Diapycnal Diffusivity Inferred from Scalar Microstructure Measurements near the New England Shelf/Slope Front

Transcription

1 Iowa State University From the SelectedWorks of Chris R. Rehmann June, 2000 Diapycnal Diffusivity Inferred from Scalar Microstructure Measurements near the New England Shelf/Slope Front Chris R. Rehmann, Woods Hole Oceanographic Institution Timothy F. Duda, Woods Hole Oceanographic Institution Available at:

2 1354 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 Diapycnal Diffusivity Inferred from Scalar Microstructure Measurements near the New England Shelf/Slope Front* CHRIS R. REHMANN AND TIMOTHY F. DUDA Applied Ocean Physics and Engineering Department, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts (Manuscript received 8 September 1998, in final form 3 August 1999) ABSTRACT Conductivity microstructure was used to estimate the diapycnal thermal eddy diffusivity K T near the New England shelf/slope front in early August Two datasets were collected with a towed vehicle. One involved several horizontal tows in and above a warm, salty layer near the seafloor, and the other was from a tow-yo transect that sampled most of the water column. In the bottom layer, K T derived from microstructure is a factor of about 5 smaller than estimates derived from tracer dispersion at the same density level, and the diffusivity decreases sharply as the buoyancy frequency N increases: K T N 3.1. With several assumptions, this behavior is consistent with laboratory results for shear-driven entrainment across a density interface. The bottom layer cools as it moves up the shelf mainly due to diapycnal mixing, and a simplified temperature budget of the layer yields a diffusivity of m 2 s 1, which is between the values derived from microstructure and tracer dispersion. In the tow-yo transect, K T and the thermal variance dissipation rate T were high in a frontal region where intrusions were observed at several depths. Averaged over the entire transect, however, K T was slightly lower in water favorable for diffusive layering than it was in either water favorable for salt fingers or diffusively stable water. The eddy diffusivity estimated throughout the water column behaved as K T N , decreasing less sharply for increasing stratification than near the bottom. 1. Introduction Diapycnal mixing is one of many important processes governing coastal ocean flow and the structure of coastal scalar fields. For example, it may play a role in determining the thicknesses of, and flow fields within, fronts such as the New England shelf/slope water front (Chapman and Lentz 1994; Houghton 1997; Houghton and Visbeck 1998). To provide a basis for evaluating the role of diapycnal mixing near fronts on the continental shelf in stratified summer conditions, we present estimates of diapycnal eddy diffusivity near the New England shelf/slope water front, derived from towed measurement of dissipative-scale structure (microstructure). Diapycnal thermal eddy diffusivity K T values are presented as functions of stratification strength and strati- * Woods Hole Oceanographic Institution Contribution Number Current affiliation: Department of Civil and Environmental Engineering, University of Illinois at Urbana Champaign, Urbana, Illinois. Corresponding author address: C. R. Rehmann, Department of Civil and Environmental Engineering, University of Illinois at Urbana Champaign, 2527 Hydrosystems Laboratory, 205 N. Mathews Ave., Urbana, IL rehmann@uiuc.edu fication type both close to and well away from the seafloor. Previous studies of diapycnal mixing have used both direct and indirect observational techniques. Direct techniques include measurements of tracer dispersion (Ledwell et al. 1993) and turbulent heat flux (Fleury and Lueck 1994; Moum 1990; Moum 1996; Yamazaki and Osborn 1993). Indirect techniques involve measuring the dissipation rates of either turbulent kinetic energy ( ) or thermal variance ( T ) and inferring K T from approximate forms of the energy or thermal variance equations. For example, Osborn and Cox (1972) estimated K T from the dissipation rate T 2D T ( T ) 2, where D T is the molecular diffusivity and the overbar indicates an average. In their model, turbulence is steady and homogeneous and the thermal variance equation becomes a balance between the production rate and the mean dissipation rate: dt 2w T T, (1) dz where w T is the turbulent heat flux and dt/dz is the temperature gradient of the mean field. Defining an eddy diffusivity by w T K T dt/dz leads to K T 2. (2) 2(dT/dz) Gregg (1984) has shown that salinity terms generate T 2000 American Meteorological Society

3 JUNE 2000 REHMANN AND DUDA 1355 The towed vehicle and instruments are described in section 2, and the sampling procedure and background conditions are described in section 3. Results on mixing near the bottom and throughout the water column are discussed in sections 4 and 5, respectively, and a summary is given in section 6. FIG. 1. The experiment was conducted on the New England shelf about 120 km south of Cape Cod. The data to be discussed were collected during 11 deep tows and one tow-yo transect, shown as heavy lines. Deep tows 7 11 were southeast of deep tows 1 6. Bathymetry contours are also shown. negligible thermal variance, even in frontal regions, and the towed measurements of Washburn (1987) suggest that the balance between production and dissipation holds in a coastal environment. We use this model throughout this paper. Indirect methods usually use vertically profiling probes to measure dissipation rates, but the cost of obtaining enough profile data for reliable statistics can be high. In contrast, towed probes can provide more data at a specific depth in a shorter time. Problems with the towed-probe technique include noise contamination (Washburn and Gibson 1984) and limited thermistor response time. Progress has been made: Osborn and Lueck (1985) reduced noise from body vibrations in their measurements, and Washburn et al. (1996) developed a method, used here, for circumventing thermistor response limitations by estimating T from conductivity microstructure. In early August 1997, we used a towed vehicle to measure conductivity microstructure on the New England shelf (Fig. 1) as part of a combined tracer dispersion and microstructure experiment in the Coastal Mixing and Optics project. Mixing near the bottom boundary layer was studied with 11 tows about 3 to 10 m above the seafloor, while mixing throughout the water column was studied with a 39-km tow-yo transect. The full dynamic range of ocean microstructure signals appears to have been captured due to a successful match between probe sensitivity, signal level, and noise level. With the system T, and thus K T, could be estimated with only a few hours of targeted data from specific sites and depths. 2. Instruments and data analysis a. Experimental apparatus The conductivity microstructure was measured with a Sea-Bird Electronics SBE 7 sensor mounted at the bow of a towed vehicle. The vehicle was towed at 1.5 to2ms 1 behind the ship. The vehicle s mass was 200 kg, and it was 1.3 m long with a 1.8-m tail, 1 m wide, and 0.6 m tall. Other instruments on the vehicle were a Sea-Bird 9 CTD, two Chelsea fluorometers, an Ocean Sensors OS200 CTD, a Marsh McBirney current meter, a Datasonics altimeter, and a Precision Navigation TCM2 three-axis attitude sensor. Two Sea-Bird CTD sensor pairs provided redundant measurements of the temperature and salinity finestructure at 6 Hz. The fluorometers measured dye concentration and chlorophyll for the tracer dispersion study. The SBE 7 sensor is a dual-needle, platinized-electrode cell with a sampling volume of a few cubic centimeters. A similar sensor is described by Meagher et al. (1982). The sensor output is preemphasized to increase signal to noise ratio at high frequencies. The output was filtered with a fourth-order, 120-Hz low-pass Butterworth filter to reduce aliasing, sampled at 400 Hz with a commercial 16-bit A/D board in a PC/104 computer assembly, synchronized with data from the other instruments and transmitted up the tow cable to the ship. The dynamic range of the system accepting the SBE 7 signals was measured to be 90.7 db. With the probe in air (i.e., zero conductivity gradient) the range was reduced to 77.4 db, and the SBE 7 output signal was 2.7 mv rms. The minimum measurable T would be K 2 s 1 in conditions of uniform salinity. The vehicle was designed to move smoothly through the water with low attack angle. The tow cable was attached to a bar reaching down to a pivoting connection behind the center of drag so that changes in drag (speed) would not change the torque around the center of mass and would not cause pitch. However, the vehicle was intended to pitch in response to forcing at the ship s A- frame so that it would follow its bow, or minimize the angle of attack. A long tail with passive fins stabilized the flow at the nose and thereby limited attack angle fluctuations induced by ship heave and pitch. An aft emergency recovery float counterbalanced the tail. b. Computations of T, dt/dz, and ds/dt Estimates of T were derived from conductivity gradient variance (CGV) measurements by accounting for

4 1356 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 the effects of salinity gradients. CGV was obtained once per second by integrating the 1-d along-track conductivity gradient spatial spectrum S C (k). A 400-point Hanning window was used in the spectral computation. After the preemphasis and the antialias filter responses were accounted for, S C was integrated and CGV was computed assuming the small-scale field to be isotropic. The method of Washburn et al. (1996, hereafter WDJ) was used to compute T from CGV. Salinity contributions to conductivity dominate at wavenumbers larger than the Batchelor wavenumber for temperature k BT 2 ( / D ) 1/4 T, but these wavenumbers are not resolved by the SBE 7 sensor. To estimate the contribution of salinity to conductivity, the WDJ method converts CGV to temperature gradient variance using a consistent and welldefined set of assumptions. In the simplest case, if salinity variations are negligible and the conductivity temperature relation is linear, then T would equal a normalized integral of S C called C, given by 2 2 C 0 0 6D T C S (k) dk, (3) CA where C 0 is a reference conductivity, A (1/C 0 ) C/ T at constant salinity and pressure, and a factor of 3 arises because of the isotropy assumption. When salinity variations are not negligible, the method gives a correction factor E that accounts for salinity contributions and allows T to be computed: T. (4) E The factor E depends on five quantities: the correlation between temperature and salinity, the slope of the T S relation, the shapes of the temperature and salinity spectra, the spatial response of the sensor, and the energy dissipation rate. The uncertainties that each may introduce into the T estimates are discussed here sequentially. The correlation between T and S and the slope ds/dt of the T S relation determine the sign and magnitude of salinity contributions to the conductivity. The T S correlation determines the contribution of the cross spectrum of T and S to the conductivity spectrum. Since the correlation is unknown, the correction factors were computed assuming both zero and perfect correlation and upper and lower bounds on T were obtained. During the deep tows, ds/dt was determined with intentional brief vertical excursions. For the tow-yo transect, ds/dt was taken as the slope of a least squares linear fit to T and S measured each second; only data with R 2 exceeding 0.5 were accepted as valid. The shapes of the temperature and salinity spectra also determine the relative contributions of temperature and salinity to S C. Our procedure assumes temperature and salinity spectra computed from one-second segments of data to be white; that is, they have constant amplitude and drop to zero at their respective Batchelor C FIG. 2. The WDJ correction factor E as a function of dissipation rate. Separate curves having reference values E 0 of 9.2 and 6.1 are plotted for perfect T S correlation (solid line) and zero correlation (dashed line), respectively. Parameter values are Wkg 1 and ds/dt 0.65 psu C 1, which are typical for the deep tows. wavenumbers k BT ( / DT 2 ) 1/4 and k BS ( / DS 2 ) 1/4. Although the Batchelor spectrum may model average spectra well, individual realizations may have other forms. For example, Dillon and Caldwell (1980) show that average spectra in weak turbulence are broader at low wavenumbers than the Batchelor spectrum. For typical of the turbulence studied here, WDJ show that correction factors computed with white spectra are smaller than those computed with Batchelor spectra by a factor of roughly 2. The spatial response of the sensor and the dissipation determine the relative portions of CGV in the measurement band contributed by salinity and temperature structure and the amount of unresolved thermal structure. The SBE 7 sensor is assumed to have the response of a single-pole filter with 3 db point at k c 100 cpm, as determined by Meagher et al. (1982). The dissipation was not measured directly, but fits of Batchelor spectra to average spectra, presented in the next subsection, suggest mean of 10 8 Wkg 1. Although can vary over several orders of magnitude, the correction factor will not vary much with because the key parameter, the ratio of k c to the Batchelor wavenumber k BT, has the weak dependence 1/4. Figure 2 shows the change in E for various values of ; E varies by a factor of 4 over the range Wkg 1 (0.1 / 0 100). The slope of the T S relation was approximately 0.65 psu C 1 and E was between 6 and 9 for all of the deep tows. In the transect, however, both the sign and magnitude of ds/dt varied, and E ranged from 0.3 to about 300. This is not a range of uncertainty for E, but a welldefined variation determined by changes in the T S relation. Finally, since E can be large, some values of T

5 JUNE 2000 REHMANN AND DUDA 1357 FIG. 3. A time series of attack angle shows low angle during ascent but high and variable angle during descent. Attack angle is shown with a solid line, and vehicle depth (pressure) is shown with a dashed line. will be less than the previously quoted noise level for conditions of uniform salinity. c. Tow vehicle performance and data quality The tow vehicle angle of attack and vehicle vibrations can also affect the data quality. The attack angle, which measures the alignment between the vehicle s longitudinal axis and the ambient flow, was small during level and upward towing, indicating undisturbed flow into the SBE 7 sensor. However, large occurred as the vehicle descended (Fig. 3). During descent the cable slackened and the angle between the tow cable and vehicle s longitudinal axis may have exceeded 90 measured from the bow, upsetting the torque balance. Data from one-second segments with any exceeding 10 were discarded. About 30% of the data from 10 of the 11 near-bottom tows was rejected, and the entire remaining deep tow was discarded because flow in the direction of the tow caused erratic vehicle behavior. Data measured as the vehicle descended during the towyo transect were also discarded. Vertical vibrational displacements of the conductivity sensor in a stratified fluid might yield a spurious fluctuating signal. An accelerometer would have helped in determining the effect of vibrations on the measurements, but one was not installed on this version of the vehicle. The vibration effect was estimated by examining the discretely sampled 2-Hz pitch. The low-pass filtered pitch was subtracted from the raw signal, and the resulting signal was assumed to represent high-frequency (2 20 Hz) vibrations aliased into the sampled frequency range. Figure 4 shows an example of the raw and filtered pitch time series. Multiplication of the rms high-frequency pitch of 0.2 by the distance from the center of mass to the sensor gives a typical sensor displacement m rms in the 2 20 Hz range. FIG. 4. Estimated high-frequency (2 20 Hz) vehicle pitch. The dashed line is the raw pitch signal. The solid line is an estimate of the high-frequency vibrations, computed as the difference of the raw and low-pass filtered pitch. The filter 3 db point is at 0.3 Hz. This displacement creates a spurious signal contributing to the Cox number, defined as C ( T ) 2 / (dt/dz) 2, and thus to K T since the Osborn Cox relation (2) can be written as C K T /D T. The vibration-induced C can be estimated by assuming a form of the probe displacement spectrum S (k). Figure 5a shows the two forms considered. One has the power-law behavior S (k) k m in the wavenumber band k min k k max. The other, which is derived from an approximation to the acceleration spectrum of the towed vehicle of Osborn and Lueck (1985), consists of two vibrational peaks superposed on a decreasing trend. Both types of spectra are normalized to have unit variance in the wavenumber band corresponding to frequencies of 2 to 20 Hz. The temperature spectrum from vibrations is dt 2 S T(k) S (k), (5) dz and, if the small scales are isotropic, the resulting temperature gradient variance is 2 k dt max ( T ) 3 ks(k) dk. (6) dz k min The vibration-induced Cox number, which is independent of the temperature gradient, is given by 2 k max T 2 2 C 3 ks(k) dk. (7) 2D T(dT/dz) 2 k min The resulting vibration-induced C is plotted as a function of the exponent m for the power-law model in Fig. 5b, and the value computed from the Osborn Lueck model is also indicated. Although the exponent m is not available for our system, vibration data from Oregon State University s MicroSoar/SeaSoar system, which

6 1358 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 5. Model probe displacement spectra and estimates of the vibration-induced Cox number. (a) Model spectra: The first type (solid line) follows a power law, S (k) k m, with the case m 3 plotted. The second type (dashed line) is derived from an approximation to the acceleration spectrum observed by Osborn and Lueck (1985). Both spectra were normalized to have unit variance in the wavenumber band corresponding to frequencies of 2 20 Hz. The rms displacement is m, and the tow speed of our vehicle is 1.5 m s 1. (b) Vibration-induced Cox number as a function of the power law exponent m for the same and tow speed. The plotted example of (a) with m 3, which is consistent the measured behavior of MicroSoar (Erofeev et al. 1998), is marked. The Cox number associated with the Osborn and Lueck (1985) spectrum is indicated with a dashed line. FIG. 6. Three examples of conductivity gradient spectra measured during the tow-yo transect are shown. The solid lines are spectra computed from about forty seconds of data, and the dashed lines are Batchelor spectra fit to the measured spectra. The dotted line is a spectrum from a period of low signal. The T values for the fitted Batchelor spectra are indicated. has a mass comparable to our vehicle s, suggest m 3 (Erofeev et al. 1998). At this value, the error in the eddy diffusivity is 0.9% of the molecular diffusivity. In the same wavenumber band, the Osborn Lueck model gives a value about half as large. As an analysis in section 4 shows, these errors are very small compared to the uncertainty due to a lack of concurrent measurements of the turbulent kinetic energy dissipation rate. One interpretation of the effect of vibrational displacements of a towed scalar sensor in a stratified fluid is that signals are created that are comparable to those from overturns of size equal to the displacement amplitude. Overturns or vibrations of less than a centimeter do not significantly exceed the Kolmogorov scale in size, do not cause strong Batchelor-scale gradients, and are not easily confused with signals from a turbulent patch having widely separated Kolmogorov and largeeddy scales. Conductivity gradient spectra are frequently used to judge data quality. Two examples of S C measured during the tow-yo transect in areas where ds/dt 0 are shown in Fig. 6. About 40 seconds of data were used to compute these spectra, whereas spectra were computed each second for the T results presented later. A spectrum from

7 JUNE 2000 REHMANN AND DUDA 1359 FIG. 7. Temperature (a) and salinity (b) fields measured during the tow-yo transect. The seafloor is shown with a solid line. the period with the lowest signal is shown to indicate an upper bound for system noise. The Batchelor spectra (Batchelor 1959; Dillon and Caldwell 1980) of Fig. 6 were computed by varying and T until they best fit the measured spectra in a least squares sense. Values of T from the fit match those computed by integrating the spectra directly. Since the fits give 10 8 Wkg 1, this value was used in the calculation of the WDJ correction factor, though we also consider the effects of letting behave as a random variable (section 4). Batchelor spectra fit well in these cases with little salinity microstructure. In cases with larger salinity variations such as in the bottom layer the spectra would not roll off at the thermal Batchelor wavenumber and a spectral model that includes salinity variations like the WDJ model would be required. 3. Sampling and background conditions These 1997 microstructure measurements accompanied two dye tracking experiments (conducted by Jim Ledwell of the Woods Hole Oceanographic Institution). In each, a two-kilometer streak of either Rhodamine WT or Fluorescein was injected along an isopycnal surface and surveyed immediately afterward. Two more surveys were done over the next four days. The first release was at about 20-m depth along the 24.6 kg m 3 surface, and the second was at about 65-m depth along the kg m 3 surface in a sharp pycnocline averaging 5 m above the bottom. Microstructure was measured between the dye surveys. N. Oakey and B. Greenan of the Bedford Institute of Oceanography measured velocity and temperature microstructure with the quasifree falling EPSONDE (Oakey 1988). We conducted three sets of tows: a set of 2 tows during the shallow dye release, a set of 11 tows during the deep dye release, and the tow-yo transect at the end of the cruise. The 11 tows during the deep dye release were done in two sets: Tows 1 6 were conducted on 8 August, while tows 7 11 were conducted on 10 August southeast of the first set (Fig. 1). Only the results from our deep tows and the tow-yo transect will be discussed in this paper. Although we will compare our eddy diffusivities to the preliminary estimates from the tracer dispersion experiment, full results of the tracer dispersion and EPSONDE measurements will be reported elsewhere. Figure 7 shows the temperature and salinity measured during the tow-yo transect. Over most of the transect there was a temperature minimum between 40 and 60 m depth. An increase in salinity rendered the bottom layer gravitationally stable. At the location of the dye this warm salty layer, a tongue extending from the shelfbreak front, moved up the shelf along the bottom. Over most of the sampled region the gradient Richardson number Ri N 2 /S 2 was high, where N is the buoyancy

8 1360 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 layer and in shallower offshore intrusive boundaries between slope water and shelf water. The T and S profiles measured between and N in Fig. 10 show that the warm salty intrusions were 5 10 m thick and several kilometers long. Diffusive layering could have occurred above each intrusion and salt fingering could have occurred below. FIG. 8. Typical T S diagram measured during several tow-yos of the transect. The data were measured from to N latitude. The dotted lines are contours of. The letters A through E mark features that are described in the text. frequency and S is the shear: Ri was measured with the ship s 300-kHz broadband Doppler sonar and the towsled CTD. It was below ¼ only in isolated spots, for example, in packets of mode-one internal solitary waves moving northward. The three properties shear, hydrography and microstructure were not sampled in manners lending to their quantitative comparison. The T S diagram in Fig. 8 from a portion of the transect was typical for the area. The warm, salty layer (A) moving up the shelf created a region favorable for diffusive layering, while the overlying water (B) was favorable for salt fingering. The water in the upper part of the water column (C) was stable to double diffusive instabilities. An intrusion (D) at 25.5 kg m 3 and the temperature minimum (E) can also be seen. Another way to differentiate the stratification is with the Turner angle Tu, defined as a four quadrant arctangent T S T S Tu arctan,, (8) z z z z where and are the thermal and saline expansion coefficients, respectively (McDougall et al. 1988). Values of Tu between 45 and 90 indicate conditions favorable for salt fingering, while values between 90 and 45 indicate conditions favorable for diffusive layering. Water with 45 Tu 45 is stable to double-diffusive instabilities. Values of Tu measured during the transect (Fig. 9) show that much of the inshore water is diffusively stable. Diffusive layering could have occurred in the warm salty intrusive bottom 4. Mixing near the bottom a. Signal characteristics The thermal variance dissipation rate T from the first deep tow is shown in Fig. 11. Figure 11a shows the vehicle depth and seafloor, while T and log 10 ( T ) are plotted in Figs. 11b,c. About half of the record is fairly quiet with T 10 7 K 2 s 1, while the other half contains 100- to 500-m long stretches of relatively high T. All of the values are well above the noise level. In this tow the apparent patchiness of T appears to be due to vertical variation since the regions of elevated T correspond roughly to denser (deeper) water. The peaky character of T (Fig. 11) suggests that T has a skewed distribution. Many previous researchers have used the lognormal distribution to describe their dissipation measurements, and dissipation distributions have also been analyzed to determine instrument noise and data quantity adequacy. Yamazaki and Lueck (1990) state that dissipation in surface and benthic boundary layers is nearly lognormal, but that dissipation in the thermocline may not be because the conventional method of estimating dissipation violates one or more of the following criteria: 1) The averaging scale must be much smaller than the scale over which the dissipation is homogeneous (the domain scale). 2) The averaging scale must be much larger than the Kolmogorov scale L K ( 3 / ) 1/4. 3) The measurements must be mutually independent and identically distributed; that is, they must be taken from a single population. The logarithms of T from three sets of data measured during the deep dye release (deep tows) are shown on probability plots in Fig. 12. Lognormally distributed data would fall on a straight line. Qualitatively, T appears to be lognormally distributed in the three cases shown, falling close to the fitted lines. Excesses of small values, like those in the smallest 10% of the data, are generally attributed to instrument noise, although the values shown should be well above the estimated noise level. Deficits of large values, usually attributed to limited Reynolds number, were not observed in these data; the excess of large values in the first tow (Fig. 12a) could indicate that two turbulent regions with different characteristics were encountered. Quantitatively, the data in Figs. 12b,c fail the Kolmogorov Smirnov (KS) test (Benjamin and Cornell 1970) for lognormality at the 5% significance level, and the data of Fig. 12a pass. The data from a single depth bin (Fig. 12b) and from all of the deep tows (Fig. 12c)

9 JUNE 2000 REHMANN AND DUDA 1361 FIG. 9. Turner angle measured during the tow-yo transect. The Turner angle indicates whether a water mass is favorable for diffusive layering or salt fingers or stable to double-diffusive instabilities. The vehicle track is shown with the white dotted line, and the seafloor is shown with a solid line. The nonuniformly spaced data were linearly interpolated onto a grid with 1-m spacing in the vertical and approximately 2-km spacing in the horizontal. Interpolated results at depths not adequately sampled during the horizontal tows are not shown. FIG. 10. Temperature and salinity profiles measured between and N. Intrusions of warm, salty water can be seen at several depths but especially at m. The temperature profiles are offset by 2.5 C, and the salinity profiles are offset by 1 psu. FIG. 11. Parameters from the first deep tow as a function of distance along the tow track. (a) Sled depth during the tow (solid line) and seafloor (dashed line). (b) T on a linear scale (solid line) and density of the sampled water (dashed line). The density scale is reversed. (c) T on a logarithmic scale.

10 1362 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 12. Lognormal probability plot of T from deep tows: (a) T from the first deep tow, (b) T in a layer with kg m 3 from all tows, (c) T from all of the deep tows. The dashed lines were fit to the central 50% of the data. The variance of the natural logarithms of T and the number of samples are shown on each plot. Only the upper bound on T is shown; the lower bound has similar behavior. probably do not satisfy the criteria of Yamazaki and Lueck (1990). The averaging scale of 1 2 m is much larger than the Kolmogorov scale ( 1 mm), and it may be smaller than the domain scale, which might be either the bottom layer thickness or the interface thickness. However, because the two sets of deep tows were conducted in different locations and on different days, their dissipation rates may not be identically distributed. The data of Fig. 12a are distributed lognormally according to this test despite the qualitative suggestion that they are from different turbulent regions or different populations. b. Averaged quantities Average profiles of temperature gradient, thermal variance dissipation rate, diffusivity, and heat flux from the deep tows are shown in Fig. 13. The valid data from the 11 tows were sorted into ten bins of equal size and averaged. The average temperature gradient is the arithmetic mean of all dt/dz in the bin, while the average T is a mean of 200 bootstrap resampled populations. The diffusivity K T is computed using the Osborn Cox relation (2). The heat flux is Q C p K T dt/dz, where is an average density and C p is the specific heat. Upper and lower bounds and bootstrap 95% confidence intervals are shown for T and K T. The right vertical axes in Fig. 13 show the corresponding to the pressure level on the left vertical axis. The mean density profile that was used to make these plots was determined by averaging the pressures from all tows in each bin. The mean density profile has a relatively sharp interface between 26.1 and 26.5 kg m 3, visible also in the temperature gradient profile of Fig. 13a. The high-gradient zone is, in fact, the shelf/slope water front, which was nearly horizontal at this site during the sampling. The average T (Fig. 13b) varies little at the top of the sampled region and then drops to a minimum near the target density kg m 3, which lies near the top of the interface region. As the depth increases, T increases sharply, as suggested by the T record from deep tow 1 in Fig. 11. Below 26.4 kg m 3, T decreases again. This decrease may be related to the decrease in dt/dz since without temperature variation T must be zero. The average T has a relatively small dynamic range of about 20. The eddy diffusivity K T (Fig. 13c) varies over three orders of magnitude, much more than the mean T.It is O(10 3 )m 2 s 1 in the bottom boundary layer and it drops sharply in the interface to O(10 6 )m 2 s 1 near the target density. The heat flux (Fig. 13d) has similar behavior; it drops from about 50 W m 2 near the bottom to about 1 W m 2 in the interface. The flux in the bottom layer is about one quarter of the solar heat flux at the surface. The sharp decrease in K T is similar to the decrease in the turbulent kinetic energy dissipation rate in the constant-stress layer of boundary layers (Dewey and Crawford 1988). However, since the bottom currents and roughness are comparable to those in the study of Dewey and Crawford (1988), our measurements at a mean distance of about 5 m above the bottom are probably well out of the constant-stress layer, estimated by Dewey and Crawford (1988) to be less than 2 m. The large value of K T in the bottom layer should be roughly equal to the eddy viscosity there. If the bottom layer is an upwelling Ekman layer (Garrett et al. 1993), then the thickness of the layer will scale as 2 t / f, where t is the eddy viscosity and f is the Coriolis parameter. A layer thickness of about 5 m implies t 10 3 m 2 s 1, comparable to the K T estimate near the bottom. Our estimates of K T can be compared to K T estimates from the tracer dispersion experiments. Here K T was observed to vary from about m 2 s 1 above the target density surface to about m 2 s 1 below the target surface (J. Ledwell 1998, personal communication). These values have been indicated on Fig. 13c with a thick line. The K T values from towed microstructure at the dye depths are about 3 to 13 times lower than values obtained using dye.

11 JUNE 2000 REHMANN AND DUDA 1363 FIG. 13. Profiles of averaged quantities measured during the deep tows: (a) Temperature gradient dt/dz, (b) thermal variance dissipation rate T, (c) eddy diffusivity K T, (d) heat flux Q. The quantities were sorted into bins of equal size and averaged as described in the text. Upper and lower bounds and bootstrap 95% confidence intervals (dotted line) for T and K T are shown in (b) and (c). The right vertical axis shows the corresponding in the mean to the pressure on the left axis. Preliminary estimates of the diapycnal diffusivity from tracer dispersion (J. Ledwell 1998, personal communication) are indicated with a thick line in (c). One limitation of our analysis, and a possible reason for the discrepancy, is the assumption that 10 8 W kg 1 everywhere. In actuality varies and its value at each point is unknown. As Fig. 2 shows, increased would reduce the WDJ correction factor E and would increase estimated T. To estimate the uncertainty in K T due to uncertainty in, K T computed with a constant value of 10 8 Wkg 1 can be compared to K T computed by treating as a random variable. For example, sampling from a lognormal distribution of mean 8 and standard deviation ⅔ gives an upper estimate of K T near the target density of m 2 s 1, which is 10% higher than the value of m 2 s 1 computed with constant. Sampling from a uniform distribution of mean 10 8 Wkg 1 and a dynamic range of 10 4 increases K T by 30% over the constant- value. Thus, although uncertainty in TKE dissipation may change individual T and K T estimates substantially, the effect on the average values is relatively small. Another possible explanation for the discrepancy is small-scale anisotropy in the scalar field. The computation of CGV assumes the small scales of the turbulence to be isotropic. However, when the stratification is strong, the assumption may not be valid, and T estimated from a single horizontal component of the gradient may be low. Field observations show anisotropy for Cox numbers less than 10 4 and suggest that T estimated from horizontal (or near horizontal) temperature gradients assuming isotropy may be low by about 15% (Sherman and Davis 1995). Laboratory experiments on unsheared stratified turbulence also indicate reduced horizontal temperature gradients and suggest that the underestimation would reach 30% one buoyancy period after the turbulence is generated (Thoroddsen and Van Atta 1996). Direct numerical simulations of sheared, stratified turbulence suggest that anisotropy occurs for / N 2 600, where is the kinematic viscosity, and that estimates of T based on the scalar gradient in one horizontal direction can be low by as much as a factor of four (Itsweire et al. 1993). To determine whether the small scales were isotropic in our case, we can estimate / N 2 near the target density with an expression for K based on the turbulent mass flux, K /N 2 (Osborn 1980). Taking 0.2 (Osborn 1980) and assuming equality, one finds / N 2 5K /. Using K from tracer dispersion gives / N 2 50, which suggests the small scales are anisotropic (Itsweire et al. 1993). Since the tracer provides an average over the experiment, instantaneous values of / N 2 will be larger, but even a factor of 10 increase would still suggest anisotropy. A third possible explanation for the discrepancy is that the turbulence could be undersampled. The towing samples only a small region over a limited time, while the tracer is continuously sampling the turbulence over the patch domain of several square kilometers.

12 1364 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 Laboratory studies of shear-driven entrainment across an interface between two layers can be used to speculate on the K T N relation. Christodoulou (1986) and Fernando (1991) reviewed many experiments where shear was due to a surface stress, a light fluid flowing over a denser fluid, a density current, or two counterflowing layers. Either the movement of the density interface or the flux across the interface was measured and expressed in terms of an entrainment velocity u e. This quantity can be related to the buoyancy flux F b by FIG. 14. Dependence of K T on N in the deep tows: K T was computed for each bin as described in the text. The width of the gray boxes represents the buoyancy frequency range for each bin, and the height of the boxes represents the range of upper and lower bounds on T from the two TS correlation choices. The lines above and below each box are the bootstrap 95% confidence intervals. The dashed line is a power law fit to the data. The range of diapycnal diffusivities estimated from tracer dispersion is shown with a thick line at N 20 cph. c. Effect of stratification on K T The sharp decrease in K T in the interface suggests a strong dependence on stratification strength (Fig. 14). The eddy diffusivity was computed in buoyancy frequency bins with an averaging procedure like that used to produce Fig. 13c. Also shown is a power law fit, which yields K T N 3.1. (9) Since K T decreases with increasing stratification strength, mixing will tend to sharpen the interface between the bottom layer and the overlying water (Phillips 1972). The buoyancy flux F b KN 2 will be smallest in the interface, where N is largest. Thus, the water at the top of the interfacial region becomes less dense as mixing proceeds, the water at the bottom becomes more dense, and the interface sharpens. Since the flux must vanish as the stratification becomes very small, a steadystate interface thickness can be achieved (Posmentier 1977). The dependence of K T on N given in (9) is quite different from that predicted for the open ocean, where analyses of mixing and turbulence due to internal waves give N 2 (Gregg 1989; Henyey et al. 1986; Polzin et al. 1995) or N 3/2 (Gargett 1990; Gargett and Holloway 1984). These are consistent with K T N a with ½ a 0. The disagreement between (9) and the open-ocean results is not surprising because mixing in the bottom boundary layer is probably driven by different mechanisms. In particular, turbulence generated at the seafloor may entrain cold water across the sharp density interface atop the layer. g 2 0 F u unh, (10) b e e where is the density difference across the interface and h is the interface thickness. Equation (10) and the definition of the buoyancy flux F b KN 2 can be combined to obtain K u e E(Ri H ), (11) uh u where u is the velocity difference across the interface. The entrainment function E u e / u depends on the stability of the flow, measured by a Richardson number Ri H g H/ 0 ( u) 2, where H is the depth of the bottom layer. The results compiled by Fernando (1991) and Christodoulou (1986) show that E approaches a constant 3/2 at low Ri H and follows a Ri H power law for Ri H 1. If the layer thickness, interface thickness, and velocity difference are roughly constant, then the large-ri H power law is consistent with K N 3. (12) This result is similar to that in (9) and Fig. 14. The entrainment analysis and its application to the field observations require several assumptions. If the flow is not strongly stratified, then the power law exponent could range from zero to 3. Data from the SuperBASS tripod of A. J. Williams III near the first set of deep tows were used to investigate the near-bottom Ri H. Data from Benthic Acoustic Stress Sensors at 3.5 and 7 m above the bottom (or median temperatures 9.5 and 8.9 C) were used to compute the velocity difference, and the density difference was computed using tripod thermistor data and the near-constant T S relation observed while towing. The velocity difference was typically below 8 cm s 1 but reached 14 to 18 cm s 1 in brief bursts, and the density difference was variable and often small. With an interface thickness of 3.5 m and a layer thickness of 7 m, Ri H always exceeded 0.24 between the beginning of the first deep tow and the end of the last. During this interval, Ri H exceeded unity 78% of the time and had mean and median values of 76 and 2.6. Thus, using the entrainment results for Ri H 1 seems justified. The entrainment analysis also requires flux due to double diffusive convection to be negligible compared to flux due to mechanically generated turbulence. Lab-

13 JUNE 2000 REHMANN AND DUDA 1365 oratory experiments (e.g., Turner 1965) show that the heat flux in a system with diffusive layering is large when 1 R* 2, where the density ratio R* ( S/ z)/ ( T/ z). Near the bottom ds/dt 0.65 psu C 1, so that R* 3. This value is relatively small but not in the supercritical range. Therefore, the assumption that bottom-generated turbulence dominates the mixing seems reasonable, given the other approximations. d. Temperature budget of the bottom layer The temperature budget of the bottom layer provides a third estimate of K T, to be compared with the microstructure and dye values. If constant coefficients K x, K y, and K z parameterize mixing in the east west, north south, and vertical directions, respectively, then the temperature budget is DT T T T Kx Ky K z, (13) Dt x y z where D/Dt is the material derivative. Precise budget evaluation would require integrating terms over the patch and comparing the rate of change of the heat content to the net flux into the path. However, we will simply estimate the magnitude of each term in (13) using data from the dye surveys to examine the relative importance of vertical mixing and horizontal dispersion. Houghton (1997) adopted a similar approach in a tracerrelease experiment in the same area. Since the dye is a Lagrangian tracer, the rate of temperature change of the tagged water can be estimated. We consider only the northernmost one-third of dye in the bottom layer during the deep experiment. During this experiment part of the dye was advected eastward and the remainder moved northward in or just above the bottom layer. We assume that the northernmost dye from the second dye survey was still the northernmost dye in the third dye survey and that we are measuring the same dye. This assumption is reasonable because the patch grew monotonically outward throughout the experiment. From the northern third, the dye-weighted mean temperatures were 8.77 and 8.30 C for the second and third dye surveys, respectively, so the water in the bottom layer was cooling as it moved up the shelf. The elapsed time was 50 hours, giving DT/Dt Cs 1. This value compares well with the Houghton (1997) value. The spatial derivatives of temperature in the bottom layer section of the dye patch were estimated using data from eight east west tracks through the patch. The derivative 2 T/ z 2 was estimated near the peaks of the vertical dye profiles. The mean value was 0.8 C m 2, consistent with vertical mixing cooling the dye in the bottom layer. The derivative 2 T/ y 2 was determined from the depth-mean temperatures at the locations of the maximum dye concentration along each track. Figure 15 shows a quadratic fit to the mean temperatures; FIG. 15. Determination of 2 T/ y 2. Solid circles denote depth-mean temperature from the positions of peak dye concentration along eight east west tracks through the dye patch. The line is a least squares second-order polynomial fit to the data. this fit gives 2 T/ y Cm 2. A similar value was obtained for 2 T/ x 2. These horizontal derivatives are a factor of about 10 less than Houghton s (1997) estimated upper bound and slightly greater than the derivatives on isopycnal surfaces. Multiplication of the horizontal derivatives by estimated K x and K y show that horizontal effects are not important to the temperature budget. The horizontal spreading rate of the northern part of the dye patch implies a value between 1 and 5 m 2 s 1 for K x and K y. Similar values were observed during dye experiments in previous years of the program (M. Sundermeyer 1998, personal communication), while Houghton (1997) estimated the cross-shelf dispersion coefficient K y to be O(10) m 2 s 1. Even with the larger value, the horizontal dispersion terms in (13) reach only O(10 7 ) C s 1, which is an order of magnitude less than the temporal rate of change. The implication is that vertical (or diapycnal) mixing is mainly responsible for changing the temperature of bottom water, with entrainment across the density interface cooling the bottom layer. Neglecting horizontal terms in the temperature budget, one can estimate the eddy diffusivity with DT/Dt Kz. (14) 2 2 T/ z The values quoted above give K z m 2 s 1, comparable to values from towed microstructure and the dye study. 5. Mixing throughout the water column a. Signal characteristics Grayscale plots of T and K T measured during the transect are shown in Figs. 16 and 17. As before, T

14 1366 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 FIG. 16. Measured T during the tow-yo transect. The nonuniformly spaced data were linearly interpolated as in Fig. 9. was computed from 1-s segments of data. Unlike the deep tow study the temperature gradient was also estimated each second using a least squares fit between temperature and pressure, and K T was computed each second with (2). Computing K T in this way is not entirely consistent with the assumptions of the Osborn and Cox model, but we will use it in this subsection to examine spatial patterns in the mixing. In the next subsection, K T is computed from averaged T as in the deep tow study. Both T and K T have large ranges, and patterns with length of about 10 km (0.1 ) and thicknesses of about 5 10 m can be seen. The thermal dissipation rate spans more than five orders of magnitude, as it did near the bottom (Fig. 12c). Because the K T of Fig. 17 are computed from instantaneous values of T rather than average T, they have a larger range than the deep towing results in Fig. 13c: K T varies from the molecular value to 10 3 m 2 s 1. The lowest values generally occurred in the deepest m sampled. The highest values occurred in the intrusion region south of 40.4 N (Fig. 10) and in a 10-m thick, 10-km long patch at about 20-m depth between and N. The high K T in the intrusion region was associated with high T, while the high K T in the patch was associated with small temperature gradients. The instantaneous T samples from the patch delineated in Fig. 17 fail the quantitative KS test for lognormality. The deviations from lognormal cannot be explained by noise or limited Reynolds number. There is a deficit of small values and an excess of large values which may indicate that different turbulent regions were sampled (Osborn and Lueck 1985). In contrast, T populations from the intrusion region and from the entire transect (with a dynamic range of eight orders of magnitude) are lognormally distributed at the 5% significance level according to the KS test, despite the potential for both turbulent and double-diffusive processes to contribute, particularly in the intrusion zone. b. Effect of stratification on T and K T In this subsection the transect data are used to study the effect of both the strength and the type of stratification on the mixing. First average K T for each stratification type are compared. Then our measurements are compared to results from a previous study in the intrusion area delineated in Fig. 17. Finally the variation of T and K T with the density ratio and buoyancy frequency is examined. Average K T values were computed for each stratification type: diffusively stable, fingering-favorable, and layering favorable (Table 1). The temperature gradients and T were binned according to Turner angle (stratification type) and averaged, and then K T was computed for each bin using (2). Diffusivities for the diffusive

15 JUNE 2000 REHMANN AND DUDA 1367 FIG. 17. Measured K T during the tow-yo transect. Interpolation is as in the previous figure. Dashed lines show the intrusion region between 40.3 and 40.4 N and a patch between and N from which data for later analysis are taken. TABLE 1. Eddy diffusivities (in m 2 s 1 ) computed from T and dt/ dz averaged according to Turner angle, or stratification type. As before, the upper and lower bounds come from the assumption of full or zero correlation between temperature and salinity. Type Lower bound Upper bound Stable Salt finger Diffusive layering Overall layering case are 5 10 times smaller than the others because of low values near the bottom, while diffusivities for the diffusively stable case are slightly higher than the others because of large values in the patch shown in Fig. 17. The overall diffusivity of to m 2 s 1 is slightly larger than the diffusivity measured in the open-ocean thermocline (Ledwell et al. 1993). Voorhis et al. (1976) studied water mass evolution near the shelfbreak with neutrally buoyant floats. They observed that 10-m thick layers exchanged both salt and heat in about two days while maintaining a density difference of 0.25 kg m 3. From these observations a simple diffusion estimate gives K T m 2 s 1. They attributed the mixing to double diffusive processes rather than mechanically generated turbulence because, from the measurements of Turner (1968), the stratification was too strong for salt and heat to be transported with comparable time scales at the rates they observed. In contrast, flux calculations based on results from laboratory experiments on diffusive convection were consistent with the observations. Profiles of average T and K T from the intrusion region are shown in Fig. 18. K T tends to exceed the transect average (Table 1) in this region. The minimum value of K T is near 10 5 m 2 s 1. The maximum approaches the value estimated by Voorhis et al. (1976) at about 20-m depth, but values of K T in the intrusions at depth m (Fig. 10) are an order of magnitude lower than the value estimated by Voorhis et al. (1976). One reason for the discrepancy could be that our computation of K T from T using the thermal Osborn Cox relation does not properly account for salinity contributions to density variance and density gradients that drive double-diffusive convection (Osborn and Cox 1972; Schmitt and Evans 1978). To assess the possible importance of double-diffusive processes, the dependence of T on the density ratio was examined. Figures 19a and 19b show T as a function of the density ratios R ( T/ z)/ ( S/ z) and R* R 1 for stratification favorable for salt fingering and stratification favorable for diffusive layering, respectively. Two differences between the fingering-favorable and layeringfavorable cases are evident: The mean T in fingering water