## 1. Introduction

The vertical mixing coefficient *D* controls the diapycnal component of the meridional overturning circulation (Munk and Wunsch 1998). The lack of knowledge of vertical mixing, its magnitude and spatial variation, leads to large uncertainties in our estimates of the volume, heat, freshwater, nutrient, and tracer transport and the role of the ocean in climate sensitivity and climate feedbacks (Manabe and Stouffer 1988). The along-isopycnal mixing coefficient *K* is used for mixing tracers such as temperature and salinity or potential vorticity along isopycnal surfaces. The value of *K*, like *D*, is thought to strongly control the overturning circulation and climate sensitivity (Gnanadesikan 1999; Sijp et al. 2006).

Munk (1966) pointed out the role of vertical mixing in the global overturning circulation using simple theoretical arguments. He discusses the balance between upwelling and vertical mixing in the thermocline. Given 25 Sv (Sv ≡ 10^{6} m^{3} s^{−1}) of diapycnal upwelling, Munk showed that the globally averaged vertical mixing coefficient *D* must be approximately 10^{−4} m^{2} s^{−1}.

Zika et al. (2009), similar to the analysis of Munk (1966), explore both the along-isopycnal and diapycnal components of the Southern Ocean meridional overturning circulation. The along-isopycnal component of the overturning is “upwelled” along isopycnal layers to the surface boundary layer of the Southern Ocean. Zika et al. found that, to upwell 20–50 Sv of dense water, the along-isopycnal mixing coefficient *K* must be 150–450 m^{2} s^{−1} and the vertical mixing coefficient must be 0.5–1.5 × 10^{−4} m^{2} s^{−1} below the thermocline at latitudes of the Antarctic Circumpolar Current. What fraction of the meridional overturning circulation is upwelled through isopycnals below the thermocline or upwelled along isopycnals in the Southern Ocean is unknown.

Most observations of *D* have been much smaller than those predicted by Munk (1966). From the distribution of the tracer SF_{6}, released below the thermocline in the eastern North Atlantic during the North Atlantic Tracer Release Experiment (NATRE), Ledwell et al. (1993) estimates *D* to be *O*(10^{−5} m^{2} s^{−1}). Many studies have used microstructure to estimate vertical mixing (e.g., Gregg 1987). These have tended to reveal weak mixing *O*(10^{−5} m^{2} s^{−1}) in most regions. Other in situ measurement approaches, while confirming weak mixing over much of the World Ocean, have identified stronger mixing, *O*(10^{−4} to 10^{−3} m^{2} s^{−1}), close to rough topography and in energetic regions such as the Southern Ocean (Naveira-Garabato et al. 2004; Kunze et al. 2006; Sloyan 2005). Since direct estimates of diapycnal mixing are sparse and infrequent, it is not clear whether vertical mixing in these energetic regions is sufficient to induce global diapycnal upwelling on the order of 25 Sv.

Few direct estimates exist for *K* in the oceans. Ledwell et al. (1998) estimated that the SF_{6} tracer, released in the eastern North Atlantic at an approximate depth of 300 m, mixed horizontally at a rate on the order of 1000 m^{2} s^{−1} over the 30 months of their experiment. Horizontal mixing at the sea surface has been estimated to be equal to or greater than that determined by Ledwell (i.e., 1000–10 000 m^{2} s^{−1}) (Zhurbas and Oh 2004; Marshall et al. 2006; Sallée et al. 2008).

A depth dependence of the mixing coefficient, particularly that which mixes potential vorticity or interface height [commonly referred to as *κ* in the Eulerian coordinate parameterization of Gent et al. (1995), with interface height being the depth of isopycnals], is reinforced by adjoint inversions (Ferraira et al. 2005) and eddy resolving models (Eden and Greatbatch 2008). It should be noted that calculations of *K*, in eddy resolving simulations, are not trivial, even with complete knowledge of a model’s full velocity and density fields (Eden et al. 2007).

Inverse methods are tools used to diagnose the ocean circulation from hydrographic data. Such methods seldom give insight into mixing processes. The most well known method is the box inverse method (Wunsch 1978). In the box method, oceanic sections bound regions of the oceans or entire ocean basins. In each region mass, heat, salt, and other properties are conserved in isopycnal layers. Inverse box models simultaneously solve a set of equations for various unknowns, including reference level velocities and diapycnal fluxes of properties or the vertical mixing coefficient *D*. Box inversions are always underdetermined, and the final solution depends largely on the choice of model and model variance. This sensitivity is particularly strong when mixing coefficients are considered as unknowns in inverse studies (Tziperman 1988). Inverse models have tended to perform better when individual diapycnal property fluxes are considered (Sloyan and Rintoul 2000) or when observed mixing coefficients are included as “knowns” (St. Laurent et al. 2001). Recent adjoint and gridded methods (Wunsch and Heimbach 2007; Herbei et al. 2008) have shown promise but have, so far, been unable to resolve the spatial structure of *D* and *K*.

Zika et al. (2010, hereafter ZMS10) present a new inverse method called the tracer-contour inverse method. This inverse method is less sensitive to error than existing methods and has been validated against the output of a numerical model. With these advances and the presence of new observations that give a better representation of the upper 2000 m of the ocean, it may now be possible to accurately infer the ocean circulation and rates of along-isopycnal and vertical mixing directly from hydrographic data. The purpose of this study is twofold: first, to demonstrate the utility of this new inverse method by applying it to ocean observations in the North Atlantic (Fig. 1) where direct estimates of vertical and along-isopycnal mixing exist and, second, to reveal the depth dependence of the mixing coefficients in that region.

This article is structured as follows: In section 2, we briefly describe the tracer-contour inverse method and the ocean climatology used. In section 3, we present the results of the tracer-contour inverse method, as applied to a region of the North Atlantic. Conclusions are given in section 4. In appendix B, we test the sensitivity of the method to the adjustment of various parameters and differing data sources.

## 2. The tracer-contour inverse method and hydrographic data

Here we describe the tracer-contour inverse method of ZMS10 and the data used in this study. We then explain how the inverse method is applied in the eastern North Atlantic.

### a. The tracer-contour inverse method

**v**is the mean lateral velocity ([

*u*,

*υ*, 0]),

*w*is the diapycnal velocity component,

^{γ}*S*is salinity, and Θ is conservative temperature. All variables are temporal and thickness-weighted means, averaged on neutral density surfaces (

*γ*

^{n}), except

**v**, which is split into a temporal mean on isopycnals [lhs of Eq. (1)] and a bolus term [

*K*

_{PV}/

*λ*, the rhs of (1)]. The direction of the along-isopycnal temperature (and salinity) gradient is

^{h}**n**(

**n**=

**∇**

_{γ}Θ/|

**∇**

_{γ}Θ|). The scale lengths

*λ*,

^{h}*λ*and scale heights

^{γ}*η*and

^{γ}*γ*), along-isopycnal mixing of tracers (denoted with the superscript ⊥), and along-isopycnal mixing of layer thickness (denoted with the superscript

*h*).

The density variable used is neutral density (McDougall 1987), and the temperature variable used is conservative temperature (McDougall 2003). The previous two points are not necessarily critical to the analysis presented here and the variables potential temperature *θ* and potential density, *σ*_{1} or *σ*_{2}, could be used instead.

**∇**

*·*

_{γ}K**n**); vertical mixing acts to restore the anomaly by mixing on the Θ −

*S*curvature

*λ*=

^{h}**∇**

*·*

_{γ}h**n**/

*h*); or the temperature contour simply moves in space (Θ

*|*

_{t}*/|*

_{γ}**∇**

_{γ}Θ|). That is, some or all of the terms on the rhs of (1) must be nonzero to balance the lhs. Nonlinear effects owing to cabbeling and thermobaricity are included in the

*K*and

*D*.

In (2), if there is mean advection across a density surface, *w ^{γ}*, there must be a compensating effect of vertical mixing (1/

*η*≈

^{γ}*γ*/

_{zz}*γ*and/or

_{z}*D*) or nonlinear effects such as cabbeling and thermobaricity (1/

_{z}*η*). Equation (1) differs from (2) in that the vertical coordinate moves with the isopycnal surfaces, whereas the lateral coordinate is fixed in space; hence, there is no

^{γ}*γ*/

_{t}*γ*term in (1) as there would be if depth

_{z}*z*was the vertical coordinate. Equation (2) is similar to that used by Munk (1966), as it relates diapycnal upwelling to vertical mixing but also includes nonlinear processes (i.e., the

*K*term on the right-hand side).

*w*=

^{γ}*Dγ*/

_{zz}*γ*and

_{z}Equation (3) is the tracer-contour equation and relates mixing on an isopycnal *γ* to the flow on a reference pressure *p*_{0}. In (3), Ψ^{p0} is the geostrophic streamfunction on *p*_{0} (Fig. 2). The Coriolis frequency is *f*. The coordinate *x*_{Θ} is that running perpendicular to **n** on the isopycnal; that is, ∫*dx*_{Θ} is the integral along a temperature contour on an isopycnal. The tracer-contour specific volume anomaly is *δ*_{contour}(*p*) = 1/*ρ*(*S*, Θ, *p*) − 1/*ρ*(*S*_{contour}, Θ_{contour}, *p*), where *S*_{contour} and Θ_{contour} are the conservative temperature and salinity values of the contour between (*x*_{2}, *y*_{2}) and (*x*_{1}, *y*_{1}); that is, *δ*_{contour} is redefined for each contour.

Equation (4) is the layer conservation equation for volume and any conservative tracer *C* and, like (3), relates mixing on *γ* to flow at *p*_{0} (Fig. 2). For conservation of volume *C* = 1, for conservation of temperature anomaly *C* = Θ − *C* = *S* − *S**S***v**^{p0} is the lateral velocity on the reference pressure and **m** is the direction normal to the lateral boundaries of the isopycnal layer. The thickness of the isopycnal layer is *h* = Δ*γ*^{n}/*γ _{z}*

^{n}for some arbitrary Δ

*γ*

^{n}about the surface

*γ*, where []

*is the difference between the upper and lower interfaces of that layer (Fig. 2). The diapycnal advection*

_{l}^{u}*w*is explicitly related to mixing using (2). We explicitly relate the lateral advection terms in (4) (i.e., the first term on the lhs and the rhs) to the geostrophic streamfunction Ψ

^{γ}^{p0}using a form of the depth-integrated thermal wind equation, as shown in appendix C.

The tracer-contour inverse method combines aspects of the three major inverse modeling concepts of modern oceanography, the box, beta spiral, and Bernoulli methods, and the advective–diffusive balance concept is motivated by Munk (1966). By assuming the mixing coefficients are constant on each layer or have some spatial form, the tracer-contour inverse method is overdetermined. There is a direct relationship between lateral advection and diffusion, and the diffusion terms are leading order. In the method, there is a linear relationship between advection and vertical mixing through the Θ − *S* curvature (*d*^{2}Θ/*dS*^{2}|_{x,y}, a quantity less susceptible to noise due to heave than Θ* _{zz}* and

*S*are individually). Down-temperature and down-salinity gradient transports along isopycnals are explicitly considered in the tracer-contour inverse method, and it is these transports that constitute the along-isopycnal component of the thermohaline overturning circulation. In the tracer-contour inverse method, a streamfunction variable is solved for, along “sections,” allowing for direct integration with the box inverse method and the computation of section transports for comparison with, or constraint by, shipboard observations.

_{zz}### b. Data and setup of the inversion

The climatology of Durack and Wijffels (2010, hereafter DW10) is used in this study. Hydrographic data from shipboard observations and Argo floats, up to and including 2008, have been used in their analysis. Annual and interannual cycles as well as a linear trend have been locally fit to the data, minimizing temporal aliasing. The data was averaged by DW10 on approximate neutral density surfaces (*γ*^{rf}, Jackett and McDougall 2005), spaced by 0.025 kg m^{−3} on a 1° latitude × 2° longitude grid. The tracer-contour inverse method has also been applied to the climatology of Gouretski and Koltermann (2004, hereafter GK04), and these results are discussed in appendix B. The inversion is carried out for the NATRE region in the North Atlantic (20°–30°N, 40°–25°W) shown in Fig. 1.

Regularly spaced points on the boundary of the domain (between 20° and 30°N, 40° and 25°W), given by the grid of the climatology, are referred to as “casts,” and the eastern, northern, western, and southern boundaries of the domain are referred to as “sections.” There is a total of 34 casts. Tracer contours (contours of constant Θ and *S* in this study) are defined on *γ*^{rf} surfaces between 200 and 1800 m. As in ZMS10, contours are defined, on each surface, by the temperature and salinity at the cast locations; that is, there is about one tracer contour per cast per surface. At each point on a contour the flow across the contour, in the direction **n**, is related to along-isopycnal and diapycnal mixing through the scale lengths *λ ^{h}*,

*λ*(1). So, integrating

^{γ}*f*/

*λ*,

^{h}*f*/

*λ*along each contour (green line in Fig. 2) gives the difference in geostrophic streamfunction, between the ends of the contour, as a function of mixing (3).

^{γ}Maps of the lateral mixing coefficient at the sea surface from Zhurbas and Oh (2004) suggest gradients of the mixing coefficient are very small in the latitude bands considered here. For this reason, we choose to find solutions where *K* does not vary laterally and hence ignore the **∇*** _{γ}K* ·

**n**term in (1). It is likely that at the boundary between different mixing regimes this term will become important and the sensitivity of the method to this choice is left to future work. By default we assume a statistically steady state, hence assuming the unsteady term Θ

*|*

_{t}_{γ}/|

**∇**

_{γ}Θ| is negligible. This choice, and sensitivity to it, are discussed in appendix B.

Volume, conservative temperature anomaly, and salinity anomaly are conserved on 28 isopycnal layers between 200 and 1800 m (4). Each “layer” in which properties are conserved may encompass more than one of the “surfaces” on which tracer contours are defined. The density range of each layer is chosen such that at least 40 tracer contours are within each layer, allowing *K* and *D* to be well resolved within each layer. The diapycnal fluxes of properties due to advection *w ^{γ}* are related to mixing through (2). The geostrophic advection across a section is

**v**·

**m**, where

**m**is the unit vector normal to the section (Fig. 2). This advection is related to the geostrophic streamfunction along the section. The difference in geostrophic streamfunction is related, at each pair of casts, to the reference level streamfunction Ψ

^{p0}at 1000 db (C2). A streamfunction is solved for, on the reference pressure, at each of the 34 cast locations. The diffusivities

*K*and

*D*are solved for on each density layer.

Contours with very small along-isopycnal gradients (**∇**_{γ}Θ < 2 × 10^{−8} K m^{−1}) are excluded. The results are generally insensitive to minor changes of the choice of reference level, depth range, minimum number of contours per layer, and the minimum along-isopycnal temperature gradient. The inversion is more sensitive to the factors considered in appendix B.

*D*variables and 28

*K*variables. There are 34 reference level streamfunction variables and 1608 tracer-contour equations. All equations (contour and box equations) are scaled to be

*O*(1). The contour equations are additionally scaled by the normalized mean isopycnal gradient of the contour,

**∇**

_{γ}Θ. As in ZMS10, the relative weighting of the tracer-contour and box equations [i.e. the weight given to (3) relative to (4)] is determined by minimizing the condition number of the matrix 𝗔 in the linear system, 𝗔

**x**=

**b**. Sensitivity analysis of the changes to the equation weighting are given in appendix B.

## 3. Results

### a. Vertical mixing

The tracer-contour inverse method gives direct estimates of the mixing coefficients *K* and *D* (Figs. 3 and 4). The vertical mixing coefficient *D* in the upper 200–1000 m of the water column is *O*(10^{−5} m^{2} s^{−1}). There is a modest increase in *D* toward the deepest layers at 1800 m. The deepest value of *D* is 2.6 ± 0.8 × 10^{−5} at 1800 m.

A SF_{6} tracer has been released in the eastern North Atlantic at an approximate depth of 300 m, and its dispersion used to infer a vertical mixing coefficient (Ledwell et al. 1993, 1998). Two observation of the effective mixing coefficient for that tracer were made: 1.2(±0.1) × 10^{−5} m^{2} s^{−1} in the spring of 1992 and 1.7(±0.1) × 10^{−5} m^{2} s^{−1} in the fall of 1993 (Fig. 3). As there are only two measurements, a suitable error range for the “long term mean” *D* is not possible. Promisingly, the error range of *D* in the upper 200–400-m depth range, as estimated in this study, almost overlaps with the spring 1992 measurements of Ledwell et al. (1993).

Microstructure measurements of a vertical mixing coefficient in the NATRE region at depths of 200–400 m are generally *O*(10^{−5} m^{2} s^{−1}). For example, St. Laurent and Schmitt (1999) observed mixing of 0.1 − 1.3 × 10^{−5} ± 10^{−5} m^{2} s^{−1} considering diffusivities for density (*D _{ρ}* = 0.1 × 10

^{−5}± 0.3 × 10

^{−5}m

^{2}s

^{−1}), potential temperature (

*D*= 0.8 × 10

_{θ}^{−5}± 0.1 × 10

^{−5}m

^{2}s

^{−1}), and salinity (

*D*= 1.3 × 10

_{S}^{−5}± 0.1 × 10

^{−5}m

^{2}s

^{−1}). Unlike St. Laurent and Schmitt, we do not consider double-diffusive convection since we have taken the vertical mixing coefficient

*D*to be the same for both Θ and

*S*. Ferrari and Polzin (2005) find an equally low turbulent diffusivity close to 0.7 × 10

^{−5}m

^{2}s

^{−1}. There are large uncertainties in microstructure estimates of

*D*owing to the assumption of a turbulent mixing efficiency. Despite this uncertainty, the estimates of

*D*presented here agree very strongly with the microstructure estimates of

*D*of Ferrari and Polzin (2005) and St. Laurent and Schmitt (1999). The increase with depth of

*D*toward 1800 m, observed in this study, is also consistent with Ferrari and Polzin (2005).

Microstructure estimates of *D* are near instantaneous, representing turbulent motions evolving over minutes and hours. The estimates of *D* by Ledwell et al. (1993, 1998) represent mixing of a tracer integrated over a 6–18-month time period. The estimate of *D* presented here represents the long-term mean effective mixing of heat and salt across isopycnals. That the three approaches (microstructure, tracer release, and tracer-contour inverse method)—each with their respective time scales, spatial scale, and uncertainties—agree on a value of *D O*(10^{−5} m^{2} s^{−1}) is encouraging. Our results support the hypothesis that, below the thermocline in the eastern North Atlantic, the *D* estimated by microstructure studies is equivalent to the long-term mean *D* and is thus an appropriate value for use in ocean circulation models. This canonical “background” value is consistent with the hypothesis that mixing is low away from boundary layers and is unable to sustain the upwelling required in the balance of Munk (1966).

### b. Along-isopycnal mixing

The along-isopycnal mixing coefficient *K* is estimated to be *O*(1000 m^{2} s^{−1}) within the upper 200–400 m of the water column, consistent with the North Atlantic Tracer Release Experiment studies for the same region (Ledwell et al. 1998; Sundermeyer and Price 1998). The mixing coefficient smoothly decreases with depth. Between 500- and 1000-m depth, *K* reduces to around 0–200 m^{2} s^{−1} (Fig. 4). These values are broadly consistent with nearby estimates from float trajectories (Joyce et al. 1998; Spall et al. 1993); ^{3}He, ^{3}H, and tritium tracer studies (Jenkins 1987, 1998); and other inverse studies (Armi and Stommel 1983; Zika and McDougall 2008). Ferrari and Polzin (2005) are able to derive a vertical profile of the along-isopycnal mixing coefficient using a mixing length argument and combining CTD and mooring data from NATRE and the Subduction Experiment (Joyce et al. 1998). There is uncertainty in the mixing efficiency that Ferrari and Polzin (2005) choose; thus, they are able to adjust the overall magnitude of *K* to fit the observations shown in Fig. 4. Our vertical profile, which cannot be adjusted and is a direct output of the inverse method, is very similar to that of Ferrari and Polzin (2005). There are no scaling parameters used in this study.

The vertical profile of *K* presented here is consistent with the findings of Ferraira et al. (2005), which suggested a depth dependence to the mixing coefficient *K* that is mostly *O*(1000 m^{2} s^{−1}) in the upper 200–500 m of the water column, decreasing to *O*(100 m^{2} s^{−1}) at 1000–2000-m depth. However, in the NATRE region, the estimates of Ferraira et al. (2005) are mostly negative. Recent parameterization efforts have focused on the need for a depth dependence of the along-isopycnal mixing coefficient (Eden and Greatbatch 2008; Danabasoglu and Marshall 2008), motivated largely by inferences from eddy resolving models and theoretical arguments. This study represents one of the few demonstrations of a strongly depth-dependent along-isopycnal mixing coefficient, inferred directly from observational data.

### c. Geostrophic flow

Because the geostrophic streamfunction at the reference level is an output of the tracer-contour inverse method, the full depth geostrophic velocity and transport are easily inferred (Fig. 5). The mean geostrophic velocity is to the southwest throughout most of the study region. The velocity is intensified in the upper 400 m of the water column with velocities of up to 2–3 cm s^{−1}. The flow is generally stronger in the zonal direction and intensifies in the southwest corner of the domain. This pattern is consistent with the largely westward migration of the SF^{6} tracer throughout the NATRE study of approximately 1.3 cm s^{−1} (Ledwell et al. 1998, see their Table 1). The flow is also consistent with the southeastward flowing limb of the wind-driven North Atlantic subtropical gyre in Sverdrup balance (Sverdrup 1947). As deep velocities are estimated to be small, *O*(0.2 cm s^{−1}), the inferred circulation is consistent with studies that use atlas data and a deep level of no motion (e.g., Reid 1978). The typical columnar nature of the velocity field, seen in conventional inverse calculations using one-time hydrographic surveys, is not seen here partly because individual eddies are not present in the climatology.

The deep flow is in no way minimized or assumed small in this analysis. Despite this, the solution for the geostrophic velocity at 1800 m is small, *O*(0.2 cm s^{−1}), relative to the upper 500 m *O*(2 cm s^{−1}). However, the inferred contribution to the section transport between 1800 m and the sea floor across each section is significant, *O*(2–3 Sv), demonstrating the need for an appropriate estimate of the deep velocity. Some structure to deep currents is revealed, including subsurface zonal currents.

## 4. Conclusions

The tracer-contour inverse method has been applied to the mean hydrography of the eastern North Atlantic. Vertical profiles of the vertical mixing coefficient *D* and along-isopycnal mixing coefficient *K* have been diagnosed, as well as the mean geostrophic circulation. The mixing coefficients and circulation patterns are consistent with in situ measurements.

Vertical mixing is found to be weak, *O*(10^{−5} m^{2} s^{−1}), throughout most of the water column. These low mixing values are consistent with the hypothesis that abyssal mixing is small in the subtropical oceans and is insufficient, on its own, to close the meridional overturning circulation. It is likely that the methods demonstrated in this study will be able to identify regions of intense mixing and quantify the global spatial structure of the mixing coefficients.

In the NATRE region, along-isopycnal mixing is found to be strong, *O*(1000 m^{2} s^{−1}), near the thermocline and reduces with depth to background values *O*(100 m^{2} s^{−1}) below 500-m depth. Our results confirm that along-isopycnal mixing varies with depth as inferred by Ferrari and Polzin (2005) and Ferraira et al. (2005), further motivating parameterizations for this effect in coarse-resolution ocean models (Eden and Greatbatch 2008; Danabasoglu and Marshall 2008). Uniquely, we reveal a depth dependence of the along-isopycnal mixing coefficient from mean hydrographic data alone.

The eastern North Atlantic is thought to be a region of low eddy kinetic energy and small diapycnal mixing and, as such, is a suitable region for determining the background *K* and *D*. The tracer-contour inverse method accurately reproduces in situ observations there, suggesting that the method can be used to identify regions of both weak and intense vertical and along-isopycnal mixing in the world’s oceans.

## Acknowledgments

We thank Paul Durack and Dr. Susan Wijffels for the use of their climatology. We are grateful to Andrew Meijers, Frank Colberg, and two anonymous reviewers for their helpful comments. This work contributes to the CSIRO Climate Change Research Program and has been partially supported by the CSIRO Wealth from Oceans Flagship and the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems.

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## APPENDIX A

### Derivation of the Advective–Diffusive Balance Equations with a Linear Equation of State

*γ*∝

*βS*−

*α*Θ), spatially uniform along-isopycnal

*K*and vertical

*D*mixing coefficients, and an absence of sources and sinks (i.e., the tracers are conservative). In an isopycnal framework with a thickness-weighted lateral velocity vector

**v**

*h*

*h*

*w*, the conservation equations for heat Θ and salt

^{γ}*S*are then

*α*

**∇**

_{γ}Θ =

*β*

**∇**

*,*

_{γ}S**∇**

*≡*

_{γ}α**∇**

*= 0,*

_{γ}β*γ*=

_{z}*βS*−

_{z}*α*Θ

*, and*

_{z}*γ*=

_{zz}*βS*−

_{zz}*α*Θ

*. Hence, (A2) may be written as*

_{zz}*α*minus (A3) multiplied by

*β*, all terms involving Θ cancel, giving

*/Θ*

_{zz}*and*

_{z}*S*/

_{zz}*S*). Equation (A4) reveals the basic flavor of (2).

_{z}**n**=

**∇**

_{γ}Θ/|

**∇**

_{γ}Θ| and exploiting the equality

*A*

_{z}^{3}(

*d*

^{2}

*B*/

*dA*

^{2})|

_{x,y}=

*A*−

_{z}B_{zz}*B*, (A5) becomes

_{z}A_{zz}*C*[ZMS10, Eq. (C9)]:

So, for any tracer, the component of the flow across contours on isopycnals is related to along-isopycnal mixing through the isopycnal curvature of the tracer (∇_{γ}^{2}*C*/|**∇*** _{γ}C*|) and to vertical mixing through the curvature of the tracer with respect to density

*d*

^{2}

*C*/

*dγ*

^{2}|

_{x,y}. Note that the curvature of Θ or

*S*with respect to

*γ*is proportional to the curvature of Θ with respect to

*S*.

**v**(not thickness weighted) and the eddy-induced velocity

**v**′

*h*′

*h*

*S*. Hence, for our purposes the advective diffusive balance equations are, following ZMS10,

*λ*for the case of a general tracer

*C*.

## APPENDIX B

### Uncertainties and Sensitivity Analysis

Mixing parameters and diapycnal fluxes determined using inverse methods are known to be highly sensitive to the model details and the data sources used (Tziperman 1988; Lux et al. 2001; St. Laurent et al. 2001). This sensitivity is not evident in standard statistical parameters [i.e., the singular values of the SVD solution] and is best explored through sensitivity tests. It is thus pertinent to test whether the results presented here, for the vertical and along-isopycnal mixing coefficients, are robust in terms of their sensitivity to changes to key assumptions and parameterizations, equation weighting, and data sources. To test the robustness of our results, we conduct a series of sensitivity tests, in each case, assessing the quantitative and qualitative changes made to the vertical and along-isopycnal mixing rates inferred. We show that the choice of equation weighting, parameterizations, and data source has a largely negligible effect, the last being the most influential.

In this study, the bolus velocity **v*** is represented as a downgradient diffusion of the thickness of isopycnal layers *h*. The rate of diffusion of thickness is controlled by the mixing coefficient *K*_{PV} such that **v****h* = −*K*_{PV}**∇**_{γ}(*h*). In the analysis presented in section 3, we assume that the coefficient for thickness is the same as that for temperature and salinity (i.e., *K*_{PV} = *K*). These choices are arbitrary, and one could equally mix potential vorticity *f*/*h* instead of *h* and/or allow *K* and *K*_{PV} to vary independently. To test the influence of this parameter in the eastern North Atlantic, we remove the thickness term from the inverse method. The effect on the mixing coefficients and their standard error is negligible (dotted line, Figs. B1a,b). This suggests that the choice of parameterization is inconsequential in this region and the method would be unable to diagnose *K*_{PV} independently here. However, the bolus velocity is likely to be of leading order importance in the Southern Ocean and regions of steeply sloping isopycnals (Gent et al. 1995; McDougall and McIntosh 2001).

Here a statistically steady balance is assumed such that the trend in Θ and *S*, on isopycnal surfaces, is negligible when compared to the mean advection, along-isopycnal mixing, and diapycnal mixing terms (i.e., we have assumed Θ* _{t}*|

_{γ}= 0). In the climatology of DW10, trends have been quantified and are discussed in DW10. We test the importance of the trend term by retaining the Θ

*|*

_{t}_{γ}/

**∇**

_{γ}Θ term in Eq. (1). This term effectively represents the gradual movement of temperature contours on isopycnals. The effect on the inverse calculation in the eastern North Atlantic is negligible (dashed line, Figs. B1a,b), suggesting a steady-state balance is a valid assumption in the region considered. This assumption may break down in regions where strong changes are detected on isopycnals and/or along isopycnal gradients are particularly low.

In ZMS10 and here, the relative weighting of the contour equation (3) versus the large-scale conservation, or box equation (4), is chosen by minimizing the condition number of the matrix 𝗔 in the system 𝗔**x** = **b**. It is conventional, in box inverse modeling studies, to weight equations by a prior error estimate. As there is significant uncertainty in the estimation of prior error, the choice is still largely arbitrary. To test the sensitivity of our mixing estimates to the weighting of contour versus box equations, we simply multiply and divide by 2, the weighting given by the minimum condition number criterion. This change in the weights gives a negligible impact on the along-isopycnal mixing coefficient profile in the upper 2000 m and a small impact on the vertical mixing profile in the upper 1000 m (Figs. B1c,d). There is a nonnegligible effect on the vertical mixing coefficient *D* in the 1000–2000-m depth range, although they show an increase with depth for all choices of weights. There is a small amount of sensitivity to the choice of reference level, partly because this choice affects the magnitude of **b**, which in turn affects the relative weights of each equation.

The scale lengths, *λ _{γ}* and

*η*and

_{γ}We have mapped the GK04 climatology onto neutral density surfaces (*γ*^{n}, Jackett and McDougall 1997). We interpolate onto the same grid as DW10 so that a clear comparison can be made between the two. The velocity and mixing coefficients generated by the tracer-contour inverse method are generally more noisy when using the GK04 climatology. When using GK04, the results are more sensitive to the proximity of the analysis to the mixed layer. For this reason, we avoid using data shallower than 250 m. The vertical mixing coefficient, inferred using GK04, is weak, *O*(10^{−5} m^{2} s^{−1}), throughout all of the water column, increasing somewhat in the 1000–2000-m depth range (Fig. B1e). The along-isopycnal mixing coefficient *K*, inferred using GK04, decreases from highs *O*(1000 m^{2} s^{−1}) in the upper 500 m of the water column to *O*(100 m^{2} s^{−1}) below 1000 m (Fig. B1f). These results are very similar to those derived using DW10 and earlier estimates summarized in Figs. 3 and 4.

## APPENDIX C

### Equivalent Form of the Thermal Wind Equation

*i*and

*i*+ 1 are closely spaced “casts” moving clockwise (outward to the left) along a bounding “section” of the domain (Fig. 2). Values at the midpoint between cast

*i*and

*i*+ 1 on the isopycnal

*γ*are given the subscript

*i*+ ½: for example, the Coriolis frequency

*f*

_{i+1/2}and pressure

*p*

_{i+1/2}. The specific volume anomaly is

*δ*(

*p*), with reference values 0°C and 35 psu. In (3)

*δ*

_{contour}(

*p*) is used, where the reference values are the conservative temperature and salinity of each contour on each isopycnal. Some improved accuracy in (C1) may be possible with the use of a more locally referenced form of

*δ*and additional terms (McDougall and Klocker 2010), although the difference is negligible in this study as isopycnals are not steeply sloped.