Quantifying Flow Speeds by Using Microstructure Shear and Temperature Spectral Analysis
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
b. Dissipation subrange of the shear spectrum
In Eq. (7), the frequency shear spectrum Du(f) could be converted by observed time series shear data with Eq. (1), but fυ is still unknown.
Another method for obtaining fυ is directly by fitting the observed shear spectrum Du(f) with the frequency Nasmyth spectrum Du_nasmy(f) [Eq. (9)]. The detailed processes for shear spectrum fitting will be described in section 3b.
c. Inertial subrange of the temperature spectrum
From Eq. (19) the speed can be directly calculated with the frequency temperature spectrum ET(f), which is obtained with the observed time series temperature data by using Eq. (11).
d. Dissipation subrange of the temperature spectrum
The Batchelor wavenumber kB is larger than the Kolmogorov wavenumber kυ for seawater [see Eq. (5)], which means that the temperature dissipation occurs at scales smaller than the smallest turbulent eddy size in oceans.
Another method to obtain fB 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 Ui = (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 U1, U2, U3, and U4 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.
Map of the field observation locations. The VMP casts are shown as red squares filled with green. The depth contour lines represent 200, 1500, and 3000 m.
Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0103.1
b. U1 and U2
We calculate U1 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 Du(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 Hs(f) = 1/[1 + (f/fc)2], where the sensitive frequency is fc = kcUaverage with Uaverage = 0.8 m s−1 for our dataset. The sensitive wavenumber kc depends on the probe’s geometry and dimensions (Bluteau et al. 2016a; Macoun and Lueck 2004), and we use kc = 49 cpm for Rockland Scientific’s shear probes (Macoun and Lueck 2004).
3) Step 3: Identifying the inertial subrange
(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 f1 = 1.2 Hz to f2 = 12 Hz with the power scaling αc = 0.31 for the fitting of Du(f) ~ fα. fcut-off = 141 Hz is the upper-limit frequency that has not been contaminated by the noise. (c) Calculated value of U1 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 U1
We calculate the flow speeds with Eq. (4) in the inertial subrange and take their median value as U1 for each vertical segment (as shown in Fig. 2c).
Figure 2 shows an example of how we obtain U1 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 fcut-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 f1 = 1.2 Hz to f2 = 12.0 Hz with the power scaling αc = 0.31 for the fitting of Du(f) ~ fα. Figure 2c shows the results calculated for the passing-probe speeds U1 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 αc and calculated values of U1 for all segments of the example profile shown in Fig. 2a. In Fig. 3a it is found that αc ranging between 0.27 and 0.4 (red circles) mainly locates in the upper 60 m for this profile. In the deeper depths, αc generally deviates away from the theoretical power scaling of 1/3. The ratio of the theoretical upper and lower limits of the inertial subrange,
(a) Power scaling αc closest to 1/3 for all vertical segments of the example profile shown in Fig. 2a. The three vertical dashed lines denote αc = 0.27, 1/3, and 0.4 from left to right, respectively. The red circles denote the segments where αc satisfies the acceptance criterion [Eq. (27)]. (b) Calculated values of U1 for all vertical segments of the example profile. The values of U1 in the segments where αc 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 U2
There are three preprocessing steps before the calculation of U2 with Eq. (7). The first two steps are the same as those applied for obtaining U1. The third step is determining fυ. As mentioned in section 2b, two methods could be used to obtain fυ, one based on the peak of the observed frequency shear spectrum, fobs_peak, with Eq. (10), and the other by using the frequency Nasmyth spectrum Du_nasmy(f) [i.e., Eq. (9)] to fit the observed shear spectrum Du(f).
Figure 4 shows the processes to obtain U2 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, fpeak_obs = 14.1 Hz and fpeak_nasmy = 18.3 Hz. fpeak_nasmy is more accurate than fpeak_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 fpeak_obs are more scattered than those of fpeak_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 U2 calculated with fpeak_obs and fpeak_nasmy for all segments of the whole example profile are shown in Fig. 4c. It is found that the values of U2 calculated with fpeak_obs are highly scattered, whereas those obtained with fpeak_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 fpeak_obs = 14.1 and fpeak_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 U2 calculated with fυ obtained from fpeak_obs (black circles) and fpeak_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. U3 and U4
The values for U3 and U4 are respectively calculated based on the inertial and dissipation subranges of the frequency temperature spectrum ET(f) with Eqs. (19) and (22), and their preprocessing steps are some similar to those of U1 and U2. Here we emphatically describe two points, which do not exist or are different for the calculation of U1 and U2.
The first one is the background parameter C. As described in sections 2c and 2d, U3 and U4 are derived under the condition that the turbulent diffusivity of the temperature and density are equal, which introduces a background parameter,
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 HT(f) = 1/[1 + (f/fh)2], where the half-attenuation frequency,
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 αc = 1.63 for ET(f) ~ fα denotes the inertial subrange ranging from f1 = 2.2 Hz to f2 = 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 fcut-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 f1 = 2.2 Hz to f2 = 7.4 Hz, with the power scaling of αc = −1.63 for the fitting of ET(f) ~ fα. fcut-off = 115 Hz is the upper-limit frequency below which the spectrum is not contaminated by noise. (b) Reduced frequency temperature dissipation spectrum of
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,
Figure 7 shows U3 (red circles) and U4 (black triangles) for the whole profile at the example site (18.00°N, 113.02°E). U3 and U4 are respectively calculated with Eqs. (19) and (22) for the inertial and dissipation subranges of the temperature spectrum ET(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 U4 obtained with the peak frequency of the fitted Kraichnan spectrum. It is found that U3 and U4 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, U1 and U2, are generally in better agreement with the VMP falling speed than those obtained from the temperature spectrum, U3 and U4.
Values of U3 (red circles) and U4 (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, U1, U2, U3, and U4, for all the 30 VMP profiles (Fig. 1), and compare them statistically.
Figure 8a shows the probability density functions (PDFs) of the normalized speeds, U1/UVMP (black circles), U2/UVMP (red circles), U3/UVMP (green circles), and U4/UVMP (blue circles) in a semilog coordinate system. Here the denominator UVMP 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 Ui/UVMP (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 Ui/UVMP are all close to 1, implying that the present approach for quantifying the flow speeds is statistically valid. By comparison, U2 agrees best with the falling speeds of the VMP.
(a) Probability density functions (PDFs) of the normalized speed, U1/UVMP (black circles), U2/UVMP (red circles), U3/UVMP (green circles), and U4/UVMP (blue circles) in a semilog coordinate system. The denominator UVMP 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 Ui/UVMP (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 Ui/UVMP = 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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