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  • View in gallery

    Example spectral observations spanning from to 10−4 W kg−1. Colored squares are placed at , i.e., where 10% of the variance in the viscous range is resolved, while colored circles are placed at . The thick lines delineate the wavenumber range that needs to be resolved to apply the integration method (). The secondary axis is for the spectral correction factor (dashed) resulting from the shear probe’s dimensions and geometry.

  • View in gallery

    Flowchart delineating the fitting algorithm used to derive from spectral shear observations.

  • View in gallery

    (a)–(d) Example corrected shear spectra with increasing ϵ. Both the fitted and values obtained from these spectra are shown in each panel’s titles. The secondary axis (green) shows the cumulative integrated spectra nondimensionalized by ϵ, which represents the proportion of variance resolved. The thin and thick green lines are for the observed and theoretical integration, respectively. The violet dotted–dashed are the Nasmyth empirical spectra for ϵ rounded to the nearest order of magnitude above and below the fitted result. For completeness, the identified dip is shown, along with the maximum integrated wavenumber used to derive . In each panel, is located at (a) , (b) , (c) , and (d) , while , except in (d), where given the instrumentation limit of cpm used by the algorithms.

  • View in gallery

    (a) Term derived from our fitting algorithm and when the fit is confined to the inertial subrange, i.e., is plotted against the integration algorithm estimates . (b) Scatterplot of the initial ϵ guesses used for setting theoretical limits from integrating () and from fitting () to the lowest decade (or if this range is narrower). (c) Histogram of the different ϵ estimates for the dataset. For (a),(b) the sorted results have been binned evenly along the x axis, provided there were more than ≈1% (or 100) data points in each bin. The error bars contain 90% of the data within each bin. The 95% CL for associated with the significance interval of the estimated spectra is also shown. The CLs were approximately a factor of 1.7 of .

  • View in gallery

    (a)–(d) As in Fig. 3, but for large segment bins of the order of 10–12 m to introduce the influence of the mean flow at low k—particularly apparent for the low ϵ example in (a). (e)–(h) The MAD misfit measure and rejection criteria plotted at the midpoint of each decade investigated are shown for the example spectra illustrated above them in (a)–(d). The secondary orange axis shows the sensitivity of the estimated , nondimensionalized by , fitted over a subset of the spectra. The corresponding orange circles represent the result obtained if the entire spectrum was fitted, and the result is also centered at the median k of the spectral observations used in the fit.

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Estimating Turbulent Dissipation from Microstructure Shear Measurements Using Maximum Likelihood Spectral Fitting over the Inertial and Viscous Subranges

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  • 1 School of Civil, Environmental and Mining Engineering, and Oceans Institute, University of Western Australia, Perth, Western Australia, Australia
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Abstract

A technique is presented to derive the dissipation of turbulent kinetic energy ϵ by using the maximum likelihood estimator (MLE) to fit a theoretical or known empirical model to turbulence shear spectral observations. The commonly used integration method relies on integrating the shear spectra in the viscous range, thus requiring the resolution of the highest wavenumbers of the turbulence shear spectrum. With current technology, the viscous range is not resolved at sufficiently large wavenumbers to estimate high ϵ; however, long inertial subranges can be resolved, making spectral fitting over both this subrange and the resolved portion of the viscous range an attractive method for deriving ϵ. The MLE takes into account the chi-distributed properties of the spectral observations, and so it does not rely on the log-transformed spectral observations. This fitting technique can thus take advantage of both the inertial and viscous subranges, a portion of both, or simply one of the subranges. This flexibility allows a broad range of ϵ to be resolved. The estimated ϵ is insensitive to the range of wavenumbers fitted with the model, provided the noise-dominated portion of the spectra and the low wavenumbers impacted by the mean flow are avoided. For W kg−1, the MLE fitting estimates agree with those obtained by integrating the spectral observations. However, with increasing ϵ the viscous subrange is not fully resolved and the integration method progressively starts to underestimate ϵ compared with the values obtained from fitting the spectral observations.

Corresponding author address: Cynthia E. Bluteau, School of Civil, Environmental and Mining Engineering, University of Western Australia, MO15, 35 Stirling Highway, Crawley, Perth WA 6009, Australia. E-mail: cynthia.bluteau@uwa.edu.au

Abstract

A technique is presented to derive the dissipation of turbulent kinetic energy ϵ by using the maximum likelihood estimator (MLE) to fit a theoretical or known empirical model to turbulence shear spectral observations. The commonly used integration method relies on integrating the shear spectra in the viscous range, thus requiring the resolution of the highest wavenumbers of the turbulence shear spectrum. With current technology, the viscous range is not resolved at sufficiently large wavenumbers to estimate high ϵ; however, long inertial subranges can be resolved, making spectral fitting over both this subrange and the resolved portion of the viscous range an attractive method for deriving ϵ. The MLE takes into account the chi-distributed properties of the spectral observations, and so it does not rely on the log-transformed spectral observations. This fitting technique can thus take advantage of both the inertial and viscous subranges, a portion of both, or simply one of the subranges. This flexibility allows a broad range of ϵ to be resolved. The estimated ϵ is insensitive to the range of wavenumbers fitted with the model, provided the noise-dominated portion of the spectra and the low wavenumbers impacted by the mean flow are avoided. For W kg−1, the MLE fitting estimates agree with those obtained by integrating the spectral observations. However, with increasing ϵ the viscous subrange is not fully resolved and the integration method progressively starts to underestimate ϵ compared with the values obtained from fitting the spectral observations.

Corresponding author address: Cynthia E. Bluteau, School of Civil, Environmental and Mining Engineering, University of Western Australia, MO15, 35 Stirling Highway, Crawley, Perth WA 6009, Australia. E-mail: cynthia.bluteau@uwa.edu.au

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 . Many indirect velocity measurement methods exist to estimate ϵ in aquatic environments as we cannot (as yet) directly measure all nine turbulent velocity gradients (e.g., ) with 3D particle imagery in the field (see Doron et al. 2001; Steinbuck et al. 2010). Each indirect method requires a number of assumptions and has its own advantages, limitations, and thus applications. Examples of indirect methods include high-frequency point measurements of velocities (e.g., Voulgaris and Trowbridge 1998; Geyer et al. 2008; Walter et al. 2014), high-frequency profile velocity measurements (e.g., Wiles et al. 2006; Lucas et al. 2014), and microstructure (airfoil) shear probes that measure one turbulent velocity gradient, for example, (e.g., Oakey 1982; Lueck et al. 2002; Fer and Paskyabi 2014).

Shear probes provide the cross-stream velocity gradients in the direction they move through the water. Typically, two probes are mounted on a microstructure shear instrument such that a vertical profiler moving in the direction provides the turbulent velocity gradients and . Osborn (1974) was the first to use a microstructure shear profiler in the ocean and popularized integrating the viscous range of the observed turbulent velocity gradient spectra ,
e1
to obtain 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 ,
e2
applied to the measured shear spectra (Oakey 1982; Macoun and Lueck 2004). The half-power wavenumber depends on the shear probe’s geometry and dimensions (Macoun and Lueck 2004). For Rockland Scientific’s shear probes cycles per minute (cpm), which leads to a correction factor for the spectrum that is more than an order of magnitude at cpm (Fig. 1), effectively creating a maximum wavenumber (and hence ϵ) for reliable integration of the shear spectrum (Lueck 2015). For example, at W kg−1, 90% of the variance is resolved for cpm, but at W kg−1, most of the variance in Eq. (1) lies beyond cpm (Fig. 1). To overcome these challenges, the shear spectra can be fitted with an expected form, such as the Nasmyth empirical spectrum, over the resolved portion of the viscous subrange or even the inertial subrange (see Baumert 2005). Here, we demonstrate how to use the maximum likelihood estimator (MLE) to obtain 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.
Fig. 1.
Fig. 1.

Example spectral observations spanning from to 10−4 W kg−1. Colored squares are placed at , i.e., where 10% of the variance in the viscous range is resolved, while colored circles are placed at . The thick lines delineate the wavenumber range that needs to be resolved to apply the integration method (). The secondary axis is for the spectral correction factor (dashed) resulting from the shear probe’s dimensions and geometry.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0218.1

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 W kg−1 the viscous rolloff occurs at 10 cpm compared to 100 cpm at W kg−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 (cpm) using the drop speed 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 , that is, . A representative value for can be derived from the rms of the turbulent velocity (Lumley 1965) or alternatively from (e.g., Fer et al. 2014). Note that unusually high W kg−1 and mixing rate m2 s−1 imply a turbulent overturn size m via the mixing length model. For these high values, m s−1, and so Taylor’s hypothesis is usually satisfied with vertical shear profilers since their drop speeds are m s−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 in Eq. (2) (Macoun and Lueck 2004). These processed shear spectral observations can now be used to derive ϵ.

b. Spectral fitting algorithm

To obtain ϵ we use the MLE to fit a model spectrum to our shear spectral observations over the resolved inertial and/or viscous subranges. Ruddick et al. (2000) used the MLE to obtain ϵ from the Batchelor spectrum, after integrating the viscous range to get the thermal variance dissipation rate , since both ϵ and are unknowns to be derived from the spectral observations. The MLE requires knowledge of the expected statistical distribution of the observations being fitted. For our application, the ratio follows the distribution (Emery and Thomson 2001). Here, 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 relationship:
e3
which is valid for fitting any model spectrum to a number n of spectral observations . Here, is the probability density function for each sample .
For our model spectrum , we use the Nasmyth (1970) empirical spectrum,
e4
as tabulated by Oakey (1982) and subsequently fitted to Eq. (4) by Wolk et al. (2002). The units of k are cpm and are . The Kolmogorov length scale is the smallest expected turbulent length scale. This empirical spectrum was derived from field observations for a nondimensional k range spanning three decades from (Nasmyth 1970). This empirical model can thus be used to fit any portion of the resolved inertial and/or viscous subrange with the MLE. For very high 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 model
e5
for the turbulent velocity gradients . Term is the Kolmogorov universal constant (see Sreenivasan 1995), while the constant , since we are measuring the gradients perpendicular to the direction of mean advection (Pope 2000). The factor is required when working in units of cycles per minute rather than in radian per meter (rad m−1). For fitting the model spectrum to our observations, we specify a most likely range of W kg−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 to fit. Ideally, should be in the viscous range, where the spectral observations roll off (Fig. 1), as these length scales are less affected by the mean flow and more likely to be isotropic (see, for instance, Baumert 2005; Bluteau et al. 2011). Commonly, the spectral minimum—the highest 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 of the spectra after excluding the local minima found at the highest wavenumbers, where the noise suddenly “drops” off (e.g., cpm in Fig. 1). These local minima can be excluded either by band averaging the spectra at high k or by specifying a fixed limit (e.g., cpm). Instead of specifying a fixed limit, the global local maximum can also be identified from the spectra to exclude local minima at higher wavenumbers. The changes of slope in the spectra are used to identify minima and maxima by differencing the spectral observations . We opted to band average up to 21 bands at and to prescribe a fixed limit of 250 Hz to exclude the local minima at high wavenumbers from the search for .

Before setting , the identified was compared with the theoretical and instrument wavenumber limits as shown in the methodology flowchart (Fig. 2). For the vertical microstructure profiler (VMP) data discussed below, the instrument limit is set at cpm, which corresponds to the order of magnitude spectral “boost” from the spatial averaging correction (Fig. 1) and is what the instrument’s supplier recommends (Lueck 2015). Comparison of with the theoretical limits ensures the fit is not in the noise-dominated portion of the spectra, while taking advantage of the more isotropic characteristics of the viscous range. These limits rely on an initial guess , which can be obtained by integrating () the shear spectra for with Eq. (1) or by fitting () over the low k that are likely in the inertial subrange. For , we fit over the lowest decade or if this k range was shorter than a decade. However, after assessing both techniques in section 4, we retained the integrated value for our initial estimate of . We then used to calculate an upper theoretical k, where most of the variance is resolved, and set this limit to , where 95% of the variance is resolved. Note the superscript refers to the ϵ estimate used to estimate the theoretical k, while the subscript refers to the percent variance resolved. If , then we set to to avoid the noise floor and to remain in the viscous subrange. With , we reestimate ϵ with the MLE, which we denote (Fig. 2). If remains below , then is our final estimate . Otherwise, we adjust by setting it to the smallest of and to converge toward our final fitted estimate .

Fig. 2.
Fig. 2.

Flowchart delineating the fitting algorithm used to derive from spectral shear observations.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0218.1

c. Spectral integrating algorithm

To assess from this fitting, we applied the integration method [Eq. (1)] to our shear spectra to derive an independent estimate for ϵ that we denote as . Unlike the fitting method, the integration technique relies on resolving the viscous range and thus the spectral “dip” . The smallest of and 150 cpm is the upper wavenumber for the integration, which must be compared with theoretical limits and thus requires the initial estimate . The integration limit is then adjusted (if needed) so that it lies between the wavenumbers where theoretically 90% and 95% of the variance should be resolved. This is done iteratively by comparing the spectral observations to the Nasmyth empirical spectra estimated for using the nth iteration’s . If the Nasmyth fit corresponding to lies mostly above the observations, then increases, otherwise decreases.

d. Misfit criteria

To discard spectra that deviate excessively from the shape of the Nasmyth empirical spectrum, we use the mean absolute deviation (MAD),
e6
estimated for n spectral observations. Ruddick et al. (2000) successfully used this misfit criterion to discard temperature gradient spectra when the MAD . This rejection criterion was determined through Monte Carlos simulations, since for perfect fits MAD . This misfit measure can also help avoid fitting the model spectrum to the low 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 was effectively identified from the band-averaged shear spectral observations, with the exception of the example at W kg−1, where no clear viscous rolloff was identifiable (Fig. 3d). Over the entire dataset, was identified within and for 90% of the spectra when an ϵ estimate was returned by our algorithms. Term was slightly lower than (Fig. 3), since the algorithm limits the integration to (Fig. 2). The fitting algorithm constrains the maximum wavenumber even further to ensure the noise floor is excluded. For the examples shown in Fig. 3, except for the high ϵ example, where (Fig. 3d). Integrating beyond becomes more problematic for decreasing ϵ; the inclusion of the noise-dominated k overpredicted ϵ by two orders of magnitude for the low W kg−1 example (thin green line in Fig. 3a).

Fig. 3.
Fig. 3.

(a)–(d) Example corrected shear spectra with increasing ϵ. Both the fitted and values obtained from these spectra are shown in each panel’s titles. The secondary axis (green) shows the cumulative integrated spectra nondimensionalized by ϵ, which represents the proportion of variance resolved. The thin and thick green lines are for the observed and theoretical integration, respectively. The violet dotted–dashed are the Nasmyth empirical spectra for ϵ rounded to the nearest order of magnitude above and below the fitted result. For completeness, the identified dip is shown, along with the maximum integrated wavenumber used to derive . In each panel, is located at (a) , (b) , (c) , and (d) , while , except in (d), where given the instrumentation limit of cpm used by the algorithms.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0218.1

We compare the final ϵ estimates obtained with our integration algorithm to our MLE fitting algorithm from the entire dataset (Fig. 4a). The majority of our estimates were between and W kg−1 with a median value of W kg−1 (Fig. 4c). With the degrees of freedom of our estimated spectra, the error bound on was about a factor of 1.7, while more than 95% of are within a factor of 1.5 of . At W kg−1, however, progressively becomes larger than as the viscous range becomes underresolved (Fig. 4a). Figure 4a also compares the integrated estimates to those obtained when the fit is forced outside the viscous subrange, that is, the decade before , which we denote as , as this range spans the inertial subrange and perhaps even the energy-containing range influenced by the mean flow. The inertial estimates are slightly more variable than with larger percentile bars, but the majority (90%) of were within a factor of 2 of (Fig. 4a). Thus, fitting the inertial subrange is a viable means to derive 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].

Fig. 4.
Fig. 4.

(a) Term derived from our fitting algorithm and when the fit is confined to the inertial subrange, i.e., is plotted against the integration algorithm estimates . (b) Scatterplot of the initial ϵ guesses used for setting theoretical limits from integrating () and from fitting () to the lowest decade (or if this range is narrower). (c) Histogram of the different ϵ estimates for the dataset. For (a),(b) the sorted results have been binned evenly along the x axis, provided there were more than ≈1% (or 100) data points in each bin. The error bars contain 90% of the data within each bin. The 95% CL for associated with the significance interval of the estimated spectra is also shown. The CLs were approximately a factor of 1.7 of .

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0218.1

Figure 4b illustrates how close the initial integrated and the initial fitted estimates are from those obtained from our fitting algorithm. Both and were close to , particularly for W kg−1 given their percentile bars (containing 90% of the data) are generally within a factor of 2 of . With increasing ϵ, the percentile bars for remained within a factor of 2, while those for progressively increased up to a factor of 4–5 from the estimates (Fig. 4b). This better performance of at high ϵ was probably because the lowest k of the spectra used to derive was outside the inertial and viscous subranges, and was subject to the influence of the flow.

To investigate the sensitivity of to the fitted wavenumber range, we reestimated 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 found with our fitting algorithm (Figs. 5a,e). At these low k, the MLE yielded ϵ that were more than an order of magnitude larger than . As k increased, the MLE returned lower ϵ, dropping below once the fit excluded the low k affected by the mean flow, and then again increasing to as the fit was more weighted toward the larger number of spectral observations in the viscous range (e.g., Figs. 5a,e). However, the downside of this bias to the highest k in the fit is that once the noise-dominated portion of the spectra was included in the fit, the MLE progressively overpredicted (Fig. 5). This overprediction became more severe with decreasing ϵ as the noise levels increased above the low “environmental” spectral energy levels. For the W kg−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 over the fitted estimate for determining the theoretical limits. Nevertheless, the MLE returned robust estimates of ϵ, provided the fit was confined to the inertial and/or viscous range.

Fig. 5.
Fig. 5.

(a)–(d) As in Fig. 3, but for large segment bins of the order of 10–12 m to introduce the influence of the mean flow at low k—particularly apparent for the low ϵ example in (a). (e)–(h) The MAD misfit measure and rejection criteria plotted at the midpoint of each decade investigated are shown for the example spectra illustrated above them in (a)–(d). The secondary orange axis shows the sensitivity of the estimated , nondimensionalized by , fitted over a subset of the spectra. The corresponding orange circles represent the result obtained if the entire spectrum was fitted, and the result is also centered at the median k of the spectral observations used in the fit.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-15-0218.1

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 , since the fit (wrongfully) includes the lowest 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 W kg−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 to obtain the first 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.

Acknowledgments

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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