## 1. Introduction

Estimating the dissipation of turbulent kinetic energy *ϵ* is core to the study of turbulence in aquatic flows as the rate of mixing *K* is related to the size of the largest turbulent overturns *L* and *ϵ* via, for example, Richardson’s mixing law *ϵ* in aquatic environments as we cannot (as yet) directly measure all nine turbulent velocity gradients (e.g.,

*ϵ*. Here,

*ν*is the kinematic viscosity of water and the overbar denotes a time-averaged quantity. In practice, the integration limits in Eq. (1) must be identified using an initial estimate of

*ϵ*, derived from the spectral observations. This first estimate

*ϵ*ensures the integration takes place over the viscous range while minimizing contributions from the noise-dominated portion of the spectra (Moum et al. 1995). More importantly, Eq. (1) becomes increasingly difficult to apply with increasing

*ϵ*as the viscous subrange moves to higher

*k*(Fig. 1). The finite spatial dimension of the shear probe leads to spatial averaging of the smallest eddies, for which the lost variance at high

*k*is typically corrected with a single-pole transfer function

*ϵ*) for reliable integration of the shear spectrum (Lueck 2015). For example, at

^{−1}, 90% of the variance is resolved for

^{−1}, most of the variance in Eq. (1) lies beyond

*ϵ*by fitting a model spectrum to field observations of turbulence shear. This technique was successfully applied to fit a model spectrum to turbulence temperature gradient spectral observations (e.g., Ruddick et al. 2000) and to the inertial subrange of turbulence velocity spectra (e.g., Bluteau et al. 2011). Unlike other techniques, such as linear least squares techniques that are confined to fitting the +1/3 slope of the inertial subrange, the MLE can take advantage of information available in both the inertial and viscous subranges, a portion of both, or simply one of the subranges.

## 2. Data analysis procedures

### a. Processing raw data

We briefly summarize the processing steps applied to the raw turbulent velocity gradient data prior to estimating *ϵ* from the shear spectral observations (see Moum et al. 1995; Lueck et al. 2002, for more details). The same preprocessing applies whether the spectra are fitted or integrated to obtain *ϵ*, except that with fitting the spectral averaging must result in constant degrees of freedom across the spectra. The shear signal is usually despiked before estimating the “raw” spectra and portions of the signal are discarded when the angle of attack of the sensor’s tip with respect to the mean flow is large (Macoun and Lueck 2004). Segments where the mean flow is highly variable (nonstationary) are also excluded.

To compute the spectra, a segment length from the profile (or time series if the shear probes are moored) must be chosen so that it is sufficiently long to ideally resolve the low *k* of the viscous subrange or the highest (less anisotropic) *k* of the inertial subrange. With decreasing *ϵ*, the turbulence range moves to progressively lower *k*, such that at ^{−1} the viscous rolloff occurs at ^{−1} (Fig. 1)—the higher the *ϵ*, the smaller the segment length can be, which in turn increases the spatial (or temporal) resolution of the *ϵ* estimates. The segment length is also limited by the lowest frequencies that can be reliably measured by the shear probe (e.g., ≈0.1 Hz for Rockland Scientific). The choice of segment length also depends on the desired statistical accuracy (degrees of freedom) of the computed spectra, which provides an error bound for the *ϵ* estimates obtained through integrating the spectrum [Eq. (1)]. The 95% confidence levels (CLs) of a spectrum with 6 degrees of freedom obtained, for example, by block averaging three 50% overlapping segments (with no windowing) yields an integrated *ϵ* within a factor of 4–5. For advection speeds past the sensor of 0.8 m s^{−1} (i.e., drop speeds) and a sampling rate of 512 Hz, a chosen bin size of about 2 m (1024 samples) could yield a spectrum of over 10 degrees of freedom if computed from three segments overlapping by 50% after applying a Hanning window. In theory, this spectrum should be sufficiently accurate to obtain an integrated *ϵ* of a factor of 3. However, the decontamination of the spectra with the multivariate technique of Goodman et al. (2006) relies on spectral coherence calculations between each of three accelerometers and the shear probes, and so requires more spectral averaging to ensure the coherence calculation is statistically significant (Emery and Thomson 2001).

With Taylor’s frozen turbulence hypothesis, the motion-corrected spectra can be converted from frequency to wavenumber space *W* or the mean fluid velocity past the sensor on a fixed platform. Taylor’s hypothesis is satisfied when *W* is larger than the turbulent velocity scale ^{−1} and mixing rate ^{2} s^{−1} imply a turbulent overturn size ^{−1}, and so Taylor’s hypothesis is usually satisfied with vertical shear profilers since their drop speeds are ^{−1}. The spectra are then corrected for the spatial averaging that resulted from the shear probe’s geometry and size by dividing the measured shear spectra with the transfer function *ϵ*.

### b. Spectral fitting algorithm

*ϵ*we use the MLE to fit a model spectrum

*ϵ*from the Batchelor spectrum, after integrating the viscous range to get the thermal variance dissipation rate

*ϵ*and

*d*is the degrees of freedom of the shear spectral estimates and thus the MLE takes account of the statistical significance of the spectral observations, effectively removing any dependency of the returned

*ϵ*on the smoothness of the spectra. To derive the most likely

*ϵ*over a specified wavenumber range, we use the log-likelihood

*n*of spectral observations

*k*are cpm and

*k*range spanning three decades from

*ϵ*, when the fit should be confined to the inertial subrange of the spectral observations, Nasmyth’s empirical spectrum can be replaced with the theoretical modelfor the turbulent velocity gradients

^{−1}). For fitting the model spectrum to our observations, we specify a most likely range of

^{−1}, with the minimum value set well below the quoted detection limit of the instrumentation and the maximum value set at the observed highs in the Mediterranean outflow (Price et al. 1993). We search for a fit starting with the lowest available

*k*and, if that does not return a result within the prescribed

*ϵ*range, then smaller subsets of the spectra (about 0.5–0.7 of a decade) are searched from the smallest to the largest

*k*until a valid

*ϵ*is found. If a fit is not found, then the spectra are completely discarded.

For short segment lengths (small bin size), such that the lowest *k* are in the inertial subrange, the wavenumber range identification is equivalent to the integration method, in that it amounts to determining the maximum wavenumber *k* before noise dominates spectra—is identified using a low-order polynomial fit to the log-transformed spectral observations, for which an order must be selected that varies with the shape of the shear spectra observations (see Lueck 2015). To avoid choosing the polynomial order, we identify the global local minima *k* or by specifying a fixed limit (e.g.,

Before setting *k* that are likely in the inertial subrange. For *k* range was shorter than a decade. However, after assessing both techniques in section 4, we retained the integrated value *k*, where most of the variance is resolved, and set this limit to *ϵ* estimate used to estimate the theoretical *k*, while the subscript refers to the percent variance resolved. If *ϵ* with the MLE, which we denote

### c. Spectral integrating algorithm

To assess *ϵ* that we denote as *n*th iteration’s

### d. Misfit criteria

*n*spectral observations. Ruddick et al. (2000) successfully used this misfit criterion to discard temperature gradient spectra when the MAD

*k*, which may be adversely affected by the mean flow.

## 3. Field data sources

An oceanographic field study was undertaken on the Australian North West shelf from November 2011 to April 2012. From 5 to 11 April, over 300 vertical microstructure shear profiles (VMP-500, Rockland Scientific) were collected throughout the study region. The VMP recorded data at 512 Hz from many channels: two airfoil velocity shear probes, 3D accelerometers, a pressure sensor, and high-accuracy temperature and conductivity sensors (SBE-3F and SBE-4C from Sea-bird Electronics) and one fast-response temperature sensors (FP07). Drop speeds were generally of order 0.5–0.8 m s^{−1} with lower drop speeds near the surface and seafloor. For the MLE fitting assessment, we used a subset of 121 profiles collected over a 24-h period on 10 April 2012 close to a site, where a 34-m-long mooring was anchored to the seafloor in 105 m of water (19°41.6′S, 116°06.6′E). Each profile was split into segments of 2048 samples (4 s) that overlapped by 50%. The FFT was estimated on subsets of 512 points with a 50% overlap (seven blocks), yielding spectra with 21 degrees of freedom. The mooring was instrumented with a mixture of velocity and temperature sensors, in addition to a conductivity–temperature sensor. The time-averaged currents were of order 0.25–0.30 m s^{−1} over the extent of the mooring, while the background stratification was nearly linear with a buoyancy period of about 9 min.

## 4. Results and assessment

Figure 3 shows example shear spectra that have been both integrated and fitted to obtain independent estimates of *ϵ*. The viscous dip ^{−1}, where no clear viscous rolloff was identifiable (Fig. 3d). Over the entire dataset, *ϵ* estimate was returned by our algorithms. Term *ϵ* example, where *ϵ*; the inclusion of the noise-dominated *k* overpredicted *ϵ* by two orders of magnitude for the low ^{−1} example (thin green line in Fig. 3a).

We compare the final *ϵ* estimates obtained with our integration algorithm ^{−1} with a median value of ^{−1} (Fig. 4c). With the degrees of freedom of our estimated spectra, the error bound on ^{−1}, however, *ϵ* from spectral shear estimates provided the influence of anisotropy at low *k* is considered when choosing the wavenumber range to fit [see Bluteau et al. (2011) for a review on anisotropy in the context of velocity measurements].

Figure 4b illustrates how close the initial integrated ^{−1} given their percentile bars (containing 90% of the data) are generally within a factor of 2 of *ϵ*, the percentile bars for *ϵ* was probably because the lowest *k* of the spectra used to derive

To investigate the sensitivity of *ϵ* over decade-long subsets of spectra computed from unusually large and inappropriately chosen segment lengths (4 times larger, i.e., 10–12 m). These long segments were purposely chosen to introduce mean flow effects at low wavenumbers (Fig. 5). The lowest *ϵ* examples were the most impacted by the mean flow, since the spectral observations over the lowest decade lay consistently above the empirical Nasmyth spectrum for the final *k*, the MLE yielded *ϵ* that were more than an order of magnitude larger than *k* increased, the MLE returned lower *ϵ*, dropping below *k* affected by the mean flow, and then again increasing to *k* in the fit is that once the noise-dominated portion of the spectra was included in the fit, the MLE progressively overpredicted *ϵ* as the noise levels increased above the low “environmental” spectral energy levels. For the ^{−1} example in Fig. 5a, the overprediction was about three orders of magnitude (Fig. 5e). Note that integrating the noise-dominated portion of this spectra also overpredicts *ϵ* by orders of magnitude but the overprediction is less severe than with the fitting technique. This is the primary reason we chose the integrated initial estimate *ϵ*, provided the fit was confined to the inertial and/or viscous range.

We use these excessively long 10–12-m segments (bin sizes) to illustrate the variation in the MAD misfit criterion as a function of the fitted wavenumbers (Figs. 5e–h). For our low *ϵ* example—the most impacted by the mean flow and instrument noise—the MAD misfit criterion was violated for almost all subsets fitted (Fig. 5e). Only a few fits around 10 cpm did not violate the rejection criteria. These fits also yielded lower *ϵ* than predicted with our fitting algorithm *k* impacted by the mean flow. The MAD can thus potentially be used to exclude these wavenumbers adversely influenced by the mean flow. In the two other *ϵ* examples (Figs. 5f,g), the MAD misfit criterion is only violated once the noise-dominated portion is reached. Over the low *k* range, where the criterion is respected, the estimated *ϵ* is also fairly constant with *k* suggesting the MAD predicts well the goodness of fit. For the high *ϵ* example, the criterion is violated at all *k* except the first few fits, implying that the higher *ϵ* obtained at these low *k* should be retained as the final fitted *ϵ* (Fig. 5h). Although the MAD identified poor sections of these example spectra, we recommend using smaller segments (bin size) to reduce the problems associated with anisotropy at low *k*, which in turn would increases the spatial (or temporal) resolution of the *ϵ* estimates. The MAD can still be used to discard spectra for which all subsets (in our case half a decade long) violate the rejection criteria.

## 5. Conclusions

The maximum likelihood estimator was used to reliably estimate *ϵ* by fitting a model spectrum, in this case the Nasmyth spectrum, to shear spectral observations. Our estimates agreed with those obtained with the more conventional integration method for ^{−1} (Fig. 4), but for higher *ϵ* the fitting method becomes progressively more reliable as the viscous subrange becomes underresolved. With the fitting method, *ϵ* was successfully derived if portions of the inertial subrange and/or viscous range were resolved. The fitting method is more sensitive than the integration method to the inclusion of the noise-dominated wavenumbers (Fig. 5); hence, we choose to integrate *ϵ* estimate used to assess the fitted *k* range against theoretical limits. The main advantage of the fitting method is that it allows *ϵ* to be estimated at times when the integration method cannot be used, thus allowing a broader range of *ϵ* to be derived from field turbulent velocity shear observations in the ocean.

An Australian Research Council Discovery Project (DP 120103036) and an Office of Naval Research Naval International Cooperative Opportunities project (N62909-11-1-7058) funded this work. We thank the people from the Australian Institute of Marine Science, the Naval Research Laboratory, and the University of Western Australia who aided in the collection of the data and the crew of the R/V *Solander*. In particular, we thank Jeff Book and Ana Rice for reviewing a draft of this manuscript. Anouk Messen also helped in the initial data analysis of the VMP.

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