## 1. Introduction

Turbulent mixing controls the distribution of contaminants, nutrients, temperature, and small organisms in the ocean. For example, the vertical flux of nutrients across the pycnocline can control the input of nutrients to the euphotic zone, thereby governing ocean productivity (Gargett 1997). Prescription of the global ocean circulation and vertical heat transport depends critically on the diapycnal mixing (Wunsch and Ferrari 2004; De Lavergne et al. 2016). Circulation models rely on turbulence closure schemes to parameterize subgrid-scale mixing, and these parameterizations are derived from idealized engineering flows or from controlled experiments at length scales and turbulence intensities that are very different from those observed in the ocean (e.g., Mellor and Yamada 1982; Ivey et al. 2008; Salehipour et al. 2016). Thus, field observations of turbulent mixing remain essential to assessing and developing these models and to understanding the connections between large-scale physical processes and small-scale mixing.

*ϵ*and, in turn, the eddy diffusivity for density

*N*is the background buoyancy frequency (Osborn 1980). This relationship assumes a balance between shear production, buoyancy flux, and dissipation, and it requires an independent estimate of

Ongoing work attempts to describe the variability of *S* is the mean shear, *ν* is the molecular kinematic viscosity, *η* is the Kolmogorov length scale. Despite these efforts the exact *ϵ*, remains uncertain.

*χ*, the vertical diffusivity of heat

*T*is the temperature, and the overbar denotes a temporal average. By equating the Osborn model [(1)] to the Osborn–Cox model [(2)], an indirect measure of

*χ*estimates. As discussed in detail below,

*χ*has traditionally been estimated with information from the high-wavenumber end of the temperature gradient spectrum using two methods: either by integrating over these high wavenumbers (e.g., Sherman and Davis 1995; Ruddick et al. 2000) or by Batchelor spectrum fitting techniques at very high wavenumbers (e.g., Luketina and Imberger 2001; Merrifield et al. 2016). Both of these methods, however, are challenging because of the temperature sensor’s poor response at high frequencies (e.g., Sommer et al. 2013, see section 2.1). Having the profiler move as slowly as 0.05–0.1 m

In this paper, we demonstrate how to use existing technology and/or datasets to determine *χ* from the more easily resolved inertial-convective subrange portion of the temperature gradient spectrum. Studies involving moored (time series) measurements have exploited the inertial-convective subrange (e.g., Bluteau et al. 2013; Holleman et al. 2016), but this practice has yet to be adapted to profiling microstructure turbulence measurements such as gliders, and autonomous and ship-based profilers from which we have historically the most measurements. We describe the theoretical and practical aspects of the method, and demonstrate its application to observations from a site known for its energetic internal waves and hence high mixing rates. The presented techniques also provide an indirect measure for

## 2. Methods

### a. Theoretical background

#### 1) Traditional methods for determining *χ*

*χ*has traditionally been determined by integrating the temperature gradient spectrum to the high-wavenumber end of the viscous-diffusive subrange, thus requiring resolution of the highest wavenumbers of the temperature spectra. Term

*χ*can be directly obtained fromwhere

*κ*is the molecular diffusivity of heat. For isotropic turbulence,

*χ*can be estimated from the turbulent temperature gradient measurements collected in one direction (Oakey 1982):and is equivalent to integrating the one-sided temperature gradient spectral observations

*χ*estimate can be obtained only when both the viscous-convective and viscous-diffusive subranges are resolved. Resolving the entire viscous-diffusive subrange is, however, difficult, since the measured spectra must be substantially corrected (sometimes up to a factor of

Many have attempted to fit the lower wavenumbers of the viscous-diffusive subrange with a model of the temperature gradient spectrum (e.g., Batchelor 1959; Kraichnan 1968; Ruddick et al. 2000; Sanchez et al. 2011; Merrifield et al. 2016) in order to “resolve” the high wavenumbers in the viscous subrange (Fig. 1). However, the universality of the constants used in these models are debated, particularly the root-mean-square of the rate of strain that is undetermined to a factor of 2 (e.g., Luketina and Imberger 2001). More importantly though, the viscous-diffusive subrange moves to higher wavenumbers *k* with increasing *ϵ* (Fig. 1), making it more difficult to resolve. For example, at typical profiling speeds of 0.6–1 m *χ* in energetic environments.

#### 2) Inertial-convective subrange method for obtaining *χ*

*k*is the wavenumber (rad

To differentiate between the *χ* estimates obtained by integrating and fitting the temperature gradient spectrum, we use the subscripts *I* and *F*, respectively, in the discussion below. Fitting the observed spectrum to obtain *ϵ* from the simultaneous velocity measurements. However, in practice, the value of *ϵ* because of the *ϵ* from the inertial-convective subrange of temperature spectra located over similar *k* as the inertial-subrange of velocity spectral observations.

Observationally, the inertial subrange of velocity measurements extends to

### b. Analysis procedures

*f*) are corrected for the thermistor’s frequency response using a double-pole transfer functionwith a time constant

*k*of the inertial-convective subrange for large

*f*) into wavenumber spectral observations

*k*) using the mean vertical profiling speed

#### 1) Spectral fitting procedures to obtain

To derive *ϵ* has been determined from the inertial and/or viscous subranges of turbulent velocity or velocity gradient measurements (e.g., Bluteau et al. 2011b, 2016). The segment length from the profile used to compute the temperature gradient spectra must be sufficiently long to resolve the wavenumbers within the inertial-convective subrange, which moves to lower wavenumbers with decreasing *ϵ* (Fig. 1). For low

*d*representing the number of degrees of freedom of the estimated spectra

*k*, we rely mainly on the information from the higher and hence more isotropic wavenumbers

*k*. The lower

*k*, potentially more adversely impacted by the mean flow, are excluded by applying the mean absolute deviation (MAD) misfit criteria proposed by Ruddick et al. (2000),to short subsets of the spectra approximately 0.5–0.7 of a decade long. Here,

*n*refers to the number of individual

*i*spectral observations

#### 2) Spectral integration procedures to obtain

Ideally, *ϵ*, these subranges move to higher wavenumbers and so we use the Batchelor wavenumber

For low *ϵ* cases, a spectral rolloff is often observed prior to instrument noise dominating the spectral observations. To avoid the noise-dominated wavenumbers, we identify the local minima of the smoothed (band averaged) spectra between 0.2 *χ* estimate for the given segment if the flow is not energetic enough (i.e., low

## 3. Field measurements

The abovementioned methods for estimating *χ* and *ϵ* from the methods described by Bluteau et al. (2016); 3D accelerometers were used to remove motion-induced contamination from the velocity gradient spectra (Goodman et al. 2006); and a pressure sensor was used to determine the instrument’s profiling speed *ϵ* estimates from these measurements ranged between *N* and temperature gradients *χ*.

The temperature gradient spectra were obtained from the FP07 measurements over the same segments used to determine *ϵ*. Each of the 118 vertical profiles were split into segments of 2048 samples (4 s) that overlapped by 50%. The temperature gradient spectra were estimated by applying an FFT on 512 point-long (1 s) subsets with a 50% overlap. A Hanning window was applied to each subset in the time domain, which resulted in temperature gradient spectra with more than 21 degrees of freedom. For each segment, these spectra were used to obtain *χ* with either the integrated or fitted technique, and for some intermediate energy segments with both techniques. We determined mixing rates *χ* estimates and the background temperature gradient *ϵ* and *χ*.

## 4. Results and assessment

The observed temperature gradient spectra, for a range of *ϵ* covering almost four decades, are compared in Fig. 2 against the Kraichnan model (Kraichnan 1968; Bogucki et al. 2012) and the inertial-convective subrange models [(6)]. We corrected the spectral observations for the FP07’s frequency response using two separate relationships, but neither can recover the viscous-diffusive subrange (Fig. 2). With the gentle frequency correction *ϵ* examples (Figs. 2e,f).

Similar conclusions can be reached by comparing the model cumulative integrated spectra to the observed spectra obtained after applying both the *χ* (Figs. 2c,d), while the corrected spectra with *k* range (Fig. 2f). Integrating over these *k* thus overestimated *χ* (Fig. 2h). In this high *ϵ* example, avoiding the noise-dominated *k* by integrating up to *k*, while the frequency response correction was almost two orders of magnitude (Fig. 1). The observations were, however, in good agreement with the inertial-convective subrange model and could thus be fitted with (6) to determine

Integrating the spectra to obtain *χ* was thus confined to periods of relatively weaker turbulence (i.e., low *ϵ*), while fitting the spectra was generally confined to periods of more energetic turbulence (i.e., high

From the 4800 segments that yielded a valid estimate for *ϵ*, more than 4000 segments (83%) returned valid *χ* by combining both the integration and fitting techniques. Of the 4800 segments, 2200 (45%) segments returned a valid *χ* and hence mixing rates

The combined dataset shows *χ* varied between *ϵ* (Fig. 4a). The mixing rates determined using the Osborn–Cox model [(2)] with just the

The estimated

## 5. Conclusions

Without prescribing a value for *ϵ* is known from the velocity measurements, *χ* is obtained by either integrating the viscous-diffusive subrange [(5)] or by fitting the inertial-convective subrange of the temperature gradient spectra [(6)], thus allowing mixing rates to be obtained via the Osborn–Cox model [(2)]. Determining *ϵ* (see Fig. 1), and so any temporal or spatial averages of *ϵ* also carry the fast-response thermistors necessary to measure *χ*, our methods can be applied immediately to existing field datasets to better understand mixing processes in our oceans.

## Acknowledgments

Funding was provided by the Australian Research Council Discovery Projects (DP 140101322), the Australian Research Council Linkage Project (LP110100017), and the Office of Naval Research (ONR) Naval International Cooperative Opportunities (N62909-11-1-7058). ONR projects “AUV Data Analysis for Predictability in Time-Evolving Regimes” and “Propagation and Dissipation of Internal Tides on Coastal Shelves” contributed funding for this work. This work was also facilitated by an Institute of Advanced Studies Distinguished Visiting Fellowship awarded by the University of Western Australia. We thank staff from the Australian Institute of Marine Science, the U.S. Naval Research Laboratory and the University of Western Australia, and the crew of the R/V *Solander*, who aided in the collection of the data.

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