A Spatial Correlation Identification Model for Coherent Structure Extraction and Three-Dimensional Visualization

Abstract

Multi-scale coherent structures have been observed in ocean currents, which are induced by the interaction of shear flows with different velocities. Understanding the spatial configuration and scale characteristics of coherent structures will promote the explanation of physical ocean phenomena. Considering the self-similarity, we propose a spatial correlation identification model for coherent structure extraction and three-dimensional visualization based on the wavelet transform and time-dependent intrinsic correlation method. The spatial and scale distributions of coherent structures are related to the dissipation rate variation. Most large-scale coherent structures, with the largest length scale of 13 m, are found to exist in stable fluid, such as the water column below 50 m. However, small-scale structures are found in chaotic fluids, such as the upper layer. Furthermore, we found that coherent structures of different scales coexist simultaneously in the same depth range, indicating a simultaneous multi-scale structure pattern for turbulent flow investigations.

1. Introduction

Ocean currents consist of multiple scales of coherent structures, recognized as interconnected fluid parcels with concentrated vorticity over spatial regions [1]. Understanding the structures is a critical task for ocean engineering, such as tidal power generation and voyage friction reduction. Meanwhile, the coherent structure is a critical contributor to fluid entrainment, mass transfer, momentum exchange, and heat mixing across scales [2,3,4,5,6]. Therefore, it is important for turbulence research to identify the coherent structure at different scales and to study its spatial characteristics and scale properties in ocean flows.

Coherent structures are typically classified as large-scale or small-scale according to their size in space [7,8]. The large-scale coherent structure is the skeleton of fluid flows [9]. It has relatively slow dynamics and is responsible for current evolution trends. The small-scale event has a faster evolution time scale, driving energy dissipation [10]. They coexist in the water column [11,12], but extracting these structures and analyzing their properties are not trivial. Previous research has developed several approaches to detect these structures, including the proper orthogonal decomposition method [13,14,15], wavelet approach [16,17,18], Fourier analysis [19,20], Eulerian methods [21,22], and Lagrangian method [23,24,25]. The main limitation of these methods is the relatively arbitrary selection of a threshold. The cut-off frequency must be manually adjusted to separate the coherent structure and noncoherent residue from the raw signal [26,27,28], which is not completely objective. Eulerian methods, such as the Q criterion [29], ${\lambda }_2$ criterion [30], and Δ criterion [31,32], provide a perspective for identifying coherent structures in the Eulerian velocity field. Similarly, an appropriate threshold is required in advance, and the choice of Eulerian reference frame would additionally affect detection results. Lagrangian methods detect coherent structures by tracking the fluid trajectory. However, only the coherent structures at the ocean surface can be detected, and the coherent events in deep ocean currents are ignored. Moreover, most of these data-driven approaches rely on single-point measurements [33,34,35], PIV [36,37,38,39], and direct numerical simulations [40,41]. The single-point measurements can provide a one-dimensional signal according to Taylor’s frozen hypothesis. Coherent events are three-dimensional structures, and the narrow information hinders spatial scale investigation, which is a critical task for chaotic turbulence interaction. PIV provides detailed particle movement trajectory for structure detection, but it is not practical for flexible observations due to deployment limitations. Meanwhile, in addition to simulation data, the practical signals measured from oceans and lakes are significant for turbulence analysis. In addition, a large array of instruments can be used to sample spatial signals over a long period. However, this deployment would result in a heavy financial burden [42,43,44].

Considering the above limitation, we propose a spatial correlation identification (SCI) model for a pair of shear data measured in an orthogonal pattern to identify coherent structures and provide a three-dimensional visualization based on the wavelet transform and time-dependent intrinsic correlation (TDIC) method. The wavelet transform provides the advantage of decomposing turbulence shear signal scale by scale since the turbulent fluid is composed of multi-scale coherent structures and because energy and other quantities tend to cascade from large to smaller events [17,45,46,47]. Extensive research has been conducted to evaluate the local energy density, such as studies by Lucia and Gassmann [46], Gong et al. [17], and Protzko et al. [47], providing a potential for the extraction and characterization of multi-scale structures. Meanwhile, TDIC offers a major advance in evaluating the spatial correlation variation between a pair of turbulence shear signals since the spatial information is highly self-similar in a coherent structure. It was first proposed in the study by Chen [48] and has been applied in various nonlinear research [49,50,51]. Furthermore, the adaptive sliding window size makes it applicable to detect correlation information from nonlinear turbulence shear signals.

In the proposed method, characteristic fluctuations with energy peaks are extracted to identify the coherent structures. The self-similarity is evaluated according to the spatial correlation between a pair of shear data collected in an orthogonal pattern, and coherent structures are detected by the highly correlated region. Unlike conventional methods, we do not rely on the prior information of structure scale. We investigate the length scale characteristics based on identification results, which provide practical scale knowledge for turbulence research. Furthermore, the data orthogonally measured in the vertical water column drives a three-dimensional structure identification way for coherent eddy visualization.

In this paper, a pair of shear data is measured from the South China Sea (SCS) using a MicroRider equipped with a pair of orthogonal shear probes. To identify coherent structures in the fluid field, we propose the SCI model to extract the coherent structures and investigate their spatial properties. In Section 2, the experimental measurements and identification methodology are presented in detail. In Section 3, the extraction results and physical properties of coherent structures are investigated. The conclusions are given in Section 4.

2. Materials and Methods

2.1. Measurements

In this section, we give a brief introduction to experimental measurement in the South China Sea (SCS). The experimental observation was conducted at the station (19°39′ N, 115°27′ E) using the MicroRider on 21 December 2020 (Figure 1). The total depth of this station is nearly 2000 m, but only the water column from 0 to 285 m was observed due to some experimental limitations. In this observation, the high-resolution MicroRider freely fell through the water column at an average speed of 0.6 m/s, and it provided a pair of shear data collected by the orthogonal shear probes installed at the head of the MicroRider at a sampling rate of 512 Hz.

2.2. Methods

It is assumed that a coherent structure is composed of self-similar particles with extremely close evolution characteristics, and it is generally surrounded by incoherence regions with different evolution tendencies [25,52]. To extract coherent structures, it is necessary to evaluate the spatial correlation in the fluid field. In this section, a spatial correlation identification (SCI) model is introduced to extract coherent events from a pair of shear data collected in an orthogonal pattern and to visualize a three-dimensional framework for these structures based on the wavelet transform method and the time-dependent intrinsic correlation (TDIC) method. An overview of the proposed SCI model is shown in Figure 2.

Before detecting coherent structures, it is necessary to define a pair of shear data $s_1\left(\mathrm{x}\right)$ and $s_2\left(\mathrm{x}\right)$ collected along the orthogonal direction,

$$\begin{gathered} s_1\left(\mathrm{x}\right)=\left[s_1\left(x_1\right),s_1\left(x_2\right),\cdots ,s_1\left(x_k\right)\right] \\ {s}_2\left(\mathrm{x}\right)=\left[s_2\left(x_1\right),s_2\left(x_2\right),\cdots ,s_2\left(x_k\right)\right] \end{gathered}$$

(1)

where $x_k$ is the depth of the sampling point. Shear data collected at the same depth allow the instantaneous horizontal characteristic of the flow structure to be evaluated based on the sectional slice. As this pair of signals varies with depth $x_k$, the instantaneous horizontal characteristics in a slice can be tracked along the vertical column, providing a three-dimensional fluid field for coherent structure extraction.

It is assumed that the dominating turbulent energy is preserved in coherent structures. Characteristic fluctuations that contain dominant energy and drive turbulent flow evolution must be extracted to identify the spatial coherent structure. Here, the wavelet transform is introduced to decompose shear into wavelet coefficients $wt\left(\alpha ,\beta \right)$ scale by scale, which are given by

$$wt\left(\alpha ,\beta \right)=\underset{-\infty }{\overset{\infty }{\int }}s\left(x\right){\left|\alpha \right|}^{-\frac{1}{2}}{\phi }^{*}\left(\frac{x-\beta }{\alpha }\right)$$

(2)

where $\alpha$ is scale dilation, $\beta$ represents space translating, and ${\phi }^{*}\left(\alpha ,\beta \right)$ denotes the complex conjugate of the mother wavelet function. $x$ denotes space parameter, and the wavelet coefficient $wt\left(\alpha ,\beta \right)$ represents an $\alpha$-scaled vortex structure generated at location $\beta$.

In addition, the wavelet energy variation of each wavelet coefficient is obtained by

$$\frac{E\left(\alpha \right)}{{\alpha }^2}=\frac{2}{c}\underset{-\infty }{\overset{\infty }{\int }}{\left[wt\left(\alpha ,\beta \right)\right]}^{2}d\beta$$

(3)

where $E\left(\alpha \right)/{\alpha }^2$ represents turbulent energy at scale $\alpha$, and $c$ is the reconstruction coefficient. ${\alpha }^{*}$ is the value of scale parameter for which the energy of the corresponding wavelet coefficient reaches a peak. The characteristic fluctuation $s^{*}\left(x\right)$ containing dominant energy is reconstructed [53] based on the selected wavelet coefficients at scale ${\alpha }^{*}$.

The three-dimensional coherent structure is a self-similar block, and each particle in this block has extremely close evolution properties. The classical global correlation expression applicable to nonlinear series may give confusing cross-correlation results between two series. Therefore, the TDIC method is introduced to evaluate the correlation between these characteristic fluctuations, e.g., $s_1^{*}\left(x\right)$ and $s_2^{*}\left(x\right)$, and to detect coherent events from the vertical fluid field. The TDIC between $s_1^{*}\left(x\right)$ and $s_2^{*}\left(x\right)$ at depth $x_k$ is given by

$$R\left(x_k^m\right)=corr\left(s_1^{*}\left(x_w^m\right),s_2^{*}\left(x_w^m\right)\right) \mathrm{at} \mathrm{any} \mathrm{depth} x_k$$

(4)

where $corr$ represents the local correlation calculation of two series $s_1^{*}\left(x_w^m\right)$ and $s_2^{*}\left(x_w^m\right)$ in a sliding window $x_w^m$, and $m$ is any positive number. The sliding window size is given as $x_w^m=\left[x_k-m{x}_d/2:x_k+m{x}_d/2\right]$, and the minimum window size is $x_d=max\left[D_1\left(x_k\right),D_2\left(x_k\right)\right]$. Here, $D\left(x\right)$ is the instantaneous depth cycle of $s^{*}\left(x\right)$ and is obtained using the zero-crossing method. The adaptive criterion ensures that at least one depth cycle is covered in the local correlation computation. In addition, a student’s t-test is introduced to ensure validation of the correlation coefficient.

If $R\left(x_k^m\right)$ is above 0.8, these two characteristic fluctuations in the sliding window $x_w^m$ follow a correlated trend, and this continuous depth region with a correlated trend implies the presence of coherent structures. On the contrary, if $R\left(x_k^m\right)<0.8$, it represents an uncorrelated trend in this continuous depth range, indicating a disorganized fluid field and the absence of coherent structures. According to the correlation characteristics, the three-dimensional spatial framework of coherent structures in fluid flow is constructed. In the three-dimensional visualization procedure, a coherent structure is constructed as a three-dimensional spheroid covering a continuous correlated depth region. Meanwhile, its vertical length scale is given as the length of this continuous region. To visualize the three-dimensional framework, the horizontal length scale of a spheroid is set equal to the vertical length scale, and the true horizontal scale will be investigated in future research.

3. Results

We have applied the SCI model to a practical marine fluid, namely the SCS. The observation configuration is described in Section 2. Based on the free-fall observation, turbulence data and fluid field characteristics of marine water column are measured and investigated. Prominent peaks can be observed in the surface depth range due to inevitable wave beating. Therefore, only turbulence records below 5 m are used to investigate the microstructure properties. Pitch variation and fall velocity variation in the effective depth range indicate a steady descent of the instrument without any violent disturbances (Figure 3a,b). Figure 3c shows a sharp temperature drop indicating a pycnocline at 105 m, and the water temperature dropped from 26 °C to 11 °C. To improve data quality, the Goodman coherence noise reduction method [54] is applied to suppress noisy contamination contained in the raw shear signals. From the wavenumber spectrum, the corrected shear spectrum agrees with the Nasmyth spectrum [55], indicating a compelling data set for the following investigation (Figure 3e). Meanwhile, turbulence mixing signatures are evaluated by the dissipation rate (Figure 3d) [56,57]. The average magnitude of dissipation rate is nearly 10⁻⁷ in the surface layer between 5 m and 90 m. Relatively weak mixing intensity is detected below 90 m, and the average magnitude order decreases to 10⁻⁹. At a depth of 85 m, a sharp increase is detected, which may be caused by the undesired release and drag during operation. This can also be observed by the rise and subsequent decrease in fall velocity within the same depth range.

The entire effective depth range is divided into 14 depth segments with an average length of 20 m. At each depth segment, the shear signal is decomposed into multi-frequency wavelet coefficients using the wavelet transform method, and their energy variations are evaluated. The wavelet spectrum between 165 m and 185 m is shown in Figure 4 as an example. Multi-frequency energy blocks are detected; so, energy contributions of each wavelet coefficient are evaluated to extract the characteristic fluctuations $s_1^{*}\left(x\right)$ and $s_2^{*}\left(x\right)$ with maximum energy (Figure 5).

The spatial correlation of characteristic fluctuations is evaluated. To present cross-correlation results in more detail, only the graphic trapezoid part of the correlation triangle is shown in Figure 6. Figure 6 shows a trapezoid: the vertical axis represents the observation depth, and the horizontal axis indicates the sliding window size. Highly correlated regions in the vertical fluid field can be seen from the graphic results. Correspondingly, coherent structures are marked by these continuous highly correlated regions (R ≥ 0.8), which are shown in red. Meanwhile, the left voids in Figure 6 represent local correlation regions that do not satisfy the student’s t-test. The leftmost vertical curve indicates the minimum size of the sliding window, such as 0.2 m in Figure 6a and 0.6 m in Figure 6h. Assuming Taylor’s frozen field, with a mean fall velocity of 0.6 m/s in this profile, the length of the high correlation region along the minimum window size curve gives a length scale of coherence events. Figure 6f shows three significant regions, indicating three coherent events at 107 m, 112 m, and 115 m, with length scales of 1.3 m, 1.9 m, and 2 m, respectively. There are eight highly correlated regions in Figure 6h, showing coherent structures at 146 m, 151 m, 155.5 m, 157.3 m, 157.6 m, 159.1 m, 161 m, and 163.2 m, with length scales of 0.5 m, 2.6 m, 1 m, 0.07 m, 0.07 m, 0.2 m, 0.7 m, and 0.7 m, respectively.

Subsequently, three-dimensional coherent structure spheroids are constructed (Figure 7) according to these graphic trapezoids. Each coherent structure is isolated by uncorrelated regions (white surround space in Figure 7). Meanwhile, these three-dimensional spheroids give the possible barriers between coherent and noncoherent structures, which are represented as spheroid edges.

For detailed illustration, structures with a vertical length scale below 0.1 m are ignored. Above 50 m, most of the small-scale coherent structures are detected, with an average length scale of 0.47 m (Figure 7a,b). Only two relatively large coherent structures are found at depths of 32 m and 34.9 m, with vertical length scales of 0.8 m. On the contrary, most large-scale coherent structures are found below the depth of 50 m. Figure 7c shows two apparent large-scale coherent structures. The upper one is located at a depth of 54.3 m, with a vertical scale of 4.5 m. The lower one covers a depth range from 59.4 m to 64.3 m, with a vertical scale of 4.9 m. Figure 7i shows several large-scale coherent structures with vertical scales of 1.3 m, 1.2 m, 1.8 m, 1.2 m, and 3.1 m, respectively. Similarly, two large-scale coherent structures are observed in Figure 7j. The upper one is located at a depth of 188.3 m, with a length scale of 5.2 m. Meanwhile, the lower one is located from 191.3 m to 204.3 m, with the largest vertical scale of 13 m.

According to the three-dimensional visualization (Figure 7) and the dissipation rate profile (Figure 3d), we can see that there are few large-scale coherent structures in the strong mixing depth range. However, as the intensity of turbulent mixing decreases and the water column becomes relatively stable, coherent structures appear, some of which have particularly large-scale structures. Thus, large-scale coherent structures with dominating turbulence energy (Figure 5) maintain the skeleton of flow [10], and small-scale coherent structures are responsible for turbulence energy dissipation [12].

Except for the most energetic peaks at 0.5 Hz, other weaker energy peaks are observed at 1.8 Hz, 7.4 Hz, and 68 Hz (Figure 8). Characteristic fluctuations at these frequencies are extracted to evaluate spatial correlations. For a detailed presentation, Figure 9 shows the correlation results in a refined depth range between 175 m and 185 m. The three-dimensional frameworks are constructed in Figure 10. As can be seen from these trapezoids, the size of coherent structures decreases sharply with frequency. There are two significant events in the 0.5 Hz trapezoids (Figure 10a), with the dominant structure at 181 m having a maximum vertical length scale of 3.1 m and the weaker event at 177.5 m having a length scale of 1.3 m. Ten significant high correlation regions are extracted in 1.8 Hz trapezoids (Figure 9b), one of which is <0.1 m. The length scale for these nine effective structures is smaller than that in 0.5 Hz detection, with an average vertical length scale of 0.7 m. Meanwhile, the most significant event is detected at 182.9 m, with a length scale of 0.9 m. There are 13 high correlation regions in minimum sliding window detection at 7.4 Hz (Figure 9c). Coherent structures extracted at 7.4 Hz are extremely tiny, four being <0.1 m length scale, with the most powerful at 177.7 m and 180.9 m having vertical length scales of 0.4 m. Meanwhile, the average vertical length scale for these nine tiny events in Figure 10c is nearly 0.2 m. Furthermore, almost no apparent correlation region can be found in the 68 Hz graphic trapezoid (Figure 9d), and the most significant structure here is still smaller than 0.1 m (Figure 10d). Compared to coherent structures extracted from the characteristic fluctuation at 0.5 Hz, detection results obtained at slightly higher frequencies (1.8 Hz or 7.4 Hz) are much denser and smaller, indicating simultaneous multi-scale coherent structures at the same depth range (Figure 11). Furthermore, these shear data at 68 Hz are considered noisy fluctuations and chaotic disturbances in fluid flow.

In addition to the length scale, the fluctuation frequency indicates the speed of motion and the time scale. Large structures at lower frequencies change slowly and stay longer [1]. Meanwhile, small events at higher frequencies have a faster time scale [58] and dissipate more intensely. Thus, large coherent structures in Figure 10a preserve more energy (Figure 5) and stay longer than smaller ones in Figure 10b,c. In addition, structures in Figure 10c have very little energy and would dissipate instantly. It indicates a simultaneous multi-scale pattern of coherent structures in the fluid field (Figure 11). Meanwhile, multiple time-scale coherent structures occur according to the peaks at different frequencies, covering both energetic, inertial, and dispersive ranges.

The wavelet cross-correlation method introduced in the study by Li [59] is applied to the same dataset and examines the validity of the proposed model. Figure 12 indicates that the coherent structures of 0.5 Hz, 1.8 Hz, and 7.4 Hz pass through the shear probes because the local core fluctuation has unusually large positive and negative peaks when an organized eddy moves close to the center line [60]. Multi-scale periodic oscillations in wavelet cross-correlation spectrum support the identification results and the simultaneous multi-scale pattern of coherent structures. Moreover, the spatial configuration and length scale characteristics of coherent structures can be investigated in a three-dimensional framework, but it cannot be obtained in Figure 12, which proves the validity and the advantage of the identification model in this paper.

4. Conclusions

In this paper, we introduced a spatial correlation identification model for coherent structures based on the wavelet transform and TDIC method. Our method is based on two shear signals measured by a pair of orthogonal probes. Characteristic fluctuations with energetic peaks are extracted to evaluate the spatial correlation. According to these correlation results, we detect coherent structures in the practical fluid column and construct a three-dimensional framework for coherent structure visualization. Based on identification results, the spatial configuration and scale characteristics are found to be associated with the dissipation rate variation. Large-scale structures are concentrated in the stable column, but most structures with smaller size are found in chaotic depths where turbulence mixing is very intensive. Furthermore, the visualization framework also suggests the coexistence of multi-scale structures in the same region and a simultaneous multi-scale coherent structure pattern, which improves the physical understanding of turbulence microstructure. Our results show that the three-dimensional coherent framework is an effective approach to extracting coherent structures, and it provides a potential way for spatial characteristic investigation.

Although our study detects multi-scale coherent structures in the fluid flow and investigates their spatial configuration and scale properties, there are still some limitations we need to improve in future research. The wavelet energy is determined according to the sum amplitude of the wavelet coefficients at the selected depth segment. Relatively weak structures may be ignored due to the strong coherent event at the same depth segment. To circumvent this limitation, we examine the wavelet spectrum at the whole depth, detect the scale and location information of each energetic block, and finally determine the applicable length of the depth segment, which is not suitable for extensive dataset analysis and needs to be optimized in future research. Meanwhile, for three-dimensional visualization, the horizontal length scale of a coherent structure is ideally defined as the vertical length scale. In the practical column, the horizontal length scale is variable in different column depths and marine areas. Therefore, multiple points are needed to complete the horizontal spatial information of coherent structures and provide thorough data for structure extraction.