1. Introduction
Vertical exchange driven by turbulent mixing is a key process in determining momentum and heat fluxes and material transport pathways in the marine environment. In recent years, our ability to measure a turbulence parameter, the rate of dissipation of turbulent kinetic energy, has led to major advances in our understanding of the vertical exchange processes and their parameterization (Burchard et al. 1998; MacKinnon and Gregg 2003; Sharples et al. 2001; Simpson et al. 1996). These advances have largely been based on profile measurements made using free-falling microstructure profilers. The major drawback of such measurements is that they are labor intensive and require a dedicated ship. Datasets thus tend to be sparse and intermittent and rarely exceed one or two days duration.
In recent years, acoustic Doppler current profilers (ADCPs) have been increasingly applied to the measurement of turbulent parameters. In the “variance method,” the shear stress in the flow above a bottom-mounted ADCP is estimated from the difference of velocity variance in the opposing ADCP beams (Stacey et al. 1999; Lu et al. 2000; Rippeth et al. 2002; Howarth and Souza 2005; Williams and Simpson 2004) in a technique analogous to that used in stress measurements in radar meteorology. The rate of turbulent energy production is obtained from the product of the shear stress and the vertical shear of velocity, which is also measured by the ADCP. A restriction in the application of the variance method is that the analysis requires that the ADCP be mounted on the seabed or a rigid platform and carefully levelled with the axis of instrument aligned to ~1° of the vertical, a requirement that can usually be met by the use of a gimbal mounting in the ADCP frame.
In the structure function (SF) method (Wiles et al. 2006), differences in the along-beam velocities between bins are used to derive an estimate of ε, the rate of TKE dissipation. The analysis requires that the differences are taken over bin separations that lie within the inertial subrange of the turbulence spectrum (i.e., the range of scales within which the turbulence can be considered isotropic). This concentration on motions that are isotropic means that there is no requirement for a particular orientation, which is an important advantage over the variance method. The range of scales in the inertial subrange is set at the lower end by the Kolmogorov microscale LK, which is usually <1 cm, and at the upper end in stratified turbulence by the Ozmidov length 
The structure function method has been validated by comparisons between measurements made from bed-mounted ADCP and shear probe profile instruments in energetic tidal flows with weak stratification (Wiles et al. 2006). Mohrholz et al. (2008) have applied the SF method to data from a pulse-coherent Doppler (pc-ADP) observations of near-bed flows in descending plumes in the Baltic. Their results provide convincing evidence of close agreement between dissipation measured (i) by the structure function, (ii) by fitting of the inertial subrange spectra from acoustic Doppler velocimetry (ADV) measurements, and (iii) from shear profiles with a mean square slope (MSS) microstructure probe. The use of the pulse-to-pulse coherent technique greatly reduces noise levels but restricts the working range to a few meters. Applying the SF method to coherent Doppler measurements allows, for example, the determination of the low dissipation rates driven by seiche motions in the bottom boundary of lakes (Lorke 2007; Lorke et al. 2008; Simpson et al. 2011).
This contribution looks at the possibility of taking the SF method one step further by using an ADCP in pulse coherent mode to measure dissipation in midwater. In this case, the instrument has to be mounted on a buoyant tether and will move around in the flow and so experience relative motions that could contaminate the structure function. After briefly reprising the basis of the SF analysis in section 2, we consider the effect of the relative motions of a tethered instrument (section 3) before describing the observational methods and the environment of the chosen site (4). The results are presented in section 5, which is followed by a concluding section that interprets the results and discusses their implications.
2. The structure function approach
ADCPs measure the water velocity along the direction of their acoustic beams by determining the Doppler shift of the returned signal from range-gated cells. Employing the single-pulse technique, ADCPs measure the Doppler frequency shift of back-scattered acoustic pings to estimate velocities of scattering particles. In the pulse–pulse coherent mode utilized here, the phase shift between two pings is correlated. This results in much improved range resolution and low standard deviations of velocity estimates but at the expense of profiling range, which is restricted by the classical range–velocity ambiguity relation 
3. Tethered ADCP operation
In contrast to ADCP measurements from fixed platforms or moorings located on the seabed, instruments tethered in midwater are free to move, thus inducing relative velocities that might be expected to compromise the determination to the structure function. We need to consider the effects of (i) translation of the ADCP “beam center” (see Fig. 1), both horizontally and vertically, and (ii) all forms of rotation of the instrument about the beam center. Underlying assumptions here are that the turbulence is spatially homogeneous on the scale of the beam separation and is statistically stationary over the sampling period.
ADCP relative velocities components induced by instrument motion ui.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
For an instrument on a long tether, the principal motion will be horizontal displacements in x and y from an equilibrium position. If the instrument is moving through the water at a velocity 
While the above arguments suggest that the motions of a tethered instrument should not seriously compromise the structure function approach to the determination of dissipation, there may be contamination of the data close to the instrument due to the local generation of turbulence in flow past the ADCP and its supporting buoy (Gartner and Ganju 2002). Reduced correlation and velocity bias near the transducer is well documented in acoustic research and is thought to be influenced by the complex field distribution in the near field, causing lateral beam modulation in amplitude and phase (Zedel et al. 1996; Lacy and Sherwood 2004; Li et al. 1997). Zedel et al. (1996) show that this can be attributed to the phase structure of the near-field beam introducing additional phase variance as targets are advected across the beam. This near-field zone boundary is expected for ranges less than 
4. Observations
The measurements reported here were made in the Clyde Sea, which is the largest of the Scottish fjords; a deep partially enclosed basin connected to the adjacent shelf sea across a shallow (45 m) entrance sill. The fjord undergoes thermal and haline stratification in the summer (Inall and Rippeth 2002) and low levels of dissipation. There is evidence of enhanced dissipation within the thermocline region (Inall and Rippeth 2002; Jackson and Elliott 2002) driven by an internal wave generated by the tidal flow over the entrance sill. Velocity data were collected by two tethered ADCP units on a single mooring at 55°47.13′N, 5°12.42′W, in a water depth of ~100 m (Fig. 2). The upper and lower ADCPs were mounted on in-line frames at mean depths of 22.5 and 35.7 m. The mooring (Fig. 3) was also instrumented with four Microcat sensors (measuring temperature and salinity) as well as temperature loggers at 2-m intervals.
Map and bathymetry of the observation area. Contours are shown in 20-m spacing from 10 to 130 m. Illustrated are the mooring position (red square) and VMP transects from each day, 14 (YD 257; black) 15 (YD 258; blue), and 16 September (YD 259; magenta).
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
Mooring configuration. Both ADCPs are mounted upward looking with Microcats positioned ~2 m above each. There were 15 temperature loggers at 2-m intervals and a further two Microcats centered at 2-m spacing.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
The RD Instruments (RDI) Workhorse 1.2-MHz ADCPs were operated in high-resolution pulse–pulse coherent mode. Data were recorded in Earth coordinates using 5-cm depth bins, blanking distances of 44 cm, and ambiguity velocities of 0.05 m s−1; the upper ADCP was sampling at a rate of 2 ping pairs per 1-s ensemble, while the lower ADCP was sampling at a rate of 3 ping pairs per 1.5-s ensemble. Two RDI algorithms were applied to the data during deployment: 1) a data rejection analysis was applied to the raw ADCP data to exclude data that displayed a difference in echo intensity of >50 from the highest beam value, to account for losses due to fish interference in the beam path, and 2) RDI bin mapping was applied, which, when the instrument is tilted, uses the beam velocities from the bin nearest to the nominal (20°) bin center. Subsequent to this all data in the measurement volume that displayed a ping-to-ping correlation coefficient of below 0.7 were rejected, eliminating data collected with poor signal quality (Zedel et al. 1996). Finally, the structure function algorithm rejects data when there are less than three values for the r2/3 fit.
There were a total of 98 VMP casts spread over 3 days indicated in Fig. 2. Casts were made with the vessel moving slowly ahead at ~0.25 m s−1 and staying close to the mooring; most profiles were taken within 1 km of the mooring with a maximum separation of ~2 km (see Fig. 2b).
5. Results
a. Surface forcing and density field
The mooring observations extended over a period of 7 days [year days (YD) 252–259] during which there were large changes in surface forcing (Fig. 4) due to the passage through the region of a low pressure system (the remains of Hurricane Katia) from the early hours of 12 September (YD 255) through to the evening of 13 September (YD 256). During this period, wind stress (Fig. 4b) increased to peak values of ~1 Pa with corresponding increases in wave height. Because of small fetch (~10 km or less), the development of the local sea was limited to relatively short waves with the dominant period increasing to 4.1 s during the storm (Fig. 2b). Net surface heat flux (Fig. 4c) was minimal during the storm period when solar radiation was reduced. In the calm period (from YD 257) following the storm, heat flux into the ocean increased considerably. This heat input together with a horizontal inflow of fresher water is reflected in the density contours (Fig. 4a), which show renewed stratification in the surface layers from YD 257 after the strong wind mixing during the storm.
The local conditions during the sampling period. (a) The color contoured thermal profile from 2-m spaced thermistors with overlaid interpolated isopycnals from Microcats centered at approximate depths of 15, 20.5, 24.1, and 33.7 m (black lines). The black triangles at the top of the plot represent the timings of the VMP casts. Approximate ADCP positions are shown (green and blue lines). (b) The wind stress (green), significant wave height (red), and wave period (blue, y axis on right-hand side). (c) The net surface heat flux calculated using Simpson and Bowers (1984).
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
b. Mean flow field
Contoured velocity data in Earth coordinates from the upper ADCP (Fig. 5) indicate the presence of semidiurnal tidal flows with amplitudes of ~7 cm s−1 prior to the storm period. During the storm, wind forced motions with peaks in horizontal current speed of up to ~30 cm s−1 predominated and regular semidiurnal reversal of the flow was not apparent. This situation continued for several days after the storm.
The u, υ, and w velocities for the upper ADCP (~22.5 m); note the different scale for the vertical velocities.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
The upper limit of tilt during the worst of the storm reached 4.5° and 8.4° for the upper and lower ADCP respectively, which equates to a bin mapped translation at the extremities of the sampled range of two and three bins. However, the median tilts were 0.74° and 0.08° for the upper and lower ADCP, respectively; these values are below the threshold to trigger the RDI bin mapping algorithm. The vertical excursions of the ADCPs, plotted in the upper panels of Figs. 6 and 7, were estimated by removing the tidal signal from the pressure readings. The ADCP pressure signal was differentiated to obtain 
Displacements for the upper ADCP. (top) The vertical excursion of the ADCP with the tidal signal removed. (bottom) The relative vertical velocities of the flow (red) and the mooring (blue), with data averaged over 5 min; note the different scales.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
As in Fig. 6, but for the lower ADCP.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
The vertical velocity 
Power spectral density of vertical velocity w from the upper ADCP (~22.5 m) and vertical motion of the buoy from the pressure sensor; w has an offset of 0.004 and 0.04, respectively, for plotting clarity: (a) in equal variance form, 
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
c. Structure function and dissipation
The error velocities, from which b′ and hence the structure function is derived, are plotted in Fig. 9 for the two ADCP instruments. The near-field zone boundary for these ADCPs, with transducer diameter of 61 mm and frequency of 1.2 MHz, would reach 74 cm, and thus regions up to this height above transducer were blanked. With the in-line mounting of the ADCPs adopted here, there is a possibility of interference from turbulence generated in the wake of instruments located higher up the mooring line. In the present case the upper and lower ADCPs were located ~2.3 m below the base of the Microcat instruments, which have clearly generated spurious turbulence contributions in bins adjacent to these levels. Thus, the regions ≥2.19 m and ≥2.14 m for the upper and lower ADCP, respectively, were blanked to remove this interference signal.
The error velocities of the ADCP Earth coordinate output from which b′ is derived: (a) the upper ADCP at ~22.5 m and (b) the lower ADCP at ~35.7 m. Both ADCP datasets are blanked from 0.74 to just above 2 m to remove the spurious data in the near and far field induced by phase variance and Microcat wake, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
The b′ velocity data were used to calculate the structure function D(z, r) [Eq. (1)] for each available bin by the application of a centered difference technique, squaring the velocities and then averaging over 5 min, a time scale deemed long enough to provide an adequate sample while the turbulence can be assumed to remain statistically stationary. Examples of the structure function at different dissipation levels are shown in Fig. 10 together with the fitted 
Structure function data from 1-s ensembles of residual velocity data from the upper ADCP. The dotted lines are the fitted polynomials from Eq. (3) exhibiting an r2/3 fit at three mean dissipation estimates: 2.93 × 10−6 W kg−1 (black), 4.80 × 10−8 W kg−1 (red), and 2.54 × 10−9 W kg−1 (blue).
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
The RDI software quotes the noise estimation based on separate pulse pairs. The SF noise estimations are obtained from samples separated along-beam, but using the same transmitted pulse pair and much of the same propagation path and thus are not completely independent. Consequently, the variance of the SF is somewhat reduced relative to the RDI figure. Turbulent kinetic energy (TKE) dissipation values derived from the structure function from each of the ADCP are presented in black in Fig. 11. In both plots, dissipation is seen to increase from low values (ε ~ 10−9 W kg−1) before the storm to maximum values (ε ~ 10−5 W kg−1) when the wind stress was at its peak on YD 255. A second maximum at the upper level (Fig. 11a) is apparent on YD 256 but is absent from the lower level. This difference reflects the fact that, whereas on YD 255 both instruments were in the same mixed layer, on YD 256 they were separated by a region of density stratification (see Fig. 4a), which seems to have inhibited the downward penetration of turbulence. After YD 256, the decline in wind stress led to diminishing levels of turbulence of similar magnitude to those at the start of the observation period.
Average energy dissipation ε (W kg−1) from ADCP (black stars) and VMP ε (red crosses) (a) at the level of the upper ADCP, ~22.5 m, and (b) at the level of the lower ADCP, ~35.7 m.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
d. Comparison with shear probe measurements
To compare dissipation derived from the structure function with values from the VMP shear probe, we have used vertically averaged values of ε from the SF and averages over the equivalent depth interval from the VMP profiles. The two independent estimates of ε, plotted together in Fig. 11, show a good degree of consistency over two decades of variation in dissipation in both the mean values and the variability of ε over short time scales. In comparing these estimates of dissipation, it is important to remember that the ADCP mooring and shipborne ADCP measurements were separated horizontally by a distance of ~1 km, as indicated in Fig. 2.
As an alternative comparison of the two ε measurements, we show the corresponding VMP and ADCP-SF values plotted against each other in Fig. 12. Here we have subtracted from the VMP measurements the instrument noise limit, quoted to be 3 × 10−10 W kg−1 (Rockland Scientific 2007). A neutral regression analysis (Garrett and Petrie 1981) of log10 ε gives a fit with a slope close to unity of 1.037 ± 0.01; mean ±2 standard deviations (std) with considerable scatter (rms deviation ~0.50) from the regression line that reflects the short-term variability in both ε estimates (see Fig. 11).
TKE dissipation from ADCP structure function and VMP profiler for both upper and lower ADCPs. The 1:1 line (blue) and neutral regression line (black) (Garrett and Petrie 1981) are shown with statistical variables. The VMP values have the instrument noise level of 3 × 10−10 W kg−1 removed.
Citation: Journal of Atmospheric and Oceanic Technology 31, 8; 10.1175/JTECH-D-13-00198.1
6. Summary and discussion
The rate of dissipation of TKE ε has been determined by applying the structure function method to velocity data from pulse–pulse coherent ADCP instruments mounted in-line on a taut wire mooring for a period of 7 days. The resulting time series illustrates the variation of energy dissipation over ~4 decades at two levels in the water column of the Clyde Sea before, during, and after the passage of a severe storm through the area. The validity of estimates of ε based on SF analysis of data from tethered ADCPs has been investigated by comparison with ε estimates from a VMP shear profiler operating nearby from a research vessel. Comparison of dissipation rates indicates generally good consistency between the two ε estimates over ~2 decades. There is, however, considerable scatter, with similar variance in both datasets, which represents the inherent short-term variability of turbulence.
The noise floor of the SF method in our results does not appear to be significantly affected by motions of the mooring and is similar to that found in operation of ADCPs on fixed bed mountings (Mohrholz et al. 2008) and lower than the apparent noise characteristic of the VMP (~3 × 10−10 W kg−1). Because it is based on difference measurements over small separations, we hypothesized that the SF method should not be degraded by the mooring motions. This hypothesis would seem to be confirmed by the good agreement of the dissipation estimates. For the same reason, the SF method minimizes contamination by the effects of surface waves (Trowbridge and Elgar 2003) because of the large vertical scale of such motions relative to the SF separation.
This comparison provides, for the first time, a clear illustration of the potential use of a pulse-to-pulse coherent ADCP instrument for the measurement of turbulent dissipation in the water column away from the bottom boundary. The technique is likely to be valuable for obtaining time series data in extreme weather conditions when, as in the storm documented here, operation of a shear profiler from a research vessel would not be possible. As originally implemented (Wiles et al. 2006), the structure function was based on data from each of the beams but where data have been recorded in Earth coordinates, as in the data used here, the error velocity may be used as representative turbulent velocity component in calculating the SF.
A possible concern in using the SF method is whether the results are influenced by the effects of stratification. Increased stratification will decrease the Ozmidov length 
To achieve sufficiently low noise levels to determine dissipation in low energy flows, it is necessary to use pulse–pulse coherent modes of operation, which severely restrict the working range of the ADCP (in the present case to a few meters). In our study, we encountered a further limitation in range due to turbulence generated by other instruments attached to the mooring line above the ADCP. An ideal arrangement for avoiding this problem in future studies would be a mounting above the buoyancy at the top of the mooring string so that only uncontaminated turbulence is sampled.
Acknowledgments
This work was funded in part by the NERC OSMOSIS project (NE/1020083/1).
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