## 1. Introduction

An abyssal mixing experiment has been carried out near the seafloor at 3500–4500-m depth on the western slope of the the very rough slowly spreading Mid-Atlantic Ridge (MAR) in the eastern Brazil Basin (Ledwell et al. 2000). One of the goals of that project, the Brazil Basin Tracer Release Experiment (BBTRE), was to determine the mechanism responsible for heat transfer into the coldest class of Antarctic Bottom Water (AABW). Cold water flows into the basin at the south but does not exit at the north. Neglecting some leakage through gaps in the Mid-Atlantic Ridge, the warming can be estimated using observations at the southern passage, the northern passage, and within the basin. Studies have shown the heat transfer into the bottom water to be consistent with diapycnal diffusivities *K* greater than 2 × 10^{−4} m^{2} s^{−1} (Hogg et al. 1982; Morris et al. 2001). These values are far in excess of the so-called background eddy diffusivity expected in much of the ocean volume, measured to be *K* = 1.5 × 10^{−5} m^{2} s^{−1} at 300 dbar in the main thermocline during the North Atlantic Tracer Release Experiment (NATRE) injection study (Ledwell et al. 1998), and also inferred from many microstructure and internal wave shear studies (Gregg 1989, 1998; Kunze and Sanford 1996). The BBTRE, using the same techniques as NATRE, showed depth-dependent *K*(*z*) equal to or greater than those from the AABW budget analyses, ranging from 1 to 8 (×10^{−4} m^{2} s^{−1}) at levels of 500 to −500 m above the MAR ridge tops. The rough MAR does not underlie the entire basin and much of the basin exhibited low levels of mixing (Polzin et al. 1997), but the observed *K* values appear to be high enough to explain the basin-average heat transfer.

The appreciable mixing observed in BBTRE, compared to NATRE, also offers a conceptual solution to the decades-old puzzle of background interior mixing levels, believed to be of order 1 × 10^{−5} m^{2} s^{−1} as measured in NATRE, being far less than bulk values derived from thermocline structure (Munk 1966) and semi-enclosed basin studies (Luyten et al. 1993; Whitehead 1987; Whitehead and Worthington 1982), including the previously cited Brazil Basin studies (Hogg et al. 1982; Morris et al. 2001). Enhanced mixing at boundaries (e.g., at the bottom), conveyed to the interior thermocline by quasi-isopycnal processes, has been proposed to explain this discrepancy. In this scenario, diffusivity far in excess of the apparent basin value at actively mixing boundaries having only limited extent could explain the observations (Armi 1978; Ivey 1987). Interaction of deep flow with sloping bathymetry, generating internal waves which may then dissipate, offers a mechanism for the enhancement, and there is substantial interest in this topic (Balmforth et al. 2002; Llewellyn Smith and Young 2002; St. Laurent and Garrett 2002).

In agreement with the boundary mixing hypothesis, vertical profile measurements collected over the rough fracture-zone-laden BBTRE area showed more microstructure activity throughout the entire water column than similar measurements collected over the abyssal plain (Polzin et al. 1997). The BBTRE profile measurements also showed increasing dissipation rate with increasing proximity to the bottom (St. Laurent et al. 2001), which was in agreement with dye behavior. Furthermore, a time series of profiles collected in the BBTRE over two periods of approximately 20 days each showed temporal microstructure variability crudely resembling the envelope of predicted local tidal current speed (St. Laurent et al. 2001), with depth-averaged turbulent kinetic energy dissipation rate varying between about 0.2 and 4.5 mW m^{−2}. The measurements reported here corroborate those results and show the temporal variability of shear at time scales of days to months.

An interpretation of the BBTRE results is that tides and currents passing over the rough MAR area produce internal waves, which then break and produce higher diapycnal flux than the low levels associated with a smooth seafloor (Kunze and Sanford 1996; Polzin et al. 1997); that is, the results support the boundary mixing hypothesis. A hidden assumption is that the energy of the bottom-generated internal wave field is sufficient to reduce the gradient Richardson number (the ratio of buoyancy frequency squared to shear squared) to the low levels associated with wave breaking. An implication is that spatial variability of buoyancy forcing, resulting from variable dissipation of internal wave energy, can be a significant contributor to circulation (St. Laurent et al. 2001). We present some evidence for vertically structured shear statistics in the BBTRE area, supporting both ideas.

Temporal variability of dissipation and fine structure observed in the area during the first few years of BBTRE motivated additional measurement of fine structure and microstructure conditions in a manner that would allow reliable mean values to be computed over the time scale of the tracer measurements, which is years. A new class of drifting floats was developed to perform long-term fine-structure sampling, the Shearmeter floats (Duda and Webb 1997). These floats are capable of measuring the magnitude of the very small shear vectors in the abyssal ocean while drifting on missions lasting months to years. (In this paper we sometimes take the grammatical shortcuts of referring to the *magnitude of the vector of vertical shear* as *shear magnitude* or *shear,* where the vector of vertical shear is the difference per unit meter vertically of the horizontal velocity vector.) In theory, the floats can measure smaller shear than Doppler acoustic profilers, which work best under conditions of strong and coherent backscatter of limited Doppler bandwidth, or than rotor-type current meters that can stall (Luther et al. 1991). They can also be tracked acoustically in the manner of RAFOS floats (Rossby et al. 1986). Six Shearmeters have been deployed at various depths in the area of the BBTRE (Fig. 1). In this paper we present mean levels of statistics of shear magnitude in the BBTRE area, show that shear normalized against stratification varies with height above the seafloor, and show that shear variability shows a bit of correlation with tidal forcing. The increased shear at depth supports the idea that shear-producing internal waves originate at the bottom. The marginal correlation with tide does not preclude tidal currents at the bottom from being the source of the waves but, rather, it indicates that more refined and plentiful data are needed to sort out the possibly multiple wave sources.

Not all of the floats were fully functional. Data from only four floats are discussed in this paper (Table 1). Only two floats provided shear data. These recorded time series of hourly mean values of shear magnitude, denoted *S*_{h}, where *S*_{h} = 〈|**S**|〉_{h} = 〈|**i***u*_{z} + **j***υ*_{z}|〉_{h}, where 〈 〉_{h} indicates 1-h average. One float operated for 100 days at roughly 1660-m depth where the buoyancy frequency *N* was about 1.1 cph based on BBTRE CTD data, and one operated for 365 days at 2850-m depth where *N* was 0.5 cph. The deep float that recorded shear and two other floats provided trajectories at depths of 2850, 3130, and 3550 m, respectively (Fig. 2, Table 1). Further discussion of shearmeter functionality can be found in Duda et al. (2002). Information about other float operations in the Brazil Basin can be found in Hogg and Owens (1999).

All shear and trajectory data from the four floats are presented here. Section 2 describes the floats and the parameters of the missions. Section 3 gives model expressions for shear and diffusivity in the ocean. Section 4 shows the shear data. Section 5 discusses the float trajectories. Section 6 compares deep shear with the monthly tidal cycle and with height above the bottom. Section 7 summarizes the results.

## 2. Methods

The Shearmeter floats are described in a paper (Duda and Webb 1997) and in a report (Duda et al. 2002). Briefly, they are drifting spar-buoy-shaped floats of approximately 10-m length. Each float has vanes rigidly mounted at either end that are similar to cup anemometer vanes. The vanes cause the entire float to rotate in response to currents at the ends. In a vertically invariant current the flow will be zero at each end, resulting in no rotation, whereas in a *sheared* flow the current *will not* be zero at each end, but will instead be approximately equal and opposite, enabling rotation. The vanes have six arms in an equally spaced spoke pattern, each extending to a radius of 45 cm. Each arm has a 14-cm half-cylinder drag element at the end, with vertical cylinder axis. The vanes measure the magnitude of the flow, thus making the flow directions (relative to the float) irrelevant but also undetectable. The ratio of vane drag to drag on the rotating cylinder is sufficiently high to ensure rapid response: The assembly has been observed to respond to varying shear in a few seconds. Temperature *T* and pressure *P* are also recorded. This description of Shearmeter operation disregards the effects of vertical currents on the vanes. Such currents will exist because these are quasi-isobaric floats. Such floats respond to isopycnal displacements by also moving, but with amplitude less than the isotherm displacement(1/10 to 1/3; see, e.g., Goodman and Levine 1990). The correlation is small between *dT*/*dt* measured by the deep float four times per day and its rotation rate (*R*^{2} = 0.0011), suggesting that response to vertical flow represents no more than a small part of the signal. A planned second-generation modification to record tilt induced by the very weak shear for the purpose of measuring the shear direction has not been implemented.

**S**| was obtained from tests in a tow tank and in Seneca Lake, New York (Duda and Webb 1997). A nonlinear function composed of two line segments passes nicely through the calibration points, each with the form

*r*=

*ms*+

*b,*where

*r*is rotation rate in revolutions per hour,

*s*is speed (at one end) in meters per second,

*m*is the slope, and

*b*is the

*r*axis intercept. The line for low speeds passes through the origin (i.e.,

*b*= 0). The other line connects to it at (

*r*

_{k},

*s*

_{k}) = (1.65, 0.0112). Parameters for the lines are

*m*

_{1}= 146.9,

*b*

_{1}= 0,

*m*

_{2}= 264.4, and

*b*

_{2}= −1.3222. The data are divided into values above and below the kink at

*r*=

*r*

_{k}. Shear values for the BBTRE floats of length

*L*(in meters) are then given bywhere subscript

*n*= 1 or 2 and

*L*for these floats was 9.5 m. The calibration parameters correspond to the arms moving with tangential speeds of 10% and 18% of the flow past the ends for the low- and high-speed regimes, respectively.

The procedure used to estimate shear was as follows: The two-axis electromagnetic compass was interrogated once per second and the sector that the compass lubber line was pointed toward was recorded; bearings were divided into eight 45° sectors. If the sector number changed from one sample to the next, then the sector counter was incremented (with sign) accordingly. Once per hour the sector counter was recorded modulo 2^{12}. The hourly averaged values of rotation rate were computed on shore using these data, with a least count of 1/8 rotation per hour, and then converted to shear using the appropriate leg of the two-leg calibration curve (1). With linear response, this would provide an unaliased and unbiased measure of the average shear magnitude over the sampling interval. However, this is not true for this instrument. Because of the nonlinear calibration curve, there can be an over estimation of *S*_{h} by order 30% if the rotation rate shifts between the slow and fast regimes during the one-hour sampling intervals, which might affect *S*_{h} values near 2 × 10^{−3} that occur throughout the shallow record and less often in the deep record (about 6% of the time), and/or might affect measurements that show a transition from one regime to the other from one sample to the next (about 9% of the deep record, 31% of the shallow record). Last, note that, because variations of shear direction are not taken into account when averaging, this value of mean shear magnitude is likely to exceed the magnitude of the mean shear vector.

The sensing vanes can respond to vertical flow, giving rotation that is not linked to shear, as mentioned before. Tests suggest that this does not happen for vertical velocity less than 0.3 cm s^{−1}. However, if this is happening, which is a worst-case scenario, then the floats would still be good internal wave gauges, measuring a combination of shear from low-frequency internal waves and vertical flow from high-frequency internal waves. Whether this happens may be determined with a future (expensive) calibration mission. If the vertical response is appreciable, it might be possible to separate the shear and the vertical flow variances by using temperature and pressure data that would indicate the isotherm heaving.

The shallower of the two floats that returned shear data (SN 001) was launched from the RV *Seward Johnson* in early 1998 and collected data for 100 days at roughly 1660 dbar (Table 1). The deployment position was 21°36.79′S, 17°42.19′W. This was a prototype float and was not trackable. The other three floats yielding data were launched from the RV *Knorr* in May 2000 and operated for 365 days. Float launch positions were 21°42.64′S, 18°37.93′W; 21°50.03′S, 18°18.50′W; and 21°50.03′S, 18°18.50′W for floats 099, 100, and 101, respectively. Table 1 lists other float mission parameters.

The floats were designed to drop a ballast weight upon completion of a programmed data-collection mission, subsequently rising to the surface and returning data via the ARGOS satellite radio system. As for the failures, one of the total of five year-2000 floats did not transmit any signals at all (termed a “no show”), three did not not produce shear records because of compass system failures, and one of those three did not return useful acoustic tracking data. A sixth year-2000 float was damaged during deployment and was returned home. The report contains complete information about float functionality (Duda et al. 2002).

## 3. Models

Analysis of pelagic shear recorded in the last two decades has usually involved comparison with expressions derived from the Garrett–Munk (GM) empirical internal wave spectrum (Gregg 1989; Gregg and Kunze 1991; Munk 1981). We use the same approach. Theoretical and numerical studies of internal wave fields following the GM spectrum have led to expressions for the energy dissipation rate of internal waves and associated diapycnal mixing. These expressions allow our *S*_{h} measurements to be compared with temporally intensive microstructure measurements at the BBTRE site and with tracer-derived diffusivity estimates.

*E*= 6.3 × 10

^{−5}is the dimensionless energy parameter,

*N*

_{0}= 5.2 × 10

^{−3}(3 cph),

*f*is the Coriolis frequency,

*ω*is internal wave frequency,

*j*is mode number,

*H*(

*j*) = (

*j*

^{2}+

*j*

^{2}

_{∗}

^{−1}/

^{∞}

_{1}

*j*

^{2}+

*j*

^{2}

_{∗}

^{−1}, and

*j*∗ = 3 is the internal wave bandwidth parameter. The bandwidth of mode index

*j*[where

*j*≈

*k*

_{z}

*bN*

_{0}/(

*πN*),

*b*= 1300 m is the stratification length scale, and

*k*

_{z}is the vertical wavenumber] is a function of

*N*as stated by Munk (1981), and so the relevant integral of this spectrum, which could be divided by

*N*

^{2}to produce the inverse Richardson number, is not necessarily proportional to

*N*

^{3}. Unfortunately, the spectrum (2) is not immediately comparable with our data because we are not analyzing the shear vector

**S**=

**i**

*u*

_{z}+

**j**

*υ*

_{z}, but rather

*S*= |

**S**|, which is the variable analyzed by Kunze et al. (1991a,b). The derivation of the spectrum of

*S*for waves consistent with the GM spectrum is beyond the scope of this paper, but simulated spectra of

*S*generated from ensembles of synthetic wavefields obeying the GM model will be compared to those observed.

*S*

^{4}

_{10}

*S*

^{4}

_{GM}

*C*is equal to 2. The remainder of the expression is based on internal wave wavenumber evolution considerations (Henyey et al. 1986).

*V*

_{S}given by the integral from low wavenumber to wavenumber

*k*

_{υu}of the vertical wavenumber shear spectrum,

*V*

_{S}

*k*

_{υu}

*πEbj*

*N*

^{2}

_{0}

*k*

_{υu}

*N*

*N*

_{0}

^{2}

*S*

^{4}

_{GM}

*V*

_{S}(

*k*

_{υu})

^{2}] with

*k*

_{υu}= 2

*π*/10. The observed quantity

*S*

_{h}must be scaled to be used in (3): 〈

*S*

^{4}

_{10}

*S*

^{2}

_{h}

^{2}〉. Choice of this scaling makes these results directly comparable to the shear-derived dissipation values of Gregg (1989), Gregg and Kunze (1991), Kunze and Sanford (1996), Polzin et al. (1995), Duda and Jacobs (1995), and perhaps others. The factor 2.11 arises from the attenuation of shear at wavelengths slightly longer than 10 m when sampling discrete shear at 10-m separation (Gregg and Sanford 1988). The factor 2 for the GM quantity arises because the shear vector components have identical statistics; this typically holds true for our data and for simulated GM internal wavefields.

Diapycnal diffusivity is often expressed in terms of dissipation as *K* ≤ 0.2*ϵN*^{−2} (Gregg and Kunze 1991; Osborn 1980), so diffusivity can be estimated from observed 〈*S*^{4}〉 by substituting in *ϵ*_{G} from (3) and following the usual practice of assuming equality. For a wave field consistent with the GM model, 〈*S*^{4}_{10}*S*^{4}_{GM}*K*(*N*) would vary slightly around the value that *K*_{GM} (*N,* *ϕ*) would have for *N* = 3.0 cph at the BBTRE latitude *ϕ*_{B}, which is 3.6 × 10^{−6} m^{2} s^{−1}.

## 4. Shear data

Floats 001 and 100 returned 2337 and 8443 hourly *S*_{h} values, respectively. The mean, median, and mode values of *S*_{h} are given in Table 2. The mean values correspond to mean velocities of 1.1 and 0.5 cm s^{−1} at each end of the shallow and deep floats, respectively. Table 3 contains higher *S*_{h} moments and ratios of *S*_{10} moments with associated GM model values. Both higher moments exceed GM model values. The hourly *S*_{h} data from the deep float are shown in Fig. 3. The time series of *S*_{h} is quite variable with no characteristic frequencies. The filtered time series in the graph shows the variability that exists on time scales of days and weeks. Figure 4 shows the spectrum of *S*_{h} from each of the two floats, each divided by *N*^{2} to account for expected stratification scaling. The deep spectrum exceeds the shallow spectrum at all frequencies. A shear field dominated by motions from near-inertial and tidal frequency waves would exhibit peaks at these frequencies in the rotary spectrum or in spectra of Cartesian components, but the spectrum of *S* from such a field would have attenuated peaks because shear from individual waves rotates with frequency-dependent attenuation of magnitude oscillation. The magnitude oscillation approaches zero as frequency approaches *f.* Therefore, the absence of peaks in the spectrum of *S*_{h} does not preclude the dominance of waves of a particular low-frequency band, such as near-inertial, diurnal tidal, or semidiurnal tidal.

The measured spectra are compared to simulated spectra, also shown in Fig. 4. The simulated spectra were computed from fields of waves exhibiting the GM spectral form. Plane waves of 140 equally spaced frequencies *ω* between 1.2*f* and *N* having mode indices *j* = 1 to 1000 for each *ω* were generated for *N* = 0.5 and *N* = 1.1 cph, respectively. Two waves were included for each (*ω,* *j*) case, one with upward wavenumber and one downward. Each wave had azimuthal direction and phase chosen at random from uniform distributions, each had amplitude taken from the GM spectrum, and each met the condition that vertical wavelength be greater than 10 m. Note that, although mode index was used, the waves were vertically propagating and were not mode like. The velocities of the plane waves were summed to yield wavefield velocity time series. The summation was performed twice, yielding time series of velocity *U*(*t,* *z*) = **i***u*(*t,* *z*) + **j***υ*(*t,* *z*) at two depths separated by 10 m. The spectra of shear magnitude *S* are shown, which typically possess about 25% of the variance of **S**. The simulated *S* time series looked surprisingly like the data. The depicted simulated spectra are ensemble averages of 10 realizations.

In comparing the observed and simulated spectra for the shallow case, the observed spectrum is slightly above the simulated spectrum at frequencies below 0.15 cph, but it is far above it at higher frequency. For the deep case, the observed spectrum exceeds the simulated spectrum at all frequencies, again with greatest discrepancy at high frequency. The simulated spectra have steeper slopes than the observed, leading to the divergence of the curves at high frequency. The simulated/ observed discrepancies are evidence of either measurement noise (response to vertical flow of high-frequency internal waves is the most likely candidate) or of a difference between the ocean wave field and the simulated field of steady-state GM-amplitude plane waves. For example, substituting a frequency dependence of *ω*^{−1.5} in place of the GM dependence of *ω*^{−2} gives simulated spectra much closer to those observed. Departure of the internal wave spectrum from GM is a rich subject, and frequency spectra differing from GM have been observed. One example was recorded above the slope of a seamount, and strikingly shows that boundaries can have an effect on internal wave spectra (Eriksen 1998). The seamount spectra show spectral enhancement in the band around the frequency where the bottom slope (well defined and single-valued in the region of the moored measurements) is equal to the slope of characteristics (internal wave ray paths). Departure from GM has an important consequence here: If ocean internal wave velocity time series were not to exhibit the GM *ω*^{−2} dependence, then the use of expressions relating *K,* *ϵ,* and *S* that depend on the GM spectrum, such as (3), would be questionable. That uncertainty may be the case for our dataset, but we note this caveat and proceed with the use of (3) nonetheless. In defense of the use of (3), note that work by Polzin et al. (1995), which rests on work by Henyey et al. (1986), suggests that the refraction behavior of short vertical waves (and thus their tendency to dissipate) is a weak function of the frequency spectrum. [They estimated the function, and so corrections to (3) are available.]

Estimates of *K* were computed from the records using *K* = 0.2*ϵ*_{G}*N*^{−2}, with *ϵ*_{G} computed in two ways. It was computed on a daily basis using 1-day averages of *S*_{10}, and it was computed using the full-deployment values of 〈*S*^{4}_{10}*K* values are equal to the full-deployment values and are 8.3 × 10^{−5} m^{2} s^{−1} for the deep float and 3.3 × 10^{−5} m^{2} s^{−1} for the other. The standard deviations of the deep and shallow daily results are 4.5 × 10^{−5} and 1.7 × 10^{−5}, respectively. Note here that a scaling of *K* ∝ (〈*S*^{2}_{10}*S*^{2}_{GM}^{2} can be used as in Kunze and Sanford (1996). This would increase *K* slightly, as can be deduced from Table 3. All of our reported diffusivities exceed the *K*_{GM} values of 2.4 × 10^{−6} m^{2} s^{−1} for the deep float and 2.9 × 10^{−6} m^{2} s^{−1} for the shallow float.

## 5. Float trajectories

Floats 099, 100, and 101 were each tracked for one year. The trajectories are shown in Fig. 2, superimposed on detailed bathymetry. Velocities computed from the trajectories can be divided into mean and fluctuating components. The zonal and meridional means and the rms values of the fluctuations are reported in Table 1. Two of the deep floats tended to move to the east-southeast at over 1 cm s^{−1}, which disagrees with the inverse solution presented by St. Laurent et al. (2001) showing flow to the southwest at these depths of about 0.2 cm s^{−1}. A deviation of the geostrophic conditions between their measurement period and this one is not unexpected.

The probability density function of meridional velocity is very close to normal. The zonal velocities also have a normal distribution for the most part, with the exceptions that 5% of the data show excessively high speed to the west and 2% show excessively high speed to the east. The largest-amplitude topographic features in this area are the zonal fracture zones (Fig. 2), and guiding of flow by these features crudely explains the observed anisotropy.

## 6. Shear versus height above bottom and tidal forcing

To more fully describe shear behavior, the records can be directly compared with other oceanographic parameters measured within or inferred for the region. Here, we show that deep *S*_{h} varies with height above the bottom. We also show that deep *S*_{h} has a pattern of variability comparable to monthly modulation of the tide.

The comparison of shear with height above the bottom is straightforward. The locations of float 100 at the hourly *S*_{h} measurements are interpolated from the trajectory. The depths at these locations are then computed from the Smith and Sandwell (1997) bathymetric database using linear interpolation. Figure 6a shows the resultant time series of the local water depth at the float locations, along with the float depth. These two are combined to give height above the bottom. The height above bottom cannot go negative, of course. The crossing of the lines in Fig. 6a is due to error in the float trajectory and/or in the database.

Figure 7 shows float height above bottom versus time along with estimates of 〈*S*_{h}〉 and of *K* (computed from means of 〈*S*^{4}_{10}*K* values are near 1.0 × 10^{−4} m^{2} s^{−1} in the three bins, spanning from 400 to 1600 m above the bottom, and decrease above and below those heights. The possibility that the low value near the bottom is due to the float being grounded is not supported by the data: The reduced mean *S*^{4}_{h}*S*^{4}_{h}

These *K* values derived from the lengthy periods of sampling in each bin agree reasonably with estimates obtained from profiler observations shown by St. Laurent et al. (2001). In our interpretation, their results show *K* < 10^{−4} m^{2} s^{−1} above 3000-m depth (our approximate float depth) in areas that are at least 1500 m above the bottom and show elevated *K* closer to the bottom, exceeding 1.5 × 10^{−3} m^{2} s^{−1} very close to the bottom. Our slight reduction in the bin nearest the bottom (bin 1) is unexplained and may be the result of poorly sampled temporal variability. It is also possible that *N* is not equal to our assumed value of 0.5 cph when bin-1 data were taken. In particular, it is possible that the float was in a canyon (or some other zone) with low *N,* which would imply a larger *K* for the observed 〈*S*^{4}_{h}*N* was anomalously low (*T* anomalously uniform) when bin-1 data were taken.

Next, we compare the time series of shear with estimated time series of tidal currents at the bottom. Figure 8a shows the time series of *S*^{4}_{h}*U*^{2}_{T}*U*^{2}_{T}*U*^{2}_{T}*M*_{2}, *S*_{2}, *N*_{2}, *K*_{2}, *K*_{1}, *O*_{1}, *P*_{1}, *Q*_{1}). From these numbers, zonal and meridional current times series [*u*(*t*), *υ*(*t*)] at the BBTRE site were synthesized for twenty tidal constituents using the tide program *tidhar.* The envelope (*U*_{T}) of tidal current speed was computed by low-pass filtering the speed (the magnitude of *u* + *iυ*), passing periods longer than 3 days.

The lower panel of Fig. 8 shows that the filtered time series of *S*^{4}_{h}*S*^{4}_{h}*S*^{4}_{h}

A second method was also used to evaluate coherence of tides and *S*_{h}. This method used the procedure of shifting portions of the record and summing them to synthesize a time series lasting one month, discriminating against noise. This process is referred to as stacking in seismic data processing. The stacking process is analogous to the way annual cycles are sometimes computed from atmospheric or oceanic data collected at regular or irregular intervals. Because the tides show a monthly cycle containing two unequal spring tides (major and minor) and two neap tides, 1-month sections were stacked to produce “mean monthly cycles” of the complete *S*^{4}_{h}*U*_{T} were first identified, then intervals of 30 days beginning 6 days before each peak were defined. Then *U*_{T} in the resulting intervals was stacked to create a mean tidal cycle *U*_{TS}, shown in Fig. 9a. Hourly *S*^{4}_{h}*S*^{4}_{S}*S*^{4}_{S}*S*^{4}_{S}*S*^{4}_{S}

The cycle of *S*^{4}_{S}*U*_{TS} by about 6 days. The amplitude of *S*^{4}_{S}*U*_{TS}. Figure 9b shows 6-day advanced *S*^{4}_{S}*U*_{TS}. The correlation between the two is weak but evident. The least squares regression line has *R*^{2} = 0.38, suggesting that over one-third of the *S*^{4}_{S}

The shallow float was not tracked, so the height above bottom analysis could not be performed with its shear record. Tidal stacking and coherence computations also showed no correlation between tide and *S*^{4} for this float. Two of the four peaks of *S*^{4}_{h}*K*) that are evident in the top panel of Fig. 5 can be nonrigorously associated with atmospheric low pressure events culled from global reanalysis datasets: the mid-April and early-May peaks (Duda et al. 2002). The late May and late June peaks are not accompanied by low pressure events or by high wind events, however.

## 7. Summary

Two Shearmeters have returned time series of hourly shear magnitude *S*_{h} in the ocean at the site of the Brazil Basin Tracer Release Experiment (Ledwell et al. 2000). These quasi-isobaric floats of the RAFOS type effectively sampled the very small shear values typical of the ocean, recording mean velocity differences over their 10-m apertures of 2.2 and 1.1 cm s^{−1} at the float depths of 1660 and 2850 dbar, respectively. Being drifters, the floats were intended to provide unbiased sampling of shear over the rough and heterogeneous bathymetry of the site, which is on the western side of the Mid-Atlantic Ridge.

Both of the *S*_{h} records display variability at periods from hours to months. The shallow record, of 100 days in duration, shows four prominent peaks. These do not correlate with spring tides, but two of them can be loosely associated with atmospheric low pressure events. The deep record, of 365 days duration, shows about 11 prominent peaks. These are irregular and are probably the result of complex temporal and spatial heterogeneity.

Mean *S*_{h}, mean *S*^{2}_{h}*S*^{4}_{h}

The diapycnal mixing expected from the observed shear fields, undoubtedly due to internal waves, can be computed using existing theory. The wavefield may not be precisely that used in the theoretical arguments, but the theory is applied nonetheless. The resulting diapycnal diffusivities are 3 and 8 × 10^{−5} m^{2} s^{−1} at 1660 and 2850 dbar, respectively. Concerning the applicability of the theory, the frequency spectra of *S*_{h} have a shallower negative slope than we believe would be consistent with the GM spectrum. Polzin et al. (1995) point out the the internal wave dissipation expected from waves refracting to high wavenumber is a function of a characteristic frequency of the wavefield. The measured spectra imply a higher characteristic frequency than GM in our data, and higher dissipation. Since expression (3) is written for GM, it possibly underestimates the dissipation for a given level of *S*^{4}_{h}*K* estimates may also be low.

Tracking of the deep float also provided an opportunity to evaluate the behavior of *S*_{h} versus height above the bottom. Some accentuation of shear and inferred mixing is seen at 400–1600 m above the bottom relative to 1600–2400 m above the bottom, with a peak of about *K* = 9 × 10^{−5} m^{2} s^{−1}, which is about one-third of the tracer-deduced *K* at these heights above the bottom (Ledwell et al. 2000). Slightly reduced *S*_{h} is evident during the short period of time (889 h) that the float was estimated to be within 400 m of the bottom. This is unexplained and may be due to differing *N* conditions or temporal variability.

Tidal currents have predictable patterns that may possibly be exploited to verify whether tidal currents are responsible for bottom-generated internal waves at this site, and by extension at other rough sites. A tide-to-shear comparison was attempted here. The mean monthly cycles of predicted tidal current and of a gappy set of *S*^{4}_{h}*S*^{4}_{h}*N* = 0.5 cph at the BBTRE latitude, which propagate at 8.7° from the horizontal, are consistent with tidally generated internal waves rising to those heights above the bottom (or above the highest neighboring ridges) in a few days. High-shear shorter-wavelength waves (vertical wavelengths of 10– 60 m) have vertical group velocities of 60–300 m day^{−1}. These waves of small vertical wavelengths are not expected to contain much of the energy of an internal wavefield generated by flow over rough bathymetry, but they contribute an important fraction of the total shear (St. Laurent and Garrett 2002). Velocity profiles and vertical wavenumber spectra of abyssal internal waves over flat and sloping bottoms provide evidence that long vertical wavelengths tend to dominate, but also that high-wavenumber variance is present (Johnson and Sanford 1980; Sanford 1991).

The degree of correlation between *S*^{4}_{h}^{−1} (with standard deviation 1.5 cm s^{−1}), which is very close to the mean tidal current. This is evidence that subtidal near-bottom flows may be generating internal waves in this area that are comparable to those created by tidal flow. Subtidal currents of this strength may also interrupt the generation of oscillatory waves from the tidal currents since the resultant currents will be asymmetrical.

These two records of shear magnitude collected above rough bathymetry provide a baseline for evaluation of the mean levels and variability of future shear records. The clear distinctions between the two records collected at two different depths in the same area exemplify the heterogeneity of ocean internal waves and finestructure. The vertical structure of mean shear magnitude and of the higher moments of shear magnitude presented here support the general idea of upward internal wave energy flux that diminishes with height above the rough bottom.

## Acknowledgments

The support of James Ledwell and John Toole enabled these floats to be included in BBTRE. Thanks are given to Webb Research Corporation and Seascan, Inc., for float development and manufacture. Clayton Jones was particularly helpful during the procurement phase. Brian Guest did a wonderful job helping to prepare the floats, and he deployed them. This work was funded by the National Science Foundation under Grants OCE9416014 and OCE9906685.

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Float mission parameters. Speeds are listed in centimeters per second, *N* is in cycles per hour, and depth is in meters. Speeds determined from deployment position and initial satellite transmission position are indicated with *

Measured deployment-duration shear statistics for the two floats

Higher moments of shear for the two floats computed over the duration of the records

^{*}

Woods Hole Oceanographic Institution Contribution Number 10892.