## 1. Introduction

The transport of tracers within the ocean plays an important role not only in ocean dynamics, thermodynamics, and biogeochemistry, but also as a method with which to observe the ocean and infer circulation properties. Because, by its nature, tracer transport integrates over both spatial and temporal scales, it allows us to measure the large-scale, integral impact of a range of smaller-scale processes that are difficult to observe directly. However, in order to correctly interpret and use tracer measurements, a good understanding of tracer transport and its relation to circulation properties is needed.

Tracer transport has been particularly useful for understanding the dynamics of the ocean’s deep overturning circulation, where small-scale turbulent mixing plays a key role (Watson and Ledwell 2000). Since the pioneering study of Munk (1966) and the realization that mixing in the interior is generally weak (e.g., Gregg 1989; Ledwell et al. 1993), mixing near the ocean’s boundaries has been thought to play a critical role in closing the global diapycnal circulation (Armi 1978; Ivey 1987b; Wunsch 1970; Phillips 1970; Thorpe 1987; Garrett 1991; Munk and Wunsch 1998; Wunsch and Ferrari 2004). Observations suggest that turbulent mixing is bottom intensified due to internal wave breaking within the stratified fluid above the boundary layer (Toole et al. 1994; Polzin et al. 1997; Ledwell et al. 2000; St. Laurent et al. 2012; Waterhouse et al. 2014). The bottom intensification of mixing within so-called “abyssal mixing layers” implies a downward mass transport across isopycnals in what some authors have termed the stratified mixing layer (SML; McDougall and Ferrari 2017). To balance the formation of dense waters at high latitudes there must therefore be a somewhat larger upwelling transport within thin bottom boundary layers (BBLs) along the sloping seafloor where the turbulent buoyancy flux converges (Kunze et al. 2012; de Lavergne et al. 2016, 2017; Ferrari et al. 2016; McDougall and Ferrari 2017). The requirement for net upwelling and the near compensation between the net diapycnal transports in the BBL and SML imply a complex balance between factors such as topographic geometry (McDougall and Ferrari 2017; Holmes et al. 2018), variations in stratification (Ferrari et al. 2016; Callies and Ferrari 2018; Banyte et al. 2018), and the lateral structure in the intensity of turbulent mixing near the ocean floor (Kunze 2017a,b). How this balance is achieved at both global and regional scales remains an open question that observations of tracer behavior may help to answer.

Field tracer release experiments (TREs) have provided many insights into the dynamics of diapycnal ocean circulation. TREs have shown that mixing away from boundaries is weak (Ledwell et al. 1993; Ledwell 1998), and highlighted the importance of strong boundary mixing for closing basin-scale budgets (Goudsmit et al. 1997; Inall 2009; Ledwell and Bratkovich 1995). The Brazil Basin TRE (BBTRE; Ledwell et al. 2000) in particular demonstrated the presence of intensified mixing in the abyss above the rough seafloor of the Mid-Atlantic Ridge, corroborating microstructure measurements (Polzin et al. 1997). The BBTRE tracer was released well above the ocean bottom, and its centroid showed a tendency to descend across isopycnals in the eastern basin consistent with the expected diapycnal downwelling in the SML. However, the observations close to the boundary suggested that in ridge canyons the tracer moves upslope, likely toward less dense waters. These observations pointed to the importance of the near boundary region, though BBTRE was not designed to study this region.

The importance of intense boundary mixing was further highlighted by the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES; Ledwell et al. 2011; Watson et al. 2013). DIMES also exposed a common discrepancy whereby diffusivities inferred from tracer measurements often exceed those estimated from microstructure surveys. These discrepancies are often attributed to sampling issues (e.g., Wüest et al. 1996; Voet et al. 2015) and can be reconciled if the full temporal and spatial distribution of the tracer is taken into account (Mashayek et al. 2017). Studies such as Mashayek et al. (2017) and Ledwell et al. (2000) highlight the importance of tracer-boundary interactions, but do not discuss the details of tracer transport within any particular boundary-driven flow. Idealized studies have examined near-boundary tracer dispersion above a horizontal boundary (e.g., Saffman 1962; Csanady 1969). Here we examine a similar problem near a sloping boundary.

Our tracer study will be conducted in the context of one-dimensional boundary layer theory (e.g., Wunsch 1970; Phillips et al. 1986; Thorpe 1987; Garrett 1990). The theory considers the one-dimensional problem of flow over a uniform slope driven by an isotropic diffusivity. To satisfy the no-flux boundary condition, isopycnals slope down as they approach the boundary, leading to a buoyancy-driven upslope flow in a weakly stratified BBL where friction is important (Garrett et al. 1993). When the diffusivity is bottom intensified a corresponding downslope flow appears in the outer portion of the abyssal mixing layer (the SML). Recently, Callies (2018) has shown that for realistic abyssal ocean parameters boundary layer theory predicts a much weaker stratification than typically observed. He suggests that restratification by submesoscale eddies, generated by baroclinic instability of the resulting flow field (Wenegrat et al. 2018), is necessary to maintain the stratification and therefore permit significant near-boundary water-mass transformation. Here, we overcome this limitation of the one-dimensional theory by using a large viscosity in the along-slope momentum equation, which acts as a simple parameterization for eddy-driven restratification (see section 2; e.g., Greatbatch and Lamb 1990).

Eddies, along with tides, intrusions, and other processes, can also drive strong along-isopycnal tracer transports (e.g., Ivey 1987a; McPhee-Shaw 2006; Wain and Rehmann 2010; Winters 2015; Dell and Pratt 2015). This along-isopycnal exchange (here captured at first order by an along-isopycnal tracer diffusivity) provides one means in which the strong boundary mixing can communicate with the interior.

In this article we aim to address the following questions: 1) What processes within abyssal mixing layers are most important for determining tracer transport? 2) Will a tracer be transported toward lighter (upslope) or heavier (downslope) density classes, on average, when released near a sloping boundary? 3) What can be inferred about the properties of mixing and circulation from bulk measurements of the tracer cloud dispersion?

To approach these questions we analyze the behavior of a passive tracer released within a flow governed by one-dimensional boundary layer theory (described in section 2). The tracer evolves in two dimensions due to the effects of diapycnal diffusion, isopycnal diffusion, and advection (section 3). We introduce an analytical framework based around the tracer moments in buoyancy space (the tracer center of mass and variance) to understand the contribution of different processes to tracer dispersion (sections 4–6). We find that the presence of the boundary can slow the rate of diapycnal tracer dispersion below that expected from averaging the diapycnal diffusivity over the tracer patch. Due to asymmetries between the upslope BBL and downslope SML flows (the BBL upwelling being strong and narrow compared to the more diffuse SML downwelling), and the partitioning of the tracer between them (near-boundary tracers readily occupy the whole BBL but a smaller fraction of the SML), the tracer tends to move upslope on average. The extent of upslope movement and tracer dispersion depends on the initial release position of the tracer, as well as the decay scale of bottom-intensified mixing and along-isopycnal diffusion (section 7). Our results have implications not only for our understanding of tracer behavior near the ocean floor (section 9), but also for what can be learned about ocean dynamics from field TREs (section 8).

## 2. Boundary layer theory

*θ*and use a coordinate system aligned with the bottom (i.e.,

*z*is bottom normal with origin at the boundary,

*y*is upslope, and

*x*is along slope; Fig. 1). Everything is uniform in the upslope and along-slope directions except pressure and buoyancy (and in the next section the tracer). Buoyancy is characterized by a constant far-field vertical stratification

*N*

^{2}

*b*is the buoyancy field relative to a reference density

*ρ*

_{0}. The steady, rotating equations of motion for this system are (e.g., Garrett et al. 1993)

*U*(

*z*),

*V*(

*z*) are the along-slope and upslope velocities;

*ν*

_{u}(

*z*),

*ν*

_{υ}(

*z*) are along-slope and upslope eddy viscosities (which may differ for reasons discussed shortly);

*κ*(

*z*) is an eddy diffusivity;

*P*(

*y*,

*z*) is the pressure field; and

*f*is the (vertical) Coriolis parameter. These equations can be combined into (e.g., Garrett 1991)

*κ*

_{∞}is the far-field diffusivity and the scalar streamfunction Ψ(

*z*) is given by

*d*Ψ/

*dz*=

*V*with Ψ(0) = 0 and

*ν*

_{u}(

*z*) =

*ν*

_{u0},

*ν*

_{υ}(

*z*) =

*ν*

_{υ0},

*κ*(

*z*) =

*κ*

_{0}(=

*κ*

_{∞}), the solution to Eq. (7) with no-slip boundary conditions is characterized by a BBL of reduced stratification and upslope flow (Wunsch 1970; Thorpe 1987; dashed black lines in Fig. 2). There is an along-slope flow in the interior in the direction opposite to Kelvin wave propagation, or upwelling-favorable in a bottom-Ekman sense (Garrett et al. 1993). The BBL width is

*S*

^{−1}=

*f*

^{2}cos

^{2}

*θ*/

*N*

^{2}sin

^{2}

*θ*is the inverse slope Burger number, and Pr

_{u0}=

*ν*

_{u0}/

*κ*

_{0}, Pr

_{υ0}=

*ν*

_{υ0}/

*κ*

_{0}are Prandtl numbers. For typical abyssal parameters of

*N*

^{2}= 10

^{−6}s

^{−2},

*f*= 10

^{−4}s

^{−1},

*ν*

_{u0}=

*ν*

_{υ0}=

*κ*

_{0}= 10

^{−3}m

^{2}s

^{−1}, tan

*θ*= 1/400 then

*S*

^{−1}= 1600 is large and the BBL thickness corresponds to an Ekman layer thickness of

*κ*

_{0}is the diffusivity near the boundary and

*d*is a decay scale. For the parameter space of interest here the BBL thickness

*d*. Following Callies (2018), for

*q*

_{0}

*d*≫ 1 we can construct an approximate analytic solution to Eq. (7) for the bottom-intensified case by patching together a solution in the BBL, where the mixing coefficients can be assumed to be constant and equal to their near-boundary values

*ν*

_{u0},

*ν*

_{υ0}, and

*κ*

_{0}, to a solution in the interior, where the influence of friction through the fourth derivative term in Eq. (7) can be neglected. This procedure is presented in appendix A.

However, as shown by Callies (2018) [see Eqs. (A4) and (A7) in appendix A], for the typical large inverse slope Burger number *S*^{−1} and order one Prandtl number regime in the abyssal ocean such a solution predicts very weak stratification over the abyssal mixing layer. Callies (2018) attributes this weak stratification to the lack of representation of baroclinic instability and its associated eddy-driven restratification in the one-dimensional system. A realistically stratified one-dimensional solution requires small values of the parameter (*S*Pr_{u0})^{−1} (see appendix A for more details) and can therefore be recovered using a large Pr_{u0}. Such a choice of large vertical momentum mixing in the along-slope momentum equation can be physically interpreted as a parameterization for restratification by baroclinic eddies based on the thickness-weighted average formalism (Rhines and Young 1982; Greatbatch and Lamb 1990; Gent and McWilliams 1990; Gent et al. 1995; McDougall and McIntosh 2001). For simplicity, and to avoid the need to resolve eddies or include an explicit Gent and McWilliams (1990) type parameterization, throughout most of this article we will consider the limit (*S*Pr_{u0})^{−1} → 0 obtained for large Pr_{u0}. This choice assumes that eddies maintain the stratification in the SML at its far-field value.^{1} In section 7c and appendix C we discuss the impact of reduced stratification in the SML through a nonzero (*S*Pr_{u0})^{−1}.

Note that the representation of eddy-driven restratification through enhanced vertical momentum mixing should have less impact on the BBL solution, which is determined by nongeostrophic frictional turbulent boundary layer physics. The eddy-driven restratification and frictional boundary layer processes can be conveniently isolated (at least in our two-dimensional context) by enhancing only the along-slope viscosity. Thus we maintain the choice Pr_{υ0} = 1 in the upslope momentum equation.

*S*Pr

_{u0})

^{−1}→ 0 in Eqs. (A4) and (A7) in appendix A]

^{2}Outside the BBL the vertical stratification is equal to the far-field value

*N*

^{2}and there is a weak downslope flow which largely compensates the upwelling within the BBL [the imbalance between SML and BBL transports is governed by Eq. (8)]. These compensating upwelling and downwelling flows are equivalent to the water-mass transformation dipole discussed recently by a number of authors (e.g., de Lavergne et al. 2016; Ferrari et al. 2016; McDougall and Ferrari 2017). It is this flow that will be used to advect and diffuse a passive tracer in a two-dimensional

*y–z*plane, as discussed in the next section.

## 3. Two-dimensional tracer dispersion in a one-dimensional slope flow

### a. The tracer conservation equation

^{−2}) which varies in the upslope and slope-normal directions (Fig. 1) given an initial distribution

**V**= [

*V*(

*z*), 0] from boundary layer theory [

*V*(

*z*) is given by the

*z*derivative of Eq. (11)] and diffusive fluxes associated with a small-scale isotropic diffusivity

*κ*(

*z*) and an along-isopycnal diffusivity

*A*

_{H}. The along-isopycnal diffusion is specified using the symmetric second rank tensor (Redi 1982)

*b*indicates differentiation. The coefficients of

*b*

_{y}=

*B*

_{y}=

*N*

^{2}sin

*θ*split into background (

*B*

_{z}and

*B*

_{y}) and perturbation components, where from Eq. (12)

*b*replaces

### b. Numerical model setup

The tracer conservation equation (13) cannot be solved analytically in the general case (although we will consider simplified cases that can be solved analytically) and so we resort to numerical simulations. We use the spectral code Dedalus (Burns et al. 2019; http://dedalus-project.org/) in the rotated *y*, *z* coordinates (Fig. 1) with periodic boundary conditions in *y*. We consider a control parameter set with a *y*-domain length *L*_{y} = 1500 km, a slope of *α* = 1/400 (roughly the western side of the Mid-Atlantic Ridge) and a *z*-domain height of *L*_{z} = 3000 m. The far-field stratification *N*^{2} = 10^{−6} m^{2} s^{−1}. The diffusivity will either be bottom-intensified with an exponential profile [Eq. (10)] or constant, with base parameters *d* = 500 m, *κ*_{0} = 10^{−3} m^{2} s^{−1} and *κ*_{∞} = 10^{−5} m^{2} s^{−1}. For those simulations with no along-isopycnal diffusion (*A*_{H} = 0) we use 384 Fourier modes in *y* and 192 Chebyshev modes in *z*, corresponding to an average Δ*y* ~ 4 km and an average Δ*z* = 15 m. The Chebyshev basis allows finer resolution of the small-scale *z* gradients in the BBL (see Fig. 2). Equation (13) is solved implicitly in time using a time step of 8 days. However, because of small-scale variations in the along-isopycnal tensor coefficients [Eq. (15)] near the boundary, those simulations with nonzero along-isopycnal diffusion were instead performed with 576 modes in *y* and 768 modes in *z* and a time step of 4 days. The simulations match analytic results where such results are obtainable (e.g., section 5). The results are robust to numerical choices, as confirmed by doubling the number of modes in *z* and running with a time step 4 times smaller, which gave almost identical results (not shown).

## 4. Bulk diffusivity and tracer moments

*s*) due to a constant diffusivity

*κ*evolves as

*t*is time,

*A*is a constant,

*σ*

^{2}is the variance with initial value

*μ*is the centroid or center of mass. The variance increases linearly with time at a rate of 2

*κ*. Thus for any tracer distribution, which may be spreading due to complex advection–diffusion processes in multiple dimensions, we can define an equivalent or

*bulk diffusivity*in any dimension

*s*in terms of the rate of increase of the variance

*σ*

_{s}(also see Wüest et al. 1996; Goudsmit et al. 1997)

*s*

*σ*

_{s}and the center of mass

*μ*

_{s}in terms of the

*s*moments of the tracer distribution

*bulk diapycnal diffusivity*

*κ*

_{bk}). In this case a mean stratification profile

*N*

^{2}, which in our two-dimensional context will be the constant [in the limit (

*S*Pr

_{u0})

^{−1}→ 0] interior stratification, is also required in order to convert the spreading rate into the units of a diffusivity. We define

*y*and

*z*

The bulk diffusivity [Eq. (23)] will be used to characterize the overall rate of spreading of the tracer across buoyancy surfaces. However, it should be emphasized that *κ*_{bk} should not necessarily be interpreted directly as a diffusivity, as such an interpretation masks the potentially complex processes that are leading to the spreading of that tracer distribution in the first place (here, two-dimensional advection and diffusion). Comparisons of *κ*_{bk} to the actual isotropic diffusivity *κ* will prove useful, as is commonly done for field TREs where microstructure measurements are also available (e.g., Ledwell et al. 2000; Watson et al. 2013; Mashayek et al. 2017). We will also compare *κ*_{bk} to the diffusivity calculated using a one-dimensional model commonly applied to field data (Ledwell and Watson 1991; Ledwell 1998; see appendix B).

## 5. Boundary restriction: Tracer behavior in *z*

*z*. Integrating the two-dimensional tracer conservation equation [Eq. (13)] across all

*y*yields

*z*= 0 and

*z*→ ∞. Thus in the slope-normal coordinate

*z*the tracer diffuses according to a one-dimensional diffusion equation that does not depend on the upslope velocity

*V*. We are not aware of any analytic solutions to Eq. (27) for the case where

*κ*is exponential,

*A*

_{H}is nonzero, and there is a boundary at

*z*= 0 (see Zamani and Bombardelli 2014, and references therein). However, when

*K*

^{z}is a constant (where here for simplicity we set

*K*

^{z}=

*κ*with

*κ*constant) a solution is easy to obtain and, despite the simple governing equation, shows some interesting behavior as a consequence of the boundary at

*z*= 0.

### a. Constant diffusivity

*κ*is constant and

*A*

_{H}= 0 the tracer evolution for an initial Gaussian distribution with center of mass at

*z*=

*μ*

_{0}and spread

*σ*

_{0}(provided the initial tracer distribution is isolated from the boundary,

*σ*

_{0}≪

*μ*

_{0}) is given by the sum of two Gaussians centered at

*z*=

*μ*

_{0}and

*z*= −

*μ*

_{0}

*z*= 0, this solution satisfies the no flux boundary condition at

*z*= 0, while the individual Gaussians satisfy the one-dimensional diffusion equation without a boundary. This analytic solution matches the numerical two-dimensional solution averaged in

*y*(shown at three different times by the orange curves in Fig. 3a).

*z*moments of this solution [defined for

*z*> 0 using Eq. (26) for the domain average] are

*μ*

_{z}≫

*σ*the center of mass is stationary

*μ*

_{z}=

*μ*

_{0},

*σ*~

*μ*

_{z}, the center of mass moves away from the boundary, eventually increasing like the square root of time

*σ*

_{z}reduces [due to the

*κ*(cf. orange solid and dashed lines in Fig. 3c). Hence the effect of the boundary is to reduce the bulk diffusivity by limiting downward tracer spreading. In the limit of long time

*z*> 0, resulting in an effective accumulation of tracer near the boundary.

### b. Bottom-intensified diffusivity

*κ*

_{∞}≠

*κ*

_{0}, with

*A*

_{H}= 0) the solution cannot be obtained analytically. However, the influence of the boundary on the tracer diffusion can still be understood by deriving equations for the evolution of the

*z*moments directly from the tracer conservation equation (this technique will also be used in later sections to analyze the tracer behavior in buoyancy coordinates). Multiplying Eq. (27) by

*z*and integrating over the domain

*z*= 0 to eliminate the first term. Equation (36) simply states that the tracer center of mass will move according to the domain-averaged tracer flux. Furthermore, using the chain rule to move the

*z*derivative from

*κ*and the boundary condition to rewrite

*z*direction (once tracer accumulates there) and thus the flux in the positive

*z*direction dominates, driving the center of mass away from the boundary. A

*z*-dependent diffusivity can also drive net tracer movement through the second term in Eq. (37). For a bottom-intensified diffusivity, this term enhances the downward tracer diffusion below the initial tracer patch driving the center of mass toward the boundary (dotted blue line in Fig. 3b). However, this downward motion is transient and the boundary effect dominates once enough of the tracer accumulates on the boundary (solid blue line in Fig. 3b).

Note that the isotropic diffusivity *κ* should strictly go to zero over some finite (rather than infinitesimal) distance approaching the boundary, which would remove the boundary effect term in Eq. (37). However, such a change in the diffusivity near the boundary plays a very similar role to the boundary effect by preventing tracer fluxes toward the boundary. If the tracer concentration is assumed to be constant over this layer in which the diffusivity reduces near the boundary then the component of the second term in Eq. (37) associated with this change in *κ* reduces to the same form as the first term, with *κ*_{0} replaced by the diffusivity at the top of the layer.

*z*tracer moment can be derived using a procedure analogous to that employed for Eq. (36)

*z*−

*μ*

_{z}. Furthermore, by shifting the

*z*derivative from the

*κ*, and absorbing the factor (

*z*−

*μ*

_{z})

*κ*over the tracer patch (or the in situ diffusivity

*reduces*this spreading rate, once tracer encounters the boundary (second term on the rhs of Eq. (40), compare solid and dashed lines in Fig. 3c). This occurs because the boundary prevents tracer spreading in the negative

*z*direction, and constricts the tracer to remain closer to its center of mass [Eq. (39)]. On the other hand, the vertical diffusivity gradient, when present, drives a modest enhancement of the bulk diffusivity through the last term in Eq. (40) (dotted blue line in Fig. 3c). It should be noted that the terms in Eqs. (40) and (37) are not independent; each term affects the tracer distribution

## 6. Diapycnal spreading: Tracer behavior in *b*

*V*does not influence the spreading of the tracer in the slope normal direction

*z*, it does affect the tracer spreading rate in the upslope direction

*y*and across buoyancy surfaces. The two-dimensional evolution of tracer patches released at

*z*

_{0}=

*d*/2 = 250 m in flows with constant (Figs. 4a–c) and bottom-intensified (Figs. 4d–f) diffusivities clearly show the impacts of the boundary layer flow. With a constant diffusivity there is only upslope flow within the BBL. This flow drives tracer up the slope after which it diffuses vertically out of the BBL, forming a characteristic upslope tracer tongue (Figs. 4a–c). The center of mass (with upslope position

*y*and integrating over the domain], and slightly away from the boundary (

*μ*

_{z}increases, see closed circles in Figs. 4a–c). Thus, the advective tracer transport in the BBL is significant despite its thinness. The expanding tracer patch continues to supply tracer to the BBL through downward diffusion, accounting for the large amount of tracer advected upward within the BBL.

When the diffusivity is instead bottom intensified (Figs. 4d–f), the upslope spreading of tracer is reduced due to both the restriction of diffusion to the region near the boundary and the presence of downslope transport in the SML (cf. closed circles in Figs. 4c and 4f). Due to the shear in the SML velocity, the expanding interior tracer patch acquires a tilt.

The evolution of the tracer concentrations binned into buoyancy classes also indicates a strong upward net motion of the tracer distribution, in both cases (orange and blue distributions in Fig. 5a), reflected in the upward motion of the center of mass (solid lines in Fig. 5b). In the following subsections we develop a framework based on the tracer moments to evaluate the contributions of various advective and diffusive tracer fluxes to the diapycnal tracer spreading.

### a. Advective and diffusive tracer fluxes and the buoyancy budget constraint

*κ*

_{bk}[Eq. (44)], is positive if the diapycnal fluxes move tracer on average away from the tracer center of mass

*μ*

_{b}. Noting that the along-isopycnal flux drops out yields contributions from diapycnal advection and isotropic diffusion

*b*|

^{2}≈

*N*

^{4}(true except in the case of reduced SML stratification considered in section 7c and appendix C) then the additional second term is the in situ diffusivity

### b. Boundary and diffusivity gradient contributions to center-of-mass motion

*B*= (

*B*

_{y},

*B*

_{z}) =

*N*

^{2}(sin

*θ*, cos

*θ*) and perturbation

*B*

_{z}]. The final term is associated with variations in the buoyancy gradient in the BBL and is negligible in our context (not shown). The first advective term, as discussed above, is equal to the sum of the three other diffusive terms. We can go even further by identifying the advective tracer fluxes in the BBL and SML with the boundary and diffusivity gradient terms respectively.

### c. BBL and SML contributions to center-of-mass motion

*S*Pr

_{u0})

^{−1}→ 0, Eqs. (A3) and (A4)], and one for the BBL where the diffusivity is constant and equal to

*κ*

_{0}[Eq. (A6)]. Therefore,

*μ*

_{b}is equivalent to the influence of the boundary term plus a minor correction due to the BBL buoyancy perturbation. In turn, this implies that diapycnal tracer advection in the SML is equivalent to the diffusivity gradient term plus an even smaller correction

*b*whenever there is tracer within the BBL (by limiting tracer diffusion toward denser fluid), as does the upslope BBL diapycnal flow. The diffusivity gradient drives the tracer center of mass toward denser fluid (

*κ*

_{z}is negative), as does the downslope SML diapycnal flow.

For a constant diffusivity there is no SML flow or diffusivity gradient, and so the center of mass moves toward lighter water due solely to the influence of BBL advection and the boundary effect, which contribute equally (orange dashed line in Fig. 5b, which when multiplied by 2 provides the full center-of-mass evolution *μ*_{b}, the solid line). In the bottom-intensified case, both the diffusivity gradient and the downslope SML transport drive the tracer center of mass downward initially (dotted blue line in Fig. 5b, which again should be multiplied by 2 to account for influence of the SML transport), but the boundary effect and advection within the BBL quickly dominates driving the tracer toward less dense fluid on average (solid blue line in Fig. 5b).

### d. BBL and SML contributions to the bulk diffusivity

^{−5}m

^{2}s

^{−1}, in all cases considered here, although they may be significant when

Equation (58) provides a useful diagnostic for examining the tracer dispersion. However, the terms are not mutually independent as they all affect and depend on the tracer concentration

*κ*

_{bk}[second term inside the curly brackets in Eq. (58)] can have either sign due to the (

*b*−

*μ*

_{b}) weighting factor. In fact, this term can be further manipulated to yield

*b*(0) are shorthand for the tracer and buoyancy on the boundary

*b*(

*y*, 0), and

*κ*

_{bk}if

*μ*

_{b}(the usual case, compare closed circles and crosses in Fig. 4).

^{3}Its magnitude depends on the separation between

*μ*

_{b}and the amount of tracer on the boundary

The reduction in the bulk diffusivity *κ*_{bk} below the in situ diffusivity (cf. dashed and solid lines in Fig. 5c) due to the boundary effect occurs as soon as the tracer encounters the boundary. The behaviors of the constant and bottom-intensified diffusivity cases are similar, apart from the magnitude of the initial, and in situ, diffusivities (cf. blue and orange lines in Fig. 5c). This is because the influence of SML advection and the diffusivity gradient are weak (dotted blue line in Fig. 5c). These terms act to drive a modest enhancement of the diapycnal spreading rate as the diapycnal SML velocity is *divergent* around the tracer patch due to the positive curvature in *κ* (since *N*^{2} is constant). It is the curvature of *κ* that matters, despite the appearance of only the first-derivative of *κ* in Eq. (58), because of the weighting factor (*b* − *μ*_{b}).

### e. Summary

To summarize, the main effect highlighted by these near-boundary point-release tracer experiments is a strong reduction in the diapycnal spreading rate, below that which would be expected from in situ measurements of the isotropic diffusivity averaged over the tracer patch,^{4} due to the presence of the boundary which limits tracer diffusion toward denser water. Accompanying this reduced spreading rate is a tendency for the overall tracer patch, quantified by its center of mass, to move up the slope toward less dense fluid. We also showed that due to the buoyancy budget constraint the advective and diffusive fluxes have equivalent influences on the tracer dispersion [Eqs. (49) and (53)], as do their individual BBL and SML contributions. We will discuss these results in the context of field TREs in section 8. However, in the next section we first examine the sensitivity of the tracer dispersion to several other parameters.

## 7. Sensitivity to other parameters

### a. Release location

In the previous two sections we showed that the presence of the boundary significantly reduces the diapycnal spreading rate of the tracer, and alters the tracer center-of-mass motion. The time scale over which these effects appear depends on where the tracer is released. When released further from the boundary (e.g., at *z*_{0} = 3*d*/4, Figs. 6a–c) the tracer initially spreads according to the isotropic diffusivity, with enhanced downward diffusion due to bottom-intensification and the downslope SML flow (orange solid line in Fig. 7b). There is an initial slight increase in the bulk diffusivity (solid orange line in Fig. 7c before day ~250), due to the fact that the SML flow is divergent (dotted orange line in Fig. 7c, which represents the equivalent influence of the diffusivity gradient). However, once the tracer encounters the boundary the diapycnal spreading rate is significantly reduced.

The reduction in the diapycnal spreading rate occurs earlier for tracers released closer to the boundary (e.g., at *z*_{0} = 62.5 m, Figs. 6d–f and blue lines in Fig. 7, effectively equivalent to a release at the boundary). While tracers released very close to the boundary visually appear to experience more overall dispersion (cf. Figs. 6f and 6c), most of this dispersion manifests in the lateral direction (despite the absence of any along-isopycnal diffusion) with less spreading across buoyancy surfaces (cf. solid blue and orange lines in Fig. 7c). For the near-boundary release the bulk diffusivity is initially more than 5 times smaller than the in situ diffusivity (cf. blue solid and dashed lines in Fig. 7c). Coupled with the large transport of tracer toward lighter fluid (blue line in Fig. 7b), almost the entire tracer distribution lies at buoyancies above its initial buoyancy after ~100 days (blue lines in Fig. 7a, also see Figs. 6e,f). This net movement toward less dense fluid is a consequence of the upslope BBL flow and the asymmetric distribution of tracer between the BBL and SML. The latter asymmetry is due to the boundary, which tends to trap tracer in the BBL. The net upward movement of tracer is strongest when the tracer is released next to the boundary, while tracers released away from the boundary in the SML experience a net downward movement (Fig. 8). Field TREs conducted very close to a sloping boundary may be expected to show a similar net diapycnal upward movement of tracer (e.g., Inall 2009). However, in section 7e we show that the introduction of along-isopycnal diffusion can limit this upward movement.

### b. Topographic slope

Though the position of the initial release point of the tracer clearly impacts the initial behavior of the tracer, the slope of the boundary does not in these point release experiments. Altering the slope of the boundary has a number of effects on the boundary layer flow. First, it influences the thickness of the BBL *θ* in Eq. (11); also compare orange and blue solid lines in Fig. 2a and see Dell and Pratt (2015)]. The change in the upslope and downslope flow strength arises because the local diapycnal advective flux of buoyancy must remain the same (as the diffusion has not changed), but the separation of buoyancy surfaces in the direction parallel to the boundary changes. Comparing tracer releases performed over a slope of 1/100 versus 1/400 shows significant visual differences in the rate of tracer dispersal (Fig. 9). However, the rate of tracer spreading across buoyancy surfaces is in fact almost identical (not shown), because the magnitude of the diapycnal velocity does not change, being set by diffusion. Instead, the spacing of buoyancy surfaces in the *y* direction (*B*_{y} = *N*^{2} sin*θ*) increases in proportion to the decrease in upslope flow (*V* ~ cot*θ*), such that the total advective flux of tracer

### c. Reduced mixing layer stratification

Throughout most of this article we have taken the limit *d* log(*κ*_{0}/*κ*_{∞})] (Callies 2018), with a similar reduction in the upslope transport Ψ [Eq. (A7) in appendix A, where there is also a small factor of 3^{1/4} modification to the BBL scale *q*_{0}, Eq. (9)]. For tracers released near the boundary, which do not extend into the far-field, this reduction in upslope transport reduces the upslope spreading of tracer (cf. Figs. 9d–f and 9g–i) and the net movement of tracer toward lighter fluid (as the upslope buoyancy gradient *b*_{y} remains the same). The rate of increase of the variance of the tracer in buoyancy space also reduces, but there is little effect on the bulk diffusivity if the stratification normalization factor [Eq. (23)] is instead chosen as the SML stratification *N*^{2}) has only a minor impact on the diapycnal tracer dispersion. However, as for the boundary slope above, this may no longer hold for tracer clouds that extend across the SML into the far-field, such as the BBTRE tracer (see section 8 and appendix C).

### d. Decay scale of bottom-intensified mixing

The tracer dispersion is also sensitive to the decay scale used for the isotropic diffusivity [*d* in Eq. (10)]. When the tracer is released at the same distance from the boundary but this *d* scale is reduced to 200 m (a value that may be more representative of the Brazil Basin; St. Laurent et al. 2001; Callies 2018) then the tracer remains restricted closer to the boundary and the net upslope movement of the tracer toward lighter buoyancy is reduced (cf. Figs. 9j–l with Figs. 9a–c). This reduction in the center-of-mass movement in buoyancy space is due to the increase in the vertical gradient of the isotropic diffusivity *κ*_{z}, or equivalently an increase in the SML transport, that drives more tracer downward (cf. blue and orange dotted and dashed lines in Fig. 10b). This increase in the vertical gradient of the diffusivity also acts to enhance the tracer spreading rate in buoyancy space such that the bulk diffusivity does not reduce as much as would be expected from the reduced in situ diffusivity (cf. orange and blue lines in Fig. 10c).

### e. Influence of along-isopycnal diffusion

Finally we consider the impact of along-isopycnal diffusion on the tracer behavior. Large along-isopycnal diffusion may be expected in abyssal mixing layers due to processes such as intrusions (e.g., McPhee-Shaw 2006) or baroclinic instability setup by boundary mixing (e.g., Callies 2018; Wenegrat et al. 2018). Along-isopycnal diffusion can rapidly mix tracer between the SML and BBL, an effect which is clear when comparing TREs with large (*A*_{H} = 100 m^{2} s^{−1}, Figs. 11d–f) and small (*A*_{H} = 10 m^{2} s^{−1}, Figs. 11a–c) along-isopycnal diffusivities. The along-isopycnal diffusivity also appears to influence the spreading of the tracer in buoyancy space, despite the inability of along-isopycnal diffusion to directly flux tracer across isopycnals (Fig. 12). At early times, the enhanced along-isopycnal diffusion limits the initial downward motion of the center of mass by mixing tracer more rapidly into the BBL, enhancing the boundary effect (cf. orange and blue solid and dashed lines in Fig. 12b before day 500). However, at later times along-isopycnal diffusion limits the motion of the center of mass up the slope toward less dense fluid as the tracer is mixed between the regions of upslope and downslope motion in the BBL and SML, respectively. In terms of the framework introduced in section 6, the limited upslope motion of the center of mass arises because of a reduction in the influence of BBL advection (or equivalently the boundary effect, compare dashed lines in Fig. 12b after day 1000), as there is less tracer in contact with the boundary.

Similarly, the reduction of the influence of the boundary, or BBL advection, on the tracer distribution under strong along-isopycnal diffusion results in an enhanced rate of diapycnal spreading as quantified by *κ*_{bk} (cf. solid blue and orange lines in Fig. 12c). The magnitude of the boundary effect responsible for the reduction in *κ*_{bk} is dependent on two factors [Eq. (62)]: 1) the amount of tracer on the boundary *μ*_{b} and that on the boundary

A secondary effect of along-isopycnal diffusion is to reduce the in situ diffusivity averaged over the tracer patch (cf. dashed blue and orange lines in Fig. 12c) by driving a net lateral movement of tracer toward the weakly mixing interior. This net lateral movement toward the far-field with along-isopycnal diffusion is once again due to the presence of the boundary that prevents tracer transport through it.

The results of these experiments are summarized in Fig. 13, where three measures of the time-averaged or cumulative diffusivity up to day 800 are shown for a range of along-isopycnal diffusivities.^{5} For all values of *A*_{H} the total diapycnal spreading rate of tracer across isopycnals, as quantified by the bulk diffusivity, is smaller than the in situ diffusivity (cf. circles and crosses in Fig. 13). As along-isopycnal diffusion increases the diapycnal spreading rate increases and the in situ diffusivity decreases. This occurs despite the inability of along-isopycnal diffusion to directly drive diapycnal tracer fluxes, due to changes in the tracer gradients and the degree to which the tracer interacts with the boundary. Finally, a more sophisticated estimate of the tracer derived diffusivity discussed in appendix B, based on a least squares fit to the results of a one-dimensional advection–diffusion equation, gives similar results to the bulk diffusivity (cf. triangles and circles in Fig. 13), meaning that if this estimate was interpreted as a measure of the in situ diffusivity it would be an underestimate.

## 8. Relation to field experiments

Our study has focused on tracer behavior in the immediate boundary region, as opposed to past field TREs such as BBTRE and DIMES. Nevertheless, it is useful to discuss our results in the context of these past experiments.

### a. BBTRE

The BBTRE tracer (Ledwell et al. 2000) was released along a target isopycnal that sat about 1000 m above the bottom of a fracture zone trough. The majority of the tracer cloud thus lay in the outer SML, although because of topographic variability some tracer was located much closer to the boundary (Fig. 4 of Ledwell et al. 2000). The initial BBTRE dispersion was well modeled with a one-dimensional advection–diffusion equation and showed a tendency to move downward toward denser water (Fig. 2a of Ledwell et al. 2000), consistent with our results for bottom-intensified mixing (e.g., Figs. 6a–c). A smaller portion of the tracer was also drawn eastward and more strongly downward toward the MAR boundary. Once significant amounts of tracer came into contact with the boundary Ledwell et al. (2000) point out that the tracer tended to get mixed back toward lighter density levels, consistent with the boundary effect presented here. Ledwell et al. (2000) do not attempt to use a one-dimensional model for these later periods of BBTRE, as they suggest it may underestimate the diffusivity. We have applied such a one-dimensional model to our idealized near-boundary TREs and found that it provides a similar estimate of the diapycnal tracer spreading rate to the bulk diffusivity *κ*_{bk} (see appendix B), and therefore indeed underestimates the in situ diffusivity

### b. DIMES

Our results have less immediate applicability to DIMES, given the topographic complexity and strong mean flows characterizing the Southern Ocean. In particular, we note that the order-of-magnitude mismatch between diffusivities estimated from the DIMES tracer and microstructure observations cannot be explained using the physics discussed here. Mashayek et al. (2017) suggested that this mismatch arose from the enhanced tracer residence time in the vicinity of near-bottom mixing hot spots where bottom flows were weak (which would be associated with a larger value of the in situ diffusivity

### c. Future near-boundary TREs

*Db*/

*Dt*, averaged over the tracer patch to be easily obtained.

*f*(0) = 1, where

*κ*

_{bk}, along with the shape function correlations

*κ*

_{0}taking into account the boundary and

*κ*

_{z}effects. Utilization of similar equations for higher tracer moments may allow for more free parameters [such as the decay scale of

Finally, it may also be possible to use the knowledge and relations gained from this study to build an intermediate complexity prognostic model of the tracer dispersion that is simplified relative to the full 2D or 3D problem but takes into account some of the boundary effects neglected by the 1D advection–diffusion model considered in appendix B. A more comprehensive inverse model along these lines may allow better estimates of the properties of the small-scale turbulent diffusivity to be obtained from the sparsely sampled tracer data. The development of such a model is, however, outside the scope of this article.

## 9. Summary

We have examined the behavior of a passive tracer released near a sloping boundary within an idealized flow governed by one-dimensional boundary layer theory. Results can be summarized as follows:

For isolated near-boundary tracer releases the presence of the

*boundary reduces the net diapycnal tracer spreading rate*below that which would be expected from averaging the in situ diffusivity over the tracer patch (Figs. 5c and 13).The dipole of diapycnal flow, upward in the BBL and downward in the SML, also influences the tracer dispersion. In particular, when the tracer is released close to the boundary it tends to move

*upslope toward less-dense fluid on average*(e.g., Figs. 4d–f) due to the asymmetric distribution of diapycnal flow between the BBL and SML (the SML flow being spread over a much wider region). However, the extent of this upslope motion depends on the proximity of the tracer release point to the boundary (Fig. 6); tracers released further from the boundary in the outer SML experience a net downward diapycnal motion (as summarized by Fig. 8).As a consequence of the advection–diffusion balance in the buoyancy equation, there is an

*equivalence between the diffusive and advective tracer fluxes*; they both contribute equally to the domain-averaged diapycnal tracer flux [Eq. (48)]. Further, the slowing of the diapycnal tracer spreading rate due to the presence of the boundary is equivalent to the influence of tracer advection within the BBL, while the modest enhancement in tracer spreading associated with the divergent diapycnal flow in the SML is equivalent to the diffusive influence of the gradient in the isotropic diffusivity.For tracers released at a point the boundary slope has little influence on the diapycnal spreading rate.

The introduction of reduced stratification and sloping isopycnals in the SML (Callies 2018) reduces the rate at which tracer spreads across isopycnals. However, as a diffusivity measures the rate of tracer spreading in physical space the bulk diffusivity is not affected.

When the vertical decay scale of bottom-intensified mixing is decreased the diapycnal tracer spreading rate is reduced as the in situ diffusivity decreases. However, an increase in the SML transport divergence acts to partially compensate this reduction in spreading rate.

*Along-isopycnal diffusion enhances the diapycnal tracer spreading rate*for near-boundary point releases by reducing the damping influence of the boundary and BBL advection (Fig. 13). Along-isopycnal diffusion also drives more tracer away from the boundary to where mixing is weak, reducing the in situ diffusivity averaged over the tracer patch.

These results have implications not only for abyssal circulation, water-mass transformation, and how we observe these processes, but also for the transport of benthic-origin chemical tracers and sediments (e.g., Lampitt et al. 2003; McPhee-Shaw 2006; Baskaran 2016).

While the differences between tracer-derived and in situ diffusivities arising from the additional boundary effects exposed in this study are generally less than an order of magnitude, our results nevertheless highlight the complexity of tracer behavior near sloping boundaries. Additional complexities present in the observational context likely make matters worse. Variations in topographic slope, mixing intensity and stratification are likely to drive additional tracer transports. Submesoscale to large-scale currents will also drive interactions between the tracer patch and different boundaries at a range of depths, buoyancies, and times. It is possible that averaging over these complexities will result in smooth Gaussian tracer spreading across buoyancy surfaces with a well-defined central diffusivity. However, our results suggest that relating this tracer-derived diffusivity to the small-scale turbulent diffusivity important for buoyancy and mass transport must be done with care. In section 8 we discussed several avenues that may help with the analysis of future near-boundary TREs.

Further modeling work is also needed to assess the impact of additional complexities. For example, a large *net* upslope or downslope mass transport associated with upslope variations in topography, stratification or mixing may influence the tracer behavior. The role of intrusions of boundary layer fluid into the interior (e.g., Gloor et al. 2000; Lampitt et al. 2003; McPhee-Shaw 2006; Kunze et al. 2012) is another area that deserves further attention. In this article, the influence of such intrusions on tracer transport was captured at first-order through an along-isopycnal diffusivity. Such an approximation may be appropriate for the statistical average tracer behavior, but is less applicable to a tracer released, for example, at a particular phase of a passing eddy. Eddy and internal wave induced strain may also influence the diapycnal tracer transport (Wagner et al. 2019). Finally, the relationship between the along-isopycnal diffusion and the eddy-driven overturning required to maintain a stratified SML in rotating boundary layer theory (e.g., Callies 2018), and whether this influences the tracer transport, is worthy of further exploration. A deep understanding of these processes is needed in order to represent them appropriately in general circulation models.

## Acknowledgments

We gratefully acknowledge support from the Australian Research Council through grant FL150100090. We thank Geoff Vasil for help with the Dedalus simulations. We thank Jörn Callies and an anonymous reviewer for useful comments. This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government.

## APPENDIX A

### The Approximate Analytic Solution

*x*and

*y*directions]. We nondimensionalize the boundary layer equations using

*κ*

_{0},

*ν*

_{u0}, and

*ν*

_{υ0}represent the near-boundary diffusivity and viscosity. Equation (7) then becomes

*r*=

*κ*

_{∞}/

*κ*

_{0},

*q*

_{0}is the BBL width parameter based on the mixing coefficients near the boundary [Eq. (9)] and we have assumed constant Prandtl numbers Pr

_{u0}and Pr

_{υ0}. For reasonable deep ocean parameters

*S*Pr

_{u0})

^{−1}where

*q*

_{0}is determined by upslope friction only, see Eq. (9)]. Therefore with

*d*~ 500 m,

*dq*

_{0}~ 20 and the fourth derivative term in Eq. (A2) can be ignored outside the thin BBL. Following Garrett (2001), a solution outside the BBL is therefore (restoring the dimensions)

*S*Pr

_{u0})

^{−1}, this gives much reduced stratification over the entire SML until

*κ*approaches

*κ*

_{∞}(Callies 2018).

*z*= 0 at

*z*= 0. To satisfy these boundary conditions we introduce an inner BBL solution where the fourth derivative term in Eq. (A2) becomes important. For this inner solution we can make the assumption that the mixing coefficients are constant

*κ*~

*κ*

_{0},

*ν*

_{u}~

*ν*

_{u0},

*ν*

_{υ}~

*ν*

_{υ0}. With Ψ = Ψ

^{BBL}+ Ψ

^{SML}, where Ψ

^{SML}is assumed constant in the BBL, substitution into Eq. (7) yields

^{BBL}(0) = −Ψ

^{SML}(0) and ∂Ψ

^{BBL}/∂

*z*(0) = 0 (neglecting the small ∂Ψ

^{SML}/∂

*z*at

*z*= 0) is given by

*S*Pr

_{u0})

^{−1}→ 0 Eq. (A7) reduces to Eq. (11), which will be used to determine the upslope velocity

*V*= ∂Ψ/∂

*z*and buoyancy field

*b*

_{z}=

*N*

^{2}sin

*θ*Ψ/

*κ*for the 2D tracer release experiments.

## APPENDIX B

### A One-Dimensional Model of the Tracer Dispersion across Isopycnals

*h*above a target buoyancy surface. A mean stratification profile is used to convert between

*h*and

*b*. The area-averaged diffusivity is assumed to be a linear function of

*h*,

*h*. Here

*t*

_{n},

*L*is a matrix operator representing the centered-in-space spatial discretization of Eq. (B1), and

^{B1}

The model matches the evolution of the buoyancy tracer profiles from the full two-dimensional simulations very well for both constant (not shown) and bottom-intensified (cf. solid lines and open circles in Fig. B1a) diffusivity cases. The least squares fit provides an estimate of a single diffusivity and its linear gradient over the full evolution of the tracer up to a given time. This diffusivity, at the center-of-mass position of the fitted tracer distribution, compares well with the time-averaged bulk diffusivity measured from the tracer moments (cf. solid blue and orange lines in Fig. B1b). In particular, the fitting method reproduces the rapid initial decrease in the tracer spreading rate below that expected from the in situ diffusivity due to the presence of the boundary.^{B2}

*h*shows that the rate of change of the center of mass

*μ*

_{h}and the one-dimensional bulk diffusivity in the one-dimensional model are given by

*μ*

_{h}[Eq. (B4)]. Comparing Eq. (B4) to Eq. (58), the boundary effect, the advective term and the diffusivity gradient term in Eq. (58) all drop out because

*κ*

_{h}are constants and can be removed from the averaging leaving the factor

## APPENDIX C

### Interior TRE Based on BBTRE

In this appendix we examine an idealized TRE based on BBTRE, which combines a number of parameters considered individually in section 7, namely, along-isopycnal diffusion and sloping isopycnals where the tracer is released well into the interior. We base this simulation on the fit to one-dimensional boundary layer theory performed by Callies (2018) for the Brazil Basin. The parameters are a slope of 1/500, *A*_{H} = 100 m^{2} s^{−1}, *N*^{2} = 1.69 × 10^{−6} s^{−2}, *d* = 230 m, *κ*_{0} = 1.8 × 10^{−3} m^{2} s^{−1}, *κ*_{∞} = 5.2 × 10^{−5} m^{2} s^{−1}, and a *u*-momentum Prandtl number of Pr_{u0} = 230 [corresponding to a reduction in the stratification in the SML by a factor of (*S*Pr_{u0})^{−1} = 1.95]. We release the tracer 1000 m above the bottom, corresponding approximately to the release height of the BBTRE tracer (Fig. 2 of Ledwell et al. 2000). Note that this height corresponds roughly to the transition between the sloping isopycnals in the outer abyssal mixing layer and the flat isopycnals in the interior (see Fig. C1).

*B*

_{z}in Eqs. (54) and (58) is no longer constant. Equation (54) for the center of mass becomes

*S*Pr

_{u0})

^{−1}]. As

*B*

_{z}now varies with

*z*, it must be included inside the averaging brackets in the third term on the rhs of Eq. (C1), and the fourth term, dependent on the curvature of the interior buoyancy field

*B*

_{zz}, appears.

As expected, the center of mass of the tracer is drawn toward denser fluid during the initial stage of the TRE due to the bottom-intensified diffusivity and downslope SML flow (solid and dotted lines in Fig. C2b), consistent with BBTRE (Ledwell et al. 2000). However, as the tracer approaches the boundary the BBL flow (and boundary effect) begin to drive tracer toward lighter fluid (dashed line in Fig. C2b). In the later part of the experiment, the movement of tracer toward lighter fluid driven by the BBL flow and boundary effect largely compensate for the driving of tracer toward denser fluid by the bottom-intensified diffusivity. The new term dependent on interior buoyancy curvature makes only a minor contribution that drives tracer toward lighter fluid as *B*_{zz} > 0 (dot–dashed line in Fig. C2b).

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