## Abstract

Turbulence intensity estimated from fast-response thermistors is compared between conductivity–temperature–depth (CTD)-attached and free-fall microstructure profilers, conducted at the same location within 2 h. The agreement is generally good but anomalously overestimated values, deviating from a lognormal distribution, appear sporadically in the CTD-attached method. These overestimated outliers are evident as spiky patches in the raw temperature gradient profiles. They often occur when the fall rate of the CTD frame *W* (m s^{−1}) is small and its standard deviation *W*_{sd} is large. These overestimated outliers can be efficiently removed by rejecting data with the criteria of *W*_{sd} > 0.2 *W* − 0.06, where *W* and *W*_{sd} are computed for a 1-s interval. After the data screening, thermal and energy dissipation, *χ* and *ε*, from CTD-attached and free-fall profilers are consistent within a factor of 3 in the ranges of 10^{−10} < *χ* < 10^{−7}°C^{2} s^{−1} and 10^{−10} < *ε* < 10^{−8} W kg^{−1}, respectively, for 50-m depth-averaged data. Energy dissipation from the CTD-attached method tended to be underestimated in the higher turbulence range of *ε* > 10^{−8}. This could be due to insufficient correction of the thermistor response for the faster fall rate (~1 m s^{−1}) of CTD frames. Since *ε* < 10^{−8} in most parts of the intermediate and deep ocean, use of the CTD-attached fast-response thermistors provides an efficient way to expand the presently sparse turbulence observations.

## 1. Introduction

Vertical turbulent mixing is a key process in the global ocean circulations, affecting the diapycnal transport of heat and salt as well as biogeochemical substances, such as nutrients, carbon, and trace metals. Turbulent mixing is estimated from measurements of the microstructure in velocity or temperature, with spatial scales of a few centimeters. As the measurements are susceptible to contamination by the vibration of an instrument, free-fall or free-rise profilers have been designed to minimize vibration (e.g., Lueck et al. 2002). Since the microstructure measurements require extra ship time for using such profilers, which need special skills for operations, conducting widespread observations is difficult. Hence, the ocean microstructure has been sparsely sampled.

In this paper we evaluate a new method for obtaining turbulence data via direct microstructure measurements using an internally recording profiler attached to a conductivity–temperature–depth (CTD) frame. Using this method, we can obtain much more microstructure data to the ocean floor from every CTD cast in a vast area than we would use a free-fall profiler, which would require extra ship time and special instruments.

A problem of this method is the motion of the CTD frame. It is connected to a steel cable and stretched throughout the deployment, and it does not fall freely. Therefore, acceleration or deceleration of the frame resulting from the rolling or pitching of a ship, as well as the variability of the winch feeding speed, causes instrument vibration that could affect microstructure measurements. Although microtemperature measurements conducted using a fast-response FP07 (Fastip Probe model 07), thermistor are less sensitive to vibration than velocity measurements with an airfoil shear probe, there is no consensus on appropriate usage of an FP07 thermistor attached to the CTD frame. To achieve CTD-attached microstructure measurements, the influence of frame movement on turbulence estimation and the limitations of this measurement technique are quantified in this paper.

To our knowledge, only Holmes et al. (2016, hereafter HMT16) have described the use of a microtemperature sensor attached to a CTD frame, but they did not provide validation of the method. They used data only from times when the CTD package was descending faster than 0.4 ms^{−1}, probably to exclude bad data caused by the “not free fall” CTD-attached observation. Earlier, Moum and Nash (2009) used thermistors attached to moorings and demonstrated that the estimated turbulence intensity is comparable to that from a free-fall profiler (Perlin and Moum 2012), suggesting that CTD-attached thermistors would allow us to measure turbulence fields more frequently and efficiently. Here, we evaluate CTD-attached measurements through a comparison with data measured with a standard free-fall profiler, and we identify objective editing criteria for removing abnormal data caused by not-free-fall measurements.

There are concerns regarding FP07 fast-response thermistor measurements independent of the abovementioned not-free-fall issues. As the time constant of the thermistors (~7 s; ms, for half attenuation) is insufficient for resolving relatively strong turbulence, various correction procedures have been proposed (Lueck et al. 1977; Gregg and Meagher 1980). In our previous companion study (Goto et al. 2016, hereafter GYN16), a double-pole (DP) low-pass filter method of correction, as proposed by Gregg and Meagher (1980), with a quarter attenuation time constant of 3 ms was demonstrated to be the most appropriate for a wide range of turbulent energy dissipation rates (10^{−10} < < 3 × 10^{−7} W kg^{−1}). This was achieved through comparing from thermistors with from concurrently measured shear probes attached to a free-fall profiler.

The objective of this paper is to establish a method of using the CTD-attached microstructure profiler for practical use. For this, we compare estimates of turbulence intensity from CTD-attached thermistors to estimates from nearly simultaneous free-fall microstructure profiles. The results of the comparison, including derivation of editing criteria, are presented in section 3. Uncertainties of estimation derived from CTD-attached thermistors are then discussed in section 4.

## 2. Data and analysis

### a. Observational data

A MicroRider 6000 (MR), manufactured by Rockland Scientific International, is an internally recording microstructure profiler to be attached to an observational platform, such as a CTD frame. FP07 microtemperature sensors were set near the bottom of the frame (Fig. 1). During three cruises 72 profiles of CTD-attached MR observations were performed (Fig. 2); 12 profiles were conducted near the Aleutian Passage from August to September 2009 using the R/V *Hakuho-Maru* (KH-09-4), 20 profiles around the Tokara Strait in June 2015 using R/V *Shinsei-Maru* (KS-15-5), and 40 profiles in the Pacific Ocean in June 2016 using R/V *Hakuho-Maru* (KH-16-3). In KH-16-3, an AFP07 microstructure profiler, manufactured by Rockland Scientific International, was used (Fig. 1b). Henceforth, AFP07 is also referred to as “MR” for simplicity. For every MR observation, a microstructure observation was performed using a free-fall Vertical Microstructure Profiler 2000 (VMP) just before or after the CTD-attached MR observations, within a period of 2 h. Turbulence intensity from the FP07 thermistors was examined by comparing the MR data with the VMP data.

The most notable difference between the CTD-attached MR and the free-fall VMP is in the fall rates. The fall rate *W* of the MR were usually larger (~1.0 m s^{−1}) than those of VMP (0.6–0.8 m s^{−1}). The fall rates of MR were slow near the surface and the bottom.

### b. Analysis method of turbulence intensity

Turbulent mixing causes temperature fluctuation on a scale of a few centimeters. At this scale, molecular thermal diffusion affects decreasing temperature variance. An indicator of this process is the temperature dissipation rate, represented as > under isotropy, where is the molecular thermal diffusivity and is the small-scale vertical temperature gradient [angle brackets () are the spatial mean over a certain vertical length]. The microtemperature vertical gradient was obtained by dividing the time derivative by *W*, interpolated from 64-Hz pressure data into 512 and 470 Hz, which are the sampling frequencies of MR and VMP, respectively.

In isotropic turbulence, the kinetic energy dissipation rate is , which is another indicator of turbulence intensity. Here, is the kinematic viscosity and is the small-scale vertical shear of horizontal velocity. Term is usually estimated by measuring velocity shear with airfoil shear probes (Lueck et al. 2002). In the absence of double diffusion, *ε* may be estimated from temperature microstructure fields by fitting a universal spectrum to an observed temperature gradient spectrum, based on the theory that *ε* is related to the Batchelor length scale , where the molecular diffusion of temperature becomes effective (Batchelor 1959). Energy dissipation is given by

where (cpm) is a reciprocal of and is known as the Batchelor wavenumber.

In this paper *ε* is determined by the best-fitted Kraichnan universal spectrum (Kraichnan 1968) to the observed temperature gradient spectrum following the maximum likelihood estimate (MLE) method introduced by Ruddick et al. (2000, hereafter R00). The universal spectrum determines , and is derived via Eq. (1).

Before using the MLE method, a time series of microtemperatures were grouped into 512 samples (about 1-s bins) and then fast Fourier transform (FFT) was performed. This segment setting is similar to that used by HMT16. After FFT, temperature gradient spectra were amplified by a double-pole low-pass filter function (Gregg and Meagher 1980) with a time constant = 3 ms following GYN16: , where *f* is frequency. The estimated from thermistors by applying this form of correction to temperature spectra was confirmed to be comparable with from the shear probes within a factor of 3 in the range of 10^{−10} < *ε* < 3 × 10^{−7} W kg^{−1} (GYN16). The dependence of the time constant on fall rate^{1} is not included in this paper.

Estimation of was conducted following R00 and GYN16. The was computed by integrating the temperature gradient spectrum between the wavenumbers of *k*_{min} and *k*_{max} after removing instrument noise via Eq. (9) in R00. The noise spectra of MR and VMP were determined from a bench test in the laboratory. Note that the noise spectra differ between MR and VMP because the cutoff frequency preventing aliasing is 98 and 165 Hz in MR and VMP, respectively. Examples of the signal and noise spectra of VMP are presented in Figs. 3a and 3b. The *k*_{min} is the minimum wavenumber of each temperature gradient spectrum and was determined by dividing the minimum frequency (Hz) by the fall rate (m s^{−1}). The *k*_{min} is, hence, approximately 1–10 cpm. Term *k*_{max} is the maximum wavenumber at which the observed signal-to-noise spectra ratio is >1.5.

The universal temperature gradient spectrum, first introduced by Batchelor (1959) and revised by Kraichnan (1968), was fitted to the observed spectrum to estimate . We used the Kraichnan form, which assumes that the intermittency of the strain rate on the spatial scale is smaller than the Kolmogorov scale. It is represented by the following equation (Roget et al. 2006):

where is the Kraichnan universal spectrum, is the Kraichnan constant, and . The has been demonstrated as = 3.4–7.9 (3.41: Antonia and Orlandi 2003; 5.26 ± 0.25: Bogucki et al. 1997, 2012; 7.9 ± 2.5: Sanchez et al. 2011). The = 5.26 is used throughout this study. The best-fitted theoretical spectrum was determined by estimating through identifying the maximum likelihood, C11, between the observed and theoretical spectra:

where *P* is probability; and are the observed and noise spectra, respectively; and is the *χ*-square distribution based on distributed as a probability density function, with degrees of freedom *d*. The *d* = 2 in our analysis. The Σ is the sum of each wavenumber component over *k*_{min} and *k*_{max}. By substituting for the largest C11 in Eq. (1), is obtained. See R00 for further details on the MLE method.

### c. Data rejection

Poorly fitted spectra were discarded by three quality tests introduced by R00. First, the mean absolute deviation [MAD, defined as Eq. (24) in R00] between the observed and theoretical spectra must be small. The threshold was set at MAD < 2 [= 2(2/*d*)^{1/2}]. Second, the signal-to-noise ratio of spectra (SNR) must be sufficiently large, where the signal is defined as *S*_{obs} − *S*_{noise}. The threshold in this paper was set at SNR > 1.5 according to Peterson and Fer (2014, hereafter PF14), which is stricter and larger than the 1.3 in R00, to ensure reliable data were obtained. Third, the observed spectra must have a sharp rolloff on the higher wavenumber side of the spectrum peak. To ensure this, the likelihood ratio [LR; *P*_{(Kraichnan spectrum)}/*P*_{(straight line)}] is used. The likelihood *P*_{(straight line)} from using a straight line fitted to the observed spectrum in log–log space (the first-order power-law fit; the blue line in Figs. 3a and 3b) should be smaller than the likelihood *P*_{(Kraichnan spectrum)} from using universal spectra. If the likelihood ratio is small, then the theoretical curve is rejected, as it does not provide a significantly better fit than the power-law fit. Further details are described in R00. In this paper, the threshold value of LR is set at LR > 100, similar to R00 and PF14. According to PF14, the LR criterion is unsuitable for a power law fitted with a positive slope (e.g., Fig. 3b). Thus, we apply the LR criterion only when the power-law fit has a negative slope, such as Fig. 3a. Examples of temperature gradient spectra passed and rejected by these tests are presented in Fig. 3. The 10-m (50 m) mean log_{10}(LR^{−1}) in Fig. 6d (Fig. 8) was computed by averaging 1-s LR^{−1} for 10-m (50 m) intervals and then taking the logarithm. This is because LR varies exponentially, and the features of small LR are masked when the LR themselves are averaged arithmetically. In the following analyses, only the data that have passed these tests will be shown.

In the case of the CTD-attached method, the fall speed is dependent on winch feeding speed and ship rolling because the frame is pulled by a wire throughout casts. Thus, when the frame is moving down at a slower speed, there would be a moment when the frame rises. This creates artificial turbulence at the bottom of the frame, where the microstructure sensors are located. The 1-s binned data with decreased pressure readings were not used, even when the decrease was instantaneous. Furthermore, only data of the new water are measured; that is, data with pressure during which the CTD rises and measurements after having the direction reversed are not used to exclude wake contamination.

## 3. Results

### a. Comparison between MR and VMP

First, *χ* and *ε* at the same locations and depths were compared between VMP and MR (Fig. 4), where *χ* and *ε* were arithmetically averaged over 50-m-depth segments. Data [especially with the red dots in Fig. 4 representing higher (>0.9 m s^{−1}) fall rates] are distributed roughly along the *y = x* line, whereas excessively large *χ* and *ε* from MR [blue dots representing lower (<0.5 m s^{−1}) fall rates in Fig. 4] are identified. Turbulence intensity from MR that is more than 10 times larger than that from VMP is referred to as “MR/VMP > 10” or “overestimated data.” For *χ* and *ε*, 12.3% and 15.9% of data are distributed above the *y =* 10*x* lines, while 5.8% and 6.0% are distributed below the *y =* 1/10*x* lines, respectively. The overestimation of MR data is thus greater than the underestimation in both *ε* than *χ*.

The pairs of MR and VMP observations used for comparison in Fig. 4 were not performed at the same time but separately within about 2 h. Considering the intermittent characteristics of turbulence and its spatial variability, the data from MR and VMP do not need to be identical; even two shear probes separated by several centimeters yield scattered results with a factor of 3 (GYN16). Similarity with some scatter is expected for the pairs of MR and VMP observations taken within about 2 h, considering that microscale (on the order of 1 cm–1 m) turbulence is related to the larger vertical (on the order of 10–100 m) and time scales (on the order of hours–days) of internal wave fields (e.g., Henyey et al. 1986). Here, we use log_{10}(*χ*_{MR}/*χ*_{VMP}) and log_{10}(*ε*_{MR}/*ε*_{VMP}) distributions (Fig. 5) to check the similarity and scatter between MR and VMP by comparing them with consecutive free-fall VMP observations performed within about 2 h (refer to the appendix).

A large portion of the MR data is consistent with VMP data, based on the log_{10}(*χ*_{MR}/*χ*_{VMP}) and log_{10}(*ε*_{MR}/*ε*_{VMP}) distributions (Fig. 5). The 10-m mean *χ*_{MR}/*χ*_{VMP} and *ε*_{MR}/*ε*_{VMP} (Figs. 5a and 5d, respectively) show lognormal distributions, with medians of around 1 [log_{10}(χ_{MR}/χ_{VMP}) = −0.02 and log_{10}(ε_{MR}/ε_{VMP}) = −0.04] and scatters, indicating that MR ≈ VMP. Scatter is represented by the standard deviation, SD, and the mean ± 1.96 × SD covers 95% of data. For the 10-m mean *χ* and *ε*, SD *=* 1.07 and 1.3, respectively; the data are thus approximately within a factor of 100. This scatter is similar to that of the repeated free-fall VMP observations at the same locations within about 2 h, as shown in the appendix, where the natural temporal variability of turbulence is examined and the width of the distribution is represented by 1.96 × SD = 1.96 × 1.01 (Fig. A1a).

With larger averaging bin length, scatters of the distributions decrease and data within factors of 3 and 10 increase for both the comparisons of *χ* and *ε* between MR and VMP (Figs. 5g and 5h) and among the repeated VMP observations (Fig. A1; Table A1). Data within a factor of 3 increased from 39% for the 10-m mean *χ* to 61% for the 200-m mean (Figs. 5a–c).

From the histograms for the 50- and 200-m mean comparisons (Figs. 5b and 5c, and 5e and 5f), the data that deviated from the lognormal distributions become noticeable in the range of MR/VMP > 10, which corresponds to the overestimated MR measurements (Fig. 4). These data that deviated from symmetric lognormal distributions are regarded as unusual, considering that the distributions of repeated VMP observations are symmetrical and 95% of the data are within a factor of about 10 (SD = 0.36–0.70) (Figs. A1b and A1c, and A1e and A1f). Data satisfying MR/VMP > 10 in the 50-m mean are then defined as overestimated data.

### b. Overestimation and disturbed spectra related to fall-rate variability

The MR overestimates could be related to the not-free-fall measurements. The CTD frame to which the MR was attached connected to a winch on the deck through a steel wire, and ship rolling and pitching changed the lowering rate of the CTD (referred to as the fall rate), thus the moving CTD frame may generate disturbance. The fall rate is the most noticeable difference between the MR and VMP observations. In this subsection, the influence of fall-rate variability on the overestimates and temperature gradient spectra are described with an example of vertical profiles (Fig. 6).

Fall-rate variability may generate disturbance in the temperature microstructure. The *χ* estimated from MR (red curves in Fig. 6a) is much larger than *χ* from VMP (blue) at 0–100, 400–600, and 1000–1100 dbar. At these depths, *W* (Fig. 6e) is small, and the standard deviation *W*_{sd} (Fig. 6f) is large. This correspondence between *χ* and (*W*_{sd}, *W*) suggests that MR measurements are bad because of the fall-rate variability. The microtemperature temporal variability is large at these depths, indicating that raw microtemperature is influenced even before performing spectral analysis.

The enlarged view of and the raw fall rate before averaging (Fig. 7) show the impact of fall-rate variability on microtemperature. Even when the fall rates were positive and the thermistors entered into new water, large microtemperature variations occurred in the period during which the fall rates took minima. This correspondence suggests wake generation from the CTD frame when the frame slows and then accelerates, even though spectra have passed quality tests with the PF14 criteria (Fig. 7e).

The dependence of overestimated *ε* of MR on the spectrum shape (MAD, LR) and on the fall-rate variability (*W*, *W*_{sd}) was quantified by compiling all of the 50-m depth-averaged data (Fig. 8). The overestimated data with *ε*_{MR}/*ε*_{VMP} > 10 increases with increasing LR^{−1}, especially for log_{10}(LR^{−1}) > −10 (Fig. 8a), where about 40% of the data are overestimated. This indicated that the spectrum shape tended to be disturbed for the overestimated data. MAD, however, is not sensitive to the overestimation, and overestimated data are distributed uniformly in the MAD range of 0.6–1.0.

Overestimates are better separated by *W* and *W*_{sd} (Fig. 8b) than by the spectrum shape presented by LR^{−1} and MAD (Fig. 8a). Overestimated data appear for relatively small *W* and large *W*_{sd} (Fig. 8b), as similarly seen in Figs. 6 and 7. Fall rate and its variability are thus good indicators for detecting and eliminating overestimated MR data. These results are from the 50-m depth-averaged data, which include about fifty 1-s spectrum data computed from 512 microtemperatures and 64 pressure raw data. If the 50-m data were found to be overestimated and rejected, then a large gap in the 50-m data would appear. To reduce such gaps as much as possible, it is better to judge and exclude data averaging as little as possible.

For the 1-m mean data (which is similar to the 1-s original spectrum data for *W* ~1 m s^{−1}), overestimated data (*ε*_{MR}/*ε*_{VMP} > 10) are also found for small *W* and large *W*_{sd} (Fig. 9a), as found for the 50-m mean data (Fig. 8b), with some scatters, as indicated by the color of the dots. To clearly identify the parameter range where the overestimates prevailed, the data in Fig. 9a were further averaged in the parameter space (*W*, *W*_{sd}) with 0.1 by 0.01 m s^{−1} intervals to indicate correspondence between *ε*_{MR}/*ε*_{VMP} and fall-rate parameters (*W*, *W*_{sd}) (Fig. 9b). Figure 9b suggests a simple criterion that separates the overestimated data from the form of *W*_{sd} > *aW* + *b,* where *a* and *b* are determined as follows: The histogram of log_{10}(*ε*_{MR}/*ε*_{VMP}) (Fig. 9c), based on the data in Fig. 9b, shows that log_{10}(*ε*_{MR}/*ε*_{VMP}) largely deviated from a lognormal distribution for log_{10}(*ε*_{MR}/*ε*_{VMP}) > 0.5. A first-order regression through points with 0.4 < log_{10}(*ε*_{MR}/*ε*_{VMP}) < 0.5, which separates a border of overestimation, is *W*_{sd} = 0.2*W* − 0.06 (circles and solid line in Fig. 9b).

### c. MR data after screening overestimated data

By removing 1-s data with *W*_{sd} = 0.2*W* − 0.06, the consistency between VMP and MR is improved, based on the better lognormal distributions (Fig. 10). Comparing Fig. 10 with Fig. 5 shows that most of the overestimation in Fig. 5 is removed. The percentage of overestimated data from MR/VMP > 10 was reduced from 12.3% to 5.8% and from 15.9% to 4.1% for the 50-m-bin *χ* and *ε*, respectively. Data within a factor of 10 increased from 82% to 93% and from 75% to 94% for the 200-m-depth mean *χ* and *ε*, respectively. The ratio of data within a factor of 10 for the 50-m-depth mean also increased. The scatters, represented by SD, for the 50- and 200-m-depth mean *ε* data are now comparable with the scatters of the repeated VMP observations with the natural temporal variability of turbulence, as shown in the appendix.

Improvement of the consistency between MR and VMP is noticeable in the scatterplots based on the 50-m depth-averaged data after screening (Fig. 11) compared to those before screening (Fig. 4). Most of the data from MR are within a factor of 10, and the regression lines based on the principal component analysis are along the *y* = *x* line (black thick lines in Fig. 11), especially for the comparison between *ε*_{MR} and ε_{VMP}, which were measured with velocity shear probes attached to the free-fall VMP (Fig. 11c).

In contrast, a slight underestimation of *ε* from MR became noticeable after removing overestimated data. It is noted that the median and mean of log_{10}(*ε*_{MR}/*ε*_{VMP}) in Fig. 10 decrease compared with those in Fig. 5. They decrease from −0.09 and 0.57 (Fig. 5f) to −0.21 and −0.18 (Fig. 10f), respectively, for the 200-m depth-averaged *ε*.

This *ε*_{MR} underestimation is also seen in scatterplots (Figs. 11b and 11c), indicating that MR tends to be underestimated in a strong turbulence environment for higher fall rates (*W* > 0.9 m s^{−1}). The *W* of a large part of the underestimated data seen in *y* < *x* for *ε*_{VMP} > 10^{−9} W kg^{−1} in Figs. 11b and 11c is greater than 0.9 m s^{−1} (represented by the red dots in Fig. 11), which is also indicated by the slope of the regression line for data with *W* > 0.9 m s^{−1} (red lines in Fig. 11b), which is less than 1. In the range of *y* > *x* for *ε*_{VMP} > 10^{−9} W kg^{−1}, in contrast, the green-blue dots representing *W* < 0.7 m s^{−1} prevail (Figs. 11b and 11c). This dependence on *W* is also seen in the thermal dissipation rate *χ* (blue dots in *y* > *x* and red dots in *y* < *x* for *χ*_{VMP} > 10^{−9} in Fig. 11a), although the overall underestimation of *χ* is not noticeable from the regression lines in Fig. 11a.

The comparison of *χ* and *ε* between the free-fall VMP and not-free-fall CTD-attached MR measurements after screening overestimated abnormal data is summarized in Table 1. Acceptable measurements here (the means plus/minus their 95% confidence interval) are within a factor of 3 of the VMP data. The turbulence intensity ranges of acceptable measurements depend on depth-averaging length. The *χ* is acceptable from 10^{−10} to 10^{−7} °C^{2} s^{−1}, in that the average of log_{10}(*χ*_{MR}/*χ*_{VMP}) is within a factor of 3 (log_{10}3 = 0.4771) for

the 10–200-m averaged data. For the 10- and 20-m mean, *χ* is acceptable but slightly underestimated [log_{10}(*χ*_{MR}/*χ*_{VMP}) = −0.411 and −0.242, respectively]. The *ε* is acceptable from 10^{−10} to 10^{−8} W kg^{−1} for the 20–200-m averaged data, and from 10^{−10} to 10^{−9} W kg^{−1} for the 10-m data. Term *ε* from MR is underestimated for strong turbulence of *ε >* 10^{−8} (10^{−9}) W kg^{−1} for the 20–200-m (10 m) averaged data.

## 4. Discussion

### a. Underestimates of ε from MR in strong turbulence fields

As shown in section 3c, *ε* from the CTD-attached MR measurements tends to be underestimated for faster fall rates and in a relatively strong turbulence environment. Here, we discuss the possible reasons of underestimation. As stated in the introduction, turbulence intensity measured with fast-response thermistors tends to be underestimated in strong turbulence environments if the correction to amplify the high-frequency part of the signal is not applied. This is because the response of the thermistors, with half attenuation time constant of ~7 ms for a single-pole (SP) function (Lueck et al. 1977), is not sufficient to resolve the high-frequency (and thus high wavenumber) part of the temperature gradient spectra in strong turbulence regions. By amplifying the high-frequency part of the spectra with the double-pole correction function (Gregg and Meagher 1980) with a quarter attenuation time constant of 3 ms, *ε* from the thermistors is reported to be acceptable, within a range of 10^{−10}–3 × 10^{−7} W kg^{−1}, by comparing simultaneously measured *ε* from shear probes attached to free-fall VMPs with a fall rate of 0.6–0.7 m s^{−1} (GYN16).

The underestimation of MR is partly explained by the higher *W* of MR (~1 m s^{−1}) than the *W* of free-fall VMPs (~0.65 m s^{−1}). The *ε* from thermistors is determined practically by detecting the frequency at the spectrum peak, , through (GYN16). As peak increases with *W* and , the frequency spectrum with larger *W* and *ε* shifts to a higher frequency where insufficient thermistor response attenuates the spectra; the degree of underestimation hence increases with increasing *W* and turbulence intensity. According to the above formulas, the acceptable upper limit of 3 × 10^{−7} W kg^{−1} of *ε* in the case of *W* ~ 0.65 m s^{−1} for the free-fall VMP (GYN16) would be reduced to 5 × 10^{−8} W kg^{−1} [~3 × 10^{−7} × (1/0.65)^{−4}] in the case of *W* ~ 1 m s^{−1} for the CTD-attached MR.

Another possible explanation for the underestimation of MR is that the temporal response of the MR thermistors is slower than that of VMP. Turbulence intensity is variable within a factor of 3 even by changing the frequency response functions and time constants within the uncertainty of manufacturer specification; single-pole half attenuation time constant τ = 4–10 ms, or corresponding double-pole quarter attenuation time constant of 2–4 ms (Fig. 12). The acceptable upper limit of *ε* would be reduced to 1 × 10^{−8} W kg^{−1} [~ 5 × 10^{−8}/(3^{2} + 3^{2})^{1/2}], which is consistent with that for the 20–200-m depth average in this study (Table 1).

It is also noted that *ε* (Fig. 12b) is more sensitive to variable correction functions and time constants than *χ* (Fig. 12a). The *ε* is estimated by detecting the wavenumber *k*_{p} at the temperature gradient spectrum peak, which would be sensitive to spectrum attenuation. The *χ*, however, is estimated by integrating spectra (section 2b). The difference in procedures for estimating turbulence intensity with spectrum attenuation could cause the sensitivity difference.

The combined impacts of insufficient correction and variable *W* on the reduction rates of *ε* and *χ* are evaluated (Fig. 13b) in the case where a relatively strong turbulence with *χ* = 10^{−7}°C^{2} s^{−1} and *ε* = 10^{−7} W kg^{−1} is measured using a thermistor, with a quarter time constant of 3.5 ms, and is then insufficiently corrected using the faster time constant of τ = 3 ms under variable fall rates from 0.2 to 2 m s^{−1} (Fig. 13). This situation was reproduced by the following procedures: the temperature gradient Kraichnan spectrum (outermost curve in Fig. 13a) with *χ* = 10^{−7}°C^{2} s^{−1} and *ε* = 10^{−7} W kg^{−1} is first attenuated by the double-pole low-pass filter with *τ* = 3.5 ms, and then amplified by double-pole correction with *τ* = 3 ms under variable fall rates from 0.2 to 2 m s^{−1} (Fig. 13a). The wavenumber of the spectrum peak decreases as the fall rate increases, and estimated *ε* and *χ* become smaller than their true values (Fig. 13b). Reduction of *ε* is more significant than that of *χ.* The difference in reduction between *ε* and *χ* influences the estimation of the mixing coefficient, which is defined as (e.g., Gregg et al. 2018). Insufficient thermistor corrections and a large *W* could lead to a larger mixing coefficient than the true values, as shown in Fig. 13b.

Although nominal uncertainty of the time constant yields uncertainty of turbulence intensity within a maximum factor of 3, it is required for reducing uncertainty under large fall-rate situations. To reduce the uncertainty of the estimated CTD-attached turbulence measurements, it is necessary to know the accurate correction function and time constant for individual thermistors, especially for a fall rate of 1 m s^{−1}.

### b. Potential reasons for overestimation

The correspondence between the overestimates of MR and a large *W*_{sd} and a small *W* indicates that the fall-rate variability of CTD influences turbulence estimation. The CTD frame descends at varying fall rates, which depend on winch feeding speed and ship motions. A descending CTD frame and attached instruments (CTD, lowered acoustic Doppler current profiler, or bottles) drag adjacent water and generate turbulent wake, usually above the frame and instruments (on the downstream side). On the other hand, when the CTD package decelerates more rapidly than the wake, which has downward momentum, the wake may be sampled at lowered *W* by the microtemperature sensor as the large in Fig. 7. These could be reasons why the overestimation observed in this study corresponds to large *W*_{sd} and small *W*. Such an artificial turbulent wake would depend on the space where the thermistors are located: a greater impact is expected for a narrow and crowded space. Thus, the threshold of *W*_{sd} = 0.2*W* − 0.06 may not be always a typical threshold.

### c. Availability and issues

Figure 7 suggested that *W* alone could be a good indicator for detecting artificial turbulence due to the wake generated from the CTD frame. When *W* is used as a sole criterion, overestimated data can be reduced (the bottom three rows of Table 2). It is close to the notation of HMT16, who used only data with fall rates larger than 0.4 m s^{−1}, although they did not discuss the validity of CTD-attached measurements and thresholds. For the present dataset, a threshold of *W* = 0.4 m s^{−1} (HMT16) reduces overestimation to 6.1% by removing 1-s-binned data (11 643 data points), whereas the criterion used (*W*_{sd} > 0.2 *W* − 0.06) reduces it to 4.1% by removing 14 134 data points. The ratio of data between the factor of 10 is 91% (hence, 9% is beyond the factor of 10) from the consecutively performed VMP in Fig. A1e; thus, the half 4.5% on the overestimated side is the required ratio for rejecting overestimated data. The ratio (4.1%) with the criteria in this study is consistent, whereas the criteria with the ratio greater than 4.5% in Table 2, including the criteria of *W* < 0.4 m s^{−1}, could be insufficient for rejecting overestimated data. For stricter criteria, overestimated data decreases but removed data increases. The choices of criteria hence depend on scientific purposes and measurement environments, such as the locations and space of thermistors, and the fall speed regulation system of CTD winches.

Another difference between the present study and HMT16 is the estimate of ε in the present study, whereas HMT16 χ is based on the procedures of Moum and Nash (2009). The present study confirmed the availability of the CTD-attached thermistor method by comparing ε from thermistors with free-fall VMP performed within 2 h, whereas HMT16 did not discuss the validity of the CTD-attached method. Further comparisons between HMT16 and the present study could contribute to turbulence estimation from fast-response thermistors attached to a CTD frame.

The CTD-attached method may not be appropriate at depths near the surface or bottom because fall rates are usually set to be slow and sensors easily suffer wake contamination. In contrast, an advantage of the CTD-attached method is easier microstructure measurement in the deep ocean, where *ε* < 10^{−8} W kg^{−1} (upper limit using the present CTD-attached method) in areas of the ocean with depths greater than 500 m (Kunze et al. 2006). The performance of this CTD-attached method depends on oceanic states (calm or rough sea) and the type of winch. To improve performance, using high-tech heave-motion winches that feed cables at constant rates, in addition to setting probes away from other instruments, such as CTD and LADCP, would be suitable.

Another issue is the thermistor response (functional form and time constant) of individual FP07 thermistors. Calibrating thermistors by simultaneously measuring with shear probes, thermistors with known responses (GYN16), and thermocouples (Nash et al. 1999; Moum and Nash 2009) in wide ranges of turbulence intensity and fall rates could contribute to further improvement of turbulence measurements with the CTD-attached thermistors.

In terms of performance of the fast-response thermistor during the faster fall rate, further quantification should be done by comparing them with shear probes, because the upper limit of measurable turbulence from the thermistor must depend on the fall rate as well as the frequency response of individual probes.

## Acknowledgments

The authors thank an anonymous reviewer for helping to improve the manuscript. The authors thank Drs. Sachihiko Itoh and Takahiro Tanaka for performing the turbulence observations during the cruises. The authors thank Prof. Mike Gregg for suggesting the importance of fall-rate variability. The authors also thank the captain, officers, and crews of R/V *Hakuho-maru*, R/V *Tansei-maru*, and R/V *Shinsei-maru*. This study is partially supported by KAKENHI 25257206, JP15H05818, JPH05817, JP15K21710, and 23710002.

### APPENDIX

#### Natural Temporal Variability in Turbulence Intensity

Histograms (distributions) of the ratios of turbulence intensity obtained from repeated VMP casts within 2 h are presented to show distributions of natural variability of turbulence fields (Fig. A1), where the medians and log mean are less than 0.05, approximately 95% of data are within a factor of 100 for the 10-m data (1.96 × SD ~2), and 95% data are within a factor of 10 for the 50- and 200-m data (1.96 × SD ~1). By comparing these distributions (Fig. A1) with those in Fig. 5, the overestimated data in Fig. 5 are unusual beyond natural variability.

The variability (scatter represented by the ratio of data within a factor of 3 or 10) of *χ* and *ε* distributions differs at each site (Table A1). Large percentages in the table indicate small variability. The variability (scatters) in Sagami Bay is lower than those of the Aleutians and the Kerama Gap. This could be because bottom topography in the low variability sites is less rough than in sites with large variability. The time intervals of observations in Sagami Bay are less than those in the Aleutian Islands and the Kerama Gap. This could be another reason for the difference in variability.

## REFERENCES

*Annu. Rev. Mar. Sci.*,

## Footnotes

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

^{1}

Gregg and Meagher (1980) showed that depends on the sensor speed as , where , while Hill (1987) reported that .