On the Variation of Dissipation Flux Coefficient in the Upper South China Sea

a. Data

The data used in this study were collected from six cruises in the SCS from April 2004 to April 2011 (Fig. 1). The profiles of microscale velocity fluctuation and fast temperature were obtained by using the Turbulence Ocean Microstructure Acquisition Profiler (TurboMAP; Wolk et al. 2002), which was loosely tethered and nearly free-falling with a speed between 0.5 and 0.7 m s⁻¹ at a sampling rate of 512 Hz (256 Hz at a few stations). Microstructure measurements were obtained at 175 stations, some of which were repeatedly occupied, and a total of 267 casts were available (red squares in Fig. 1). Among these stations, continuous measurements were performed over two days (from 2130 LT 10 May to 0200 LT 13 May 2010) at a station in central SCS with a sampling interval of approximately 1 h (black circle in Fig. 1); 50 casts were obtained at this station after excluding two invalid measurements.

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Station locations (red squares) and bathymetry of the SCS. The black contour marks the isobath of 200 m, which distinguishes the stations used in this study (solid red squares) from those excluded (hollow red squares). The gray box marks the area west of the Luzon Strait, representing region E of the SCS, and the rest of the stations represent region Q. The repeatedly occupied station in the central SCS is marked by a black circle. Topography data are from the ETOPO1 dataset (https://ngdc.noaa.gov/mgg/global/global.html).

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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Microstructure measurements at all stations were limited in the upper layer, with the maximum sampling depth approximately 550 m. At shallow waters, the measurements may be influenced by various dynamic processes near the seabed, and the definition of Γ may not hold near the bottom boundary layer (Mashayek et al. 2017). Therefore, the stations with water depth shallower than 200 m were excluded from data analysis, which included 17 stations and 34 casts (hollow red squares in Fig. 1).

b. Turbulent patch identification

Pioneered by Moum (1996a,b), turbulence-related analyses focus on clear turbulent patches, and many identification techniques were developed. Here, following Mater et al. (2015) and Ijichi and Hibiya (2018), we use the cumulative Thorpe displacement ${\displaystyle \sum {\delta }_T}$ to identify turbulent patches, where the Thorpe displacement δ_T is the depth difference of a water parcel between the original and sorted potential temperature profiles. An example of turbulent patch identification is presented in Figs. 2a–c; the original and sorted potential temperature profiles are coincident at depths of 64.0–65.6 and 73.4–75.0 m, respectively, where the water columns are statically stable, and the corresponding δ_T and ${\displaystyle \sum {\delta }_T}$ remain zero. In contrast, the water column between 65.6 and 73.4 m is significantly overturned, and the values of δ_T and Σδ_T become nonzero and show strong vertical variation. Although temperature, δ_T, and ${\displaystyle \sum {\delta }_T}$ can all indicate the occurrences of overturns, in some complicated cases (e.g., Figs. 2d–f), it is not easy to determine the bounds of a turbulent patch using temperature and δ_T (Figs. 2d,e). Fortunately, ${\displaystyle \sum {\delta }_T}$ remains nonzero within the patch and becomes zero outside the patch (Fig. 2f); as a result, it is adequate to identify turbulent patches.

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Two examples illustrating how the turbulent patch is identified by ${\displaystyle \sum {\delta }_T}$. (a),(d) The vertical profiles of the original and sorted potential temperature θ, (b),(e) δ_T, and (c),(f) ${\displaystyle \sum {\delta }_T}$.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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The profiles of fast temperature are used to identify turbulent patches. Before identifying turbulent patches, the vertical resolution of the fast-temperature profiles is lowered to 0.1 m to remove fake overturns due to instrument vibration and sensor electronic noise. The measurements in the upper mixed layer are not taken into account, since the falling speed of TurboMAP is not yet stable, and the active mixing events within that layer are mostly convection driven due to the loss of buoyancy flux, and the definition of Γ does not hold anymore. We identify and then exclude the upper mixed layer by employing a threshold method with a relatively large temperature threshold of 0.7°C. This threshold may overestimate the thickness of the upper mixed layer for some stations/casts, but it guarantees the upper mixed layer is thoroughly removed.

To make the results more robust, two criteria are applied to remove some turbulent patches. First, the turbulent patches with vertical size smaller than 1 m are excluded to avoid data distortion. Second, the turbulent patches with −1/2 ≤ R_ρ ≤ 2 are excluded to avoid possible contamination by a strongly salinity-stratified layer and density-compensated intrusion (St. Laurent and Schmitt 1999). Here, $R_{\rho }=\alpha ⟨{\theta }_z⟩/\beta ⟨S_z⟩$ is the density ratio, where α is the thermal expansion coefficient, β is the saline contraction coefficient, θ_z and S_z are vertical gradients of potential temperature and salinity, respectively, and the angle bracket denotes an average over each turbulent patch. As R_ρ and Turner angle Tu (an indicator of water-column stability to double diffusion) are related by R_ρ = −tan(Tu + 45°) (Ruddick 1983), this criterion also avoids the influences of diffusive convection (−90° < Tu < −45°, 0 ≤ R_ρ ≤ 1) and strong salt finger (75° < Tu < 90°, 1 ≤ R_ρ ≤ 2; You 2002).

As a result, a total of 9641 turbulent patches are identified by examining ${\displaystyle \sum {\delta }_T}$ calculated from the fast-temperature profiles. By applying the two criteria, 54.7% of turbulent patches are excluded (corresponding to 5271 patches, among which 5068 were excluded for inappropriate R_ρ, 44 for patch size smaller than 1 m, and 159 for both), and 4370 patches remain for analysis.

c. Calculations of Γ, Re_b, and R_OT

The most widely used method to estimate Γ is based on the assumption that K_ρ is equal to the thermal diffusivity K_T. Using the fast-temperature measurements, K_T is inferred from $K_T={\chi }_T/(2{\theta }_z^2)$ (Osborn and Cox 1972). By combining K_ρ = Γε/N² and $K_T={\chi }_T/(2{\theta }_z^2)$ and considering an average over a turbulent patch, an instantaneous patch-averaged Γ is obtained as Γ = 〈χ_T〉〈N²〉/(2〈ε〉〈θ_z〉²) for each turbulent patch. The two parameters, Re_b and R_OT, are subsequently estimated as Re_b = 〈ε〉/(ν〈N²〉) and R_OT = L_O/L_T, where L_O = 〈ε〉^1/2/〈N〉^3/2 is the Ozmidov scale and $L_T={⟨{\delta }_T^2⟩}^{1/2}$ is the Thorpe scale.

The method we use to calculate ε and χ_T is different from previous studies. To simplify, we take the calculation of ε as an example. Assuming isotropic turbulence and using the Taylor’s frozen assumption, the vertical gradient of velocity u_z was derived from a temporal gradient of velocity u_t with a falling speed w as u_z = u_t/w. The vertical wavenumber spectra of u_z, φ(k), was then calculated every 2 s with an overlapping of 1 s. In this study, we did not use the often-employed iterative method (e.g., Wolk et al. 2002) to determine the upper integration limit, the Kolmogorov wavenumber k_K, and hence ε, but provided a series of Nasmyth spectra φ_N(k) corresponding to ε_N = 10^e W kg⁻¹ (besides dissipation rate, it requires kinematic viscosity to determine a Nasmyth spectrum, which can be calculated by temperature and salinity; Nasmyth 1970; Oakey 1982; Wolk et al. 2002), where the exponent e varies from −12 to −4 at an interval of 0.1 (gray and blue curves in Fig. 3a). Next, the difference between the observed spectra φ(k) and each Nasmyth spectrum φ_N(k), D_φ, was integrated in the wavenumber range lower than k_K, where k_K was determined by the corresponding ε_N as k_K = (ε_N/ν³)^1/4/2π for each φ_N(k). And ε was finally chosen as ε_N of the specific φ_N(k) with the minimum D_φ. The procedure to calculate χ_T using temporal gradient of fast temperature is similar to that of calculating ε, except that we provided a series of Batchelor spectra ψ_B(k) corresponding to χ_TB = 10^e °C² s⁻¹ (Fig. 3b). The upper integration limit of D_ψ, the Batchelor wavenumber k_B = (ε/νκ²)^1/4/2π, is fixed when ε is determined, and κ is the thermal molecular diffusivity.

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Examples of (a),(e) vertical wavenumber spectra of velocity gradient φ(k) (red curve) and of (c) temperature gradient ψ(k) (red curve), together with Nasmyth spectra φ_N(k) [gray, blue, and yellow curves in (a) and (e); blue has minimum D_φ and yellow is the result estimated by the iterative method] and Batchelor spectra ψ_B(k) [gray, blue, and yellow curves in (c); blue has minimum D_ψ and yellow is the result estimated by the iterative method]. (b),(d),(f)The integrated differences D_φ (or D_ψ) between φ(k) and φ_N(k) [or ψ(k) and ψ_B(k)] with varying ε_N (χ_TB). The gray dots in (a) and (e) mark k_K of given ε_N, whereas k_K and k_B of observed ε and χ_T are denoted by the cyan and green lines, respectively.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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As suggested by the examples shown in Fig. 3, the values of D_φ and D_ψ are sensitive to the variations of ε_N and χ_TB (Figs. 3b,d): they reach their minima when φ(k) and ψ(k) are well fitted by specific φ_N(k) and ψ_B(k), respectively, suggesting ε = 2.0 × 10⁻⁹ W kg⁻¹ and χ_T = 1.0 × 10⁻⁸°C² s⁻¹ in this case (Figs. 3a,c); when ε_N and χ_TB deviate from these values, D_φ and D_ψ grow exponentially to show the increasing bias between the observed ε (χ_T) and given ε_N (χ_TB). This new method, namely, the difference integration method, gives more accurate estimates than the iterative method when φ(k) cannot be well fitted by φ_N(k). As shown in Fig. 3e, φ(k) in a wavenumber range lower than 7 cpm is significantly inconsistent with any given φ_N(k). The iterative method is highly impacted by the uplift of φ(k) in the low-wavenumber range, resulting in an overestimate of ε as 7.2 × 10⁻⁹ W kg⁻¹. While the difference integration method takes all wavenumbers smaller than k_K into account and gives a smaller estimate as 1.3 × 10⁻⁹ W kg⁻¹, whose spectrum fits much better with φ(k) in the wavenumber range between 10 cpm and k_K, 33 cpm; D_φ of difference integration method (31) is also smaller than that of iterative method (67), suggesting result of the new method is closer to the truth. Considering the calculation procedure is performed twice for ε and χ_T, and the estimate of χ_T is on the basis of the calculation of ε, using the difference integration method could significantly improve the accuracy of the results of “anomalous” spectra. A detailed comparison between these two methods and their influences on the results can be found in the appendix.

The terms N² and θ_z are also estimated by the fast-temperature measurements after the resolution is lowered to 0.1 m. As demonstrated in Smyth et al. (2001) and reaffirmed in Ijichi and Hibiya (2018), 〈θ_z〉 should be examined in different ways for proper estimates of Γ and L_O (Figs. 2a,d): the average one 〈θ_z〉_mean inferred from the linear fitting of the sorted potential temperature should be used to calculate Γ, while the bulk one 〈θ_z〉_bulk = 〈θ′²〉^1/2/L_T, where θ′ is the difference between the original and sorted potential temperatures, should be used to calculate L_O. For turbulent patches consisting of a single, large overturn, as in the example shown in Fig. 2a, 〈θ_z〉_mean and 〈θ_z〉_bulk are almost identical, being 0.040 and 0.041°C m⁻¹, respectively. However, when a turbulent patch contains several overturns separated by thin stable layers (Fig. 2d), those thin stable layers are included to calculate 〈θ_z〉_mean but are largely ignored to obtain 〈θ_z〉_bulk (θ′ is much smaller), thus 〈θ_z〉_mean turns out to be greater than 〈θ_z〉_bulk. In the case presented in Fig. 2d, 〈θ_z〉_mean is twice as large as 〈θ_z〉_bulk, being 4.6 × 10⁻³ and 2.3 × 10⁻³ °C m⁻¹, respectively.

a. Horizontal variation of Γ

Near the Luzon Strait, a large number of vigorous internal tides are generated (Klymak et al. 2011), and those of higher modes break and dissipate locally to furnish elevated mixing (Niwa and Hibiya 2004; Tian et al. 2009; Yang et al. 2016; Shang et al. 2017; Lu et al. 2021). The low-mode internal tides continue to propagate southwestward and enter the SCS, becoming very weak when arriving at the southwestern SCS (Zhao 2014), accompanied by low-level turbulent mixing (Yang et al. 2016; Lu et al. 2021). In this view, we divided the SCS into two regions, the area west of the Luzon Strait (114°–121°E, 17°–23°N), featured with energetic turbulent mixing (region E; marked by a gray box in Fig. 1), and the rest of the SCS deeper than 200 m as the quiescent region (region Q). We explore the features of Γ in these two regions based on 1227 turbulent patches identified from 74 casts in region E and 3143 patches from 145 casts in region Q.

In both regions Γ varies over several orders of magnitude (Fig. 4a); Γ scatters in a larger range in region Q than in region E, with the former from 1 × 10⁻³ to 15.0 and the latter from 1 × 10⁻² to 6.9. This observed variation range of Γ between 10⁻³ and 10¹ is similar to various observations conducted in different regions (e.g., Smyth et al. 2001; Ijichi and Hibiya 2018; Ijichi et al. 2020; Vladoiu et al. 2021; Masunaga et al. 2022). The median (mean) value of Γ is 0.23 (0.39) in region E, larger than in region Q, 0.17 (0.30). The distributions of Γ in both regions are slightly positively skewed with skewness of 0.11 in region E and 0.09 in region Q. In region E, Γ follows a nearly mesokurtic distribution with excess kurtosis of 0.09, while that in region Q is a little leptokurtic, with excess kurtosis of 0.71. The maximum, median, and mean values of ε in region E (2.7 × 10⁻⁷, 4.8 × 10⁻¹⁰, and 3.4 × 10⁻⁹ W kg⁻¹, respectively) are all greater than those in region Q (1 × 10⁻⁷, 8.1 × 10⁻¹¹, and 7.9 × 10⁻¹⁰ W kg⁻¹, respectively; Fig. 4b). The probability distribution of ε in region Q is clearly positively skewed (skewness 1.05) and leptokurtic (excess kurtosis 0.66), and it presents a distinct peak at 4.6 × 10⁻¹¹ W kg⁻¹, while that in region E is mildly positively skewed (skewness 0.39) and platykurtic (excess kurtosis −0.36), with high probability of ε ranging from 4.2 × 10⁻¹¹ to 2.5 × 10⁻⁹ W kg⁻¹, indicating region E experiences a stronger dissipation than region Q. The difference of χ_T between the two regions is similar to that of ε (Fig. 4c), and the maximum, median, and mean values of χ_T are all greater in region E (7.0 × 10⁻⁷, 1.4 × 10⁻⁹, and 9.0 × 10⁻⁹°C² s⁻¹, respectively) than those in region Q (5.1 × 10⁻⁷, 2.0 × 10⁻¹⁰, and 2.0 × 10⁻⁹°C² s⁻¹, respectively). The distributions of χ_T in regions E and Q are both platykurtic (excess kurtosis −0.35 and −0.34, respectively), whereas the latter is more positively skewed than the former, with skewness of 0.57 and 0.13, respectively.

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Histograms normalized by probability of (a) Γ, (b) ε, (c) χ_T, (d) N², (e) Re_b, and (f) R_OT in regions E (red curve) and Q (blue curve). The dots and triangles mark the median and mean values of the corresponding curve, respectively. The gray dashed line in (a) denotes Γ = 0.2.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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In contrast to the distinct features of ε and χ_T between regions E and Q, the stratification distributions in both regions are similar (Fig. 4d), and their median and mean values are nearly identical, 3.3 × 10⁻⁵ and 4.7 × 10⁻⁵ s⁻², respectively, in region E and 3.1 × 10⁻⁵ and 4.6 × 10⁻⁵ s⁻², respectively, in region Q, although showing slight differences in probability distribution. The probability distribution of N² in region E is slightly negatively skewed and platykurtic (skewness −0.18, excess kurtosis −0.16), whereas that in region Q is positively skewed and leptokurtic (skewness 0.20, excess kurtosis 0.13). Vladoiu et al. (2021) found that energetic and quiescent regions in the western Mediterranean Basin are contrasted in both dissipation and stratification, while our results suggest the energetic and quiescent regions in the upper SCS are distinguished by distinct dissipation with similar stratification. The probability distributions of Re_b and R_OT in region E are distinct from those in region Q (Figs. 4e,f), as they are directly derived from distinct ε and similar N². In region Q, Re_b scatters in a range from 1.6 × 10⁻¹ to 3.6 × 10³, with median and mean values of 4.0 and 31.8, respectively, which are all smaller than those in region E (from 3.1 × 10⁻¹ to 6.7 × 10⁴, and 19.4 and 234.6). The distributions of Re_b in both regions are positively skewed, with skewness of 0.38 in region E and 0.83 in region Q; the excess kurtoses are −0.18 and 0.49, respectively, suggesting the former is platykurtic while the latter is leptokurtic. The scatters of R_OT are from 2.7 × 10⁻³ to 5.8, with median and mean values of 0.21 and 0.35 in region Q versus from 9.7 × 10⁻³ to 11.0, with 0.42 and 0.69 in region E. And their skewnesses (excess kurtoses) are 0.09 and −0.04 (1.63 and 0.30), respectively. The variation ranges of Re_b and R_OT in the upper SCS are generally consistent with those found in previous observations (Smyth et al. 2001; Ijichi and Hibiya 2018; Ijichi et al. 2020; Lu et al. 2021; Vladoiu et al. 2021; Masunaga et al. 2022), except that some small values (Re_b < 1 and R_OT < 0.1) are captured in our study. All these differences between regions E and Q validate that turbulent mixing in region E is more energetic and efficient than that in region Q.

b. Vertical variation of Γ

Ijichi and Hibiya (2018) observed a clear tendency of Γ increasing from surface to 3000 m in the western and central North Pacific Ocean and the Southern Oceans. Smyth (2020) suggested Γ should be around 0.2 away from the upper and lower boundaries, mingled with high-efficiency coherent structures. The variation of Γ with depth is related to the prevalence of coherent structure, which encourages us to explore the vertical variation of Γ in the upper SCS.

The vertical variations of N² in regions E and Q are characterized by three layers (Fig. 5): the thermocline with strongest stratification N² > 1.6 × 10⁻⁴ s⁻² (black contours), a moderate layer with stratification 1.6 × 10⁻⁴ s⁻² ≥ N² ≥ 4.0 × 10⁻⁵ s⁻² (white contours), and the deeper less-stratified layer with N² < 4.0 × 10⁻⁵ s⁻². The thermocline can be observed in all casts in region E, while it is shallow and weak in some casts in region Q. In general, the thermocline depth varies between 100 and 200 m, and the moderate layer depth varies between 200 and 400 m; as a result, we use 150 and 300 m to separate the observed water column into three layers.

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Vertical variation of N² (s⁻²) within the upper 500 m in region (a) E and (b) Q. The black and white contours denote the N² values of 1.6 × 10⁻⁴ and 4 × 10⁻⁵ s⁻², respectively.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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Figure 6 shows the vertical variations of Γ in regions E and Q, along with other turbulence-related variables. The median values of Γ, binned by depth, and their linear fitting present a clear increasing trend with increasing depth in both regions (Fig. 6a). The median values of Γ at 75 m are 0.1 for both regions, and they increase to 0.25 and 0.48 at 500 m in regions Q and E, respectively. The increasing rate of Γ with depth in region E is 2 times higher than that in region Q, being 1.4 × 10⁻³ and 8.2 × 10⁻⁴ m⁻¹ (in logarithmic space), respectively. There are some detailed vertical structures of Γ in different layers: in region Q, Γ increases in a slightly weaker rate within the moderate layer than that in the thermocline and less-stratified layer; in region E, Γ keeps increasing in the thermocline and the moderate layer but shows a somewhat decreasing trend in the less-stratified layer, such as at depth deeper than 450 m. The variation range of Γ (represented by the 5th and 95th percentiles) also shifts to larger magnitude with increasing depth. Within the thermocline, Γ in region Q varies over a wider range than that in region E, with the 5th percentile being smaller and the 95th percentile being larger in region Q than in region E. Beneath the thermocline, Γ in region E varies in a range of larger magnitude than that in region Q, as suggested by the 5th and 95th percentiles in region E being both lager than those in region Q. In general, turbulent mixing tends to become more efficient with increasing depth in both regions; however, in the water column deeper than 150 m, it has a stronger ability to elevate the potential energy of seawater in region E than in region Q. This increasing trend of Γ with increasing depth is similar to the result of Ijichi and Hibiya (2018); Ijichi et al. (2020) also suggested young instabilities with higher Γ are more common at depth due to the weaker stratification. Moreover, Smyth (2020) revealed that highly efficient, long-lasting coherent structures are responsible for intermittent higher Γ in the interior.

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Vertical variations of (a) Γ, (b) ε, (c) χ_T, (d) N², (e) Re_b, and (f) R_OT of each turbulent patch in regions E (small magenta dots) and Q (small cyan dots). The median values binned by depth and their associated linear fittings are marked by big dots and lines, respectively, with red representing region E and blue representing region Q. The 5th and 95th percentiles are marked by dash–dotted lines to outline the variation ranges. The horizontal gray dashed lines in each panel mark the bottom depths of the thermocline and the moderate layer, 150 and 300 m, respectively, and the vertical gray dashed line in (a) denotes Γ = 0.2. The black dots denote those patches with R_OT smaller than 0.05 and depth shallower than 150 m. The turbulent patch numbers of each depth bin are presented in (e) for region Q and (f) for region E.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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The terms ε and χ_T share similar vertical variations (Figs. 6b,c). In general, they decrease with depth in both regions. In region Q, ε and χ_T weaken sharply within the thermocline and the moderate layer, with ε varying from O(10⁻⁹) at 50 m to O(10⁻¹¹) W kg⁻¹ at 220 m and χ_T varying from O(10⁻⁹) at 50 m to O(10⁻¹⁰) °C² s⁻¹ at 260 m, respectively. It contrasts sharply with the scenario in the less-stratified layer in region Q, where ε and χ_T vary slightly and have the same order; ε is basically identical to 7 × 10⁻¹¹ W kg⁻¹, and χ_T is mostly O(10⁻¹⁰) °C² s⁻¹. The vertical variability of ε and χ_T in region E is similar to that in region Q, although the magnitude is 3–5 times larger in region E than in region Q at different depths. The variation range of ε in region Q becomes narrower with increasing depth, whereas that in region E shows no clear change.

The stratifications in both regions decrease logarithmically with depth from O(10⁻⁴) to O(10⁻⁵) s⁻² (Fig. 6d), presenting no substantial difference. The vertical variation of median values of Re_b suggests it generally decreases with increasing depth (Fig. 6e), which varies from 13.3 to 3.6 between 70 and 500 m in region Q and from 27.5 to 16.6 in region E. In region Q, the decrease of Re_b is more significant in the thermocline, which varies from 13.3 to 4.0, and it remains at that value beneath the thermocline. Note that the slight decreasing tendency of Re_b with increasing depth here is due to the relatively faster decreasing of ε (about two orders in magnitude from 50 to 550 m) compared with the relatively slower decreasing of N² (about one order of magnitude; Figs. 6b,d). However, the decreasing of Re_b in the upper 550 m is opposite to the general variation of Re_b when considering the full water depth, where Re_b increases several orders of magnitude with increasing depth (Mashayek et al. 2017; Ijichi et al. 2020). The larger Re_b in region E than in region Q clearly suggests a higher-level turbulence intensity near the Luzon Strait.

The median values of R_OT in region Q (region E) show a consistent decreasing tendency between 70 and 500 m from 0.37 to 0.13 (from 0.72 to 0.22; Fig. 6f). Beneath the thermocline, R_OT in region Q varies in a narrower range than that in region E, both of which have lower limit of about 10⁻¹. However, the lower bounds of variation range in the thermocline in both regions bend obviously toward smaller R_OT as a result of the intensive occurrences of turbulent patches with R_OT smaller than 0.05, which are very rare beneath the thermocline. These extremely small R_OT may be related to the life-span of turbulence and are discussed in section 3c in detail. In summary, there is a clear decreasing variability for ε, χ_T, N², Re_b, and R_OT with increasing depth, although at different rates; while Γ increases with depth at a greater increasing rate in region E compared with that in region Q.

c. Variation of Γ related to the life-span of turbulence

We found 75 turbulent patches with R_OT smaller than 0.05 within the thermocline in both regions (black dots in Fig. 6; green dots with black edges in Fig. 7), which have the smallest magnitude of R_OT compared with the other turbulent patches at all depths. As the Kelvin–Helmholtz instability is believed to be the main mechanism of turbulence generation in the stratified, sheared ocean (Smyth et al. 2001; Mashayek et al. 2017) and R_OT can be used to indicate the life-span of such turbulent patch (Smyth et al. 2001; Mater et al. 2015), we infer these patches are in an “initial” stage of turbulence evolvement. We noticed that turbulent patches with extremely small R_OT were also captured in Mater et al. (2015), which were driven by different mechanisms away from the thermocline. They demonstrated that R_OT can be used as an indicator of turbulence age not only for Kelvin–Helmholtz instability but also for other large-scale instabilities such as convectively driven turbulence.

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Differences of Γ and ε between the young (R_OT < 0.5; green dots) and mature stage (R_OT > 0.5; yellow dots) of turbulent patches in regions (a) E and (b) Q. The green and yellow squares with black edges are the median values of Γ and ε from the patches in the young and mature stage, respectively, and the dots with black edge are those patches with R_OT < 0.05 and depth shallower than 150 m.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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The definition of an initial stage of turbulence life-span is put forward on the basis of the “young” and “mature” stages (Smyth et al. 2001; Mashayek et al. 2017); therefore, it is necessary to verify if the turbulence behaviors in young and mature stages are valid for our observations. Turbulent patches with R_OT < 0.5 are regarded as the young stage, whereas patches with R_OT > 0.5 are considered as the mature stage (Smyth et al. 2001). Although this criterion is based on numerical simulations with a low Reynolds number, in which the evolution of turbulence might differ from that in the real ocean, we tried to use it to check if the young and mature turbulences can be distinguished in our field observations. As shown in Fig. 7, the observed turbulent patches with R_OT < 0.5 are typically distinguished from those with R_OT > 0.5 by low ε and high Γ. In region E, the median values of ε and Γ from the turbulent patches with R_OT < 0.5 and R_OT > 0.5 are (1.5 × 10⁻¹⁰ W kg⁻¹, 0.30) and (1.9 × 10⁻⁹ W kg⁻¹, 0.16), respectively, while those in region Q are (6.0 × 10⁻¹¹ W kg⁻¹, 0.18) and (1.2 × 10⁻⁹ W kg⁻¹, 0.12), respectively. These differences are consistent with previous conclusions on the young and mature stages of turbulence, that is, the young-stage turbulence is featured with high efficiency and low dissipation because it represents the two-dimensional Kelvin–Helmholtz billow that overturns the density interfaces (high efficiency) before the turbulent, three-dimensional secondary instabilities are fully developed (low dissipation). This consistency suggests young and mature stages of turbulence can be well identified in our observations.

The turbulent patches with extremely small R_OT in the thermocline are highlighted by black edges in Fig. 7. Compared with the other turbulent patches in the young stage, they present significantly higher mixing efficiency, with Γ ≈ 1 in region E and Γ ranging from 0.32 to 6.85 in region Q, which are remarkably greater than the median values of young-stage turbulent patches of 0.30 and 0.18, respectively. As for ε, its median values (1.7 × 10⁻⁹ W kg⁻¹ in region E and 5.7 × 10⁻¹⁰ W kg⁻¹ in region Q) are higher than those of young-stage turbulent patches [O(10⁻¹⁰) W kg⁻¹]. However, if examined in the same depth range, the ε values of those patches are smaller than the median values of ε (2.5 × 10⁻⁹ W kg⁻¹ in region E and 1.3 × 10⁻⁹ W kg⁻¹ in region Q; Fig. 6b). That is to say, these patches are featured with smaller R_OT, smaller ε, and greater Γ; therefore, they are definitely “younger” than those turbulent patches in the young stage and hence are defined as initial stage. The initial stage refers to the very beginning stage of turbulence, when the relatively large Kelvin–Helmholtz billow is just formed, and upcoming smaller secondary instabilities are yet to be developed. This is confirmed by the temperature examples shown in Figs. 8a–d. The unsorted temperature profiles suggest the occurrence of overturns. Compared with those of young and mature turbulent patches (e.g., Fig. 2d), these profiles are significantly uncomplicated, containing only several eddies with relatively large vertical scales (represented by “S” structures). Although these profiles are obtained at different stations (Fig. 8e) and dates, the initial Kelvin–Helmholtz billow shed more eddies (from 1 to 3) with R_OT increasing from 0.009 to 0.026 (Figs. 8a–d). These structure evolvements, together with the increasing R_OT, are consistent with the beginning stage of the evolution of Kelvin–Helmholtz billows (Fig. 1 of Smyth et al. 2001).

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(a)–(d) Examples of temperature profile of the turbulent patches in the initial stage, and (e) the station locations where these patches are observed. The black curves in (a)–(d) are the sorted temperature profiles, whereas the colorful ones are the original temperature profiles; the station locations of these examples are marked by corresponding colors in (e). In (e), the number to the left of the station indicates the number of initial-stage patches at that station.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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It seems that these initial-stage turbulent patches have a unique spatial distribution. Vertically, they are observed within the thermocline, suggesting their energy source is potentially the near-inertial energy input from the upper mixed layer induced by surface wind. Besides, among the total 75 initial-stage patches, only 7 are observed in region E, indicating the initial stage is more easily captured in region Q than in region E. A possible explanation for this regional difference is that the turbulence in region E may require much shorter time to experience its whole lifespan compared with that in region Q, because very energetic internal waves in region E promote turbulence evolving much faster, leading to a relatively shorter transient initial stage and hence is not frequently observed.

d. Dependences of Γ on R_OT and Re_b

Because more turbulent patches were detected in region Q and hence can give a more representative and solid conclusion, we demonstrate the relationships of Γ with Re_b and R_OT in region Q in detail (Fig. 9). The results in region E are also given in Fig. 10, showing a similar pattern, except for some quantitative discrepancies.

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Variation of Γ (a),(b) with Re_b and (c),(d) R_OT in region Q. The general variations of Γ with Re_b and R_OT are examined by considering all turbulent patches as a whole in (a) and (c), although they are colored by R_OT and Re_b, respectively. The black dots in (a) and (c) are the median values of Γ binned by Re_b (R_OT), and the black vertical bars represent the corresponding 10th–90th percentile ranges. The red curves in (a) and (c) represent the parameterizations proposed by Mashayek et al. (2017) and Ijichi and Hibiya (2018), respectively. The solid black line in (c) is the linear fitting of the median values of Γ binned by R_OT, while the dashed lines are linear fittings in different R_OT ranges. The variations of Γ with Re_b and R_OT within specific ranges of R_OT and Re_b are presented in (b) and (d), respectively. The dots with black edges and overlying lines denote the median of Γ binned by Re_b or R_OT and the corresponding linear fittings, respectively, with colors indicating different ranges of R_OT and Re_b; fitting presenting slope distinct from 1/2 or −4/3 is marked by dashed line. The red line in (d) represents the parameterization proposed by Ijichi and Hibiya (2018). The gray line in each panel marks Γ = 0.2.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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As in Fig.9, but for region E.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0127.1

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The general variations of Γ with Re_b and R_OT are evaluated separately (Figs. 9a,c). The median values of Γ vary in a range of 0.13–0.25, with Re_b increasing from 0.5 to 400, which can almost be taken as a constant when compared with Γ spanning over several orders of magnitude (Fig. 9a). When Re_b is greater than 400, Γ drops to about 0.1, which may suggest the decrease of Γ for very strong turbulence intensity, as reported in previous studies (Mashayek et al. 2017; Monismith et al. 2018; Vladoiu et al. 2021), although the insufficient sample size brings bias to this result. This decrease of Γ is not seen in region E, even when Re_b exceeds 1000 (Fig. 10a). There is a nonmonotonic relationship between Γ and Re_b. The variation of Γ within the range of Re_b between 1 and 100 fits especially well with the parameterization formula $\Gamma ({\mathrm{Re}}_b)=2{\Gamma }^{*}{({\mathrm{Re}}_b/{\mathrm{Re}}_b^{*})}^{1/2}/[1+({\mathrm{Re}}_b/{\mathrm{Re}}_b^{*})]$ proposed by Mashayek et al. (2017), where Γ^* is the maximum of Γ at ${\mathrm{Re}}_b={\mathrm{Re}}_b^{*}$. The fitting does not work well when Re_b < 1 and Re_b > 100, which may be due to sample size being inadequate in these ranges. Ιn region Q, Γ^* and ${\mathrm{Re}}_b^{*}$ are 0.22 and 16, and Γ^* = 0.22 is comparable to the lower bound of the datasets reported in Mashayek et al. (2017), while ${\mathrm{Re}}_b^{*}$ = 16 is much smaller than O(10²) in Mashayek et al. (2017). The median of Γ decreases by an order of magnitude with increasing R_OT (Fig. 9c). This negative dependence of Γ on R_OT has been reported in DNS and field observations (e.g., Smyth et al. 2001; Mashayek et al. 2017; Ijichi and Hibiya 2018; Ijichi et al. 2020). Some of those results match well with a simple relationship derived from theoretical scaling (Ijichi and Hibiya 2018), $\Gamma \propto R_{\text{OT}}^{-4/3}$, whereas our results indicate $\Gamma \propto R_{\text{OT}}^{-0.4}$, presenting a gentler decreasing trend. However, if we focus on the ranges for R_OT < 0.2 and R_OT > 1, Γ decreases with R_OT with a slope close to −4/3. It is notable that for 0.2 < R_OT < 1, Γ changes little with R_OT with a slightly increasing tendency. The scenario in region E is similar to that in region Q (Figs. 10a,c), but the quantitative discrepancies exist, such as the fitting curve is with (Γ^*, ${\mathrm{Re}}_b^{*}$) = (0.3, 40), $\Gamma \propto R_{\text{OT}}^{-0.5}$ in the whole R_OT range, and slightly varied slope in different ranges of R_OT.

What should be noted is that the turbulent patches with greater R_OT correspond to higher Re_b (Figs. 9a,b), and also patches with greater Re_b correspond to higher R_OT (Figs. 9c,d), implying Re_b and R_OT are not independent but positively correlated. A greater R_OT indicates the turbulence is in an “elder” stage of its life-span, which suggests stronger dissipation and weaker stratification, and hence greater Re_b. However, some earlier studies suggested no correlation between Re_b and R_OT (Mater et al. 2015). The covariation of Re_b and R_OT here suggests they both have an impact on the variation of Γ; therefore, the turbulent patches are classified into different R_OT (Re_b) groups to explore the dependence of Γ on Re_b (R_OT) (Figs. 9b,d). A clear monotonic dependence of Γ on Re_b or R_OT appears when only considering turbulent patches within fixed R_OT and Re_b groups (Figs. 9b,d), and both of them can be well approximated by power-law relations in the forms as $\Gamma \propto c_1{\mathrm{Re}}_b^a$ and $\Gamma \propto c_2{R}_{\text{OT}}^b$, where c₁ and c₂ are scaling factors; the fitted values of c₁, c₂, a, and b are presented in detail in Tables 1 and 2.

Table 1

Coefficients for least squares fitted relations between Γ and Re_b within different ranges of R_OT in regions Q and E. Fittings presenting slopes distinct from 1/2 or −4/3 are marked by bold and italic font.

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Table 2

Coefficients for least squares fitted relations between Γ and R_OT within different ranges of Re_b in regions Q and E. Fitting presenting slope distinct from 1/2 or −4/3 is marked by bold and italic font.

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In contrast to Fig. 9a, Γ presents a remarkable increasing trend with increasing Re_b for R_OT < 0.2, 0.5 ≤ R_OT ≤ 1 and 0.5 ≤ R_OT ≤ 1 (Fig. 9b). This positive dependence of Γ on Re_b is consistent with the results reported by Mashayek et al. (2017) and Ijichi et al. (2020) when Re_b is relatively small but contrast to the conclusions given by Monismith et al. (2018) and Vladoiu et al. (2021). The exponent a ≈ 1/2 for the three groups with R_OT < 0.2, 0.2 ≤ R_OT ≤ 0.5 and 0.5 ≤ R_OT ≤ 1, same as Γ ∝ ${\mathrm{Re}}_b^{1/2}$ when Re_b < ${\mathrm{Re}}_b^{*}$ reported by Mashayek et al. (2017), and c₁ values are 0.16, 0.06, and 0.04, decreasing gradually with increasing R_OT. The variability with R_OT > 1 deviates from this trend largely due to the relatively few patches in that range.