Abstract

Throughout much of the ocean interior, the diapycnal buoyancy flux is maintained by both mechanical and double-diffusive processes. Assessing the relative roles of each is a challenge, particularly in complex coastal environments. During February–March 1995, a repeat-profiling CTD system, equipped with a dual-needle microconductivity probe, was deployed off the central California coast (35°N, 121°W) from the research platform FLIP. The probe’s vertical resolution (8 cm) appears sufficient to resolve the low wavenumbers of the turbulent inertial subrange. This paper presents depth–time maps, spanning 12 days and 100–400 m, of temperature dissipation rate χ̂, and Cox number Ĉ. High χ̂ and Ĉ values tend to occur in layers, on a variety of spatial scales.

Simultaneously, finescale (6.4-m) Richardson number, effective strain rate, and Turner angle are measured. The occurrence of intense microstructure fluctuations is correlated with all three quantities, affirming that both mechanical turbulence and double diffusion are active at the site.

Depth-averaged dissipation rate ɛμ is inferred from the χ̂ records under the assumption that a Batchelor spectrum for scalars obtains and that the buoyancy flux Jb and dissipation ɛ are related through a constant mixing efficiency Γ, Jb = Γɛ. Time series of ɛμ are highly correlated with dissipation rate computed from Thorpe scales (ɛT), estimated from large (2 m and greater) density overturns (except during periods when large portions of the water column are double-diffusively unstable: ɛμ ≫ ɛT in these regions, suggesting enhanced fluxes due to double diffusion.

1. Introduction

In regions of the ocean where temperature and salinity contributions to the density gradient are in opposition, double-diffusive fluxes may equal or exceed those due to mechanical turbulence caused by breaking internal waves (Schmitt 1994; Mack and Schloberlein 1993; St. Laurent and Schmitt 1999). While detailed observations of both phenomena have been made, comprehensive measurements that can identify both processes and distinguish their differing roles in the evolution of the thermocline are rare. Such insights are required in order to advance mixing parameterizations in large scale numerical models.

Alford and Pinkel (2000, henceforth AP) present observations of mechanical mixing processes (the occurrence of 2-m overturns) in the Marine Boundary Layer Experiment (MBL). These are related to 6.4-m shear (S), 2-m strain (γ = N2/N2, where N is the buoyancy frequency), 2-m effective strain rate

 
formula

and 6.4-m Richardson number (Ri = N2/S2). They observe an increased likelihood of overturning in the presence of small (5–20-m vertical scale) internal waves, indexed by high (∂ŵ/∂z)2. High strain (weak stratification) and low 6.4-m Ri also tend to accompany overturns.

In the present work, the focus is broadened to consider double-diffusive phenomena. Intensive depth–time measurements of temperature, salinity, density, and 8-cm microconductivity fluctuations were sampled every 4 minutes during MBL over a 300-m vertical aperture. Estimates of traditional microstructure quantities are formed from these directly measured microconductivity data, and related to the overturns and the background finescale T–S fields. Mixing structures are repeatedly sampled as they drift past the Research Platform FLIP. The ability to track individual “microstructure” patches in depth/density and time greatly facilitates the effort, as does the ability to examine patch evolution in an isopycnal-following frame.

Centimeter-scale conductivity gradient fluctuations and meter-scale density overturns represent distinctly different phenomena. To oversimplify, overturns initiate turbulent motion, while conductivity microstructure can represent the “aftermath” (AP), or indicate double-diffusive activity. Overturns are restored to gravitational stability within several buoyancy periods (15–30 min in the present observations). The resultant conductivity signals (which are dominated by temperature) can persist much longer. Once formed, passive temperature fluctuations at the 8-cm limit of our probe’s resolution would be diffused away in τ = L2K−1T ≈ 12 h.

While microstructure can result from either mechanical turbulence or double-diffusive processes, large (>2 m) density overturns are not likely to be formed by double diffusion, whose net effect is always to increase density stratification.

We approach the present observations with the hypothesis that both mechanical and double-diffusive fluxes are present. The corresponding conductivity signals are termed “turbulent” and “double-diffusive” microstructure.

Turbulent microstructure and overturning should display generally similar relationships with the Ri and (∂ŵ/∂z)2 fields, which index internal wave instabilities. Double-diffusive microstructure, on the other hand, should occur in regions where the density ratio Rρ and (equivalently) Turner angle Tu (Ruddick 1983) indicate that the water is unstable to either salt fingering or diffusive convection.

As in AP, we attempt to use dense depth–time sampling to gain a view of the evolution of upper-ocean processes. We anticipate the presence of mixing “patches,” “puffs,” and “wisps,” (Gregg 1987) as well as zones of salt fingering and diffusive regime that accompany thermohaline intrusions. The various structures presumably result from modulations in the various finescale fields, which index shear, convective, salt fingering, and other instabilities. Given the multiplicity of phenomena at MBL, the discussion takes the tone of a survey, rather than an exhaustive account.

Descriptions of the experiment (section 2), and the instruments and methods used (section 3), follow. We then present depth–time maps of the entire 12-day cruise (section 4), and 12-h close-ups (section 5). In section 6, we undertake a statistical summary of the variety of phenomena associated with microstructure. Section 7 compares dissipation rate estimates based on microconductivity observations to those inferred from overturning scales. We summarize and conclude in section 8.

2. Experiment

Phase I of the Marine Boundary Layer Experiment was conducted from the Research Platform FLIP, 30 km west of Point Argüello, California (35°N, 121°W), during February–March 1995. Data used in this study were collected over the 12-day period yearday 56–68. During the first 10 days, FLIP was secured with a one-point mooring in 1.5 km of water. On 6 March (yearday 66), the mooring line was released, and FLIP drifted freely. An eight-beam Doppler sonar system and a repeat-profiling CTD sampled the upper 400 m of the water column. An SBE dual-electrode microconductivity cell was mounted in a special housing below the CTD package. Details of the experiment, and a schematic of the CTD/sonar array, may be found in AP.

High shears and large baroclinic tidal displacements characterized the MBL site throughout the experiment (AP). Rms 6.4-m shear is about 1.4 times the Garrett–Munk 1975 (GM) value, for depths above 250 m. Below this, mean shear increases with depth (rms shear is 3 times the GM value at 400 m), suggesting a deep source of wave energy at the site. Rms baroclinic tidal displacements are typically 20–30 m. Larger (40–50 m) displacements occurring in a limited depth range near 150 m during yearday 60–62 are associated with the passage of baroclinic energy along a tidal ray, presumably stemming from nearby topography.

This region of the California coast is known to contain T–S variations (fronts, “squirts,” “jets,” and intrusions) on a variety of time/space scales. These modulate the double-diffusive stability of the water column (e.g., Gregg 1975). Of particular relevance to the present study is the passage of three sloping T–S structures ∼50 km wide over yeardays 59–62 (section 4). Layers of double-diffusively unstable water bound these features above and below. In addition, smaller thermohaline intrusions continually advect through the MBL site. Bursts of heightened temperature dissipation rates χ̂ coincident with these features suggest double-diffusive processes are active.

3. Instruments and methods

a. Sensors

During MBL, velocity profiles (0–400 m) are obtained from a Doppler sonar mounted on the stern of FLIP at a depth of 87 m. Every minute, estimates of velocity (3.2 m vertical resolution) are recorded. Averages are subsequently computed over 4 minutes to facilitate merging with the CTD data. Use of repeat sequence codes (Pinkel and Smith 1992) improves precision of the system.

An automatic CTD profiles temperature, conductivity, and pressure every 4 min from the surface to 415 m. The CTD, a modified Sea-Bird Instruments SBE-9 unit (Fig. 1a), falls at approximately 3.6 m s−1. At this speed, the 24-Hz sample rate corresponds to a measurement every 16 cm. Conductivity data are corrected for thermal inertia effects (Lueck and Picklo 1990; Morison et al. 1994). A cross-spectral technique, described in Anderson (1993), is then employed to match the responses of the temperature and conductivity probes. Data are low-pass filtered with a 2-m cutoff prior to calculating salinity and density profiles.

Fig. 1.

(a) The CTD package used during MBL. The microconductivity probe is mounted approximately one meter below the rest of the package to avoid wake effects. (b) Microconductivity probe close-up.

Fig. 1.

(a) The CTD package used during MBL. The microconductivity probe is mounted approximately one meter below the rest of the package to avoid wake effects. (b) Microconductivity probe close-up.

Microconductivity data are obtained from a Sea-Bird Instruments SBE 7-02 microconductivity sensor (Fig. 1b). The probe consists of two electrodes separated by approximately 1 mm. It is mounted 80 cm below the main CTD package in a special protected unit. At the 3.6 m s−1 fall speed, a 96-Hz sample rate yields data every 4 cm. Prior to digitizing, the analog signals are passed through two antialiasing filters (4 pole Bessel with a time delay of 19.1 msec). These provide a minimum of 20-dB signal reduction for frequencies above the Nyquist.

Ten sample microconductivity profiles from 100 to 400 m near yearday 64.1 (Fig. 2a) demonstrate the probe’s ability to detect large- and small-scale signals. Fifty-meter close-ups of microconductivity (Fig. 2b) and microconductivity gradient (Fig. 2c) data surrounding the energetic “patch” (Fig. 2a) show irregular features, steps, and inversions over a 30-m depth range. Numerous 2-m overturns in density (not shown) accompany the patch. Since the region is stable to double diffusion, the patch presumably results from mechanical turbulence.

Fig. 2.

Cascade plots of conductivity (a). Close-ups of conductivity (b) and conductivity gradient (c) surrounding an active “patch.” Ten profiles are plotted, spanning 40 minutes near yearday 64.1. Traces are offset by (a) 0.02 S m−1 in (a), 0.01 S m−1 in (b), and 0.1 S m−2 in (c). Since the region is stable to double-diffusive processes, the patch is assumed to be due to mechanical turbulence. (d) Inferred temperature gradient spectrum from inside the patch (black line). The 20 m of data (512 points) from each of the 10 profiles centered on the patch is tapered with a triangle window. The spectrum is computed by averaging the 10 individual spectral estimates and is smoothed over Δ(log10k) = 0.2 before plotting. When divided by the modeled microconductivity cell response T(k) = sinc2(πk/kN) (gray line), observed slope is roughly +⅓ (dotted line), consistent with the inertial subrange of the turbulent spectrum. The observed signal is well above the noise, in this active region.

Fig. 2.

Cascade plots of conductivity (a). Close-ups of conductivity (b) and conductivity gradient (c) surrounding an active “patch.” Ten profiles are plotted, spanning 40 minutes near yearday 64.1. Traces are offset by (a) 0.02 S m−1 in (a), 0.01 S m−1 in (b), and 0.1 S m−2 in (c). Since the region is stable to double-diffusive processes, the patch is assumed to be due to mechanical turbulence. (d) Inferred temperature gradient spectrum from inside the patch (black line). The 20 m of data (512 points) from each of the 10 profiles centered on the patch is tapered with a triangle window. The spectrum is computed by averaging the 10 individual spectral estimates and is smoothed over Δ(log10k) = 0.2 before plotting. When divided by the modeled microconductivity cell response T(k) = sinc2(πk/kN) (gray line), observed slope is roughly +⅓ (dotted line), consistent with the inertial subrange of the turbulent spectrum. The observed signal is well above the noise, in this active region.

To investigate the response of the microconductivity probe, the conductivity gradient spectrum is estimated from 10 profiles inside the “patch” (Fig. 2d, heavy black line). A 512-point (20 m) section of each profile, centered on the instantaneous location of the patch, is tapered with a triangle window and Fourier transformed. The 10 resulting spectral estimates are averaged together, and smoothed over Δ(log10k) = 0.2 before plotting. The spectrum is expressed as an equivalent temperature gradient spectrum ϕ̂TG(k) by multiplying conductivities μ by b ≡ ∂T/∂μ = 9.16 [°C/(S m−1)].

Significant attenuation is observed for k > 6 cpm, due to the finite sampling rate of the conductivity probe [based on measurements in a stratified tank, Meagher et al. (1982) estimate that the inherent spatial response of the SBE 7-02 probe is uniform to 100 cpm]. It is possible to correct the spectrum for degradation associated with finite sampling. Formally, the inverse of the antialiasing filter applied in the instrument’s A/D processing, expressed in terms of wavenumber, is the appropriate correction. However, T(k) = sinc2(πk/kN) [where sinc(x) = sin(x)/x, and kN = 13.3 cpm is the Nyquist wavenumber] is a good approximation to the Bessel filter (Fig. 3). The sinc2 filter is the appropriate spectral correction if each sample represents a uniformly weighted average over an interval 1/(2kN). When the sinc2 correction is applied to the wavenumber spectrum, a spectral slope consistent with +⅓ (Fig. 2d, dotted line) is evident for 2 cpm > k > 9 cpm. This is consistent with the inertial subrange of the scalar dissipation spectrum, suggesting that the probe is in fact detecting turbulent motions.

Fig. 3.

Probe response functions (a) T(k) = sinc2(πk/kN) compared with Sea-Bird’s Bessel antialiasing filter (b).

Fig. 3.

Probe response functions (a) T(k) = sinc2(πk/kN) compared with Sea-Bird’s Bessel antialiasing filter (b).

To quantify sensor noise, spectra of microconductivity gradient are computed in a quiescent region. A noise variance of σ2N = 5 × 10−10 (S m−1)2 is estimated. Assuming that noise is white in microconductivity, noise in the gradient should exhibit a k+2 slope:

 
formula

Revisiting the active turbulent patch (Fig. 2d), the modeled noise spectrum is found to be well below the observed spectral levels for k < 9 cpm, increasing until it approaches the measured spectral level near 9 cpm (SN = 0.62 × 10−4 °C2 cpm−1 at 9 cpm). For k > 9 cpm, the corrected spectrum begins to rise above k+1/3, indicating that noise may be significant at the highest wavenumbers. Still, in the most active regions such as this one (which dominate contributions to χ̂ and Ĉ), noise appears to be a minor effect.

b. χ̂ and Ĉ

Temperature-fluctuation dissipation rate, χ̂, and Cox number, Ĉ, can be estimated from the microconductivity profiles. We emphasize that the 4-cm sampling of our data is inadequate to resolve most of the variance in χ and C. These quantities are underestimated by a dissipation-rate-dependent factor of 6–100 (appendix). Use of the caret (χ̂, Ĉ) distinguishes estimates from the fully resolved quantities. We optimistically assume that space/time patterns in χ̂ and Ĉ are not significantly different those of the true χ and C.

Given a microconductivity profile μ(z), the Cox number estimate is given by

 
formula

Overbars represent 1-m vertical averages centered about zi. The factor of 3 accounts for the contribution from horizontal gradients, under the assumption (suspicious at these scales) that the conductivity fluctuations are isotropic.

In regions of small background mean gradient the denominator in (2) may be unduly influenced by measurement noise. Very large, spurious, Ĉ estimates can result. To minimize this influence on Cox number variability, a threshold denominator Dmin = 10−10 (S m−1)2 is enforced. Occurrences of denominator D < Dmin are replaced by Dmin before Cox number is computed. The appearance of the depth–time maps is not influenced by the exact choice of Dmin.

Temperature fluctuation dissipation rate, χ̂, is given by

 
formula

Isotropy is again assumed. In computing χ̂ and Ĉ it is assumed that conductivity variability is dominated by temperature at the scales of interest; that is, ∂T/∂z = bμ/∂z.

c. Semi-Lagrangian reference frame

Interpretation of Eulerian records is complicated by the vertical advection of the thermocline by internal waves. We choose to examine the time evolution of χ̂ and Ĉ in an isopycnal following, or semi-Lagrangian (s-L), reference frame. To compute s-L quantities, a set of reference isopycnals, separated on average by one meter, is formed using the SBE-9 density profiles. Linear interpolation is used to map the Eulerian observations onto isopycnal surfaces. “Depth–time” s-L maps are presented as functions of the mean depth of each reference isopycnal, and time. On these plots, the cross-isopycnal migration rate is given by the apparent slope of isosurfaces.

d. Ri and (∂ŵ/∂z)2

The Richardson number Ri = N2/S2 is a measure of the potential for Kelvin–Helmholtz instability. Parallel, stratified shear flow is stable to small perturbations for Ri > ¼ (Miles 1961; Howard 1961). Following AP, 6.4-m Richardson number estimates are formed using combined CTD and sonar data. Shear-squared profiles (S2) are computed from 6.4-m vertically differenced velocity. Profiles of buoyancy frequency N are formed by differencing the density profiles after “re-sorting” the overturns to enforce monotonicity. These are then smoothed with a 6.4-m Butterworth filter. The 6.4-m Ri is computed from these instantaneous S2 and N2 profiles. Here ∂ŵ/∂z is computed from CTD data as in AP. Briefly, between any two time series of isopycnal depth z(t, ρi) and z(t, ρj), the normalized depth difference is given by

 
formula

where z ≡ [z(ρi) + z(ρj)]/2 is the mean depth of the pair, and Δzz(ρi) − z(ρj) is the mean separation. The effective strain rate is then defined by

 
formula

We distinguish between this “effective strain rate,” ∂ŵ/∂z, and the true strain rate or vertical divergence, ∂w/∂z, since advection of horizontal structure can contribute significantly to ∂ŵ/∂z.

Depth–time maps of ∂ŵ/∂z reveal wavelike features (AP, Fig. 13). Individual crests within well-defined groups can often be seen. The effective strain rate magnitude, (∂ŵ/∂z)2, highlights the “envelopes” of these groups and is strongly correlated with the location of overturning events (AP).

e. Turner angle

The double-diffusive stability of the water column is indexed by the Turner angle (Ruddick 1983)

 
formula

where

 
formula

A one-to-one mapping exists between Tu and the commonly measured density ratio,

 
formula

For hydrostatically stable water, |Tu| < 90°, where |Tu| < 45° indicates double-diffusive stability. Values lower than this range (−90° < Tu < −45°, cold, fresh water overlying warm, salty) indicate susceptibility to the “diffusive regime.” Higher values (45° < Tu < 90, warm salty overlying cold fresh) indicate the possibility of salt fingers.

To estimate Tu, temperature and salinity are computed on isopycnals. Differences are computed between isopycnals whose mean separation is 4 m. Here Tu is then calculated using Eq. (6).

4. Twelve-day maps of scalar dissipation and Cox number

a. General features

Semi-Lagrangian depth–time maps of log10(χ̂) (Fig. 4a) and log10(Ĉ) (Fig. 4b) are presented for the 12-day MBL record. Since signals above 87 m (the bottom of FLIP) are potentially contaminated by FLIP’s wake, we restrict our study to a 100–400-m range. Prior to plotting, the logarithm of each quantity is averaged over 1 h in time and 15 m in depth. Black vertical bars represent periods where the CTD and/or the microconductivity probe was out of service.

Fig. 4.

Twelve-day semi-Lagrangian depth–time series of (a) temperature fluctuation dissipation rate χ̂ and (b) Cox number Ĉ. The ordinate is the isopycnal whose mean depth is indicated. The logarithm of each quantity, smoothed over 1 h in time, and 15 m in depth, is plotted. The color scale for χ̂ is marked above the color bar, and that for Ĉ is marked below.

Fig. 4.

Twelve-day semi-Lagrangian depth–time series of (a) temperature fluctuation dissipation rate χ̂ and (b) Cox number Ĉ. The ordinate is the isopycnal whose mean depth is indicated. The logarithm of each quantity, smoothed over 1 h in time, and 15 m in depth, is plotted. The color scale for χ̂ is marked above the color bar, and that for Ĉ is marked below.

Here χ̂ activity is apparent on a variety of time and space scales. We associate persistent patches of elevated χ̂ and Ĉ with double-diffusive processes. Shorter-lived patches (5–20 m thick, duration several hours), such as the one at yearday 64.1, depth 200 m (shown in Fig. 2) appear to be of mechanical origin.

The scalar dissipation χ̂ [Eq. (3)] is weighted relative to Ĉ by the mean conductivity gradient squared [Eq. (2)]. Thus, χ̂ is much higher in the highly stratified region near 100 m, yearday 67–68, and in other regions of strong mean gradient.

Heightened χ̂ and Ĉ levels are present at all depths beginning at yearday 59. Subsequently, three features (beginning at about 100, 220, and 280 m at yearday 59) appear to migrate downward. The middle, most intense one may be followed for a day and a half, when it disappears near 300 m. The top one disappears and reappears several times, but may be tracked until yearday 64. Each extends approximately 20 m in the vertical and appears to migrate downward at approximately 50 m day−1. These features are associated with the advection of sloping, double-diffusively unstable lenses of water past the experimental site. The interpretation of observed temporal evolution as advected horizontal structure is discussed in section 4b. The double-diffusive stability of the lenses is discussed in section 4c.

b. Advection

Advection affects the observed temporal variability significantly in these moored observations. A progressive vector diagram for the 12 days (Fig. 5) provides a useful index of the flow of water past FLIP.

Fig. 5.

MBL progressive vector plot. Velocity profiles are averaged over 120–350 m in depth and integrated in time. FLIP was moored from the beginning of the experiment until yearday 66. Then, the mooring cable was released, after which FLIP drifted freely. During the moored portion, FLIP continually sampled “new” water.

Fig. 5.

MBL progressive vector plot. Velocity profiles are averaged over 120–350 m in depth and integrated in time. FLIP was moored from the beginning of the experiment until yearday 66. Then, the mooring cable was released, after which FLIP drifted freely. During the moored portion, FLIP continually sampled “new” water.

While moored (yeardays 59–66), the current trajectory reflects the superposition of strong westward flow and weaker inertial motions (evident as daily “wiggles” in the progressive vector). During this moored period, no loops are present, indicating that inertial motions are not strong enough to cause the same water to pass FLIP twice. Nevertheless, the time coordinate of our depth–time maps indicates a mix of intrinsic and advective variability. We can model the persistent structures as “frozen” in time,1 and we can use observed advective velocities to estimate lateral scales. Using a nominal value of 20 cm s−1, one day corresponds to 17 km.

c. T–S structure

Coincident with the large, slanted χ̂, Ĉ features beginning at yearday 59, three sloping frontal features pass underneath the experiment site, evidenced by distinct changes in T and S along isopycnals. The observed heightened microconductivity levels appear due to double-diffusive processes occurring at the interfaces.

We investigate this hypothesis by examining records of isopycnal temperature anomaly (Fig. 6a). Density is primarily a function of temperature and salinity alone, given the limited depth range of these isopycnal surfaces. An analogous plot of isopycnal salinity anomaly therefore shows identical structure.

Fig. 6.

Twelve-day semi-Lagrangian depth–time series of isopycnal temperature anomaly (a). Temperature is computed relative to its 12-day mean value on each isopycnal surface. Relatively warm (and salty) water appears as red, cold (and fresh) water appears as blue. In (b), the Turner angle Tu is plotted. Blue and red regions indicate the potential for salt finger and diffusive convective instability, respectively. Overplotted on both (a) and (b) are regions where χ̂ > 10−9 (°C)2 s−1. Elevated χ̂ levels are associated with unstable Tu values due to sloping lenses of water advecting past FLIP (yearday 59–63).

Fig. 6.

Twelve-day semi-Lagrangian depth–time series of isopycnal temperature anomaly (a). Temperature is computed relative to its 12-day mean value on each isopycnal surface. Relatively warm (and salty) water appears as red, cold (and fresh) water appears as blue. In (b), the Turner angle Tu is plotted. Blue and red regions indicate the potential for salt finger and diffusive convective instability, respectively. Overplotted on both (a) and (b) are regions where χ̂ > 10−9 (°C)2 s−1. Elevated χ̂ levels are associated with unstable Tu values due to sloping lenses of water advecting past FLIP (yearday 59–63).

Temperature variation along an isopycnal is not sensitive to internal wave straining. Variability in Fig. 6a is therefore due to the horizontal advection of water of varying T–S characteristics past the experimental site. Prior to yearday 59, warmer, saltier water (relative to the 12-day mean) overlies colder, fresher. The water undergoes a sharp transition to colder, fresher (yearday 59), and back again (yearday 59.5). Above 230 m, all isopycnals experience the change simultaneously; below this, the transitions slope downward.

The layered structures are reminiscent of interleaving layers typically observed near fronts. An idealized interpretation of the layered structures in terms of frontal interleaving is possible. A vertical boundary between cold, fresh water (to the left in Fig. 6) and warm, salty water (to the right) is encountered at yearday 59.5. In accordance with the laboratory results of Ruddick and Turner (1979), a “nose” of cold, fresh water penetrates into the warmer water to the right. This interleaving process is assumed to have occurred at a previous time. The resultant interface–nose structure then sweeps past FLIP.2

Assuming all observed time changes are due to advection, the cold layer slopes downward to the southwest at about (50 m day−1)/(17 km day−1) = 0.3% or 0.26°. The protruding layer’s downward slope away from the interface is consistent with higher buoyancy fluxes due to salt fingering at the top of the layer than by the diffusive regime at the bottom. With a net buoyancy flux out of the layer, it sinks into the surrounding fluid as it advances (Stern 1967). This behavior is observed in laboratory experiments with salt and sugar (Turner 1978), and in the ocean (Gregg 1980; Joyce et al. 1978; Gregg and McKenzie 1979; Ruddick 1992; Anderson and Pinkel 1994).

The vertical scale of interleaving layers across a sharp T–S gradient has been estimated by Ruddick and Turner (1979). They consider the potential energy in a layer, before and after the T–S gradients have been “run down” by double-diffusive fluxes. This vertical scale is given by their Eq. (11)

 
formula

where n = αFT/βFs is the flux ratio, expressed in terms of density (n = 0.56 for T–S fingers). Along an isopycnal, βS| = αT|. Substituting into (7) and expressing the density gradient in terms of N2,

 
formula

Using observed values ΔT = 0.4°C, N = 2.53 cph, and α = 1.5 × 10−4 °C−1, g = 9.8 m s−2 yields H ≈ 20 m, consistent with the observed vertical scale of the lenses. Based upon their downward slope away from the vertical interface, and their vertical scale, the layers appear to be double-diffusively driven.

The protruding cold, fresh layers in Fig. 6b (e.g., yearday 60, 250 m) are associated with the potential for salt fingering (blue, Tu > 45°) on top, and diffusive regime (red, Tu < −45) below. The opposite holds for hot, salty layers. Regions where χ̂ > 9.0 × 10−9 (°C)2 s−1 are contoured with black lines in both 6a and 6b. It is clear from Figs. 4, 6 that the protruding lenses are associated with large χ̂ values.

Almost all of the persistent regions of elevated χ̂ visible in the 12-day records occur where |Tu| > 45°. Both regions with potential for salt-fingering (e.g., yearday 61–64, 250–350 m) and the diffusive regime (e.g., yearday 65–68, 100–250 m) display high χ̂ values, though correspondence is somewhat better with salt-fingering regions.

Of course, |Tu| > 45° only implies potential for double diffusion. The occurrence of mechanical turbulence is presumably unaffected by double-diffusive phenomena. Patches of turbulence often occur in double-diffusively unstable regions, as appears to be the case in some of the shorter-lived patches apparent in the 12-h maps (next section). But in the 12-day records, the patterns in Tu and χ̂ are so similar that we conclude that double-diffusive processes are in fact responsible. The downward tilt and vertical scale of the layers support this conclusion.

5. Twelve-hour maps of scalar dissipation and Cox number

The relationship between the microstructure and the fine-scale fields can be examined in more detail in 12-h maps. Three pairs of (χ̂,Ĉ) maps are presented for yearday 61.5–62.0, centered on one of the slanted T–S lenses. The first (Fig. 7) is in an Eulerian frame. The last two (Figs. 8, 9) are in a s-L frame. In these, regions are contoured where extreme values of the finescale fields Ri, (∂ŵ/∂z)2, and Tu occur. Viewing the three presentations together, one can gain an appreciation of the variety of instability processes that appear to be co-occurring.

Fig. 7.

Eulerian χ̂ (a) and Ĉ (b) from yearday 1995 61.5–62.0 (12 h). The logarithm of each quantity is smoothed over 4 m in depth and 12 min in time prior to plotting. Black dots mark the locations of overturns whose maximum Thorpe displacement Lmax >2 m. Solid black lines are isopycnal depths (20-m mean spacing).

Fig. 7.

Eulerian χ̂ (a) and Ĉ (b) from yearday 1995 61.5–62.0 (12 h). The logarithm of each quantity is smoothed over 4 m in depth and 12 min in time prior to plotting. Black dots mark the locations of overturns whose maximum Thorpe displacement Lmax >2 m. Solid black lines are isopycnal depths (20-m mean spacing).

Fig. 8.

Semi-Lagrangian χ̂SL (a) and ĈSL (b) for the same depth range and period as in (7). The ordinate is the isopycnal whose mean depth is shown. Black and white dots indicate overturn locations, as in (7). Solid black lines surround regions where (∂ŵ/∂z)2 > twice its mean value over the region, and solid gray lines indicate 6.4-m Ri < 1.

Fig. 8.

Semi-Lagrangian χ̂SL (a) and ĈSL (b) for the same depth range and period as in (7). The ordinate is the isopycnal whose mean depth is shown. Black and white dots indicate overturn locations, as in (7). Solid black lines surround regions where (∂ŵ/∂z)2 > twice its mean value over the region, and solid gray lines indicate 6.4-m Ri < 1.

Fig. 9.

As in Fig. 8 but solid black lines surround regions where Tu > 45° (salt fingering), and solid gray lines indicate Tu < −45° (diffusive regime).

Fig. 9.

As in Fig. 8 but solid black lines surround regions where Tu > 45° (salt fingering), and solid gray lines indicate Tu < −45° (diffusive regime).

a. Eulerian representation of χ̂ and Ĉ fields

Twelve-hour Eulerian records of log10(χ̂) and log10(Ĉ) are presented in Fig. 7. Prior to plotting, the logarithm of each quantity has been averaged over 4 m in depth and 12 min in time. Isopycnal surfaces, separated in the (12-day) mean by 20 m, are plotted as thin black lines. Locations of density overturns >2 m (AP) are marked with black dots.

The most prominent feature in both fields extends from the start of the record to the finish and is associated with one of the T–S lenses (Figs. 4 and 5). Midway through the record, a second, shallower, lens appears and eventually converges with the first.

Elsewhere, both χ̂ and Ĉ display the “patchy” nature of microstructure. Thin (<10 m) layers of elevated scalar dissipation, χ̂, are common. Many of these patches occur in highly stratified regions, resulting in high χ̂ but average Ĉ (e.g., yearday 61.77, 150 m). Features in both fields typically last several hours, implying length scales of several kilometers assuming advection at 20 cm s−1.

In general, layers of χ̂ and Ĉ demonstrate a strong tendency to follow isopycnal surfaces. Even rapid displacements are often tracked by both microstructure and overturning, as in the feature beginning at yearday 61.75, depth 300 m. Over the course of two hours, the high Ĉ feature remains between two isopycnals, which complete two up–down cycles of about 20 m amplitude. High χ̂ layers near 150-m depth also closely track isopycnals. Some cross-isopycnal migration is seen, however. This becomes more evident in the semi-Lagrangian maps presented in the next section.

Observed 2-m overturns display a tendency to lie within high χ̂ and Ĉ regions. Since salt fingering is not expected to generate large density overturns, mechanical turbulence is likely present in these regions. Note even the double-diffusively unstable lens discussed in the last section displays overturning, implying the simultaneous presence of double-diffusive and mechanical turbulent microstructure.

Better correspondence with overturning is seen in the Ĉ field than in χ̂, a result of the different N scaling of the two quantities. Here χ̂ tends to be large in highly stratified regions; Ĉ has this tendency removed. It has been found that large overturns are more likely in weakly stratified regions (AP). This emphasizes the importance of comparing quantities that scale similarly with N.

A few overturns are seen in low Cox number regions (e.g., 150 m and yearday 61.95, 150 m and yearday 61.72, 170 m and yearday 61.8). These may be sporadic, isolated events in which the overturn occurs below FLIP but the aftermath is “swept away.”

b. Mechanical turbulence: Comparison with Ri and (∂ŵ/∂z)2

Semi-Lagrangian quantities χ̂s-L and Ĉs-L for the same period and depth range are plotted in Figs. 8a,b. Observed 2-m overturns are overplotted as in Fig. 7. Regions where 6.4-m Ri < 1 are contoured with heavy gray lines. Heavy black lines correspond to regions where (∂ŵ/∂z)2 exceeds twice its mean value over the 12-h, 300-m domain.

Horizontal traces (where χ̂ or Ĉ layers remain on specific isopycnals) dominate the record. However, some cross-isopycnal migration is also present (e.g., yearday 61.6, 200 m). Without replenishment, turbulent patches are restricted to remain on isopycnals. In some circumstances, microstructure appears to follow disturbances as they propagate past isopycnals. Such features are reminiscent of a spilling breaker, wherein continually generated whitewater follows the passage of the surface bore.

Strong correspondence is observed between both (Ĉ,χ̂) and (∂ŵ/∂z)2 (heavy black lines). Apparently, microstructure patches, like overturns (AP), are more likely where (∂ŵ/∂z)2 is high.

Here χ̂ and 6.4-m Ri (Fig. 8a) are not strongly correlated. Gregg et al. (1986) observed that 10-m Ri was correlated with small “puffs” of turbulence but not with larger, more persistent mixing patches. Since observed high χ̂ regions may be patches, puffs, or double-diffusive regions (unrelated to Ri), one-to-one correspondence with 6.4-m Ri is not expected. Further, χ̂ is high in stratified regions, whereas 6.4-m Ri tends to be low in weakly stratified regions (AP).

Better agreement is seen between Ĉ and 6.4-m Ri. Low Richardson number regions often display high Cox number (e.g., yearday 61.55, 350 m; yearday 61.9, 175 m). However, examples exist of low 6.4-m Ri regions that persist for hours without a Ĉ signature (yearday 61.8, 180 m). Analogous persistent zones of low 6.4-m Ri that demonstrate no 2-m overturns are documented in AP and are also seen in these records. Evidently, instability does not always develop when 6.4-m Ri is low.

Likewise, Ĉ may also be large where the 6.4-m Ri > 1. The region near 200 m, yearday 61.55 is an example, as is the feature that migrates from 240 to 280 m near yearday 61.8. High (∂ŵ/∂z)2 contours and 2-m overturns track the Ĉ signatures, indicating that instability of small-scale internal waves may have led to the microstructure. As discussed in AP, shear at unresolved horizontal and vertical scales causes the “true” Ri to be lower than that reported here. Alternatively, such features may be persistent “patches” [shown by Gregg et al. (1986) to be poorly correlated with low 10-m Ri] or double-diffusive structures (for which Ri is irrelevant). The Turner angle for this region (next section) is examined to isolate the latter possibility.

c. Double diffusion: Comparison with Turner angle

The largest Ĉ and χ̂ features in the 12-day maps (Fig. 4) appear well described by Tu. To investigate this codependence on shorter time (space) scales, the 12-h χ̂ and Ĉ fields are replotted in Fig. 9. Black lines now enclose regions unstable to salt fingering (Tu > 45°). Gray lines correspond to the diffusive regime (Tu < −45°). This 12-h period is coincident with the passage of the T–S lenses; therefore, much double-diffusive activity is anticipated. Indeed, double-diffusively unstable layers 10–25 m thick pervade the record below 220 m. The sloping features evident in the 12-day maps (Fig. 6) are visible here, though the apparent aspect ratio is altered. High χ̂ and Ĉ levels are associated with most layers. However, some layers exhibit no microstructure (300 m, yearday 61.5 and 320 m, yearday 61.9).

Here χ̂ is better correlated with Turner angle than is Ĉ. Quite strong correspondence is seen with the lenses, as expected from section 4. For example, the Tu > 45° layer near 230 m, yearday 61.6 bifurcates. Associated microstructure tracks the split. Several regions of Tu > 45° exhibit high χ̂ and average Ĉ (indicating that the feature occurred in a high gradient region). Examples are at (230 m, yearday 61.5–61.7), and (330 m, yearday 61.8–62). The remaining high Ĉ regions appear better correlated with Ri and (∂ŵ/∂z)2 (Fig. 8b). The suggestion is that Ĉ is a better indicator than χ̂ of mechanical turbulence. This may be simply because double-diffusive processes tend to occur at high gradients (high χ̂, lower Ĉ), while overturns, low Ri, and high (∂ŵ/∂z)2 all occur in weak stratification (AP).

Salt fingering and diffusive structures associated with the passage of the lenses appear to make up a large portion of the microstructure in this 12-h record, suggesting that double-diffusive fluxes may be important in depth-averaged budgets. Indeed, estimates of kinetic energy dissipation rate, ɛ, based on the assumption that all observed χ̂ is due to mechanical turbulence are much larger than those based on the observed overturns during this period (section 7). Away from the lenses, double diffusion plays a lesser role. There, microconductivity fluctuations produce more reliable estimates of kinetic energy dissipation.

d. Double-diffusive processes and overturning

Salt fingering generates stabilizing density fluxes, and therefore cannot produce large overturns in density. On the other hand, mechanical turbulence is generated by density and shear fluctuations, and should therefore occur approximately independently of the T–S characteristics of the water column.

As a simple test of the independence of overturning and double-diffusive processes, the probability density of Turner angle (Fig. 10, black line) is computed for all observations between 200 and 350 m, yeardays 59–68 (the same range used in other PDFs computed in section 6, and in AP). The probability density is then computed using only data exhibiting 2-m overturns (dotted line).3 The technique allows statistical evaluation of the degree to which overturning is influenced by the value of local Tu.

Fig. 10.

Probability density function of the Turner angle. The PDF is computed using all data from 200–350 m and yearday 59–68 (solid line), and using only observations at overturns (gray line). To a first approximation, overturning appears to occur independently of Tu.

Fig. 10.

Probability density function of the Turner angle. The PDF is computed using all data from 200–350 m and yearday 59–68 (solid line), and using only observations at overturns (gray line). To a first approximation, overturning appears to occur independently of Tu.

The two distributions differ, but not greatly. The largest differences in the two PDFs appear in the double-diffusively stable regime, |Tu| < 45°. The two populations demonstrate nearly identical percentages of the water occupied by the various diffusive regimes (Table 1), suggesting that large-scale overturning occurs independently of double-diffusive processes.

Table 1.

Double-diffusive percentages.

Double-diffusive percentages.
Double-diffusive percentages.

e. Profiles and spectra inside double-diffusive regions

The small-scale structure within the intrusive, double-diffusive regions is of interest. Close-up profiles of two cold, flesh intrusions are presented in Figs. 11, 12. In the first (yearday 61.75, 310 m), elevated microstructure occurs in the salt-fingering-favorable region above the intrusion. In the second (yearday 61.8, 265 m), the more intense activity is in the diffusive-favorable region below. These displays are direct analogs of Fig. 2 (mechanical turbulence), spanning 40 min in time.

Fig. 11.

As in Fig. 2, but centered on a section of one of the sloping lenses near yearday 61.8. The cold, fresh intrusion is evident. Microconductivity fluctuations appear above and below, in salt-fingering- and diffusive- unstable layers, respectively. The salt-fingering signals appear stronger. Spectra taken inside the salt-fingering region demonstrate steeper slope than k+1/3.

Fig. 11.

As in Fig. 2, but centered on a section of one of the sloping lenses near yearday 61.8. The cold, fresh intrusion is evident. Microconductivity fluctuations appear above and below, in salt-fingering- and diffusive- unstable layers, respectively. The salt-fingering signals appear stronger. Spectra taken inside the salt-fingering region demonstrate steeper slope than k+1/3.

Fig. 12.

As in Figs. 2 and 11 but the focus is on fluctuations within the diffusive regime below the cold, fresh intrusion. In this case, diffusive regime fluctuations (below) are stronger than those in the salt-fingering zone (above). The spectrum inside the diffusive region exhibits a slope of approximately k+1/3.

Fig. 12.

As in Figs. 2 and 11 but the focus is on fluctuations within the diffusive regime below the cold, fresh intrusion. In this case, diffusive regime fluctuations (below) are stronger than those in the salt-fingering zone (above). The spectrum inside the diffusive region exhibits a slope of approximately k+1/3.

Whereas the conductivity profiles in Fig. 2a decrease monotonically on large scales (implying double-diffusive stability), the passage of a cold, fresh “nose” is evident in Fig. 11a near 310 m. Salt fingering is expected when warm, salty water overlies cold, fresh at the top of the layer. Irregular features are seen here, on the strong gradient above the cool nose (11b). Large fluctuations in microconductivity gradient (11c) are present above a relatively quiescent region in the center of the intrusion. Weaker fluctuations below the layer (where cold, fresh overlies hot, salty) appear to be associated with the diffusive regime.

The characteristic staircase signature of double diffusion is absent. Indeed, more “steps” appear in the “turbulent” patch (Fig. 2) than in Fig. 11. However, the spectrum (computed as in Fig. 2d) exhibits slope steeper than k+1/3 (11d), as is expected for spectra of salt fingering fluctuations (Gargett and Schmitt 1982). We caution that these spectra are pushing the probe’s vertical resolution limit.

Figure 12 also shows a cold, fresh intrusion. Again, activity is detectable above and below the intrusive lens. However, in this case the stronger, thicker signal appears below, in the diffusive regime. Here ϕTG for this region exhibits a slope closer to k+1/3. Perhaps convective motions within the diffusive cell have become turbulent.

In spite of the differences in spectral slope, no obvious differences are discernible by eye between the fluctuations within mechanically turbulent, salt fingering, and diffusive regime patches. As noted by Linden (1971), weak background turbulence may disrupt the double-diffusive structures.

6. C and χ: A statistical summary

Following the approach taken by AP, general trends in the depth–time maps are summarized using probability density functions (PDFs). These are formed using the same data as in AP; that is, the range 200–350 m, over a 9-day period from yearday 59–68. PDFs of χ̂ and Ĉ are formed in the Eulerian frame, using all 334 012 observations in this depth–time range. Then, dependence upon other variables is examined by forming conditionally sampled PDFs, using subsets of data that satisfy certain conditions in these other variables. These include situations where overturning is observed, unstable Turner angle, low 6.4-m Ri, and high (∂ŵ/∂z)2. Table 2 summarizes the various PDFs computed.

Table 2.

Conditionally sampled variables.

Conditionally sampled variables.
Conditionally sampled variables.

Heavy black lines (Fig. 13) represent full-population probability densities of χ̂ and Ĉ. The PDF of log10χ̂ is nearly lognormal with a peak at 1.99 × 10−10 °C2 s−1. Lognormal χ distributions are often observed (e.g., Gregg et al. 1986). Recall that χ̂ estimates are underestimates of the true χ. Still, the apparent lognormal distribution of observed χ̂ gives confidence that the observed signals, though underresolved, reflect true ocean microstructure.

Fig. 13.

PDFs of log10χ̂ (a) and log10Ĉ (b). Heavy black lines are computed using all observations from 200–350 m and yearday 59–68. Light lines are computed using subsets of data that demonstrate Tu > 45° (blue solid line), Tu < −45° (blue dotted line), Ri < 1 (red line), (∂ŵ/∂z)2 > 2 × 10−5 s−2 (green line), and overturns with Lmax > 2 m (black line).

Fig. 13.

PDFs of log10χ̂ (a) and log10Ĉ (b). Heavy black lines are computed using all observations from 200–350 m and yearday 59–68. Light lines are computed using subsets of data that demonstrate Tu > 45° (blue solid line), Tu < −45° (blue dotted line), Ri < 1 (red line), (∂ŵ/∂z)2 > 2 × 10−5 s−2 (green line), and overturns with Lmax > 2 m (black line).

The PDF of log10Ĉ displays a central tendency, but is skewed. The true Cox number often has a lognormal distribution (J. Sherman 1998, personal communication), though it is not strictly expected to (Davis 1996). Were Ĉ fully resolved, the PDF would be shifted to the right. Since the right side of the distribution would be shifted more than the left, accurate C estimates might appear more nearly lognormal than the underresolved quantity, Ĉ.

Subsampled PDFs (light lines) are shifted relative to the general population PDFs. The overturning population (black line) demonstrates a very slightly shifted χ̂ PDF. Correspondence is seen between χ̂ and overturning in the maps (Figs. 7–9). The PDF sampled using overturning locations should therefore be shifted to higher values relative to the general population PDF. However, the general tendency is for χ̂ to be high in strong stratification and for overturns to grow large in weak stratification. This reduces the expected shift. In addition, high χ̂ values in salt-fingering regions (which occur independently of overturns) are responsible. A more sensitive study (not attempted here) would involve a comparison of the PDF of χ̂ sampled only in double-diffusively stable regions.

By contrast, the PDF of Cox number, considering overturns only, displays a marked right shift relative to the overall population PDF. This is consistent with visual conclusions drawn from the maps.

As expected from the maps, PDFs of χ̂ and Ĉ are both shifted toward higher values when considering the salt fingering (blue lines) and diffusive (blue dotted lines) populations. However, the effect is much stronger on χ̂ than Ĉ, again because Ĉ is less sensitive than χ̂ to microstructure occurring in high-gradient regions.

For both χ̂ and Ĉ, the PDF of the diffusive subsample is right shifted slightly relative to that of salt fingering. Based on this result and close-ups of individual regions (Figs. 11, 12), both the fingering and diffusive forms of double diffusion appear able to generate strong conductivity fluctuations.

Considering only regions of 6.4-m Ri, the distribution is slightly left shifted relative to the general population. The poor correspondence of Ri with persistent mechanical turbulence patches (Gregg et al. 1986), and with double-diffusive structures, is responsible. In addition, the different dependence upon stratification of χ̂, and Ri play a role. For Ĉ, where this problem is avoided, differences are seen in the expected sense: the low Ri population exhibits a higher Ĉ distribution.

The strongest shift in both curves is seen in the subsample exhibiting (∂ŵ/∂z)2 > 2 × 10−5 s−2. Evidently, (∂ŵ/∂z)2 (which indexes the presence of small-scale wave groups) is a robust indicator of microstructure, as it is of overturns (AP).

7. Depth-averaged dissipation rate

Microconductivity fluctuations in MBL are due to a combination of turbulent and double-diffusive processes. Time series of a depth-averaged “effective dissipation rate” (ɛμ) can be computed by assuming that all fluctuations are due to mechanical turbulence. We anticipate that ɛμ will be a gross overestimate of true dissipation if double-diffusive processes, rather than turbulence, produce the fluctuations. By contrast, the dissipation rate inferred from observed density overturns (ɛT) is not affected by double-diffusive processes. By comparing these two different estimates of dissipation rate, the relative importance of the two sources of microstructure can be assessed.

The ɛμ estimate assumes that all fluctuations are due to turbulence. Assuming that the conductivity spectrum takes the Batchelor form and that the dissipation rate is simply related to potential energy production through a constant mixing efficiency Γ = 0.2 (Oakey 1982), ɛμ may be estimated in terms of observed χ̂. Though approximate, the estimate is not ad hoc; that is, there are no adjustable parameters in the calculation. The method is described in detail in the appendix.

We emphasize that ɛμ is an approximate measure based on numerous assumptions, some of which do not strictly hold in the present case. For example, the cruise-averaged Rρ, rather than its in situ value, is used in the calculation. Given the variability of Rρ (Fig. 6), this assumption is clearly suspect. The hope in computing ɛμ is that these and other errors do not bias the depth and time averages significantly.

Here ɛT is computed from the overturning (Thorpe) scale LT. The method is based on Dillon’s (1982) observation that the Ozmidov scale LO = ɛ/N3 = 0.8LT. The technique is described in AP. Briefly, for each 4-min CTD profile, the mean dissipation rate is evaluated by computing the volume average of the dissipation produced by the overturns:

 
formula

where ɛi = 0.64L2TN3 is the Thorpe-inferred dissipation rate accomplished by the ith overturn. Here LP,i is the vertical extent (patch size) of the ith overturn, and H is the depth range sampled. The method produces estimates that demonstrate the same magnitude and dependence upon shear and stratification as the Gregg (1989) parameterization (AP), which seems to be quite robust (e.g., Polzin et al. 1995). Since no “ground truth” dissipation rate measurements are available for MBL, we settle upon ɛT as a reasonable proxy.

Time series of ɛμ and ɛT (Fig. 14a) show extremely strong correspondence in spite of the numerous assumptions made. Here ɛμ nearly always exceeds ɛT, but during most of the record the difference is less than 40% of ɛT. No ad hoc scaling of either series has been made. The implication is that during regions of close agreement, observed microstructure is mainly turbulent and not double diffusive. This agreement increases our confidence that the microconductivity probe is indeed responding to turbulent signals.

Fig. 14.

(a) Time series of depth-averaged dissipation rate estimated from Thorpe scales of observed overturns (ɛT, black line) and microconductivity data (ɛμ, gray line). Here ɛμ is computed by assuming that conductivity fluctuations obey the Batchelor spectrum for turbulent scalars, and that a constant mixing efficiency applies. No ad hoc correction has been applied to either time series. Each time series is computed using data from 200 to 350 m and is smoothed with a 16-h convolution filter prior to plotting. (b) Percentages of the water column between 200 and 350 m, which are subject to salt fingering (pSF, dotted line), diffusive regime (pDR, dash–dot line), and either double-diffusive process (pDD = pSF + pDR, solid black line). The stable percentage pstable = 1 − pDD is plotted with a heavy black line. The ratio of the two dissipation rate estimates ɛTμ in (a) is plotted in gray. Where large portions of the water were double-diffusively unstable (e.g., yearday 59–60), ɛTμ ≪ 1, suggesting that some of the microconductivity fluctuations during these times were double-diffusive and not turbulent. By contrast, strong agreement is seen in regions (e.g., yearday 63–65) where pDD is small.

Fig. 14.

(a) Time series of depth-averaged dissipation rate estimated from Thorpe scales of observed overturns (ɛT, black line) and microconductivity data (ɛμ, gray line). Here ɛμ is computed by assuming that conductivity fluctuations obey the Batchelor spectrum for turbulent scalars, and that a constant mixing efficiency applies. No ad hoc correction has been applied to either time series. Each time series is computed using data from 200 to 350 m and is smoothed with a 16-h convolution filter prior to plotting. (b) Percentages of the water column between 200 and 350 m, which are subject to salt fingering (pSF, dotted line), diffusive regime (pDR, dash–dot line), and either double-diffusive process (pDD = pSF + pDR, solid black line). The stable percentage pstable = 1 − pDD is plotted with a heavy black line. The ratio of the two dissipation rate estimates ɛTμ in (a) is plotted in gray. Where large portions of the water were double-diffusively unstable (e.g., yearday 59–60), ɛTμ ≪ 1, suggesting that some of the microconductivity fluctuations during these times were double-diffusive and not turbulent. By contrast, strong agreement is seen in regions (e.g., yearday 63–65) where pDD is small.

Poor agreement (ɛμ ≫ ɛT) is expected if observed χ̂ is due to salt fingering and not turbulence. To investigate this possibility, the ratio ɛTμ is plotted in 14b as a thick gray line. Also plotted are percentages of the water column that are subject to salt fingering pSF, dotted line), diffusive regime (pDR, dash–dot line), and either double-diffusive process (pDD = pSF + pDR, solid black line). The stable percentage pstable = 1 − pDD is plotted with a heavy black line. Comparing these series to Figs. 4 and 6, the sloping lenses (yearday 59–62) occupy large fractions (∼40%) of the sampled water. The greatest differences between ɛμ and ɛT (yearday 59–60.5, yearday 67–68) occur when much of the water column is double-diffusively unstable (Fig. 14b). The best agreement (yearday 64–65) occurs when less than 10% of the water column is double-diffusively unstable. The agreement in this region is interpreted as evidence that both the Thorpe method and the microconductivity method are reasonable estimates of depth-averaged dissipation rate, provided that double-diffusive processes do not dominate. Large overestimates in the lens regions suggest that buoyancy fluxes supported by double-diffusive processes can be significant.

8. Summary and conclusions

Eight-centimeter scale microconductivity fluctuations have been monitored over a 12-day, 300-m domain in the California Current. The site demonstrates frequent overturns, large tidal displacements, and strong shears. These increase with depth below 250 m, implicating a deep or bottom energy source.

Between yearday 59 and 62, three sloping thermohaline intrusions drift past. Their vertical scale and sense of slope are consistent with laboratory models of double diffusion (Ruddick and Turner 1979). Elevated χ̂ and Ĉ are observed at the boundaries of these features, in regions where |Tu| > 45. The lenses and associated microstructure appear to be double-diffusively driven.

Close-up maps of χ̂ and Ĉ over a 12-h period centered on these features reveal that while the most persistent χ̂, Ĉ patches are associated with double-diffusively unstable Tu values, short-term correspondence is also found between Ĉ and both 6.4-m Ri and (∂ŵ/∂z)2: χ̂ is correlated with (∂ŵ/∂z)2 as well, but not with 6.4-m Ri (because high stratification, persistent mixing patches, and double-diffusive processes, which are poorly or anticorrelated with Ri, all lead to high χ̂). The 2-m overturns are also found to co-occur with both χ̂ and Ĉ. Since overturning, Ri, and (∂ŵ/∂z)2 are associated with mechanical turbulence, it appears that both mechanical turbulence and double-diffusive processes are active at the site.

This interpretation is supported by comparison of time series of depth-averaged dissipation rate inferred from overturning scales (ɛT; Dillon 1982) and from the microconductivity fluctuations (ɛμ; appendix) using a model that assumes a Batchelor scalar spectrum and a constant mixing efficiency. Good correspondence is observed between the two dissipation rate estimates during periods when only a small fraction of the water column is double-diffusively unstable. In the lens regions, where much of the water is double-diffusively unstable, ɛμ ≫ ɛT, consistent with microconductivity variance stemming from double-diffusive processes.

Fig. A1. Theoretical inertial and Batchelor subrange temperature gradient spectra [Eq. (A1)]. The spectra correspond to χ = 10−8 °C2s−1 and Re = ɛν−1N−2 = 22.5, 225, 2250 and 22 500 (ɛ = 2.74 × 10−10, 2.74 × 10−9, 2.74 × 10−8, 2.74 × 10−7 W kg−1). As dissipation rate and Batchelor wavenumber kB = (ɛ/υK)4 increase, more turbulent variance is shifted to wavenumbers >12 cpm (vertical line), the resolution of the probe.

Fig. A1. Theoretical inertial and Batchelor subrange temperature gradient spectra [Eq. (A1)]. The spectra correspond to χ = 10−8 °C2s−1 and Re = ɛν−1N−2 = 22.5, 225, 2250 and 22 500 (ɛ = 2.74 × 10−10, 2.74 × 10−9, 2.74 × 10−8, 2.74 × 10−7 W kg−1). As dissipation rate and Batchelor wavenumber kB = (ɛ/υK)4 increase, more turbulent variance is shifted to wavenumbers >12 cpm (vertical line), the resolution of the probe.

Fig. A2. The fraction of total temperature gradient variance resolved by the microconductivity probe, assuming a Batchelor spectrum, plotted vs ɛ over typical oceanic ranges. As the spectrum shifts to higher wavenumber, the probe resolves a smaller fraction of total χ. Depending on ɛ, only 1%–16% of variance is resolved. Nevertheless, reasonable inferences about turbulent parameters are possible.

Fig. A2. The fraction of total temperature gradient variance resolved by the microconductivity probe, assuming a Batchelor spectrum, plotted vs ɛ over typical oceanic ranges. As the spectrum shifts to higher wavenumber, the probe resolves a smaller fraction of total χ. Depending on ɛ, only 1%–16% of variance is resolved. Nevertheless, reasonable inferences about turbulent parameters are possible.

Acknowledgments

This work was funded by the Office of Naval Research under the Marine Boundary Layer Program. The authors wish to thank Eric Slater, Lloyd Green, Mike Goldin, and Chris Neely of the Ocean Physics Group at SIO for their help in the design, construction, deployment, and operation of the sensors used during MBL. The experiment would not have been possible without the expertise and cooperation of the captain, Tom Golfinos, and crew of FLIP. Discussions with Mike Gregg, Chip Cox, Eric D’Asaro, Myrl Hendershott, and Lawrence Armi were helpful.

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APPENDIX

Estimating Dissipation Rate from Underresolved Conductivity Gradient Fluctuations

Dissipation rate estimates are often based on underresolved shear measurements. In situations where the measured fluctuations exhibit an inertial subrange, then ɛ can be inferred from the spectral level in that range. The assumption is that the unresolved scales exhibit predicted spectral forms.

Underresolved scalar gradient data cannot be used in this manner since the spectral level in the inertial subrange is determined by both ɛ and χ. Though ɛ and χ are related through the physics of stratified turbulence, they are independent parameters in the Batchelor spectrum formulation. As a result, neither can be determined precisely without fully resolved spectra.

Nonetheless, approximate ɛ estimates can be made from the MBL microconductivity data. If the fluctuations are described by the Batchelor spectrum for scalars, then the fraction of total conductivity variance resolved by the probe may be expressed as a function of ɛ (as ɛ increases, more variance is shifted to higher wavenumbers, and the fraction at resolvable wavenumbers decreases). Assuming quasi-steady conditions, a constant mixing efficiency and constant density ratio, ɛ is a simple function of χ. The value of ɛ consistent with both constraints may then be determined. Observed correspondence between ɛμ and ɛT during double-diffusively stable periods (Fig. 14) implies that in the depth mean, the numerous assumptions made are reasonable (or at least the errors cancel out).

The resolution problem is illustrated in Fig. A1. Theoretical equilibrium range temperature gradient spectra ϕTG(k) are plotted, given by

 
formula

k* = cPr1/2kB is the wavenumber that separates the inertial and Batchelor subranges. Here Pr = νK−1T = 7 is the Prandtl number,

 
formula

is the Batchelor wavenumber, and c∗ = 0.0968 is a constant (Gibson and Schwarz 1963). The Batchelor subrange

 
formula

is given as a function of nondimensional wavenumber α = (2q)1/2k/kB. Here q is a universal constant, taken to be 2(3)1/2 (as argued theoretically by Gibson and Schwarz 1963). The parameter β = qc2/3 is not to be confused with the haline contraction coefficient used earlier in the text. Dillon and Caldwell (1980) provide a more complete discussion.

Representative Batchelor spectra are plotted in Fig. A1. Each has equal variance (χ ≡ 6K 0ϕTG(k) dk = 10−8 °C2 s−1 for all curves). As ɛ increases, kB increases, shifting more variance out to unresolved scales. Integrating out to the probe’s Nyquist wavenumber kN = 13.3 cpm yields measured χ̂ = 6K kN0T(k)ϕTG(k) dk. Here T(k) is the probe response function (section 2b), taken to be T(k) = sinc2(πk/kN) (if Sea-Bird’s antialiasing response function (Fig. 3) is used instead, there is little difference). We define r(ɛ) ≡ χ̂/χ as the ratio of turbulent temperature gradient variance at wavenumbers less than kN (which are resolved by the probe) to the total.

The ratio r(ɛ) (Fig. A2) is well fitted by a power law

 
r = cɛp = 1.99 × 10−5 ɛ−0.4 [W kg−1].

The dissipation rate ɛ determines the amount by which the conductivity variance is underestimated. This factor ranges from 1/4 to 1/100 over the range of dissipation rates shown. The stronger the dissipation rate, the smaller the percentage of resolved variance.

To estimate ɛ from χ̂, we compute χ̂pe, the rate at which potential energy fluctuations are dissipated:

 
formula

The density ratio Rρ represents the relative contributions of temperature and salinity to density. Strictly, χpe should be estimated using in situ Rρ. However, to avoid introducing too much complexity, a constant value of Rρ = −1 is used, consistent with the 12-day mean value.

Since the factors distinguishing χ̂pe from χ̂ are not affected by probe resolution, χ̂pe and χ̂ are underestimated by the same amount. That is,

 
formula

where c = 1.99 × 10−5 (if ɛ is expressed in W kg−1), and p = −0.4.

Here ɛ may be estimated in terms of χpe. On average, the buoyancy flux Jb is expressible in terms of kinetic energy dissipation and a mixing efficiency Γ (taken to be 0.2, Oakey 1982):

 
Jb = Γɛ.
(A4)

In a quasi-steady state, Jb = ½χpe (Seim and Gregg 1994). Thus ɛ is determined in terms of χpe:

 
formula

Combining (A3) and (A5), and solving for ɛ, we obtain ɛ in terms of the measured χ̂pe:

 
ɛ = (2Γc)−1/(p+1)χ̂1/(p+1)pe.
(A6)

Substituting (A2) yields, finally, ɛ in terms of the measured χ̂:

 
formula

Note that for a perfect probe (p = 0, c = 1) (A7) reduces to the familiar (A5), which results from the Osborn–Cox (1972) assumptions, with KT = Kρ and Jb = Γɛ.

The method assumes that the Batchelor spectrum obtains. In practice, the Batchelor subrange in temperature gradient spectra is usually observed in high Cox number situations (Dillon and Caldwell 1980). Additional assumptions are made, that is, that reasonable estimates of χpe are given by using in situ N2 but cruise-mean Rρ, that χpe yields the buoyancy flux (i.e., quasi-steady state), and that buoyancy flux and ɛ are related by a constant mixing efficiency Γ = 0.2. In spite of all these assumptions, remarkable agreement (Fig. 14) is obtained between ɛμ (which involves centimeter-scale microconductivity fluctuations) and ɛT (which involves 2-m overturns and greater). The tight agreement suggests that, during periods where double-diffusive processes are unimportant, both methods produce reliable depth-averaged dissipation rate estimates.

Footnotes

Corresponding author address: Dr. Matthew H. Alford, Ocean Physics Department, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA 98105-6698.

1

A simple test of the validity of this model is possible. Alford and Pinkel (2000) observe that |∂ŵ/∂z| is reduced immediately following overturning events, relative to values in advance. Since overturning represents irreversible processes, the observed differences imply that overturning alters the ∂ŵ/∂z field. Lacking a preferred spatial direction, some variability in ∂ŵ/∂z is therefore due to intrinsic temporal evolution. The same technique is applied to the Ĉ and χ̂ fields. Inasmuch as microstructure represents the aftermath of overturning events, χ̂ and Ĉ should be higher following overturns. However, we detect no difference in χ̂ and Ĉ after overturns relative to before them. Apparently, intrinsic temporal evolution in χ̂, Ĉ is more difficult to detect than in ∂ŵ/∂z, given their longer decay times.

2

That is, the observed layer slope does not represent sinking with time, but rather downward spatial slope away from the interface of a previously formed structure. Were the mean currents reversed, the structure would appear inverted, but the sense of cold, fresh water sloping downward into warmer water would remain.

3

Within a statically unstable region, |Tu| > 90°. Calculation of Tu on isopycnals, which are computed from sorted density profiles and therefore cannot cross, removes this possibility. Here Tu “at” an overturning point represents the relative T–S contributions to the background (4-m) stratification that the overturn is overcoming.