## 1. Introduction

Turbulent mixing is a dominant process to transfer and redistribute matter and energy in the global ocean, such as nutrients, suspended matter, carbon, dissolved gases, and cross-scale energy cascades (Polzin et al. 1997; Schmitt et al. 2005). It plays a vital role in maintaining the global meridional overturning circulation and ocean stratification, thereby significantly affecting atmospheric movements and global climate changes (Wunsch and Ferrari 2004). The direct method employed for observing oceanic turbulent mixing is measuring the turbulent dissipation rates. The turbulent kinetic energy dissipation rate *ε* and temperature dissipation rate *χ* are usually measured by using fast-response shear and temperature probes, respectively. Generally, *ε* and *χ* are directly inferred by simply integrating the shear and temperature gradient wavenumber (*k*) spectra using frequency cutoff defined by the noise of the instrument (Gregg et al. 2018; Lueck et al. 1997; Moum and Nash 2009; Wolk et al. 2002), or indirectly inferred by fitting the observed shear and temperature wavenumber spectra with the corresponding theoretical spectra in the inertial or dissipation subranges (Batchelor 1959; Bogucki et al. 2012; Kraichnan 1968; Nasmyth 1970; Oakey 1982). Time series shear and temperature data can only be directly converted into frequency (*f*) spectra; therefore, the flow speeds (*U*) past the measuring probe are required to translate the frequency spectra into wavenumber spectra with the Taylor frozen hypothesis, *k* = *f*/*U* (Dillon and Caldwell 1980; Peterson and Fer 2014). Thus, the flow speeds past the probe should also be known or measured in addition to the microstructure velocity shear and temperature. For microstructure measurements from a profiling, towed, or propelled instrument, this speed is well defined as the sink, rise, or tow speed of the instrument through the water, and usually measured by self-contained pressure probe (Bluteau et al. 2016a; Fer and Paskyabi 2014; Goto et al. 2018; Lueck et al. 2002; Peterson and Fer 2014; Wolk et al. 2009). For a moored observational instrument, however, this speed relies upon the ambient current, and its acquirement needs extra instruments, such as acoustic Doppler current profiler (ADCP), acoustic Doppler velocimetry (ADV), rotor current meter, or pitot-static tubes (Becherer and Moum 2017; Bluteau et al. 2016b; Fer and Paskyabi 2014; Lueck et al. 1997; Moum 2015; Perlin and Moum 2012; Song et al. 2013; Tian et al. 2014). Occasional failed work of these extra instruments may lead to potential risk of the speed data not being available; for example, when using the ADCP or ADV, the low concentration of suspended particles in relatively clear water may lead to a low signal-to-noise ratio and unsuccessful speed measurement (Habersack et al. 2011; Zhang et al. 2001). Under these circumstances, the turbulent dissipation rate *ε* and *χ* could not be estimated due to the lack of flow speeds. Therefore, here we proposed an alternative approach for quantifying the flow speeds by only using the observed time series microstructure shear and temperature data. This approach may be complementary and useful in the absence of directly measured speeds.

The remainder of this paper is organized as follows. In section 2, we explain how to derive the flow speeds from the inertial and dissipation subranges of the frequency shear spectrum (sections 2a and 2b) and temperature spectrum (sections 2c and 2d). In section 3, we describe the vertical microstructure profiler (VMP) observational dataset collected in the South China Sea in 2017 (section 3a), as well as the procedures for calculating the flow speeds using an example profile (sections 3b and 3c). We use all the microstructure profiles of the dataset to test the inferred flow speeds and compare them with the actual passing-probe speeds, i.e., the falling speed of the VMP (section 3d). In section 4, we summarize our findings.

## 2. Theoretical derivation

### a. Inertial subrange of the shear spectrum

*ε*in isotropic turbulence,

*ν*is the kinematic viscosity, ∂

*u*′/∂

*z*is the small-scale velocity gradient along

*z*direction, the angle brackets represent averaging,

*D*

_{u}(

*f*) is the shear spectrum in frequency (

*f*) domain, and

*f*

_{cut-off}is the upper limit frequency that has not been contaminated by noise.

*k*) domain are typically divided into two subranges, where the first is the inertial subrange and the second is the dissipation subrange. In the inertial subrange the shear spectrum is independent of the viscosity

*ν*and it is given as

*A*

_{1}depends on the velocity component employed. In the streamwise direction of the main flow,

*A*

_{1}= 0.49, and in the other two directions,

*A*

_{1}= 0.65 (Bluteau et al. 2011). Based on the Taylor frozen hypothesis, the wavenumber shear spectrum

*D*

_{u}(

*k*) and frequency shear spectrum

*D*

_{u}(

*f*) can be converted to each other as

*D*

_{u}(

*k*) =

*D*

_{u}(

*f*)

*U*, where

*U = f*/

*k*is the flow speed past the shear probe.

*U*past the shear probe can then be deduced from Eq. (3) as

*U*(1 here and 2, 3, and 4 in the subsequent subsections) denotes the

*i*th method for deriving the flow speed

*U*. From Eq. (4) the speed can be directly calculated with the frequency shear spectrum

*D*

_{u}(

*f*), which is obtained with the observed time series shear data by using Eq. (1).

### b. Dissipation subrange of the shear spectrum

*k*

_{υ}, defined as (Batchelor 1959)

*ε*, and then transforming it into the frequency domain, we obtain

*f*

_{υ}=

*k*

_{υ}

*U*is the Kolmogorov frequency. The flow speed past the shear probe is then deduced as

In Eq. (7), the frequency shear spectrum *D*_{u}(*f*) could be converted by observed time series shear data with Eq. (1), but *f*_{υ} is still unknown.

*D*

_{u_nasmy}(

*f*) satisfies

*D*

_{u_nasmy}(

*f*) =

*D*

_{u_nasmy}(

*k*)/

*U*, we can get

*D*

_{u_nasmy}(

*f*) by using Eqs. (7) and (8),

*d*[

*D*

_{u_nasmy}(

*f*)]/

*df*= 0, which yields

*f*

_{υ}= 37.04

*f*

_{peak_nasmy}, where

*f*

_{peak_nasmy}is the peak frequency of the Nasmyth spectrum. So

*f*

_{υ}can be approximately obtained from the peak of the observed shear spectrum as

*f*

_{peak_obs}is the peak frequency in the dissipation subrange of the observed shear spectrum

*D*

_{u}(

*f*).

Another method for obtaining *f*_{υ} is directly by fitting the observed shear spectrum *D*_{u}(*f*) with the frequency Nasmyth spectrum *D*_{u_nasmy}(*f*) [Eq. (9)]. The detailed processes for shear spectrum fitting will be described in section 3b.

### c. Inertial subrange of the temperature spectrum

*T*′ is the microstructure temperature variance and

*E*

_{T}(

*f*) is the frequency temperature spectrum. Based on the Taylor frozen hypothesis,

*E*

_{T}(

*f*) can be transformed into the wavenumber temperature spectrum as

*k*) domain can also be divided into two parts: the inertial subrange (i.e., the inertial–convective subrange) and the dissipation subrange (i.e., the higher-wavenumber part composed of the so-called viscous–convective and viscous–diffusive subranges). In the inertial subrange, the temperature is a passive scalar convected with the fluid, and the viscosity

*ν*and thermal diffusivity

*κ*

_{T}can be neglected, so this subrange is also called the inertial–convective subrange. Due to the purely passive character of the temperature in the inertial subrange, the wavenumber temperature spectrum

*E*

_{T}(

*k*) is independent of

*ν*and

*κ*

_{T}, and it has a well-known −5/3 power scaling with the wavenumber

*k*,

*A*

_{2}= 0.4 is a nondimensional constant called Obukhov–Corrsin constant (Sreenivasan 1996), and

*χ*is the temperature dissipation rate. In isotropic turbulence,

*χ*is defined as

*D*

_{T}(

*f*) has following relationship with the frequency temperature spectrum

*E*

_{T}(

*f*) as:

*D*

_{T}(

*f*) =

*E*

_{T}(

*f*)

*k*

^{2}=

*E*

_{T}(

*f*)

*f*

^{2}

*U*

^{−2}.

*N*[

*g*and

*ρ*

_{0}are respectively the gravitational acceleration and reference density. Γ is the mixing efficiency and generally treated as a constant, Γ = 0.2 (Osborn 1980). Despite extensive evidence that Γ varies over a wide range in different turbulent environments (Caulfield 2021; Ijichi and Hibiya 2018; Khani 2018; Monismith et al. 2018), most of oceanic observation estimations of Γ still tend to statistically agree with Γ ≈ 0.2 within a reasonable tolerance because of the ubiquity of marginal instability in geophysical turbulence (Gregg et al. 2018; Smyth 2020). In the case where the turbulent diffusivities of the temperature and density are equal, we can obtain an expression for

*ε*in terms of

*χ*as

From Eq. (19) the speed can be directly calculated with the frequency temperature spectrum *E*_{T}(*f*), which is obtained with the observed time series temperature data by using Eq. (11).

### d. Dissipation subrange of the temperature spectrum

*ν*and then thermal diffusivity

*κ*

_{T}. Thus, in this part, the temperature spectrum actually contains two subranges: the viscous–convective and viscous–diffusive subranges. Here we collectively refer to this part as the dissipation subrange. In this subrange, the smallest length scale of the temperature fluctuations before domination by molecular diffusion is the Batchelor scale, which corresponds to the Batchelor wavenumber

*k*

_{B}(Batchelor 1959),

The Batchelor wavenumber *k*_{B} is larger than the Kolmogorov wavenumber *k*_{υ} for seawater [see Eq. (5)], which means that the temperature dissipation occurs at scales smaller than the smallest turbulent eddy size in oceans.

*f*

_{B}=

*k*

_{B}

*U*is the Batchelor frequency. The flow speed past the temperature probe is then deduced as

*E*

_{T}(

*f*) can be obtained with the observed time series temperature data by using Eq. (11), but the Batchelor frequency

*f*

_{B}is still unknown. To get

*f*

_{B}, we define reduced frequency temperature dissipation spectrum

*D*

_{T}(

*f*) is frequency temperature dissipation spectrum.

*q*

_{k}is estimated as ranging from

*q*

_{k}= 3.4 − 7.9 (Antonia and Orlandi 2003; Bogucki et al. 2012; Sanchez et al. 2011). We use

*q*

_{k}= 5.26, which was proposed in some recent studies (Bogucki et al. 2012; Goto et al. 2016; Peterson and Fer 2014). With Eq. (23) we can derive the reduced frequency Kraichnan spectrum as

*D*

_{T_krai}(

*f*) satisfies

*D*

_{T_krai}(

*f*) =

*D*

_{T_krai}(

*k*)/

*U*, and

*d*[

*D*

_{T_krai}(

*f*)]/

*df*= 0, which gives

*f*

_{B}= 5.62

*f*

_{peak_krai}. Thus,

*f*

_{B}can be approximately obtained from the peak of the observed spectrum

Another method to obtain *f*_{B} is by fitting the observed reduced frequency temperature dissipation spectrum

## 3. Context with VMP observations

### a. Field observation data

In this section, we first describe the dataset of microstructure profiles, which is used to explain the calculation and test processes of *U*_{i} = (*i* = 1, 2, 3, 4). This dataset was collected during a physical oceanography research cruise in the northern South China Sea by the Research Vessel *Shiyan 3* between 22 and 24 September 2017. The observation sites are shown in Fig. 1. The dataset contained 30 profiles collected by a loosely tethered microstructure profiler called VMP (VMP250, Rockland Scientific). The water depths at all sites range between 2100 and 2500 m. The VMP was equipped with fast-response velocity shear and temperature probes (sampling rate: 512 Hz), as well as conventional conductivity, temperature, and pressure probes (sampling rate: 64 Hz), to measure the hydrological data in the upper 400 m. The VMP’s falling and rising speeds ranged between 0.5 and 1.0 m s^{−1}. The accuracy and resolution of the shear probe were 5% and 10^{−3} s^{−1}, respectively, and those of the fast-response temperature probe were 0.005° and 10^{−5}°C. For a more detailed description of VMP, please see the official production website of Rockland Scientific (https://rocklandscientific.com/products/profilers/vmp-250/). We use this dataset because the speeds could be calculated by using its high-resolution shear and temperature data according to the formulas derived above [Eqs. (4), (7), (19) and (22)], and conveniently compared with the actual passing-probe speeds, i.e., the falling speeds of the VMP (the data from the rising stage are not used because the water is disturbed by the instrument’s body). In the following sections, we first describe the calculation steps and results for the flow speeds *U*_{1}, *U*_{2}, *U*_{3}, and *U*_{4} using an example profile (site coordinates: 18.00°N, 113.02°E), and then analyze and compare the statistical results for all of the profiles.

### b. U_{1} and U_{2}

We calculate *U*_{1} based on Eq. (4) in the inertial subrange of the frequency shear spectrum according to the following steps 1 to 4.

#### 1) Step 1: Converting time-series shear data into frequency shear spectrum

The time series shear data for a profile are divided into 10 s segments, corresponding to about 8 m vertical segment length. A fast Fourier transform (FFT) with a half-overlapping window of 4 s is applied and averaged to obtain the frequency shear spectrum *D*_{u}(*f*) for each segment.

#### 2) Step 2: Correcting the shear spectrum

The finite size of airfoil shear probe means that eddies smaller than the probe could not be accurately measured, which may result in averaging in high frequency part and underestimation of the turbulent kinetic energy dissipation rate (see Fig. 1 in Macoun and Lueck 2004). This underestimation of turbulent kinetic energy dissipation rate may result in an underestimation of the inferred speed. Macoun and Lueck (2004) proposed a single-pole correction function as *H*_{s}(*f*) = 1/[1 + (*f*/*f*_{c})^{2}], where the sensitive frequency is *f*_{c} = *k*_{c}*U*_{average} with *U*_{average} = 0.8 m s^{−1} for our dataset. The sensitive wavenumber *k*_{c} depends on the probe’s geometry and dimensions (Bluteau et al. 2016a; Macoun and Lueck 2004), and we use *k*_{c} = 49 cpm for Rockland Scientific’s shear probes (Macoun and Lueck 2004).

#### 3) Step 3: Identifying the inertial subrange

*L*

_{o}= (

*ε*/

*N*

^{3})

^{1/2}, and the smallest turbulent scale characterized by the Kolmogorov scale

*L*

_{υ}= (

*ν*

^{3}/

*ε*)

^{1/4}. These upper and lower bounds of the inertial subrange are directly related to the undetermined dissipation rate

*ε*, which means that it is a challenge to explicitly identify the inertial subrange. We fit all possible frequency portions spanning a decade with

*D*

_{u}(

*f*) ~

*f*

^{α}. The portion where the power scaling is closest to 1/3 is recorded with the power scaling

*α*

_{c}, as well as the start and end frequencies

*f*

_{1}and

*f*

_{2}(as shown in Fig. 2b). This method follows the routine for identifying the inertial subrange proposed by Bluteau et al. (2011). If the

*α*

_{c}for a segment differs significantly from 1/3, this may mean that an inertial subrange is not available for this vertical segment. We employ the following acceptance criterion:

*α*

_{c}< 0.4, to determine the available inertial subrange, and we reject the data segments if the criterion is not satisfied.

(a) Example profile of velocity shear at the site (18.00°N, 113.02°E). (b) Shear spectrum (black line) corrected from the original spectrum (gray line) for the vertical segment depth ranging from 12 to 21 m in (a). The red line denotes the identified inertial subrange from *f*_{1} = 1.2 Hz to *f*_{2} = 12 Hz with the power scaling *α*_{c} = 0.31 for the fitting of *D*_{u}(*f*) ~ *f*^{α}. *f*_{cut-off} = 141 Hz is the upper-limit frequency that has not been contaminated by the noise. (c) Calculated value of *U*_{1} for the inertial subrange in (b) with median value of 0.81 m s^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Example profile of velocity shear at the site (18.00°N, 113.02°E). (b) Shear spectrum (black line) corrected from the original spectrum (gray line) for the vertical segment depth ranging from 12 to 21 m in (a). The red line denotes the identified inertial subrange from *f*_{1} = 1.2 Hz to *f*_{2} = 12 Hz with the power scaling *α*_{c} = 0.31 for the fitting of *D*_{u}(*f*) ~ *f*^{α}. *f*_{cut-off} = 141 Hz is the upper-limit frequency that has not been contaminated by the noise. (c) Calculated value of *U*_{1} for the inertial subrange in (b) with median value of 0.81 m s^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Example profile of velocity shear at the site (18.00°N, 113.02°E). (b) Shear spectrum (black line) corrected from the original spectrum (gray line) for the vertical segment depth ranging from 12 to 21 m in (a). The red line denotes the identified inertial subrange from *f*_{1} = 1.2 Hz to *f*_{2} = 12 Hz with the power scaling *α*_{c} = 0.31 for the fitting of *D*_{u}(*f*) ~ *f*^{α}. *f*_{cut-off} = 141 Hz is the upper-limit frequency that has not been contaminated by the noise. (c) Calculated value of *U*_{1} for the inertial subrange in (b) with median value of 0.81 m s^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

#### 4) Step 4: Calculating *U*_{1}

We calculate the flow speeds with Eq. (4) in the inertial subrange and take their median value as *U*_{1} for each vertical segment (as shown in Fig. 2c).

Figure 2 shows an example of how we obtain *U*_{1} by using the example profile at the site (18.00°N, 113.02°E). The velocity shear profile measured by VMP is shown in Fig. 2a. This profile exhibits relatively large fluctuations in the upper 70 m. Figure 2b shows the original (gray line) and corrected (black line) frequency shear spectra for the data segment depth ranging from 12 to 21 m (corresponding to 35–45 s in the falling period). It is found that the original shear spectrum is clearly underestimated in the high frequency range. The spectrum is acceptable below the frequency of *f*_{cut-off} = 141 Hz, above which the spectrum is unusable and rejected due to noise contamination. Using the method proposed above for identifying the inertial subrange, we determine the inertial subrange from *f*_{1} = 1.2 Hz to *f*_{2} = 12.0 Hz with the power scaling *α*_{c} = 0.31 for the fitting of *D*_{u}(*f*) ~ *f*^{α}. Figure 2c shows the results calculated for the passing-probe speeds *U*_{1} for the inertial subrange with Eq. (4). The values vary between 0.75 and 0.85 m s^{−1}, and their median is 0.81 m s^{−1}.

Figure 3 shows the power scaling *α*_{c} and calculated values of *U*_{1} for all segments of the example profile shown in Fig. 2a. In Fig. 3a it is found that *α*_{c} ranging between 0.27 and 0.4 (red circles) mainly locates in the upper 60 m for this profile. In the deeper depths, *α*_{c} generally deviates away from the theoretical power scaling of 1/3. The ratio of the theoretical upper and lower limits of the inertial subrange, _{b} with stronger dissipation rate *ε* and weaker stratification *N* in the upper mixed layer could lead to more recognizable inertial subrange than that in the deeper layer, where Re_{b} is buoyancy Reynolds number. Figure 3b shows the calculated values of *U*_{1} for all segments of the example profile. The red circles, corresponding to those in Fig. 3a, are markedly closer to the actual passing-probe speeds, i.e., the falling speeds of the VMP (green line).

(a) Power scaling *α*_{c} closest to 1/3 for all vertical segments of the example profile shown in Fig. 2a. The three vertical dashed lines denote *α*_{c} = 0.27, 1/3, and 0.4 from left to right, respectively. The red circles denote the segments where *α*_{c} satisfies the acceptance criterion [Eq. (27)]. (b) Calculated values of *U*_{1} for all vertical segments of the example profile. The values of *U*_{1} in the segments where *α*_{c} ranging between 0.27 and 0.4 are accepted and shown as red circles. The falling speeds of VMP are shown with the green line for comparison.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Power scaling *α*_{c} closest to 1/3 for all vertical segments of the example profile shown in Fig. 2a. The three vertical dashed lines denote *α*_{c} = 0.27, 1/3, and 0.4 from left to right, respectively. The red circles denote the segments where *α*_{c} satisfies the acceptance criterion [Eq. (27)]. (b) Calculated values of *U*_{1} for all vertical segments of the example profile. The values of *U*_{1} in the segments where *α*_{c} ranging between 0.27 and 0.4 are accepted and shown as red circles. The falling speeds of VMP are shown with the green line for comparison.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Power scaling *α*_{c} closest to 1/3 for all vertical segments of the example profile shown in Fig. 2a. The three vertical dashed lines denote *α*_{c} = 0.27, 1/3, and 0.4 from left to right, respectively. The red circles denote the segments where *α*_{c} satisfies the acceptance criterion [Eq. (27)]. (b) Calculated values of *U*_{1} for all vertical segments of the example profile. The values of *U*_{1} in the segments where *α*_{c} ranging between 0.27 and 0.4 are accepted and shown as red circles. The falling speeds of VMP are shown with the green line for comparison.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

#### 5) Step 5: Calculating *U*_{2}

There are three preprocessing steps before the calculation of *U*_{2} with Eq. (7). The first two steps are the same as those applied for obtaining *U*_{1}. The third step is determining *f*_{υ}. As mentioned in section 2b, two methods could be used to obtain *f*_{υ}, one based on the peak of the observed frequency shear spectrum, *f*_{obs_peak}, with Eq. (10), and the other by using the frequency Nasmyth spectrum *D*_{u_nasmy}(*f*) [i.e., Eq. (9)] to fit the observed shear spectrum *D*_{u}(*f*).

*D*

_{u}(

*f*) with the frequency Nasmyth spectrum

*D*

_{u_nasmy}(

*f*). This method was first proposed by Ruddick et al. (2000) for fitting the temperature dissipation spectrum with theoretical Batchelor spectrum, and then wildly employed to fit the velocity shear spectrum and temperature dissipation spectrum (e.g., see Bluteau et al. 2016a; Goto et al. 2016). MLE is advantageous for a non-Gaussian error distribution, and more unbiased than the least squares method (Goto et al. 2016). For a detailed description of MLE for the shear spectrum fitting, please refer to Bluteau et al. (2016a). To reject bad fittings, the mean absolute deviation (MAD) between the observed and theoretical spectra is employed as an acceptance criterion (Ruddick et al. 2000),

*n*is the number of data points of the dissipation subrange; ⟨ ⟩ and | | denote calculating the average and absolute values, respectively.

Figure 4 shows the processes to obtain *U*_{2} for the example profile (Fig. 2a). In Fig. 4a, the corrected observed shear spectrum and its fitted Nasmyth spectrum are shown by black and red lines, respectively, with their corresponding peak frequencies, *f*_{peak_obs} = 14.1 Hz and *f*_{peak_nasmy} = 18.3 Hz. *f*_{peak_nasmy} is more accurate than *f*_{peak_obs} because of the fluctuations of the observed spectrum. Figure 4b shows the peak frequencies for the observed spectrum (black circles) and fitted Nasmyth spectrum (red circles) of the whole example profile. The values of *f*_{peak_obs} are more scattered than those of *f*_{peak_nasmy}, especially in the mixed layer (approximately the upper 70 m). The peak frequencies generally decrease with the increasing depth, which implies that the turbulent kinetic energy dissipation rate also decreases with increasing depth [see Eq. (5)]. The values of *U*_{2} calculated with *f*_{peak_obs} and *f*_{peak_nasmy} for all segments of the whole example profile are shown in Fig. 4c. It is found that the values of *U*_{2} calculated with *f*_{peak_obs} are highly scattered, whereas those obtained with *f*_{peak_nasmy} are generally more focused and they agree better with the VMP falling speed (green line in Fig. 4c). The values of MAD (Fig. 4d) calculated for all segments of the whole example profile show that most of the data points satisfy the acceptance criterion [Eq. (28)], except for the first two points. Baumert et al. (2005) argued that most of turbulence profilers equipped with shear probes should fall faster than 0.5 m s^{−1}, to make sure the time that the profiler traverses over turbulent overturn is much shorter than the turbulent characteristics dissipation time. Under this condition the Taylor frozen hypothesis is generally true. For the example profile the low falling speeds (<0.5 m s^{−1}) of the VMP for the two initial segments results in large MAD values for the Nasmyth spectrum fitting, and these two data points are rejected in further statistical analyses.

(a) Original (gray line) and corrected (black line) observed frequency shear spectrum for the vertical segment depth ranging from 12 to21 m of the example profile (Fig. 2a). The fitted frequency Nasmyth spectrum is shown by the red line. The arrowheads show the peak frequencies of the observed spectrum and the fitted Nasmyth spectrum as *f*_{peak_obs} = 14.1 and *f*_{peak_nasmy} = 18.3 Hz, respectively. (b) Peak frequencies of the observed spectrum (black circles) and fitted Nasmyth spectrum (red circles) for all segments of the example profile. (c) Values of *U*_{2} calculated with *f*_{υ} obtained from *f*_{peak_obs} (black circles) and *f*_{peak_nasmy} (red circles) for the example profile. The green line denotes the falling speeds of VMP. (d) MAD values calculated for all segments of the example profile. The vertical dashed line denotes MAD = 1.2.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Original (gray line) and corrected (black line) observed frequency shear spectrum for the vertical segment depth ranging from 12 to21 m of the example profile (Fig. 2a). The fitted frequency Nasmyth spectrum is shown by the red line. The arrowheads show the peak frequencies of the observed spectrum and the fitted Nasmyth spectrum as *f*_{peak_obs} = 14.1 and *f*_{peak_nasmy} = 18.3 Hz, respectively. (b) Peak frequencies of the observed spectrum (black circles) and fitted Nasmyth spectrum (red circles) for all segments of the example profile. (c) Values of *U*_{2} calculated with *f*_{υ} obtained from *f*_{peak_obs} (black circles) and *f*_{peak_nasmy} (red circles) for the example profile. The green line denotes the falling speeds of VMP. (d) MAD values calculated for all segments of the example profile. The vertical dashed line denotes MAD = 1.2.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Original (gray line) and corrected (black line) observed frequency shear spectrum for the vertical segment depth ranging from 12 to21 m of the example profile (Fig. 2a). The fitted frequency Nasmyth spectrum is shown by the red line. The arrowheads show the peak frequencies of the observed spectrum and the fitted Nasmyth spectrum as *f*_{peak_obs} = 14.1 and *f*_{peak_nasmy} = 18.3 Hz, respectively. (b) Peak frequencies of the observed spectrum (black circles) and fitted Nasmyth spectrum (red circles) for all segments of the example profile. (c) Values of *U*_{2} calculated with *f*_{υ} obtained from *f*_{peak_obs} (black circles) and *f*_{peak_nasmy} (red circles) for the example profile. The green line denotes the falling speeds of VMP. (d) MAD values calculated for all segments of the example profile. The vertical dashed line denotes MAD = 1.2.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

### c. U_{3} and U_{4}

The values for *U*_{3} and *U*_{4} are respectively calculated based on the inertial and dissipation subranges of the frequency temperature spectrum *E*_{T}(*f*) with Eqs. (19) and (22), and their preprocessing steps are some similar to those of *U*_{1} and *U*_{2}. Here we emphatically describe two points, which do not exist or are different for the calculation of *U*_{1} and *U*_{2}.

The first one is the background parameter *C*. As described in sections 2c and 2d, *U*_{3} and *U*_{4} are derived under the condition that the turbulent diffusivity of the temperature and density are equal, which introduces a background parameter, *T* and potential density *ρ* for the example site (18.00°N, 113.02°E), which were sampled by the conventional probes equipped on the VMP. It is found that the depth of the upper mixed layer is about 60 m. Figures 5c and 5d show the profiles of the temperature gradient *T*_{z} and buoyancy frequency *N*. The values of *T*_{z} and *dρ*/*dz* are calculated with linear fitting for each of the 10 s vertical segments of the temperature and density profiles. These segments are the same as those of the shear and temperature data transformed into corresponding frequency spectrum. The profiles of *T*_{z} and *N* have similar patterns, implying that the density in the upper layer is mainly dominated by the temperature.

Profiles of the (a) potential temperature, (b) potential density, (c) potential-temperature gradient, and (d) buoyancy frequency at the example site (18.00°N, 113.02°E). The dashed lines in (c) and (d) denote the values of 0.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

Profiles of the (a) potential temperature, (b) potential density, (c) potential-temperature gradient, and (d) buoyancy frequency at the example site (18.00°N, 113.02°E). The dashed lines in (c) and (d) denote the values of 0.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

Profiles of the (a) potential temperature, (b) potential density, (c) potential-temperature gradient, and (d) buoyancy frequency at the example site (18.00°N, 113.02°E). The dashed lines in (c) and (d) denote the values of 0.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

The second one is the correction of the temperature spectrum. The fast-response temperature probe (FP07) equipped on the VMP has a response time of about 12 ms, which is mainly caused by diffusion and attenuation of the signal in the boundary layer around the probe (Peterson and Fer 2014). This response time could attenuate the temperature spectrum in high frequency part. For practical applications, the temperature spectrum usually needs to be corrected by a single-pole or double-pole low-pass correction function (Gregg and Meagher 1980; Goto et al. 2016; Lueck et al. 1977; Peterson and Fer 2014). Here we correct it with a wildly used single-pole correction function *H*_{T}(*f*) = 1/[1 + (*f*/*f*_{h})^{2}], where the half-attenuation frequency, *τ*_{0} = 0.012 s and average speed *U*_{average} = 0.8 m s^{−1} for our dataset (Peterson and Fer 2014).

Figure 6a shows the observed frequency temperature spectrum (black line), corrected with the original spectrum (gray line), for the temperature segment over the depth ranging from 12 to 21 m. It is found that the original spectrum is clearly underestimated in the high frequency range. The fitted red straight line with a power scaling of *α*_{c} = 1.63 for *E*_{T}(*f*) ~ *f*^{α} denotes the inertial subrange ranging from *f*_{1} = 2.2 Hz to *f*_{2} = 7.4 Hz, which is identified by using basically the same method employed for identifying the inertial subrange of the shear spectrum, as described in section 3b. The only difference is that the fitting portion spanning half a decade, instead of one decade used for the shear spectrum. The data are acceptable below the frequency of *f*_{cut-off} = 115 Hz, above which the spectrum is unusable and rejected due to contamination by noise.

(a) Corrected frequency temperature spectrum (black line) and the original spectrum (gray line) for the segment depth ranging from 12 to 21 m of the microstructure temperature profile at the example site (18.00°N, 113.02°E). The red line denotes the identified inertial subrange from *f*_{1} = 2.2 Hz to *f*_{2} = 7.4 Hz, with the power scaling of *α*_{c} = −1.63 for the fitting of *E*_{T}(*f*) ~ *f*^{α}. *f*_{cut-off} = 115 Hz is the upper-limit frequency below which the spectrum is not contaminated by noise. (b) Reduced frequency temperature dissipation spectrum of *f*_{peak_obs} = 20.1 and *f*_{peak_krai} = 23, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Corrected frequency temperature spectrum (black line) and the original spectrum (gray line) for the segment depth ranging from 12 to 21 m of the microstructure temperature profile at the example site (18.00°N, 113.02°E). The red line denotes the identified inertial subrange from *f*_{1} = 2.2 Hz to *f*_{2} = 7.4 Hz, with the power scaling of *α*_{c} = −1.63 for the fitting of *E*_{T}(*f*) ~ *f*^{α}. *f*_{cut-off} = 115 Hz is the upper-limit frequency below which the spectrum is not contaminated by noise. (b) Reduced frequency temperature dissipation spectrum of *f*_{peak_obs} = 20.1 and *f*_{peak_krai} = 23, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Corrected frequency temperature spectrum (black line) and the original spectrum (gray line) for the segment depth ranging from 12 to 21 m of the microstructure temperature profile at the example site (18.00°N, 113.02°E). The red line denotes the identified inertial subrange from *f*_{1} = 2.2 Hz to *f*_{2} = 7.4 Hz, with the power scaling of *α*_{c} = −1.63 for the fitting of *E*_{T}(*f*) ~ *f*^{α}. *f*_{cut-off} = 115 Hz is the upper-limit frequency below which the spectrum is not contaminated by noise. (b) Reduced frequency temperature dissipation spectrum of *f*_{peak_obs} = 20.1 and *f*_{peak_krai} = 23, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

Figure 6b shows the reduced frequency temperature dissipation spectrum, *E*_{T}(*f*) (Fig. 6a). The dissipation subrange is fitted with the reduced frequency Kraichnan spectrum *f*_{peak_obs} = 20.1 Hz and *f*_{peak_krai} = 23 Hz, respectively.

Figure 7 shows *U*_{3} (red circles) and *U*_{4} (black triangles) for the whole profile at the example site (18.00°N, 113.02°E). *U*_{3} and *U*_{4} are respectively calculated with Eqs. (19) and (22) for the inertial and dissipation subranges of the temperature spectrum *E*_{T}(*f*) for each vertical 10 s segment of the microstructure temperature profile. The acceptance criteria for identifying the inertial and dissipation subranges of the temperature are the same as those of the shear spectrum [Eqs. (27) and (28)], that is, the data points which do not satisfy the acceptance criteria are unusable and rejected. Moreover, as mentioned in section 3b, in the dissipation subrange the peak frequency of the observed spectrum and its inferred speed are much more uncertain than those of the fitted theoretical spectrum (can also see Figs. 4b and 4c), so we only show the values of *U*_{4} obtained with the peak frequency of the fitted Kraichnan spectrum. It is found that *U*_{3} and *U*_{4} are mostly between 0.5 and 2 times the falling speed of VMP (green line). By comparing Fig. 7 with Figs. 3b and 4c, it can be seen that the passing-probe speeds obtained from the shear spectrum, *U*_{1} and *U*_{2}, are generally in better agreement with the VMP falling speed than those obtained from the temperature spectrum, *U*_{3} and *U*_{4}.

Values of *U*_{3} (red circles) and *U*_{4} (black triangles) calculated from all segments of the temperature profile at the example site (18.00°N, 113.02°E). The falling speed (green line) of the VMP is shown for comparison.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

Values of *U*_{3} (red circles) and *U*_{4} (black triangles) calculated from all segments of the temperature profile at the example site (18.00°N, 113.02°E). The falling speed (green line) of the VMP is shown for comparison.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

Values of *U*_{3} (red circles) and *U*_{4} (black triangles) calculated from all segments of the temperature profile at the example site (18.00°N, 113.02°E). The falling speed (green line) of the VMP is shown for comparison.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

### d. Statistical comparison

In this section, we show the results of the calculated passing-probe speeds, *U*_{1}, *U*_{2}, *U*_{3}, and *U*_{4}, for all the 30 VMP profiles (Fig. 1), and compare them statistically.

Figure 8a shows the probability density functions (PDFs) of the normalized speeds, *U*_{1}/*U*_{VMP} (black circles), *U*_{2}/*U*_{VMP} (red circles), *U*_{3}/*U*_{VMP} (green circles), and *U*_{4}/*U*_{VMP} (blue circles) in a semilog coordinate system. Here the denominator *U*_{VMP} denotes the falling speeds of the VMP. It is found that all the four PDFs can be fitted well by using the lognormal model, with corresponding determination coefficient 0.96, 0.93, 0.93, and 0.90, respectively. The statistical results of *U*_{i}/*U*_{VMP} (*i* = 1, 2, 3, 4) are shown in Fig. 8b. Their mean values are respectively 1.32, 1.03, 1.56, and 1.43, corresponding standard-error ranges (68% probability range) are [0.99, 1.74], [0.88, 1.21], [0.81, 3.01], and [0.56, 3.64], and double-standard-error ranges (95% probability range) are [0.75, 2.29], [0.75, 1.42], [0.42, 5.76], and [0.22, 9.24]. The mean values of *U*_{i}/*U*_{VMP} are all close to 1, implying that the present approach for quantifying the flow speeds is statistically valid. By comparison, *U*_{2} agrees best with the falling speeds of the VMP.

(a) Probability density functions (PDFs) of the normalized speed, *U*_{1}/*U*_{VMP} (black circles), *U*_{2}/*U*_{VMP} (red circles), *U*_{3}/*U*_{VMP} (green circles), and *U*_{4}/*U*_{VMP} (blue circles) in a semilog coordinate system. The denominator *U*_{VMP} denotes the falling speeds of the VMP. The PDFs are fitted with lognormal model, as shown with corresponding colored lines. (b) Box-and-whisker plots of the normalized speeds *U*_{i}/*U*_{VMP} (*i* = 1, 2, 3, and 4). In each box-and-whisker plot, the bar through the box denotes the mean value, the bottom and the top ends of the box denote the standard error range (68% probability range), and the whiskers extend to the double standard error range (95% probability range). Their corresponding values are shown near the plots. The dashed line denotes *U*_{i}/*U*_{VMP} = 1.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Probability density functions (PDFs) of the normalized speed, *U*_{1}/*U*_{VMP} (black circles), *U*_{2}/*U*_{VMP} (red circles), *U*_{3}/*U*_{VMP} (green circles), and *U*_{4}/*U*_{VMP} (blue circles) in a semilog coordinate system. The denominator *U*_{VMP} denotes the falling speeds of the VMP. The PDFs are fitted with lognormal model, as shown with corresponding colored lines. (b) Box-and-whisker plots of the normalized speeds *U*_{i}/*U*_{VMP} (*i* = 1, 2, 3, and 4). In each box-and-whisker plot, the bar through the box denotes the mean value, the bottom and the top ends of the box denote the standard error range (68% probability range), and the whiskers extend to the double standard error range (95% probability range). Their corresponding values are shown near the plots. The dashed line denotes *U*_{i}/*U*_{VMP} = 1.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

(a) Probability density functions (PDFs) of the normalized speed, *U*_{1}/*U*_{VMP} (black circles), *U*_{2}/*U*_{VMP} (red circles), *U*_{3}/*U*_{VMP} (green circles), and *U*_{4}/*U*_{VMP} (blue circles) in a semilog coordinate system. The denominator *U*_{VMP} denotes the falling speeds of the VMP. The PDFs are fitted with lognormal model, as shown with corresponding colored lines. (b) Box-and-whisker plots of the normalized speeds *U*_{i}/*U*_{VMP} (*i* = 1, 2, 3, and 4). In each box-and-whisker plot, the bar through the box denotes the mean value, the bottom and the top ends of the box denote the standard error range (68% probability range), and the whiskers extend to the double standard error range (95% probability range). Their corresponding values are shown near the plots. The dashed line denotes *U*_{i}/*U*_{VMP} = 1.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1

## 4. Summary

In this study, we propose an approach to quantify the flow speeds past measurement probes by using the inertial and dissipation subranges of the shear [Eqs. (4) and (7)] and temperature [Eqs. (19) and (22)] frequency spectra, based on the turbulent spectral theories. Using the microstructure profiles collected from a cruise in the South China Sea during 2017, we introduce the calculation steps of the speeds in detail, and compare the calculated speeds with the actual passing-probe flow speeds, i.e., the falling speeds of VMP. The PDFs of the normalized speeds, i.e., the ratios of the calculated speeds to the actual ones, are fitted well with the lognormal model. The mean values of the normalized speeds are 1.32, 1.03, 1.56, and 1.43, with corresponding standard error ranges [0.99, 1.74], [0.88, 1.21], [0.81, 3.01], and [0.56, 3.64], respectively. The statistical results show that the present approach for quantifying the flow speeds is valid, and the speeds inferred from the dissipation subrange of shear spectrum agreed much better with the actual ones than those from the inertial subrange of shear spectrum and the inertial and dissipation subranges of temperature spectrum. The present work indicates that the flow speeds could be inferred using only the microstructure shear or temperature time-series data, and the proposed speed-quantification approach may be useful for determining the turbulent dissipation rate in the absence of directly measured speeds.

## Acknowledgments

This study was supported by the National Natural Science Foundation of China (91952106, 41776033, 91752108), the Science and Technology Foundation of Guangzhou (201904010312, 201804020056), the Key Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (GML2019ZD0304), and grant from the Institution of South China Sea Ecology and Environmental Engineering (ISEE2018PY05) and LTO (LTOZZ2001). We thank all the scientists and staff members on the Research Vessel *Shiyan 3* for their assistance in the Open Research Cruise of South China Sea during 2017. We also thank the HPC managers of the High Performance Computing Division in the South China Sea Institute of Oceanology.

## Data availability statement

The microstructure data used in this paper are available at https://figshare.com/articles/The_dataset_of_turbulent_microstructure_in_South_China_Sea_during_2017/9917255.

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