1. Introduction
Dissipation rates of thermal variance (χ) and turbulent kinetic energy (TKE, ϵ) are used to infer diapycnal fluxes of heat, salt, and density in the ocean. The divergence of the density flux will dictate the strength of the diapycnal advection. Further dynamics are set by the vertical divergence of the diapycnal advection, which provides the vorticity forcing in the ocean’s interior. Thus, in a fundamental way, the general circulation of the ocean is influenced by the second derivatives of the dissipation rates with respect to the vertical. This work will examine the problem of estimating the diapycnal advection from observations of χ and ϵ in a region influenced by the salt-finger form of double-diffusive convection.
Many regions of the World Ocean are characterized by haline stratification that is top heavy with respect to density. Such regions include the Mediterranean outflow, the western tropical North Atlantic, and the “central waters” of the subtropical gyres (Schmitt 1994). The unstable potential energy stored in the top-heavy salinity stratification may be released in small-scale convection cells known as “salt fingers.” The vertical fluxes produced by the finger instability act to diffuse concentrations of heat and salt down their mean gradients. While convecting plumes (fingers) generally preserve their salinity variance, thermal variance is lost to the molecular conduction of heat across individual cell boundaries. Hence, fingers act to vertically transport salinity more efficiently than heat. The haline component of the buoyancy flux exceeds the thermal component, resulting in an upgradient flux of density. Mixing by salt fingers may be modeled in terms of separate eddy diffusivities for heat (kθ), salt (ks), and density (kρ). While the finger fluxes of heat and salt are both down their respective gradients, with ks > kθ > 0, the finger flux of density is upgradient so that kρ < 0. This upgradient flux of density has been shown to form and maintain a thermohaline staircase, in which a series of well-mixed layers are separated by sharp interfaces (Stern and Turner 1969). Such staircases have been observed at the Mediterranean outflow, the Tyrrhenian Sea, and in the tropical Atlantic near Barbados (Schmitt et al. 1987). The existence of a thermohaline staircase is taken to be a strong indicator of salt-finger mixing.
However, staircases are not always found in open-ocean regions having finger-favorable stratification. In general, the upgradient density flux induced by salt fingers will be opposed by downgradient turbulent fluxes produced by internal wave breaking. Unlike salt-finger mixing, the high Reynolds number turbulence occurring in the ocean produces fluxes of heat and salt that are uninfluenced by differences in molecular properties, resulting in uniform diffusivities among scalars (i.e., kθ = ks = kρ). Thus, in a region experiencing both salt-finger and turbulent fluxes, the magnitude and direction of the net buoyancy flux is then determined by the competition between the two processes. To properly determine the buoyancy flux in these regimes, a means of assessing both the relative occurrence and magnitude of the two processes is needed. The need for a procedure allowing calculation of the net kρ in a region with both turbulence and salt fingers serves as the primary motivation for this study.
The work presented here will rely on observations of microscale dissipation obtained from the free falling high-resolution profiler (HRP; Schmitt et al. 1987), as a means of assessing the diapycnal fluxes occurring in the thermocline at the North Atlantic Tracer Release Experiment (NATRE) site. A previous comparison of microstructure-derived diffusivities (Toole et al. 1994) and tracer-derived diffusivities (Ledwell et al. 1993) has indicated general agreement. This paper will strive to identify the turbulent and salt-finger contributions to the net diffusivities. Furthermore, we will allow for the possible elevation of the haline diffusivity over the thermal diffusivity, as well as the possibility for a negative density diffusivity. We will establish a means of identifying both the relative frequency and magnitude of salt-finger and turbulent dissipation events utilizing several nondimensional parameters available from quantities measured by the HRP. The density ratio Rρ = (αΘz)(βSz)−1 is the parameter that dictates a system’s susceptibility to double-diffusive instability. The salt-finger instability is permitted if 1 < Rρ < 100 (Schmitt 1979a). However, in an ocean constantly perturbed by internal wave strain, modes of instability with characteristic period longer than that of the wave field (2πN−1) may fail to grow. For this reason Schmitt and Evans (1978) suggest only modes with growth rates near N will become strongly established, this being true if 1 < Rρ < 2. A second parameter, the gradient Richardson number Ri = N2
We will begin by describing some qualitative evidence for salt fingers at the NATRE site (section 2). Quantitative evidence for salt-finger mixing in terms of dissipation data is presented in section 3. We give a description of HRP data (section 3a), present a model of the dissipation ratio (section 3b), discuss the method of analysis (section 3c), discuss the statistical treatment of dissipation data (section 3d), and present our results (section 3e). The results of the (Rρ, Ri, Γ) parameter analysis are then applied to the NATRE profile data to estimate the net diffusivities of heat and salt (section 4a) and the diapycnal advection (section 4b). We present a discussion of this material in section 5.
2. Qualitative evidence for salt fingers at the NATRE site
The North Atlantic Tracer Release Experiment was conducted in the Canary Basin of the eastern North Atlantic. A 150-station microstructure survey of the release site was completed prior to the tracer release phase of the experiment, discussed by Ledwell et al. (1993, 1998). Measurements were made using a free falling, autonomously profiling instrument, the high-resolution profiler. The HRP produces measurements of temperature, conductivity, and velocity at both finescale [O(1 m)] and microscale [O(1 cm)] resolution. Measurements of optical microstructure were also made using a shadowgraph imaging system. Descriptions of the HRP and all of its component instruments, including the shadowgraph, are given by Schmitt et al. (1988).
The NATRE site rests within the central waters of the North Atlantic subtropical gyre (Fig. 1). Several studies (Schmitt and Evans 1978; Schmitt 1981; Schmitt 1990) have argued that salt fingers contribute to the vertical mixing in this region. In particular, these studies argue that finger mixing should dominate turbulent mixing if Rρ ⩽ 2. Indeed, this describes the density ratio structure in the upper 500 m of the NATRE site thermocline (Fig. 2). However, no permanent thermohaline staircase occurs at the NATRE site. This suggests that the finger contribution to the vertical buoyancy flux is insufficient for maintaining a series of well-mixed layers and sharp interfaces. If turbulent fluxes produced by shear instability of internal waves are large enough (and frequent enough) to overcome the finger fluxes, the net density flux may be downgradient. Polzin (1996) has found a clear connection between mixing and shear instability at the NATRE site. His work indicates that exceptional dissipation generally occurs when the local Richardson number (Ri = N2/
Despite the lack of a thermohaline staircase, optical structures recorded during two shadowgraph profiles give qualitative evidence of salt-finger activity. Thin filament-like optical structures (Fig. 3) occurred from the bottom of the mixed layer (z ∼ 150 m) to the base of the thermocline (z ∼ 1000 m). These features are most abundant just below the mixed layer, occurring in patches with several meter vertical extent, with gaps between patches of 5–10 m. Patches containing filament structures become more sparse at at greater depth, but generally occur with a frequency of 2–4 patches for every 40 m. While all possible orientations were encountered, filaments in the form of laminae tilted 10°–20° from horizontal were most frequently observed (Fig. 3a). These structures are identical to those previously observed by Kunze et al. (1987) in a thermohaline staircase. As was the case with the staircase observations, the laminae at the NATRE site are characterized by cross-filament wavelengths of 0.5–1 cm. Kunze (1990) identified these structures as salt fingers that have been tilted by shear. Other classes of optical structures observed include sharp interfaces, isotropic features, and billows.
We regard the abundance of thin tilted laminae at the NATRE site as suggestive evidence for salt fingers. To quantitatively assess the frequency and strength of salt fingers at the NATRE site, we rely on estimates of χ and ϵ derived from HRP microstructure measurements.
3. Quantitative assessment of salt-finger mixing
a. Description of data
Data from the NATRE HRP survey will provide the foundation for this study. However, with the intent of making our study more general, we have supplemented the NATRE data with data from a second HRP survey. These data come from field work conducted in the northeast subtropical Pacific at Fieberling Guyot. Data from the Fieberling survey (TOPO) is discussed by Toole et al. (1997) and Kunze and Toole (1997). In the present study, TOPO data are included as a means of introducing data from a double-diffusively stable (hereafter, doubly stable) stratification regime. While dissipation occurring in the salt-finger regime may be attributable to a combination of turbulence and fingers, dissipation in the doubly stable regime can only be attributed to turbulence. By regarding the features of turbulent dissipation in the doubly stable regime as a null hypothesis, we can objectively assess the dissipation observed in finger-favorable data.
The profile data from NATRE typically extends to 2000 m. These profiles are characterized by a deep mixed layer (80–150 m thick) capping the finger-favorable thermocline. Below the thermocline, intrusive features exist with both “diffusive” favorable (the form of double diffusion with cold fresh water over warm salty) and doubly stable character. The TOPO data can be broken into two classes. The data collected above the seamount summit are characterized by high shear and weak stratification in the presence of thermohaline interleaving. The data collected on the seamount flanks are characterized by lower levels of shear and stronger stratification. In particular, these two classes have heterogeneous shear statistics, with shear levels at the summit exceeding those at the flank by a factor of 2. For this reason, we will treat these two classes of TOPO data separately in the analysis that follows. While the stratification at the TOPO site was generally doubly stable, some double-diffusive favorable patches were also present.
Initial processing of all HRP data results in estimates of all conventional (e.g., Θ, S, U, and V) and microstructure quantities at 0.5-m intervals. A detailed description of the algorithms used for this initial stage of data analysis can be found in Polzin and Montgomery (1996). Dissipation rates are calculated from observations of thermal and velocity microstructure using the relations χ =
Finestructure gradient quantities, particularly Rρ and Ri, will be used extensively in the analysis that follows. To estimate the vertical gradients of scalars, we have used the slope of a linear fit over a 5-m segment, centered at each 0.5-m interval. The 5-m scale was chosen as a suitable trade-off between the need for high vertical resolution and statistically reasonable regression estimation. The magnitudes of all 5-m scalar gradients were compared to their associated standard error. Gradient quantities with standard errors larger than twice their magnitude were excluded from the analysis. This resulted in roughly a 5% data loss, mostly from noisy N2 estimates. Figure 4 shows typical profiles from the two HRP surveys used for this study. Data from the TOPO survey are shown in Figs. 4a (a seamount summit profile) and 4b (a seamount flank profile). A profile from NATRE is shown in Fig. 4c.
b. The dissipation ratio model
Figure 5 presents the plausible range of the nondimensional parameters of the salt-finger and turbulence models. Results from laboratory studies were used to plot the mixing efficiency of turbulence (5a) and the buoyancy-flux ratio of salt fingers (5b). These numbers were used to compute the dissipation ratio models for turbulence and salt fingers (5c). The value Γ(t) = 0.2 is shown as representative of the turbulence model, with a plausible range shown as 0.05 < Γ(t) < 0.25. The plausible range of the salt-finger dissipation ratio is shown with a 99% confidence band determined from the density–ratio binned statistics of r.
c. Method of analysis
Our primary investigation of the dissipation rates will be done using the dissipation ratio Γ. The existence of simple models for Γ in cases of turbulent and salt-finger dissipation give this parameter merit. However, two issues detract from this parameter’s apparent usefulness. First, differences as small as a factor of 2 distinguish the value of Γ between the two processes. This obstacle can be overcome by incorporating large numbers of dissipation observations into each estimate of Γ to reduce error bars enough to resolve such a subtle parameter range. This is a serious consideration since Γ is the ratio of four noisy variables. Second, given an ensemble of dissipation observations, only a fraction of the data may be representative of mixing events appropriately modeled by (1) and (2). This issue must receive careful attention.
The dissipation rates ϵ and χ are computed by integrating the observed shear and thermal variance residing at scales between about 1 and 50 cm. Variance at these scales originates from diabatic, irreversible processes. The processes of turbulence and salt fingering are among these, producing fluxes of heat, salt, and buoyancy that irreversibly alter the local temperature, salinity, and density fine structure. However, it is plausible that internal wave and molecular processes may also account for variance at small enough scales to influence dissipation estimates, and these set the oceanic background levels of the dissipation rates. This type of oceanic “noise” is not appropriately modeled by (1) and (2). Similarly, the noise of the sensors and associated electronics, though low for the HRP (Polzin and Montgomery 1996), will contribute to uncertainty in ϵ and χ.
In the work presented here, we seek to attribute observations of irreversible microstructure to salt fingers or turbulence. We have attempted to rule out the influence of noisy dissipation estimates by only examining dissipative events of higher magnitude, while still retaining enough data to uncover a potentially subtle signal of salt fingers. To do this, we have used the combined dataset involving observations from both NATRE and TOPO surveys. This combined dissipation record was then examined in terms of various upper thresholds of dissipation rate magnitude, this being done separately for observations occurring in doubly stable and finger-favorable patches.
We have found that exceptional levels of finger-favorable χ are associated with bimodal ϵ statistics. In particular, we have examined the distribution of ϵ data using a threshold defined as χ > χ75 ∼ 1 × 10−9 K2 s−1: the upper 75th percentile of the combined χ record. The statistical distribution for ϵ(χ > χ75) is shown in Fig. 6. The finger-favorable data (Fig. 6a) seems to have a primary mode at ϵ ∼ 2 × 10−10 W kg−1, with the secondary mode occurring near ϵ ∼ 1 × 10−9 W kg−1. A simple statistical test for bimodality (Haldane 1952) indicates that the apparent antimode at ϵ ∼ 5 × 10−10 W kg−1 is significant to the 0.15 level (i.e., significant at the 85% confidence level). In contrast to the finger-favorable data, the associated distribution of doubly stable ϵ data lacks bimodal character (Fig. 6b). Both turbulence and salt fingers act as dissipative mechanisms in the finger-favorable regime, while only turbulence acts in the doubly stable regime. The existence of bimodal ϵ in only the finger-favorable regime suggests that the two modes are associated with the two processes.
To investigate the character of the finger-favorable data more thoroughly, we have examined the ϵ(χ > χ75) population in terms of the stability parameters Rρ and Ri. Figure 7 shows the distribution of ϵ(χ > χ75) data after being partitioned into four subsections of (Rρ, Ri) data space. When the portion of parameter space having Ri > 1 is considered, the low-ϵ mode dominates. This is particularly true when Rρ < 2 (Fig. 7a), this low density ratio range having about twice the number of χ > χ75 events as Rρ > 2 (Fig. 7b). In the portion of parameter space with Ri < 1 (Figs. 7c and 7d), the high-ϵ mode is apparent at both large and small values of Rρ. However, in the case where (1 < Rρ < 2, Ri < 1), a low-ϵ mode is also apparent, with a population comparable to the high-ϵ mode. The documented trend is consistent with an association of the two modes with turbulence and salt fingering. In particular, we associate the low-ϵ mode, dominant at low Rρ, with salt fingers. We associate the high-ϵ mode, dominant at low Ri, with turbulence.
Ruddick et al. (1997) used microstructure observations from a different instrument to examine dissipation at the NATRE site. The noise level of their measurements limited their analysis to observations having ϵ > 7 × 10−9 W kg−1 Their observations clearly fall into the high-ϵ mode that we have associated with turbulence. Ruddick et al. (1997) use a Reynolds number Re = ϵ/(νN2), and they find no clear evidence of salt fingers in the parameter range 101 < Re < 104. In a salt-finger regime, the parameter ϵ/(νN2) is equivalent to the Stern number Jb/(νN2). Stern (1969) argued that this parameter must be O(1) for fingers to be active, while McDougall and Taylor (1984) found experimentally that the Stern number can be O(10) for Rρ < 2. A characteristic Stern number for our low-ϵ mode is O(10). Thus, the Ruddick et al. (1997) Reynolds numbers are too large to admit the possibility for fingers.
In examining only an upper threshold of our dissipation data, we have attempted to filter out weak dissipation events not likely associated with salt fingers or turbulence, as well as those sites where signals are weak relative to instrumental noise levels. Additionally, we will assume that a subset of observations with χ > χ75 can be well modeled by the production–dissipation balances of (1) and (2). In doing so, we will expect the dissipation ratio analysis of this data to provide valuable information on the mixing processes to which the dissipation is attributable. By examining Γ in the (Rρ, Ri) parameter space, we may capitalize on the association between the two dissipative processes and their stability parameters. We expect Γ to be consistent with the turbulence model at low Ri and with the salt-finger model at low (finger favorable) values of Rρ. There are regions of the (Rρ, Ri) parameter space that do not favor either process. These include the large Ri region of the doubly stable regime, and the region of the finger-favorable regime where both Ri and Rρ are large. In these regions where the finger and shear instabilities are not favored, exceptional dissipative events should be rare.
d. Statistical treatment of dissipation data
Mean and variance estimation of dissipation rate data has been discussed by many authors. The apparent tendency for the statistical distributions of χ and ϵ to be lognormal has produced arguments in favor of maximum likelihood estimation (MLE; Baker and Gibson 1987). The breakdown of lognormality assumptions has also been documented, and Davis (1996) concludes that arithmetic estimation is the most robust form of analysis. Also, we have evaluated the different estimation methods through Monte Carlo exercises involving lognormally distributed random data. For random data distributions with characteristics reflecting those of our dissipation data, the discrepancy between MLE and arithmetic methods becomes less than 10% for as few as 200 degrees of freedom. For these reasons, arithmetic estimation was adopted as the analysis procedure for this study.
e. Results
We begin our examination of exceptional dissipation data within the doubly stable regime. The available datasets were subsampled in favor of all 0.5-m dissipation observations with corresponding density ratio in the doubly stable range −100 < Rρ < −1. Data in the density ratio range between −1 and 0 were held from the analysis, as these data are characterized by weak thermal stratification where Θz → 0, making Γ ∝ N2/
Analysis of the exceptional doubly stable dissipation data is shown in Fig. 8. The data were classified into six overlapping populations of Ri. In each panel, the symbol and error bar denote the mean with 95% confidence interval for a 1000-element bin of data, with the symbol centered at the bin’s mean density ratio. Richardson numbers less than 1 are considered in Figs. 8a, 8b, and 8c. These data are characterized by Γ between 0.1 and 0.25, in good agreement with the results of previous studies (Moum 1996; Ruddick et al. 1997). Additionally, Γ has no discernible density ratio dependence. We note that 75% of all exceptional dissipation data within the doubly stable regime occur when Ri < 1. The 25% of exceptional TOPO χ data that occur in patches where Ri > 1 are shown in Figs. 8d, 8e, and 8f. Data at Ri > 5 are too sparse to yield a single 1000-element estimate. However, available data occurring at Ri > 1 show Γ between 0.15 and 0.25, numbers like those observed at lower values of the Richardson number.
Exceptional dissipation occurring in the finger-favorable regime was investigated by conditionally sampling the available data for patches with Rρ > 1. As was done for the doubly stable data, these data were conditionally sampled into six overlapping populations of Ri and then sorted by Rρ in bins of 1000 elements. Data from the NATRE survey contributes most of the finger-favorable observations, with bins containing 400–700 degrees of freedom.
A formal statistical examination of the dissipation ratio signal was done after assembling a gridded (Rρ, Ri) data space for both doubly stable and finger-favorable regimes. As in the analysis described above, we have selected data from the upper quartile χ population of the combined TOPO and NATRE dataset. This data space consisted of discrete data bins uniformly spaced in the set of transformed coordinates [log(|Rρ| − 1), log(Ri)]. This choice of coordinates is favorable because it maps a wide range of parameter values, while emphasizing the parameter range (Rρ < 2, Ri < 1) where most of the data lies. Figure 11 shows contour maps of Γ for both stratification regimes. The shaded bins in each map contain 100 or more degrees of freedom. Bins with fewer degrees of freedom were excluded from the analysis. We find that the doubly stable regime is well characterized by a constant value of the dissipation ratio of Γ = 0.16 ± 0.04. In contrast, while much of the finger regime is characterized by 0.2 < Γ < 0.3, elevated values of Γ dominate the upper left (small Rρ, large Ri) quadrant of the map.
To evaluate the significance of the finger-regime results, we have implemented a simple statistical test. The dissipation ratio result from the doubly stable regime is taken as indicative of turbulent mixing. Specifically, we take Γ(t) = 0.16 ± 0.04 as the null hypothesis of mixing attributable to turbulence. We have tested the finger-regime results against this null hypothesis, seeking evidence that the dissipation ratio estimates of the finger-favorable data are different from those of the doubly stable data. The results of the two-tailed hypothesis test are shown in Fig. 12a. The standardized variable Z = (Γ − Γ(t))(δΓ2 +
By testing the original null hypothesis of turbulence at the 0.01 significance level, we have allowed for very generous uncertainty in the value of Γ(t). This also makes the rejection of the null hypothesis very difficult. In this manner we have been very conservative in labeling any binned χ > χ75 data as “nonturbulence.” Similarly, by testing the “nonturbulent” observations against a salt-finger hypothesis at the 0.01 significance level, we have allowed generous uncertainty in the Γ(f) model. For these reasons, we view this work as a consistency check. The upper quartile of χ observations having stability parameters in the range (Rρ < 2, Ri > 1) can be attributed to salt fingers in a manner consistent with available information from theory and laboratory work.
4. Mixing at the NATRE site
a. Vertical mixing influenced by salt fingers
We seek estimates of the net diffusivities kθ and ks that dictate the vertical fluxes over O(100 m) scales. In such estimates, the presence of salt-finger mixing will tend to enhance the haline diffusivity over the thermal diffusivity. Since there is a high occurrence of patches with low Rρ and stable Ri at the level of the tracer injection in NATRE, we find much of the dissipation can be ascribed to salt fingers.
To derive estimates of the net diffusivities, we employ 100 uniformly spaced stations from a 400 km × 400 km survey grid (Fig. 1). This April 1992 survey took 18 days to complete. Each profile includes a full record of dissipation data to 2000 m. Our goal is to estimate survey scale diffusivities applicable to the synoptic snapshot represented by our survey. We make the assumption that the variance budgets (1) and (2) are valid within the context of (i) survey scale lateral averaging, (ii) O(100 m) vertical averaging, and (iii) the time average invoked by treating the 18-day survey as a snapshot.
The mean field was derived with reference to a neutral-density coordinate system (σn; Jackett and McDougall 1997). Epineutral (i.e., along neutral surface) averaging yielded mean profiles of
The vertical diffusivities kθ and ks were calculated using a scheme that follows from the results of our (Rρ, Ri) analysis of Γ. The general form of the model we will employ has similarities to the model suggested by McDougall and Ruddick (1992). The differences between our approach and that of McDougall and Ruddick (1992) will be described in section 5. Our approach is motivated by the results of the dissipation ratio analysis of exceptional χ data (section 3e). Within each layer, all dissipation data are partitioned into two groups. All 0.5-m ϵ and χ estimates associated with (1 < Rρ < 2, Ri > 1) are presumed salt-finger dissipation. All dissipation associated with Rρ and Ri outside the finger-favorable range [i.e., the complement of (1 < Rρ < 2, Ri > 1)] are attributed to turbulence. In this manner, we attribute some percentage of the dissipation observations to fingers (P(f)) and turbulence (P(t) = 1 − P(f)). In using this approach, we assume that the regime partition found for exceptional χ is more generally applicable to the ensemble averages of dissipation profiles. We will further discuss the applicability of this model in section 5.
b. Estimates of diapycnal advection
Estimates of the diapycnal advection are given in Table 2 and plotted in Fig. 14. The epineutral contribution to w i.e.,
5. Discussion
We have shown that elevated dissipation events are related to stability parameters characterizing the locale where the mixing is occurring. Elevated χ (χ > 1 × 10−9 K2 s−1) can be associated with either turbulence or salt fingers based on the local values of the density ratio or Richardson number. Dissipation rates χ and ϵ were examined with respect to the two processes through the use of the nondimensional dissipation ratio Γ. Approximately 70% of the elevated χ data was consistent with the dissipation ratio of turbulence. These observations were generally associated with Ri < 1 (65% of the elevated χ). Data with (Rρ > 2, Ri > 1) were also characterized by Γ consistent with the turbulence model. The remaining elevated χ occurred when (Rρ < 2, Ri > 1). The dissipation ratio of these observations was inconsistent with the turbulence model. Instead, Γ of the (Rρ < 2, Ri > 1) data was found to be consistent with the model for fingers. Thus, when the Richardson number exceeds unity, there is a general transition at Rρ ∼ 2 between a turbulent-dominated and finger-dominated regime. Only 3% of elevated χ observations fall outside the generalizations referred to above. Observations binned in the parameter range (1.1 < Rρ < 1.3, 0.3 < Ri < 1) are consistent with salt fingers, while the (2 < Rρ < 4, 10 < Ri < 30) observations were inconsistent with either model.
The dissipation ratio analysis clearly rejects the notion that salt fingers are important throughout the general range of finger-favorable stratification (1 < Rρ < 100). On the contrary, most elevated χ data meet the dissipation ratio criterion for turbulence. However, evidence for fingers is strong when the stability parameters fall in the range (1 < Rρ < 2, Ri > 1). Our finding, that fingers are constrained to this narrow range of density ratio and Richardson number, supports conclusions of previous studies. Schmitt and Evans (1978) argued that growing modes of the finger instability overcome wave-strained gradients of Θ and S only when Rp < 2. Schmitt (1981) pointed out that the occurrence of thermohaline staircases is confined to Rρ < 1.7. Additionally, Linden (1974) used theory and experiment to show that weak shear permits fingers in the form of sheets aligned with the flow. Kunze (1994) has considered time-dependent shear and found that finger sheets will be tilted as the shear vector turns with time. However, at Rρ < 2, vertical finger fluxes remain strong.
The formal examination of the parameter space (Rρ, Ri, Γ) presented here included only elevated dissipation rate observations of the form χ > 1 × 10−9 K2 s−1. For elevated levels of χ, there is a partition finger and turbulence regimes. However, since the distributions of ϵ and χ are roughly lognormal, the exceptional events dominate the ensemble average. These ensemble averages dictate the magnitude of the vertical diffusivities. Thus, we expect the conclusions drawn about the association of salt fingers and turbulence with the stability parameters to apply not only to exceptional dissipation, but more generally to average dissipation.
We have implemented a method for estimating the diapycnal fluxes driven by both turbulent and salt-finger mixing events. The calculations involve partitioning observations of ϵ and χ into finger-favorable and turbulent conditions using the local values of Rρ and Ri. For each class of dissipation, we calculate the thermal diffusivity using the production–dissipation balances (1) and (2). The haline diffusivity is set equal to kθ for the ensemble of data not favorable to fingers. However, ks for the finger-favorable ensemble is calculated by enhancing the thermal diffusivity by the factor Rρ/r, with r = (αkθ
Several models have been proposed that parameterize salt-finger fluxes in terms of the density ratio. The Schmitt (1981) model is ad hoc, with a power-law dependence that allows a considerable enhancement of ks at low density ratios. One aspect of the Schmitt (1981) model is the tendency for finger fluxes to maintain a tight Θ–S relation. Supporting evidence for this is found in the comparison of 5-m Rρ with the O(100 m) Rρ in Fig. 13. These data suggest that the Θ–S relation is tightest in the depth range 200 m <
In terms of the vorticity budget for the upper thermocline, the diapycnal advection we have documented would play a limited role. Specifically, “stretching” of fluid columns in the upper ocean given by f∂zw(total) would be dominated by the adiabatic component of vertical velocity u · ∇zn, which is set by the Ekman pumping through the mixed layer. Joyce et al. (1998) estimate the Ekman downwelling applicable to the NATRE region to be ∼40 m yr−1. Thus, the diabatic forcing we have discussed makes up ⩽10% of the total vertical velocity in the ventilated thermocline.
However, at depths not influenced by Ekman pumping, diabatic forcing will dominate the total vertical velocity and the vortex stretching. In particular, the underside of the Mediterranean property core is strongly favorable to salt fingers. Salt-finger fluxes in this region would drive downward advection beneath the core. Spall (1999) has examined the circulation in this region and found that vortex stretching by salt-finger driven advection would produce a circulation consistent with observations. Furthermore, the diapycnal fluxes we have documented must be of consequence to the long timescale density budget. Models of the thermohaline circulation suggest that the efficiency of the oceanic heat transport is strongly influenced by differences between the vertical diffusivities kθ and ks (Gargett and Ferron 1996; Zhang et al. 1998). In particular, these models find that salt fingering in the upper thermocline decreases the strength of the meridional overturning cell. This is because salt fingers drive an upgradient density flux, and upgradient density flux is detrimental to the vertical advective–diffusive balance that closes the thermohaline cell. As demonstrated by the NATRE data, the microstructure observations allow for estimation of the salt-finger component of the mixing. These salt-finger estimates are crucial in explaining the downward diapycnal–advection of tracer, as turbulent mixing alone cannot account for an upgradient density flux. Thus, we suggest that expanded use of the techniques developed here will be important in determining the influence of salt fingering on the general circulation.
Acknowledgments
The authors wish to thank J. Ledwell, K. Polzin, J. Toole, E. Kunze, and an anonymous reviewer for valuable comments on the manuscript. We thank D. Koehler, E. Montgomery, T. Bolmer, D. Wellwood, D. Gloss, and P. LeGrand, and the officers and crew of the R/V Oceanus for support of the field program. The NATRE HRP work was supported by NSF and ONR under ONR Contract N00014-92-1323.
REFERENCES
Baker, M. A., and C. H. Gibson, 1987: Sampling turbulence in the stratified ocean: Statistical consequences of strong intermittency. J. Phys. Oceanogr.,17, 1817–1836.
Bevington, P. R., and D. K. Robinson, 1992: Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, 328 pp.
Davis, R. E., 1996: Sampling turbulent dissipation. J. Phys. Oceanogr.,26, 341–358.
Efron, B., and G. Gong, 1983: A leisurely look at the bootstrap, the jackknife, and cross-validation. Amer. Stat.,37, 36–48.
Gargett, A. E., and B. Ferron, 1996: The effects of differential vertical diffusion of T and S in a box model of thermohaline circulation. J. Mar. Res.,54, 827–866.
Haldane, J. B. S., 1952: Simple tests for bimodality and bitangentiality. Ann. Eugen.,16, 359–364.
Hamilton, J. M., M. R. Lewis, and B. R. Ruddick, 1989: Vertical fluxes of nitrate associated with salt fingers in the world’s oceans. J. Geophys. Res.,94, 2137–2145.
Huq, P., and R. E. Britter, 1995: Turbulence evolution and mixing in a two layer stably stratified fluid. J. Fluid Mech.,285, 41–67.
Jackett, D. R., and T. J. McDougall, 1997: A neutral density variable for the world’s oceans. J. Phys. Oceanogr.,27, 237–263.
Joyce, M. T., J. R. Luyten, A. Kubryakov, F. B. Bahr, and J. S. Pallant, 1998: Meso- to large-scale structure of subducting water in the subtropical gyre of the eastern North Atlantic Ocean. J. Phys. Oceanogr.,28, 40–61.
Kunze, E., 1987: Limits on growing, finite-length salt fingers: A Richardson number constraint. J. Mar. Res.,45, 533–556.
——, 1990: The evolution of salt fingers in inertial wave shear. J. Mar. Res.,48, 471–504.
——, 1994: A proposed flux constraint for salt fingers in shear. J. Mar. Res.,52, 999–1016.
——, and J. M. Toole, 1997: Tidally driven vorticity, diurnal shear, and turbulence atop Fieberling Seamount. J. Phys. Oceanogr.,27, 2663–2693.
——, A. J. Williams, and R. W. Schmitt, 1987: Optical microstructure in the thermohaline staircase east of Barbados. Deep-Sea Res.,34, 1697–1704.
Ledwell, J. R., A. J. Watson, and C. S. Law, 1993: Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment. Nature,364, 701–703.
——, ——, and ——, 1998: Mixing of a tracer released in the pycnocline of a subtropical gyre. J. Geophys. Res.,103, 21 499–21 529.
Linden, P. F., 1974: Salt fingers in a steady shear flow. Geophys. Fluid Dyn.,6, 1–27.
Lueck, R., 1987: Microstructure measurements in a thermohaline staircase. Deep-Sea Res.,34, 1677–1688.
McDougall, T. J., 1991: Water mass analysis with three conservative variables. J. Geophys. Res.,96, 8687–8693.
——, and J. R. Taylor, 1984: Flux measurements across a finger interface at low values of the stability ratio. J. Mar. Res.,42, 1–14.
——, and B. R. Ruddick, 1992: The use of ocean microstructure to quantify both turbulent mixing and salt fingering. Deep-Sea Res.,39, 1931–1952.
Moum, J. N., 1996: Efficiency of mixing in the main thermocline. J. Geophys. Res.,101, 12 057–12 069.
Oakey, N. S., 1985: Statistics of mixing parameters in the upper ocean during JASIN phase 2. J. Phys. Oceanogr.,15, 1662–1675.
Osborn, T. R., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr.,10, 83–89.
——, and C. S. Cox, 1972: Oceanic fine structure. Geophys. Fluid Dyn.,3, 321–345.
Pedlosky, J., 1996: Ocean Circulation Theory. Springer-Verlag, 453 pp.
Polzin, K. L., 1996: Statistics of the Richardson number: Mixing models and finestructure. J. Phys. Oceanogr.,26, 1409–1425.
——, and E. T. Montgomery, 1996: Microstructure profiling with the High Resolution Profiler. Proc. ONR Workshop on Microstructure Sensors, Mt. Hood, OR, 109–115.
Robinson, A. R., and H. Stommel, 1959: The oceanic thermocline and the associated thermohaline circulation. Tellus,11, 295–308.
Rohr, J. J., E. C. Itsweire, and C. W. Van Atta, 1984: Mixing efficiency in stably-stratified decaying turbulence. Geophys. Astrophys. Fluid Dyn.,29, 221–236.
Ruddick, B., D. Walsh, and N. Oakey, 1997: Variations in apparent mixing efficiency in the North Atlantic Central Water. J. Phys. Oceanogr.,27, 2589–2605.
Schmitt, R. W., 1979a: The growth rate of super-critical salt fingers. Deep-Sea Res.,26, 23–40.
——, 1979b: Flux measurements on salt fingers at an interface. J. Mar. Res.,37, 419–436.
——, 1981: Form of the temperature-salinity relationship in the Central Water: Evidence for double-diffusive mixing. J. Phys. Oceanogr.,11, 1015–1026.
——, 1990: On the density ratio balance in Central Water. J. Phys. Oceanogr.,20, 900–906.
——, 1994: Double diffusion in oceanography. Annu. Rev. Fluid Mech.,26, 255–285.
——, and D. L. Evans, 1978: An estimate of the vertical mixing due to salt fingers based on observations of North Atlantic Central Water. J. Geophys. Res.,83, 2913–2919.
——, H. Perkins, J. D. Boyd, and M. C. Stalcup, 1987: C-SALT: An investigation of the thermohaline staircase in the western tropical North Atlantic. Deep-Sea Res.,34, 1655–1665.
——, J. M. Toole, R. L. Koehler, E. C. Mellinger, and K. W. Doherty, 1988: The development of a fine- and microstructure profiler. J. Atmos. Oceanic Technol.,5, 484–500.
Shen, C. Y., 1993: Heat-salt finger fluxes across a density interface. Phys. Fluids A,5, 2633–2643.
——, 1995: Equilibrium salt-fingering convection. Phys. Fluids A,7, 706–717.
Spall, M. A., 1999: A simple model of the large scale circulation of Mediterranean Water and Labrador Sea Water. Deep-Sea Res.,46, 181–204.
Stern, M. E., 1969: Collective instability of salt fingers. J. Fluid Mech.,35, 209–218.
——, 1975: Ocean Circulation Physics. Academic Press, 246 pp.
——, and J. S. Turner, 1969: Salt fingers and convecting layers. Deep-Sea Res.,16, 497–511.
Taylor, J., and P. Bucens, 1989: Laboratory experiments on the structure of salt fingers. Deep-Sea Res.,36, 1675–1704.
Toole, J. M., K. L. Polzin, and R. W. Schmitt, 1994: Estimates of diapycnal mixing in the abyssal ocean. Science,264, 1120–1123.
——, R. W. Schmitt, K. L. Polzin, and E. Kunze, 1997: Near-boundary mixing above the flanks of a midlatitude seamount. J. Geophys. Res.,102, 947–959.
Turner, J. S., 1967: Salt fingers across a density interface. Deep-Sea Res.,14, 599–611.
Yamazaki, H., and T. R. Osborn, 1990: Dissipation estimates for stratified turbulence. J. Geophys. Res.,95, 9739–9744.
Zhang, J., R. W. Schmitt, and R. X. Huang, 1998: Sensitivity of the GFDL Modular Ocean Model to the parameterization of double-diffusive processes. J. Phys. Oceanogr.,28, 589–605.
Location of the NATRE HRP survey. Over 150 dives were completed over 26 days. The survey consisted of the 100 station grid spanning the (400 km)2 region shown. An additional 50 stations were tightly centered about (26°N, −28°W). The HRP survey was completed two weeks prior to the tracer-release phase of the experiment.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
A section showing 10 profiles of the density ratio Rρ, = (α〈Θz〉)(β〈Sz〉)−1. The gradients were calculated using a 5-m scale, smoothed using a 50-m running average. Stations 3–12 constitute the meridional section at the western edge of the survey. Each successive station is offset two Rρ units, and the reference value Rρ = 2.0 is shown for each profile. There is large variability in the mixed layer and beneath z = 600 m due to intrusive features.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Shadowgraph images of optical microstructure that were obtained during the NATRE HRP survey. (Shadowgraph image intensity is proportional to the Laplacian of the refractive index of light. The negative of each original image is shown here.) The tilted laminae shown here are observed throughout the thermocline. The circular window has a diameter of 10 cm, and the optical features have a characteristic wavelength of 0.5–1.0 cm. Laminae tilted 10°–20° from the horizontal (a) were the most frequently observed orientation, although filaments with vertical alignment (b) were also observed. The images shown here were obtained near 300-m depth.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Characteristic profiles of Rρ, Ri, ϵ, and χ from the three HRP data groups used in this study. (a) Data from above the summit of Fieberling Guyot collected as part of the TOPO HRP survey. (b) Data from the flank of Fieberling Guyot, about 20 km off the axis of the summit. (c) Data from the NATRE HRP survey. The TOPO site is characterized by predominantly doubly stable stratification (Rρ < 0). In each profile of Ri, a reference value of 0.5 is shown as the dashed vertical line. The high occurrence of low Ri above the seamount distinguishes the summit profiles (a) from those above the seamount flanks (b).
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
The plausible range of three nondimensional parameters for finger-favorable stratification (Rρ > 1). The logarithmic axis for Rρ − 1 is useful because the distribution of finger-favorable Rρ is approximately lognormal about Rρ = 2. (a) The mixing efficiency of turbulent mixing (Rf). Laboratory data supports the range 0.05 < Rf < 0.20. (b) The buoyancy-flux ratio of salt fingers (r) from laboratory data. (c) The range of the dissipation ratio for turbulence and salt fingers are shown. The turbulence model was calculated using Γ(t) = Rf(1 − Rf)−1, with Γ(t) = 0.2 taken as the nominal value. The finger model was computed from the laboratory data for r, with average values of r binned by Rρ. The 99% confidence band is shown for the finger model, while the range for the turbulence model is dictated by 0.05 < Rf < 0.20.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Histogram of the TKE dissipation rate for the subset of observations having χ > χ75 for (a) all finger-favorable data and (b) all doubly stable data. The upper quartile of the combined TOPO and NATRE thermal dissipation record is χ75 ∼ 1 × 10−9 K2 s−1. The bimodal structure in the finger-favorable histogram is statistically significant at the 0.15 level. The modes are located at ϵ ∼ 2 × 10−10 W kg−1 and ϵ ∼ 1 × 10−9 W kg−1.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
The finger-favorable distribution of ϵ(χ > χ75) is broken into four distinct regions of (Rρ, Ri) data space. In each histogram, the ϵ bins corresponding to the modes of the distribution shown in Fig. 6a are emphasized by darker shading.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
The dissipation ratio of doubly stable observations with χ > χ75. The data were grouped into six different Ri populations (a)–(f). Estimates were derived separately for each of the three HRP data groups: NATRE (circles), TOPO seamount flank (triangles), and TOPO seamount summit (squares). Each symbol is the estimate Γ = (〈N2〉〈χ〉)(2〈ϵ〉〈Θz〉2)−1, where the ensemble average was computed for a bin of 1000 dissipation estimates. The error bars give the 95% confidence interval for a reduced number (<1000) of degrees of freedom. In each panel, a reference value of Γ = 0.2 is shown.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
The dissipation ratio of finger-favorable observations with χ > χ75.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Estimates of the salt finger buoyancy-flux ratio r = (αkθΘz)(βksSz)−1. The results of four laboratory experiments are shown: Turner (1969, diamonds), Schmitt (1979b, circles), McDougall and Taylor (1984, upward triangles), and Taylor and Bucens (1989, squares). Also shown are the numerical results of Shen (1993, downward triangles) and the theoretical models of Stern (1975, light curve), and Schmitt (1979a, dark curve). Buoyancy-flux ratio estimates were derived from HRP observations by assuming that the NATRE (χ > χ75, Ri > 5) dissipation ratio signal (Fig. 9f) was entirely attributable to salt fingers. The relation r = RρΓ(RρΓ + Rρ − 1)−1 was used. The HRP data-derived estimates are shown with a 99% confidence band.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
A contrast between Γ(Rρ, Ri) of doubly stable and finger-favorable exceptional χ observations. Observations were grouped into bins of the [log(|Rρ| − 1), log(Ri)] parameter space, and Γ was estimated in bins where the data density exceeded 100 degrees of freedom (all shaded regions). In each panel, the estimate Γ = (〈N2〉〈χ〉)(2〈ϵ〉〈Θz〉2)−1 is contoured. Regions of darker shading denote bins where the uncertainty of Γ exceeds 25% of the mean. The doubly stable regime (a) is characterized by a broad parameter range having Γ = 0.16 (inner three contours). The finger-favorable regime (b) is characterized by larger values of Γ, particularly in the (low Rρ, large Ri) region of parameter space [quadrant II, (Rρ < 2, Ri > 1)].
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Statistical classification of the exceptional χ observations in the finger-favorable regime. (a) The observations were tested against the null hypothesis that Γ was attributable to turbulence. The finger-favorable regime can be broken into the four quadrants defined by the [log(Rρ − 1), log(Ri)] data space (I–IV as labeled). Most of quadrants I, III, and IV are consistent with the turbulence hypothesis at the 0.01 significance level (99% confidence), and these bins are identified by dark shading. (b) The observations were tested against the null hypothesis that Γ was attributable to salt fingers. Data consistent with the finger hypothesis are indicated by dark shading. All of the quadrant II data meet consistency with the finger model at the 0.01 significance level, as do several bins already found to be consistent with turbulence. (c) The percentage of total data represented by each bin is reported. Those data with degenerate classification are assigned to turbulence. This results in the classification of nearly 70% of the data as turbulence (darkest shading) and 30% as salt fingers (medium shading). Ntotal = 119 570.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Results from the analysis of neutral density layers at the NATRE site. The mean depth of the neutral surfaces are used as the reference vertical coordinate. (a) The density ratio structure of each layer is shown. The shaded box is the interquartile range of the 5-m Rρ, with the median value shown as the vertical line. The large-scale density ratio is computed from the layer-averaged
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
The diapycnal advection w∗ computed using the profiles of kθ and ks shown in Fig. 13b. (a) The epineutral component of the diapycnal advection. This small contribution comes mainly from the cabbeling term. (b) The diapycnal-flux component of the diapycnal advection. Only layers at
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
The diapycnal advection w∗ when no salt-finger enhancement of the haline diffusivity is used. (a) The scalar diffusivity of turbulence k0 = 0.2〈ϵ〉(
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Comparison of the finger diffusivities for salt as estimated by the NATRE microstructure and salt-finger models. The Kunze (1987, 1994) models are based on stability considerations. The Zhang et al. (1998) model is based on the the Schmitt (1981) ad hoc model.
Citation: Journal of Physical Oceanography 29, 7; 10.1175/1520-0485(1999)029<1404:TCOSFT>2.0.CO;2
Net diffusivity estimates.
Components of the dianeutral advection.
Flux components of the dianeutral advection equation.
