates ε ~ N2, and horizontally localized mixing in the de Lavergne et al. parameter-space exploration. Previously published papers’ mixing parameterizations are based on high-quality microstructure measurements of limited extent, while the mixing used here is based on data with more widespread coverage that, though agreeing with microstructure measurements where available (Waterhouse et al. 2014) and suggesting that microstructure sampling is biased to regions of strong tidal forcing (Kunze 2017), is nevertheless based on a parameterization with factors of 2–3 uncertainties (Polzin et al. 2014; Whalen et al. 2015). Boundary transport inferences were arrived at from similar assumptions as the previous papers, though McDougall and Ferrari (2017) required a very high average Kb of 50 × 10−4 m2 s−1 compared to the Kb = 10−4 m2 s−1 used in this paper based on near-bottom values from Kunze (2017), which are independent of buoyancy frequency N and bottom slope s on average (Fig. 5). McDougall and Ferrari and Ferrari et al.’s choices led to a strong up/downwelling (+100 vs −80 Sv) cell near the boundaries, which is not found in the zonal averages here. However, McDougall and Ferrari’s net near-boundary transport of ~20 Sv is similar to that reported here and apparently insensitive to the magnitude of near-boundary mixing, so perhaps the smaller near-boundary mixing used here is reasonable. Though there are known sites with interior downwelling (and isopycnal squeezing) compensated by boundary upwelling such as the Brazil Basin (St. Laurent et al. 2001), the results here (Figs. 6, 7) suggest that these may be compensated by regions with interior upwelling and stretching. Despite differences, this paper concurs with the previous conclusions that topographically controlled boundary transports are crucial for closing the abyssal meridional overturning circulation. In this case, average near-bottom diffusivities of 10−4 m2 s−1 (Fig. 5) are sufficient to accomplish consistent boundary water-mass transformations between bottom and lower deep waters.
The original abyssal recipes’ story (Munk 1966; Munk and Wunsch 1998) of interior diapycnal mixing maintaining the abyssal ocean’s stratification and driving the upwelling limb of the meridional overturning circulation is only one contributor and not where intended. Above 2000-m depth, interior diapycnal diffusivities are weaker than 10−4 m2 s−1, and mixing at Southern Hemisphere high-latitude density outcrops also appears to be important. As shown here (Fig. 8) and by other recent work (McDougall and Ferrari 2017; de Lavergne et al. 2016; Ferrari et al. 2016), boundary transports driven by basin hypsometry rather than interior mixing dominate water-mass transformation from bottom to deep water in the abyss. Shared by these recent attempts is uncertainty in the global distribution of turbulent buoyancy flux 〈w′b′〉 ~ −γε = −KN2 and the need for improved abyssal sampling to better constraint mixing parameterizations.
For Walter Munk on his 100th birthday. Barry Ma and Fiona Lo assisted with data extraction and the neutral density calculation. Questions about the distinction between interior and boundary transports raised by John Toole and discussions with Trevor McDougall and Casimir de Lavergne benefited revision of the manuscript immeasurably. Conversations with Susan Hautala, Pascale Lelong, Cimarron Wortham, and Jeffrey Early were much appreciated. This research was funded by U.S. National Science Foundation Grant OCE-1153692.