## 1. Introduction

The statistics of turbulence in the open ocean and coastal seas are inherently difficult to estimate due to the sporadic and intermittent nature of turbulence. The patchiness of turbulence coupled with coarse sampling that is characteristic of conventional microstructure profiling leads to a spatial and temporal undersampling of the turbulent field (Baker and Gibson 1987). In this paper, a technique is described that yields robust estimates of quasi-continuous vertical profiles of turbulent kinetic energy dissipation rate (*ϵ*), acquired using a commercial off-the-shelf (COTS) acoustic Doppler current profiler (ADCP), from a vessel underway.

*u*/∂

*z*or ∂

*υ*/∂

*z*) is computed, with a single probe measuring one component of the shear. The dissipation rate of turbulent kinetic energy, assuming isotropy, is estimated as (Osborn 1974)

*ψ*(

*k*) is the one-dimensional spectrum of velocity shear,

*k*is the wavenumber, and

*ν*is the kinematic viscosity.

As Eq. (1) suggests, the actual calculation of the mean-square shear is normally performed in wavenumber space by integrating the spectrum. While this has become the most widely accepted method for quantifying ocean turbulence, there are a number of difficulties associated with the technique. The shear probes themselves are difficult to manufacture and are inherently fragile, making them expensive and prone to failure. Also, because the probes must be sensitive to small wavelength velocity fluctuations, they are also responsive to platform vibrations and other noise sources. And although not explicit in Eq. (1), the integration limits are quite important when processing shear-probe data. Judicious selection of the upper-integration limit is required because the spectra can be contaminated with noise at high wavenumbers, causing an overestimate of the shear variance (Wesson and Gregg 1994; Moum et al. 1995). As a result, shear-probe data require careful analysis, usually by a trained analyst who is extremely familiar with the process and who can select quality segments of data to analyze or, just as important, remove contaminated portions of the data record. It is beyond the scope of this paper to discuss the intricacies of estimating turbulence levels with shear probes, and the reader is referred to Wesson and Gregg (1994) and Wolk et al. (2002), which provide detailed discussions of the procedure.

*ϵ*provide an alternative to measuring the centimeter-scale velocity shear variance with microstructure profilers. The large-eddy dissipation estimate can be constructed using dimensional analysis by assuming that there is a balance between turbulent production and dissipation (Taylor 1935). If the largest-scale turbulent eddies with velocity

*q*′ are dissipated over the duration of an eddy overturn time scale

*τ*=

*l*/

*q*′, where

*l*is the eddy size, then the rate of kinetic energy dissipation can be stated functionally as

*C*

_{ϵ}is an empirically determined order one constant. Since the energy-containing scales are much larger than the dissipative scales, especially at high Reynolds number, they may be inherently easier to measure and quantify (Moum 1996). Moum (1996) argues that, since the energy-containing scales of turbulence are easier to quantify, methods based on measurements of the energy-containing scales are of practical importance.

The large-eddy method can be implemented with a modified ADCP operating in beam coordinates. The use of an ADCP as a standoff turbulence sensor can be traced back to the work of Ann Gargett in the mid-1990s. Gargett (1994) outlined a computational procedure for estimating quasi-continuous vertical profiles of *ϵ* with a ship-mounted, modified ADCP that she called the Doppler turbulence (DOT) system. This technique was tested in the waters of British Columbia, Canada, in a region between the Strait of Georgia and Haro Strait. The combination of strong tidal flows, sills, and the formation/decay of tidal fronts produces vigorous turbulence, sometimes encompassing the entire water column and creating surface “boils.” The observed eddy structures had strong vertical velocities associated with them exceeding 40 cm s^{−1}, making the test location ideal for evaluating large-eddy methods for estimating turbulence with an ADCP. A major advantage of the large-eddy method with a ship-mounted ADCP is that it provides continuous horizontal coverage, affords vertical aperture, and can sample large areas quickly. Gargett (1999) contended that ADCP estimates of *ϵ* provide a favorable balance between quantity and quality.

In Gargett (1994), a narrowband ADCP was fitted with a custom sensor head such that one beam was oriented vertically, allowing for direct vertical velocity (*w*) measurements, while the other three beams retained their original orientation. The estimate of *w* from a single vertical beam is different than conventional estimates using the four-beam solution from a Janus-style configuration of transducers. The four-beam solution represents the average vertical velocity over a circle with a diameter roughly equal to the distance from the ADCP to the range bin in which *w* is calculated. This averaging area is much too large to resolve turbulent structures. However, a single vertical beam can measure the vertical velocity averaged over the width of a single beam, which is typically a few percent of the range or *O*(1 m) at a range of 50 m.

The vertical velocity field has several practical aspects associated with it that make it more appealing than the other velocity components for characterizing the turbulence. First, if particular care is taken to ensure the beam is vertical, then *w* does not contain horizontal motion associated with the surface craft moving through the water. This is important because the ship speed can be 2–3 orders of magnitude larger than the turbulent velocities. Second, as will be discussed later, the single-ping uncertainty (*σ*) of an ADCP velocity estimate is directly related to the maximum range of velocities that can be measured, a quantity called ambiguity velocity. Ambiguity velocity can be set low for the vertical beam, decreasing noise and leading to better turbulence measurements, if the canted beams containing a component of the mean ship speed are not required. If they are required, then wrapping can be mitigated for 1–2 phase wraps with standard techniques. Last, as pointed out by Gargett (1994), if the flow is turbulent, then it must have vertical velocity; and since this component is generally weak in other flow regimes, it becomes the most distinctive velocity component for identifying regions of turbulence.

There are two other approaches used to estimate turbulence levels with ADCPs, usually referred to as the variance and structure function methods. The variance method has been used with ADCPs on stationary platforms in pulse-coherent (Lohrmann et al. 1990) and broadband (Rippeth et al. 2003) modes to estimate turbulence levels. These methods require long time averages to produce statistically robust estimates and force an assumption of homogeneity (if the ADCP is on a moving platform) and stationarity over the averaging period. For example, Rippeth et al. (2003) used 10-min time averages to balance the trade-off between having a low-error Reynolds stress estimate and a statistical average that could be considered stationary. Structure functions also typically make the same stationarity assumption as mentioned above and require long time averages. In Wiles et al. (2006) the structure function was computed using a 10-min average. In summary, these techniques are excellent and well suited for estimating turbulence that is persistent—for example, boundary layers—but are not applicable to regions with intermittent and sporadic turbulence with scales less than the averaging period. The advantage of the large-eddy method is that it allows for time windows as short as 30 s compared to several minutes or longer of averaging for the above-mentioned techniques and the time between successive microstructure profiles. The large-eddy method affords the horizontal resolution required to resolve the structure of patchy turbulence. In addition, deployments from mobile platforms enable high-resolution surveys of oceanographic features with horizontal structure, such as coastal and tidal fronts or flows influenced by topography.

This paper evaluates the large-eddy method for estimating turbulence levels in estuaries and near coasts using a broadband ADCP and introduces a correction procedure for random noise contamination of velocity and length-scale estimates. This work is a direct extension of the work described in Gargett (1994) with her narrowband DOT system; however, here we exploit the decreased noise levels present in modern broadband ADCPs to increase the sensitivity to weaker turbulence. The turbulent velocity (*w*′) and the eddy length scale (*l*) are also calculated differently than Gargett (1994), and the effects of various parameters in the resultant estimates of rms velocity, internal length scales, and *ϵ* are discussed. The analysis presented here is intended to update the oceanographic community with regard to using broadband ADCPs for estimating turbulence levels in the coastal ocean. ADCP estimates of dissipation rate (*ϵ*_{ADCP}) are compared with over 100 quasi-coincident vertical profiles of dissipation rate taken with a traditional microstructure profiler, the Modular Microstructure Profiler (MMP) (Gregg et al. 2012). In subsequent discussions, the MMP estimates of dissipation rate will be designated *ϵ*_{MMP}.

## 2. Data and instrumentation

Testing took place aboard the University of Washington, Applied Physics Laboratory, 15-m utility boat the Research Vessel (R/V) *Miller*, on 23–25 May 2006 in and around Colvos Passage in Puget Sound, Washington (Fig. 1). Colvos Passage was chosen because it is a region where there is regular and intense turbulent mixing (Seim and Gregg 1997). Strong tidal currents, 1–2 m s^{−1}, coupled with channel curvature and shallow sills create highly turbulent patches. Since a broad range of turbulent intensities can be found in this region, including strong turbulence, it was an ideal location for testing ADCP techniques for quantifying turbulence levels. All the data shown were collected either in Colvos Passage or near the Tacoma Narrows Bridge.

ADCP estimates of the dissipation rate are compared to measurements of turbulent microstructure from MMP casts. The MMP is a 2-m-long loosely tethered, free-falling instrument used to measure vertical profiles of temperature, conductivity, centimeter-scale vertical shear of horizontal currents, and centimeter-scale vertical gradients of temperature and conductivity (Gregg et al. 2012). The MMP was deployed from the stern of the R/V *Miller* while underway and maintained a fall rate of 0.6–0.7 m s^{−1}. When the vehicle approached the bottom, as sensed by an altimeter, the tether was pulled taught and the profiler was reeled back to the surface and subsequently released again for another vertical profile.

Data from each profile was segmented into ≈0.8-m vertical bins, and *ϵ*_{MMP} was estimated from two shear probes using the spectral form of Eq. (1). The MMP has two shear probes oriented to be sensitive to changes in orthogonal components of horizontal velocity. For each depth interval in a profile, the average dissipation rate was calculated from the two probes unless one estimate was less than one-quarter of the larger, in which case the smaller was used. MMP profiles were collected every 4–5 min while the boat was underway, resulting in a profile every 300 m in the horizontal when the vessel speed through the water was approximately 1 m s^{−1}. Approximately 100 individual vertical profiles were obtained for comparison with the ADCP.

*U*) were

*υ*

_{2}was canted 60° from the vertical and perpendicular to the ship track.

Orientation and position of ADCP beams relative to the ship.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Orientation and position of ADCP beams relative to the ship.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Orientation and position of ADCP beams relative to the ship.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

A cross section of MMP and ADCP data collected on a pair of nearly reciprocal courses is shown in Fig. 3 collected over a 75-min time interval. The data are from Colvos Passage as the ship transited down and back across a 20-m sill. The coincident MMP and ADCP profiles extend to approximately 60 m with water depths 90–110 m. Density, vertically unsorted, is highly variable in both the vertical and horizontal directions with buoyancy periods ranging from 5 to 20 min (Fig. 3a). Large density overturns are observed throughout the cross section with ±10-m inversions (Fig. 3b). The density inversions were calculated as the vertical vector displacement of a range bin required to return the density profile into a gravitationally stable, monotonic profile. The density inversions are highly correlated with *ϵ*_{MMP} (Fig. 3c), with turbulence levels exceeding 10^{−6} m^{2} s^{−3}. The buoyancy Reynolds number (*ϵ*/*νN*^{2}) tends to exceed 4.3 × 10^{4} in the regions of strong turbulence and density overturning. This suggests that the turbulence is fully developed and nearly isotropic (Gargett et al. 1984). The root-mean-square velocity from the vertical beam (*w*′)—see section 3 for the definition—is also shown (Fig. 3e). Regions of strong vertical velocity variability are collocated with overturning and intense turbulence.

(a)–(d) Section comprised from two nearly reciprocal courses of MMP and (e) ADCP data that are representative of the environmental conditions in Colvos Passage; (a)–(e) extend over the same horizontal domain (≈10 000 m) and vertical extent. (f) Geographical location of the transect with bathymetry contours superimposed (CI = 10 m). The MMP drop order (MMP drop numbers 14848–14860) indicated at the top of (a) correspond to those locations in (e). Panels show (a) potential density (*σ*_{θ}) from the MMP, (b) vertical overturning scale, (c) directly measured *ϵ*_{MMP}, (d) buoyancy Reynolds number, and (e) rms vertical velocity with vertical overturning profiles from (b) superimposed.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

(a)–(d) Section comprised from two nearly reciprocal courses of MMP and (e) ADCP data that are representative of the environmental conditions in Colvos Passage; (a)–(e) extend over the same horizontal domain (≈10 000 m) and vertical extent. (f) Geographical location of the transect with bathymetry contours superimposed (CI = 10 m). The MMP drop order (MMP drop numbers 14848–14860) indicated at the top of (a) correspond to those locations in (e). Panels show (a) potential density (*σ*_{θ}) from the MMP, (b) vertical overturning scale, (c) directly measured *ϵ*_{MMP}, (d) buoyancy Reynolds number, and (e) rms vertical velocity with vertical overturning profiles from (b) superimposed.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

(a)–(d) Section comprised from two nearly reciprocal courses of MMP and (e) ADCP data that are representative of the environmental conditions in Colvos Passage; (a)–(e) extend over the same horizontal domain (≈10 000 m) and vertical extent. (f) Geographical location of the transect with bathymetry contours superimposed (CI = 10 m). The MMP drop order (MMP drop numbers 14848–14860) indicated at the top of (a) correspond to those locations in (e). Panels show (a) potential density (*σ*_{θ}) from the MMP, (b) vertical overturning scale, (c) directly measured *ϵ*_{MMP}, (d) buoyancy Reynolds number, and (e) rms vertical velocity with vertical overturning profiles from (b) superimposed.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

^{−1}and in the present study vertical velocities are 5 cm s

^{−1}or less. To make measurements of weaker turbulence with an ADCP, it is essential to minimize the single-ping uncertainty of a velocity estimate. Functionally,

*σ*has the form

*V*

_{a},

*f*

_{0}, and Δ

*z*are the ambiguity velocity, system acoustic frequency, and range cell size, respectively. The system frequency is a characteristic of the instrument and is usually chosen based on a trade-off between maximum range and

*σ*. In practice, increasing the frequency to reduce

*σ*will result in a proportional decrease in maximum range.

The ambiguity velocity represents the range of radial velocities that can be measured by the system before the value overscales. Overscaling occurs because the Doppler measurement with broadband ADCPs is calculated as a phase shift between two closely spaced acoustic pulses with ±*V*_{a} corresponding to ±*π*. As the velocity increases to *V*_{a} and beyond, the apparent velocity will abruptly transition from *V*_{a} to −*V*_{a} as the phase shifts from *π* to −*π*. Therefore, it is important to set *V*_{a} as small as possible to minimize *σ*, yet be large enough to avoid phase wrapping. In practice, single phase wraps can usually be detected and corrected, allowing the time series to be reconstructed. The vertical beam and beam 2, *υ*_{2}, are perpendicular to the mean velocity of the boat through the water and are less prone to wrapping. The speed of the boat moving through the water usually sets the lower threshold for *V*_{a}, since it affects those ADCP beams for which there is a projection of the boat velocity in the direction of the individual beams. In the setup used here (Fig. 2), the projection of the boat speed *U* onto beams 3 and 4 is ±*U* sin(30°) = ±*U*/2 and zero for beams 1 and 2.

It is also clear from Eq. (5) that it is important to use the largest bins possible in order to reduce noise levels. In practice, this is the most effective way to minimize *σ*. The choice of Δ*z* warrants careful consideration though. If the energy-containing scales or, as discussed later, the Thorpe scales, are too small relative to the bin size, then they will effectively be filtered out, since they cannot be resolved in the vertical. If Δ*z* is set to a small value, then the resultant ADCP noise level could be too large to measure the velocities associated with weak turbulence. Noise due to small bin size can be mitigated by averaging adjacent bins in postprocessing. Averaging *N* bins in the vertical will reduce *σ* by *N* will reduce the noise by 1/*N*. For example, if 50-cm bins were averaged to form 1-m bins, then *σ* would only be reduced by *σ* would be a factor of 2 less. Therefore, it is important to use the largest bins that will give the vertical resolution required to characterize the turbulent structures being investigated.

*M*pings together and faster sampling rates permit finer temporal/spatial resolution for a desired effective noise level. Although there are a number of ways to increase the ADCP ping rate, there are essentially three factors that determine the minimum time between pings. These factors are sound travel time Δ

*t*

_{TT}, time for the instrument to process the individual binned returns Δ

*t*

_{B}, and system overhead Δ

*t*

_{OH}. The minimum time between pings can be written as their sum,

The travel time component Δ*t*_{TT} = 2*R*/*c*, where *c* is the speed of sound and *R* is the along-beam range to the last bin, while Δ*t*_{B} is proportional to the number of range bins *N* and can be written Δ*t*_{B} = *C*_{B}*N*, where *C*_{B} is a scale factor. The overhead time is a constant that is characteristic of the system. From these relationships, it is clear that the maximum range and the number of bins will be the primary factors in determining the sample rate. Another benefit from using large bins is that, for a given maximum range, Δ*t*_{B} will be smaller because *N* is smaller, resulting in a faster sample rate. Recording in beam coordinates, as opposed to Earth or ship coordinates, also has advantages with regard to increasing the sample rate. In beam coordinates internal calculations within the ADCP are reduced, thereby reducing *C*_{B}. The time it takes for the ADCP to read its internal compass is one of its most time-consuming operations. If an external compass can be used, then the ADCP internal compass can be disabled, decreasing the time between pings by approximately 40 ms. Other ADCP configuration settings, such as recording each individual ping without ensemble averaging, also improve the data quality by allowing more effective screening of individual samples for wild points.

## 3. Method: Estimating *ϵ*_{ADCP}

*ϵ*

The large-eddy method for calculating *ϵ*_{ADCP} requires a robust procedure for estimating *w*′ and *l*. The broadband noise component inherent to ADCP measurements introduces a systematic overestimate of *w*′ and an underestimate of *l*. Essentially, the random noise is added to *w*′ and, because it is uncorrelated, it reduces the apparent value of *l*, which is estimated as a correlation length scale. Both of these effects combine to produce an overestimate in *ϵ*_{ADCP} = *C*_{ϵ}*w*′^{3}/*l*. In this section, the data processing procedures that compensate for the ADCP sensor noise will be described.

There are numerous ways to calculate the fluctuating, or turbulent, velocity component. Conceptually, the Reynolds decomposition (Tennekes and Lumley 1972) separates the flow into mean and fluctuating components by suitable time averaging, with the average representing the mean, while the remainder is the fluctuating component, whose mean value is zero. Shipboard ADCP data are a spatial series, rather than a time series at a fixed location, and, in the dynamic environment being sampled, it is clearly not stationary. In spite of these nonideal conditions, a procedure has been adopted whereby *w*′ is calculated as the root-mean-square (rms) vertical velocity in a series of horizontal windows, one for each range cell.

After wild point removal, velocity data in each range cell are broken up into a sequence of windows of length *L* that are typically overlapped by 50%. The average boat speed through the water (*U*) over the duration of the ADCP record and the ADCP sample rate *f*_{s} determine the number of points *n* in each window. A linear trend is removed from the windowed vertical velocity and the root-mean-square is computed to form *w*′. Since the mean speed in the file was used to convert the time series to a spatial series, a condition was established to ensure that the speed was nearly constant within a run; if the standard deviation of *U* was less than 0.3*U*, then *U* was considered quasi constant in magnitude. The choice of window length is clearly an important parameter for the procedure and will be discussed in detail later.

*σ*

_{n}is that its spectrum is white, that is, the spectrum is constant at all frequencies; this feature can be exploited to calculate

*σ*

_{n}from the data. Generally, the spectrum of water velocities in the ocean tends to be red over a broad range of frequency, with larger scales containing more energy. However, the observed spectrum of ADCP velocity will have a flattened appearance at higher frequencies due to instrument noise if the instrument is sampled rapidly enough. An example of frequency spectra from beams 1–4 are shown in Fig. 4. The average value of the along-beam velocity has been removed for each ping before calculating the spectra to remove the effects of boat motion. At frequencies above 0.5 Hz, the spectra from all beams flatten to nearly the same spectral level. In this band, the value of the spectrum at the highest frequency will be representative of the

*σ*

_{n}spectrum and is computed as

*f*

_{N}is the Nyquist frequency, and

*f*

_{t}is the inverse of the time window length used to compute

*S*. Typically

*f*

_{N}≫

*f*

_{t}and the formula reduces to

*σ*

_{n}for each of the beams calculated using Eq. (8).

Average power spectral density of velocity for beams 1–4. For each beam, the spectrum is calculated in each bin and then subsequently averaged. The single-ping uncertainty of a velocity estimate, using Eq. (8), is shown in the top-right-hand corner. In this example, 3-m bins were used.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Average power spectral density of velocity for beams 1–4. For each beam, the spectrum is calculated in each bin and then subsequently averaged. The single-ping uncertainty of a velocity estimate, using Eq. (8), is shown in the top-right-hand corner. In this example, 3-m bins were used.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Average power spectral density of velocity for beams 1–4. For each beam, the spectrum is calculated in each bin and then subsequently averaged. The single-ping uncertainty of a velocity estimate, using Eq. (8), is shown in the top-right-hand corner. In this example, 3-m bins were used.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

*l*, defined as

*ρ*

_{w}is the autocorrelation coefficient of detrended

*w*, within the same window that is used to compute

*w*′, and

*L*

_{0}is the lag distance of the first zero crossing in

*ρ*

_{w}. According to Tennekes and Lumley (1972), the transverse integral scale

*l*= Λ/4, where Λ is the true integral scale, which is related to the low-wavenumber cutoff (

*k*

_{0}) in the inertial subrange by Λ = 0.29/

*k*

_{0}. In the formal definition, the integral is carried out to infinity, which, of course, cannot be realized in practice. The effect of using different limits for the integral in Eq. (9) was studied by O’Neill et al. (2004), and the above-given definition, integrating from the origin to the first zero crossing, was found to yield the most stable results.

It was argued by Gargett (1999) that a vertical length scale is more suitable than a horizontal length scale in the presence of stratification when Eq. (2) is used with vertical velocity. However, the computational procedure described above yields a horizontal length scale for vertical velocity and therefore we are explicitly assuming the turbulence is isotropic. In our case this could lead to an overestimate of the scale *l* in Eq. (2), though the direct comparisons of *ϵ* measured via this method and MMP data (section 5) do not indicate this is the case. Evidence for the applicability of a horizontal length scale can also be seen by considering the velocity data from beam 2, which is also transverse to the ship track but canted 60° from vertical. Because of its orientation, the radial velocity measured by this beam is primarily horizontal. Figure 5 compares the spectra of the 60° canted beam and vertical beam under a range of conditions and turbulence levels. The average along-track speed *U* was used to transform from frequency to wavenumber space. The vertical black line denotes the wavenumber *k* = 1/*L* of the 30-m window used to calculate *w*′ and *l* in the following sections, and the close agreement between the spectra of the two components suggests that the flow is isotropic within that band and that the horizontal correlation scale is a reasonable surrogate for the vertical.

Power spectra for the vertical (red line) and 60° canted (blue line) ADCP beams. Window length used to calculated *w*′ and *l* is shown for reference (black vertical line). Average MMP estimated turbulent dissipation rate is (top left) 10^{−8}, (top right) 3 × 10^{−7}, (bottom left) 10^{−6}, and (bottom right) 3 × 10^{−6} m^{2} s^{−3}.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Power spectra for the vertical (red line) and 60° canted (blue line) ADCP beams. Window length used to calculated *w*′ and *l* is shown for reference (black vertical line). Average MMP estimated turbulent dissipation rate is (top left) 10^{−8}, (top right) 3 × 10^{−7}, (bottom left) 10^{−6}, and (bottom right) 3 × 10^{−6} m^{2} s^{−3}.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Power spectra for the vertical (red line) and 60° canted (blue line) ADCP beams. Window length used to calculated *w*′ and *l* is shown for reference (black vertical line). Average MMP estimated turbulent dissipation rate is (top left) 10^{−8}, (top right) 3 × 10^{−7}, (bottom left) 10^{−6}, and (bottom right) 3 × 10^{−6} m^{2} s^{−3}.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

It is also typical for the buoyancy Reynolds number (e.g., Fig. 3d) to exceed 4.3 × 10^{4} in regions of strong turbulence. In those instances the turbulence should be fully developed and isotropic (Gargett et al. 1984). This evidence, along with the results cited above, justifies the use of a transverse, horizontal length scale in Eq. (2) for the environmental conditions in Colvos Passage and Tacoma Narrows.

*l*when the signal-to-noise (SNR) is large, where we define

*l*is systematically underestimated. In the case of uncorrelated noise, the effect of noise on the

*ρ*

_{w}can be calculated directly. Note that Eq. (9) uses

*ρ*

_{w}, the autocorrelation coefficient function, and should not be confused with the autocorrelation function (

*R*

_{w}), but the relationship is straightforward; that is,

*ρ*

_{w}(

*L*) =

*R*

_{w}(

*L*)/

*R*

_{w}(0), where

*R*

_{w}(0) is the correlation at zero lag or variance of

*w*. Consider the autocorrelation function of the measured signal including noise:

*m*is the sum of the signal,

*s*, and the noise

*n*, and

*E*{ } indicates the expected value. The independent variable is

*x*and the lag length, or spatial sample interval, is Δ

*x*. Expanding in terms of the component autocorrelation functions [

*R*

_{ss}(Δ

*x*) and

*R*

_{nn}(Δ

*x*)] and cross-correlation functions [

*R*

_{ns}(Δ

*x*) and

*R*

_{sn}(Δ

*x*)],

*ρ*

_{w}, estimates of

*l*will be underestimated when

*σ*

_{n}is large compared to

*σ*

_{s}or, equivalently, when SNR is small.

Autocorrelation function as a function of lag index. Solid and dashed curves represent *R* for time series without and with noise, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Autocorrelation function as a function of lag index. Solid and dashed curves represent *R* for time series without and with noise, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Autocorrelation function as a function of lag index. Solid and dashed curves represent *R* for time series without and with noise, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

*R*

_{ss}as

*A*

_{s}, the integral scale without noise

*l*

_{s}is defined as

*R*

_{mm}, which represents the measurement, is

*A*

_{s+n}and

*l*estimated by Eq. (9) using the measured signal variance, the correlation length based on the measured signal, the ADCP noise variance derived from the spectrum, and the sampling interval, which are all measured quantities.

*w*′,

*l*, and

*ϵ*. Following Tennekes and Lumley (1972), the signal spectrum is assumed to be the one-dimensional transverse velocity spectrum for inertial turbulence, namely,

*k*is the wavenumber and

*α*= 0.5. As previously discussed, the limiting wavenumber

*k*

_{0}is related to the transverse integral scale by 4

*l*= 0.29/

*k*

_{0}(Tennekes and Lumley 1972) and for these simulations

*k*

_{0}was set to 1/20 m

^{−1}. At wavenumbers less than

*k*

_{0}, the spectrum has a parabolic shape given by

^{1}Similarly, the noise component is computed by inverse transform of a white spectrum, and the composite spatial series is obtained by adding the noise to the turbulence series in the spatial domain.

The effects of sensor noise on the windowing algorithm for estimating the rms velocity and the integral scale are illustrated in Fig. 7, left column. The rms velocity and *l* were calculated using 10-, 20-, 30-, and 50-m-long windows. As expected, *w*′ increases with increasing window size and *σ*_{n}, while *l* also increases with window size but decreases with *σ*_{n}, confirming the original assertion that the addition of noise will decrease the apparent integral scale. The net effect of noise and window length on the calculated dissipation rate using Eq. (2) can be represented by *C*_{ϵ}, where *C*_{ϵ} is the value necessary for *C*_{ϵ}*w*′^{3}/*l* to equal the corresponding value of *ϵ* in the spectra generated using Eqs. (15) and (16). Here *C*_{ϵ} decreases with increasing noise level due to the combined effect of *w*′ and *l* being overestimated and underestimated, respectively. However, note that even in the presence of noise, the value of *C*_{ϵ} does not appear to be very sensitive to the window length, indicating that the large-eddy estimates of *ϵ* are robust with respect to this parameter.

(top) Rms velocity *w*′, (middle) integral scale *l*, and (bottom) *C*_{ϵ} as a function of rms noise *σ*_{n}. (left) Noise-containing and (right) noise-compensated calculations. Black dashed, black solid, gray dashed, and gray solid lines denote 10-, 20-, 30-, and 50-m-long windows, respectively. Noise-compensated calculations are essentially independent of *σ*_{n}, and *C*_{ϵ} is nearly independent of window length.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

(top) Rms velocity *w*′, (middle) integral scale *l*, and (bottom) *C*_{ϵ} as a function of rms noise *σ*_{n}. (left) Noise-containing and (right) noise-compensated calculations. Black dashed, black solid, gray dashed, and gray solid lines denote 10-, 20-, 30-, and 50-m-long windows, respectively. Noise-compensated calculations are essentially independent of *σ*_{n}, and *C*_{ϵ} is nearly independent of window length.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

(top) Rms velocity *w*′, (middle) integral scale *l*, and (bottom) *C*_{ϵ} as a function of rms noise *σ*_{n}. (left) Noise-containing and (right) noise-compensated calculations. Black dashed, black solid, gray dashed, and gray solid lines denote 10-, 20-, 30-, and 50-m-long windows, respectively. Noise-compensated calculations are essentially independent of *σ*_{n}, and *C*_{ϵ} is nearly independent of window length.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

The correction procedure can be tested by using the net rms velocity from Eq. (7) and the noise-free integral length scale from Eq. (14) on the simulated data (Fig. 7, right column). For each window length, a consistent estimate of *w*′, *l*, and *C*_{ϵ} is obtained for a wide range of noise values, even when SNR approaches unity. Again, although *l* increases significantly with window length, the changes are compensated by a corresponding increase in *w*′^{3} to yield a value of *C*_{ϵ} that is essentially independent of window length. The results of these calculations suggest that *C*_{ϵ} ≈ 0.35 with only a weak dependence on window length.

Figure 8 shows the effect of window length on *w*′, *l*, and *C*_{ϵ} explicitly. In this plot, the value of each parameter is normalized by the value at the longest window length used in these simulations, 200 m. Window length, on the abscissa, has been normalized by 20 m, the inverse low-wavenumber cutoff in the inertial subrange that was used to generate the synthetic time series. The rms velocity and *C*_{ϵ} are 90% and 95% of their final value, respectively, when the window length is twice the largest length scale in the inertial subrange, with all quantities reaching their asymptotic value by a normalized window size of four. Therefore, the window length should be at least twice the size of the largest-scale eddies in the flow, as defined by the low-wavenumber cutoff, to ensure robust estimates of *ϵ* using the large-eddy method and 4 times the size for accurate estimates of *l*.

Normalized values of

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Normalized values of

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Normalized values of

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Figure 9 illustrates the mechanics of calculating *w*′ and *l* using measured ADCP data. The top row shows a segment of single-beam vertical velocity data for a single range bin. The gray dashed-line box denotes a 30-m-long window in which *w*′ and *l* are computed. The middle row is vertical velocity within the gray box with a linear trend removed. In this example the net rms velocity, or rms velocity with noise removed, is 18.3 mm s^{−1} and this will be referred to as the turbulent velocity *w*′ henceforth. The bottom row shows the autocorrelation coefficient function of *w* within the 30-m window as the function lag distance, where the corrected integral scale computed using Eqs. (9) and (14) is 1.35 m. Note that *l* is related to the low-wavenumber cutoff via 4*l* = 0.29/*k*_{0} (Tennekes and Lumley 1972) and therefore the scale of the energy-containing eddies are ≈14 times larger, consistent with the large-scale structure observed in the middle panel of Fig. 9. Using Eq. (2), with *C*_{ϵ} = 0.35, the calculated dissipation rate is 1.6 × 10^{−6} m^{2} s^{−3}. This procedure is employed to yield values of *w*′, *l*, and *ϵ*_{ADCP} for each range bin in a window that progressively moves horizontally.

Example of *w*′ and *l* calculation. (top) Single-beam velocity from 10-m depth of a vertically oriented beam as a function along-track distance. Gray dashed box denotes the 30-m window in which *w*′ and *l* are computed. (middle) Velocity within gray box with linear trend removed. Noise-compensated value of *w*′ is noted. (bottom) Autocorrelation coefficient function of *w* inside the window. Area of the curve between the origin and the first zero crossing is the uncorrected integral scale. Indicated value of *l* has been corrected using Eq. (14). Each panel’s time was converted into distance using *U*.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Example of *w*′ and *l* calculation. (top) Single-beam velocity from 10-m depth of a vertically oriented beam as a function along-track distance. Gray dashed box denotes the 30-m window in which *w*′ and *l* are computed. (middle) Velocity within gray box with linear trend removed. Noise-compensated value of *w*′ is noted. (bottom) Autocorrelation coefficient function of *w* inside the window. Area of the curve between the origin and the first zero crossing is the uncorrected integral scale. Indicated value of *l* has been corrected using Eq. (14). Each panel’s time was converted into distance using *U*.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Example of *w*′ and *l* calculation. (top) Single-beam velocity from 10-m depth of a vertically oriented beam as a function along-track distance. Gray dashed box denotes the 30-m window in which *w*′ and *l* are computed. (middle) Velocity within gray box with linear trend removed. Noise-compensated value of *w*′ is noted. (bottom) Autocorrelation coefficient function of *w* inside the window. Area of the curve between the origin and the first zero crossing is the uncorrected integral scale. Indicated value of *l* has been corrected using Eq. (14). Each panel’s time was converted into distance using *U*.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

## 4. Results

The central issue in this paper is the fidelity of *ϵ*_{ADCP} in representing turbulence levels that are consistent with more established techniques such as MMP. It is important that ADCP estimates faithfully reproduce both the magnitude and spatial distribution of turbulence levels, albeit acknowledging the differences in spatial resolution and dynamic range of each sensor. In this section, ADCP and MMP measurements will be compared over a broad range of turbulent intensities and dynamic environments, but specific generation mechanisms or sources will not be addressed. The primary objective is to examine and compare *ϵ*_{ADCP} to *ϵ*_{MMP}.

The overall level of qualitative and quantitative agreement between MMP and ADCP data is demonstrated in Fig. 10. ADCP vertical velocities were used to estimate *ϵ*_{ADCP} using a 30-m window for computing *w*′ and *l* and *C*_{ϵ} = 0.35. Both sections were collected in Colvos Passage; however, the average turbulence levels in the right column of Fig. 10 are an order of magnitude greater than in the left column. These particular examples were chosen because they demonstrate the ability of the ADCP to capture both the spatial structure and magnitude of *ϵ*. In each example the turbulence is patchy, with quiescent regions separating areas of vigorous mixing. For example, in the right column of Fig. 10, strong turbulence can been seen in both MMP and ADCP data at 0–1000-m along-track distance at all depths except for a small pocket of weak turbulence near depth 38 and 200–600 m. Similar levels of agreement can be seen throughout the transects in both examples. Given the discrepancy of the intrinsic length scales that each method is sensitive to and the difference in sampling schemes, the similarities in the images are quite encouraging.

MMP–ADCP comparison: (top) MMP profiles and (bottom) ADCP estimates of *ϵ* using 30-m windows and *C*_{ϵ} = 0.35. Aspect ratio of (top) and (bottom) are equal, and horizontal axes have been aligned. Note that (left) and (right) have different color axes.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

MMP–ADCP comparison: (top) MMP profiles and (bottom) ADCP estimates of *ϵ* using 30-m windows and *C*_{ϵ} = 0.35. Aspect ratio of (top) and (bottom) are equal, and horizontal axes have been aligned. Note that (left) and (right) have different color axes.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

MMP–ADCP comparison: (top) MMP profiles and (bottom) ADCP estimates of *ϵ* using 30-m windows and *C*_{ϵ} = 0.35. Aspect ratio of (top) and (bottom) are equal, and horizontal axes have been aligned. Note that (left) and (right) have different color axes.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Statistical properties of the turbulence, as defined by the mean and standard deviation of log_{10}(*ϵ*), are also well represented by the large-eddy technique with the ADCP (Fig. 11). Again, in these examples *C*_{ϵ} = 0.35 and a window length of 30 m was used for computing *ϵ*_{ADCP}. In each panel of Fig. 11, all the values of log_{10}(*ϵ*) from the cross sections shown in Fig. 10 are represented by histograms. The mean (log) dissipation rate is nearly identical in the MMP and ADCP estimates, but the standard deviation of *ϵ*_{ADCP} is much smaller. The discrepancy in standard deviation can be explained by differences in sampling volumes and noise levels. Since the ADCP technique computes *ϵ* within a much larger volume, ≈40–60 m^{3}, compared to a one-dimensional measurement over a length of 0.8 m, there is less variance in *ϵ*_{ADCP}. This behavior is consistent with Kolmogorov’s third hypothesis (Kolmogorov 1962), which states *A* is related to the large-scale character of the flow, *k* is a universal constant, *r* is the characteristic dimension of the averaging volume. Also note, the *ϵ*_{MMP} distribution has a longer tail at the weak dissipation rate end, which is most evident in the section with the weaker turbulence in the left-hand panels, due to lower noise levels present in the MMP estimates of *ϵ*. The minimum observed levels in the ADCP measurements indicate a noise level that corresponds to a dissipation rate of approximately 10^{−8} m^{2}s^{−3}.

Statistical properties of the data used in Fig. 10. Legend in each panel indicates the mean plus or minus standard deviation of log_{10} *ϵ*.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Statistical properties of the data used in Fig. 10. Legend in each panel indicates the mean plus or minus standard deviation of log_{10} *ϵ*.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Statistical properties of the data used in Fig. 10. Legend in each panel indicates the mean plus or minus standard deviation of log_{10} *ϵ*.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

## 5. Quantitative comparison

To test the procedures and parameters for estimating *ϵ*_{ADCP} established in the previous sections, ADCP measurements will be compared against all available coincident MMP data. In the following analysis, the ADCP averaging window at each depth bin will be centered on the position of the MMP at the same depth level. The combination of strong currents, up to 1 m s^{−1}, and the time to take an MMP cast caused considerable lateral drift of the MMP position during a profile. Measured water velocities were used to reconstruct the position of the MMP, assuming it moved exactly with the current. Therefore, lateral drifts of the MMP along the ship track were accounted for and the ADCP window at depth could be collocated with the MMP position. Although the MMP drifts perpendicular to the ship track were calculated, it was not possible to account for this drift because ADCP measurements were collected directly under the boat.

As previously mentioned, ADCP cell sizes were varied between 2 and 4 m and, for the following comparisons, the MMP data were reprocessed and regridded in the vertical to match the size and vertical position of the ADCP cells. The rms Thorpe scale, which represents the average (not the maximum) vertical length scale of the overturning eddies, was calculated from the CTD on the MMP. The rms Thorpe scale was calculated as the rms vertical displacement of a density (*ρ*) measurement at depth such that ∂*ρ*/∂*z* < 0 over the profile. Thorpe scales, like *ϵ*_{MMP}, were calculated over bin sizes and centered about cell positions consistent with the ADCP measurement. Rms Thorpe scales as large as 5 m were observed in regions of intense turbulence and are highly correlated with strong values of rms vertical velocity, indicating we are measuring overturns and not internal waves. Since it is impossible to differentiate internal waves from turbulence with a vertical velocity measurement alone, we recommend that a small subset of the ADCP measurements be accompanied by microstructure measurements to ensure strong rms vertical velocities are associated with overturning. This can now be achieved with small hand-deployable microstructure equipment that is commercially available at relatively modest cost.

Figure 12 shows scatterplots of *ϵ*_{ADCP} versus *ϵ*_{MMP} for all MMP bins and all casts. Again, all *ϵ*_{ADCP} estimates were calculated in a window centered on the vertical and horizontal position of an MMP estimate of *ϵ*. The data were organized and broken up into groups according to the ratio of the Thorpe scale (*z*_{th}) to ADCP bin size (*z*_{bin}) (Fig. 12). The degree of scatter is not surprising when examining *ϵ* measured by two different techniques that are inherently sensitive to different length-scale fluctuations. Furthermore, the two measurement volumes are not collocated in space and time exactly (though a correction was attempted). The level of variability is consistent with nearly coincident measurements of two microstructure profilers (Moum et al. 1995). However, as shown in section 4, when the datasets are averaged over long spatial/temporal windows (e.g., Fig. 11), the means of the logarithms from the two estimates are statistically indistinguishable.

Comparison of MMP and ADCP estimates of *ϵ*. Data have been broken up into groups according to the ratio of the Thorpe scale to the ADCP cell size, as indicated in the lower-right corner of each panel. Black line is a one-to-one line. Correlation between the MMP and ADCP estimates of *ϵ* are shown in the top-left-hand corner of each panel.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Comparison of MMP and ADCP estimates of *ϵ*. Data have been broken up into groups according to the ratio of the Thorpe scale to the ADCP cell size, as indicated in the lower-right corner of each panel. Black line is a one-to-one line. Correlation between the MMP and ADCP estimates of *ϵ* are shown in the top-left-hand corner of each panel.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Comparison of MMP and ADCP estimates of *ϵ*. Data have been broken up into groups according to the ratio of the Thorpe scale to the ADCP cell size, as indicated in the lower-right corner of each panel. Black line is a one-to-one line. Correlation between the MMP and ADCP estimates of *ϵ* are shown in the top-left-hand corner of each panel.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Figure 13 illustrates the effect of the ratio of the Thorpe scale to ADCP bin size in more detail. The data were sorted by *z*_{th}/*z*_{bin} and averaged in *z*_{th}/*z*_{bin} windows 0.2 in size. The error bars are the 95% confidence limits on the correlation (*ρ*) values. In general, correlations are highest when *z*_{th}/*z*_{bin} 0.5–2. The effectiveness of the correction procedure for removing ADCP noise artifacts on *w*′ and *l* via Eqs. (7) and (14) can also be seen by comparing the two panels in Fig. 13. When *w*′ and *l* are corrected for noise (top panel), the correlation between *ϵ*_{MMP} and *ϵ*_{ADCP} improves by 0.1–0.2 and the size of the 95% confidence limits decreases.

Correlation between of MMP and ADCP estimates of *ϵ* as function of the ratio of Thorpe scale to ADCP cell size. Terms *ϵ*_{MMP} and *ϵ*_{ADCP} were broken up into groups of *z*_{th} and *z*_{bin} 0.2 in size, and a least square regression was performed on the data and the correlation was computed. Correlations (top) with and (bottom) without noise correction. Error bars denote the 95% confidence limits on the correlation value.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Correlation between of MMP and ADCP estimates of *ϵ* as function of the ratio of Thorpe scale to ADCP cell size. Terms *ϵ*_{MMP} and *ϵ*_{ADCP} were broken up into groups of *z*_{th} and *z*_{bin} 0.2 in size, and a least square regression was performed on the data and the correlation was computed. Correlations (top) with and (bottom) without noise correction. Error bars denote the 95% confidence limits on the correlation value.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Correlation between of MMP and ADCP estimates of *ϵ* as function of the ratio of Thorpe scale to ADCP cell size. Terms *ϵ*_{MMP} and *ϵ*_{ADCP} were broken up into groups of *z*_{th} and *z*_{bin} 0.2 in size, and a least square regression was performed on the data and the correlation was computed. Correlations (top) with and (bottom) without noise correction. Error bars denote the 95% confidence limits on the correlation value.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

*C*

_{ϵ}= 0.35 visually represents a reasonable fit to the data and results from the numerical simulation in section 3, a least squares fit will provide quantitative confirmation. To minimize the error between

*ϵ*

_{ADCP}and

*ϵ*

_{MMP}, a least squares fit was used to estimate

*C*

_{ϵ}, consistent with

*ϵ*is, theoretically, lognormally distributed, it makes sense to rewrite Eq. (17) as log

_{10}

*C*

_{ϵ}= log

_{10}

*ϵ*

_{MMP}− log

_{10}

*w*′

^{3}/

*l*and an empirical value of

*C*

_{ϵ}is computed as

*y*intercept of zero. Figure 14 shows

*C*

_{ϵ}as a function of

*z*

_{th}/

*z*

_{bin}. The empirical value is within a factor of 2 relative to the predicted value and agrees remarkably well considering the observed spectrum deviates from a true −

Data were binned and sorted as in Fig. 13. The value of *C*_{ϵ} was determined by Eq. (18). The gray line represents the predicted value from the simulations in section 3.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Data were binned and sorted as in Fig. 13. The value of *C*_{ϵ} was determined by Eq. (18). The gray line represents the predicted value from the simulations in section 3.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

Data were binned and sorted as in Fig. 13. The value of *C*_{ϵ} was determined by Eq. (18). The gray line represents the predicted value from the simulations in section 3.

Citation: Journal of Atmospheric and Oceanic Technology 32, 2; 10.1175/JTECH-D-13-00207.1

## 6. Conclusions

An ADCP has been evaluated as a low-cost, easily deployed instrument for measuring nearly continuous profiles of turbulent dissipation rate in the sheltered coastal ocean environment where seas are calm and adverse boat motion does not contaminate the velocity measurements. By tilting a commercially available ADCP in a mounting bracket to align one beam vertically, the vertical velocity from a single beam could be measured to obtain horizontal resolution comparable to the beam diameter. Although broadband ADCPs have been commercially available for 20 years, they have not been routinely employed for dissipation rate measurements using the large-eddy technique because noise levels were thought to be prohibitive. By carefully selecting the ADCP operating parameters to minimize instrument noise and using a simple, yet rigorous, procedure for correcting the ADCP velocity measurements for the effects of noise, we have been able to provide estimates of *ϵ* with mean values that are nearly identical to coincident measurements with the microstructure profile MMP over three orders of magnitude in *ϵ*.

The procedure for compensating for sensor noise is based on the fact that the noise is predictable in amplitude and uncorrelated with the signal or itself. This property allows one to remove the noise variance by subtraction and to correct the integral scale derived from the autocorrelation coefficient function. This procedure has been tested using model turbulence and noise spectra and was found to be effective over a wide range of window lengths with signal-to-noise ratios up to order one. Consistent estimates of *ϵ* are achieved for window lengths on the order of the integral scale or greater. The constant of proportionality in Eq. (2) is *C*_{ϵ} ≈ 0.35 when the length scale is the scale derived from the transverse autocorrelation coefficient function. Theoretically, this length scale *l* = Λ/4, but in practice the computed value depends on window length, approaching the theoretical value when the window length is several times longer than the length scale at the spectral peak. Observationally, *C*_{ϵ} agreed well with the predicted value and varied between 0.2 and 0.8 depending on the ratio of the Thorpe scale to ADCP bin size.

This agreement has been achieved without the need for any scaling parameters to fit the data other than the requirement that the length of the averaging window is of order 2 times the largest-scale eddy in the flow field, as represented by the low wavenumber cutoff in the turbulent inertial subrange, and that the ADCP bin size is within a factor of 2 of the Thorpe scale. The limiting value of dissipation that can be measured by this technique occurs when the sensor noise is approximately equal to the signal, which for the dataset considered here, occurs when *ϵ* ≈ 3 × 10^{−8} m^{2} s^{−3}. This threshold value could be improved by decreasing the noise level, for example, by increasing the acoustic frequency. Doubling the frequency should result in nearly an order of magnitude decrease in the noise floor of *ϵ*_{ADCP} due to the cubic dependence of *ϵ* on *w*′.

## Acknowledgments

The authors thank the crew of the R/V *Miller*. We also thank Jesse Hansen for his help with ADCP processing and Derrick Custodio for his thorough review of the manuscript. This analysis was supported by Section 219 funding under Contract 40000011560/0010.

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^{1}

Technically, the amplitudes of the simulated spectrum should also be random variables, but for the purposes of this exercise, this feature is not essential.