a. The diapycnal velocity

We begin with the Boussinesq buoyancy equation,

where b is buoyancy and $\mathcal{B}$ is a turbulent, nonadvective, buoyancy flux per unit area. Throughout most of this article we will consider a diffusive flux,

where κ is a turbulent eddy diffusivity. However, it should be noted that within a fully mixed bottom layer (as used in MF17) the turbulent buoyancy flux is not related to the local buoyancy gradient.

The diapycnal velocity ω (or fluid volume flux across an isopycnal toward lighter densities per unit area of the isopycnal) can be written (Marshall et al. 1999)

Diapycnal upwelling occurs when the turbulent buoyancy flux per unit area is convergent resulting in buoyancy gain. The factor of |∇b|⁻¹ is present because fluid parcels move more rapidly across widely spaced isopycnals given a similar rate of buoyancy gain or loss. A finite value of |∇b| is needed to define the diapycnal velocity as otherwise the location of the given isopycnal across which ω transports volume is ambiguous.

Equation (3) suggests that one would expect diapycnal upwelling to occur within some layer immediately adjacent to the seafloor. To see this, consider a nonzero turbulent diffusivity κ and a stable stratification some distance H away from the boundary. The buoyancy flux $\mathcal{B}$ will have a component that is directed toward the boundary at this distance. The zero normal buoyancy flux condition at the seafloor (or an inward pointing geothermal buoyancy flux) then implies a convergence of the turbulent buoyancy flux and upwelling, ω > 0. This intuition will be put on a firmer footing in section 2c. However, we first present several simple decompositions of Eq. (3).

b. A geometric decomposition of the diapycnal velocity

By writing the diffusive buoyancy flux $\mathcal{B}=\mathcal{B}\widehat{\mathbf{n}}$ in terms of its magnitude $\mathcal{B}$ and the isopycnal normal $\widehat{\mathbf{n}}$ and using the chain rule the diapycnal velocity in Eq. (3) can be decomposed into two terms,

Equation (4) shows that spatial variations in the buoyancy flux magnitude, associated with ${\omega }_{\mathcal{B}}$, are not the only contributor to diapycnal transport. The additional factor ω_g, which for a diffusive flux where $\mathcal{B}=\kappa \hspace{0.17em}\left|\nabla b\right|$ can also be written

arises from isopycnal curvature. The factor ω_g is zero when isopycnals are flat even if the stratification varies. This decomposition has been discussed by Garrett [2001, his Eq. (11)].

The diapycnal velocity can also be decomposed into components associated with variations in the diffusivity and the buoyancy gradient, respectively [substitute Eq. (2) in Eq. (3) and use the chain rule],

Equations (6) and (4) are related through ${\omega }_{\mathcal{B}}={\omega }_{\left|\nabla b\right|}+{\omega }_{\kappa }$, where

is associated with gradients in isopycnal layer thickness. The decomposition in Eq. (6) will prove useful as the first term is largely responsible for downwelling in the SML (where κ decreases with height) while the second term is largely responsible for upwelling in the BBL (where the slope-normal buoyancy gradient must go to zero at the boundary). These decompositions will be discussed in more detail in section 3 using solutions from one-dimensional boundary layer theory.

c. Net diapycnal transport across an isopycnal near the boundary

Returning to the total diapycnal velocity [Eq. (3)], in this section we revisit the MF17 derivation of the net diapycnal transport across an isopycnal near the boundary in order to establish under what conditions net upwelling should be expected. Here we will generalize the MF17 result by considering an arbitrarily curved isopycnal geometry (accounting for any diapycnal transport associated with ω_g and lifting the assumption of a fully mixed BBL). We also include the diapycnal transport term associated with convergences or divergences of the buoyancy flux across isopycnals inside the BBL that was neglected by MF17. For convenience, we will consider a coordinate system aligned with a sloping abyssal boundary (with slope to the horizontal of tanθ) where z is the slope-normal coordinate (with origin at the boundary), y is the upslope direction and x is the along-slope direction (see Fig. 1). The net volume transport across an isopycnal with buoyancy value b within a distance h of the boundary, $\mathcal{E}\left(b,h,t\right)$, can be derived by considering the region between that isopycnal and the isopycnal with buoyancy value b − db [$d\mathcal{R}\left(b,h,t\right)$, gray region in Fig. 1], where db is an infinitesimal buoyancy difference. Integrating Eq. (3) over the isopycnal out to a slope-normal distance z = h away from the boundary,

where $\mathcal{A}\left(b,h,t\right)$ is the area of the isopycnal within a distance h of the boundary and $d\mathcal{S}$ is an area element on that isopycnal. We then make use of the mathematical identity (Marshall et al. 1999)

for any field a(x, t), where $\mathcal{R}\left(b,h,t\right)$ corresponds to the region where buoyancy is less than b below the height h. This identity allows the division by the potentially problematic buoyancy gradient magnitude |∇b| to be replaced in favor of a buoyancy derivative. Applied to Eq. (8) this yields

Using the divergence theorem and the insulating boundary condition for $\mathcal{B}$ at the boundary then yields

where

is the area-integrated turbulent buoyancy flux exiting the infinitesimal isopycnal layer across the height z = h (where the normal vector $\widehat{\mathbf{n}}$ points outward away from the region $\mathcal{R}$). By introducing the (positive) area-integrated turbulent buoyancy flux over the b isopycnal,

and noting that along the dA surface $d\mathcal{S}/db=\hspace{0.17em}dx/b_y$, Eq. (11) can be rewritten as

where the integral in the along-slope x direction is performed along the b isopycnal at the height z = h and ${\mathcal{B}}^z$ is the slope-normal component of the buoyancy flux per unit area. An alternative derivation of Eq. (14), following the same approach as MF17, is provided in appendix A. Equation (14) holds generally for any isopycnal at any height h even in the presence of unsteadiness.

A schematic illustrating the rotated coordinate system used in this article, where z is the slope-normal direction (where the slope is tanθ), y is the upslope direction, and x is the along-slope direction. In the one-dimensional solutions the far-field interior stratification is N². The top of the BBL is defined by the height z = H^BBL at which the diapycnal velocity ω is zero (black dotted line in A). The gray shaded region denotes an infinitesimal region between two isopycnals and within a distance h of the boundary (black dashed line in B) over which the net diapycnal transport is calculated (also see appendix A). This yields a relationship [Eq. (14)] between the diapycnal transport across the isopycnal $\mathcal{E}$, the diffusive buoyancy flux across that isopycnal F, and the diffusive buoyancy flux toward the boundary at the height h, ${\mathcal{B}}^z$, that is valid for an arbitrary isopycnal geometry (C).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

A schematic illustrating the rotated coordinate system used in this article, where z is the slope-normal direction (where the slope is tanθ), y is the upslope direction, and x is the along-slope direction. In the one-dimensional solutions the far-field interior stratification is N². The top of the BBL is defined by the height z = H^BBL at which the diapycnal velocity ω is zero (black dotted line in A). The gray shaded region denotes an infinitesimal region between two isopycnals and within a distance h of the boundary (black dashed line in B) over which the net diapycnal transport is calculated (also see appendix A). This yields a relationship [Eq. (14)] between the diapycnal transport across the isopycnal $\mathcal{E}$, the diffusive buoyancy flux across that isopycnal F, and the diffusive buoyancy flux toward the boundary at the height h, ${\mathcal{B}}^z$, that is valid for an arbitrary isopycnal geometry (C).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

A schematic illustrating the rotated coordinate system used in this article, where z is the slope-normal direction (where the slope is tanθ), y is the upslope direction, and x is the along-slope direction. In the one-dimensional solutions the far-field interior stratification is N². The top of the BBL is defined by the height z = H^BBL at which the diapycnal velocity ω is zero (black dotted line in A). The gray shaded region denotes an infinitesimal region between two isopycnals and within a distance h of the boundary (black dashed line in B) over which the net diapycnal transport is calculated (also see appendix A). This yields a relationship [Eq. (14)] between the diapycnal transport across the isopycnal $\mathcal{E}$, the diffusive buoyancy flux across that isopycnal F, and the diffusive buoyancy flux toward the boundary at the height h, ${\mathcal{B}}^z$, that is valid for an arbitrary isopycnal geometry (C).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

When the turbulent buoyancy flux per unit area $\mathcal{B}$ is directed perpendicular to isopycnals (e.g., a diffusive buoyancy flux), then the first term in Eq. (14) will always be positive if the fluid is stably stratified at the height z = h (i.e., ${\mathcal{B}}^z=-\kappa \hspace{0.17em}{b}_z$ is negative if b_z is positive, while b_y will always be positive at the last point at which the isopycnal crosses the height h before encountering the boundary given a large-scale stable background stratification). Hence, our intuition that net upwelling should occur near the boundary will hold providing that upslope variations in the turbulent buoyancy flux across isopycnals between the height h and the boundary [the second term in Eq. (14)] do not overwhelm the convergence of the turbulent buoyancy flux in the slope-normal direction. This is clearly true for small h where F(b, h, t) is small, while for larger h the rate at which F(b, h, t) must vary with b in the upslope direction to achieve net downwelling turns out to be quite large, as examined in section 5. This second term in Eq. (14) is the abyssal ocean analog of water-mass transformation driven by the convergence of the lateral diffusive buoyancy flux across near vertical isopycnals in the mixed layer (Marshall et al. 1999). To continue further, we examine the diapycnal velocities and transport that arises in simple solutions taken from one-dimensional boundary layer theory.

a. One-dimensional boundary layer theory

We consider the flow near an abyssal boundary with constant slope and far-field interior stratification N² such that

Throughout most of the article we will use a bottom-intensified diffusivity with exponential form,

with far-field value κ_∞, near boundary value κ₀ a decay scale $\widehat{d}$. However, in section 3e we briefly discuss the sensitivity of our results to other choices of the form for the diffusivity in the BBL.

An approximate one-dimensional (uniform in the upslope and along-slope directions) solution to the steady, rotating equations of motion (e.g., Garrett et al. 1993), discussed in more detail in Holmes et al. (2019) and Callies (2018), can be derived assuming that the decay scale $\widehat{d}$ of the diffusivity is much larger than the thickness scale $q_0^{-1}$ of the layer in which friction is important in the constant κ case,

In Eq. (18), S = N² tan²θ/f² is the slope Burger number, f is the Coriolis parameter and Pr_u0 = ν_u0/κ₀ and Pr_υ0 = ν_υ0/κ₀ are the Prandtl numbers near the boundary associated with the viscosity in the along-slope (x) and upslope (y) momentum equations, respectively. The approximate analytic solution is given by (see appendix A of Holmes et al. 2019),

where Ψ(z) is the net upslope transport (per along-slope meter) below the slope-normal height z such that the upslope velocity V = ∂Ψ/∂z. In steady state Ψ(z) is equivalent to the net diapycnal transport below the height z, or indeed the net transport across a horizontal surface between the height z and the boundary (Garrett et al. 1993; Garrett 2001).

As highlighted by Callies (2018), for typical abyssal ocean parameters and order one Prandtl number this solution predicts unrealistically weak stratification throughout the SML where κ is enhanced [Eq. (20)]. This weak stratification is due to the lack of representation of baroclinic-instability-driven restratification. A realistic stratification can be recovered using a large Pr_u0, 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). Here, as we will mainly discuss the flow within the BBL, we will focus on the limit SPr_u0→∞ where eddies maintain the stratification in the SML at its far-field value N². However, in section 3d we briefly discuss the impact of reduced SML stratification through a finite choice for Pr_u0.

Example profiles of the buoyancy gradient [Eq. (20)], the buoyancy flux and the various diapycnal velocity components in the limit SPr_u0→∞ are shown in Fig. 2 for the parameters tanθ = 1/100, N² = 10⁻⁶ s⁻¹, κ_∞ = 10⁻⁵ m² s⁻¹, κ₀ = 10⁻³ m² s⁻¹, $\widehat{d}$ = 200 m, Pr_υ0 = 1, SPr_u0→∞. While these solutions do not capture the full complexity present in more realistic BBLs (e.g., Allen and Newberger 1998; Umlauf and Burchard 2011; Winters 2015; van Haren 2017; Wenegrat et al. 2018; Ruan et al. 2019), they serve as a useful starting point.

Profiles from one-dimensional boundary layer theory. (a) The magnitude of the turbulent buoyancy flux in the z direction κb_z, the magnitude of the turbulent buoyancy flux $\mathcal{B}=\kappa \left|\nabla b\right|$, and the slope-normal buoyancy gradient multiplied by the near-bottom diffusivity κ₀b_z. (b),(c) Various contributions to the total diapycnal velocity ω. Note that the x scale has been split into two sections. The parameters used are a slope of 1/100, N² = 10⁻⁶ s⁻², κ_∞ = 10⁻⁵ m² s⁻¹, κ₀ = 10⁻³ m² s⁻¹, and $\widehat{d}$ = 200 m. The solid black line marks the top of the BBL where ω = 0, and the dashed black line marks the height $z=\pi q_0^{-1}$. Note that the y axis here is z and does not correspond to the distance along an isopycnal, meaning that the relative contribution of different components of the diapycnal velocity to the total diapycnal transport cannot be assessed from these profiles (see Fig. 3).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Profiles from one-dimensional boundary layer theory. (a) The magnitude of the turbulent buoyancy flux in the z direction κb_z, the magnitude of the turbulent buoyancy flux $\mathcal{B}=\kappa \left|\nabla b\right|$, and the slope-normal buoyancy gradient multiplied by the near-bottom diffusivity κ₀b_z. (b),(c) Various contributions to the total diapycnal velocity ω. Note that the x scale has been split into two sections. The parameters used are a slope of 1/100, N² = 10⁻⁶ s⁻², κ_∞ = 10⁻⁵ m² s⁻¹, κ₀ = 10⁻³ m² s⁻¹, and $\widehat{d}$ = 200 m. The solid black line marks the top of the BBL where ω = 0, and the dashed black line marks the height $z=\pi q_0^{-1}$. Note that the y axis here is z and does not correspond to the distance along an isopycnal, meaning that the relative contribution of different components of the diapycnal velocity to the total diapycnal transport cannot be assessed from these profiles (see Fig. 3).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

Profiles from one-dimensional boundary layer theory. (a) The magnitude of the turbulent buoyancy flux in the z direction κb_z, the magnitude of the turbulent buoyancy flux $\mathcal{B}=\kappa \left|\nabla b\right|$, and the slope-normal buoyancy gradient multiplied by the near-bottom diffusivity κ₀b_z. (b),(c) Various contributions to the total diapycnal velocity ω. Note that the x scale has been split into two sections. The parameters used are a slope of 1/100, N² = 10⁻⁶ s⁻², κ_∞ = 10⁻⁵ m² s⁻¹, κ₀ = 10⁻³ m² s⁻¹, and $\widehat{d}$ = 200 m. The solid black line marks the top of the BBL where ω = 0, and the dashed black line marks the height $z=\pi q_0^{-1}$. Note that the y axis here is z and does not correspond to the distance along an isopycnal, meaning that the relative contribution of different components of the diapycnal velocity to the total diapycnal transport cannot be assessed from these profiles (see Fig. 3).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

b. Contributions to diapycnal flow within the BBL

The slope-normal buoyancy gradient from the one-dimensional solutions [Eq. (20) in the limit SPr_u0→∞] shows a maximum at $z=\pi \hspace{0.17em}{q}_0^{-1}$, slightly increased from its far-field value of N², before decaying to zero as the boundary is approached (blue line in Fig. 2a, where the dashed black line marks $z=\pi \hspace{0.17em}{q}_0^{-1}$). The magnitude of the slope-normal component of the buoyancy flux κb_z, whose divergence determines the diapycnal velocity in the one-dimensional case [Eq. (3) yields ω = ∂_z(κb_z)/|∇b|], shows a similar structure with the additional effect of the gradual reduction in κ over its 200-m decay scale (green line in Fig. 2a). The strong convergence of the slope-normal component of the buoyancy flux κb_z drives diapycnal upwelling (blue line in Figs. 2b,c) which transitions to diapycnal downwelling away from the boundary at the top of what we define as the BBL (where ω = 0, solid black line in Fig. 2). This highlights the fact that much of the BBL must be well stratified, as the top of the BBL corresponds to the maximum in the buoyancy flux and therefore cannot have a stratification much reduced from that in the SML.

While the slope-normal component of the diffusive buoyancy flux goes to zero at the boundary there is still a small but nonzero upslope component −κb_y. This means that the magnitude of the buoyancy flux $\mathcal{B}=\kappa \hspace{0.17em}\left|\nabla b\right|$ is not actually zero at the boundary (although it is too small to distinguish from zero, orange dashed line in Fig. 2a). However, in this one-dimensional context the upslope component of the buoyancy flux has no convergence, meaning that ω is zero at the boundary. If upslope variability were permitted, then ω could be nonzero at the boundary (although in a steady solution ω must be zero at the boundary to satisfy a no-slip boundary condition). However, as discussed in more detail in section 5, upwelling would still be expected to dominate in the BBL due to the difference in length scales between along-boundary and slope-normal gradients.

The BBL upwelling is strongly peaked in the lower portion of the BBL where the convergence of the slope-normal buoyancy flux is maximum. This structure is a consequence of the structure in b_z, with ω closely following the term ${\omega }_{{\nabla }^{2}b}$ (orange dashed line in Fig. 2b). The term ${\omega }_{{\nabla }^{2}b}$ is positive below the height $\pi \hspace{0.17em}{q}_0^{-1}$ (dashed black line in Fig. 2). If κ were constant then this height would correspond to the top of the BBL where ω = 0. Instead, with a bottom-intensified diffusivity, ω_κ is negative (green line in Fig. 2b) shifting the top of the BBL (and the height of maximum upslope transport Ψ) toward the boundary. The ω_κ drives weak but extensive downwelling except close to the boundary where the isopycnal normal points along the boundary $\widehat{\mathbf{n}}=\widehat{\mathbf{y}}$ in which direction κ is constant [Eq. (6)]. Therefore, in the one-dimensional solutions (and assuming no reduction in SML stratification), ω_κ captures the “SML downwelling” referred to by MF17 and Holmes et al. (2018, 2019), while ${\omega }_{{\nabla }^{2}b}$ captures the BBL upwelling. However, ${\omega }_{{\nabla }^{2}b}$ also contributes a small amount of downwelling above $z=\pi \hspace{0.17em}{q}_0^{-1}$ because of the slight enhancement of stratification there.

The diapycnal transport can also be split into components arising from variations in the buoyancy gradient magnitude ${\omega }_{\mathcal{B}}$ and isopycnal curvature ω_g (orange dashed and green solid lines, respectively, in Fig. 2c). The buoyancy gradient magnitude ${\omega }_{\mathcal{B}}$ dominates throughout most of the BBL. However, in the lower portion of the BBL where isopycnal curvature is maximum ω_g also contributes. The isopycnal curvature drives upwelling here because isopycnals must curve downward (on average) to encounter the boundary at right angles and thus are concave with respect to the positive buoyancy coordinate. For small slopes, this term is only significant close to the boundary in the lowest part of the BBL as it is only here that the isopycnals depart significantly from the horizontal.

c. Contributions to net upwelling across the BBL

Figure 2c suggests that ω_g may make a contribution to net diapycnal upwelling within the BBL. However, ω is defined as the diapycnal transport per unit isopycnal area, not per unit height z. For the 1D solutions where gradients are constant in y, the net diapycnal transport [Eq. (8)] associated with a particular diapycnal velocity component ω_c within a distance h of the boundary can be written in terms of an integral in the slope-normal coordinate z (also see Garrett 2001),

where ds is a distance along the isopycnal which has been rewritten in terms of the buoyancy gradient magnitude as $ds/dz=\left|\nabla b\right|/b_y$. This metric term converts diapycnal transport per unit area of an isopycnal to diapycnal transport per unit z. For the steady one-dimensional solutions a diapycnal transport per unit z is equivalent to an upslope velocity V_c (i.e., ω_c|∇b| = V_cb_y), or equivalently the diapycnal transport below the height h is equivalent to the upslope transport below the height h [${\mathcal{E}}_c\left(h\right)={\displaystyle \int {\Psi }_c\left(h\right)\hspace{0.17em}dx}$].

For the full diapycnal velocity ω_c = ω [Eq. (3)], Eq. (22) reduces to the expression derived in section 2c for $\mathcal{E}\left(h\right)$ [Eq. (14)], where dF/db = 0 for the one-dimensional solutions. By definition $\mathcal{E}\left(h\right)$ peaks at the top of the BBL where ω = 0. Much of this BBL upwelling is associated with flow in the lower portion of the BBL (Fig. 2c). In fact, in the case of a constant diffusivity it can be shown that the bottom π⁻¹ of the BBL hosts half of the net upwelling (see appendix B).

Integrating through the SML into the far-field yields the net diapycnal transport,

This transport, associated with the far-field diffusivity, is small in this one-dimensional context. Similar expressions can be derived for the net diapycnal transports due to the individual contributions to ${\mathcal{E}}^{\infty }$ (see appendix C). In the limit $q_0^{-1}\ll \widehat{d}$ these reduce to

with ${\mathcal{E}}_{\mathcal{B}}^{\infty }={\mathcal{E}}^{\infty }-{\mathcal{E}}_g^{\infty }$. As ${\mathcal{E}}_{{\nabla }^{2}b}^{\infty }$ corresponds to the total BBL upwelling in the one-dimensional solution, the “amplification factor” quantifying the ratio of BBL upwelling to total upwelling in this solution is given by

This can be large, but only because the net transport in the one-dimensional solution is small.

More interesting is the fraction of BBL upwelling associated with isopycnal curvature,

This function is plotted in Fig. 3a as a function of the slope tanθ. Counterintuitively, the contribution of isopycnal curvature to BBL upwelling increases as the slope increases even though the isopycnal slope experiences less change approaching a steep slope than a shallow slope. This is because the transport due to isopycnal curvature is relatively constant at slopes less than 1/10 [arctan(x⁻¹) does not change much for small x], while the BBL transport ${\mathcal{E}}_{{\nabla }^{2}b}^{\infty }$ decreases significantly as the slope increases [Eq. (25)]. Profiles of the contributions to net BBL upwelling (ω_g and ${\omega }_{\mathcal{B}}$, which sum to give ${\omega }_{{\nabla }^{2}b}$ in the limit $q_0^{-1}\ll \widehat{d}$) which include the metric term |∇b|/b_y indeed show that the BBL transport decreases as the slope increases, while the isopycnal curvature term remains relatively constant (Fig. 3b). Thus, for small slopes the geometry term does not make a significant contribution to the total diapycnal transport. However, it may be significant or even dominant for steep slopes, such as the sides of fracture zone canyons thought to host much of the mixing that converts Antarctic bottom water back to lighter waters (Thurnherr et al. 2020). For example, the northern wall of the fracture zone canyon shown in Fig. 5 of Thurnherr et al. (2020) has a very steep slope of roughly 1/2. For this slope the isopycnal curvature term accounts for more than 70% of the diapycnal upwelling in the BBL (Fig. 3a). This is because for such steep slopes the overall BBL transport, associated with the convergence of the slope-normal buoyancy flux, is much weaker. In terms of the decomposition of ω into ${\omega }_{\mathcal{B}}$ and ω_g in Eq. (4), the magnitude of the buoyancy flux at the boundary (associated with only the along-boundary component) is not significantly less than that in the interior, meaning that ${\omega }_{\mathcal{B}}$ (through $\nabla \mathcal{B}$) is much weaker (green dashed line in Fig. 3b).

Structure and properties of the isopycnal curvature contribution to diapycnal transport ω_g in the one-dimensional boundary layer solutions. (a) The fraction of diapycnal upwelling in the BBL due to isopycnal curvature ${\mathcal{E}}_g^{\infty }/{\mathcal{E}}_{{\nabla }^{2}b}^{\infty }=\tan \theta \arctan \left(\cot \theta \right)$ as a function of boundary slope tanθ. (b) The upslope velocity, or diapycnal transport per unit x distance along-slope and per unit z (in contrast to Fig. 2 where the diapycnal velocity is per unit area of an isopycnal) for slopes of tanθ = 1/100 (orange lines), tanθ = 1/20 (blue lines), and tanθ = 1/2 (green lines) associated with isopycnal curvature ω_g (solid lines) and the buoyancy flux magnitude ${\omega }_{\mathcal{B}}$ (dashed lines). The top of the BBL is shown in dotted lines.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

Structure and properties of the isopycnal curvature contribution to diapycnal transport ω_g in the one-dimensional boundary layer solutions. (a) The fraction of diapycnal upwelling in the BBL due to isopycnal curvature ${\mathcal{E}}_g^{\infty }/{\mathcal{E}}_{{\nabla }^{2}b}^{\infty }=\tan \theta \arctan \left(\cot \theta \right)$ as a function of boundary slope tanθ. (b) The upslope velocity, or diapycnal transport per unit x distance along-slope and per unit z (in contrast to Fig. 2 where the diapycnal velocity is per unit area of an isopycnal) for slopes of tanθ = 1/100 (orange lines), tanθ = 1/20 (blue lines), and tanθ = 1/2 (green lines) associated with isopycnal curvature ω_g (solid lines) and the buoyancy flux magnitude ${\omega }_{\mathcal{B}}$ (dashed lines). The top of the BBL is shown in dotted lines.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

Structure and properties of the isopycnal curvature contribution to diapycnal transport ω_g in the one-dimensional boundary layer solutions. (a) The fraction of diapycnal upwelling in the BBL due to isopycnal curvature ${\mathcal{E}}_g^{\infty }/{\mathcal{E}}_{{\nabla }^{2}b}^{\infty }=\tan \theta \arctan \left(\cot \theta \right)$ as a function of boundary slope tanθ. (b) The upslope velocity, or diapycnal transport per unit x distance along-slope and per unit z (in contrast to Fig. 2 where the diapycnal velocity is per unit area of an isopycnal) for slopes of tanθ = 1/100 (orange lines), tanθ = 1/20 (blue lines), and tanθ = 1/2 (green lines) associated with isopycnal curvature ω_g (solid lines) and the buoyancy flux magnitude ${\omega }_{\mathcal{B}}$ (dashed lines). The top of the BBL is shown in dotted lines.

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

d. The impact of reduced SML stratification

The solutions discussed above assume that isopycnals remain flat in the SML. However, observations from the Brazil Basin instead show that the stratification in the SML is reduced by a factor ~3 relative to the far-field (Callies 2018). To evaluate the impact of reduced SML stratification and sloping isopycnals on the diapycnal transport in the BBL we therefore compare a solution with the choice SPr_u0 = 1/2 to the limit SPr_u0→∞. Such a solution has a factor of 3 reduction in the slope-normal buoyancy gradient in the SML [Eq. (20)] along with a reduced BBL thickness by a factor ~3^−1/4 ≈ 0.76 [Eq. (18), Fig. 4a]. The net diapycnal or upslope transport within the BBL is reduced by a similar factor of 3 [Eq. (19), the area under the curve of Fig. 4b]. Throughout most of the BBL the diapycnal velocity and its various components show little change as the reduction in the convergence of the buoyancy flux associated with the reduction in b_z is compensated by the factor |∇b|⁻¹ in Eq. (3) (Fig. 4c). However, in the bottom ~5 m the diapycnal velocity (and its contributions from both ω_g and ${\omega }_{\mathcal{B}}$, dotted and dashed lines in Fig. 4c) does show a reduction due to the fact that the upslope component of the buoyancy gradient b_y, which does not depend on SPr_u0, makes a significant contribution to |∇b| [which itself is in the denominator of Eq. (3)]. We conclude that while eddy-driven restratification, associated in the one-dimensional solutions with the parameter Pr_u0, is an important process that influences the net diapycnal transport in the BBL and SML, the BBL thickness and the slope of isopycnals in the SML, with respect to the details of the flow in the BBL it acts largely as an overall scaling.

A comparison of the (a) slope-normal buoyancy gradient b_z, (b) upslope velocity, and (c) diapycnal velocity between solutions in the limit SPr_u0→∞ (as used elsewhere in the article, blue) and with a Prandtl number chosen such that the SML stratification is reduced by a factor of 3. Panel (c) (note the reduced vertical scale to focus on the near boundary region) also shows the components of the diapycnal velocity arising from changes in isopycnal curvature (dashed lines) and buoyancy flux magnitude (dotted lines).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

A comparison of the (a) slope-normal buoyancy gradient b_z, (b) upslope velocity, and (c) diapycnal velocity between solutions in the limit SPr_u0→∞ (as used elsewhere in the article, blue) and with a Prandtl number chosen such that the SML stratification is reduced by a factor of 3. Panel (c) (note the reduced vertical scale to focus on the near boundary region) also shows the components of the diapycnal velocity arising from changes in isopycnal curvature (dashed lines) and buoyancy flux magnitude (dotted lines).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

A comparison of the (a) slope-normal buoyancy gradient b_z, (b) upslope velocity, and (c) diapycnal velocity between solutions in the limit SPr_u0→∞ (as used elsewhere in the article, blue) and with a Prandtl number chosen such that the SML stratification is reduced by a factor of 3. Panel (c) (note the reduced vertical scale to focus on the near boundary region) also shows the components of the diapycnal velocity arising from changes in isopycnal curvature (dashed lines) and buoyancy flux magnitude (dotted lines).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

e. The impact of the BBL diffusivity structure

The one-dimensional boundary layer solutions discussed above assume a specific profile for the turbulent diffusivity that is largely constant within the BBL [Eq. (17), which will be referred to as κ_exp in this section, blue line in Fig. 5a]. In contrast, frictional boundary layer theory suggests that the turbulent diffusivity and viscosity should reduce linearly toward zero within a frictional log layer of thickness estimated as ranging from a few centimeters to a few meters (e.g., Wüest and Lorke 2003; Holtappels and Lorke 2011). In this section we examine the sensitivity of our results to an alternative choice for the turbulent diffusivity profile in the BBL where the diffusivity decays toward zero within the bottom ~2 m of the BBL,

where κ_m = 0.95 and z_m = 1 m (${\kappa }_{\text{dec}1}$, orange line in Fig. 5). For completeness, we also consider a case with z_m = 10 m where the diffusivity decays over a length scale more comparable to the BBL thickness (κ_dec10, green line in Fig. 5). Equation (29) does not satisfy the criteria needed for the approximate analytic solution [Eqs. (19), (20)] and thus we must resort to numerical solutions of the boundary layer equations. Following Callies (2018) we solve the one-dimensional steady-state boundary layer equations numerically by projecting them onto 2048 Chebyshev polynomials using the software package Dedalus (Burns et al. 2020). In the exponential diffusivity case we note that the numerical solution closely matches the approximate analytic solution used elsewhere in the article (cf. blue solid and dot–dashed lines in Figs. 5b,c). Parameters used here are identical to those used in Fig. 2.

A comparison of the BBL structure between 1D boundary layer solutions using two forms for the diffusivity in the boundary layer, an exponential decay (blue lines as used elsewhere in the article), and a simple representation of a frictional log layer where the diffusivity decays toward zero within either ~2 m (orange lines) or ~20 m (green lines) of the boundary [see Eq. (29) where z_m = 1 m for κ_dec1 or z_m = 10 m for κ_dec10]. Otherwise parameters are the same as used in Fig. 2. Shown are the (a) diffusivity, (b) slope-normal buoyancy gradient b_z, (c) upslope velocity, and (d),(e) diapycnal velocity and its components from numerical solutions to the one-dimensional boundary layer equations [note the differing vertical scale for (d) and (e)]. The thin dotted lines in (a)–(c) mark the top of the BBL. The approximate analytic solution in the exponential diffusivity case used elsewhere in the article is also shown in (b) and (c) (dot–dashed lines, see section 3a).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

View Full Size

A comparison of the BBL structure between 1D boundary layer solutions using two forms for the diffusivity in the boundary layer, an exponential decay (blue lines as used elsewhere in the article), and a simple representation of a frictional log layer where the diffusivity decays toward zero within either ~2 m (orange lines) or ~20 m (green lines) of the boundary [see Eq. (29) where z_m = 1 m for κ_dec1 or z_m = 10 m for κ_dec10]. Otherwise parameters are the same as used in Fig. 2. Shown are the (a) diffusivity, (b) slope-normal buoyancy gradient b_z, (c) upslope velocity, and (d),(e) diapycnal velocity and its components from numerical solutions to the one-dimensional boundary layer equations [note the differing vertical scale for (d) and (e)]. The thin dotted lines in (a)–(c) mark the top of the BBL. The approximate analytic solution in the exponential diffusivity case used elsewhere in the article is also shown in (b) and (c) (dot–dashed lines, see section 3a).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

A comparison of the BBL structure between 1D boundary layer solutions using two forms for the diffusivity in the boundary layer, an exponential decay (blue lines as used elsewhere in the article), and a simple representation of a frictional log layer where the diffusivity decays toward zero within either ~2 m (orange lines) or ~20 m (green lines) of the boundary [see Eq. (29) where z_m = 1 m for κ_dec1 or z_m = 10 m for κ_dec10]. Otherwise parameters are the same as used in Fig. 2. Shown are the (a) diffusivity, (b) slope-normal buoyancy gradient b_z, (c) upslope velocity, and (d),(e) diapycnal velocity and its components from numerical solutions to the one-dimensional boundary layer equations [note the differing vertical scale for (d) and (e)]. The thin dotted lines in (a)–(c) mark the top of the BBL. The approximate analytic solution in the exponential diffusivity case used elsewhere in the article is also shown in (b) and (c) (dot–dashed lines, see section 3a).

Citation: Journal of Physical Oceanography 50, 11; 10.1175/JPO-D-20-0066.1

• Download Figure
• Download figure as PowerPoint slide

The slope-normal buoyancy gradient is slightly enhanced for the κ_dec1 case throughout the BBL, which results in a shift in the upslope transport profile toward the boundary and a slight reduction in the BBL depth (cf. orange and blue lines in Figs. 5b,c). Upwelling transport is strongly enhanced in the bottom ~2 m due to the convergence of the slope-normal buoyancy flux associated with the reduction in diffusivity. This has a knock-on effect throughout the rest of the BBL as the total upwelling transport across the BBL must remain the same. While overall these changes across the majority of the BBL are not large, the differences in the diffusivity near the boundary have a more dramatic influence on the diapycnal velocity, and its various components, within the bottom 5 m (Figs. 5d,e, note the focused vertical scale). The larger near-boundary slope-normal buoyancy gradient shifts the diapycnal velocity profile peak to within half a meter of the boundary (cf. solid orange and blue lines in Fig. 5d) and significantly reduces the contribution of the isopycnal curvature term ω_g (cf. solid orange and blue lines in Fig. 5e). The diapycnal velocity associated with changes in the buoyancy flux magnitude ${\omega }_{\mathcal{B}}$ now dominates the total diapycnal velocity (cf. orange dotted and solid lines in Fig. 5d). However, the convergence of the buoyancy flux associated with the decay in the diffusivity in the bottom ~2 m now contributes significantly to the upward diapycnal velocity (ω_κ, orange dotted line in Fig. 5e), in contrast to the κ_exp case where ω_κ drove a very weak downward diapycnal velocity throughout the BBL (e.g., green line in Fig. 2b). Increasing the length scale over which the diffusivity decays to a significant fraction of the BBL thickness results in a further increase in the BBL stratification, shifts the upslope velocity profile closer to the boundary and further decreases the magnitude of the diapycnal velocity (cf. green and orange lines in Fig. 5).