Buoyancy Forcing by Turbulence above Rough Topography in the Abyssal Brazil Basin

a. Spatial variability

The spatial patterns of enhanced dissipation apparent in Fig. 2 suggest that variability of dissipation is associated with the different sloping regions that characterize the crest–canyon system of the FZs. This also suggests that height above bottom (h_ab) may serve as a better coordinate than depth for examining the spatial distribution of ϵ. This approach was adopted, and the available profiles of ϵ were classified according to the meridional slope of the local bathymetry Z_b(x, y). HRP stations were categorized as being FZ crests (∂_yZ_b ≅ 0, ∂_yyZ_b < 0), canyon floors (∂_yZ_b ≅ 0, ∂_yyZ_b > 0), or sloping regions (∂_yZ_b ≠ 0). The profile data, classified in this manner, are shown as functions of h_ab in Fig. 3. Each ensemble profile results from data that has been vertically averaged into 100-m bins. Auto-lag-correlation analysis of ϵ profiles shows zero crossings at 20-m vertical separations, so each 100-m bin contains five degrees of freedom. The 95% confidence interval of each bin was calculated using the “boot strap” Monte Carlo method (Efron and Gong 1983). Dissipation decreases with height in each of the three classes, and all profiles reach background levels of dissipation, ϵ ≅ 3 × 10⁻¹⁰ W kg⁻¹, at heights above bottom greater than 1000 m. The slope profile is notably different from the other two, in both the magnitude of the maximum dissipation and the scale of decay. The slope profile has a bottom value of ϵ ≅ (3–9) × 10⁻⁹ W kg⁻¹ and decays with an e-fold scale of (150 ± 50) m. The crest and canyon profiles peak at ϵ ≅ (2–5) × 10⁻⁹ W kg⁻¹ and decay over a larger scale, (500 ± 100) m.

b. Temporal variability

In addition to this spatial structure, temporal variability also exists in the dissipation data. Daily averages of dissipation data roughly modulate with the spring–neap cycle of tidal currents. This correlation is most evident when the dissipation profiles are vertically integrated, and a comparison between integrated ϵ and tidal current speed is made using estimates of barotropic tidal velocity produced by the TPXO model of Egbert et al. (1994) (Fig. 4). For the time intervals where dissipation data were available, the TPXO model was used to produce velocity estimates at the latitude and longitude appropriate for each HRP station.

In both 1996 and 1997 comparisons, profiles of ρϵ were integrated up from the local bottom to h_ab = 2000 m, and all profiles were time binned in 24-h intervals and averaged. The modulation of integrated dissipation follows the phase of the spring–neap tidal cycle closely, and a one day lead of the tidal signal is consistent within the statistical spread of the data. While the correlation between the two records is quite low for the 1996 data, near 0.3, the correlation coefficient for the 1997 data is 0.7. Furthermore, this comparison uses only the tidal record reconstructed along the survey track, though the dissipation is possibly influenced by tidal beams that have radiated from other sites. We regard the temporal fluctuations in the dissipation record as suggestive of tidal forcing, and explore this possibility further in section 3b.

a. Model equations

We focus here on the ocean abyss and make the a priori assumption that the diabatic terms in the governing equations are significant. All dynamical equations will be cast in terms of flow along and across isopycnals. The dynamically relevant density variable for this analysis is neutral density (γ_n: Jackett and McDougall 1997): the neutral density coordinates are everywhere parallel to the locally defined potential density. Preliminary examination of multiple-year hydrographic data revealed that the temporal variation of the isopycnal-height field was negligible (Ledwell et al. 2000). Therefore, we will assume that the density field can be regarded as steady over the 14-month period spanned by the observations.

The flow regime is divided into a series of density layers, each characterized by a layer thickness h(x, y). In these layers, geostrophic advection is related to isobaric gradients of geopotential anomaly (φ) by

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where u₀ is a reference level velocity. Thus, the advection of potential vorticity through a density layer, as previously stated in (2), may be expressed in the form

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where (u_r, υ_r) are the components of relative geostrophic velocity given by (u − u₀, υ − υ₀) in (6). Thus, the problem of determining the lateral flow u is reduced to determining the reference level velocity, that is, the vertical integration constants (u₀, υ₀) of a thermal-wind balance.

In layers where flow encounters topography, the use of geostrophic balance is questionable. A modified momentum balance, such as that of the bottom Ekman layer, may be called for in regions where a density layer comes within h_ab ≃ 100 m of the bottom. Additionally, the difficulty of defining the geopotential anomaly along pressure surfaces interrupted by bathymetry severely limits the practical use of (6) and (7) in deep layers. For these reasons, it is necessary to employ additional dynamical constraints on the flow. This is achieved through the use of integrated advective budgets for mass,

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and for potential temperature

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The integrated expressions (8) and (9) are bounded in a control volume of zonal and meridional extent (γ, Δ), and vertically bounded in a density layer by the surfaces z_u(x, y) and z_l(x, y). In the temperature advection equation, Θ′ is the temperature anomaly relative to the mean temperature in each cell, and κ and k_ρ are the respective isopycnal and diapycnal diffusivities. Since (8) and (9) assume nothing about the nature of the momentum balance, their applicability is not limited to conditions of geostrophic balance. However, without any model for the momentum balance, the components of the lateral flow in (8) and (9) must be treated as unknowns at all depths, in contrast to (7), which relates the velocity at all depths to the single component pair (u₀, υ₀).

The diffusivity for isopycnal mixing, κ, is treated as an unknown parameter in the inverse model. Relations for the diapycnal diabatic terms are expressed using the dissipation rate with the use of (4) and (5). For use in the inverse model, the mixing efficiency parameter Γ is treated as constant, Γ₀ = 0.20 ± 0.04. This allows us to specify the diapycnal diffusivity for turbulence as

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and the diapycnal advection as

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We emphasize that (10) and (11) are approximations, where a constant mixing efficiency has been assumed. While oceanic observations of turbulence from well-stratified regions suggest Γ₀ ≅ 0.2 (Moum 1996; St. Laurent and Schmitt 1999), observations from fjords suggest Γ ≅ 0.05 (Stigebrandt and Aure 1989). Additionally, laboratory measurements of turbulence suggest 0.05 < Γ < 0.25 characterizes the variation of mixing efficiency over a wide range of Reynolds, Richardson, and Froude numbers (Ivey and Imberger 1991; Huq and Britter 1995). In our inverse model estimates, we accept that using Γ₀ = 0.20 ± 0.04 may lead to a specification of k_ρ that is biased high. The use of a constant mixing efficiency in (11) is more problematic, as variations in Γ with depth will not only influence the magnitude of the w∗ estimate, but also the sign (w∗ > 0 for upwelling, w∗ < 0 for downwelling). For this reason, w∗ is regarded as an unknown in the inverse model, and (11) serves as a “constraint” in the inversion. In this manner, (11) provides an a priori estimate, while full inversion of (7), (8), and (9) determines the best estimate of w∗. The use of constraints in inverse models is discussed by Menke (1989) and Wunsch (1996).

In expressions (10) and (11), ϵ is meant to denote some suitably averaged function of the dissipation data. Averaging over multiple profiles is necessary to achieve statistical stability in the mean dissipation estimates. Since our inverse model utilizes the steady-state forms of the advective budgets, we seek an averaging procedure that yields an estimate of the time-mean dissipation rate. Additionally, the averaging procedure should preserve the dependence of ϵ on bathymetric class, as previously described in section 2. We now discuss such a model for the time-mean dissipation.

b. Model for the turbulent dissipation

Spatial and temporal variations in the observed dissipation data were described in sections 2a and 2b. These variations are not random, and simple arithmetic averaging may alias modes of variability in a manner that will produce biased estimates of mean dissipation. Estimates of spatially averaged dissipation are sensitive to the layout of bathymetric crests, slopes, and canyons in that ϵ levels above slopes can exceed levels above crests and canyon floors. Similarly, estimates of time-averaged dissipation are sensitive to the phase of the spring–neap tidal cycle, as ϵ levels are maximum near spring tide and minimal near neap tide. Due to the logistics of the tracer component of the experiment, canyon sites were favored during data collection, and three of five weeks of sampling occurred during neap tides. Thus, simple averages of the observed dissipation data are biased low relative to the true temporal and spatial mean. We therefore rely on simple modeling to recover an unbiased estimate of mean dissipation.

First, we consider the fortnightly modulation of the dissipation with the barotropic tidal current. Traditionally, frictional boundary-layer dynamics are assumed as the mechanism for tidally driven dissipation. The energy dissipated in the frictional boundary layer is estimated as ϵ = ρ₀c_d|u|u², where c_d ≃ 0.0025 is the drag coefficient and u is the barotropic velocity of flow. Taking u = 0.02 m s⁻¹, the tidal dissipation in the frictional boundary layer is W_f ≃ 0.02 mW m⁻². This is two orders of magnitude smaller than the integrated dissipation occurring between the bottom and a height of 2000 m. It is further noted that HRP profiles seldom extend close enough to the bottom to sample the frictional boundary layer, which is typically 1 to 10 m thick.

In contrast to bottom friction, the internal waves generated by tidal flow over rough bathymetry radiate energy upward, away from the bottom. Instabilities in the internal wave field act to transfer some of this energy to the small scales where it is dissipated by viscosity. Thus, internal waves provide an additional, though indirect, mechanism by which tidal energy is dissipated. The internal wave generation problem of flow over bottom bathymetry is well known, and a general discussion of steady flow “lee wave” generation is given by Gill (1982). Internal waves produced by tidal flow are termed “internal tides,” and unlike lee waves, internal tides have intrinsic frequencies equal to those of the barotropic tides. Internal tide generation models are discussed by Baines (1982), who shows that the energy flux associated with the internal tide radiated from linear slope topography scales as E_f ≃ Kρ₀Nu²h², where N is the buoyancy frequency, u is the barotropic tidal velocity, h is the depth, and K is a parameter that depends on the height of the topography relative to the water depth and the geometry of the slope. This scaling is valid for linear slopes of arbitrary steepness, though the particular value of K depends on whether the slope is supercritical (slope steeper than the beam of radiated waves) or subcritical (slope less steep than waves). For the case of subcritical topography, Bell (1975) shows that the energy flux for topography of arbitrary shape scales as E_f ≃ kρNu²d², where (k, d) are the wavenumber and height of a topographic obstacle. Using this scaling as a lower bound, an energy flux of E_f ≃ 1–10 mW m⁻² is likely to characterize the internal tide generated across the FZs. This value for energy flux is the same scale as the observed integrated dissipation values to a height of 2000 m above the topography.

Assuming that the energy available for dissipation scales as the energy flux of the internal tide, a model for the dissipation was developed in several steps. First, the data were temporally dealiased assuming that ϵ ≃ αu(t)², where u(t)² is the squared barotropic tidal speed that has been smoothed with a one day running average. The long-term average for u² was computed over the 14-month period between the 1996 and 1997 surveys using tidal velocity estimates at the central site (21°S, 18.5°W). The resulting temporally dealiased dissipation data are a function of spatial variables only. The second procedure in the modeling was to simplify the spatial dependence. This was done by grouping the data into the three bathymetric classes: FZ crests, canyons, and the slopes between them. The resulting model for dissipation is dependent only on h_ab(x),

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where angle brackets represent the average over an ensemble of profiles associated with a bathymetric class.

c. Treatment of data

The dynamics described in section 3a can be employed in an inverse model where the observed density field (h(x, y), N²), geopotential-anomaly field φ(x, y, p), temperature field Θ(x, y, z), and modeled dissipation ϵ(h_ab(x)) are used to solve for the unknowns (u, w∗, κ). The geometry of the inversion grid needs to accommodate both the finite element nature of the beta-spiral vorticity balance (7) and control-volume cells for (8) and (9). This is done using a staggered grid of nested control-volume cells. For each density layer, cells are assigned to serve as control volumes with the integrated components of lateral velocity constrained on each of the lateral walls and diapycnal velocities constrained on the upper and lower bounding density interfaces.

The 1997 HRP survey data provided the hydrographic fields. The temperature and salinity data were first used to calculate isobaric maps of geopotential anomaly relative to 2000 dbar. These fields were modeled using a polynomial fit over the survey domain with the form φ(X, Y) = ϕ₀ + ϕ_xX + ϕ_yY + ϕ_xxX² + ϕ_xyXY + ϕ_yyY², where (X, Y) are Cartesian coordinates and ϕ_( · ) are the coefficients of the fit. The second-order polynomial fit was found to filter tidal displacements in the density field (St. Laurent 1999). The gradient of the fit is used with (6) to give estimates of the geostrophic flow relative to 2000 m. This reference level was chosen based on ideas about basin-scale water masses, with 2000 m being roughly between northward flowing intermediate water and southward flowing North Atlantic Deep Water (NADW). The hydrographic data were sorted into layers of neutral density, and polynomial fits were used to give quasi-analytic expressions for Θ(x, y), Θ_z(x, y), h(x, y), and N²(x, y) in each layer. This layered system comprises the static stratification field, which is used for the inversion. Vertical derivatives for N² and Θ_z and their associated uncertainty were calculated using the slopes from linear fits to the data. For each density layer, the linear fits were computed about the level of the bounding isopycnal over a depth interval comparable to the density layer thickness. Assignments of geostrophic shear for each layer were made through explicit reference to the isobaric surfaces used in the geopotential-anomaly calculation. Thus, the inverse domain is a layered system of hydrographic estimates on a staggered grid of nested cells. Various domain dimensions were explored, but the calculations discussed here cover 19°–25°S, 14°–22°W and six density layers, with γ_n = 27.6 kg m⁻³ (density units are suppressed hereafter) serving as the uppermost interface and the seafloor bathymetry serving as the lower boundary of the inverse domain. Each cell of the staggered grid has Mercator dimensions of (0.5°)², and the altimetry-derived bathymetry data of Smith and Sandwell (1997) were subsampled onto this grid. The modeled dissipation data (12) were incorporated into the inverse domain by identifying each estimation point as a bathymetric crest, slope, or canyon. This was done through a classification scheme that utilizes both the first and second meridional derivatives of bathymetry, which were computed at two arc minute resolution.

Residuals between the fitted and observed density field are typically 20–50 m, assumed to be the result of tidal heaving, and an assessment of hydrographic noise was done using Monte Carlo methods. In particular, the influence of 50-m amplitude vertical displacements of the isopycnal-height field arising from tidal frequency motions was examined. To accomplish this, the observed isopycnal-height field was perturbed by a random field of Gaussian noise with 50-m standard deviation. The perturbed density field was then treated with the polynomial smoothing discussed above, and this smoothed but perturbed hydrography was used for the inversion. In this manner, 10 alternate realizations of the density field were produced, and an ensemble of inverse solutions was calculated.

Since the survey data were concentrated above several FZs along an irregular grid (Fig. 1), some regions of the inverse domain correspond to regions that were sparsely sampled. There is no survey data in the northeast and southeast corners of the 19°–25°S, 14°–22°W inverse domain, so the mapped properties in these regions are poorly constrained. The inverse estimates in these sparsely sampled regions will receive minimal attention.

d. Linear algebraic equations

The fitted hydrographic fields and the dissipation data were used to assemble a linear system of coupled equations of the form E_ijx_j + n_i = d_i, where E_ij is the coefficient matrix of the terms in (7), (8), and (9) involving the unknowns x_j = [u₀, υ₀, w∗, κ]^T, plus the estimates w∗ ≅ N⁻²Γ₀∂ϵ/∂z, which act as constraints. The vector d_i is the vector of observable terms, and n_i explicitly denotes the residuals of the inverse model fit. This linear system is solvable by least square methods that generally minimize the residuals of the model fit. A variety of inversion procedures are discussed by Wunsch (1996), and the Gauss–Markov estimator was chosen for the calculations needed here. This form of inversion produces an estimate of x_j that utilizes a priori knowledge of the statistics for x_j and n_i, which is done through a specification of covariance matrices. The covariance matrix of unknowns C_jj is taken to be the diagonal matrix composed of the variance quantities. A priori variance levels for the lateral flow and the lateral diffusivity were taken as 〈(u − u)²〉 ≃ (0.01 m s⁻¹)² and 〈(κ − κ)²〉 ≃ (100 m² s⁻¹)². The assumed scale for the lateral flow is justified by float observations (Hogg and Owens 1999). The scale for the lateral diffusivity was derived from a mixing length estimate. Using an eddy kinetic energy of 1 cm² s⁻² (Hogg and Owens 1999) and a deformation radius length scale of order 10 km suggests u′l′ ≃ 100 m² s⁻¹. The variance levels for the diapycnal advection came directly from the statistics of the dissipation constraint (5). Along density surfaces well above topography where mixing rates are weak, a typical variance level for diapycnal advection was 〈(w∗ − w∗)²〉 ≃ (1 m yr⁻¹)². Along density surfaces closer to topography, variance levels were 〈(w∗ − w∗)²〉 ≃ (10–30 m yr⁻¹)². A diagonal covariance matrix for the model residuals R_ii was constructed using an uncertainty analysis of the elements of d_i. For example, the scale for the residuals of the beta-spiral equations involves the error propagation of terms like (δh)², (δh_x)², and (δh_y)² in (7). The δ( · ) terms are calculated using the standard errors of the mapped hydrography, where the expressions for error propagation were derived using standard techniques (Bevington and Robinson 1992). Typical signal to noise ratios of the observed terms of the geotrophic model equations are (|d_i|/δd_i) ≃ 1/5 for the beta-spiral and heat equations and (|d_i|/δd_i) ≃ 1 for the mass and diapycnal advection constraints. The statistical uncertainties assigned to the ageostrophic advective budgets are 0.01 × 10⁶ m³ s⁻¹ and 1 × 10² m³ °C s⁻¹ for the mass and heat.

In addition to the statistical scalings provided by the two covariance matrices, several nonstatistical scalings are used to improve the rank of the inversion. The linear model is nondimensionalized by column scaling with a diagonal matrix of dimensional scales D_jj so that all elements of x_j are O(1). Additionally, the rows of E_ij are scaled by the equation norms N_ii and the thermal advection equations are row scaled by the temperature contrasts T_ii of each control-volume cell. The later scaling accounts for the general decrease in temperature contrast as layers get deeper and is discussed by Zhang and Hogg (1992). Thus, the Gauss–Markov estimate for the nondimensional variables x^′_j and the associated uncertainty δx^′_j are given by

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where the scaled matrices are E^′_ij = N⁻¹_iiT⁻¹_iiE_ijD_jj, C^′_jj = D⁻²_jjC_jj, R^′_ii = N⁻²_iiT⁻²_iiR_ii, and d^′_i = N⁻¹_iiT⁻¹_iid_i.

a. Shallow flow

Diapycnal mixing in the layers shallower than 3000 m was weak and diffusivity estimates were k_ρ = (Γ/N²)ϵ < 1 × 10⁻⁴ m² s⁻¹. Consequently, the diapycnal advection through these layers was also weak, with |w∗| < 1 m yr⁻¹. Diabatic forcing of this magnitude plays a rather insignificant role in the determination of the lateral flow, as the diabatic term in (7) is typically less than 10% of the adiabatic stretching term represented by u · (∇h/h). Therefore, the weak diabatic flow in these shallow layers is not discussed further.

Within the domain treated by the inversion, the bottom topography does not shoal less than 3000 m with the exception of a seamount in the southeast corner of the region. Thus, geostrophic dynamics were assumed valid everywhere in layers shallower than 3 km, and the combined use of the beta spiral with mass and temperature advection yielded an overdetermined system for the barotropic flow at each grid point. The estimated geostrophic flow fields for levels shallower than 3000 m, including the reference level, are shown in Fig. 5. The largest current speeds in these layers are 2 mm s⁻¹, and the mean speeds are around 0.5 mm s⁻¹. The flows in these layers are generally consistent with ideas about water mass circulation. Above and below a level of very weak flow (γ_n = 27.83, z ≅ 2200 m), the intermediate water layer (γ_n = 28.01, z ≅ 1400 m) flows in a northward direction, while the NADW layer (γ_n = 28.06, z ≅ 2900 m) flows in a southward direction. The local nature of these flow domains makes detailed comparisons with previous results difficult, as most past work has considered basin scales. Durrieu De Madron and Weatherly (1994) describe the Brazil Basin circulation of NADW and AABW water masses, but their observations do not cover the region considered in our study. Reid (1989) estimated net northward flow at z = 2000 m near 20°S, 20°W in his study of large-scale South Atlantic circulation. Also, Speer et al. (1995) describe a section at 22°S, 15°W with net eastward transport in the depth range between 1300 and 3200 m as part of the Namib Col Current, a feature not apparent in our data. However, the flows found here have qualitative similarity to flow estimates discussed by Zhang and Hogg (1992).

In Fig. 5, an ensemble of 10 velocity estimates is plotted at each grid point. The spread of vectors at each point is the result of uncertainty in the determination of the reference-level flow due to ±50 m noise in the isopycnal-height field. In many cases, the estimated variations are small and not apparent in the figures. The flow estimates are most sensitive in the northeast and southeast corners of the domain, where no stations were occupied. This sensitivity amounts to an uncertainty in flow direction not greater than π/12 radians. Statistical uncertainty in the magnitude of the flow field is typically 0.1–0.2 mm s⁻¹, as assessed from standard error propagation techniques used with the polynomial fits to the geopotential-anomaly fields.

b. Deep flow

In the layers deeper than 3000 m, bottom bathymetry has a clear influence on the flow dynamics. In particular, proximity to the bottom is the controlling factor for the magnitude of the diabatic forcing. The vertical diffusivity characterizing these layers is k_ρ ≃ 1 × 10⁻⁴ m² s⁻¹ or greater, and this corresponds to an increased importance of the diabatic terms in the governing equations. The estimated lateral flow fields, with associated vertical diffusivities, are shown in Figs. 6, 7, and 8. Flow in these layers is generally southward and westward, with average current magnitudes of 2–4 mm s⁻¹ and the faster currents occurring in the western portion of the region exceeding 5 mm s⁻¹. The influence of ±50 m noise in the isopycnal-height field has also been evaluated, and the lateral flow is very stable to perturbations of this magnitude.

The neutral density range 28.12 γ_n < 28.18 (45.88 < σ₄ < 45.96) is of particular interest, because this density layer bounds the target isopycnal used for the tracer release, γ_n = 28.16 (σ₄ = 45.94). The tracer inventory made during the 1997 survey indicated that the tracer's center of mass was advected 130 km southwest over a period of 14 months, as reported by Ledwell et al. (2000). This is equivalent to a time average speed of 3.6 mm s⁻¹. The average velocity in the density layer 28.12 < γ_n < 28.18, as estimated by the inversion for the region 22°–25°S, 18.5°–22°W, is |u| = (3 ± 0.3) mm s⁻¹ in the direction 220°–230°. Thus, the lateral flow estimate given by the inversion is consistent with the rate of lateral advection measured by the tracer. The calculated flow in these layers is also consistent with float observations made in this region during 1994–96, as reported by Hogg and Owens (1999). These estimates of flow are not consistent with the estimates of Reid (1989), who found northward flow near 20°S, 20°W at 4000 and 4500 m.

A detailed discussion of tracer- and microstructure-derived diffusivities for the target isopycnal is given by Ledwell et al. (2000). The tracer-derived result of k_SF6 ≅ 3 × 10⁻⁴ m² s⁻¹ compares well with the 95% confidence interval of the microstructure estimate, k_ρ = (2.3 ± 1) × 10⁻⁴ m² s⁻¹. The later estimate was made using the 14-month tracer concentration field as weighting in the spatial average, and is thus not representative of the larger diffusivities of (5–10) × 10⁻⁴ m² s⁻¹ that occur east of the tracer patch. As is apparent in Figs. 7 and 8, there is an order of magnitude increase in vertical diffusivity over the longitudinal extent of the inverse model region.

Diapycnal advection estimates were made on density surfaces, and these estimates are presented in Figs. 9, 10, and 11. The γ_n = 28.08 (σ₄ = 45.83) surface is characterized by downward advection with mean w∗ = −3±1 m yr⁻¹; larger downward pumping occurs in the east. The deeper surfaces, γ_n = 28.16 (σ₄ = 45.94) and γ_n = 28.20 (σ₄ = 45.98) are characterized by larger values of advection, |w∗| ≃ 10 m yr⁻¹, with individual estimates having associated uncertainty of roughly 20%–40%. The γ_n = 28.16 surface was the target isopycnal for the tracer release experiment, and downward advection is estimated over the part of the domain occupied by the tracer patch. This is consistant with the 14-month tracer inventory, and a later 28-month inventory, both of which indicate that the tracer's center of mass sank relative to the target isopycnal (Ledwell et al. 2000).

Diapycnal advection through the eastern range of the tracer surface is upward, as it is for nearly all of the surfaces deeper than the γ_n = 28.20 surface. Figures 10 and 11 show contoured fields of w∗ that have been highly smoothed. Where ridge topography occupies each density layer there is no flow, and the contoured maps show estimates of w∗ that have been averaged in 1° × 1° cells, which often include regions blocked by topography. Estimates of upward advection in canyons exceed w∗ ≅ 30 m yr⁻¹ and are as large as w∗ ≅ 100 m yr⁻¹ in some local regions. In FZ canyons, eastward flow is estimated with magnitude 0.4 ± 0.2 mm s⁻¹. It is this eastward canyon transport that supplies fluid for the diapycnal upwelling.

c. Mixing rates and flow in an abyssal canyon

A fundamental feature of the abyssal Brazil Basin is the system of crests and canyons associated with the FZs of the MAR. Many canyons run zonally for hundreds of kilometers, and each canyon is bounded to the north and south by crests extending 500 to 1000 m above the canyon floors. The meridional wavelength of the canyon–crest system is roughly 30–50 km, and because of this the flow in these FZs was difficult to resolve in the inverse model calculations. Instead, each canyon is better modeled locally so that the special geometry of the system can be considered.

Consider the integral equations for advection of potential temperature and mass in a density layer bounded latitudinally by canyon walls. If the density layer is bounded by a density surface that intersects the canyon bottom to the east, then the control-volume budgets for heat and mass are

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Here, the dydz integrals are evaluated at the western edge of the control volume, and the dxdy integrals are taken along a surface of constant density. We can algebraically solve expressions (14) and (15) for an along-canyon flow rate u if we express the cross-canyon integrals in simplified forms. We take ∫∫ dy dz u = uA_(yz) and ∫∫ dy dz uΘ = uΘ_(yz)A_(yz), where A_(yz) is the area of the cross-canyon section and Θ_(yz) is the cross-section weighted average of potential temperature. We may also treat the bounding isopycnal as a surface of nearly constant temperature Θ_(xy) of area A_(xy). This allows us to express the flow rate as

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where the thermal flux has been expressed as F_θ = −ΓϵN⁻²(∂Θ/∂z).

This expression was evaluated using data from the section of HRP profiles shown in Fig. 12 for the density layer with γ_n > 28.20. The diffusivities along the γ_n = 28.20 isopycnal were estimated as the ensemble average k_ρ = 〈Γϵ(h = h_σ)N⁻²〉 across the canyon width, down the length of the canyon. Error analyses for the k_ρ estimates were done using Monte Carlo methods incorporating an uncertainty of ±100 m in the bottom depth, along with the statistical uncertainty in ϵ(h_ab). Heat fluxes across the γ_n = 28.20 surface are roughly −1 × 10⁻⁶ m°C s⁻¹, with the largest values at the eastern end. While the easternmost edge of the γ_n = 28.20 surface experiences the largest mixing rates, the stratification there was very weak (N² = 6 × 10⁻⁸ s⁻²), keeping the buoyancy flux in this area comparable to fluxes occurring where mixing rates are weaker. These mixing rates are used in (16) and yield an up-canyon flow rate of u = (3.5 ± 1) mm s⁻¹ at the western edge of the control volume. This result can be compared with the inverse estimates of (0.4 ± 0.2) mm s⁻¹ up-canyon flow. The apparent discrepancy is resolved if volume transports are compared rather than flow rates. The inverse model estimate utilized a subsampled bathymetric grid, such that the dimensions of canyons are only coarsely resolved. The canyon width was 55 km over the entire canyon's depth, giving 3 × 10⁷ m² as a typical cross section. The estimate from (16) of canyon flow relied on more careful estimates of roughly 5.5 × 10⁶ m² for canyon cross section. Therefore, the inverse estimates of volume transport are (0.012 ± 0.006) × 10⁶ m³ s⁻¹, while the section-derived estimate is (0.019 ± 0.005) × 10⁶ m³ s⁻¹. The two estimates are indistinguishable at the significance level of one standard error.

Net diapycnal upwelling is required across the γ_n = 28.20 surface between 18° and 16.5°W. From mass conservation (15), the average diapycnal advection across γ_n = 28.20 in the canyon is w∗ = (A_(yz)/A_(xy))u = (230 ± 70) m yr⁻¹.

d. Deep upwelling and mixing efficiency

As presented above, strong diapycnal upwelling is required to close the heat and mass budgets in the canyons. Figure 13 presents a comparison of the a priori estimates of w∗ to the inverse solutions for w∗ on the γ_n = 28.20 neutral surface. Inverse estimates are generally consistent with the a priori estimates of w∗ at sites where the γ_n = 28.20 surface is more than 300 m above the bottom. However, the inverse estimates suggest diapycnal upwelling is favored at sites where the γ_n = 28.20 surface is within 300 m of the bottom.

Diapycnal upwelling must be occurring in regions where turbulence supports a convergent buoyancy flux: w∗N² ≅ −∂J_b/∂z > 0. This condition is satisfied by turbulence occurring close to the bottom since the buoyancy flux must decrease with depth to meet a no-flux condition at the seafloor. Within the context of (3), the buoyancy flux is related to the dissipation rate by a mixing efficiency parameter, J_b = −Γϵ. We believe that a reduction in mixing efficiency near the bottom allows the no-flux condition to be met at the seafloor. This reduction may occur in a 1 to 10 m thick boundary layer (i.e., the “log layer”), which was unresolved by our measurements. However, variations in mixing efficiency may extend to greater heights in the canyons. The physical mechanisms controlling the efficiency of mixing near topography are not well understood, but turbulent mixing in regions enclosed by topography may favor mixing with reduced efficiency. Stigebrandt and Aure (1989) found that mixing in fjords occurred at 0.05 efficiency. Given our observations of dissipation and stratification along canyon slopes, a decrease in mixing efficiency from 0.2 to 0.05 in the bottom-most 100 m above FZ slopes would account for diapycnal upwelling as large as w∗ ≅ +300 m yr⁻¹.

a. Local consequences of buoyancy forcing

The results of the inverse calculation suggest that, while diabatic flow is weak in much of the interior, stronger levels of turbulence above rough bathymetry lead to enhanced levels of diabatic flow. Divergence of the diabatic flow acts as a mechanism of buoyancy forcing for the lateral circulation though the vortex stretching term (f/h)(w∗(z_u) − w∗(z_l)) in the vorticity budget. The significance of diabatic vortex stretching as a forcing mechanism must be assessed relative to other forcing agents for the flow. From the inverse estimates, we may compare the strength of buoyancy forcing to the adiabatic mechanism of vortex stretching, represented by (f/h)u · ∇h in the vorticity budget.

In the deep density layers, adiabatic vortex stretching is cyclonic, while diabatic vortex stretching is anticyclonic. In the region where mixing rates are largest, these two mechanisms are comparable in magnitude, and the vortex stretching terms in (2) nearly cancel. Since βυ ≅ 0, there is little meridional flow just above the level of FZ crests (Figs. 6 and 7). In addition to the stretching mechanisms, friction imposed by topography may also be a significant forcing mechanism near the bottom, though we have not attempted to assess this contribution. Farther west of the Mid-Atlantic Ridge where mixing rates are weaker, the adiabatic stretching exceeds the diabatic stretching, resulting in the southward flow where the largest lateral velocities occur.

Spall (2001) has examined the planetary geostrophic dynamics for a circulation driven by buoyancy forcing over sloping topography. In his model, Spall represents only the large-scale zonal slope of FZ topography, with diapycnal upwelling from FZ canyons represented as flow out of the sloping bottom. Spall finds that bottom pressure torque tends to balance the diabatic vortex stretching in the deepest density layer, keeping the flow along the bottom weak even when the buoyancy forcing is strong.

To produce a clearer summary of the deep circulation near canyons, a zonal streamfunction (ψ) was defined by latitudinally integrating the velocity so that uδ_y = −ψ_z and w∗δ_y = ψ_x, where the latitudinal limits of integration were taken over the inverse domain (19°–25°S),

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The inverse model grid is coarsely spaced in depth, so an objective analysis technique (Bretherton et al. 1976) was used to interpolate the inverse solution and compute the streamfunction. Prior to the objective analysis, the grid coordinates were nondimensionalized by the assumed correlation scales. For the vertical dimension, the advective–diffusive scale ẑ = k/w∗ ≃ 300 m was used, while twice the baroclinic deformation radius was used for a lateral scale, x̂ = 2Rd ≃ 30 km. The autocorrelation of the streamfunction was specified to e fold over three nondimensional correlation radii, r̂² = (z/ẑ)² + (x/x̂)². The resulting smoothed streamfunction is shown in Fig. 14 relative to representative bathymetry and the density field. The circulation shallower than 3500 m is characterized by westward flow and diapycnal flow that is downward, while deeper flow is characterized by diapycnal upwelling at the eastern ends of canyons feeding westward flow at the level of the FZ crests. The streamfunction estimates were not extended deeper than 5000 m, as the objective mapping loses skill near the canyon floors. However, the diapycnal upwelling east of 18°W indicated by ψ at 5000 m is consistent with the eastward canyon flow resolved coarsely in the inverse model solution.

The net transport through a density surface (T_σ) can be expressed in terms of the streamfunction as T_σ = ψ(x_w) − ψ(x_e), where (x_e, x_w) are the eastern and western limits of the surface area integral. The transport calculation is simplified if the eastern limit is taken to be the longitude where the surface incrops along the bottom, so that ψ(x_e) = 0. For the γ_n = 28.20 (σ₄ = 45.98 and θ ≅ 0.8°C) surface, the net diapycnal transport between 19° and 25°S is (0.34 ± 0.14) × 10⁶ m³ s⁻¹. This is the integrated upwelling over 6° of latitude; the actual upwelling occurs in the canyons. Assuming that an up-canyon flow of 3.5 mm s⁻¹ is typical of canyons that are 500 m deep and 30 km wide, then six or seven canyons would carry the required mass for upwelling. This is consistent with the number of canyons in the region.

b. Consequences for the basin-scale mass budget

Maps of seafloor bathymetry show a network of 20–30 FZs occupying the entire meridional extent of the Brazil Basin. Additionally, the character of barotropic tidal energy does not vary significantly along the axis of the MAR. Treating these as proxies for turbulent dissipation, the spatial distribution of mixing depicted in Fig. 2 may be characteristic of the FZs throughout the greater basin. This would imply that buoyancy forcing plays an important role in driving the abyssal circulation. Indeed, O'Dwyer and Williams (1997) have produced basin-scale maps of q = f/h that show that the steamfunction for adiabatic flow in the Brazil Basin is blocked by boundaries at depths greater than 3500 m. Flow in regions of blocked q contours occurs only under the action of buoyancy forcing and friction.

Some basin-scale consequences of abyssal buoyancy forcing are examined by considering the mass budget for water of temperature class θ < 0.8°C in the Brazil Basin. This budget was considered by Hogg et al. (1982) and Morris et al. (2001). Morris et al. (2001) find that (3.7 ± 1.7) × 10⁶ m³ s⁻¹ of net upwelling occurs across the θ = 0.8°C isotherm surface in the latitude range between 30°S and the equator. If the circulation shown in Fig. 14 is characteristic of other FZ regions, a simple scaling of our localized estimate (6° of latitude) for net upwelling to the basin scale (30° of latitude) suggests an upwelling across θ = 0.8°C isotherm of (1.7 ± 0.7) × 10⁶ m³ s⁻¹. This number is in the general range needed to close the basin-scale mass budget.

The various regions of bottom water circulation in the Brazil Basin are shown in Fig. 15. Bottom water enters the basin through the Vema Channel, and a boundary current carries this deep water northward along the western portion of the basin (Reid 1989; Speer and Zenk 1993; Durrieu De Madron and Weatherly 1994). However, over much of the zonal band between 35° and 25°W, FZs are absent and mixing levels and isopycnal slopes are likely to be small, so the mean flow should satisfy βυ ≅ 0. Indeed, float observations in this region suggest that the flow is dominantly zonal (Hogg and Owens 1999). Farther to the east, mixing levels and isopycnal slopes increase above FZ topography. In this region, our inverse results serve as an example, and diapycnal advection will influence the lateral flow through the vorticity relation βυ = (f/h)u · ∇h + f(∂w∗/∂z). Net diapycnal upwelling of water masses colder than θ ≅ 0.8°C likely occurs in FZ canyons. Inversion of observations from January 1996 to April 1997 indicate a net southward transport of −(0.34 ± 0.05) × 10⁶ m³ s⁻¹ for the water class denser than γ_n > 28.08 in the boxed region (Fig. 15), though recent float observations suggest that meridional flows in this region are time varying (N. Hogg 2001, personal communication).

The work described here demonstrates that direct measures of mixing can be combined with hydrographic observations to give an estimate for circulation that is consistent with other large-scale budgets. This is in contrast to many earlier attempts to reconcile local dissipation estimates with mixing rates inferred from advective–diffusive budgets. A notable difference between the current approach and early treatments of dissipation data rests with the application of an inverse technique with both microstructure and conventional hydrographic data. While direct observations of mixing tend to give detailed information on local diapycnal exchange processes, hydrographic budgets predict only the mean of diabatic forcing over large scales. Neither approach alone would give a meaningful measure of the full three-dimensional circulation.

Buoyancy forcing provided by turbulent mixing likely provides the dominant forcing for the abyssal circulation, and the highly localized observations described here constitute the only available observations of deep dissipation. The localized measurements presented here suggest that the spatial structure of abyssal buoyancy forcing is far richer than Stommel and Aron's example of uniform upwelling.